Bulletin - University of Washington

AIDJEX B u l l e t i n No. 20

May 1973

TABLE OF CONTENTS

ON THE ATMOSPHERIC BOUNDARY LAYER: THEORY AND METHODS - - R . A . B r o w n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

SOME CRITICAL REMARKS ON TURBULENT D I FFUSION THEORY ON I C E DYNAMICS

- - S . I. P a i and Huon L i . . . . . . . . . . . . . . . . . . . . . . 143

DATA MANAGEMENT REPORT ON 1972 AIDJEX P ILOT STUDY . . . . . . . . . . . . 1 5 3

ABSTRACTS OF INTEREST . . . . . . . . . . . . . . . . . . . . . . . . . 158

CONTENTS OF PAST AIDJEX BULLETINS . . . . . . . . . . . . . . . . . . . 159

Front cover: NASA CV-990 remote-sensing fZight over ATDJEX main carp during 1972 p i l o t study.

Back cover: Evergreen he Zieopter removing the rotating dome which housed the CRREL laser.

AIDJEX BULLETIN No. 20

May 1973

* * * * *

FinanciaZ support for AIDJEX i s provided by the Na$ionaZ Science Foundation,

the Office of Naval Research, and other U.S. and Canadian agencies.

* * * * *

Arct ic Ice Dynamics Jo in t Experiment D i vis ion of Marine Resources

University of Washington S e a t t l e , Washington 98105

Division o f Marine Resources UNIVERSITY OF WASHINGTON

The AIDJEX Bulletin aims t o provide both a f o m for discussing AIDJEX problems and a source of infomation pertinent t o a l l AIDJEX participants. Issues--numbered, dated, and sometimes subtitled-contain technical material close l y related t o AIDJEX, informal reports on theoretical and f i e l d work, translations of relevant s c i en t i f i c reports, and discussions of interim AIDJEX resul ts .

With the exception of a few pages a t the end, Bulletin No. 20 i s an investigation, by R. A . Brown, in to planetary boundary layer dynamics. I t evaluates the analytic methods avai Zable for predicting a i r s tress for a synoptic-scale ice dynamics mode l

Bulletins No. 21 and 22 are now being prepared for printing. One w i l l report on recent developments i n arct ic data buoy systems.

Any correspondence concerning the AIDJEX Bulletin should be addressed t o

Alma Johnson, Editor AIDJEX Bul le t i n 4059 Roosevelt May N. E. Seattle, Washington 98105

ON THE ATMOSPHERIC BOUNDARY LAYER THEORY AND METHODS

by R. A. Brown

AIDJEX

CON TENTS

Preface (3)

1. In t roduct ion (5) 1.1 Background 1 .2 :Basic Approach 1.3 The Boundary Layer Concept and Underlying Assumptions

2. Discussion of t h e Equations and In t roduct ion of Boundary Layer Models (11) 2 . 1 Conservation Laws 2.2 The Ekman Layer Equations 2.3 Boundary Layer Methods

3. Development of t h e Governing Equations (20) 3 . 1 Pr imi t ive Equations 3 .2 Scal ing S impl i f i ca t ions 3.3 Geostrophic Equations 3.4 Thermal Wind 3.5 The Ekman Layer Equations

4. The Geos trophic/Ekman Layer Solu t ion (29) 4 .1 Discussion 4.2 The Geostrophic/Ekman Layer Equations

5. The Surface Layer ( 3 4 ) 5.1 D i s cussion 5.2 Surface Layer Equations 5 .3 Surface Layer Solu t ion 5.4 Solu t ion v i a Dimensional Reasoning 5.5 Mixing Length 5.6 Surface Layer Methods of Experimental Analysis

6. The Matched Ekman/Geostrophic - Surface Layer Solu t ion (42) 6 . 1 L i m i t So lu t ions 6.2 Surface Stress Rela t ions 6.3 Summary

1

7. S i m i l a r i t y Methods (51) 7.1 Discussion 7.2 7.3 Matching t h e Solut ions 7.4 Summary

Dimensional Analysis and Dynamic S i m i l a r i t y

8. Eddy Viscos i ty Theories (60) 8.1 D i s cuss ion 8.2 Rossby Model 8.3 8.4 Analyt ic Modeling of K(z) 8.5 Summary

Empirical Determination of Eddy Viscos i ty

9. Ekman I n s t a b i l i t y (77)

10. Secondary Flow Model (80) 10.1 Basic Concept 10.2 Some Aspects of t h e Secondary Flow Assumptions 10.3 Secondary Flow Equations 10.4 D i s cuss ion 10.5 Summary

11. Thermal Ef fec t s (91) 11.1 Fundamental Parameters 11.2 Convection 11.3 Convection E f f e c t s i n the Dynamic Flow Models

12. Nons t a t i o n a r i t y (9 8) 12.1 Discussion 12.2 The Diurnal Cycle 12.3 Synoptic Var i a t ions 12.4 Summary

13. Application t o S t r e s s Calculat ions (105) 13.1 Discuss ion 13.2 13.3 Ekman Layer Models 13.4 Surface Layer Measurements 13.5 S i m i l a r i t y Models 13.6 Summary

The Geostrophic Flow Boundary Condition

Appendix A: The Momentum Defect Method (117)

Appendix B: A Simple Momentum I n t e g r a l Model (121)

Appendix C: Data Processing t o Evaluate Models (129)

Bibliography and References (135)

2

PREFACE

This i n v e s t i g a t i o n i n t o p l ane ta ry boundary l a y e r dynamics eva lua te s

t h e anakyt ic methods a v a i l a b l e t o p r e d i c t t h e air stress f o r a synoptic- . -

scale ice dynamics model. A l l of t h e s e methods, from t h e s imples t (Ekman's

s p i r a l f rou a cons t an t eddy v i s c o s i t y ) t o t h e most complex (numerical

i n t e g r a t i o n with stress from mean momentum t r a n s p o r t terms), p i v o t on t h e

t reatment of t h e viscous f o r c e term; t h e r e f o r e , considerable discussion

is devoted t o t h e fundamental d e r i v a t i o n s and assumptions p e r t a i n i n g t o

t h e stress term. Although t h i s may c o n s t i t u t e o v e r k i l l f o r some simple,

s t eady , n e u t r a l cases r e a d i l y amenable t o empir ical parameter izat ion, i t

provides e s s e n t i a l background f o r any needed e x t e n s i o m t o t h e s t r a t i f i e d , uns tab le , o r heterogeneous cases, It is expected that as t h e d i v e r s e pre-

d i c t i o n s of theory are compared with t h e AIDJEX d a t a , no n e u t r a l theory

w i l l be adequate f o r a l l condi t ions found i n the A r c t i c , bu t ad hoc ve r s ions

can be ex t r ac t ed from t h e b a s i c t h e o r i e s provided h e r e (e.g. , i n Appendix B)

t o accommodate observed d a t a t rends.

The geostrophic flow equation i s discussed, as a l l stress values are

eventua1:Ly l inked t o t h i s synop t i c parameter. The synopt ic boundary l a y e r

solution---Ekman's r e s u l t f o r t h e balance between C o r i o l i s and viscous

forces--is developed, examined f o r s t a b i l i t y , and f i n a l l y modified t o r e t a i n

i t as a v i a b l e equi l ibr ium ao lu t ion . The geostrophic/Ekman l a y e r s o l u t i o n

i s presented i n a manner which complements subsequent inner-outer develop-

ments.

Ekman's constant eddy v i s c o s i t y r e s u l t s q u a l i t a t i v e l y desc r ibe many

f a c e t s of the p l ane ta ry boundary l a y e r flow. H i s b a s i c s o l u t i o n has been

modified by a d j u s t i n g the lower boundary condi t ions, matching t o an inne r

s o l u t i o n , spec i fy ing a v a r i a b l e eddy v i s c o s i t y , and adding a secondary flow.

These d i v e r s e methods p r e d i c t s u r f a c e stress more i n l i n e wi th spa r se

observat ions than t h e p l a i n Ekman theory.

3

The resul ts a l l l a c k s u f f i c i e n t experimental d a t a t o e i t h e r v e r i f y

p red ic t ed flow c h a r a c t e r i s t i c s o r determine empir ical constants needed t o

parameter ize large-scale flow c h a r a c t e r i s t i c s . To re la te s u r f a c e stress

t o synop t i c ( f reestream) flow, o t h e r methods are a v a i l a b l e which s i d e s t e p

t h e d e s c r i p t i o n of t h e boundary l a y e r flow dynamics. The d i r e c t procedure

(measurement of s u r f a c e stress and mean flow t o e s t a b l i s h empir ical co r re l a -

t i o n s ) is the most popular approach. Surface l a y e r methods of a n a l y s i s

are discussed, as they are the b a s i s f o r processing d a t a with which t o

eva lua te t h e mesoscale t h e o r i e s ,

A two-region (Ekman l a y e r / s u r f a c e l a y e r ) s o l u t i o n is developed using

t h e method of matched asymptotic expansions.

r e l a t i o n s between synop t i c flow and s u r f a c e stress based on very gene ra l

s i m i l a r i t y arguments app l i ed t o a two-layer flow. Empirical and a n a l y t i c

S i m i l a r i t y methods y i e l d

eddy v i s c o s i t y t h e o r i e s are reviewed, Thermal e f f e c t s and n o n s t a t i o n a r i t y

are introduced, F i n a l l y , t h e s p e c i f i c a p p l i c a t i o n t o s u r f a c e stress predic-

t i o n is discussed. The momentum i n t e g r a l method i s considered i n Appendix A

and app l i ed , i n Appendix B , t o some s p e c i a l AIDJEX 1972 t e the red bal loon

da ta . S p e c i f i c plans f o r evaluat ing p red ic t ed t h e o r e t i c a l cons t an t s from

t h e d a t a are ou t l ined i n Appendix C.

Since t h e material ranges from the simple t o the r econd i t e , t h e r e are

chapter summaries which should g ive t h e casua l reader a good o v e r a l l v i e w ,

wi th only a cursory glance a t t h e d e t a i l e d development.

n e c e s s a r i l y condensed, and a bibliography is provided f o r t h e s e r i o u s

user .

A l l t h e o r i e s are

R. A . Brown

4

1 . INTRODUCTION

upon the a v a i l a b l e dr iv ing fo rce represented by the geos t rophic

(which w i l l depend on t h e ice c h a r a e t e r i s t

L. the f c e motion i n response , t o wind stress, dependhg a d d i t i o n a l l y

on the ice dynamics, t h e s i d e dary condi t ions , and - the i n t e r a c t i o n

wi th ocean.

c. the ocean motion, and consequent stress i n t e r a c t i o n a t the i c e

i n t e r f a c e .

5

The boundary l a y e r flow problem w a s f i r s t approached i n the beginning

of t h i s century.

i n the t h i n l a y e r near t h e su r face in 1904,

provided an a n a l y t i c s o l u t i o n f o r t h e balance between these viscous fo rces

and the Cor io l i s fo rces .

done on the p lane tary boundary l aye r .

motivation; t he re was no p r a c t i c a l demand f o r t he s o l u t i o n , and few scien- tists des i r ed t o expend t h e i r energ ies chipping a t so immense a problem.

Data were a l s o lack ing , both i n quan t i ty and in q u a l i t y . scales of p lane tary boundary layers--1000 m i n the atmosphere, 40 m i n t h e

ocean, and g loba l h o r i z o n t a l extent--prohibited p r a c t i c a l f i e l d measurement.

And f i n a l l y , t h e extreme v a r i a b i l i t y of the parameters, when they could be

measured, f r u s t r a t e d analysis. large-capaci ty computers, and the a b i l i t y t o f o r e c a s t geophysical phenomena

has provided a p r a c t i c a l need, and subsequent funding, f o r a boundary l aye r

model.

P rand t l pointed out t h e importance of t h e viscous f o r c e s

In the same period, Ekman

Then f o r s e v e r a l decades very l i t t l e work w a s

This w a s due p a r t l y t o l a c k of

The huge vertical

Now, l a rge-sca le modeling i s poss ib l e on

It is not s u r p r i s i n g t h a t t h e m a i n e f f o r t has gone i n t o at tempting

t o deduce t h e entire l a y e r s o l u t i o n from t h e a v a i l a b l e su r face observat ions.

Experimental d a t a from above t h e lowest 10 meters are very sca rce , and

at tempts t o model t he complete boundary l a y e r have m e t wi th only l i m i t e d

success. Nevertheless , t h i s is a necessary s t e p toward l i nk ing t h e f a i r l y

ex tens ive ly measured su r face l a y e r t o t h e r e l a t i v e l y e a s i l y modeled synopt ic

flow.

The s i t u a t i o n wi th regard t o d a t a a c q u i s i t i o n has changed somewhat,

There

However,

in t h a t d ive r se and soph i s t i ca t ed measurements are now be ing taken.

is a l s o e f f i c i e n t computerized processing and s t o r a g e of t h e da ta .

most of these d a t a are s t i l l taken from t h e access ib l e po r t ion of t h e

boundary l a y e r (up t o 20 meters) , wi th only a small amount from t h e Ekman

l aye r . bal loon t racking p o i n t s i n the lower ki lometer and, more r ecen t ly , from measure-

ments made from a i r c r a f t with i n e r t i a l platform c a p a b i l i t y .

and ba l loon d a t a inc lude t h e lower po r t ion of the Ekman l a y e r , which may

extend t o a few k i lometers . Here, t h e Ekman l a y e r is considered as t h a t

The r e s u l t s from the Ekman l a y e r are from sparse radiosonde o r p i l o t

Some tower d a t a

region i n which the su r face e f f e c t s on t h e flow are s i g n i f i c a n t .

6

1.2 Bas ic Approach

Theore t i ca l a n a l y s i s of t h e p l a n e t a r y boundary l a y e r is burdened w i t h

an eighth-order of non l inea r governing equat ions, a p l e t h o r a of

a c losu re p rob l variable boundary condi t ions, and s p a r s e observat

da t a . Success i n developing a t h e o r e t i c a l s o l u t i o n o r model f o r t h e v a r i a b l e s

of i n t e r e s t (ve loc i ty , stress, h e a t and momentum fluxes, and t h e thermodynamic

state) depends upon the deduction of r e l i a b l e s i m p l i f i c a t i o n s t o t h e complete

problem. Some discussion i s devoted i n i t i a l l y t o t h e assumptions underlying

t h e development of t h e b a s i c equat ions f o r t h e flow. P a r t i c u l a r a t t e n t i

is paid t o the stress, both because i t is t h e least understood term and

because the stress value a t the s u r f a c e i s required i n t h e AIDJEX problem.

To place t h e chosen models i n perspect ive, t h i s paper w i l l systemati-

c a l l y p r e s e n t t h e d ive r se approaches t o desc r ib ing t h e p l a n e t a r y bDmdarp

l a y e r and w i l l explain why a p a r t i c u l a r model with s p e c i f i c s implifying

assumptions i s used. Exposure t o t h e d i f f e r e n t methods of handl ing t h e ,

complexities introduced with nons teady state, nonhorizontal homogeneity,

and s t r a t i f i c a t i o n may f a c i l i t a t e f u r t h e r refinement of t h e model.

The p r i m i t i v e equat ions are f i r s t discussed w i t h an o v e r a l l view of

the assumptions and s i m p l i f i c a t i o n s involved. The c u r r e n t methods of

boundary l a y e r a n a l y s i s are introduced i n t o t h i s framework of equat ions.

These equat ions are then considered i n g r e a t e r d e t a i l t o y i e l d t h e equat ions

appropriate t o each model.

experiment are b r i e f l y discussed.

t o mesoscale flow are t r e a t e d f a i r l y ex tens ive ly .

A l l models which w i l l f i g u r e in t h e AIDJEX

The models which are most l i k e l y to apply

The s i m i l a r i t y r e l a t i o n s produced when a scale a n a l y s i s is done on

t h e Navier-Stokes equat ions f o r t h e boundary l a y e r are discussed in d e t a i l .

These are based on t h e hypothesis t h a t d i f f e r e n t s e l f -similar s o l u t i o n s

exist i n each of t h e l a y e r s cha rac t e r i zed by the s u r f a c e roughness and t h e

boundary l a y e r depth.

region y i e l d s r e l a t i o n s l i n k i n g t h e su r face va lues t o t h e f r e e stream

parameters.

The condi t ion t h a t t hese s o l u t i o n s match i n some

The secondary flow model, which inc ludes a s t a b i l i t y a n a l y s i s of t h e

Ekman boundary l a y e r s o l u t i o n , is o u t l i n e d f o r t h e n e u t r a l l y s t r a t i f i e d l a y e r ,

7

with comments on the e f f e c t of s t r a t i f i c a t i o n where p e r t i n e n t . A l l models

can b e extended- t o include va r ious s t r a t i f i c a t i o n s and t i m e dependence with

an a t t endan t increase in complexity.

w i t h r e fe rences t o the corresponding d e t a i l e d s t u d i e s .

The t reatment is o f t e n q u a l i t a t i v e ,

1.3 The Boundary Layer Concept and Underlying Assumptions

The problem i s t o desc r ibe the motion of t h e f l u i d ad jacen t t o t h e

boundary. I n general , t h e boundary d e t a i l s are small i n dimension compared

t o o v e r a l l vertical dimensions of t he boundary l aye r .

ence suggest t h a t t he e f f e c t s of t hese boundary d e t a i l s w i l l n o t be f e l t

very f a r from t h e boundary due t o t h e i n a b i l i t y of t he f l u i d t o support

I n t u i t i o n and experi-

shea r stresses of s i g n i f i c a n t magnitude o r du ra t ion . Hence, f i r s t s o l u t i o n s

f o r t he atmospheric o r oceanic flow w i l l i n e v i t a b l y ignore t h e boundary

effect e n t i r e l y and so lve t h e balance equat ions f o r a f l u i d of e f f e c t i v e l y

i n f i n i t e ex ten t . This works q u i t e w e l l f o r f l u i d s of g r e a t depth as long

as d e t a i l e d knowledge of t h e flow i n t h e v i c i n i t y of t h e boundary i s no t

needed. Thus, a gene ra l c i r c u l a t i o n model f o r t h e atmosphere is f a i r l y

success fu l i n p r e d i c t i n g t h e large-scale (synoptic) flow w i t h l i t t l e regard

f o r t h e boundary l a y e r flow.

analyses.

i n p u t w i l l i n f luence the general c i r c u l a t i o n and must b e included in t h e

ana lys i s .

n o t less than 12 hours, so t h a t a synop t i c model which con t inua l ly updates

the s o l u t i o n empi r i ca l ly w i t h real d a t a (e.g., p re s su re da t a ) a t i n t e r v a l s

less than c h a r a c t e r i s t i c synopt ic v a r i a t i o n t i m e s can treat t h e boundary

l a y e r f a i r l y grossly.

This is most l i k e l y t r u e f o r short-period

The accumulative e f f e c t s of boundary l a y e r d i s s i p a t i o n and hea t

It appears t h a t the t i m e scale f o r t h i s i n f luence is t y p i c a l l y

Information about the boundary flow i t s e l f is o f t e n needed. I n t h e

arct ic sea-ice problem, the d e s i r e d quan t i ty from t h e atmospheric s o l u t i o n

is the f o r c e vec to r on the boundary. I n t h i s case, the boundary condi t ion

is t h a t the f l u i d comes t o rest a t t h e boundary. In add i t ion , some quant i -

tative measure of t h e a b i l i t y of t h e f l u i d t o support t h e stress of decelera-

t i o n from geostrophic v e l o c i t y t o no s l i p a t t h e boundary is needed t o

i n d i c a t e how deep i n t o the f l u i d t h e boundary in f luence w i l l b e f e l t . This

8

measure w i l l be a d d i t i o n a l l y complicated i f it i s not merely a property of

t he f l u i d , but depends upon t h e n a t u r e of t h e boundary.

I f t h e measure i s a c h a r a c t e r f s t i c of t h e f l u i d , then a un ive r sa l

l a w , a s i m i l a r i t y r e l a t i o n f o r t h e flow over any boundary, can be expected

t o be app l i cab le f o r a l l boundaries. The v a l i d i t y of such s i m i l a r i t y cri-

teria can be e s t ab l i shed only by c a r e f u l cons idera t ion of s ca l ing i n theory

and measurements. To develop a c l a s s i c a l , laminar theory f o r t h e p lane tary

boundary l a y e r , i t is necesaary t o assume that the u l t ima te roughness of

t he su r face i s recognized i n the no-sl ip boundary condi t ion a t the su r face

only. The actual roughness elements are assumed t o be of a s c a l e very much

smaller than t h e c h a r a c t e r i s t i c scale assoc ia ted wi th t h e f l u i d adjustment

depth. Al-so,the scale appropr ia te t o laminar flow governed by molecular

v i s c o s i t y i s i n s i g n i f i c a n t l y small in t h e p lane tary boundary l a y e r , and w e w i l l t he re fo re d e a l only wi th tu rbu len t boundary l a y e r s and seek mean

parameter izat ions of t h i s flow.

The cons idera t ion of s c a l e s i s important i n t h e geophysical boundary

l a y e r problem, p a r t i c u l a r l y when t h e eddy v i s c o s i t y assumptions are used to

e f f e c t c losu re of t h e Navier-Stokes equat ions, i .e . , when a v i s c o s i t y concept

i s employed t o approximate the stress t e r m , thereby reducing the number of

unknowns t o the number of equat ions when a value of eddy v i s c o s i t y is given.

In t h i s concept, t h e b a s i c d e f i n i t i o n of a continuum must b e expanded t o

inc lude tu rbu len t eddies .

The l i m i t a t i o n of t he Euler ian formulation of t h e conservat ion l a w s

f o r a continuum l ies i n t h e immense d i f f i c u l t y of r e l a t i n g t h e stress f i e l d

t o any measurable quant i ty . Because t h e a v a i l a b l e t h e o r e t i c a l models f o r

s implifying t h e stress f o r c e genera l ly lead t o insurmountable a n a l y t i c prob-

l e m s , t hese equat ions have been subjec ted t o severe reduct ion by sequences

of assumptions. Usually, t h e end product is s t i l l a high-order, nonl inear

set of equat ions appropr ia te only t o numerical so lu t ion .

l a y e r so lu t ion is t h e end product of one such series of assumptions.

provides one of t he rare "exact" so lu t ions of t h e conservation equat ions f o r

t h e f l u i d dynamic continuum. The sequence of equat ions and an a r b i t r a r y

confidence assessment of t he assumptions a t each s t e p areshown i n Figure 1 and disciussed i n t h e following s e c t i o n .

The Ekman boundary

It

9

WORKING EQUATIONS

I 1

ASSUMPTIONS

Decreasing Validity _____L_ 1 Invalid

I .COKSERVATION TAWS Continuum

0 - 0 + p v v 3 0

K, = K = Constant I -

2.

2.1 Conservation Laws

DISCUSSION OF THE EQUATIONS AND INTRODUCTION OF BOUNDARY LAYER MODELS

tls a f i r s t s t e p , a continuum is assumed.. It is s p e c i f i c a l l y assumed

t h a t a meaningful d e r i v a t i v e can b e def ined by t a k h g a l i m i t , small wi th

respect t o t h e c h a r a c t e r i s t i c scale of v a r i a t i o n s in t h e state def in ing

parameters y e t l a r g e enough t o exclude i n h e r e n t l y random processes underlying

the mean flows.

underlying random process , b u t i t should be reconsidered when eddy v i s c o s i t y

i n t e r p r e t a t i o n s are involved. With these r e se rva t ions , one can s tar t wi th

the conservat ion equat ions ax iomat ica l ly . This produces one vec to r equat ion

and two scalar equat ions f o r t h e unknown vec to r v and s c a l a r s dens i ty A d

energy, provided t h e body fo rces , h e a t sources , and stress t enso r are speci-

f i e d . The body fo rces , g rav i ty , and C o r i o l i s fo rces are r e a d i l y expressed

i n terms of known q u a n t i t i e s and/or b a s i c unknowns o r derivatives the reo f ,

Energy sources o r s i n k s can a l s o b e r e l a t e d t o decoupled parameters o r

ignored e n t i r e l y f o r flows considered here. One is l e f t w i th t h e problem

of desc r ib ing t h e stress tensor , an opera tor which acts on the u n i t normal

to a s u r f a c e t o produce t h e f o r c e s on t h e su r face .

t i v e equat ions express ing t h i s term by known o r measurable parameters and

the b a s i c unknowns is t h e p r i n c i p a l preoccupation of geophysical boundary

l a y e r f l u i d dynamics, and is r e f e r r e d t o as t h e c losu re problem.

This i s a secu re assumption when molecular motion is t h e

The search f o r cons t i t u -

One concept which m e t wi th cons iderable success is t h a t of a p e r f e c t

f l u i d . Here, t h e stress is t h e p re s su re f o r c e only, and t h e stress tensor

is simply a diagonal matrix wi th equal elements, t h e p re s su re a t a point.

The a d d i t i o n a l unknown p res su re r equ i r e s a c o n s t i t u t i v e equat ion of state

t o c lose t h e set of equat ions. This approximation w a s s o success fu l i n

allowing s o l u t i o n s which descr ibed w e l l t h e observed flow p a t t e r n s t h a t

hydrodynamicists were moved t o des igna te one of t h e obvious f a i l u r e s as a

paradox.

accura te ly p r e d i c t i n g t h e observed flow around t h e body. This f a l s e paradox

w a s explained by in t roducing a t h i n l a y e r i n which the p e r f e c t f l u i d concept

f a i l s . ode1 of t h e f l u i d stresses,

producing s k i n f r i c t i o n .

'This w a s t h e p red ic t ion of no drag on an immersed body whi le

It must be replaced by a more real is t i

This w a s accomplished by inc luding t h e f l u i d stress

11

i n proport ion t o t h e rate of s t r a i n , i n accordance w i t h observations.

t h e s imples t r e l a t i o n ( t h e f i r s t terms i n a Taylor expansion of stress as a

funct ion of v e l o c i t y ) , one ob ta ins a l i n e a r viscous f l u i d .

i n the expansion are c a l l e d v i s c o s i t i e s and are a measure of t he stress

f o r c e on a p a r t i c l e i n a f i e l d o f deformation (i.e., i n t h e presence of a

v e l o c i t y g rad ien t ) . The r e s u l t i n g equat ions are c a l l e d t h e Navier-Stokes

equat ions (see Fig. 1 ) .

Taking

The constants

With the add i t ion of a c o n s t i t u t i v e equat ion f o r t h e eddy v i s c o s i t y

c o e f f i c i e n t s K , and K,, t hese eighth-order, non l inea r equations can be i n t e -

g r a t e d numerically, with d i f f i c u l t y and under l i m i t e d condi t ions. The

assumptions f o r t h e K c o e f f i c i e n t s ( including t h e eddy thermal c o e f f i c i e n t

Kh = K / p c U ) are many and d ive r se (Sec. 4 ) .

K , = K = constant and K2Vv = 0 are as good assumptions as any f o r t h e Ekman

l a y e r s o l u t i o n . They r e s u l t in s i g n i f i c a n t l y s i m p l i f i e d ve r s ions of t h e

Navier-Stokes equat ions.

d i r e c t i o n , p a r t i c u l a r l y nea r t h e surface. This v a r i a t i o n becomes important

when the s u r f ace l a y e r is considered.

It c u r r e n t l y appears t h a t

However, K does vary s i g n i f i c a n t l y i n t h e ver t ical

2.2 The Ekman Layer Equations

The approximations c o l l e c t i v e l y c a l l e d t h e boundary l a y e r assumptions

are those asymptotic r e l a t i o n s obtained when a t h i n region is considered.

A closed form s o l u t i o n is p o s s i b l e when s t eady state and h o r i z o n t a l homogene-

i t y are assumed, e l imina t ing t h e non l inea r i n e r t i a l terms.

equat ions were solved by Ekman i n 1905.

The r e s u l t i n g

The s o l u t i o n f o r t h e balance between

C o r i o l i s and viscous fo rces y i e l d s an a e s t h e t i c a l l y s a t i s f y i n g exponent ia l

s p i r a l i n g v e l o c i t y p r o f i l e . It extends from t h e ze ro v e l o c i t y boundary value

t o t h e constant geostrophic v e l o c i t y i n t h e atmosphere (see Fig. 2 ) .

t h e ocean, i t occurs i n exact analogy a t t h e bottom boundary.

at the ocean s u r f a e e , from t h e s u r f a c e v e l o c i t y t o t h e depth of n e g l i g i b l e

s u r f a c e inf luence. The s o l u t i o n w a s found e x p l i c i t l y f o r t h e atmospheric

l a y e r by Akerblom [1908] and Taylor [1915].

I n

It a l s o occurs

Unfortunately, although a s p i r a l i n g v e l o c i t y p r o f i l e is gene ra l ly

observed i n t h e atmospheric boundary l a y e r , i t seldom corresponds t o t h e

1 2

I

I

' I

+ * * X

on Surface is Hodograph of Ekman Spiral (height in meters)

Fig. 2 . Sketch showing typical wind values in an Ekman layer.

13

a n a l y t i c Ekman p r o f i l e w i t h cons t an t K. The f a i l u r e of t h i s s o l u t i o n t o b e

v e r i f i e d by observat ion has been gene ra l ly a t t r i b u t e d t o i n a c c u r a t e c l o s u r e

assumptions; t he re fo re , boundary l a y e r i n v e s t i g a t i o n s have been usua l ly

confined t o empi r i ca l approaches.

More r ecen t ly , t he Ekman s p i r a l was found t o b e uns t ab le t o i n f i n i t e s -

imal per tu rba t ions under many condi t ions. This f a c t has been incorporated

i n t o a model producing a modified s p i r a l (Sec. 10).

2.3 Boundary Layer Methods

2.3.1 Surface Layer Methods

One approach which recognizes the s t a t i s t i ca l n a t u r e of t u rbu len t

f l u c t u a t i o n s and at tempts t o measure them in the immediate v i c i n i t y of t h e

boundary is t h e eddy c o r r e l a t i o n method.

C o r i o l i s f o r c e i s n e g l i g i b l e , t h e governing equat ions become parameter less ,

and l o c a l p a r a l l e l flow s i m i l a r i t y l a w s may b e hypothesized f o r t h e mean

flow. I n t h i s case, t h e Navier-Stokes equat ions are reconsidered. The

molecular v i s c o s i t i e s are neglected and t h e boundary l a y e r assumptions are

made.

and stresses appear as mean momentum f l u x terms only.

i s assumed. The o b j e c t is t o f i n d empi r i ca l parameters, such as an aero-

dynamic roughness c o e f f i c i e n t , which relate t h e measured f l u x e s and s u r f a c e

stress t o t h e measured mean wind, f o r a broad range of su r faces . This lower

region of t h e boundary l a y e r , wherein the f l u x e s remain f a i r l y constant , i s

c a l l e d t h e s u r f a c e l aye r . It is general ly 10-30 m th i ck .

In such a shallow region, t h e

The flaw i s then separated into a mean p lus a tu rbu len t component,

A m e a n s teady s ta te

Measurements can vary from d i r e c t stress measurements on a drag p l a t e

t o d e t a i l e d v e l o c i t y f i e l d s . I n the eddy c o r r e l a t i o n method, d i r e c t measure-

ments o f f l uxes are r e l a t e d t o t h e mean wind a t a f i x e d level (usua l ly 10 m). I n the p r o f i l e method, t h e f l u x and stress are determined i n conjunction

w i t h several s i m i l a r i t y constants der ived from measurements of v e l o c i t y and

temperature p r o f i l e s i n t h e s u r f a c e l a y e r .

14

2 . 3 . 2 S i m i l a r i t y Methods

Dynamic s i m i l a r i t y a n a l y s i s can b e employed by using t h e equat ions

and boundary condi t ions appropr i a t e t o d e f i n i t e regions. The s u r p l u s of

v a r i a b l e s f o r t h e enclosed set of equations is then el iminated by p o s t u l a t i n g

t h a t they occur only i n s p e c i f i c combinations. Idhen this technique is used

i n conjunction w i t h a two-layer concept, t h e s u r f a c e l a y e r o r i nne r s o l u t i o n

can be patched t o t h e corresponding l i m i t form of the Ekman l a y e r , o r o u t e r

so lu t ion . This provides a r e l a t i o n between the s u r f a c e and freestream

boundary condition. This approach i s discussed i n Sections 6 and 7.

2 . 3 . 3 K Theory Methods

Another line of approach considers t h e entire v e l o c i t y p r o f i l e as

the boundary l a y e r and includes C o r i o l i s fo rces and three-dimensionality i n

the ana lys i s . The b a s i c search i n t h i s approach i s f o r an empi r i ca l

parameter izat ion of t he eddy viscous f o r c e s t o balance t h e Cor io l i s fo rce .

The unknawn p r o p o r t i o n a l i t y c o e f f i c i e n t between stress and rate of s t r a i n

is o f t e n designated K , from t h e o r i g i n a l eddy v i s c o s i t y concept, and these

methods can b e c o l l e c t i v e l y ca l l ed K t h e o r i e s .

When t h e m e a n momentum t r a n s p o r t terms are approximated using an eddy

v i s c o s i t y , t h e assumptions and equat ions are similar t o Figure 1 f o r the

Ekman equat ions, except t h a t the stress terms take t h e form (KUz)z wi th a

v a r i a b l e K ( z ) i n t h e closure assumption. (Subscr ipt x denotes p a r t i a l

d i f f e r e n t i a t i o n . )

2.3.4 Numerical Method

Addit ional approaches, made p o s s i b l e by t h e advent of l a r g e computer

c a p a b i l i t i e s , o f f e r new p o s s i b i l i t i e s f o r boundary l a y e r ana lys i s . One i s

the d i r e c t , numerical i n t e g r a t i o n of t h e entire time-dependent equat ions.

In t h i s case, t h e complete Navier-Stokes equat ions can be considered,

including t i m e dependence. This approach appears t o b e l i m i t e d only by the

computer capaci ty , b u t t h e broad range of scales appearing i n the geophysical

boundary layer problem makes t h i s p r a c t i c a l l i m i t a t i o n c r i t i ca l f o r t h e next

1 5

f e w generat ions of computers.

dimensions of t h e box t o be ca l cu la t ed and a p r a c t i c a l f i n i t e d i f f e r e n c e

s t e p s i z e .

box" n a t u r e of t h i s method a l s o r e q u i r e s extensive computer t i m e f o r numeri-

c a l experiments t o e s t a b l i s h a cause-effect r e l a t i o n s h i p , by varying i n p u t s

and boundary condi t ions and by analyzing d i f f e r e n c e s i n the prodigious d a t a

generated.

A balance m u s t be made between the e x t e r i o r

Subgrid s i z e phenomena must then be parameterized. The %lack

The numerical methods are n o t discussed i n t h i s paper.

2 . 3 . 5 Secondary Flow Model

The f i n a l method discussed h e r e r e t u m s t o t h e large-scale Ekman

s o l u t i o n , b u t modifies i t by including secondary flows,

both a numerical s o l u t i o n of t h e Ekman l a y e r s t a b i l i t y problem and a f i n i t e

pe r tu rba t ion equi l ibr ium hypothesis.

This model r e q u i r e s

The b a s i c concept of t h i s model i s t h a t the Ekman s o l u t i o n i s n o t

observed because i t is uns t ab le t o small pe r tu rba t ions . However, as t h e

pe r tu rba t ions grow, they w i l l modify t h e mean flow such t h a t i t becomes

s t a b l e .

Ekman s p i r a l ) p lus a f i n i t e pe r tu rba t ion secondary flow ( h e l i c a l r o l l

c i r c u l a t i o n ) . The d e t a i l s of t h e flow w i l l depend upon t h e shape of t h e

two-dimensional pe r tu rba t ion (wave length, o r i e n t a t i o n , and magnitude as a

funct ion of h e i g h t ) , an energy balance t o determine t h e equ i l ib r ium magni-

tude, and the i n t e r a c t i o n wi th t h e mean flow.

h o r i z o n t a l homogeneity assumption i s relaxed, y i e ld ing t h e modified Ekman

equat ions f o r t he mean flow.

developed i n Sect ion 10.

The r e s u l t i n g equi l ibr ium flow c o n s i s t s of a mean flow (a modified

In Figure 1, only t h e

The p e r t i n e n t equat ions t o t h i s flow are

2.3.6 Summary

Most of the a n a l y t i c progress h a s been made f o r t h e n e u t r a l l a y e r

under s t eady state and h o r i z o n t a l homogeneity.

s i m p l i f i e s t h e l i n e a r i z e d equat ions.

t i o n a n a l y s i s f o r secondary flows.

assumption is t h e numerical i n t e g r a t i o n of t h e non l inea r equat ions.

The las t assumption g r e a t l y

It is relaxed somewhat i n the per turba-

The only v i a b l e alternative t o t h i s

16

Departures from s t eady state de r ive p r i n c i p a l l y from synop t i c d i s t u r -

bances and t h e d i u r n a l cycle . Few at tempts have been made t o e v a l u a t e these

e f f e c t s .

s u c c e s s f u l s teady state model. I n the Arctic, t h e d i u r n a l cycle i s minimized,

and synopt ic v a r i a t i o n s are less dynamic than those found in t h e middle

l a t i t u d e s .

Consideration of nonsteady state can b e s t be incorporated i n t o a

S t r a t i f i c a t i o n e f f e c t s are essential t o a good working model.

value of a n e u t r a l l a y e r model w i l l b e determined in l a r g e p a r t by i ts easy

extension t o nearby s t r a t i f i e d mean states. However, the n e u t r a l state may

be simply t h e in t e rmed ia t e s ta te between two very d i f f e r e n t s o l u t i o n s , making

The

e x t r a p o l a t i o n d i f f i c u l t . The add i t ion of convective e f f e c t s g r e a t l y complicates any model.

Convection can e s t a b l i s h a boundary l a y e r when no mean flow is p resen t .

There i s abundant l i t e r a t u r e on convection, but no consensus on a p r a c t i c a l

working model. Since t h e r e is usua l ly a mean flow, convective e f f e c t s w i l l

be considered he re as modif icat ions t o a dynamic model. Tvlaximum a i r stress

w i l l coincide w i t h maximum mean flow condi t ions.

The boundary l a y e r regions mentioned are shown i n Figures 3 and 4 .

The c h a r a c t e r i s t i c s c a l e s w i l l be discussed i n later s e c t i o n s .

18 krr THEWAL

EDDY MIND S I Z E

1

CONSTART I t NEAT

fLUX COUPLED

TO yf(v-v,)= - Jt; - SURFACE

y f ( U - U )= k k Y STRESS

LAYER TROPICAL

+ !;;;;L

EDDl ES

- COiiVCCT 1 V r

TObJERS 9 J z

1 kn

- OOUKIIOV - - - A HAVELENGlM OF ROLL Y

K 2 S IM1 LARl TY

TIIEORIES or CONSTANT RC,.IOi,

-- u, L s =(L/f ) f

.- -- -I- L:::: - ROUGHNESS HEIGIIT

20 Y r Y ?,

l O U n

Y

10 1?7

1 lr;

IOCrr;

lU7

Fig. 3 . Some c lass i f icat ions of boundary layer regions and characteristic lengths.

1 3

TropoPause ___I t -

10-20 km

Geostrophic Flow - 1 km -L

-T- ‘\I IN-

-- ~

Freauent Inversion Height - - - - ~ _ I

~ e . 5 - 3 1 km Ekman Layer f

Smooth Surface

-

Fig. 4. Schematic of scales i n t h e atmospheric boundary l a y e r .

19

3. DEVELOPMENT OF THE GOVERNING EQUATIONS

3.1 P r i m i t i v e Equat ions

It is unwieldy and confusing t o include e x p l i c i t l y a l l of t he terms

i n t h e general d i scuss ion of t h e equat ions.

f o r example, t h e t o t a l v e l o c i t y v e c t o r i n t h e boundary is f i n a l l y sepa ra t ed

i n t o a b a s i c mean component; a m e a n modif icat ion component; a zero mean,

secondary flow; and t h e random v e l o c i t i e s of t u rbu len t flow.

convenient t o d i scuss t h e i n i t i a l development of t h e equat ions in t h e i r most

p r i m i t i v e form, where t h e terms are a l l - i n c l u s i v e bu t very general .

prel iminary s i m p l i f i c a t i o n of t h e equat ions is o f t e n done i n l a r g e s t e p s ,

with only a b r i e f d e s c r i p t i o n of t h e assumptions used.

f a i l u r e of a f i n a l model f o r t h e flow can sometimes b e t r aced t o an inappro-

p r i a t e app l i ca t ion of one of t he fundamental hypotheses, c e r t a i n assumptions

which might be obvious i n classical f l u i d dynamics may b e t r e a t e d more care-

f u l l y he re w i t h t h e h inds igh t provided by experience in geophysical appl ica-

t i o n s .

In t h e secondary flow model,

It i s t h e r e f o r e

The

However, s i n c e t h e

We have the axiomatic conservation equat ions

pV = d i v g - F (momen tum) e

- . p + v . v = 0 (mass)

pcvP = V KVT - PV V + Qr + Qp + (9 (energy)

(& = dWk - dQ)

p = Pm (state)

(3-1)

- u i s t h e stress t enso r ,

F are body fo rces , t o include g, the uniform g r a v i t y f i e l d , and t h e -

v i r t u a l f o r c e f o r t h e e a r t h as a r o t a t i n g frame of reference, f = 21;2sin(lat),

E is t h e net energy, Wk is work done on the system, and Q is h e a t added.

20

The fmdamental problem i n s o l v i n g t h i s set of equat ions is t o properly

desc r ibe the stress f o r c e . The stress t enso r is t h e ope ra to r which acts on

the unit normal t o provide the stress f o r c e vec to r .

Force = /n gdA = /. dA = d i v g d V A r e a Area Vol .

$n a dA = ip:i -& 1 d i v 2 dv = d i v g- Force - l i m i t 1 - -- 6V+O -87 -

Unit A r e a

6A 6V

We assume t h a t t h e major scale p r o p e r t i e s which de f ine t h e kinematic

and thermodynamic state of t h e p l ane ta ry boundary l a y e r may b e def ined on

scales small enough t o b e considered i n f i n i t e s i m a l . make t h e assumptions f o r t h e stress which c o l l e c t i v e l y de f ine t h e Newtonian

f l u i d :

Prom observat ions, w e

1. The s t a t i c f l u i d stress reduces t o t h e p re s su re f o r c e only.

2. The stress is independent of h e a t f l u x , depending on l o c a l

kinematic and thermodynamic states only.

There exis t no p r e f e r r e d d i r e c t i o n s . 3. 4 . The stress is p ropor t iona l t o t h e v e l o c i t y gradient .

The f i n a l assumption, based on experimental observat ions, sugges t s

expanding

0 -_ -

Using the

19 70) 0 - -

t h e stress i n a Taylor series i n v: 1 A + grad V _ - assumptions above, t h i s expression can be w r i t t e n (e.g., s a t c h e l o r ,

where I i s t h e i d e n t i t y t enso r and * denotes t h e a d j o i n t ope ra to r .

K2 are scalar c o e f f i c i e n t s which may depend on the l o c a l state of the f l u i d .

K, and

The stress f o r c e is then written:

d i v 1- -- = grad (K1 d i v V) + d i v [K2(grad V + grad* v)] By means of va r ious v e c t o r i d e n t i t i e s , t h i s expression becomes

d i v - = K,V(V V) + K,[V2V + V(V v)] + VK,(V V) + VK2(Vv + V*v)

21

Ant ic ipa t ing t h e eddy v i s c o s i t y analogy, w e wish t o r e t a i n t h e g e n e r a l i t y

of K ( z ) . Thus, when K i s assumed t o be only a func t ion of z, d i v 1 - = KlV2V + K2V2V + KZV x V x V + KI V V + Kl(VV + V*V)

This expression i s s t i l l burdensome i n i t s d where t h e prime denotes - dz complexity.

t i o n f o r t h e boundary l a y e r , V v = 0, reducing t h e stress term t o a

workable form,

Fortunately w e s h a l l be able t o make t h e incompressible assump-

d iv - = K2V2v + K;(w, + u,, W Y + V,, 2 ~ ~ ) . (3-2)

For t h e last case, t h e s u b s c r i p t can be dropped from t h e parameter K, which is

t h e v i s c o s i t y c o e f f i c i e n t f o r molecular motion and t h e eddy t r a n s f e r coef f i -

c i e n t f o r t u rbu len t motion.

3.2 Sca l ing Simp1 i f i c a t i o n s

Consider t h e con t inu i ty equat ion

Pb/P + (UP, + VPY + WP,)/P + v v = 0

For c h a r a c t e r i s t i c t i m e per iods much g r e a t e r than one second, pt/p << V V . This i s p a r t i c u l a r l y t r u e f o r t h e near s teady state condi t ions considered

here . A s long as t h e v e r t i c a l e x t e n t

of t h e region i s very much less than t h e scale height H = p g / p = 12 km,

w p , / p is small.

Thus, t h e r e l a t i v e change i n dens i ty of a p a r c e l of a i r following t h e flow

is n e g l i g i b l y small compared t o t h e cross l a y e r change when t h e flow i s

confined t o a thin h o r i z o n t a l l a y e r , such as t h e boundary l a y e r region.

In t h e atmosphere, p, >> py o r p,.

The f l u i d can then b e approximated as nondivergent, V*V = 0.

One can show t h a t dens i ty changes i n t h e boundary l a y e r are s m a l l and

t h a t they r e s u l t p r imar i ly from thermal e f f e c t s compared t o p re s su re e f f e c t s

( t h e Boussinesq approximation; see Boussinesq [1903], Spiegel and Veronis

[1960], Calder [ 19681, and Brown [1970a].

These observat ions, p lus t h e assumption of a constant

v i s c o s i t y , K, produces

G = -2n x v - g - vp/p + KV2V

v*v = 0 (3-3)

22

0 -

T = KhW2T

d p / p = -- dT/T

According t o classical f l u i d dynamic boundary l a y e r ana lys i s , t h e

laminar flow s o l u t i o n of these equat ions w i l l b e replaced by completely

turbulen t flow on t h e l a r g e scales cha rac t e r i z ing t h e geophysical problem.

The molecular v i s c o s i t i e s and conduct ivi ty are neg l ig ib ly s m a l l on these

sca l e s . The equat ions may thus be w r i t t e n i n component form,

Ut + (UV)I, + (VV) + (UW), = -P,/P + fv V t + (VUE, + (“VIy + (VW), = -P,lP - fv (3 -4 )

w, + (WE, + (WY + (W, -p,m - g

Y

=

u, + V -I- w, = 0 Y where i s t h e mean dens i ty f o r t h e region, so t h a t thermal (buoyancy) e f f e c t s

are neglec ted here . Eddy viscous e f f e c t s remain i n the non-linear advection terms.

Although t h i s is a set of four equat ions i n t h e fou r unknowns U, V ,

W, and P, i t is highly nonl inear and unsolvable except by numerical m e a n s .

3 . 3 Geostrophic Equations

The top of t h e boundary l a y e r is def ined as the he igh t a t which the

f r i c t i o n fo rce becomes n e g l i g i b l e , r e s u l t i n g i n a geostrophic balance (o r

grad ien t balance i f t h e i n e r t i a term due t o curva ture i n t h e flow i s included) .

Since t h i s he igh t i s needed t o guide observat ions, and t h e geostrophic flow

c o n s t i t u t e s t h e upper boundary condi t ion, t h e na tu re of t h i s flow is of

i n t e r e s t .

When hor i zon ta l homogeneity is added t o t h e assumptions included i n

( 3 - 4 ) , we have

Ut + (UT), = -PJT + fv V t + maz = - p y m - fv (3-5)

Wt = -P,/P - g - pw, = 0

where the b a r denotes a ho r i zon ta l average.

23

Continui ty implies W 0 when W 0 a t a boundary, leaving t h e hydro-

s ta t ic r e l a t i o n . For s teady state, t h e geostrophic balance between Cor io l i s

and pressure grad ien t fo rces results i n

P,/P = -g (hydros t a t i c pressure grad ien t )

P Z l P = fv py/p = -fV

(geostrophic balance) (3-6)

These equat ions descr ibe t h e flow away from t h e boundary, where the ho r i zon ta l

components of v e l o c i t y are much g r e a t e r than t h e v e r t i c a l component, and vary

much less hor i zon ta l ly than v e r t i c a l l y .

genera l ly small i n the f r e e stream flows, so t h a t convection and buoyancy

e f f e c t s are n e g l i g i b l e . Since t h e geostrophic wind becomes the upper

boundary condi t ion f o r the boundary flow, it m u s t b e an i n p u t t o a boundary

l a y e r model. This measure of t h e f rees t ream momentum i s then t h e d r iv ing

force which is t r a n s f e r r e d t o t h e su r face i n some proport ion depending upon

the c h a r a c t e r i s t i c s of t h e f l u i d and t h e boundary sur face .

are determined d i r e c t l y by pressure grad ien t measurements under t h e geo-

s t r o p h i c balance. For large-scale determinations (1000 km), t he cen t r i fuga l

fo rce a r i s i n g from t h e curvature of t h e flow is genera l ly neg l ig ib ly small

(gradient wind component). However, i f t h e i soba r s e x h i b i t l a r g e curvature

and t h e geostrophic flow i s s t rong (g rea t e r than 16 m/sec) , a co r rec t ion t o

the geostrophic flow can be made.

Thermal e f f e c t s on dens i ty are a l s o

nese v e l o c i t i e s

These e f f e c t s are i l l u s t r a t e d by consider ing r ep resen ta t ive va lues

i n (3-3) of L = l o 3 m, V = 15 m/sec, IT = lo4 s e c and f s 5 0 = 1.46 X 10-4/sec,

t V VV + f(k x V) + VP/p = O "t 1.0 2.2 22.0 22 .o m/sec2.

The values f o r t h e f i r s t two terms are t y p i c a l max ima , so t h a t t he synop t i c

equat ion genera l ly contains only t h e last two terms, t h e geostrophic balance.

In t h e Arc t i c , t he f i r s t two terms, which arise from d i f f e r e n t p re s su re

systems moving through the area, are genera l ly smaller, about m / s e c 2 .

Although the magnitude of t h e fo rces ind ica t e s t h a t the geostrophic

balance e x i s t s a t a p o i n t , t h e r e may e x i s t l a rge-sca le divergence as t h e

pressure f i e l d diverges o r converges, d i sp lays curvature , o r changes wi th

24

time. These f a c t o r s can be e x t r a c t e d from a ca l cu la t ed flow f i e l d which

has been cor rec ted f o r t hese extremely small geostrophic devia t ions .

When s f g n i f i c a n t h o r i z o n t a l temperature grad ien ts are p resen t , t h e i r

e f f e c t s must be considered, s ince the su r face pressure f i e l d may d i f f e r

from t h e geos t:rophic l e v e l pressure f i e l d ( t h e thermal wind) .

3 . 4 Thermal Wind

The geostrophic flow varies i n the v e r t i c a l , as pressure g rad ien t s

genera l ly vary v e r t i c a l l y . From Figure 5 showing cons tan t pressure l i n e s

i n two b a r o c l i n i c s i t u a t i o n s , t h e d i f f e r e n t v e r t i c a l s t r u c t u r e of t h e

pressure f i e l d (and hence V ) i s seen f o r similar su r face temperature

grad ien ts . The gradien ts are magnified considerably f o r i l l u s t r a t i o n . 9

A 1 1

I @-

Warm Low

Fig. 5. Two d i f f e r e n t vertical pressure (hence geostrophic flow) f i e l d s f o r similar su r face temperature g rad ien t s .

The equat ions f o r t h i s "thermal wind" v a r i a t i o n i n t h e geostrophic wind can

be obtained d i r e c t l y from t h e geostrophic balance by d i f f e r e n t i a t i n g and

using

The f

the equat ion of state

u = (Ug/T>Tz - (g/fr>Ty 93

rst terms on t h e right-hand s,..2 of

second terms. For l imi t ed v e r t i c a l e x t e n t

(3-7)

3-7) are much smaller than t h e

(e. g. , Ekman l a y e r depth) , the

h o r i z o n t a l temperature grad ien t is near ly constant , and v e r t i c a l v a r i a t i o n

25

of t h e geostrophic flow is l i n e a r with he ight . The magnitude of t h i s e f f e c t

For horizon t a l temperature g rad ien t s of: (OC/106km)

0.25

0.5

0.75

1.0

1.25

1.5

1.75

2.0

i s shown i n the following t a b l e f o r t he Arc t ic .

TABLE 1

THERMAL WIND AT 75'N LATITUDE AND O°C AIR TEMPERATURE

There are v e r t i c a l g rad ien ts of geostrophic wind: (m/sec/km)

0.65

1.28

1.92

2.57

3.21

3.85

4.49

5.14

Thus, i n condi t ions of s t rong thermal g rad ien t s , the top of t h e boundary

inf luence may be determined by the occurrence of a l i n e a r shear with height

r a t h e r than a constant geostrophic flow. Furthermore, t h i s v e r t i c a l varia-

t i o n i n t h e v e l o c i t y p r o f i l e w i l l be superimposed on t h e f i n a l mean

boundary l aye r so lu t ion . Some examples are shown i n Figure 6 .

3.5 The Ekman Layer Equations

To ob ta in t h e boundary e f f e c t , one must r e t a i n the m e a n momentum

t r anspor t terms.

i z e the equat ions by consider ing the t o t a l v e l o c i t y t o be composed of a

mean p a r t p lus a zero mean per turba t ion .

t a l l y averaged vers ion of (3-5) a f t e r s u b s t i t u t i n g V = V + V ' , etc., and

then dropping the primes and ba r s on mean v e l o c i t i e s :

To obta in an a n a l y t i c so lu t ion , i t is necessary t o l i n e a r -

Consider t h e s teady-s ta te horizon- -

uwz = -Px/p + f V = f ( V - V9)

9 = -P,/P

mz = -Py/p - f U = -f(U - U9) (3-8)

where Ug and V are s u b s t i t u t e d f o r t h e cons tan t pressure grad ien ts from (3-6). 9

26

/ 1.2

26.8.

'i

I "r f

763 538

Fig. 6 . Hodographs of Ekman s p i r a l s modified by a constant thermal wind, v = v + cz (from Blackadar, e t al., 1965). 9 90

The h y d r o s t a t i c balance i s r e t a ined throughout t he boundary l a y e r .

pursue t h e viscous analogy f o r t h e tu rbu len t eddy momentum t r a n s p o r t UT, iJe in t roduce t h e eddy t r a n s f e r c o e f f i c i e n t K s o t h a t

To

- - 3 : - uw E KUz = T , v w KVz 3 (3-9)

27

(3-10)

I n the absence of a d e f i n i t i v e theory o r measurement f o r t h e va lue

of K, a constant va lue i s o f t e n assumed and r e s u l t s i n the Ekman equat ions.

This rou te t o t h e s e equat ions is s l i g h t l y d i f f e r e n t from t h e d i r e c t , postu-

la t ive eddy v i s c o s i t y assumed i n t h e d i scuss ion of Figure 1. There i s no

d i f f e r e n c e in t h e requirement f o r a properly defined continuum, s ince t h e

ex i s t ence of a mean d e r i v a t i v e without s i g n i f i c a n t change in t h e p o i n t va lue

of the momentum t r a n s p o r t term UW is implied.

t h a t t h e r e must exist only a s i g n i f i c a n t h o r i z o n t a l mean f o r on a scale

s m a l l enough t o de f ine t h e v e r t i c a l d e r i v a t i v e of U. Since t h e h o r i z o n t a l

e x t e n t is e s s e n t i a l l y i n f i n i t e , t h i s m e a n s t h a t a vertical scale which i s

c h a r a c t e r i s t i c of t h e t y p i c a l scale of t h e ind iv idua l elements which produce

uw must b e much less than t h e c h a r a c t e r i s t i c depth of t h e m e a n v a r i a t i o n .

Hence t h e eddy elements might b e as l a r g e as several decameters without

imposing a random cha rac t e r t o t h e mean v e l o c i t y d e r i v a t i v e .

ties which occur when U, o r V , i s equal t o zero i n the equat ions de f in ing K

restrict t h e usefulness of t h i s r e l a t i o n f o r a r b i t r a r y v e l o c i t y p r o f i l e s .

However, i t does po in t o u t

-

The s i n g u l a r i -

28

4. THE GEOSTROPHIC/EKMAN LAYER SOLUTION

4.1 Discussion -

For t h e AIDJEX boundary l a y e r model, we hope t o achieve some measure

of t he s u r f a c e stress based on synop t i c scale parameters.

parameter i s expected t o b e t h e geostrophic flow vec to r .

general ly has good ver t ical coherence i n t h e p re s su re g rad ien t , s o t h a t

s u r f a c e p re s su re measurements should provide a good measure of t h i s boundary

condi t ion by using the r e s u l t s of Sec. 3.3 . In o rde r t o relate t h e geo-

s t r o p h i c flow e x p l i c i t l y t o t h e Ekman flow, and even tua l ly t o t h e s u r f a c e

l a y e r flow, it i s b e n e f i c i a l t o r ede r ive a continuously v a l i d s o l u t i o n i n

the range 0 < z < 03.

The main i n p u t

The atmosphere

- - As t h e r e i s no c h a r a c t e r i s t i c l eng th i n the v e r t i c a l , a p r i o r i , a

nondimensionalization wi th r e spec t t o t h e a r b i t r a r y c h a r a c t e r i s t i c l eng th L

reveals a s i n g l e s i n g u l a r i t y parameter, E = K/fL2. l a r g e , as t h e s c a l e he igh t , Plpg 20 km, then t h e E -+ 0 l i m i t y i e l d s t h e

geostrophic equat ions. I f w e choose L = 6 = (K/f)** and the Ekman equat ions r e s u l t . These equat ions are then s e l f - s i m i l a r w i th

r e spec t t o 6 . measure i n cm, then t h e s u r f a c e l a y e r equat ions are found i n t h e r e s u l t i n g

I f L is chosen t o b e

1, = 200 m, then E = 1,

When L is chosen t o b e very s m a l l , L = z 0 , a s u r f a c e roughness

E + - w l i m i t .

This condi t ion, of d i f f e r e n t s o l u t i o n s app l i cab le wi th in a continuous

range of a f r e e independent v a r i a b l e , suggests applying the p r i n c i p l e s of

i nne r and o u t e r (and intermediate) s o l u t i o n s . The p l ane ta ry boundary l a y e r

composite s o l u t i o n is not a p a r t i c u l a r l y good, o r taxing, example of t h i s

powerful mathematical technique. This i s because the o u t e r (geostrophic)

and inne r ( su r face l a y e r ) s o l u t i o n s are h igh ly s i n g u l a r pe r tu rba t ions on t h e

high-order intermediate (Ekman) s o l u t i o n . Thus , t h e s i n g u l a r "end" so lu t ions

are t r i v i a l , and t h e matching condi t ions are a l s o degenerately simple.

Tiowever, t he b a s i c i d e a of s e p a r a t e l y so lv ing and matching t h e s o l u t i o n s

provides a d i d a c t i c framework f o r organizing t h e d i v e r s e approaches t o t h e

many a spec t s of t h e p l ane ta ry boundary l a y e r .

29

4.2 The Geostrophi c/Ekman Layer Equations

A h A

P,/p - f? - K U z z - K,U, = 0

$/p t fc - K?,, - K,?, = 0

Consider (3-lo), including t h e p re s su re g rad ien t term i n p l a c e of V 9

(4-1)

?,/P = - g

Nondimensionalize (4-1) with r e spec t t o a r b i t r a r y c h a r a c t e r i s t i c

length L , v e l o c i t y V o , p and Po ( s i n c e only p re s su re g rad ien t s occur) .

ilenceforth, a l l v a r i a b l e s are nondimensional un le s s otherwise s t a t e d .

"Hats" denote dimensional v a r i a b l e s .

The D parameter appears only i n conjunction w i t h p re s su re .

thus b e el iminated by transforming P ' = Ro D P and dropping the primes.

Eiis r e f l e c t s t he f a c t t h a t t he p re s su re gradient can assume a r b i t r a r y

va lues as a f r e e parameter.

ocean on the synop t i c scale, except ing the neighborhood of t h e s i n g u l a r

p o i n t a t the equator .

It can

Ro*D is of o r d e r un i ty in the atmosphere o r

Equations (4-2) can be combined i n t o a s i n g l e equat ion w i t h ROOD E 1 vzzzz + 2(Kz /K)Vzzz + (Kz /K)2Vzz + V = 0 (4-3)

This equation can be solved by numerical m e a n s f o r V ( z ) when an a r b i t r a r y

K ( z ) has been s p e c i f i e d .

d i s t r i b u t i o n s . A closed s o l u t i o n can b e found f o r s p e c i f i c K

Now consider t h e continuous s o l u t i o n i n the region above an i n f i n i t e

p l ane f o r t h e case of K constant:

V + E U,, - Px/p = 0

U - E V,, + P,/p = 0

30

(4-4)

g, + P,,/P - 0 - (4-4 1

wi th boundary condi t ions V ( w ) = G s i n a, U ( w ) = G cos a, V(0) = 0, and

U ( 0 ) = Uh, where uh is an a r b i t r a r y v e l o c i t y a t t h e lower boundary and G i s

the geostrophic o r f r ees t r eam v e l o c i t y .

equat ions which are app l i cab le in t h e i n v i s c i d l i m i t . This l i m i t w i l l apply

t o t h e region s u f f i c i e n t l y above t h e plane t h a t v i s c o s i t y e f f e c t s are neg l i -

g i b l e (K -+ 0, R e -+ 03):

The case E + 0 y i e l d s t h e geostrophic

v - Px/p = 0

u + Py/p = 0 (4-5)

-Pz/P =: g,

The d i r e c t "solution" t o t h e s e equat ions, Va = Px/p, Ua = -Py/p, can s a t i s f y the o u t e r boundary condi t ions i f V(m) = Px(x, y , m) = s i n a and

V ( m ) = -P,(x, y, -)/p(x, y, O) = cos a, The lower boundary condi t ions

V ( 0 ) = Uh are s a t i s f i e d only i f P(X, 3 , 0) is appropr i a t e ly spec i f i ed .

a h y d r o s t a t i c balance, only one ver t ica l boundary condi t ion nlay be s a t i s f i e d

f o r P(z, y, z ) . Thus t h e upper boundary condi t ions can b e s a t i s f i e d , b u t

i n general the lower cannot (one could choose t o s a t i s f y only the lower

condi t ions, b u t this case is n o t p e r t i n e n t f o r t h e "outer" s o l u t i o n ) .

d i f f i c u l t y i n s a t i s f y i n g t h e boundary condi t ions i s t o b e expected, s i n c e

t h e l i m i t w e have taken drops t h e o rde r of t h e equat ions by four , a s i n g u l a r

pe r tu rba t ion , and t h e r e s u l t i n g s o l u t i o n might be expected t o b e nonuniformly

v a l i d i n t h e e n t i r e region.

For

The

This deficiency can b e remedied by adding an i nne r s o l u t i o n which

w i l l s a t i s f y t h e inne r b o u n d a b condi t ions while l eav ing t h e ou te r s o l u t i o n

unchanged.

and t h e corresponding boundary condi t ions , He.nce, l e t U = Ui + Ua, V = Vi + Vu. Subs t i t u t ing i n t o (4-4)

Vi + E U; = o

ui - E vi = o zz

ZZ

wi th boundary condi t ions,

(4-6)

31

u p = v p = 0

I f we l e t < = E-'z in (4-6),

The s o l u t i o n t o (4-7) is

Vi(<) = ~xP[ -c . I (C~ COS< - C, s i n < ) + e x p [ t ] ( ~ , cos< - C, sin<)

vi(<) = exp[-<](Cl s i n < + C, cos<) + exp[<](C3 s i n < + C, cos<) . ( 4 - 8 )

Then the complete, nondimensional s o l u t i o n , uniformly v a l i d i n 0 - - < z < m, is

exp[-zl (p0 + ~ ~ > c o s z + p0 s i n z

P PO [ Y X 1 -pY + U(2) = ui. + ua = -

(4-10)

+ exp[-zI [(Po + Uh)sin z - Po X cos z 1 pX V ( Z ) = V3 + va = -

P PO Y

where t h e c h a r a c t e r i s t i c length has been chosen as L 6 5

and the c h a r a c t e r i s t i c v e l o c i t y is Vo G G (geostrophic speed).

Note t h a t w e have s t r e t c h e d t h e z coordinate by d iv id ing i t by 6 t o

obtain t h e inne r so lu t ion .

i n the inne r so lu t ion .

This then is t h e n a t u r a l s ca l ing parameter f o r z

It is t h e e-folding depth of Ui(z) .

32

The s o l u t i o n (4-10) s a t i s f i e s t h e boundary condi t ions

u9 u p ) = -P,(z, y , "(3, y, 00) = cos a =

v9 U(0) = -P (x, y , 9 ) l P b Y y , 3) + Po /Po + Uh = Uh

V ( 0 ) = Pz(X, Y , O)/P(ZY Y , 0) - Po / P o = o

V ( w ) = Px(z, y, m)/p(z, y , 03) = s i n a =

Y Y

2

This choice of boundary condi t ions implies a coordinate system chosen

with t h e z-axis p a r a l l e l t o t h e s u r f a c e l a y e r v e l o c i t y Uh ( p a r a l l e l f low),

s o t h a t a is t h e angle between t h e geostrophic flow and t h e s u r f a c e l a y e r flow.

The s o l u t i o n i s given i n terms of a p resc r ibed p res su re g rad ien t

f i e l d . To proceed a n a l y t i c a l l y , i t i s necessary t o assume a form f o r t h i s

d i s t r i b u t i o n . A constant p re s su re g r a d i e n t i s the s imples t , and n o t unreal-

i s t i c , assumption. I n t h i s case, t h e p re s su re gradient a t t h e upper boundary

s p e c i f i e s t h e geostrophic flow, and we may s u b s t i t u t e

us = -Py/po = cos a, Vg = P,/po = s i n a.

The Ekman l a y e r s o l u t i o n then becomes

U ( z ) := cos a - exp(-z) [ cos (a + z ) - U COS z ]

V(z) h

.- s i n a - exp( -z ) [ s in (a + z ) - Uhsin z ] .

(4-11)

(4-12)

Dimensionally, t h i s is

A e = G {cos a - exp(-9/6)[cos(a + 6) 2 - 7 uh cos -1) 6 (4-13)

33

5. THE SURFACE LAYER

5.1 Discussion >

Xotivat ion toward s u r f a c e l a y e r a n a l y s i s w a s provided by t h e l a c k of

obse rva t iona l v e r i f i c a t i o n of t h e c h a r a c t e r i s t i c Ekman s p i r a l of t h e ou te r

s o l u t i o n ,

a = 45".

n e a r e r t o 30" f o r n e u t r a l condi t ions, and may vary considerably f o r d i f f e r e n t

s t r a t i f i c a t i o n s . I f the v e l o c i t y i s n o t forced t o zero a t t h e lower boundary

b u t assumes a constant value, then c1 is reduced i n proport ion t o t h i s

constant , as i s t h e stress. One must then consider t h e s u r f a c e l a y e r t o

e s t a b l i s h t h e value f o r U , ( O ) = Uh.

The implied su r face stress, T,, = KdU(O)/dz , corresponds t o

Observed angles between t h e geostrophic and s u r f a c e winds are

Another remedy t o accommodate t h e ou te r s o l u t i o n t o observat ions may

be found i n t h e s t ab i l i t y / secondary flow approach discussed i n later s e c t i o n s .

Measurements i n geophysical boundary l a y e r s gene ra l ly take p l a c e i n

the lowest po r t ion of t h e l a y e r . Steady state with constant p re s su re

g rad ien t is assumed. Observations suggest t h a t t he ver t ical v a r i a t i o n s of

stress and v e l o c i t y are much g r e a t e r than t h e h o r i z o n t a l v a r i a t i o n s .

Indeed, t h e h o r i z o n t a l plane is e f f e c t i v e l y i n f i n i t e i n extent, w i th no

characteristic l e n g t h with which t o scale f i n i t e v a r i a t i o n s .

fo rce term i s sca l ed by 1/Ro = f i / V , .

e f f e c t s t o be s i g n i f i c a n t , as we are consider ing a l a r g e h o r i z o n t a l ex ten t .

In p r a c t i c e , t h e r a t i o of ver t ical t o h o r i z o n t a l range need be only ve ry

small, s o t h a t f o r very t h i n l a y e r s t h e h o r i z o n t a l e x t e n t may a l s o become

too small f o r C o r i o l i s phenomena t o b e important.

The Cor io l i s

One might expect C o r i o l i s fo rce

We are l e f t w i t h an equi l ibr ium flow such t h a t t h e sum of t h e f o r c e s

The only h o r i z o n t a l f o r c e i s t h a t due t o the stress g rad ien t . is ze ro ,

Thus, t h e stress is constant i n such a l a y e r . From Newton's experiment f o r

p a r a l l e l flows, t h e v e l o c i t y g rad ien t i s p ropor t iona l t o t h e stress divided

by the eddy v i s c o s i t y , which may vary v e r t i c a l l y .

increases i n proportion t o t h e v e r t i c a l coordinate , t h e i n t e g r a t i o n of t h i s

r e l a t i o n p r e d i c t s t h a t t h e v e l o c i t y varies as t h e n a t u r a l l og of t h e ver t ical

coordinate.

I f t h e eddy v i s c o s i t y

34

5.2 Surface Layer Equations

These assumptions can b e formalized, s t a r t i n g with t h e dimensional

p r i m i t i v e equat ion . A u + 2s1 x ^u - 6 E + v;/; = 0.

This becomes

f(v̂ - V g ) + d/dz^(?3C/p) = o f (v^ - Ug) - d/d^z(TY/p) = 0

Q(0) = 0,

(5-1)

under t h e assumptions of s t eady state, no h o r i z o n t a l v a r i a t i o n o f 3 , p = po

(constant) and constant V P = v 9'

Nondimensionalizing with r e s p e c t t o a r b i t r a r y V , , H, and T ~ , and

o r i e n t i n g the axis i n the stress d i r e c t i o n , produces

(V - V g ) + E f d / d z ( T ) = 0 (5-2)

where E

( o r f +. 0) , E" +. m, and (5-2) y i e l d s d T / d z = 0.

~ ( 0 ) = 1 implies T f T,,.

is a form of the Ekman number and equals -ro/(p0fVoH). For H +. 0

The boundary cond i t ion h

This constant stress l a y e r i s app l i cab le i n t h e region where E' i s

l a rge . 2 e c a l l i n g t h e eddy stress concept, T may be writ ten as t h e rate of

t r a n s f e r of x momentum by eddy d i f f u s i o n i n terms of t h e eddy c o e f f i c i e n t : -

:/p = - $3 = K GZ Thus, t h e equat ion f o r U i n t he constant stress l a y e r is

A

K d$/& = T o / p 0 = Uk, 2(0) = 0, U,(O) = T O / K ( O ) (5-3) - 2

5.3 Surface Layer Solution

The equation f o r t h e s u r f a c e l a y e r can a l s o b e obtained from (5-2) by s u b s t i t u t i n g t h e eddy v i s c o s i t y , producing (4 -2 ) . When t h e l i m i t E -f co is

taken i n ( 4 - 2 ) , t he equat ions appropr i a t e t o t h e region of high v i s c o s i t y

and/or s m a l l c h a r a c t e r i s t i c dimension H, are

35

U'I + ( K ' / K ) U * = 0

where t h e prime denotes d i f f e r e n t i a t i o n . The momentum equat ions have

become decoupled wi th t h e demise of t he Cor io l i s term, and only one equat ion

i s needed. We consider t h e coordinate system o r i en ted p a r a l l e l t o U ( z ) U i n t h i s p a r a l l e l flow region.

The equation ( 5 - 4 ) is of t h e general form, U" + P ( z ) U ' = 0. While no

general so lu t ion f o r a r b i t r a r y P(z) e x i s t s , so lu t ions can be found f o r

c e r t a i n P ( z ) . In p a r t i c u l a r , f o r K = Koz,

U" + l / z U' = 0 ( 5 - 5 )

This i s a form of Euler ' s equat ion, and t h e s u b s t i t u t i o n s 4 = U' and

5 = Zn z , l ead t o the so lu t ion ,

However, t h i s s o l u t i o n f a i l s t o s a t i s f y t h e lower boundary condi t ion

A s o l u t i o n which satis- U ( 0 ) = 0.

f i e s t h i s boundary condi t ion i s

This is a consequence of K ( 0 ) K , = 0.

u = B !?Jn(z + 11, with

K = K,(z + l ) , ( 5 - 7 )

c, = 0.

The constant cannot be determined without an add i t iona l boundary condi t ion

s i n c e both condi t ions i n ( 5 - 4 ) are s a t i s f i e d i d e n t i c a l l y .

5.4 Solution Via Dimensional Reasoning

Alterna te ly , t h e s o l u t i o n can be taken from (5-3):

The only c h a r a c t e r i s t i c parameters f o r t h e nondimensionalization of t h i s

set are v, = ( T , / P ) k u* (5-9)

36

The nondimensional equat ion becomes

(5-10)

I f w e choose K = ku,z0z, where k is an a r b i t r a r y cons t an t , t he governing

equat ion becomes parameter less , dU/dz = z l k , r e s u l t s :

and a s e l f similar s o l u t i o n

1 h 1 u = Rn z , o r Ulu, = kn 2 / z 0

A s be fo re , t o s a t i s f y the lower boundary condi t ion U ( 0 ) = 0, set

K = ku,z,(z + 1)

y i e l d i n g

(5 -11)

The constants k and z o must b e determined experimentally.

This implies t h a t KO = ku,z , . The phys ica l explanat ion of t h e

ex i s t ence of K O is n o t clear.

s u r f a c e l a y e r i s expected, and i t corresponds t o t h e mixing l eng th concept.

iiowever, the linear i n c r e a s e of K i n the

5.5 Mixing Length

F i n a l l y , an argument may be adduced t o relate t h e eddy mixing c o e f f i -

c i e n t t o a more fundamental parameter. One such theory is t h e mixing l eng th

concept. This carries over t h e molecular momentum exchange argument t o eddy

momentum exchange.

t i ca l fashion compared t o t h e molecular exchange process, b u t bo th r e l y on

s t a t i s t i ca l averaging.

displacement, t h e excess momentum can b e represented by a Taylor expansion:

The momentum w i l l be exchanged i n a complicated s ta t is-

I f t h e exchange takes p l a c e a f t e r only a s h o r t

U ( z ) = U(z - 2) + Z(dV/dZ)

U ( z ) = U(z + 2) + Z(dU/dz)

37

where U'(z) = Z ( d U / d z ) . cm t he eddy s c a l e f o r i s o t r o p i c turbulence,

(u ) = ( w ) ; and the Reynolds stress term i s r = uw N 22(d~/dz)2 =

and K = Z2(du/dz).

-

It appears t h a t l i t t l e progress has been made, except t o t r a n s f e r

the unknown s p e c i f i c a t i o n of K t o t h a t of 2 . may have been found i f t h e r e e x i s t s a un ive r sa l r e l a t i o n f o r t h e "fundamental"

parameter, 2 .

However, a more b a s i c r e l a t i o n

Assume Z = k [ ( P + z , ) / z , ] , then the s o l u t i o n becomes

(5-12) 1 A

u/u* = 7; An[(: + zO) /zO1

with K = Zu,z = ku,(z^ + z o ) . 0

The value of k (which could a l s o be c a l l e d Z,, the mixing length a t

the sur face) w a s found t o be 0.4 i n s e v e r a l i n i t i a l experiments, and is

ca l l ed the von K&&n constant . [Businger, et a l . , 19711 which obtained k = 0.35 have c a s t doubt on the

un ive r sa l i t y of t h i s constant .

Recent measurements i n t h e atmosphere

5.6 Surface Layer Methods of Experimental Analysis

5.6.1 Eddy Corre la t ion Method -

The d e f i n i t i o n of K = uw/U, appears convenient f o r t h e empir ica l

The su r face l a y e r is determination of K ( z ) by eddy c o r r e l a t i o n techniques.

def ined as t h a t p a r t of t h e boundary l a y e r a t which t h e fluxes are constant .

This ho lds t r u e r a t h e r w e l l f o r t h e lowest 10-20 m. In t h i s l a y e r , it is

poss ib l e t o measure the Uzj term wi th a s o n i c anemometer.

stress is r e l a t e d to the m o m e n t u m t r anspor t , t h i s measurement is a l s o a

d i r e c t measure of t h e stress.

of a drag c o e f f i c i e n t , CD = ?/e2 = - =/e2. ( o r ex t rapola ted to) some re ference level, usua l ly 10 meters.

values f o r C over ice are from 0.0014 (snow covered) t o 0.0026 [Unters te iner

and Badgley, 1964; B a k e and S m i t h , 19711.

Since t h e s u r f a c e

The r e s u l t i s genera l ly expressed in tenus

H e r e , U i s measured at

Typical

D

38

The procedure is t o p l o t the measured v e l o c i t i e s versus Rn 2, ob ta in ing a graph represented by Figure 7.

tl /U *

Fig. 7. Sketch of v e l o c i t y vs . he igh t d a t a in t h e s u r f a c e l a y e r .

The "log l aye r " is determined by a s t r a i g h t l i n e f i t t o t h i s data . The

ex t r apo la t ed c ros s ing of t h i s l i n e wi th t h e absc i s sa y i e l d s z o from (5-11).

The upper l i m i t of t h e log l a y e r h may a l s o b e ind ica t ed , b u t i t is seldom

sha rp ly defined. When several such graphs are a v a i l a b l e a t a s t a t i o n f o r

var ious v e l o c i t y p r o f i l e s , t h e s i m i l a r i t y w i t h r e spec t t o u* and z o i n

(5-11) implies that a l l graphs w i l l coincide f o r t h e properly chosen us.

The s l o p e of t h i s l i n e w i l l then y i e l d a value of k.

5.6.2 P r o f i l e Method Applied t o Diabat ic Layers

.A c e n t r a l o b j e c t i v e of micrometeorological research i s t o e s t a b l i s h

f l u x e s from a, knowledge of the mean p r o f i l e s . Many e f f o r t s i n t h i s d i r e c t i o n

have been undertaken i n the las t twenty-five yea r s o r so.

e f f o r t s t o o b t a i n a n a l y t i c a l s o l u t i o n s have n o t been success fu l , i t has been

poss ib l e t o use a s i m i l a r i t y d e s c r i p t i o n t o o b t a i n semi-empirical r e l a t i o n s .

Although t h e o r e t i c a l

A f a i r l y complete d e s c r i p t i o n has been given r e c e n t l y by Businger

e t al. [1971] using observat ions taken i n Kansas by t h e Boundary Layer Branch

of t h e A i r Force Cambridge Research Laboratory. The wind shea r w a s expressed

i n t h e dimensionless form $m = - uz, and t h e temperature gradient s i m i l a r l y k2 u+

39

A

as $h = (2/6,)qz, where 8, = w't ' /u, .

funct ions of t he dimensionless he ight 3 = $/L, where L =

Monin-Obukhov length .

30th $,,, and ($h were expressed as -u,~T

- 9 the kg T t w t

The po in t s f i t t he following a n a l y t i c a l r e l a t i o n s q u i t e w e l l :

f o r 5 < 0 @??I

= 1 + 4.75 f o r 5 > 0 4, 4, = 0.74(1 - 93)-' f o r 5 < 0

@h = 0.74 + 4.73 f o r 3 > 0

-% = (1 - 155) (5-13)

(5-14)

When 3 = 8, 4, = 1, which leads t o the logari thmic p r o f i l e f o r

z >> z o .

Equation (5-13) and (5-14) may be in t eg ra t ed t o give t h e following

e x p l i c i t expressions f o r t he wind and temperature p r o f i l e s [ see Paulson,

19701 : A

Y J - a = -(An- 1 - z O

u* k f o r 3 < 0 (5 -15)

where

Y , = 2Rn[(l + 3G)/2] + Rn[(l + x 2 ) / 2 ] - 2 tan-lz + T/2

(with x = (1 - 15<)% = 4;');

A

f o r 3 > 0; a 1 z u* 0 - = (An z + 4.73) ' (5-16)

and A

h

f o r < < 0 e - eo z 9, 0

= 0.74(Rn z - Y 2 ) (5-17)

where

Y2 = Rn[(1 + n/21

( w i t h 1 = (1 - 9g)% = 0 . 7 4 $ ~ ~ - ' ) ,

and 8, is the extrapolated temperature for z = 0. aZis is not necessarily

the actual surface temperature,

40

--

f o r 5 > 0 2 = ,0.74 Rn - + 4.75 8 - 0,

e z O (5-18)

From t h e s e equat ions i t i s clear t h a t t he d i a b a t i c p r o f i l e s depend

on two h e i g h t scales, x g and L .

t he re fo re defined by bo th za and L.

su r face layer , , t h e r a t i o za/L should be t h e same [Businger, 19551.

The s t r u c t u r e of t h e s u r f a c e l a y e r is

For f u l l s i m i l a r i t y condi t ions of t h e

Equations (5-16, 5-17, 5-18) m y b e used t o compute t h e f luxes when

observat ions of t h e p r o f i l e s are a v a i l a b l e .

the smoothed p r o f i l e s of u and

be determined.

and t h e stress from T = pu, .

By least square f i t t i n g of

t o these equat ions, u*, e,, L and z o may

The h e a t f l u x can then simply b e obtained from Fh = -epPu,B, 2

41

6. THE MATCH.ED EKMAN/GEOSTRQPHIC - SURFACE LAYER SOLUTION

6.1 L im i t Solut ions

If w e naw consider t h e geostrophic/Ekman l a y e r s o l u t i o n as t h e o u t e r

so lu t ion and the s u r f a c e l a y e r as t he inner so lu t ion , we would l i k e them

t o overlap i n some region. The two s o l u t i o n s are assoc ia ted wi th two

d i spa ra t e c h a r a c t e r i s t i c lengths , e.g, 6 and z 0 .

b a s i c s o l u t i o n t o (4-1) f o r mesoscale o r l a r g e r C h a r a c t e r i s t i c lengths is

the geostrophic balance. When the vertical coordinate is s t r e t c h e d by 6, t h e c h a r a c t e r i s t i c scale of t h e Ekman l a y e r , 'the h ighes t o rde r terms i n (4-1) are r e t a ined i n t h e E + 00 l i m i t .

boundary condi t ions is then found (sec. 3.6). This would be t h e complete,

uniformly v a l i d s o l u t i o n f o r 0 5 Z 5 00 i f it were not f o r t he f a c t t h a t i n

geophysical problems t h e region of p r i n c i p a l interest (and t h e preponderance

of measurements) is o f t e n bur ied deep i n s i d e t h e Ekman l aye r . In p a r t i c u l a r ,

we know t h a t t h e eddy v i s c o s i t y c o e f f i c i e n t is not constant i n the l a y e r

immediately ad jacent t o t h e sur face .

l i n e a r i nc rease wi th he ight f o r K i n t h i s l aye r .

so lu t ion f o r t h i s region.

We have found t h a t t h e

A complete s o l u t i o n s a t i s f y i n g t h e no-sl ip

Measurements i n d i c a t e t h a t t he re is a Thus, we need a l i m i t

When t h e vertical coordinate i s s t r e t c h e d another order of magnitude

(o r s eve ra l ) by zi, a c h a r a c t e r i s t i c scale assoc ia ted wi th t h e su r face l aye r ,

t h e su r face l a y e r s o l u t i o n , emerges (sec. 5). This s o l u t i o n is a s ingu la r

pe r tu rba t ion and thus cannot s a t i s f y a l l boundary condi t ions. It i s chosen

t o s a t i s f y the su r face condi t ion U(0) = 0, and becomes s i n g u l a r as z/z i + 00.

This c h a r a c t e r i s t i c of the "inner so lu t ion" prevents using the methods of

matched asymptotic expansions i n the usual way. However, s i n c e the re are

two s e p a r a t e s o l u t i o n s (one three-dimensional wi th cons tan t v i s c o s i t y and

one two-dimensional wi th a l i n e a r l y inc reas ing v i s c o s i t y ) , each s a t i s f y i n g

t h e lower boundary condi t ions, t h e r e exists t h e p o s s i b i l i t y of matching

asymptotic expansions of each about some inner limit.

match a coordinate expansion ( z + Q) t o a parameter expansion (E * 0 ) .

We can also terminate t h e outer s o l u t i o n a t a f i n i t e he igbt , h, where U = U(h) = Uh and V = 0 , and at tempt to pa tch the asymptotic l i m i t of this

I n p a r t i c u l a r , we can 1

42

s o l u t i o n f o r z / 6 -+ h/6 N 0 t o t h e s u r f a c e l a y e r s o l u t i o n obtained from the

l i m i t equat ions (zi/G -t 0).

I n a n t i c i p a t i o n o f t he s u r f a c e l a y e r s o l u t i o n , t h e o u t e r s o l u t i o n

The w a s made t o f i t an a r b i t r a r y lower condi t ion, U ( 0 ) = Uh a t z / 6 + 0.

second condi t ion a t the matched l a y e r w i l l be t h e con t inu i ty of stress,

or U'. We would now l i k e t o f i t t h e two s o l u t i o n s i n t o a complete s o l u t i o n

f o r t he boundary l a y e r flow. This would produce a r e l a t i o n s h i p between t h e

large-scale o u t e r s o l u t i o n , depending on v and 6 , and t h e s u r f a c e l a y e r

s o l u t i o n , depending on u* and z Q . 9

We have the o u t e r s o l u t i o n , i n dimensional form, with I V I C G: 9

U(z)/G = cos a - exp(-2/6)[cos (a + 2/6) - Uh/G cos 2/61

P(z)/G = s i n a - exp(-2/6)[s in (a + 2/61 - U ~ / G s i n 2/61

A

( 6 - 1 )

and the d e r i v a t i v e s ,

The lower boundary condi t ions are con t inu i ty of v e l o c i t y U = U and

This implies a constant h'

stress K U r h , with the p a r a l l e l s u r f a c e l a y e r flow.

d i r e c t i o n f o r t he v e l o c i t y and stress vec to r s , s o t h a t

U ' ( 0 ) = A,U(O), V'(0) = A,V(O) ( 6 - 3 )

where Ai is an a r b i t r a r y constant .

The second condi t ion i n ( 6 - 3 ) i n d i c a t e s t h a t

Uh = (cos a - s i n a)G = - v9 ( 6 - 4 )

The continuous o u t e r s o l u t i o n can now be w r i t t e n

( ̂z/ 6 1 = cos a - exp(-2/6) (cos 2/6 - s i n 2/6) s i n a G

A

- V,(5/6)

- s i n a - exp(-2/6) (cos 2/6 + s i n 2/6) s i n a G

( 6 - 5 )

4 3

This s o l u t i o n corresponds t o t h a t obtained by Taylor [1915] when a trans- formation i s made t o a coordinate system a l igned wi th geostrophic flow.

In t h e same coordinate system, t h e s u r f a c e l a y e r s o l u t i o n is

Expanding Ua in

- -

Simi la r ly ,

small 5 E z / 6 y i e l d s

cos a - s i n a(cos < - sin <) (1 - < + r 2 / 2 + . . .) cos a - s i n a(1

cos a - sin a + cos a - s i n a +

- 2 5 + c 2 - . . .> (6-7)

sin a - s i n a(cos < + s i n <) (1 - < + c2/2 - . . .I

s i n a - s i n a(1 - c2 + . . .> ( 6 - 8 )

0 + 0[r21 A

W e would l i k e t o have l i m ?a($?/6) = Ui@/zi) i n some overlap domain. f + O

6.1.1 Patching So lu t ions

The i n n e r s o l u t i o n is a l r eady i n a form appropr i a t e f o r patching t o

the asymptotic r ep resen ta t ion of the o u t e r so lu t ion : h

U a / G =

Ui/G = (Gi/u*)(u*/G) = u,/Gk Rn(%zi + 1)

(cos a - s i n a) + 2sinaRn(^z/6 + 1) A

S e t t i n g these two expressions equal in the matched l a y e r ,

cos a - s in a + 2 s i n a Ryr(h/6 + 1) = uJGk hz(h/zi + 1) or

a . cos a - sin a + 2 s i n a ( E . 1) = u,/& Rn h/z i

f o r small h / 6 .

44

This provides an expression f o r t h e pa t ch ing h e i g h t h:

h l z . = exp[ (.Gk/u,) (cos a - s i n a) 3 (6-10) 1,

For a given u,/G, 6, a, and z t h e complete s o l u t i o n including t h e i’ s u r f a c e l a y e r is

f o r cos OL - exp(-^z/6) [ cos (a + 2/61 - (cos a - s i n a)cos 2/61 a(,) = G

{ u , / G Rn(2/zi + 1) - (6-11) for

s i n a - exp(2/6) [ s i n ( a + 2/6) - (cos a - s i n a ) s i n 2/61 e(,) = G { 0

The extreme s e n s i t i v i t y of t h i s r a t i o t o k, u,/G, and a c u r r e n t l y

precludes any p r a c t i c a l determination of h / z . from measurements, z Patching d e r i v a t i v e s y i e l d s

1 1 + 6 / h N - 5 B 2 s i n cc = - Gk -

u* 2 1 + Z i / h (6-12)

This r e l a t i o n reappears i n subsequent solut ions,and a n e u t r a l l a y e r va lue

f o r B of about 5 h a s been determined from measurements of u,/G and a. corresponds t o 6/h -N 10, as expected.

This

The value of B is found t o vary

s i g n i f i c a n t l y w i t h s t r a t i f i c a t i o n .

6.1.2 Matching Solut ions

I n accordance with t h e methods of matched asymptotic expansions

[ V a n Dyke, 19641, w e w r i t e t h e o u t e r s o l u t i o n wi th r e spec t t o the i n n e r

v a r i a b l e and t ake t h e appropr i a t e i nne r l i m i t . This l i m i t w a s determined

t o b e E +- 00 from t h e equat ions.

E = </r = 6 / z i . The i n n e r v a r i a b l e i s 5 = z / z i , and

Thus, with 6 f i x e d and E -+ OJ,

= *a

V, =

cos a - s i n a + sin a [</E - %(</E)~ + . . . I s i n a - s i n a l l - ( E / E ) ~ + . . . I

(6-13)

45

Likewise, we form the

w r i t t en i n ou te r v a r i a b l e s , 1

Vi = 0 ui = E [ E < - $ ( E c ) 2

where E i s f ixed and G -t 0.

asymptotic expansion of t h e i n n e r s o l u t i o n

+ . . . I (6-14)

I f we simply match zero-order terms of (6-13) and (6-14),

cos a - s i n a E Uh = 0

impl ies a = 45'.

Matching f i r s t - o r d e r terms y i e l d s (6-11).

The zero-order so lu t ion i s an Ekman s p i r a l t o t h e su r face .

When t h e asymptotic matching p r i n c i p l e i s appl ied , t h e n t h term of

the ou te r expansion can be set equal t o

expansion ( a consequence of t h e d i spa ra t e s c a l i n g ) ,

t h e m = n + 1 term o f the i n n e r

For n = 0, m = 1,

G % - (cos a - s i n a) = u* k (6-15)

For n = 1, m = 2,

(6-16)

where E = 6 / z i is f ixed and < -N h/6 i s s m a l l .

o rde r i n the matched l a y e r .

When Z i = zo

W e see t h a t Va 0 t o second

m, then E

apply t o a very t h i n region 0[10z,].

may b e the he igh t of t h e s u r f a c e l a y e r .

B % 10/4. B e l o w t h i s he igh t , t he su r face l a y e r s o l u t i o n f o r v a r i a b l e K

p r e v a i l s . The o u t e r s o l u t i o n , which becomes inaccura t e near t h e su r face due

to t h e d e t e r i o r a t i o n of t h e constant K assumption, is matched t o t h e top of

the su r face l a y e r so lu t ion .

r e l a t i o n s h i p s between the boundary condi t ions represented by the two charac-

teristic velocities u+ and G, l a y e r match is the m u l t i p l i c i t y of v e l o c i t y p r o f i l e s f r o m the three-

dimensional o u t e r solution that are available to match to the two-dimensional

lo5 and the i n n e r s o l u t i o n would

However, t h e c h a r a c t e r i s t i c l eng th

In t h i s case E % 10, c % 0.1, and

The condi t ions f o r a smooth match s p e c i f y

An unusual f e a t u r e of the p lane ta ry boundary

46

l o g p r o f i l e . These p r o f i l e s vary as a funct ion of a, which is a funct ion

of the matching he igh t o r t h e parameter E .

6 .2 Surface Stress Relations

AIDJEX needs an expression f o r T with r e spec t t o e a s i l y measured

mesoscale parameters.

and matching, From the patching s o l u t i o n s equat ions, a t a = 45" t he s u r f a c e

l a y e r d i sappea r s and t h e Ekman/geostrophic s o l u t i o n extends t o t h e su r face .

We then have t h e following r e l a t i o n s :

We have two s l i g h t l y d i f f e r e n t r e l a t i o n s from patching

0 h = z

T/ Po := K o U r 0 ( O ) = K(2) 'G/6 = f6G/(2) '

6 = (2K/f3+ = (2u*kzo/f)~

K = KO = ku*zo

These provide t h e stress f o r a simple Ekman s p i r a l l a y e r . It depends

on f, G, and is.

condi t ions, is realist ic; however, t h e 45" angle between t h e stress and t h e

geostrophic flow i s no t o f t e n found. To use these r e l a t i o n s t o determine

stress, we need t o modify t h e Ekman s p i r a l s o l u t i o n .

consider two-:Layer s o l u t i o n s w i l l accomplish t h i s .

The magnitude of t h i s stress, 2-3 dynes/cm2 f o r t y p i c a l

The methods which

The expression f o r t h e patched v e l o c i t i e s from (6-9) may be w r i t t e n :

o r

' h 1 Rn(5 + 1) 2G b ( g + 1) kG [sin a +

u* (6-17)

where 5 = z / 6 , 5 = z / z i , U / G = cos a - s i n a. h have 5 very l a r g e ( z o i n t h e Arctic is about 0.02 cm) . (6-17) becomes u s e l e s s f o r p r a c t i c a l purposes. However, i f w e re-examine z

Note t h a t f o r zi = z o , w e

Since 5 i s small,

i

47

as determined i n (5-9), the small value x 0 is assoc ia t ed with KO, the

r e s i d u a l va lue of eddy v i s c o s i t y a t the surface!.

t i o n f o r c h a r a c t e r i s t i c K might b e t h e value a t the top of t h e s u r f a c e

l a y e r , where K equals t he constant used in the outer s o l u t i o n , KE. produces

A more appropr i a t e selec-

This

N 10 meters.

Using t h i s c h a r a c t e r i s t i c value f o r t he sur face l aye r , and the f i n i t e r a t i o s < = h/6 and

s i n a

where C, = *

5 = h / z i , we ge t

t c,] = c2

- 1 Rn(hlz; + 1 ) and C, - - 2 !?.n(h/6 + 1 )

(6-18)

The range of these constants f o r poss ib l e values of t h e r a t i o s is

i n d i c a t e d i n Table 6.1.

TABLE 6 .1

C, AND C, FOR TYPICAL BOUNDARY LAYER CHARACTERISTIC VALUES

h/ zi

0 . 1 1 0.7 3.5 3.7 0.1 i o 0.7 3.5 60.0 0.2 10 0.7 1.8 6.6 0.2 10 0.7 1.8 32 .O 0.5 10 0.7 0.7 3 .O 0.5 50 0.7 0.7 5.0 0.5 10 0.7 0.7 14 .O

In p r a c t i c e , t h e r a t i o s i n Table 6.1 cannot be measured. Rather, t h e

constants i n (6-18) must be empi r i ca l ly r e l a t e d to u,/G and a.

In t h e matched region, 5 N 1 and E = &/zi = b / h 10. If we assume

t h a t zi and 6 are c h a r a c t e r i s t i c constants f o r a given f l u i d flowing over a

p a r t i c u l a r sur face , then w e can expect E t o b e constant .

following table of values obtained from the matchhg r e l a t i o n s (6-15) and (6-16). W e then have the

48

TABLE 6.2

VALUES FROM (6-15) AND (6-16) FOR SELECTED a; E< 5 1

a u,lG E B

45 .o 0 .Q CQ 03

42.5 .02 50 14.0 40.0 .o 20 6.0 35.0 .10 10 2.5

A ske tch of the matched v e l o c i t y p r o f i l e s i s shown i n Figure 8. The

lower po r t ion of t h e o u t e r v e l o c i t y p r o f i l e s is f a i r l y i n s e n s i t i v e t o varia-

t i o n s i n a f o r 10 This i s r e f l e c t e d i n a l a r g e v a r i a t i o n i n u,/G

f o r small changes i n a, and i t hampers p r a c t i c a l app l i ca t ion . Furthermore,

t h e Ekman s p i r a l i s a v a l i d ou te r s o l u t i o n only i n the case of s t a b l y

s t r a t i f i e d f l u i d ( s e e s e c . 9 ) . Since t h i s i s o f t e n t h e condi t ion i n t h e

Arctic, t h e r e is t h e p o s s i b i l i t y of c o r r e l a t i n g t h e geostrophic drag c o e f f i -

c i e n t u,lG t o (I and a c h a r a c t e r i s t i c constant , B . The value of B i s about 5

from observat ions i n n e u t r a l t o s l i g h t l y unstable l a y e r s , corresponding t o

a = 3 8 " , u,/G "9.06.

a modified Ekm,an s p i r a l w i th secondary flow needs t o be used as t h e o u t e r

s o l u t i o n , producing s i g n i f i c a n t l y lower p red ic t ed va lues of a and u,lG.

a 5 45.

These values are l a r g e r than observed. In t h i s case,

1.0

Fig. 8. Sketch of lower p o r t i o n of two- l a y e r , matched s o l u t i o n s . U . 5

/' Yatching layer M''

T Surface 1 ayer

i i

0 0 - . 5

z 1 .o

49

- -

6.3 Summary

Nhile t h e sea rch f o r a simple, r e l i a b l e general parameter izat ion of

t he mesoscale boundary l a y e r has n o t been resolved by t h e approaches discussed

here , some progress h a s been made and a framework provided f o r f u t u r e

development.

su r f ace l a y e r ) s o l u t i o n w i t h i n the planetary boundary l a y e r , This i n n e r

s o l u t i o n provides a b e t t e r v e l o c i t y p r o f i l e p red ic t ion (proport ional to Rn z )

f o r t he n e a r s u r f a c e region by consider ing a l i n e a r increase i n eddy viscos-

i t y . It has been w e l l e s t ab l i shed by observations.

The two-layer concept provides f o r a boundary l a y e r ( t h e

The c r i t i ca l parameter governing the f i t of t h e two l i m i t s o l u t i o n s

t o the Navier-Stokes equat ions w a s found t o b e t h e r a t i o of t h e c h a r a c t e r i s t i c

length scales of t he ou te r and inne r regions, E .

s o l u t i o n p resen t s an i n f i n i t e s e l e c t i o n of p r o f i l e s t o match t h e two-

The three-dimensional o u t e r

dimensional i nne r l a y e r so lu t ion . These p r o f i l e s vary w i t h E, t he h e i g h t

of t he matching l a y e r o r t h e angle of t u rn ing i n the o u t e r s o l u t i o n .

primary d i f f i c u l t y i n app l i ca t ion arises from the i n d e f i n i t e n a t u r e of t h e

pe r tu rba t ion parameter E . There is ambiguity i n t h e determination of scale

he igh t s . Both may b e r e l a t e d t o eddy v i s c o s i t y .

e s t a b l i s h e d f o r t he atmosphere o r ocean.

The

Ne i the r has been accu ra t e ly

A s i g n i f i c a n t l i a b i l i t y t o t h i s method is t h a t t h e o u t e r Ekman s p i r a l

s o l u t i o n does n o t exis t i n an unmodified form.

numerical r e s u l t s f o r s t a b l e Ekman s p i r a l s w i th secondary flows is a possi-

b i l i t y , the general s i m i l a r i t y methods i n t h e next s e c t i o n o b t a i n corresponding

r e s u l t s without deal ing with t h e e x p l i c i t s o l u t i o n s .

While a patching t o t h e

50

7. SIMILARITY METHODS

7.1 Discussion

Most of t h e methods discussed f o r analyzing t h e boundary l a y e r have

employed t h e eddy v i s c o s i t y concept a t some p o i n t .

t h a t each method i s most o f t e n subjected t o chal lenge.

It is on t h i s assumption

I n some t h e o r i e s ,

t h e concept is less c r u c i a l than i n o the r s . It served Ekman w e l l in 1905

when he f i r s t employed t h e eddy v i s c o s i t y c o e f f i c i e n t i n the viscous-Coriolis

fo rce balance. His model gave an explanat ion f o r t h e dev ia t ion of t h e stress

f o r c e from the wind d i r e c t i o n observed i n the movement of t he arct ic ice.

Even i n t h i s i n i t i a l paper, however, t h e i d e a of a v a r i a b l e c o e f f i c i e n t w a s

considered. Ekman found l i t t l e d i f f e r e n c e i n p red ic t ed v e l o c i t y hodographs

when he used an eddy v i s c o s i t y which decreased i n proport ion t o t h e v e l o c i t y

gradient . i&en a l a rge - sca l e q u a n t i t a t i v e assessment i s needed of t h e boundary

l a y e r p r o f i l e o r of t h e s u r f a c e stress, none of t h e models has proved t o be

adequate f o r even f a i r l y general usage.

K, p r e d i c t s reasonable magnitudes of su r face stress, KU,(O).

i n v a r i a n t 45' angle p red ic t ed between t h e s u r f a c e stress and t h e geostrophic

flow i s n o t gene ra l ly observed.

a t t ach ing a su r face l a y e r to a l te r t h e lower boundary condi t ion, t h i s angle

can v a r y i n proport ion t o a d d i t i o n a l parameters determined by t h e su r face

l a y e r c h a r a c t e r i s t i c s . Hence, t h e i n p u t of a d d i t i o n a l c h a r a c t e r i s t i c

constants i n a d d i t i o n t o v of equations .

The Ekman s p i r a l , wi th t h e constant

However, t h e

When t h e Taylor s p i r a l i s considered by

K, and f is needed t o c l o s e t h e r e s u l t i n g set 9'

In view of the polemic a t t ached t o the eddy v i s c o s i t y concept, i t

seems appropriate t o i n v e s t i g a t e what progress toward a boundary l a y e r

s o l u t i o n can ble made without employing t h i s c losu re assumption.

w i l l be l e f t w i t h a nonclosed set of equat ions, and some ambiguity as t o the

c o r r e c t boundary condi t ions, t h e method of a n a l y s i s w i l l employ t h e i m p l i c i t

reasoning of dimensional ana lys i s and dynamic s i m i l a r i t y . We can expect t h e

s o l u t i o n s t o involve unknown a r b i t r a r y cons t an t s i n l i e u of e x p l i c i t boundary

conditions app l i ed t o a closed set of equat ions.

Since we

5 1

7.2 Dimensional Analysis and Dynamic Slmilari ty

To use t h e p r i n c i p l e s of dynamic s i m i l a r i t y , it is f i r s t necessary t o

reduce t h e equations as f a r as poss ib le to extract a workable number of

va r i ab le s . Although w e abandon t h e s p e c i f i c r e l a t i o n between stress and

rate of s t r a i n (grad ien t of v e l o c i t y ) , we s t i l l assume t h a t there exists

some (unknown) r e l a t i o n between stress and v e l o c i t y grad ien t which implies

t h a t v e r t i c a l v a r i a t i o n s i n stress are much g r e a t e r than ho r i zon ta l var ia -

t i ons . ‘hen the assumptions of s teady state and ho r i zon ta l homogeneity are

Surface roughness

Scale h e i g h t of atmosphere

Ekman depth

0

0 T; = 0

0 u = ug 1 u - u g - - T * z = 0

0 (7-1)

included, t h e equations become

f6 - vg> + ( ? / P ) p =

Assuming t h a t v stress forces are neg l ig ib l e , there are f i v e unknowns i n t h i s set of two

equat ions, precluding any unique so lu t ions .

information t h a t p occurs only i n conjunction w i t h ? and need n o t be

considered as a sepa ra t e va r i ab le .

unknown, the number o r type of boundary condi t ions is unknown.

i s given as the upper boundary condi t ion , where 9

The equat ions do provide t h e

Since t h e order of the equat ions i s

Nondimensionalizing wi th respec t t o a r b i t r a r y c h a r a c t e r i s t i c va lues ,

H, g, and -rg* where T* = T/P,

B e cause t h e

Note

appropr ia te

momentum equat ions are similar, only

t h a t f o r var ious l i m i t i n g magnitudes

t o t h e d i f f e r e n t regimes emerge :

Layer C h a r a c t e r i s t i c Length I L i m i t of E I

one is discussed.

of E, the equat ions

L i m i t Equations

52

- -

We see t h a t i n a region of high To o r small H ( o r & o r f), w e have

T* = constant E u* = - .9* This is t h e s u r f a c e l a y e r , cha rac t e r i zed by high u,/H and constant stress.

With t h i s r ep resen ta t ion of 'Cf the c o e f f i c i e n t becomes u*/HVo f. 4 Dimensional a n a l y s i s on t h e parameters G = (us

two p o s s i b l e c h a r a c t e r i s t i c l eng ths , G/f o r u J f , and two p o s s i b l e charac-

t e r i s t ic v e l o c i t i e s , G and (To" ) 2.t

t o provide c h a r a c t e r i s t i c values f o r t h e ver t ical scale of t h e boundary

l aye r .

-I- vg) , f and U* reveals

?- Plane ta ry values f o r G / f are too l a r g e

Therefore, choose H = (T;/f2)% = uJ f , so t h a t (7-2) becomes

0 (7-3)

>:> 1 w e ob ta in t h e s u r f a c e l aye r , where % + 0

<<: 1 w e o b t a i n geostrophic flow where % -f G

=: 1 we ob ta in the middle (Ekman) l a y e r and 5 N u*.

For u*/{

S e l e c t & = uJc f o r t he middle l a y e r problem, ob ta in ing parameter less

equat ions and hence self s i m i l a r i t y . Simultaneously, (7-3) imp l i e s t h a t w e

cannot o b t a i n the s u r f a c e l a y e r equat ions with t h i s choice of H. We know

t h a t t h e su r face l a y e r i s cha rac t e r i zed by small H, and w e can hypothesize

t h a t t he re exists a c h a r a c t e r i s t i c roughness length zi i n this l a y e r .

Since E can be made s u f f i c i e n t l y l a r g e by v i r t u e of small H = z . 2'

t h e governing equation f o r t h i s region, T* = 0 , i n d i c a t e s t h a t G is no

longer p r e s e n t i n the equations.

c h a r a c t e r i s t i c v e l o c i t y i n the s u r f a c e l a y e r .

Z

This leaves u* as t h e only choice f o r

Note t h a t w e can now w r i t e general r e l a t i o n s f o r the s o l u t i o n s t o

(7-1) i n the two regions cha rac t e r i zed by t h e d i f f e r e n t l eng ths 2

s i n c e by dimensional arguments w e have reduced t h e problem t o two equat ions and u, / f , i

U i n two unknowns, - and 1. r e l a t i o n s :

We can hypothesize t h e following func t iona l u* u*

u/u* = f', [ Z / Z $ 3 , i n n e r l a y e r (-7-4)

?The square roo t of the product of t h e s e alternate scales i s a l s o a p o s s i b l e scale. However, these are awkward c h a r a c t e r i s t i c va lues and add no q u a l i t a t i v e change t o t h e following.

53

upper l a y e r (7-5 1

The upper s o l u t i o n f a i l s n e a r the s u r f a c e where t h e r a t i o of charac-

This corresponds t o the case discussed terist ic scale . v e l o c i t i e s %/u* - t 0. i n Section 6 i n which t h e Ekman s o l u t i o n assumption of constant K breaks

dawn i n the su r face l a y e r and 6 l H , = ( / m f l / H , -f 0.

7 . 3 Matching the Solutions

The only boundary conditions s a t i s f i e d t o t h i s po in t is v = vg a t

z -+ a. The constant stress d e f i n i t i o n of t h e s u r f a c e l a y e r p l aces no require-

ment on the v e l o c i t y a t t h e lower boundary. However, i t may add a boundary

condition which must be s a t i s f i e d by t h e upper so lu t ion because of t he

constant stress c o n s t r a i n t .

The condi t ions f o r obtaining s o l u t i o n s which match ( i . e . y are t h e

same i n an overlapping region) , w i l l reveal t h e form of t h e s o l u t i o n i n the

matched l a y e r , s i n c e t h e matching condi t ions produce boundary condi t ions

t o be s a t i s f i e d by each e x t e r i o r s o l u t i o n .

In t h i s case we do n o t have e x p l i c i t asymptotic expansions wi th terms

t o match. Nevertheless, we can f o r c e a smooth j o i n i n g of t he two s o l u t i o n s

if t h e va lues of t h e v e l o c i t i e s and t h e d e r i v a t i v e s can b e made t o match i n

some region nea r t h e surface. Assuming this can be done, we have

The derivative is calculated for fixed. Taking a cue from the

method of asymptotic expansion i n different parameters used in Section 6 ,

where the inner l i m i t (l/Eo + 0, €, constaut) of the outer solution ( w r i t t e n

i n inner coordinates) was formed, we differentiate (7-6) w i t h respect to the

parameter E 0 '

54

This desc r ibes t h e v a r i a t i o n i n t h e shape o f t h e o u t e r s o l u t i o n due t o

changes i n t h e parameter E o .

scale lengths , is d i r e c t l y r e l a t e d t o t h e r a t i o of t h e two boundary values ,

U and u*. From (7-7) and (7-8),

This parameter, the r a t i o of t he c h a r a c t e r i s t i c

9

which may be written

(7-10) d(Ug/u,> 1

'0 d E , = S f ' ( 0 =

where k is a constant , s i n c e t h e l e f t s i d e is a funct ion of Eo on ly and the

r i g h t s i d e of 5 only. I n t e g r a t i o n of (7-10) and t h e corresponding U momentum

equations y i e l d s A

U/u, = (Rn ^ziz i ) /k C ( 0 ) = 0

f l u * = 0 ?(O) = 0

and

(7-11)

(7-12)

where A , B , and k are a r b i t r a r y constants .

Equations (7-11) r ep resen t t h e s o l u t i o n i n t h e matched l a y e r , where

2/zi i s nea r un i ty o r l a r g e r and $ / (u , / f ) is small.

very l a r g e .

Ug = G cos a, and u,.

lienee, Eo= u * / f z . is 2

Determination of A depends on t h e simultaneous eva lua t ion of E,,

Eq. (7-12) can b e r e w r i t t e n as

Rn u,/G = A - Rn G l f i , + kG/u,(l - s i n 2 a ) (7-13)

This provides an i m p l i c i t r e l a t i o n f o r u,(G, z o , a , A , k), which corresponds t o

(6-18), &ere t h e s u r f a c e l a y e r parameter appears as t h e he igh t r a t i o s , S/zi and h/zi r a t h e r than t h e cons t an t s k and A.

55

Equation (7-13) is cont ingent upon t h e logar i thmic p r o f i l e f o r t h e

v e l o c i t y i n the matched region.

an outer , viscous-Coriolis fo rce balance t o an i nne r logar i thmic l aye r .

Both s o l u t i o n forms have long been known f o r t h e atmosphere.

has been a r r i v e d a t less formally than presented here .

[1967] ascr ibed the der iva t ion t o Kazanskii and Monin [1961]. They presented

va lues of A and B der ived from var ious theor ies . Blackadar and Tennekes

[1970] discussed t h e inne r lou te r matching of s i m i l a r i t y so lu t ions i n essen-

t i a l l y the form presented here . They cite work by G i l l (unpublished) and

Csanady [1967] which a l s o a r r ived a t (7-13) by s i m i l a r i t y considerat ions.

A summary of values of A and B is shown i n Table 7.1. Csanady [1972] has

extended t h e ana lys i s t o t h e d i a b a t i c l a y e r wi th a commensurate inc rease i n

cons tan ts to be determined empir ical ly .

The r e l a t i o n r e s u l t s from simply patching

Hence (7-13)

Z i l i t i n k e v i c h e t al.

TABLE 7 .1

VALUES OF A AND E IN Rn u,/G = A - Rn G/fZo + (s - ~ 2 ) ~ 2 2

Source A B

Blinova and Kibel 1 Monin

Blackadar

Let tau

from t a b l e 2, Z i l i t i nkev ich e t a l . [1967]

Bobyleva e t al . ) Blackadar and Tennekes [1968]

G i l l (unpublished)

Deardorf f [ 19 70 ]

1.57 2.99

1.5 5.0

6.7 1 .8

4.6 0.9

2 .o 2.2

4.5 0.0

4.7 1 .7

3.2 0.7

The v a r i e t y i n the values may be genera l ly ascr ibed t o d i f f e r e n t

assumptions of eddy v i s c o s i t y d i s t r ibu t ion .

numerical simulation model.

Deardorff 's values are from a

There exists some anbiguity over the value of these cOnstauts. This

is p a r t l y a resul t of the nature of the matding (or patching) process.

constants arise as differences between large ntrmbers and are extremely

The

56

s e n s i t i v e t o v a r i a t i o n s i n t h e p r o f i l e s from which they are determined,

Concurrently, once t h e geostrophic d rag c o e f f i c i e n t is determined, t h e

v a r i a t i o n i s i n s e n s i t i v e t o small e r r o r s i n the constants .

o r (7-13) then o f f e r s t h e prospect of ob ta in ing u,/G as a funct ion of t h e

geostrophic flow v e c t o r , t h e s u r f a c e flow d i r e c t i o n , and empi r i ca l ly

determined constants . So f a r , t h e a n a l y s i s i s r e s t r i c t e d to the n e u t r a l

l aye r .

a f f e c t t h e value of the constants .

Equation (6-18)

The add i t ion of s t r a t i f i c a t i o n t o t h e problem w i l l , a t t h e least,

Clarke [1970] has ca l cu la t ed t h e s i m i l a r i t y constants A and B from

experimental da t a .

the energy equat ion i n s i m i l a r i t y form,

H i s a n a l y s i s adds a s t r a t i f i c a t i o n f a c t o r by inc lud ing

(7-14)

2 where s = - g H o / ( p o ep 8, fu,), 8 is t h e p o t e n t i a l temperature, e , ( z ) e ( 0 ) . This l e a d s t o

do = the eddy f l u x of p o t e n t i a l hea t , and

u p * = [Rn ( u * / f z o ) - A ( s ) l / k

w e , = U J ~ Z ~ - c ( s ) i / k

vg/u* = - B ( s ) / k (7-15)

Clarke found a dramatic v a r i a t i o n with s t r a t i f i c a t i o n i n these funct ions i n

t h e neighborhood o f n e u t r a l conditions (see Fig. 9 ) .

Fig. 9. Variat ion of s i m i l a r i t y parameters w i t h strati- f i c a t i o n (from Clarke [1970], l i n e s , and Z i l i t i n k e v i c h [1967], po in t values A and B ) .

-40 I -300 -200 -100 0 100 200 unstable I stoble I

sk I

57

One f a c t o r which complicated Clarke ' s ca l cu la t ions w a s t h e genera l

d i f f e rence bemeen the winds observed a t t h e top of t h e p l ane ta ry boundary

l a y e r and the geos t rophic v e l o c i t y as c a l c u l a t e d from the s u r f a c e p re s su re

grad ien t . This disagreement was due t o d i u r n a l e f f e c t s , ba roc ldn ic i ty , and

nonsteady s ta te e f f e c t s .

'

_ .

7.4 Summary

We have assumed t h a t t he re e x i s t two s o l u t i o n s cha rac t e r i zed b y ' t h e

and u*/ f and hence determined by d i f f e r e n t very d i f f e r e n t length s c a l e s 2

governing equat ions.

c o e f f i c i e n t of t h e stress term i n t h e complete equat ion where t h e terms have

been sca led t o o rde r un i ty by means of nondimensionalization by c h a r a c t e r i s t i c

values .

n e g l i g i b i l i t y of t he stress term f o r t h e region cha rac t e r i zed by t h e chosen

scale length.

ana lys i s and dynamic s i m i l a r i t y p r i n c i p l e s app l i ed t o t h e appropr ia te

equat ions . For values of of order un i ty , a balance between v iscous and

Cor io l i s forces w a s obtained. The s o l u t i o n t o t h i s equat ion is known t o be

inadequate i n a region nea r t h e s u r f a c e although the lower boundary condi t ions

are s a t i s f i e d .

ob ta ined as l i m i t equat ions f o r l /Eo+ 0.

l e n g t h s c a l e s f o r each region enabled f u n c t i o n a l forms of each s o l u t i o n t o

be pos tu la ted .

i The s i n g l e s i m i l a r i t y parameter Eo = u,/P occurs as a

The choice of c h a r a c t e r i s t i c l eng th H determines t h e dominance o r

The c o r r e c t s c a l i n g parameters w e r e obtained by dimensional

Equations appropr i a t e t o t h i s near s u r f a c e region were

The c h a r a c t e r i s t i c v e l o c i t y and

We assume t h a t t h e d i f f e r e n t s o l u t i o n s can be chosen t o blend smoothly

i n t o a continuous s o l u t i o n i f w e match t h e v e l o c i t i e s and f i r s t d e r i v a t i v e s

of t h e func t iona l expressions f o r each s o l u t i o n i n t h e range of common

domain. There is no a p r i o r i guarantee of t h e success of th i s matching.

Zowever, in the case of t h e p l ane ta ry boundary layer, the matching region

is found to b e the log p r o f i l e range.

p r o f i l e f requent ly inc ludes the entire surface l aye r .

tions supply new lower boundary condi t ions f o r the o u t e r so lu t ion .

are a f i n i t e v e l o c i t y and shear at a height which is small enough t o be

e f f e c t i v e l y zero w i t h r e s p e c t t o the characteristic scale of the o u t e r l a y e r .

Observations i n d i c a t e that this

The matching condi-

These

r -

This leaves t h e o u t e r s o l u t i o n e s s e n t i a l l y unchanged. However, t h e ou te r

s o l u t i o n can vary wi th r e s p e c t t o a parame,ter which reflects the v a r i a t i o n

of t h e p r o f i l e w i th changing boundary condi t ions. This a d d i t i o n a l freedom

i n choice of o u t e r p r o f i l e s t o match t h e i n n e r p r o f i l e is a consequence

of the three-dimensionality of t h e o u t e r l a y e r . The condi t ions f o r matching

provide a correspondence between V constants: z k , A , and a. 3'

and 'r0 which depends on several a r b i t r a r y i.7

Xn t h e constant eddy v i s c o s i t y method, t h e nonsingular Ekman boundary

l a y e r s o l u t i o n s a t i s f i e d boundary condi t ions on t h e dependent v a r i a b l e s and

t h e i r second d e r i v a t i v e s a t t h e top and bottom ( 4 - 3 ) .

s o l u t i o n w a s patched t o t h e su r face l a y e r , t he upper boundary condi t ions

remained the same, b u t t he lower boundary condi t ions then s p e c i f i e d va lues

Uhen t h e Ekman

of v e l o c i t y and f i r s t d e r i v a t i v e as determined by t h e su r face l a y e r .

i n n e r s o l u t i o n could s a t i s f y the condi t ions u(0) = 0, and s o determine t h e

constant z . = z o . However, t h e r e is no independent condi t ion on U ' ' ( 0 ) f o r

t h i s p a r a l l e l flow regime. Indeed, t h e second s u r f a c e boundary condi t ion

U ' ( 0 ) = ro/K, involves t h e primary unknowns i n the problem.

cons t an t s i n the s o l u t i o n manifest this indeterminacy of t h e two-layer

so lu t ions .

The

2

The a r b i t r a r y

They must b e evaluated by experiment.

59

8.

8.1

t h a t

EDDY VISCOSITY THEORIES

Discussion

The value of K = "',/Uz = k ~ , ( z + z 0 ) determined from (5-12) i n d i c a t e s

K l i n e a r l y inc reases wi th h e i g h t i n t h e constant f l u x l aye r . The

quest ion arises as t o what value of t h e eddy v i s c o s i t y c o e f f i c i e n t should

be taken as a constant f o r t h e upper boundary l aye r .

I f t h e constant K assumption i s t h e flaw i n the Ekman s o l u t i o n , then

perhaps a proper considerat ion of K ( z ) i n (3-10) w i l l y i e l d hodographs

corresponding t o measurements. The sea rch f o r an appropr i a t e K ( z ) has

continued s i n c e Ekman i n i t i a t e d i t a t t h e t u r n of t h e century. As observa-

t i o n s accumulated, i t became poss ib l e t o hypothesize K d i s t r i b u t i o n s t o

produce f acs imi l e s of observed hodographs. Such K d i s t r i b u t i o n s produce

s p e c i f i c observed wind s p i r a l s when s u b s t i t u t e d i n t o (3-lo), r e s t a t e d h e r e ,

( K U ' ) ' - f(V - V g ) = 0

(8-1) ( K V ' ) ' + f(U - Us) = 0

The prime is used i n t h i s chapter t o designate dldz.

Observations a l s o i n d i c a t e t h a t t he re i s o f t e n a r e s i d u a l va lue of K

found i n the flow above the planetary boundary l a y e r (e.g., see P r i e s t l e y

[1954]).

t h e s u r f a c e and falls to a s m a l l constant va lue at the freestream.

Thus, the search has been f o r K ( z ) which l i n e a r l y inc reases near

In add i t ion , c e r t a i n choices of K ( z ) , such as KlzrL , allow s o l u t i o n s

of (8-1) i n terms of elementary funct ions. These m d e l s are q u i t e a r b i t r a r y

and have m e r i t only because t h e s p i r a l hodograph i s q u a l i t a t i v e l y i n s e n s i t i v e

t o d ive r se K d i s t r i b u t i o n s . Conversely, K ( z ) is very sensitive t o v a r i a t i o n s

i n t h e p r o f i l e s from which i t is derived and can b e expected t o b e very

s e n s i t i v e t o d i a b a t i c e f f e c t s . As a r e s u l t , t h e K d i s t r i b u t i o n produced

when t h e empi r i ca l determination is made from t h e spa r se a v a i l a b l e d a t a i s

l i k e l y t o b e only q u a l i t a t i v e l y s i g n i f i c a n t .

The Leipzig wind p r o f i l e is a well-known example of an observed wind

s p i r a l , although better d a t a are now available. S t r a t i f i c a t i o n condi t ions

60

are included i n d a t a taken a t O ' N e i l l and thoroughly documented by Le t t au

and Davidson [1957].

p i c t u r e of the flow f i e l d . Y e t many a spec t s of t h e va r ious t h e o r i e s remain

untested. This leaves room f o r considerable l a t i t u d e i n t h e t h e o r e t i c a l

speculat ion arid necessitates e d i t i n g i n p re sen ta t ion . Several t heo r i e s have

been a r b i t r a r i l y s e l e c t e d as r ep resen ta t ive t o i l l u s t r a t e t h e approaches.

The Rossby method w a s a pioneering i n v e s t i g a t i o n and i s p a r t i c u l a r l y

i n t e r e s t i n g as i t i s a complete treatment which is c lose ly r e l a t e d t o the

s o l u t i o n i n s e c t i o n s 6 and 7.

The Aus t ra l i an d a t a have provided a more comprehensive

8.2 Rossby Model - Rossby [1932] used t h e mixing length concept f o r t he gene ra l three-

dimensional flow case.

i s w r i t t e n i n complex n o t a t i o n ,

I f t h e expression f o r t h e d i s tu rbed v e l o c i t y p r o f i l e

where a i s the angle between u and u i s c a r r i e d o u t as i n s e c t i o n 5.5 f o r t h e two-dimensional case.

The expansion f o r t he dis turbance g'

"(2 - 2) + . . . 2 q = q ( z - 2) + q ' ( 2 - 2) + g

+ . . . "(2 + 2) q = q ( z + 2) + q ' ( z + 2) + 4

q " ( z - 2) + q"(2 + 2) + . . . 4

This equation f o r the d i s tu rbed v e l o c i t y f i e l d may b e w r i t t e n

I n dynamically similar flows, q ( z ) are i d e n t i c a l . If s i m i l a r i t y i s

t o e x i s t , then t h e eddy scales must be t h e same and hence t h e pe r tu rba t ion

61

t o t h e mean must be a canstant:

This may be w r i t t e n

= constant . 4'

Vien expanded per (8-2) ,

W e use the fol lowing general ized vers ions of t h e two-dimensional

- equat ions i n s e c t i o n 5:

This set of equat ions can be solved for K(z) and Z(S) by i q t e g r a t i n g

from the top of t he su r face l a y e r t o t h e p o i n t a t which K and

consider ing t h i s as the top of t he p l ane ta ry boundary l a y e r . The s o l u t i o n

shows t h a t 2 and K decrease i n t h e Ekman l a y e r according t o

are zero,

K

and the constants C, and C, are a function of k H - . Z where 5 = - H - h The constants must be evalua ted empir ica l ly .

comparison w i t h Taylor 's [1915] data.

and q ( h ) . Rossby found kR = .065 frm a

R

The expressions f o r Z and K in the surface l a y e r ,

K Ei

together with

hlH

0.103

0.108

0.116

0.110

are patched t o the va lues i n t h e Ekman l a y e r a t t h e top of t h e sur face l aye r ,

producing

k kR

0.4 0.065

0.38 0.065

0.35 0.065

0.4 0.07

The he igh t r a t i o hlH i s an implied constant and i s shown i n Table 8.1

f o r var ious combinations over the range of observed values of t h e "constants"

k and kB.

TABLE 8.1

HEIGHT RATIOS FROM (8-6) AND (8-7)

This so lu t ion i s similar t o (6-10) and (7-14) and w a s obtained by

extending t h e mixing length hypothesis t o the Ekman l aye r .

r e l a t i o n between t h e geostrophic flow, the Cor io l i s force , sur face roughness,

and the constant angle of geostrophic devia t ion i n the su r face l a y e r . The

na ture of th is r e l a t i o n is t h a t t h e angle is r e l a t i v e l y i n s e n s i t i v e to t h e

Cor io l i s parameter f (a 2' change f o r l a t i t u d e s from 20" t o go"), o r t o

geostrophic speed G (a 4' change for 5 L G 5 5 0 ) .

q u i t e s e n s i t i v e , however, and i s shown i n Figure 10.

It provides a

The v a r i a t i o n with z is a

63

." 0. I I .o 10

zo cm

100 IO00

Fig. 10. Angle between su r face l a y e r flow and geostrophic flow versus z o from equat ion (8-7), l a t i t u d e = 4 5 O ,

k = 0.4, kR = 0.065.

/

This s o l u t i o n can be modified t o inc lude t h e observat ion that K does Rossby and Montgomery [1935] used t h e n o t vanish at the geostrophic l e v e l .

following K d i s t r i b u t i o n f o r the case of weak winds,

z 2 h

In a s i m i l a r procedure, t h e following r e l a t i o n between R,, ag, k, kR, f a n d

zo w a s obtained

% 3 2 3'4k (WS s i n a , ) (cota , - 1) = 2 Iln N - Rn(Ros s ina , ) - k2) (8-9) N%

kR

fz 0

where N = 7 .

This r e l a t i o n implies an inc rease i n a, with an increase i n geostrophic

speed.

and slowly decreases f o r G > 1Q m/sec.

o t h e r than t h i s q u a l i t a t i v e comparison.

Experiments i n d i c a t e t h a t t h e angle does inc rease f o r G < 10 dsec Data are i n s u f f i c i e n t to provlde

64

8 .3 Empirical Determination o f Eddy Viscosi ty Distribution

We would l i k e t o e x t r a c t K(.z) from (8-1) so t h a t i t can b e determined

empi r i ca l ly from an observed v e l o c i t y p r o f i l e .

p ly ing each momentum equat ion by t h e complementray v e l o c i t y component and

s u b t r a c t i n g the t w o r e s u l t i n g equat ions.

This can b e done by mult i -

This y i e l d s ,

U ( K V ' ) ' - V ( K U ' ) ' = [ K ( U V ' - V U ? ) ] ' = f(U2 + v2 - uus>

where axes are a l igned wi th us. I n t e g r a t i n g wi th r e spec t t o z and d iv id ing by (UV' - V U ' ) , w e o b t a i n

z

f 1 (u2 + v2 - uug) dz

UV? - V U ? K = - 0 ( 8-10 )

Let t au [1962] used t h e Leipzig wind p r o f i l e , an observed wind s p i r a l

i n nea r n e u t r a l conditions (Fig. 111, t o de r ive an eddy c o e f f i c i e n t d i s t r i -

bu t ion as shown i n Figure 12 . The Leipzig p r o f i l e w a s obtained by two

Fig. 11. Leipzig wind p r o f i l e from Milner [ 19321.

Y I I I I \

-B -6 -4 -2 1

20

19

"g 16

14

12

u Q

IO f 3

B

6

4

c

? *

65

- -

? I

t heodo l i t e observa t ns of 28.pilot-halloczn ascents over a five-hour period.

The n e a r e s t l a p s e rate measurement-was n e a r l y 100 d l e s away.

Fig. 12. Eddy ViscosYty d i s t r i - bu t ion der ived from Leipzig p r o f i l e .

., 0 I234567891011121314I5

K Cml/Kc)

Several o the r determinations of K d i r ibu t ions using (8-10) have

been made.

value. Maximum K occurs anywhere from 200 t o 500 meters.

In general , K maximum was found t o be about 25 times the Leipzig

The determination of accura te averaged p r o f i l e s needed to eva lua te a

characteristic K ( z ) requi res as lpng an avergging per iod as poss ib l e ,

without inc luding s i g n i f i c a n t changes i n s t r a t i f i c a t i o n . Recent r e s u l t s , as

discussed i n sec t ions 5 and 11 ind ica t e t h a t more information is needed f o r

an e f f e c t i v e eva lua t ion of K ( z ) than w a s a v a i l a b l e i n t h e Leipzig da ta .

Clarke [1970] presented c a r e f u l and complete observat ions i n the

p lane ta ry boundary l a y e r made by the Cormaonwealth S c i e n t i f i c and I n d u s t r i a l

Research Organization i n Aus t ra l ia .

t o a S t r a t i f i c a t i o n parameter.

The d a t a have been c l a s s i f i e d according

Universal p r o f i l e s were obtained f o r f o u r

s t a b i l i t y groups wi th respect t o the s i w i l a r i t y parameters u* and u, / f . The f r i c t i o n v e l o c i t y was obtained from wind measurements a t 0.5 o r 0.75 meters.

66

A drag c o e f f i c i e n t f o r t h e s i t e w a s used t o determine u* according t o u* = 0.087 Uo.s, o r U* = 0.08 Uo.75. U and V were measured by double

t h e o d o l i t e bal loon method f o r over 150 soundings which were then separated

i n t o t h e f o u r c l a s s i f i c a t i o n s of thermal s t a b i l i t y .

obtained from (8-1) and = K dV/dz . Figures 13-16 show t h e v a r i a t i o n of

U, V , &, and K f o r t h e four c l a s s i f i c a t i o n s . Note t h a t a coordinate system

al igned with t h e s u r f a c e wind w a s used.

wind from su r face p re s su re g rad ien t s was unsuccessful i n these experiments

due t o b a r o c l i n i c i t y , d i u r n a l mixing cycle , and t i m e changes i n V

d i f f e rence between class I and I1 w a s determined by an a r b i t r a r y inve r s ion

he igh t , s o t h a t class 11 i s f o r shallow convection, z < 0.3. The mean

values f o r each c l a s s i f i c a t i o n are shown i n Table 8.2.

The K d i s t r i b u t i o n w a s

The determination of t h e geostrophic

The g'

i -

Fig. 13. Vertical p r o f i l e s of V ( z ) f o r s t r a t i f i c a t i o n categories from Table 8.2. From Clarke [1970].

Z

16 18 20 22 24 26 28 30 32 34 L' CLASS I - I U .

20 30 40 50 60 70 80 90 100 110

U CLASS=.

Fig. 14. Vertical p r o f i l e s of V ( z ) f o r s t r a t i f i c a t i o n ca t egor i e s from Table 8.2. From Clarke [1970].

2 0 -2 4 -6 -8 -10 -12 -14 -16

v

67

0.4

0.3

z '0.2

0. I

C

I I I

0.02 003 0.04 0.05 0.06 0.01

Fig. 16. V e r t i c a l p r o f i l e s of K ( z ) f o r s t r a t i f i c a - t i on ca tegor ies from Table 8.2. From Clarke 119701.

Fig. 15. Vertical p r o f i l e s of R(z) f o r s t r a t i f i c a - + t i o n categolries from . Table 8 .2 . From Clarke [1970].

t

0 0.01 0.02 0.03 0.04

K

E k k n l aye r tu rn ing w a s observed i n a l l cases . There is a maximum i n

U and a minimum i n V f o r a l l cases except f o r V i n the n e u t r a l case, where

w a r m a i r advection complicates t h e p r o f i l e .

is q u i t e similar t o t h e Leipzig s p i r a l .

boundary l aye r determined by the he igh t of t h e invers ion may be much deeper

than an Ekman l aye r , as ind ica t ed by a s p i r a l i n g o r t u rn ing i n accordance

with s t eady state dynamic so lu t ions .

Nevertheless , t he n e u t r a l case

In the uns tab le cases , t h e convecting

There is a very large inc rease i n R and K f o r the case of deep ccm- The increase vec t ion compared t o shallow convection throughout the l aye r .

in these values over the n e u t r a l case w a s large in t h e lower half of the

Convecting l a y e r , b u t a significant decrease was observed in the upper h a l f

of this l aye r . The peak value of K ranges from 32.6 m 2 / s e c at about 250 meters

68

TABLE 8.2

MEAN VALUE FOR CHARACTERISTIC VALUES I N EACH CLASSIFICATION

U * l f m

3333.

4000. 5000.

1333.

Class u* m/sec

0.28

0.33

0.42

0.12

1

I1 I11

IV

2 0

cm

St r a t i f i c a t i o n A z a t 2=0.3

m

0.5

0.5

0.5

2.0

1000.

1200.

1400.

400.

unstable

uns tab l e

n e u t r a l

s t a b l e

f o r case I t o a f a i r l y constant value of 16.8 m2/sec above 200 meters f o r

case 111. This suggests a well-mixed l a y e r f o r t h e n e u t r a l case, w h i l e a d e f i n i t e d i f f e r e n c e exists i n K va lues between t h e upper and lower por t ions

of t h e convecting l a y e r s .

There is a l a r g e r e s i d u a l value of z and K f o r t h e n e u t r a l and s t a b l e

condi t ions.

h a s no t been reached i n t h e graphs f o r these cases. The d i f f e rence i n K

and from t h e Leipzig values a t he igh t s g r e a t e r than 900 m may be due t o

inaccuracies i n e i t h e r set of d a t a s i n c e both contained sources of e r r o r

i n t h i s region. Thus, t h e va lue of a r e s i d u a l K o r i s i n d e f i n i t e .

However, cases I1 and I11 i n d i c a t e t h a t = 25-40 m above these l a y e r s .

It appears l i k e l y t h a t t h e t o p of t h e dynamic boundary l a y e r

8.4 Analyt i lc Modeling of K ( z )

One can obtain expressions f o r t h e K and 2 d i s t r i b u t i o n s by f i t t i n g

curves t o t h e da t a . These expressions can then b e used in t h e viscous term

f o r numerical models of t h e boundary l a y e r flow. Since t h e v e l o c i t y d i s t r i -

but ion is apparent ly n o t extremely s e n s i t i v e t o t h e exact form of K ( z ) , one

may be tempted t o approximate these K d i s t r i b u t i o n s w i t h a n a l y t i c expressions.

The governing equat ions are suf f i e i e n t l y complicated t o l i m i t t h i s procedure

t o f a i r l y simple r e l a t i o n s .

The s b p l e s t r e l a t i o n s h i p , suggested by t h e s u r f a c e l a y e r observat ions

and ana lys i s , is t o l e t K = K,(z) f o r t h e e n t i r e boundary l a y e r . Although

69

t h i s is obviously a poor approximation i n the upper regions o f t h e p l ane ta ry

boundary l aye r , the decrease i n v e l o c i t y g rad ien t s may be s u f f i c i e n t t o

reduce t h e magnitude of the viscous terms i n t h i s range.

This K d i s t r i b u t i o n is p a r t i c u l a r l y attractive s i n c e a coordinate

transformation on the equat ions of motion i n t h e complex v e l o c i t y , q, y i e l d s

t h e Bessel equat ion, wi th a well-known general so lu t ion . Thus, i n terms of

Q E U - U 9 + i ( V - $1 ¶ (8-11)

(8.1) may be w r i t t e n

- K - d d & - i f & = 0 . dz d z

When t h e transformation

(8-12)

is made, w e g e t

( 8-1 3)

This i s the zero-order Bessel equation. Blinova and Kibe l ' used t h e tabula ted

so lu t ions t o t h i s equat ion wi th the boundary condi t ions q = 0 a t z = z o and

q = U are reasonable near t he sur face (as expected, s ince K = K,(z) i s a good

approximation i n t h i s region) .

meters he igh t , however.

a t z -t 03 t o ca l cu la t e v e l o c i t y p r o f i l e s and cxo . The ve loc i ty p r o f i l e s g

They become nea r ly constant a t a few hundred

The a, are much smaller than observed.

A more real is t ic K d i s t r i b u t i o n w a s solved by Yudin and Shvets [1970].

To improve t h e upper po r t ion of the previous so lu t ion , they l e t K-become a

cons tan t , Kh, a t the top of the su r face l a y e r , h . Bessel s o l u t i o n f o r t he su r face l a y e r and t h e Ekman so lu t ion , expressed in

complex no ta t ion , f o r the Ekman layer :

Thus, they ob ta in t h e

(8-14)

70

The s u r f a c e l a y e r r e l a t i o n s ,

u: = K1q1 '

are used f o r z -+ z,,.

The improvement i n t h i s model over t h e Rossby s o l u t i o n i s t h e inclusion

of the C o r i o l i s terns and three-dimensionality i n the s o l u t i o n f o r t he upper

po r t ion of t h e s u r f a c e l a y e r .

and q and 4') are used, a s o l u t i o n can be obtained f o r t h e flow f i e l d w i t h

r e s p e c t t o U These r e s u l t s are shown i n Table 8 . 3 .

When t h e usua l matching condi t ions ( con t inu i ty

f , z o , and one i n t e r n a l parameter s e l e c t e d from Z,, h , o r u*. 9'

TABLE 8 . 3

VELOCITY (U ,V) AND MAGNITUDE (C) FOR VALUES OF uq (m/sec), KH (m / s ed AND z,, = 0.1 m

ug = 2.9 k, = 42.3

10 21.8 50 loo 131 200 300 500 OOO 500

3.17 3.68 4.16 4.49 4.59 4.75 4.80 4.58 2.04 1-46

9.41 10.98 12.69 14.06 14 58 15.60 16.89 19.06 22.50 24.0

19" 40' 19 35 19 10 18 40 18 25 17 45 16 30 13 50 730 330

39'0' 32 40 26 40 14 0 040

0.47 0.75 0.96 1.78 1.23 2.74 0.84 3.52 0.04 4.12

8.86 10.35 11.99 13.32 13.83 14.85 16.19 18.50 22.32 23.97

1.28 0.58 101 1.50 21.8 2.45 50 3.42 lOa 4.12

I - - = 5.75; k, = 3.44 u = 6.65 I k, = 20.9

2.83 3.33 3.56 3.79 4.51 5.42

24'0' 23 30 23 10 220 m IO

10 21.8 50

100 163.5 200 300

6.05 12 40 6.26 I 12 0 6.13

6.34 6.90

14 20 1.30 0.00

6.50 6.90

I 1 20 0 6.05 9 10

6.13 2 20 I ug = 10.52; kn = 1.77

29-10' 28 50 27 40 24 25 2020 12 30 3 45

1.47 1.80 2.44 3.36 2.30 0.04

2.34 2.93 4.13 7.42 10.16 10.52

39'0' 37 55 35 10 27 10 15 10 0 20

10 4.65 2.59 5.32 21.8 5.16 2.85 5.90 50 6.35 3.32 7.17

100 8.23 3.74 904 N U 9.65 3.57 10.25 300 12.28 2.71 12.57 500 14.25 0.87 13.27

11.9 1.82 20 2.31 40 3.46

100 6.62 200 9.90 500 10:52

From Yudin and Shvets [1940] i n Matveev [1965].

71

Ekman [1905] first expressed t h e f e e l i n g t h a t K should be related t o

the wind shea r . P rand t l [19321 considered t h i s r e l a t i o n i n connection w i t h

the in t roduc t ion of t h e mixing length,

Blackadar [1962] has obtained numerical s o l u t i o n s f o r (8-1) by spec i -

f y i n g an a r b i t r a r y Z d i s t r i b u t i o n which matches s u r f a c e l a y e r theory a t

s m a l l z and at ta ins a small constant value a t l a r g e 2. Accordingly, observed

d i s t r i b u t i o n s f i t the following formula f a i r l y c losely:

kz (1 + k z ) / A z = (8-16)

where A i s some unknown funct ion of t h e e x t e r n a l parameters.

t o produce a0 = 32" i n accordance w i t h observat ions and condi t ions a t

Brookhaven, N.Y. , taken by Bernstein [1959]. Numerical i n t e g r a t i o n of t h e

equations was required f o r t h i s a r b i t r a r y Z ( Z > . r e s i d u a l value of z o , i t w a s assumed t o be unaffected by changes i n zp .

Thus A w a s taken t o be h = 0.00027 G/f from the d a t a and s i m i l a r i t y condi-

t i o n s . The two K d i s t r i b u t i o n shown i n Figure 1 7 are determined from (8-15)

and (8-16). F a l l e r [1966] has derived i n t e g r a t e d mean values of K f o r t hese

p r o f i l e s , obtaining K = 2.5 and 4.5 m 2 / s e c .

of 20 m2/sec were obtained by Duskin and Lomonosov [1963].

It is chosen

Since A r ep resen t s a

Elsewhere, "best" mean values

Blackadar 's numerical s o l u t i o n s p r e d i c t r e l a t i o n s h i p s between u,/G

o r a and log G/fz, as shown i n Figure 18.

K ( z ) o r Z(z) d i s t r i b u t i o n s and s i m i l a r i t y arguments.

These curves are based on empi r i ca l

L e t t au has suggested t h a t

where z For u,/G constant , th is

expression di f fe rs f r o m (8-16) by only the exponent i n the denominator,

providing d i f f e rences i n 2 only at large z.

= f(u*/f, is t h e h e i g h t of maximum 2. m

72

1000

900

800

700

600

Fig. 1 7 . Empirical K E 500 d i s t r i b u t i o n s N

from Blackadar (8-15).

4 00

300

200

100

I . I I I . I

V?r?ical . D i s t r i b u t i o n s o f K(z) f r o m V!ind P r o f i l e s from Blackadar IS62 ( co r rec t ed 1

.\ \ \

'\ \ .

0 0 2 4 6 8 1 0

K m2/sec

n*

25'

a

15'

Log ( G l f i , ) Fig. 18. Predic ted angle between su r face wind and U

and geostrophic drag coe f f i c i en t u,/G SQ from Blackadar [1962].

73

There are many o t h e r assumed K d i s t r i b u t i o n s with a n a l y t i c and numerical

s o l u t i o n s , The d i f f e rences between almost a l l of t hese models is less than

t h e s c a t t e r i n t h e d a t a a v a i l a b l e f o r comparison.

have been summarized i n Table 8.4. Many of these models

8.5 Summary

The K d i s t r i b u t i o n theo r i e s have recognized t h e d i s s i m i l a r i t y of

d i f f e r e n t regions of t h e p lane tary boundary l a y e r .

f o r the flow f i e l d i n the lower por t ion of t h e boundary l a y e r t o t h e top of

the s u r f a c e l a y e r .

used by Rossby, and t h e Bessel func t ion used by Yudin, may w e l l be more s u b t l e

than i s meri ted by t h e na tu re of the p lane tary boundary l aye r .

They provide good models

The d i f f e rences between t h e l o g p r o f i l e f o r t h i s region,

The myriad of proposed t h e o r e t i c a l d i s t r i b u t i o n s are matched ( v e r i f i e d ? )

i n the corresponding a r r ay of empir ica l ly der ived K d i s t r i b u t i o n s , as

presented i n Z i l i t i nkev ich e t al . [1967] and shown i n Figure 19. There

appears t o be more t o determining t h e v e l o c i t y p r o f i l e than i s represented

i n K ( z ) . I n add i t ion , t h e r e e x i s t s abundant evidence t h a t t h e p l ane ta ry

boundary l a y e r has o t h e r c h a r a c t e r i s t i c s ou t s ide of t h e ho r i zon ta l homogene-

i t y , s teady s ta te assumptions.

i n t i m e and space. It sometimes possesses a "low l e v e l jet" i n t h e upper

po r t ion of the Ekman l aye r . When t h e r e are secondary c i r c u l a t i o n s due t o

dynamic o r convective causes, t h e eddy v i s c o s i t y concept is i n v a l i d f o r

these la rge-sca le , nonrandom eddies (sec. 10) .

The wind p r o f i l e can be extremely v a r i a b l e

The d iu rna l cyc le and t h e consequent v a r i a t i o n i n S t r a t i f i c a t i o n

g r e a t l y complicate this approach.

s e n s i t i v e t o v a r i a t i o n s i n s t r a t i f i c a t i o n o r s u r f a c e roughness, the p r a c t i c a l

I f the 'concept i s v a l i d b u t K is very

value is small.

for much of t he wide d i s p a r i t i e s i n empi r i ca l ly der ived K d i s t r i b u t i o n s .

The small d i f f e rences i n s t r a t i f i c a t i o n probably account

I n the A r c t i c , however, we m i g h t spec i fy certain s t a b l y s t r a t i f i e d

condi t ions not inf luenced by the d i u r n a l cycle.

it is possible that a characteristic eddy viscosity d i s t r i b u t i o n may b e

found which is a p p l i c a b l e f o r significant ranges.

Under these circumstances,

74

TABLE 8.4

OUTLINE OF K THEORY MODEL DEVELOPMENT

Da.te

190 2

1905

1908

1915

19 30

1.9 30

19 33

19 35

19 36

1940

1940

1950

195 7

1962

Author -

Ekman

Ekman

Ackerb 1.om

Taylor

P rand t 1

von Jbrruan

Takaya

K.ohler

R.ossby,

Rossby and

Montgomery

Blinova & Kibe l

Kibel

Yudin & Shvets

L e t t au

A r i e l

Blackadar

K

Constant

Id?/ dz I Constant

Constant

C1(z+zo), z 3 h

C 2 ( 1 $ )’ h S z,LH

o r z z h K1

KIZ

ku*z

K1z z I h

Klh z 2 h

empir ical

Remarks

Three dimensional s p i r a l

f o r t he ocean

Equiangular s p i r a l

Eddy v i s c o s i t y f o r

t h e atmosphere

V ( 0 ) 2 0

Mixing l eng th concept, 1 . k.= von Karman constant

B e s s e.1 s equat ion

F i r s t two l a y e r models

Bessel equation

u*=(2e/q 1 Y.2

From Lei.zpig wind p r o f i l e

Numerical s o l u t i o n with

empir ical c o n s t a n t s , h , X ,

75

lo2

10

1

,

Fig. 19. [From Zilitinkevich et a l . , 19691. Empirical data on the vertical distribution of the coefficients of turbulent viscosity K and turbulent exchange for heat Kh, from numerous studies. (For the spec i f ic reference, see Zilitinkevich e t a l . )

76

9. EKMAN INSTABILITY

Under the assumption of constant K, (3-10) can be so lved a n a l y t i c a l l y

producing t:he Ekman l a y e r s o l u t i o n :

The pa th t o t h i s s o l u t i o n w a s paved with many assumptions which are

s u b j e c t t o quest ion i f observat ions f a i l t o v e r i f y the f i n a l s o l u t i o n . This

proved t o b e t h e case: t h e a n a l y t i c s o l u t i o n , t h e Ekman s p i r a l , is almost

never observed i n t h e p l ane ta ry boundary l a y e r , The major t h r u s t of p l ane ta ry

boundary l a y e r modeling w a s t h e r e f o r e d i r e c t e d toward semi-empirical descrip-

t i o n , b u i l d i n g from the su r face outward.

However, i t i s i n t e r e s t i n g t o r e t u r n t o t h e Ekman s o l u t i o n t o see i f

i t is a s t a b l e s o l u t i o n f o r t he Cor io l i s /v i scous balance. Observational and

experimental information i n d i c a t e s t h a t two-dimensional pe r tu rba t ions dominate

i n sheared flows. Thus, although the Ekman l a y e r i s n o t a p a r a l l e l flow,

t h e s t a b i l i t y a n a l y s i s assumes independence of t h e flow i n an a r b i t r a r y

long i tud ina l d i r e c t i o n . The r e s u l t i n g s t a b i l i t y equations are i d e n t i c a l

t o the c l a s s i c a l p a r a l l e l flow equat ions, with t h e la teral component of t h e

mean flow appearing as a parameter. Only when t h e C o r i o l i s f o r c e s on t h e

pe r tu rba t ion are considered does the long i tud ina l component of t h e mean flow

e n t e r t h e s t a b i l i t y equations. The Cor io l i s terms a f f e c t t h e i n s t a b i l i t y

mode i n t h e v i c i n i t y of t he n e u t r a l s t a b i l i t y curve, and thus may a f f e c t t h e

value of t h e minimum cr i t ical Reynolds number f o r i n s t a b i l i t y and t h e i n i t i a l

growth mode [ L i l l y , 1966; E t l i n g , 19711. However, i n t y p i c a l p l ane ta ry

boundary l a y e r condi t ions , t h e magnitude of these terms becomes n e g l i g i b l y

s m a l l , as do t h e viscous terms [Barcilon, 1965; Brown, 19723.

The i n f l e c t i o n p o i n t i n s t a b i l i t y mode i s found i n t h e second-order

i n v i s c i d equat ions, b u t i t is necessary t o r e s t o r e t h e viscous terms t o

77

explore t h e i n s t a b i l i t y modes.

n a t u r e of t h e i n v i s c i d l i m i t . The modes are modulated by t h e viscous fo rces

i n the region of t h e c r i t i ca l l a y e r (U = er) and by the add i t ion of boundary

condi t ions s a t i s f i e d by t h e higher-order viscous equat ions [Barcilon, 1965;

Brown and Lee , 19721.

This i s due t o t h e s i n g u l a r pe r tu rba t ion

When simple , harmonic-wave pe r tu rba t ions are added t o t h e mean veloci-

t i es i n (3-3) and the r e s u l t i n g equations are l i n e a r i z e d and nondimensionalized,

t h e mean terms are sub t r ac t ed o u t , and C o r i o l i s and s t r a t i f i c a t i o n (buoyancy)

e f f e c t s are neglected, then a classical pe r tu rba t ion equation, t h e O r r -

Sommerfeld equation, r e s u l t s :

( V - e) (@I ' - y2@) - V"@ = ( l l i y Re) (@"" - 2y2@" + y4@) (9 -2)

where V, = V i - z), and a stream funct ion is defined from t h e cont inui ty

r e l a t i o n

v = w = -qJy

i n which I) = $ ( z ) eiy(y - et). Here y = wave number, e = complex eigenfunct ion.

This i s an equation f o r t h e vertical s t r u c t u r e of t h e stream funct ion

$I f o r two-dimensional pe r tu rba t ions o r i e n t e d a t a r b i t r a r y angles t o t h e

geostrophic flow.

assumed t o be e

Since t h e time-dependent behavior of t h e pe r tu rba t ion i s i ct , any s o l u t i o n s a t i s f y i n g t h e boundary condi t ions and

having a negat ive imaginary component of t h e eigenvalue e w i l l i n d i c a t e

exponent ia l ly growing pe r tu rba t ions , hence unstable condi t ions. The e x i s t -

ence of such s o l u t i o n s w i l l depend upon t h e following parameters: t h e wave

number y; the Reynolds-Rossby number Re = 2Vg/f6; and t h e v e r t i c a l v e l o c i t y

p r o f i l e V , which depends upon the o r i e n t a t i o n of the pe r tu rba t ion i n t h e

mean flow (see Figure 20). The range o f each parameter m u s t b e surveyed

t o determine t h e p o i n t of maximum growth rate (maximum ye.). 2

When t h e buoyancy term is included i n the s t a b i l i t y a n a l y s i s , (9-2)

becomes a s ix th -o rde r equation wi th the Richardson number R i = g!2'2/[T(Vg/6)2] as an a d d i t i o n a l parameter.

then modified sowwhat, and the p o s s i b i l i t y of convective instabil i t ies

enters.

The dynamic i n f l e c t i o n p o i n t i n s t a b i l i t y is

78

5.0 - 4.5 -

-1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0

Fig,, 20. Lateral Ekman v e l o c i t y p r o f i l e s taken normal t o t h e r o l l axis. The curves are l a b e l l e d wi th angle E between r o l l axis and geostrophic d i r e c t i o n .

Solut ions t o (9-2) i n d i c a t e exponen t i a l ly growing pe r tu rba t ions when

V" = 0 somewhere i n t h e v e l o c i t y p r o f i l e [e.g. , see L i l l y , 1966; Brown, 19701.

Since i n f l e c t i o n p o i n t s are c h a r a c t e r i s t i c of t h e p l ana r components of t h e

tu rn ing ve:Locity s o l u t i o n , t h i s s o l u t i o n is i n h e r e n t l y uns t ab le and cannot

be expected t o b e observed. The c r i t i c a l minimum Re f o r t h e n e u t r a l l a y e r

is about 50 (20 when moderate buoyancy i s p r e s e n t ) , while t y p i c a l atmospheric

values are about 1000.

Subsequent s o l u t i o n s have shown the mean ( h o r i z o n t a l l y homogeneous)

s o l u t i o n t o be dynamically uns t ab le over a wide range of boundary l a y e r

s t r a t i f i c a t i o n [Brown and L e e , 1972; Brown, 1972a; Kaylor and F a l l e r , 19721.

When the convective i n s t a b i l i t y (Rayleigh type) is added t o t h e dynamic

( i n f l e c t i o n p o i n t ) i n s t a b i l i t y , it becomes apparent t h a t even t h e t h e o r e t i c a l

expectat ion of t he p l ane ta ry boundary l a y e r i s no t f o r a simple, s teady,

horizon tal:Ly homogenous s t r u c t u r e .

79

10. SECONDARY FLON MODEL

10.1 Basic Concept

The con junct ion of t he need f o r mesoscale and l a r g e r parameter izat ion

of t h e p l ane ta ry boundary l a y e r and t h e accumulation of d a t a i n d i c a t i n g t h e

< n i s t ence of s t eady state secondary flows i n t h e boundary l a y e r on these

scales l e d t o t h e development of a secondary flow model. When t h e secondary

flow i s s i g n i f i c a n t l y less i n magnitude than the mean flow, i t can be t r e a t e d

as a pe r tu rba t ion flow and found i n t h e l i n e a r i z e d equat ions.

mean value of the momentum t r a n s p o r t terms may be comparable t o t h e o t h e r

mean flow acce le ra t ions , r e f l e c t i n g the f a c t t h a t the secondary flow may

s i g n i f i c a n t l y a l te r the mean flow. I f these terms are included i n t h e mean

flow equat ions, a modified mean flow s o l u t i o n w i l l r e s u l t . Solutions of

t h i s type have been obtained f o r convective flow (Rayleigh problem) by

Malkus and Veronis [1958], f o r plane P o i s e u i l l e flow and t h e flow between

r o t a t i n g cy l inde r s i n a general development by S t u a r t [1958] and f o r t h e

Ekman boundary l aye r by Brown [ 19701.

However, t h e

The following model contains t h r e e f a i r l y s e p a r a t e pa r t s : (1) t h e

s t a b i l i t y ana lys i s , t o determine t h e form of t h e fastest growing per turba-

t i o n ; (2) the equi l ibr ium energy balance, t o determine i f t h e r e i s an

equi l ibr ium with a reasonable f i n i t e magnitude of t h e pe r tu rba t ion flow;

and (3) t h e modified mean s o l u t i o n , t o determine t h e mean flow s o l u t i o n

when secondary e f f e c t s are included.

10.2 Some Aspects o f the Secondary Flow Assumptions

Before discussing t h e boundary l a y e r model, i t i s probably h e l p f u l

t o p o i n t ou t t h e polemic aspec t s of t h e s o l u t i o n . The assumptions are many

and t h e treatment is coarse, so that only t h e agreement of t h e p r e d i c t i o n s

with observat ions prompts confidence i n the model.

The problems involved i n pursuing an atmospheric analogue t o molecular

v i s c o s i t y have been discussed. In the s t a b i l i t y theory used to determine

t he shape of t he secondary flow, a b a s i c mean flow v e l o c i t y p r o f i l e is used.

For t h e Ekman s o l u t i o n , t h e n e u t r a l la teral v e l o c i t y p r o f i l e s are i n t r i n s i -

c a l l y unstable , and would become s o even i n t h e developing s t a g e s from rest.

The problem i s complicated i n t h i s r e spec t when the e f f e c t o f s t r a t i f i c a t i o n

of t h e mean flow i s included, so t h a t dynamic s t a b i l i t y may vary mainly with

r e s p e c t t o s t r a t i f i c a t i o n . In the case of uns t ab le s t r a t i f i c a t i o n , t h e

i n t e r a c t i o n of dynamic and convective i n s t a b i l i t i e s must be considered.

In the p l ane ta ry boundary l a y e r , t he re is t h e p o s s i b l i t y of y e t another

complication t o the s t a b i l i t y problem. Up t o t h i s p o i n t , the e f f e c t of t h e

C o r i o l i s f o r c e , a r o t a t i n g frame of r e fe rence , has a r i s e n only i m p l i c i t l y ,

i n the establishment of the m e a n v e l o c i t y p r o f i l e . I f t h e C o r i o l i s terms are

included i n t h e s t a b i l i t y equat ions, they make poss ib l e energy t r a n s p o r t

between t h e l o n g i t u d i n a l and lateral flow components. This enormously

complicates the s t a b i l i t y a n a l y s i s , p a r t i c u l a r l y i n the case of s t a b l e

s t r a t i f i c a t i o n . I n t h i s case, t h e i n s t a b i l i t y modes may be suppressed a t

d i f f e r e n t rates, and the p o s s i b i l i t y of resonance w i t h g rav i ty wave

frequencies e n t e r s [Kaylor and F a l l e r , 19721.

Since t h e l i n e a r i z e d s t a b i l i t y equat ions are homogeneous, no magni-

tudes of t h e pe r tu rba t ion flow are involved. To o b t a i n a magnitude, i t i s

necessary t o p o s t u l a t e an equi l ibr ium flow wherein an energy balance between

mean and pe r tu rba t ion flow can be w r i t t e n . A t t h i s p o i n t , it i s assumed

t h a t t he pe r tu rba t ion h a s modified the mean flow such t h a t no n e t energy

t r a n s f e r is f e d t o the pe r tu rba t ion growth ( o r t h a t only s u f f i c i e n t energy

i s t r a n s f e r r e d t o balance d i s s i p a t i o n ) . I f t h i s equi l ibr ium i s a t t a i n e d when

the secondary flow i s l a r g e enough t o be s i g n i f i c a n t b u t s m a l l enough t o b e

a pe r tu rba t ion , then the model may succeed.

Since t h e man v e l o c i t y p r o f i l e i s being a l t e r e d by t h e pe r tu rba t ion ,

the shape of t h e p e r t u r b a t i a n might be expected t o change, r equ i r ing a

l abor ious i t e r a t i o n . However, i t w a s found t h a t t h e shape of t h e pe r tu rba t ion

(its cr i t ical wavelength f o r maximum growth rate and t h e corresponding vertical

d i s t r i b u t i o n ) i s r e l a t i v e l y i n s e n s i t i v e t o t h e r e s u l t i n g changes i n mean

v e l o c i t y . I n a d d i t i o n , t h e i n i t i a l pe r tu rba t ion mode w i l l a l r eady have

grown s i g n i f i c a n t l y by t h e time t h e mean p r o f i l e is a l t e r e d . Thus, one

81

might expect t h e nonl inear a spec t s of t h e s t a b i l i t y process t o favor

cont inual growth of t h i s e s t a b l i s h e d mode.

The f i n a l modified p r o f i l e w i l l s t i l l e x h i b i t i n f l e c t i o n p o i n t s ,

implying i n s t a b i l i t i e s t o i n f i n i t e s i m a l pe r tu rba t ions .

c r i t e r i o n implies n e t s t a b i l i t y of t h e p r o f i l e , t h e weakened i n f l e c t i o n

point i n s t a b i l i t y may be suppressed, o r balanced by pe r tu rba t ion absorpt ion

elsewhere.

If t h e energy

After t h i s b r i e f discussion of t h e p i t f a l l s of t he theory, t h e

following o u t l i n e of the secondary flow theory may be more c r i t i c a l l y read.

10.3 Secondary F l o w Equations

10.3.1 The Ekman Solut ion

When E q s . (9-1) have been nondimensionalized with c h a r a c t e r i s t i c

scales G and 6 and a coordinate system al igned with t h e r o l l i n s t a b i l i t i e s

is chosen, one ob ta ins

-2 U ( z ) = cos E - e cos(z + E)

V ( z ) = -sin E + e s i n ( z + E) -2 (10-1)

where E i s t h e angle between the geostrophic flow and the r o l l axis, p o s i t i v e

f o r leftward-oriented r o l l s . The shape of t h e la teral V ( z ) p r o f i l e i s shown

i n Figure 20. When these V ( Z ) are used i n (9-1) ( o r a modified ve r s ion of

(9-1) w i t h s t r a t i f i e d and/or C o r i o l i s e f f e c t s ) , t h e c h a r a c t e r i s t i c s of t he

pe r tu rba t ion shape are found. For n e u t r a l s t r a t i f i c a t i o n , the maximum

pe r tu rba t ion growth rate i s found a t E = 18'.

f o r s t a b l e s t r a t i f i c a t i o n and decreases f o r unstable s t r a t i f i c a t i o n , reaching

E = 0 f o r moderately nega t ive Richardson numbers.

pe r tu rba t ion is a modified h a l f wave form with a maximum a t about z = 6 and

decreasing exponent ia l ly above.

This c r i t i ca l angle i n c r e a s e s

The vertical shape of t h e

82

10.3.2 - The Energy Balance

The growing p e r t u r b a t i o n has been included i n a n energy balance

equat ion i n o r d e r t o estimate how much energy i s a v a i l a b l e f o r i t s growth

and t o e s t a b l i s h equ i l ib r ium condi t ions. The p e r t i n e n t energy equat ion,

h e n averaged over a volume element which inc ludes one wavelength of t h e

secondary f l a w , may be w r i t t e n

The l e f t s i d e r ep resen t s t h e average change i n secondary flow energy. The

f i r s t term on the r i g h t s i d e r ep resen t s t h e interchange of energy with the

mean flow; t h e second term i s t h e rate o f work done by buoyancy due t o t h e

secondary flow; and t h e las t t e r m r ep resen t s a r e d i s t r i b u t i o n of energy

wi th in the boundary l a y e r , t h e average va lue of which vanishes when i n t e g r a t e d

over t h e boundary l a y e r . The d i s s i p a t i o n term has been neglected i n (10-2).

This i s j u s t i f i a b l e s i n c e the scale of t h e secondary flow i s much l a r g e r

than the smallest eddies where viscous d i s s i p a t i o n becomes important. It

can be c a l c u l a t e d and included i n t h e energy balance when warranted.

For t h e n e u t r a l case, the requirement of equi l ibr ium, i .e . , t h a t

aE,/at = 0, may be expressed a f t e r a h o r i z o n t a l i n t e g r a t i o n as

(10 -3)

where t h e overbar i n d i c a t e s an average i n the y d i r e c t i o n . This condi t ion

cons t r a ins the amplitude of t he pe r tu rba t ion .

It i s of i n t e r e s t t o n o t e t h a t t h e energy i s t r a n s f e r r e d from t h e

mean flow t o the r o l l s i n t h e lower p a r t of t h e boundary l a y e r and vice

versa i n the upper p a r t of the boundary l a y e r . This i s a l s o apparent from

comparing the modified hodograph wi th the o r i g i n a l Ekman s p i r a l .

83

10.3.3 The Modified Mean Flow Equations

The amplitude d i s t r i b u t i o n of t h e pe r tu rba t ion a l s o determines t h e

magnitude of t he s e c w d a r y flow as given by t h e s o l u t i o n of (9-2) and (10-3). The n e t e f f e c t of che momentum t r a n s f e r from the mean flow t o t h e secondary

flow and vice versa is t h a t an extra stress term appears i n t h e equat ion f o r

t he mean component:

2v + uzz

2u - vzz

0 (10-4)

A Re

where A i s t h e momentum t r anspor t con t r ibu t ion of wTZ which is the only

nonzero m e a n con t r ibu t ion from the secondary flow.

A closed form s o l u t i o n t o (10-4) can be w r i t t e n f o r t h e mean change,

v = VE + v: Z

F(z) = C,coshz s i n z + C,sinhz cosz - 2 Re

T(z) = Clsinhz cosz - C,coshz s i n z + 2 Re IZ s inh(z - 5) s in(z - S ) A ( S ) d S

cosh(z - 5) cos(z - C)A(C)dE

(10-5) S 0

0

- where z - IMX

C, = R e $ (S cosh 5 cos 5 - T s i n h s i n F) A(E)dE 0

0

S ( a s h zmax s i n z maX ) / D

P = (sinh zmax COS z )/D

E =

maX -

2 - 5 and D = s i n h 2 z f sin2z max' max max

84

10.4 Di s c:uss i Qn

10.4.1 Solut ion Procedure

There now e x i s t s u f f i c i e n t equat ions f o r a s o l u t i o n t o the problem

of t h e flow f i e l d i n the Ekman l a y e r . The Ekman v e l o c i t i e s are obtained

from (4-12), t h e shape parameter A ( z ) from s o l u t i o n s t o (9-2), t h e e q u i l i b -

rium secondary flow magnitude' from (10-3), the mean flow from (10-5), and

f i n a l l y the secondary v e l o c i t i e s from t h e stream funct ion. The e n t i r e

procedure has been programed i n For t r an I V .

The model r e s u l t s i n a computerized concatenation which

1. Calculates the general Ekman boundary l a y e r p r o f i l e f o r given

synopt ic v a r i a b l e s p l u s a c h a r a c t e r i s t i c mean roughness parameter 6 .

parameter is Ekman's depth of f r i c t i o n a l r e s i s t a n c e , o r t h e e-folding length

of t he l a t e r a l v e l o c i t y p r o f i l e ,

eddy v i s c o s i t y .

This

It i s p ropor t iona l t o K , t h e mean tu rbu len t

2. Solves f o r t h e eigenvalues and eigenfunct ions of t h e s t a b i l i t y

equation, t.he Orr-Sommerfeld equat ions modified t o inc lude s t r a t i f i c a t i o n .

3. Solves t h e set of equat ions f o r t h e equ i l ib r ium mean flow and

the secondary flow magnitude and shape.

Each p a r t of t h i s a n a l y s i s can be r e f ined o r s i g n i f i c a n t l y changed

independent:ly . Some a d d i t i o n a l considerat ions may b e included: pure

convective i n s t a b i l i t i e s i n the presence o f s h e a r produce long i tud ina l r o l l s

with c h a r a c t e r i s t i c s similar t o t h e dynamic i n s t a b i l i t y shape; d i s s i p a t i o n

might be adde t h e energy balance ecrease t h e amount a v a i l a b l e t o t h e

secondary flow; and the mean flow might b e replaced with another p r o f i l e ,

such as a thermal wind turning. The model discussed is t h e s imples t ve r s ion

and can serve as a framework f o r f u t u r e refinements, i f t hese become needed.

10.4.2 Resu l t s -- The b a s i c r e s u l t is an a n a l y t i c s o l u t i o n %r t h e p l ane ta ry boundary

l aye r . For va r ious condi t ions of s t r a t i f i c a t i o n (hence, d i f f e r e n t angles E ) ,

t h e seconda.ry flow magnitude varies from 6% t o 12% of U The cr i t ica l g'

85

wavenumber is y = 0.5.

For a t y p i c a l atmospheric 6 , t h i s y i e l d s a la teral wavelength between 1 . 4

and 3 km. perpendicular t o t h e r o l l a x i s , a t an angle 18' t o t h e l e f t of t he geostrophic

wind f o r n e u t r a l condi t ions , o r 0' f o r moderately uns tab le s t r a t i f i c a t i o n .

The s'econdary flow appears as h e l i c a l v o r t i c e s as shown i n Figures 21 and 22.

Figure 2 1 i s a cross sec t ion normal t o t h e r o l l a x i s .

t h e v e r t i c a l corresponds t o t h e mean lateral shear .

of the secondary c i r c u l a t i o n on passive elements is schematical ly ind ica t ed

f o r eddy vo r t i ce s . Figure 22 is a perspec t ive of t he r o l l s p lus t h e mean

I t is i nva r i an t w i th Re and varAes slowly with Ri.

Thus, one expects a near -c i rcu lar secondary motion in a p lane

The phase s h i f t in

The sweeping e f f e c t

flow.

mean flow hodographs are shown i n Figure 23.

This secondary flow is added t o t h e mean flow, and t y p i c a l modified

Re=900 e e = I5O, ~ ~ 0 . 5 e R i = 0

1 :o

2 km

1 .o 2.0 Y kin

3.0

Fig. 21. Cross sec t ion of secondary flow stream funct ion nondimensionalized wi th 6V . 2 and Y are rep resen ta t ive values f o r 6 = 200 m.

Schematic representa- t i o n o f r o l l effect on s ma9 1-scale eddy vor t i ce s .

86

I tiodogroph

Fig . 22. @pica1 secondary flow i n the planetary boundary layer (modified Ekman layer) .

s. 2 0.6

0.4

0 2

I

\ - I -

- I, - '8

8 - '8 \ - '8

8. '., , 0 - " ' 1

87

13.5 Summary

Several lmpl lca t ions of t h i s s o l u t i o n merit d iscuss ion . The r o l l s

p ic tured i n Figure 21 were determined from t h e i n f l e c t i o n point i n s t a b i l i t y

mode i n the lateral v e l o c i t y p r o f i l e s of t h e Ekman s o l u t i o n f o r a n e u t r a l

l aye r .

Furthermare, i t appears t h a t even widely d i s s i m i l a r energy sources ( inc luding

The shape is q u a l i t a t i v e l y s i m i l a r f o r a wide range of s t r a t i f i c a t i o n s .

convective and perhaps viscous modes) are n o t q u a l i t a t i v e l y d i f f e r e n t i n

shape of t h e maximum growth rate pe r tu rba t ion mode.

i n t o t h e model i f i t becomes subs t an t i a t ed as a real atmospheric phenomenon.

However, t h e q u a n t i t a t i v e d i f f e rence i s unl ike ly t o be s i g n i f i c a n t compared

t o tha t between t h e flows with and without secondary flow.

Each can be incorporated

The r o l l s travel l a t e r a l l y , with a phase v e l o c i t y approximately equal

t o V a t t h e i n f l e c t i o n point . From Figure 20 t h i s i s seen t o vary from -7%

t o +20% of Us. depending on E, which depends upon s t r a t i f i c a t i o n .

phase speed (and t h e t h e o r e t i c a l shape of t h e r o l l s ) h a s been observed and

analyzed i n the atmosphere by Lemone [1972],

e t al. [1971] suggest t h a t secondary flows are present . The long wavelength

motions tend t o organize t h e smaller-scale turbulence. This cha rac t e r i s t i c

banded s t r u c t u r e is being observed wi th increas ing frequency as new observa-

t i o n methods are found. Radar, t he acous t i c sounder, and sa te l l i t e photos

are revea l ing t h e ubiqui ty of t he r o l l s t r u c t u r e .

even appear i n the Mars satell i te photographs (C. Leovy, personal communica-

t i on ) . Addit ional observat ions are found in Angell [1971, 19721, Angell e t

al. [1965, 19681, Hardy and Ot te rs ten [1969], Konrad [1968, 19701, Ot te rs ten

[1969], and Schuetz and F r i t z [1961].

This

Flux measurements by Haugen

The c h a r a c t e r i s t i c bands

In addi t ion t o t h e d i f f e rences p red ic t ed in mean wind hodographs

(Fig. 23), t h e r e is a s i g n i f i c a n t v a r i a t i o n l a t e r a l l y f o r instantaneous

(or short-period) measurements, depending upon where in the r o l l t h e observa-

t i o n is made .

d i s t r i b u t i o n s der ived from observat ions. The presence of t he low-frequency

secondary motions i n t h e == KVa d e f i n i t i o n f o r K ( z ) w i l l produce erratic r e s u l t s . Indeed, this d e f i n i t i o n of K becomes ques t ionable for t h e large-

scale eddies , where t h e scale is so large that the molecular analogy is inappropr ia te .

This is one poss ib l e explanat ion f o r t h e widely divergent

88

The convergence and divergence areas superimposed on the h o r i z o n t a l l y

homogeneous mean flow w i l l produce regions o f h ighe r humidity, turbulence

o r p o l l u t i a n .

w i l l be augmented and suppressed accordingly.

be important, they must be parameterized i n t o the p l ane ta ry boundary l a y e r

model.

The cgnvective s t a b i l i t y and t h e h e a t and momentum f l u x e s

A s these e f f e c t s are found t o

W e have e s t a b l i s h e d t h a t , much of t he t i m e , t h e o u t e r s o l u t i o n

discussed i n s e c t i o n s 6 and 7 i s r e a l l y a modified Ekman s p i r a l w i t h a

nonhorizontal ly homogeneous secondary flow component.

determine what e f f e c t t h i s w i l l have on t h e two-layer s o l u t i o n .

It is important t o

Since

matching t h e i n n e r s o l u t i o n i n t h e s u r f a c e l a y e r does no t change t h e flow

above t h i s l a y e r , t he o u t e r s o l u t i o n w i l l be the one determined i n t h i s

s ec t ion .

c a n t l y changed by t h e secondary flow, Nevertheless , t h e matching r e l a t i o n s

are q u i t e general , and t h e d i f f e rences i n the o u t e r s o l u t i o n should r e s u l t

only i n d i f f e r e n t values of t h e s i m i l a r i t y constants . Since these are

determined empir ical ly

r e l a t i o n s h i p . Data must be taken on s c a l e s l a r g e enough t o determine mean

values w i t h respect t o t h e secondary c i r c u l a t i o n . This r equ i r e s l e n g t h

scales perpendicular t o t h e r o l l s of t e n s of ki lometers , o r t i m e averages

of one-half hour o r more.

'fie asymptotic l i m i t of t h e o u t e r mean p r o f i l e shape i s s i g n i f i -

t h e r e i s no p r a c t i c a l e f f e c t on t h e geostrophic drag

89

11. THERMAL EFFECTS

11.1 - Fundamental Parameters

Up t o t h i s p o i n t , s t r a t i f i c a t i o n has been included i n t h e models whenever

i t w a s convenient and did n o t d e t r a c t from t h e understanding of t he n e u t r a l

model.

stress i s measured r a t h e r d i r e c t l y and the temperature d i s t r i b u t i o n can b e

obtained simultaneously with t h e o t h e r extensive measurements. However , we

need a broad g e n e r a l i z a t i o n of t h e stress dependence on some s p e c i f i c s t ra t i -

f i c a t i o n parameter.

d i r e c t l y r e l a t e d t o t h e temperature g rad ien t , t o t he o t h e r energies involved.

This w a s t h e case i n t h e s u r f a c e l a y e r p r o f i l e methods, where the

This ' f a c t o r should re la te the buoyant energy, which is

Under s t a t i c condi t ions t h e appropr i a t e balance is between t h e buoyant

fo rces and viscous ( d i s s i p a t i v e ) fo rces , as represented i n t h e Rayleigh

number Ra = g/B*Bzh'/KifSh. Here, Kh is t h e eddy h e a t t r a n s f e r c o e f f i c i e n t from

When shear flow i s p resen t , t h e dynamic ( k i n e t i c ) energy e n t e r s , and t h e

r a t i o t o viscous fo rces is Re = G h/K,. is r e l a t i v e l y small, and t h e appropr i a t e parameter is t h e Richardson number

Generally, t h e d i s s i p a t i v e energy

R i = Ra/PrRe2 = gBZ/eVz2

The eddy P r a n d t l number, Kh/Km, is gene ra l ly taken as constant near u n i t y ,

bu t i t may vary with s t r a t i f i c a t i o n .

Probably t h e main stumbling block t o success fu l parameter izat ion of

s t r a t i f i c a t i o n e f f e c t s i s t h e ambiguity with r e spec t t o t h e c o r r e c t param-

eter.

f o r a semi - in f in i t e f l u i d over an i n f i n i t e plane.

t h e eddy v i s c o s i t y concept, t h e characteristic l eng th 6 = ( K / f ) # occurs.

I f t h e s u r f a c e stress is given as a boundary condi t ion, u*/f provides a

c h a r a c t e r i s t i c length.

flow models.

c h a r a c t e r i s t i c scale l eng th i n t h e a n a l y s i s con t r ibu te s to a n indeterminacy

A p r i o r i , t h e r e does no t e x i s t a c h a r a c t e r i s t i c l eng th i n t h e problem

When t h e equations employ

Both of t h e s e scales w e r e employed i n t h e dynamic

For pure convection i n a nea r s t a t i c f l u i d , t h e l a c k of a

9 1

i n t h e i n s t a b i l i t y mode. I n add i t ion , t he inde f in i t eness i n appropr ia te s c a l e com- p l i c a t e s t h e establ ishment of a bulk va lue f o r Ra or R i based on a f i n i t e d i f f e rence r ep resen ta t ion of t h e v e l o c i t y o r temperature grad ien ts . One is l e f t with l o c a l values of V z , e,, and R i which vary v e r t i c a l l y .

s i o n he ight i f known) and bulk s t a b i l i t y parameters has been developed by Deardorff [1970]. It is designed t o provide f luxes compatible wi th t h e genera l c i r c u l a t i o n model requirements. Clarke [1970], i n t he Aus t ra l ian experiments, determined the invers ion he ight f o r t he uns tab le cases , apparent ly mainly from t h e vigor of con- vect ion. The Ekman l a y e r depth was ind ica ted by a turn ing of t h e v e l o c i t y p r o f i l e induced by su r face f r i c t i o n . of t h e flow was o f t e n much less than t h e invers ion depth. Clarke found t h a t using t h e dynamic c h a r a c t e r i s t i c depth u,/f reduced s c a t t e r i n t h e da t a s i g n i f i c a n t l y more than using the observed invers ion depth, thus support ing t h e hypothesis t h a t the dynamic processes dominate w e l l i n t o t h e thermally uns tab le regime.

A parameter izat ion based on a given boundary l a y e r he ight (e.g. , t h e inver-

This c h a r a c t e r i s t i c depth assoc ia ted wi th the dynamics

11 .2 Convection --- Convection has been under i n v e s t i g a t i o n s i n c e Bgnard's experiments i n 1901.

Rayleigh 119161 i n i t i a t e d t h e t h e o r e t i c a l development, e s t ab l i sh ing the c r i t i c a l (minimum) Rayleigh number cr i ter ia f o r the onse t of i n s t a b i l i t y and the inde ter - minacy i n the i n s t a b i l i t y mode. The convective i n s t a b i l i t y mode is t i e d c lose ly t o t h e c h a r a c t e r i s t i c depth through the Rayleigh number dependence. The c r i t i c a l (maximum growth r a t e ) wavelength is a l s o s e n s i t i v e t o lateral boundary condi t ions and i n i t i a l condi t ions [Ogura, 19711, and t h e r e i s a time-dependent s h i f t i n modes [Krishnamurti, 19701. The general u n a v a i l a b i l i t y of t h e necessary d a t a f o r any of these condi t ions i n geophysical problems is an impediment t o model development,

B&ard, i n h i s convective experiments , observed long i tud ina l s t r i p s , as w e l l as the f ami l i a r convective cells , when a genera l v e l o c i t y w a s p resent . The com- p l e x i t i e s inherent i n adding a general v e l o c i t y p r o f i l e t o t h e s t a b i l i t y ana lys i s have g r e a t l y r e s t r i c t e d so lu t ions [see, e.g. , Gage and Reid, 19681. Indeed, t h e complete geophysical problem inc ludes t h e i n h i b i t i n g e f f e c t s of shear and r o t a t i o n on t h e thermal i n s t a b i l i t y , p lus t h e i n t e r a c t i o n wi th t h e dynamic i n s t a b i l i t y .

S a t e l l i t e observat ions show t h e presence of mesoscale convection i n hexagonal cells and i n long i tud ina l s t r i p s , sometimes merging together . t h e labora tory and a n a l y t i c r e s u l t s which i n d i c a t e t h a t a mean shear flow suppresses pe r tu rba t ions i n t h e plane of t h e shear , as discussed i n Brown [1972b]. t akes p l ace i n h e l i c a l r o l l c i r c u l a t i o n s i n planes perpendicular t o t h e m e a n flaw. The problem becomes involved i f t h e mean flow is three-dimensional, as i n an Ekman s p i r a l . f o r c e s only as they occur i n subordinate conjunct ion wi th dynamic forces .

This corresponds t o

Convection

The convection problem is still unse t t l ed , so we s h a l l consider convective

92

\

11.3 - Convection E f fec ts i n the Dynamic Flow Models

11 .3.1 The Diaba t i c Ekm-aLLayer

The geostrophic/Ekman s o l u t i o n remains unchanged i n t h e presence of

moderate temperature g rad ien t s i n t h e h o r i z o n t a l o r ver t ical . The b a r o c l i n i c

e f f e c t , i .mplicit i n t he p re s su re terms, produces a thermal wind which may

be l i n e a r l y superimposed on t h e Ekman o r Taylor s p i r a l s .

temperature g rad ien t i s decoupled f o r t hese motions by t h e Boussinesq

approximaition and t h e assumed h o r i z o n t a l homogeneity.

The ver t ical

The extension of K theory modeling t o t h e s t r a t i f i e d case requ i r e s

t h e i n t r o d u c t i o n of a s t r a t i f i c a t i o n parameter i n t o t h e r e l a t i o n f o r X ( z ) . The d i s t r i b u t i o n of turbulence w i l l n a t u r a l l y be g r e a t l y inf luenced by t h e

e f f e c t of buoyancy 0 t h on t h e mechanism of turbulence production and t h e

d i s s i p a t i o n rate. The general form of t h i s e f f e c t can be conjectured and

incorporated i n t o t h e formulas f o r K i n s e c t i o n 8. This has been done i n

Blackadar and Ching 119651, f o r example.

t h e l i a b i l i t i e s of n e u t r a l K modeling are simply compounded.

Some t r ends emerge, bu t gene ra l ly

When t h e i n n a t e i n s t a b i l i t y a t s u p e r c r i t i c a l Reynolds numbers is con-

s i d e r e d and t h e equi l ibr ium s o l u t i o n inco rpora t ing secondary flow is

ca l cu la t ed , t h e d i a b a t i c s ta te of t he boundary l a y e r becomes important.

The i n s t a b i l i t y mode is influenced by t h e s t r a t i f i c a t i o n as represented by

a l o c a l Richardson number. The i n f l e c t i o n po in t s t a b i l i t y problem becomes

s i x t h order r a t h e r than fou r th o r d e r , and t h e R i number i n t h e v i c i n i t y of

t he i n f l e c t i o n po in t e n t e r s as a parameter. The flow is s t a b l e f o r a l l

v e l o c i t i e s i f Rig > 0.25.

wave o s c i l l a t i o n s a t more s t a b l e Richardson numbers. Recent c a l c u l a t i o n s

by Kaylor and F a l l e r [1972] i n d i c a t e the p o s s i b i l i t y of r o l l s i n very

s t a b l y s t r a t i f i e d condi t ions due t o resonance i n s t a b i l i t i e s . The r e l a t i o n -

s h i p between g r a v i t y waves and i n s t a b i l i t y waves has y e t t o be f u l l y analyzed.

There i s s t i l l t h e p o s s i b i l i t y of forced g rav i ty

The hngle between the r o l l a x i s and the geostrophic flow varies from

g r e a t e r t han 45' f o r s l i g h t l y s t a b l e s t r a t i f i c a t i o n toward Oo f o r uns t ab le

s t r a t i f i c a t i o n s .

s i g n i f i c a n t e f f e c t on the mean flow p r o f i l e .

This change i n o r i e n t a t i o n of t h e secondary flow has a For s t a b l y s t r a t i f i e d

93

condi t ions , t h e mean flow e x h i b i t s a supergeostrophic j e t ( s e e Fig. 23).

t h i s condi t ion prevails a t n igh t , i t may expla in the connnonly observed nocturnal

jet .

r o l l p l an form and its equi l ibr ium magnitude do not vary much over t h e range

of atamspheric and oceanographic parameters.

As

Under t h e assumed nonconducting boundary condi t ions , t he shape of t h e

'The problem fac ing t h e dynamic flow model of t he modified Ekman l a y e r - 1

i s t h a t o f "the change i n equi l ibr ium boundary condi t ions.

temperature p r o f i l e can b e envisioned by r e t r a c i n g the development f o r a

s t a b l e (o r d i f f e r e n t equi l ibr ium) s ta te ( see Fig. 24). The s t e p s shown

follow a d i u r n a l cyc le from morning t o evening.

involves t r a n s i t i o n from an uns tab le t o a s t a b l e s i t u a t i o n . I n the l a t e

dgternoon, t h e d i r e c t i o n of t h e hea t f l u x w i l l change and the su r face begins

The equi l ibr ium

The o the r p a r t of t he cyc le

the coolfng whfch may become very marked f o r n igh t s amenable t o r a d i a t i o n

cooling. I "Thus, a s t a b l e po r t ion of t he - sur face l a y e r develops f i r s t .

case of a s t r o n g secondary c i r c u l a t i o n , ' t h e mixing may allow t h e e n t i r e l a y e r

to cool uniformly before t h e i n f l e c t i o n po in t region, in t h e lower p a r t of

t h e Ekman l a y e r , becomes s t a b l y s t r a t i f i e d .

I n t h e

, The- produdtion terms f o r tu rbulen t k i n e t i c energy (and eddy v i s c o s i t y )

small and d i s s i p a t i o n cont inues, so t h a t t h e general l e v e l of turbu-

A l l i n s t a b i l i t i e s appear t o be'damped out when R i R > 0.25.

become

lence decreases.

produce a r e l a t i v e l y inv i sc id flow (with respect t o eddy v i s c o s i t y ) .

A s l i p w p e boundary condi t ion becomes appropr ia te , wi th a 'calm laminar

sub laye r fn the s u r f a c e l aye r . The res

s t o a lowered Rik

s t rong shear a t t h e top of

s t a b i l i t i e s occur , pro-

ducing f u r t h e r mixing.

momentum f l u x which e x i s t e d i n t h e uns tab le p lane tary boundary l a y e r disappears

The near homogeneity i n p o t e n t i a l temperature and

i n t h e stable boundary l aye r .

-Dyn&nic i n s t a b i l i t i e s a t t h e Brunt-Vaissala frequency, determined from

t h e vertical momentum equat ion balance between i n e r t i a l and buoyant fo rces ,

may develop. These i n s t a b i l i t y waves, c a l l e d g r a v i t y w a v e s , a t frequency

94

r e q u i r e a f i n i t e pe r tu rba t ion forcing.

component and be instrumental i n momentum f lux .

They may propagate with a v e r t i c a l

I n gene ra l , s t a b l e s t r a t i f i c a t i o n p l aces the eddy v i s c o s i t y concept

i n ques t ion and seve re ly complicates the a n a l y t i c approach.

dence that: the s p i r a l i n g v e l o c i t y p r o f i l e is observed even i n t h e s t a b l y

s t r a t i f i e d condi t ions, i n d i c a t i n g a Coriolis-viscous f o r c e balance p r e v a i l s .

There are a l s o d a t a taken i n a s t a b l y s t r a t i f i e d boundary l a y e r during a

d i u r n a l cyc le which show t h e e n t i r e l a y e r quickly responding t o t h e change

i n s u r f a c e temperature [Riordan, 19721. This could n o t be accomplished by

molecular d i f f u s i o n and t h e r e f o r e i n d i c a t e s an eddy ac t ion .

There is evi-

The h e a t exchanger e f f e c t of the secondary flow suggests using con-

duct ing boundary condi t ions. This w i l l r ep resen t t he e f f e c t of t h e enhanced

temperature g rad ien t s , from r a d i a t i o n hea t ing a t t h e bottom t o a d i a b a t i c

mixing a t the top. A con t inua l hea t ing of the e n t i r e mixed boundary l a y e r

can a l s o be t r e a t e d . The determinat ion of t h e h e a t t r a n s f e r c o e f f i c i e n t s

a t t h e boundaries and the subsequent i nc rease i n secondary c i r c u l a t i o n

complicates t h e model. However, t h i s concept o f f e r s an explanat ion of the

l a r g e h e a t t r a n s p o r t s and secondary c i r c u l a t i o n observed [Lemone, 19721.

The modif icat ion t o t h e e x i s t i n g models would be f a i r l y simple, b u t i t

would depend on a r b i t r a r y parameters which are not c u r r e n t l y a v a i l a b l e from

experiments.

11.3.2 Diaba t i c S i m i l a r i t y Theory

When a d i aba t i c s ta te is included i n t h e a n a l y s i s of s e c t i o n 7, the

p o t e n t i a l temperature, dens i ty , and p res su re may vary a r b i t r a r i l y i n t h e

vertical throughout t h e boundary l a y e r depth. The o u t e r s o l u t i o n is essen-

t i a l l y t h e same as t h e n e u t r a l case f o r moderate heat ing.

parameter .r/p now varies due t o p(z) i n a d d i t i o n t o ~ ( z ) .

pep w'Z' ' , t h e eddy h e a t f l u x , is r e l a t e d t o t h e p o t e n t i a l temperature g r a d i e n t

by a d i f f u s i o n r e l a t i o n ,

The s i m i l a r i t y

The new v a r i a b l e ,

95

The problem remains closed provided p(z) o r 8 ( z ) is given, s i n c e t h e pressure

g rad ien t is now va r i ab le :

-The energy equat ion is tr ivial f o r t h e ho r i zon ta l homogeneous, s teady-s ta te

condi t ions unless a r b i t r a r y h e a t source-sinks are given. For t h i s reason, t he

d i a b a t i c conducting boundary l a y e r is merely def ined i n terms of the a d i a b a t i c

dynamic flow problem and one o r more add i t iona l a r b i t r a r y parameters.

Obukhov [1971] has considered a sublayer i n which the s t r a t i f i c a t i o n

0) and su r face stress determines t h e turbulence e f f e c t is 81~111 ( R i

F . Sketch of etature-height orjf f o r a case

of su r face hearfng .

f

Inversion height

Adiabatic layer

\

. 10 Ts d. Secondory flow

i

0 . Stable T ( t l b. Surfocr hcoting

96

s t r u c t u r e . The c h a r a c t e r i s t i c he igh t of t h i s sublayer i s

This c h a r a c t e r i s t i c scale, a r r i v e d a t by purely dimensional reasoning i n

Monin and Obukhov [1954], provides an alternate c h a r a c t e r i s t i c l eng th for

t h e s u r f a c e l a y e r s o l u t i o n . The a n a l y s i s could then continue using L i n

p l a c e of ,go.

t he logari thmic s i n g u l a r i t y a t z = 0 . The choices f o r nondimensionalizing

t h e d i a b a t i c s u r f a c e l a y e r p r o l i f e r a t e . Clarke [1970] has analyzed d a t a

t o determine an appropr i a t e scale. H e chose t o have the constants A , B ,

and C vary with u,/fL, as discussed b r i e f l y i n s e c t i o n 7 .

t h a t s i m i l a r l y c o n s i s t e n t r e s u l t s may be obtained f o r t h e inne r /ou te r

s o l u t i o n of s e c t i o n 6 b y , l e t t i n g h / z o be a func t ion of s t r a t i f i c a t i o n .

However, i n gene ra l , z o must e n t e r t h e equations t o prevent

It i s p o s s i b l e

Deardorff [1972] has devised a gene ra l parameter izat ion scheme t o be

compatible with numerical general c i r c u l a t i o n models. H e employs a bulk

Richardson number, t he Monin-Obukhov l eng th , and t h e roughness l eng th t o

e s t a b l i s h t h e h e i g h t of t h e s t r a t i f i e d boundary l a y e r . S t r e s s and f luxes

are then determined i n conjunction with t h i s he igh t .

The a d d i t i o n a l c h a r a c t e r i s t i c scales i n the d i a b a t i c problem, p l u s

t h e pauci ty of d a t a wi th d e f i n i t i v e measurements of t h e parameters involved,

l i m i t t h e d i a b a t i c boundary l a y e r a n a l y s i s a t p re sen t .

97

12. NONSTATI ONARITY

12.1 Discussion

Deviations from steady s ta te i n t h e p lane tary boundary layer can

assume many forms. Such departures inev i t ab ly lead t o nonhorizontal-homo-

genei ty . The governing equations are then nonl inear , with t i m e as an

add i t iona l independent va r i ab le ; and they r equ i r e in i t ia l condi t ions which

are d i f f i c u l t t o ob ta in . The a t t i t u d e of the modeler may w e l l be: either

a s teady state model approximates the real condi t ions, o r t h e problem i s

too complicated f o r t h e o r e t i c a l models. Nevertheless, certain time

dependent v a r i a t i o n s must be considered, i f only t o determine t h e limits of

the s teady s ta te model. It may then be poss ib l e to approximate t h e d i f f e r e n t

states with combinations of s teady state , hor i zon ta l ly homogeneous models

Some deviat ions from s t a t i o n a r i t y and homogeneity can b e incorporated

i n t o the models without d i s tu rb ing t h e b a s i c flow pa t t e rn .

t i o n i n the thermodynamic parameters (ba roc l in i c i ty ) changes b u t does n o t

des t roy the ho r i zon ta l homogeneity of t h e Ekman l a y e r flow.

homogeneous secondary flow r e t a i n s a s teady state mean flow.

i n t e r r a i n , p a r t i c u l a r l y between land and sea, may b e handled wi th s t e p

changes i n eddy v i s c o s i t y and/or s t r a t i f i c a t i o n .

Horizontal var ia -

The nonhorizontal

Differences

The most obvious v a r i a t i o n s i n t h e s teady s ta te model w i l l come from

the t i m e dependent v a r i a t i o n i n r a d i a t i o n a l hea t ing i n t h e d iu rna l cycle , and from t h e s p a t i a l v a r i a t i o n from t h e movement of synopt ic pressure systems.

12.2 The Diurnal Cycle

Many of t h e phenomena discussed i n p lane tary boundary l a y e r modeling

develop from in i t ia l condi t ions, e s t a b l i s h a s teady state, and degenerate

i n a per iod s i g n f f i c a n t l y s h o r t e r than t h a t assoc ia ted with t h e d i u r n a l

va r i a t ion .

stress f i e l d f o r per iods g r e a t e r than the d iu rna l , the t i m e Variation f o r

long-term effects may be approximated w i t h a two-step model w h i c h r ep resen t s

the d i f f e r e n t mean states during night and day.

However, since one of the parameters of f requent interest is the

Ihe simplest such model

98

would merely consider a two-step K i n the n e u t r a l equat ions. This i s intended t o r ep resen t t h e obvious p e r i a d i c i t y i n t h e small-scale turbulence

induced by hea t ing .

da t a , few, i f any, experiments have been of s u f f i c i e n t durat ion t o e s t a b l i s h

Although this t r end is, ev iden t in the obse rva t iona l

a d i u r n a l p a t t e r n t o K(z,t) . The change i n flow can b e expected t o b e most

s i g n i f i c a n t i n t h e o u t e r po r t ions of t he p l ane ta ry boundary l a y e r . Indeed,

i f secondary flow i s n o t considered i n t h e a n a l y s i s , t h e Ekman l a y e r s p i r a l

would vanish f o r most of t h e cycle as t h e eddy v i s c o s i t y became zero a t

l o w levels.

e f f e c t i v e l y decoupling t h e s u r f a c e from t h e f r e e stream. This e f f e c t would

depend an the percentage of t h e small-scale turbulence which w a s produced

by h e a t i n g compared t o t h a t r e s u l t i n g from dynamic i n s t a b i l i t i e s (e.g., viscous shear).

modify t h i s p i c t u r e by providing a well-mixed deep l a y e r .

t o d i s t r i b u t e throughout t he l a y e r any turbulence produced l o c a l l y

shea r regions.

stable bulk s t r a t i f i c a t i o n , t h e e f f e c t w i l l be t o allow the well-mixed l a y e r

t o p e r s i s t even i n t o s t a b l y s t r a t i f i e d s i t u a t i o n s .

The night t ime s t a b l e s t r a t i f i c a t i o n would damp ou t t h e turbulence,

The presence of secondary flow c i r c u l a t i o n would tend t o

This would tend

Since shea r i s bound t o be high i n some region even f o r

B u a j i t t i and Blackadar [1967] solved the n e u t r a l Ekman equat ion

a t i m e dependent K ( t ) . The t i m e dependent terms become a f o r c i n g func t ion

on the right-hand s i d e of t h e Ekman equat ions. When a p e r i o d i c v a r i a t i o n

is assigned t o K ( t ) , an a n a l y t i c s o l u t i o n can be found. They a l s o obtained

numerical s o l u t i o n s f o r empi r i ca l ly der ived K ( z , t ) . The a n a l y t i c s o l u t i o n

w a s r e f i n e d by Sheih [1971]. The decay t i m e f o r a d i u r n a l l y imposed t r a n s i e n t

component i s about 2% days. This component, from t h e d iu rna l v a r i a t i o n s i n

t h e p re s su re g rad ien t , provided a b e t t e r f i t t o observat ions. The e f f e c t

of d i u r n a l v a r i a t i o n i n p re s su re g r a d i e n t on the Ekman s p i r a l w a s a l s o

computed by Pandolfo [1969] and analyzed by Ching and Businger [1968]. These

s o l u t i o n s showed i n e r t i a l o s c i l l a t i o n s appearing i n t h e wind s p i r a l s , varying

i n phase and amplitude w i t h he igh t .

In t h e Arctic, t h e d i u r n a l p e r i o d becomes longer than the s i g n i f i c a n t

pe r iods wi th r e spec t t o ice dynamics. This v a r i a t i o n w i l l then appear

as a t r e n d i n r e l a t e d parameters, such as K ( z ) , and mean va lues can be

s u c c e s s f u l l y employed. When long-term effects are needed, then time dependent

99

v a r i a t i o n s i n the eddy v i s c o s i t y can b e included, as was done by Estoque

[1963]. For longer per iods , ice dynamics and thermodynamic balance w i l l

become con t ro l l i ng f a c t o r s in the problem.

12.3 Synoptic Variations

The upper boundary condi t ion, t h e geostrophic flow, varies s p a t i a l l y

over t h e thousands of square k i lometers of t h e arctic ice.

pressure systems migrate eastward, t he upper flow w i l l a l s o vary temporally.

Again, i n the Arc t ic , the magnitude of these system v a r i a t i o n s is minimal

compared t o those of t h e middle l a t i t u d e s . S t i l l , t h i s v a r i a t i o n i n t h e

d r iv ing force , modified i n t r a n s i t i o n through t h e p lane tary boundary l a y e r

t o t h e sur face , i s responsible f o r most of t h e middle-scale i c e motion.

In addi t ion , the oceanic geostrophic flow can b e expected to r e f l e c t t h e

atmospheric synopt ic pressure d i s t r i b u t i o n .

As l a rge-sca le

Although the determination of the synopt ic s i t u a t i o n i s fundamental

t o the AIDJEX goal of r e l a t i n g stress t o synopt ic parameters, i t l i es mainly

ou t s ide p lane tary boundary l aye r ana lys i s . Here, w e consider only t h e

response of the boundary l a y e r model t o v a r i a t i o n s i n t h e upper boundary

condition.

In h i s o r i g i n a l paper, Ekman [1905] included a b r i e f d i scuss ion of

the development t i m e f o r h i s s teady state so lu t ion .

i nd ica t ed the development t i m e of the s p i r a l v e l o c i t y p r o f i l e t o be comparable

t o the i n e r t i a l period in the ocean.

H i s i n i t i a l c a l c u l a t i o n s

The synopt ic v a r i a t i o n appears i n the equat ions as a t i m e dependent

pressure v a r i a t i o n . ,This f a c t o r has been included by Pandolfo and Brown

[1967] as a pe r iod ic v a r i a t i o n of geostrophic v e l o c i t y i n t h e i r numerical

model.

by Ching and Rusinger [1968].

i ne r t i a l o s c i l l a t i o n is a p e r s i s t e n t c h a r a c t e r i s t i c of t h e s o l u t i o n , decaying

slowly.

eddy v i s c o s i t y c o e f f i c i e n t and t h e i n i t i a l state.

obtained by Ching and Businger are s h m i n Figures 25 and 26.

An a n a l y t i c s o l u t i o n f o r t h e pe r iod ic p re s su re v a r i a t i o n w a s provided A l l of t hese so lu t ions i n d i c a t e t h a t t h e

The response of t h e e n t i r e l a y e r depends on t h e assumed constant

The variable hodographs

There is a

100

24 *

STEADY EKMAN

>

0.96

SPIRAL

STEADY STATE EKMAN SPIRAL

0 , 2 4

v 16

Fig, . 25. Theoretical wind hodographs for a rotating pressure gradient with a periodic osci l lat ion of 4 days (a), and 1 day (b), From Ching and Businger [ 19681.

101

INITIAL STEADY STATE EKMAN- SPIRAL

48

A C f STATE SPiRAL

F i g . 26. Same hodographs as i n Figure 25 in a coordinate system rotating with the pressure gradient, for a period of 4 days (a), and 1 day @I. Ching and Businger [1968].

From

102

corresponding v a r i a t i o n i n geostrophic drag c o e f f i c i e n t .

due t o the change i n 2 , from t h e v a r i a b l e v e l o c i t y g rad ien t a t t h e su r face .

There i s an a d d i t i o n a l change due t o t h e changing geostrophic f l o w corre-

sponding t o t h e pressure g rad ien t change ( i s a l l o b a r i c wind) .

This i s p a r t l y

The v a l i d i t y of t h e h o r i z o n t a l homogeneity assumption is examined

by Arya [:L972] by considering a cons t an t ly varying s u r f a c e roughness.

f i n d s s i g n i f i c a n t dev ia t ions i n t h e ou te r s p i r a l f o r moderate g r a d i e n t s of

s u r f a c e stress. H i s general s i m i l a r i t y argument includes s p e c i f i c boundary

l a y e r condi t ions and assumptions which might be smoothed o u t i n a hetero-

geneous mesoscale region. Nevertheless, t h e c a l c u l a t i o n s i n d i c a t e t h a t

nonhomogerieity due t o i n e r t i a l o r advect ive a c c e l e r a t i o n s may d i s t u r b the

equ i l ib r ium p r o f i l e s under r a p i d l y changing cond i t ions . Thus, t h e advection

term, U U3: - 10 m/sec U x , i s t h e same o r d e r of magnitude as t h e C o r i o l i s

term, fv 10-4/sec*10 m/sec =

Fortunately, i n t h e Arctic, the uniform t e r r a i n of very l a r g e areas p lus t h e

moderate synopt ic and d iu rna l v a r i a t i o n s make t h i s l i m i t a t i o n t o t h e s t eady

state, h o r i z o n t a l l y homogeneous models less c r i t i c a l .

He

m/sec2 f o r an a c c e l e r a t i o n of 10 cmlseclkm.

12.4 - Sunimary

The t i m e dependent t e r m s have been neglected by n e c e s s i t y i n o rde r

t o o b t a i n a n a l y t i c so lu t ions . The scant a n a l y t i c r e s u l t s a v a i l a b l e i n d i c a t e

t h a t Ekmart l a y e r development t i m e is comparable t o t h e d iu rna l frequency.

These estimates are cont ingent on t h e value of eddy v i s c o s i t y and do not

include the e f f e c t s of secondary flow, e i t h e r dynamically o r convect ively

induced.

and decrease r e a c t i o n t i m e s i n the p l a n e t a r y boundary l a y e r . From t h e o r e t i c a l

magnitudes of secondary flows and from observat ions, Brown [1970b] estimates

the per iod of r o t a t i o n f o r a t y p i c a l two ki lometer diameter h e l i c a l c i r c u l a -

t i o n i n the atmospheric boundary l a y e r i s about % hour.

phenomenon is f r equen t ly p e r s i s t e n t f o r many hours, i n d i c a t i n g a s t eady

state s i t u a t i o n .

The presence of secondary c i r c u l a t i o n would g r e a t l y enhance mixing

The cloud street

The c h a r a c t e r i s t i c t i m e scale is of primary importance. If the

problem be ing i n v e s t i g a t e d is t h e flow i n the af ternoon o r during t h e n i g h t ,

103

then steady state i s a reasonable assumption.

are considered, t h e d i u r n a l v a r i a t i o n may be averaged ou t , b u t t h e synopt ic

may become important.

For time periods i n between, t h e t i m e dependent terms are l i k e l y t o become

important.

employed t o determine t h e s e so lu t ions .

m e n extremely long per iods

Climatic s t u d i e s may consider even these as averages.

Currently, i t appears t h a t numerical methods would have t o be

104

13. APPLICATION TO STRESS CALCULATIONS

13.1 Discussion

The t h e o r i e s , models o r experimental methods which have been discussed

are shown i n Table 13.1.

r ep resen t a l l c u r r e n t l y v i a b l e approaches, with t h e p o s s i b l e exception of

t h e numerical i n t e g r a t i o n of t h e non l inea r equat ions [Deardorff, 1970, 19721.

Boundary .layer c h a r a c t e r i s t i c s which have been f r equen t ly observed (ho r i zon ta l

v a r i a b i l i t y , s t r a t i f i c a t i o n , v a r i a b l e a) are checked i n t h i s t a b l e i f they

These methods of p l ane ta ry boundary l a y e r a n a l y s i s

are considered i n t h e model. The f a c t o r s involved i n t h e c a l c u l a t i o n of t h e

s u r f a c e s t r e s s are l i s t e d . Some o f t h e constants are l i k e l y t o vary,

p a r t i c u l a r l y those a s soc ia t ed with the s u r f a c e cha rac t e r . This s t a t e - o f -the-

a r t summaicy i s q u i t e b r i e f , p a r t i c u l a r l y f o r s t r a t i f i e d o r nonhorizontal ly

homogeneous condi t ions.

Many approaches have n o t been discussed here . They gene ra l ly repre-

s e n t refinements t o , o r combinations o f , t h e general c a t e g o r i e s t r e a t e d .

For examplle, refinements i n K t heo r i e s have been only b r i e f l y mentioned.

Their inadequacy stems p a r t l y from the frequent nonhomogeneity of t he p l ane ta ry

boundary l a y e r . Numerical i n t e g r a t i o n of t h e complete equat ions m u s t span

too many scale magnitudes t o provide p r a c t i c a l r e s u l t s at p re sen t . However,

i t is somitimes p o s s i b l e t o check t h e o r i e s using these numerical r e s u l t s .

Calculat ion of drag using the momentum de fec t method has been success-

f u l i n aerodynamics. Since t h i s method r e q u i r e s a d e t a i l e d v e l o c i t y survey,

i t i s imprac t i ca l f o r atmospheric boundary l a y e r s . However, t he ice platform

h a s allowed accura t e measurements i n the oceanic boundary l a y e r , and the

method may be of use the re . It is b r i e f l y summarized i n Appendix A . The

method has been appl ied t o a K-theory der ived a n a l y t i c p r o f i l e by Le t t au and

Dabberdt 119721.

When one assumes t h a t a l l boundary layer parameters are approximately

constant , t h e momentum de fec t method may b e appl ied t o tke atmospheric boundary

l a y e r , Although’ this assumption seems gross , kytoon measurements i n l igh t

w i n d s (< 5; m / s e c ) over t h e ice i n d i c a t e t h a t it sometimes approximates a c t u a l condi t ions. This p a r t i c u l a r a p p l i c a t i o n is discussed i n Appendix B.

105

TABLE 13.1

FACTORS I N STRESS CALCULATION FOR VARIOUS MODELS

Measurements Required Nonhorizontal S t r a t i f i - Range Constants

Model Homogeneity ca t ion of a Used once/surf ace of ten/day Comment

EKMAN LAYER

-Ekman

-Ekman wi th secondary flow

-Ekman wi th secondary flow

F -Taylor s p i r a l 0 Q\

-K theor ies

-K theor ies

TWO LAYER SOLUTIONS

-General

-Gener a1

s i m i l a r i t y

s i m i l a r i t y

-1nne r / out e r

-Inner / out e r (Ekman)

(Ekman)

X

X X

X

X

X

45 "

2 8"

0-50"

20 -45 "

20 -45 "

0-45'

0-45"

0-45 "

0-45 "

0-45"

f K P(X,Y 1 Fixed a, l a r g e

f K P(X,Y> Fixed a

K P(z,y) and a(dO/dz> e(o ( z 5 6 ) Does 8(z<10 m) -

@ ( a 4 N 100 m)?

f

f K P ( x , y ) and Needs v e l o c i t y U(h) a t top of t h e

s u r f ace l a y e r

K ( z ) P(Z,Y) No un ive r sa l f K ( z ) found

K L Z , e ( ~ ) i P(X,Y) a d e w f

k z g , a, A, C P(tc,y) and Few evaluations e w of cons tan ts , consider ab l e

CI 0 ' U

TABLE 13.1 (CONTINUED)

FACTORS I N STRESS CALCULATION FOR VARIOUS MODELS

Model Nonho r izon t a l S t r a t i f i - Range Cons t an ts i4e as ii remait s Xeq u i r ed

Homogeneity c a t i o n of a Used once/surface of ten/day Coment

SURFACE LAYER k -- -P ro f i l e method X

-Eddy correla- t i on method X X --

-Eddy correla- t i on method and p r o f i l e method

-Eddy correla- t i on method and p r o f i l e method

U(0-10 m), Surface l a y e r 3 0 e(0-10 m), methods need a

f o r T. "Direct " p o i n t - stress measure-

u 'w ' ment, co r re l a - t i o n with veloc- i t y .

o r u'w ' (6 )

' D = '/ulo 2

o r

CD = T / G ~

An at tempt t o measure drag d i r e c t l y wi th a drag p l a t e cut from t h e

i c e w a s made by Karelin and Timokhov [1972]. The results exh ib i t ed wide

scatter and served mainly t o i l l u s t r a t e t he d i f f i c u l t i e s i nhe ren t i n seeking

a cons i s t en t r e l a t i o n s h i p between various p o i n t measurements n e a r t h e su r face

a f t h e p lane tary boundary layer.

For AIDJEX, t h e des i red r e s u l t is the average stress vec to r a t t h e

su r face f o r a scale of hundreds of ki lometers .

the requi red d a i l y o r hourly input information must be f a i r l y easy t o obta in .

The da ta input from the Arc t ic w i l l be r e s t r i c t e d t o measurements from

sparse ly placed da ta buoys and pe r iphe ra l manned s t a t i o n s .

p r o j e c t demonstrated tha t f a i r l y good pressure data can be obtained from

buoys. It seems l i k e l y t h a t durable sensors f o r wind d i r e c t i o n and tempera-

t u r e up t o three meters can be designed.

sur face temperature may be adequate.

mining the upper flow boundary condi t ion.

For a &del t o b e workable,

The 1972 p i l o t

Satellite i n f r a r e d de tec t ion of

Satell i te soundings may a i d i n de t e r -

Many one-time measurements w i l l be made i n t h e manned AIDJEX experi-

Values of x,,, k , and the s i m i l a r i t y parameters can be ca lcu la ted . ments.

Direct stress measurements w i l l be made i n the constant f l u x l a y e r using

sonic anemometers and an attempt w i l l be made t o c o r r e l a t e t h e s e measurements

t o the simultaneously measured synopt ic parameters. A l l models r equ i r e t h e

geostrophic flow t o be determined from pressure measurements.

must be examined f o r compat ib i l i ty wi th these AIDJEX input c a p a b i l i t i e s .

The models

\

Chances of ob ta in ing a success fu l c o r r e l a t i o n appear b e t t e r as l a r g e r -

scale parameters are obtained.

maximum wind v e l o c i t i e s .

upper p lane tary boundary l a y e r are p r a c t i c a l l y nonexis tent .

t e n t upper boundary l a y e r measurements t o d a t e have been obtained from t h e

p i l o t balloon/double theodo l i t e t r ack ing method.

flow would g r e a t l y complicate such measurements, r equ i r ing averaging.

only mean value measurements available f o r t h e atmosphere w e r e Obtained from

towers.

f o r h a l f the yea r , and it can be expected t o b e th inner , perhaps a hundred

meters o r less at t i m e , a moderately t a l l tower is l i k e l y t o reveal mean

Measurements are needed in condi t ions of

Rel iab le means f o r ob ta in ing such d a t a from t h e

The most consis-

The presence of secondary

The

Since the s t a b l y s t r a t i f i e d boundary layer may p r e v a i l i n t h e Arctic

10 8

c h a r a c t e r i s t i c s of the p l ane ta ry boundary l a y e r in t h e Arctic.

i t would evaluate t h e a b i l i t y of three-meter temperature d a t a t o r ep resen t

general s t a b i l i t y c h a r a c t e r i s t i c s of t h e boundary l a y e r .

In p a r t i c u l a r ,

13.2 - The Geostrophic Flow Boundary Condition

The abso lu te determinat ion of ug w i l l b e done f a i r l y accu ra t e ly based

s o l e l y on t h e impressed geostrophic p re s su re g rad ien t on the surface.

main source of e r r o r is expected t o arise from thermal wind components

( l a r g e h o r i z o n t a l temperature g rad ien t s ) t u rn ing t h e geostrophic wind. The

lowest level of geostrophic balance must b e determined, as i t e s t a b l i s h e s

t h e direct:ion and s c a l i n g . When an inve r s ion i s p resen t , the wind a t t h e

inve r s ion i s a good estimate of t h e geostrophic value. I f t h e r e e x i s t l a r g e

temperature g rad ien t s , they can be observed frQm t h e synop t i c maps o r avail-

a b l e s t a t i o n s , and t h e thermal wind can be ca l cu la t ed . Some a t t e n t i o n t o

the o r ig in and i n t e r a c t i o n of i nve r s ions (a l a y e r of relative warming i n

the vertical temperature d i s t r i b u t i o n ) may b e needed t o accu ra t e ly r ep resen t

t he s t r a t i f i c a t i o n e f f e c t on secondary flows. However, as long as l a rge -

scale h o r i z o n t a l homogeneity e x i s t s , even wi th inve r s ions , t he h o r i z o n t a l

temperature gradient a t t h e s u r f a c e may r ep resen t t he thermal wind t o a f a i r

accuracy. When s t r a t i f i c a t i o n is taken i n t o account, a su r face vertical

temperature g rad ien t may b e required.

d i r e c t i o n a l o r i e n t a t i o n of s u r f a c e stress t o geostrophic flow, s i n c e t h e

secondary flow o r i e n t a t i o n (and ex i s t ence ) are dependent on t h e s t r a t i f i c a -

t i o n i n t h e lower region of t h e Ekman l a y e r .

The

This w i l l have an e f f e c t on the,

Tethered bal loon measurements may extend t h e Z’(z) and U(Z) measurements

i n t o the upper po r t ion of t he boundary l a y e r . This would allow b e t t e r del ine-

a t i o n of t h e s t r a t i f i c a t i o n near t h e i n f l e c t i o n po in t he igh t .

ments of t h e mean wind i n t h i s region can a l s o b e r e l a t e d to t h e modified

Ekman l a y e r t h e o r e t i k a l ’ p red ic t ions .

w i l l provide information on t h e h e i g h t of t h e geostrophic flow.

information is excep t iona l ly good, some measure of

i n fe r r ed .

geostrophic wind ca l cu la t ions .

Any measure-

Radiosonde d a t a from nearby l o c a t i o n s

I f t h i s

6, and hence K,cw be The Weather Bureau maps w i l l provide a s t a r t i n g p o i n t f o r t h e

This data can be r e f i n e d t o inc lude a d d i t i o n a l

109

pres su re d a t a i f i t is a v a i l a b l e and needed.

Arctic Basin i n d i c a t e s t h e t press.ure d a t a from buoys o r s t a t i o n s on a 400 km

scale can improve t h e s tandard Weather Bureau a n a l y s i s during active weather

per iods .

A comparison program f o r t h e

13.3 Ekman Layer Models

The o r i g i n a l Ekman s o l u t i o n y i e l d s an a n a l y t i c expression f o r t h e

. This can be separated i n t o components and su r face stress T~ = K V E

r e l a t e d t o p re s su re g rad ien t , z

T O Y = K VE (0) = Px(0) + Py(0) 2

( 1 3-1)

Although the stress magnitudes from t h e Ekman c a l c u l a t i o n are n o t unreason-

a b l e , t he 45" angle from the geostrophic d i r e c t i o n i s l a r g e and f ixed . When

the secondary flow and consequent mean modif icat ion are included, UE and VE

are replaced by UE + u and VE + 5. The r e s u l t i n g stress magnitude is changed

s l i g h t l y depending upon o r i e n t a t i o n E of t h e h e l i c a l secondary flow. The

s u r f a c e stress v e c t o r d i r e c t i o n is found t o l i e t y p i c a l l y between 25" and

40" depending upon s t r a t i f i c a t i o n , w i t h 28" as t h e value f o r n e u t r a l s t ra t i -

f i c a t i o n . This a n a l y t i c r e s u l t corresponds w e l l t o observat ions [e.g. see

Aagaard, 19701.

The i n p u t requirements f o r t h e modified Ekman s p i r a l models are

minimal. Some measure of t he

s u r f a c e roughness o r turbfilence production is needed, as a mean K o r 6 . When s t r a t i f i c a t i o n e f f e c t s are important, some measure of t h e temperature

gradient i n the l o w e r 100 m e t e r s is required. The v a l i d i t y of e x t r a p o l a t i n g

t h e temperature g rad ien t of t he - lower 3 meters t o g r e a t e r h e i g h t s would need

t o b e checked.

is common t o a l l models.

The geostrophic flow must b e determined.

The prablem of e s t a b l i s h i n g a good s t r a t i f i c a t i o n parameter

The secondary flow model p r e d i c t s t he stress at the surface as K V z ( 0 ) .

The t h e o r e t i c a l stress vec to r s are shown i n Figure 27. The s u r f a c e velocity

110

Fig. 27. Surface stress f o r t h e Ekman l a y e r modified by secondary flow a t va r ious angles E.

20 IO 0

-10 -20

1.03 .98 .95 .95 .97

gradient w i l l vary wi th s t r a t i f i c a t i o n s i n c e t h e s t a b i l i t y s o l u t i o n p r e d i c t s

varying r o l l angles with the geostrophic wipd. The value of K is a l s o

l i k e l y t o va ry wi th s t r a t i f i c a t i o n . This i s sometimes expressed i n empiri-

c a l l y der ived formulas involving t h e "surface Rossby number, I' Ro = Ug/fz,, and R i . The dependence on Ro is w e a k , which is f o r t u n a t e i n view of t h e

i n d e f i n i t e n a t u r e of t h i s parameter (experimental values o f z,, range over

several o rde r s of magnitude).

toward r e l a t i n g the stress v e c t o r t o the s t r a t i f i c a t i o n parameter. The mean

value of K can be found which matches Ekman l a y e r stress t o su r face measured

values. This value can be compared t o o t h e r experimental values .

Thus, t h e major e f f o r t should be d i r e c t e d

13.4 - Surface Layer Measurements.

Data obtained from AIDJEX w i l l al low c a l c u l a t i o n of stress a t the

s u r f a c e b17 t h e eddy c o r e l a t i o n method and t h e p r o f i l e method a t t h e manned

s t a t i o n s .

r e l a t i n g each t o commonly determined constants .

CD, t h e s e may inc lude R i , A and B , C, and C, or hlz , , K or 6, ad inf ini tum.

Correspondence between the var ious methods w i l l depend upon

I n a d d i t i o n t o T ~ , u*, o r

111

'fie t i m e dependent v a r i a t i o n of c h a r a c t e r i s t i c parameters must be examined

t g determine t h e lgca l (p.1-1 km) versus large-scale (lQ0-1000 km) e f f e c t s .

The b a s i c question t o b e answered h e r e concerns t h e v a l i d i t y of ex t r apo la t ing

information ac ross t h i s wide gap i n scales.

a v a i l a b l e over ice [Un te r s t e ine r and Badgley, 1965; Smith, 1972; and S e i f e r t

and Lagleben, 19721 i n d i c a t e t h a t i t w i l l be very d i f f i c u l t t o o b t a i n

memingful averages from t h e small-scale, short-term su r face l a y e r measure-

ments.

The spa r se amount of d a t a

The eddy c o r r e l a t i o n method y i e l d s an accu ra t e determination o f t h e

stress a t a p o i n t . It is o f t e n r e l a t e d t o the v e l o c i t y a t 10 meters he igh t

t o form t h e drag c o e f f i c i e n t . A 10% e r r o r i n U,, produces a 17% e r r o r i n

S imi l a r ly , a geostrophic drag c o e f f i c i e n t C = ro/pUg2 can be

g

CD = T 0 / P U l O D ca lcu la t ed i f U is a v a i l a b l e . This can be compared d i r e c t l y t o the modified

Ekman l a y e r p red ic t ion of drag c o e f f i c i e n t .

The p r o f i l e method can be appl ied t o t h e mean wind measured i n t h e

lower po r t ion of t h e s u r f a c e l aye r , t o produce yet. apother va lue of T ~ .

This ca l cu la t ion w i l l a l s o provide values of z o .

CD

The p o s s i b i l i t y of r e l a t i n g

t o zo can be i n v e s t i g a t e d f o r n e u t r a l condi t ions [see Let tau, 19591. 9

I n each o f t h e s e cases, t h e magnitude is determined f o r t h e stress.

The d i r e c t i o n must be obtained by no t ing t h e mean wind d i r e c t i o n wi th e i t h e r

t he b a s i c instrument o r a nearby vane. The stress is assumed t o be in t h e

d i r e c t i o n of t he mean wind i n t h e p a r a l l e l flow s u r f a c e l a y e r regime.

A s t r a t i f i c a t i o n parameter can b e determined from hea t f l u x measure- 3 ments, using Ri = z / L = k g T 'w ' z/!Zk,.

vec to r and s t r a t i f i c a t i o n can then b e sought. Such measurements can a l s o

b e used t o determine whether s impler d a t a (temperature, 0-2 m) can adequately

r ep resen t s t r a t i f i c a t i o n f o r large-scale modeling purposes.

Cor re l a t ions between t h e stress

The geostrophic drag c o e f f i c i e n t provides a gross m e a n va lue r e l a t i n g

t h e s u r f a c e stress t o the geostrophic flow magnitude.

be measured i n t h e s u r f a c e l a y e r t o determine t h e stress d i r e c t i o n .

p o s s i b l e that a i s a constant o r a func t ion of U v a r i a b i l i t y of C

I n add i t ion , a must

It i s

f o r uniform terrain. The 9

and a wi th s t r a t i f i c a t i o n might be found wi th s u f f i c i e n t D

112

data .

t h i s eva lua t ion .

Accurate determination of the geostrophic flow vec to r is needed f o r

The s u r f a c e layer method is purely an empi r i ca l determinat ion of any

c o r r e l a t i o n between measured stress and measured geostrophic flow, m d

possibly su r face temperature g rad ien t s . Its usefulness i n modeling w i l l

depend

13.5

s o l e l y on the success of t h i s correspondence.

- Similarity Models

These models hypothesize t h a t t h e r e e x i s t constants which relate

s u r f a c e l a y e r flow and stress t o t h e geostrophic flow.

t h e o r i e s depends on the d u r a b i l i t y of t h e cons t an t s ,

due t o s t r a t i f i c a t i o n complicates t h e sea rch f o r t h e s e va lues , Accurate

determination of t h e constants relies on measurements t o determine z o , a, u*

The value of t hese

The added v a r i a b i l i t y

and Ug.

To the p re sen t , experimental eva lua t ion of the s i m i l a r i t y constants

has been p a r t i a l l y success fu l . It i s evident t h a t s t r a t i f i c a t i o n e f f e c t s

are s i g n i f i c a n t , and the re i s a p a r t i c u l a r l y l a r g e v a r i a t i o n i n t h e s t a b l y

s t r a t i f i e d regime. Hence the re is a need f o r a n evaluat ion of t hese s i m i l a r i t y

parameters i n t h e Arctic. The p r i n c i p a l advantage of t h i s method over s u r f a c e

l a y e r c o r r e l a t i o n s arises from t h e considerat ion of t h e upper boundary l a y e r

flow dynamics, implying a l a r g e r - s c a l e domain f o r t h e s i m i l a r i t y constants .

13.6 Summary - For t h e purpose of determining the stress vector , each model u l t i m a t e l y

The d i f f e r e n c e s are relies on one o r more empi r i ca l ly determined constants .

as follows;:

(1) Surface l a y e r measurements determine the magnitude of the air

stress fo rce a t a loca l s t a t i o n .

by l o c a l topography.

improved bly averaging many measurements over d ive r se regions.

a l l y n o t aivailable i n such abundance.

o f large-s8 cale areas due to t he unusually uniform terrain.

The measurements are l i k e l y t o b e inf luenced

The a p p l i c a b i l i t y t o l a rge - sca l e domains would b e

Data is gener-

Arctic d a t a may b e more r e p r e s e n t a t i v e

113

The p o i n t measurenents can be compared w i t h the geos t roph ic flow t o

form aerodynamic drag c o e f f i c i e n t s . H.owever, t h e wide d i s p a r i t y i n scales

sugges t s a r e l i a b l e c o r r e l a t i o n i s un l ike ly .

v e l o c i t i e s may be more success fu l .

l o c a l flow t o the synopt ic .

scatter i n values of z o , CD and u,*[G.

A carrelatipn wi th loca l flow

The problem then arises t o relate t h e

P resen t ly a v a i l a b l e d a t a indicate . very l a r g e

(2) S i m i l a r i t y models a l s o relate the stress t o geostrophic flow via

constants determined i n the s u r f a c e l a y e r . The theory assumes t h a t l a rge -

scale s i m i l a r i t y i n flow condi t ions exists f o r a two-layer p l ane ta ry model,

and t h e s i m i l a r i t y v a r i a b l e s should r ep resen t correspondingly large-scale

mean values . Since t h e s e v a r i a b l e s may change w i t h S t r a t i f i c a t i o n , some

measure o € ’ t h e temperature s t r u c t u r e is required.

suggest t h a t t he v a r i a t i o n i n the s t a b l y s t r a t i f i e d regime is q u i t e l a r g e ,

P resen t ly a v a i l a b l e d a t a

(3 ) Ekman l a y e r models gene ra l i ze t h e boundary l a y e r dynamics for

l a r g e regions and provide t h e s imples t l a r g e scale r e l a t i o n s h i p between

su r face stress v e c t o r and geostrophic flow.

r e l a t i o n between geostrophic flow and a i r stress involving a s i n g l e parameter

r ep resen t ing the mean roughness. With secondary c i r c u l a t i o n , they admit

v a r i a t i o n i n stress d i r e c t i o n and magnitude with s t r a t i f i c a t i o n e f f e c t s .

They provide information on t h e boundary l a y e r flow s t r u c t u r e .

A summary t a b l e of AIDJEX models and q u a n t i t i e s t o be evaluated,

They imply a f a i r l y constant

together w i th flow cha r t s dep ic t ing data processing t o eva lua te the models,

is given i n Appendix C.

I t has been t h e i n t e n t i o n i n t h i s paper t o p r e s e n t a l l popular methods

of p l ane ta ry boundary l a y e r a n a l y s i s as o b j e c t i v e l y as poss ib l e .

assumptions i n e v i t a b l e i n each theory, coupled wi th t h e wide d i spe r s ion in

a l l d a t a a n a l y s i s of boundary l a y e r parameters has con t r ibu ted t o no clear

preference o r e l imina t ion of any method.

amount of available d a t a is a t r u e i n d i c a t o r of t h e v a r i a b i l i t y in s u r f a c e

l a y e r parameters, then t h e r e is a clear need t o g e t away from the v a g a r i e s

of measurements a t t h i s p a r t j x u l a r scale.

bombardment on a f l a t p l a t e appears as skin friction on t h e scale of the

continuum, we a n t i c i p a t e that the p res su re drag on the many forms a s s o c i a t e d

The numerous

If the scatter i n t h e moderate

J u s t as the statistical molecular - I

114

wi th r idges , l eads , and hummocks w i l l emerge as a f r i c t i o n a l su r f ace stress

on the 100 km scale.

expected t o b e on t h e largest scales, as represented in the s i m i l a r i t y

theory constants o r the modified Ekman l a y e r roughness c h a r a c t e r i s t i c .

I f success fu l c o r r e l a t i o n s do emerge, they are

The s t r a t i f i c a t i o n condi t ion, with i t s atteQdant e f f e c t on t h e

turbulenoe cha rac t e r , i s espected t o be an important parameter. It has

been found t o b e s i g n i f i c a n t i n s i m i l a r i t y theory, and i t plays an important,

a l b e i t secondary, r o l e i n modified Ekman l a y e r theory.

F i n a l l y , mesoscale p l ane ta ry boundary l a y e r a n a l y s i s i s i n need of

new i d e a s f o r parameter izat ion. This is p a r t i c u l a r l y t r u e i n t h e case of

s t a b l e s t r a t i f i c a t i o n . These ideas are f r equen t ly provoked by in spec t ing

b a s i c da t a , as t h e A I D J E X 1972 kytoon d a t a suggested a momentum i n t e g r a l

approach (Appendix B) . The accumulation and processing of t h e appropriate

d a t a i s an important i ng red ien t f o r r ap id progress i n p l ane ta ry boundary

l a y e r modeling.

Since t h e p r i n c i p a l d r i v i n g fo rce f o r t h e large-scale ice motion is

t h e wind stress, an adequate r ep resen ta t ion of the wind f i e l d i s important

t o AIDJEX. A t t h e same t i m e , under t h e assumption t h a t t h e ice moves under

the i n f luence of t h e l a rge - sca l e wind stress f i e l d , AIDJEX provides a unique

opportuni ty f o r t he i n v e s t i g a t i o n o f atmospheric boundary l a y e r dynamics.

The vast, r i g i d , r e l a t i v e l y uniform s u r f a c e provides c r e d i b i l i t y t o l a rge -

scale parameter izat ion assumptions. The slowly changing synop t i c condi t ions,

s t r a t i f i c a t i o n , and d i u r n a l cyc le are compatible with s teady-s ta te s o l u t i o n s .

The Corio:Lis fo rce i s maximized f o r t h e Ekman l a y e r t h e o r i e s .

retical advantages may be s u f f i c i e n t t o overcome t h e l i a b i l i t y of t h e inherent

d i f f i c u l t i e s involved i n d a t a a c q u i s i t i o n i n t h i s region.

These theo-

115

APPENDIX A

THE MOMENTUM DEFECT METHOD

An extension t o t h e von K&” momentum i n t e g r a l method (e.g., see

Schl icht ing , 1959) can be made f o r t he three-dimensional p l ane ta ry boundary

l a y e r . This simply relates t h e s u r f a c e stress t o t h e change i n momentum of

t h e boundary flow i n a downstream d i r e c t i o n .

The equat ions can be w r i t t e n i n a f a i r l y general form f o r s t eady state

and 8/8y E 0. The las t condition can

cep tua l change. The equat ions i n t h e

and coordinates a l igned with t h e geostrophic flow, s o t h a t V = (G, 0, 0) 9 be relaxed subsequently, with no COLI-

boundary l a y e r are then :

UV, + WZ - fu - P,/p = 8/32

w, + ww, - g - Pz/p = 8/82 (P) ux + wz = 0

The freestream flow i s geostrophic , s o t h a t G = - l / p f . P y .

the constant p re s su re g rad ien t i s impressed on t h e boundary l a y e r , s u b s t i t u t e

the geostrophic flow f o r t he p re s su re g rad ien t term, and i n t e g r a t e from t h e

su r face out t o z = h :

Assuming that

Using the con t inu i ty r e l a t i o n t o s u b s t i t u t e W = JU, dz and i n t e g r a t i n g

by p a r t s y i e l d s :

117

A t z = h , U = G, ‘rX = .cy = 0.

(A-4)

X For compact n o t a t i o n , t h e two-dimensional vec to r s (U,V) and (‘r0 y ~ o Y )

Y ‘r0 = .cox + k ~ , . can b e w r i t t e n i n complex no ta t ion :

following i n t e g r a l s are then,.def ined:

q = U + i v y The

Q) co

= J ( l - p z and 6, = J ; (1 - 5 ) d z (A-5) 0 - 0 6 l

Then (A-4) may be w r i t t e n

‘r0 = G2 6, - i f G 6 , . (A-6) X

When t h e v e l o c i t y p r o f i l e s i n the boundary l a y e r are eva lua ted at two

s t a t i o n s , a l igned wi th t h e i soba r s , t h e q u a n t i t i e s 6,, 6 When t h e flow i s h o r i z o n t a l l y homogeneous, t h e gradient of t h e momentum

thickness , 6, , w i l l vanish and t h e stress i s simply r e l a t e d t o the displace-

ment thickness , 6,, at a s t a t i o n .

i n t e g r a l of t he i n e r t i a l terms must be evaluated at a s t a t i o n .

and G y i e l d ‘r,. ,X

X I f t h e flow i s nons ta t iona ry , t hen t h e

The d i f f i c u l t y i n ob ta in ing the v e l o c i t y throughout t h e boundary l a y e r

m a k e s t h i s method f o r ob ta in ing m e a n stress imprac t i ca l f o r t h e atmospheric

boundary l a y e r . However, i t might b e u t i l i z e d in t h e oceanic boundary l a y e r .

I f a d i r e c t i o n p a r a l l e l t o t he geostrophic flow cannot b e s e l e c t e d ,

then f o r lateral v a r i a t i o n i n (A-1)

118

X uux + uu + wiJ2 - fv - Px/P = (T ) z

UVX + Y V + wz - fv - Py/P ('c Y IZ

uwx + vw + wwz - Pz/P = (T IZ

Y

= Y

Z

Y

? , + V + w z = Q Y

In a s imilar fashion, t h i s w i l l produce

(.A-7)

o r , i n complex n o t a t i o q ,

Hence, the stress may be w r i t t e n

T o = us Q 8 , + Vg Q 0, - i f Q 0 , X Y

(A-10)

00

where U,, Q 0, = Uq' dz 0

and Q0, = dz.

Equation (A-10) expresses t h e mean stress between any two su r face s t a t i o n s

i n terms of t h e d i f f e rence i n momentum between the s t a t i o n s as determined by

the v e l o c i t y p r o f i l e s .

0

Equations (A-4) can be i n t e g r a t e d f o r l i m i t i n g cases of atmospheric

boundary l a y e r flow. One such case is discussed in Appendix B.

119

It i s p o s s i b l e t h a t more than me model can p r e d i c t stress w i t h an

accuracy s u f f i c i e n t f o r t h e purposes of AIDJEX. I f t h e d a t a from t h e AIDJEX

s t u d i e s show t h i s p o s s i b i l i t y t o b e a r e a l i t y , t h e b e s t of t h e s e models

would have t o be s e l e c t e d .

120

APPENDIX B

A S I M P L E MOMENTUM INTEGRAL MODEL

The omnipresence of shear i n s t a b i l i t i e s r e s u l t i n g from t h e v e l o c i t y

change i n magnitude and/or d i r e c t i o n inhe ren t i n t h e boundary l a y e r flow

produces tu rbu len t d i f f u s i o n by small-scale eddies and advection by secondary

flows.

mixed l a y e r s might work,

geophysical a p p l i c a t i o n s . The following simple model i s a n a t u r a l develop-

ment suggested by t h e r e l a t i v e l y uniform boundary l a y e r d a t a obtained by t h e

197 2 AIDJEX t e the red bal loon (kytoon) experiment,

This suggests t h a t some gross momentum i n t e g r a l model f o r w e l l -

Several such models have been proposed f o r d i f f e r e n t

The assumed boundary l a y e r c o n s i s t s of a thmoughly mixed l a y e r with

constant v e l o c i t y , stress, and p o t e n t i a l temperature. Change i n v e l o c i t y

(magnitude and d i r e c t i o n ) is assumed t o t ake p l a c e a t t h e top of t h e boundary

l a y e r w i t h i n an inve r s ion . The inve r s ion is necessary t o suppress i n s t a b i l i t i e s

i n t h i s s t r o n g shear ing region by providing Richardson numbers g r e a t e r than

t h e c r i t i c a l . The boundary l a y e r then has an e f f e c t i v e l i d , and t h e momentum

equations can be i n t e g r a t e d t o produce a r e l a t i o n between t h i s he igh t and

t h e stress, geostrophic v e l o c i t y , and t h e boundary l a y e r v e l o c i t y .

The constant stress i n t h e boundary l a y e r can be represented i n the

drag c o e f f i c i e n t

where t h e boundary l a y e r v e l o c i t y is u' = (u,v> (geostrophic) i s U = (U,O),

and the f r ees t r eam -f

121

The s t eady- s t a t e momentum equations are then

I n t e g r a t i n g t o t h e top of t h e boundary l a y e r , h , where u = U, v = 0 , CD = 0:

f v h = CD 03-31

i- f ( u - u>h = - CD v lul

If CD is el iminated from (B-3)

u 2 + v 2 = u u = GI 03-41

This r e l a t i o n s h i p i s represented i n t h e c i rc le ( a s pointed ou t by R. Thompson

i n an unpublished paper: " S t r a t i f i e d Elanan boundary l a y e r models--bottom and

top," Woods Hole Oceanographic I n s t i t u t e , 1972):

122

From similar r i g h t t r i a n g l e s ,

- 5 U A V AU

and

AU2

From (B-3 ) and ( B - 5 ) ,

cD f h AU/lt12

From rough kYtoon d a t a taken during t h e 1972 AIDJEX p i l o t s tudy (Table B 1)

we can c a l c u l a t e values of CD using ( B - 6 ) .

constant Iu 1 and well-defined inve r s ions .

i n d i c a t e s u r p r i s i n g l y constant values of CD % 0.002 f o r near n e u t r a l condi-

t i o n s .

method, :in which CD = 0.0029.

runs, t h e l a y e r remained a d i a b a t i c and t h e h e a t apparent ly went i n t o

The p r o f i l e s possessed nea r ly

These prel iminary c a l c u l a t i o n s -f

This corresponds t o r e s u l t s by Goddard on 19 March using t h e p r o f i l e

Although hea t ing took p l ace during several

changing t h e inve r s ion he igh t and/or uniformly heat ing t h e l a y e r .

example of a d a t a run i s shown i n Figure B1.

An

-f When h i s the unknown, it can b e ca l cu la t ed from Coy U, and lul using

( B - 6 ) . If only U and CD are known, we may be ab le t o c l o s e t h e s e t by adding

a c r i t e r i o n r e l a t i n g h t o a c r i t i ca l Richardson number.

The c r i t i c a l R i t o suppress i n s t a b i l i t i e s i s found t h e o r e t i c a l l y t o

be Ri, = 1 / 4 [Brown, 1972al.

suggested i n Figure 24.

Consider an equi l ibr ium model similar t o t h a t The potentiaZ R i number ( i . e . , t h e bulk R i ) which

123

TABLE B . l

CALCULATED CD FROM PRELIMINARY KY'K)ON DATA, 1972 (unpublished)

Invers ion u lZ1 AU h Run Time height(m) m / s m/s m/s TU S t r a t i f i c a t i o n CD X l o 3

4 March

14 March 1.1 1329-1340

4 .1 1330-1345 4.2 1348-1355

15 March 6 1 1030-1040 6.2 1048-1054

19 March 7 . 1 1130-1140 7.2 1153-1204 8.1 1301-1319

20 March 11.1 0900-0930 11.2 1000-1025 12 .1 1600-1628

21 March 13.1 0840-0900 13.2 0900-0911 15.1 1535-1600 15.2 1600-1625

26 March 16.1 1300-1320

1 7 . 1 2100-2110

70

90 50

110 75

50 50 55

20 10 55

70 60

115 130

75

10

4.4 3.6 2.9 25 a d i a b a t i c 2.0

3.3 3.0 1.9 60 a d i a b a t i c 3.0 3.5 2.8 2.1 21 adiab.-s table 1.8

5.2 3.8 3.8 26 adiab.-unstable 4.0 5.1 3.8 3.92 20 a d i a b a t i c 2.9

4.0 3.0 2.6 25 a d i a b a t i c 2 .1 4.7 3.5 3.14 20 a d i a b a t i c 1.9 3 , 2 2.9 1.35 27 a d i a b a t i c 2.6

5.4 4.3 3.27 5 s t a b l e , warming 0.8 5.0 4.0 3.0 4 s t a b l e , warming 0.5 5.5 4.0 3.8 15 a d i a b a t i c 1 .9

5.0 4.0 3.0 23 a d i a b a t i c 1 .9 5.0 4.0 3.0 20 a d i a b a t i c 1.4-2.0 4.6 4.0 2.27 50 a d i a b a t i c 2.4 5.9 5.0 3.13 42 ad iab . - s l igh t ly 2.4

uns tab le

4.25 3.7 2.9 27 adt;iat;f$$ghtly 2.2

3.25 1.4 2.93 3 adiab.-s table 1.8

would be formed i n a boundary l a y e r which i s d i f -

f u s i v e only is

Bg (Tg - Ts> h R i = Q 1 / 4 = Ri, (B-7)

AU2

124

TEMPERATURE WIND SPEED O C M/S

Fig. B 1 . Kytoon d a t a , 21 March 197 2.

I- Run 13.1 (0840-0900)

*--- Run 13.2 (0900-0911)

The set (B-2) and(B-7) now expresses the unknowns U , V and h i n terms of

t h e boundary condi t ions U, Coy and A!2' = Tg - T s .

From (B-2), (B-41, and (B-51,

This cubic, together w i th

(B-9)

(B-10)

imply h , u, v from given U, CD, and AT.

Unfortunately, AT is n o t r e a d i l y a v a i l a b l e , as d a t a on Th are lacking.

I n two s p e c i a l cases, assumptions can be made t o provide 0.

shown i n Figure B2.

These are

In case a, t h e s t a b l y s t r a t i f i e d l a y e r experiences

125

Fig, B 2 . Two poss ib l e temperature p r o f i l e developments wi th an invers ion ,

i s e n t r o p i c mixing t o h , providing an ad iaba t i c l a y e r a t Ts and

AT = Th .. ITs = TB - To = I'hI2,

where I' is the l apse r a t e before mixing, The c r i t i c a l R i from t h e l a y e r

y i e l d s g I' h2

Ri, E Ts- 2 AV2'

o r

(B-11)

I n case b y t he l aye r experiences hea t add i t ion with cons tan t mixing,

providing an a d i a b a t i c l a y e r t o an invers ion he ight which i s determined

by t h e amount of hea t ing and t h e i n i t i a l l a p s e r a t e , With AT = r h , and

( B - 1 2 )

I n e i t h e r case, one can so lve f o r

H o - N ll H o

h - ( 1 + fHoly'

(B-13)

126

(E-13)

I n s e r t i n g t y p i c a l A r c t i c va lues i n (B-11) and (B-12), l e t Ri, = 1/4,

T, = 240°, g = 10 m/sec, and r = 0.006'/m:

H, = 32 f o r case a ; H , , = 22 f o r case b.

These r e l a t i o n s seem t o provide c o r r e c t o rde r o f magnitude values €or

t h e boundary l a y e r parameters.

be der ived from t h i s model w i l l depend upon a d d i t i o n a l d a t a t o eva lua te

t h e p r a c t i c a l i t y of t h e assumptions i n the model,

The ques t ion of whether more accuracy can

12 7

APPENDIX C

DATA PROCESSING TO EVALUATE MODELS

The AIDJEX ice dynamics model r equ i r e s t h a t t h e mean stress v e c t o r a t

There are five b a s i c models from which t o 100 km2 g r i d p o i n t s be predicted.

choose. These model concepts are l i s t e d i n Table C . l . The choice can be

reduced tlo simply the one which is most c o n s i s t e n t w i t h a v a i l a b l e observat ions.

The su r face l a y e r da t a and t e t h e r e d b a l l o o n d a t a from t h e 1972 AIDJEX experi-

ment should be s u f f i c i e n t t o provide some checks on each model. Surface

l aye r stress eva lua t ions w i l l be made by the p r o f i l e method ( sec t ion 5.6.2) .

The geostrophic flow i s determined from s u r f a c e p re s su re measurements a t a l l

a v a i l a b l e s t a t i o n s ( inc lud ing AIDJEX d a t a buoys) i n the Arctic Basin and i t s '

periphery (see sec. 3 . 3 ) . The reduct ion of t hese data t o the s i g n i f i c a n t

parameters which may be used t o eva lua te t h e models i s summarized i n several

flow c h a r t s i n Figures C 1 , C2, and C3.

The i n i t i a l processislg of t h e d a t a i n t o t h e c h a r a c t e r i s t i c parameters

i s shown i n Figure C 1 . Superimposed on a l l consis tency checks i s an implied

tolerance i n t h e d a t a scatter; e.g., if ca lcu la t ed va lues of zo are i n d e f i n i t e ,

the models which employ t h i s value can s t i l l b e checked, with t h e indetermi-

nacy kep t i n mind. It may be t h a t t he stress i s r e l a t i v e l y i n s e n s i t i v e t o

s c a t t e r i n some parameters. Surface l a y e r Richardson numbers, R i , , as w e l l

as l o c a l IZi f o r t h e Ekman l a y e r can be ca l cu la t ed t o eva lua te t h e degree of

s t r a t i f i c a t i o n .

Figures C2 and C 3 show t h e appropr i a t e parameters f o r each model and

the consis tency checking procedure. Previous experience suggests t h a t t h e r e

w i l l be no d e f i n i t i v e agreement w i t h any p a r t i c u l a r model. This may be due

t o assumptians made i n t h e method o f c a l c u l a t i n g t h e parameters o r t h e geo-

s t r o p h i c approximation. I n the case of no agreement hetween d a t a parameters

and models, more s o p h i s t i c a t e d d a t a may be needed.

sonic anemometer determination of u*, averaging of many measurements of u*, and accura t e soundings of V ( z > and T ( z ) .

This would inc lude a

129

TABLE C . l

BOUNDARY LAYER MODELS

lesired Resul ts

&. Stress vec tor n e u t r a l l a y e r

3 . Stress vector s t r a t i f i e d 1 aye r

- C. Stress vectol

and f luxes ol h e a t , momen- tum, vapor, etc.

[ode1

r l Ekman

Illa Hod. Ekman w/

r2 Aerodynamic

secondary flow

113 K theor ies

T 4 S imi l a r i t y

T 5 Inner /outer

S 1 Mod. Ekman w/ secondary flow

S2 Aerodynamic

S 3 K theory

S 4 Simi la r i ty

~5 Inner /outer

F1 Mod. Ekman w/ secondary flow

F2 Diffusion

Measurements Fixed jeveral /day

G .

G

G, VT

G, VT

G, VT

G, VT

G , VT

G , V T ; o r Vq, VH, e t c

etc. across boundary layer

A q , AH, AV,

Coments

a = 45" too l a r g e , un s t ab l e

a = 27"

Co = ( u , / G 1 2 , da ta s c a t t e r e d

Data incomplete, s c a t t e r e d

Scatter i n k, z o ; a, A

C , and C , constant ' Data lacking.

Near n e u t r a l

Data lacking

the0 ry

Data lacking

Data s c a t t e r e d

Data lacking

130

AIDJEX 1972 F-

4

Buoys Weather Bureau NCAR balloons \ Goddard, cup anemometers, temperature probes on mast maps, s t a t i o n da ta

i

t \ P ( X ¶ Y ,t) f

0-4m 0-4m 1 I I

Geos trophi c Flow -

3 I

56- __LI

Thermal b J/ Wind 1

I

no

J I

I

\ s t l a b i l i t y and check s t r a t i f i e d models J

Fi.g. C1. Flow cha r t f o r processing d a t a i n t o c h a r a c t e r i s t i c parameters corresponding t o theory.

131

Neutral Models

Derived da ta , near neut ra l condi t ion u*IG, a, 2 0 , f, k, G

I

No No

I (Check f o r C, and C, t o f i t

N Q

I No

Fig. C2. A flow c h a r t f o r eva lua t ion of cha rac t e r is t i c constants f o r n e u t r a l models.

~,~ (Use s i m i l a r i t y theor T4

Try co r re l a t ions w i t h R i o r l a r g e r e r r o r toleran

132

Fig.

Thermally S t r a t i f i e d Models

/ & \ k m o i r i c a l data i s i n s u f f i c i e n t . Go back

I

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1 41

[The following paper is reproduced here a t the request of t h e authors. It was d e l i v e r e d a t the AGU symposium on a i r -sea i n t e r a c t i o n in December 1972. I

SOME CRITICAL REMARKS ON TURBULENT DIFFUSION THEORY ON ICE DYNAMICS

by

S. 1 . Pai I n s t i t u t e f o r F l u i d Dynamics and Appl ied Mathematics U n i v e r s i t y o f Maryland, Col lege Park, Maryland 20742

and

Huon L i Polar Oceanography D i v i s i o n

Naval Oceanographic O f f i c e , Washington, D.C. 20390

I n t h i s paper, we discuss the a p p l i c a t i o n o f tu rbu len t d i f f u s i o n theory t o the random motion o f the pack i c e f l o e s i n c l u d i n g sea-a i r i n t e r a c t i o n i n the A r c t i c Ocean and i t s surrounding areas. F i r s t the essent ia l features o f the well-known theory o f t u r b u l e n t d i f f u s i o n r e l a t e d t o wind tunnel turbulence and atmospheric turbu- lence are discussed which inc lude ( i ) Euler ian and Lagrangian descr ip t ions o f t u r b u l e n t d i f f u s i o n and t h e i r proper s t a t i s t i c a l q u a n t i t i e s , ( i i ) T a y l o r ' s , Richardson's and Goldste in 's theor ies o f t u r b u l e n t d i f f u s i o n , ( i i i ) One p a r t i c l e d i f f u s i o n and two- p a r t i c l e d i f f u s i o n and ( i v ) two- and three-dimensional f lows o f the t u r b u l e n t f low. The s i m i l a r i t i e s and d i f fe rences between the wind tunnel turbulence, atmospheric turbulence and the t u r b u l e n t motion o f the i c e dynamics are discussed. F i n a l l y , spec ia l f a c t o r s t o the random motion o f i c e f l o e s are considered, which inc lude ( i ) meteorological cond i t ions , ( i i ) boundary e f f e c t s and exchange processes, ( i i i ) the source terms due t o me1 t i n g and f reez ing o f the ice, ( i v ) the i n t e r a c t i o n and c o l l i s i o n o f i c e f l o e s and (v) the e f f e c t s due t o the d i f f e r e n c e i n v e l o c i t y of the i c e and i t s surrounding medium.

1 . I n t r o d u c t i o n - For the lbasic research o f the dynamics o f the pack i c e i n the A r c t i c Ocean anld i t s surrounding areas, one o f t h e main purposes i s t o coord inate t h e o r e t i c a l and f i e l d inves t iga t ions so t h a t the fo recas t ing o f i c e motion

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o r the general c i r c u 1 be improved. One o f

a t i o n i n the A r c t i c Ocean and i t s surrounding areas may the major f i e l d i nves t i ga t i ons i s the measurements o f

the movement o f the i c e f l o e s , p a r t i c u l a r l y by remote sensing. l From the measured r e s u l t s of the motion o f the i c e f l oes , we would l i k e t o p r e d i c t the average motion o f the i c e and t o exp la in the observed e f f e c t s on the motion o f i ce . Since the motion o f the i c e f l oes i s usua l l y random i n nature, we should use s t a t i s t i c a l methods t o analyze the movement o f i c e f loes . Hence some t h e o r e t i c a l analyses based on the s t a t i s t i c a l method

i c e w i t h the he lp o f the f i e l d measurements o f the motion o f i c e f l oes . A t a f i r s t glance, the c l a s s i c a l theory o f t u rbu len t d i f f u s i o n based on the random walk o f p a r t i c l e s 2 - 4 may be used t o study the movement o f i c e f l o e s i n the A r c t i c Ocean and i t s surrounding areas. However, when we look a l i t t l e deeper i n t o the present s ta tus o f t u rbu len t d i f f u s i o n theory, 2-4 we f i n d t h a t a major p o r t i o n o f turbulence s tud ies i s concerned w i t h labora tory turbulence, p a r t i c u l a r l y wind tunnel turbulence w h i l e some stud ies dur ing the l a s t ten years a re concerned w i t h atmospheric turbu- lence, bu t very l i t t l e has been done w i t h spec ia l emphasis i n the turbulence o f pack i ce . There are some s i m i l a r i t i e s between the turbulence i n i c e f i e l d w i t h those o f wind tunnel turbulence and/or atmospheric turbulence. Hence some o f the r e s u l t s o f wind tunnel turbulence and atmospheric turbulence may be used t o exp la in the tu rbu len t motion o f the ice f loes . On the o the r hand, there a re some essent ia l d i f f e rences between these th ree types o f t u rbu len t f lows. Some fac to rs wich are very important t o the tu rbu len t motion o f i c e f l o e s have not been s tud ied because they are no t so important i n wind tunnel turbulence o r atmospheric turbulence. I n t h i s sho r t pdper, we are going t o l i s t these po in ts f o r f u r t h e r research i n the tu rbu len t motion o f i c e f l o e s .

re needed i n o rder t o have a b e t t e r understanding o f the dynamics o f pack

The modern concepts o f s t a t i s t i c a l theory o f t ~ r b u l e n c e ~ - ~ have been developed by f l u i d dynamicists r e l a t e d t o labora tory turbulence, p a r t i c u l a r l y the wind tunnel turbulence. I n these wind tunnel turbulence cases, the tu rbu len t i n t e n s i t y i s genera l l y small compared t o the mean wind v e l o c i t y and the mean v e l o c i t y i s usua l l y kept constant. Under these cond i t ions , the tu rbu len t f l u c t u a t i o n s can e a s i l y be separated from the mean v e l o c i t y and t h e i r s t a t i s t i c a l c h a r a c t e r i s t i c s can be de f ined w i thou t great d i f f i - c u l t y . Hence both the experimental and t h e o r e t i c a l i nves t i ga t i ons o f wind tunnel turbulence are most ly based on the c o n d i t i o n tha t a mean v e l o c i t y i s w e l l def ined. For instance, the tu rbu len t f l u c t u a t i o n s are determined by a ho t -w i re anemometer w i t h proper e l e c t r o n i c equipment so t h a t the mean v e l o c i t y i s e l im ina ted and we may ob ta in the s t a t i s t i c a l p roper t i es o f the tu rbu len t f l u c t u a t i o n s immediately. The t h e o r e t i c a l ana lys is has been mostly developed f o r the case w i t h constant mean v e l o c i t y .

The atmospheric turbulence d i f f e r s from wind tunnel turbulence i n t h a t the mean wind v e l o c i t y i s seldom w e l l def ined and the separat ion o f the turbu- l e n t f l u c t u a t i o n s from the mean v e l o c i t y becomes q u i t e d i f f i c u l t . Hence

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we have t.o choose a sca le before we could de f i ne the mean v e l o c i t y . For the turbulence i n an i c e f i e l d , the s i t u a t i o n i s s i m i l a r t o the atmospheric turbulence i n t h a t the mean f l ow v e l o c i t y i s no t w e l l def ined. Hence we have t o choose a scale before we could de f i ne the mean v e l o c i t y f o r the general nlotion o f i c e f l o e s .

There are some bas ic d i f fe rences i n the s t a t i s t i c a l features o f these three types o f turbulence. I n the wind tunnel , the s izes o f eddies are l i m i t e d . Usual ly the s izes of eddies i n a wind tunnel s t a r t from the order o f magnitude of one cent imeter and cont inuously down t o a f r a c t i o n o f a m i l l i m e t e r . For atmospheric turbulence, the range o f eddy s izes i s much l a r g e r than those o f wind tunnel turbulence and i t includes those s izes o f eddies i n wind tunnel as w e l l as l a rge eddies o f several c e n t i - meters o r meters, o r even k i lometers . Hence the spectrum o f atmospheric turbu1enc:e covers a very la rge range o f eddies. One must have a la rge amount of data on the f l u c t u a t i n g wind v e l o c i t i e s t o be able t o compute meaningful s t a t i s t i c a l q u a n t i t i e s o f atmospheric turbulence. Thus we expect scme d e f i n i t e d i f fe rences i n the phys ica l nature o f atmospheric turbulence from those i n a wind tunnel .

For the turbulence i n an i c e f i e l d , we would expect q u i t e d i f f e r e n t c h a r a c t e r i s t i c s f rom both wind tunnel turbulence and atmospheric tu rbu- lence. The s izes o f i c e f l oes i n the A r c t i c Ocean and i t s surrounding areas are i n the range o f a few f e e t i n diameter t o a thousand f e e t i n diameter o r even la rge r . Since the eddy s izes should be l a r g e r than the p a r t i c l e s izes o f i c e f l oes , we would expect t h a t the range o f eddy s izes o f wind tunnel turbulence i s no t important i n i c e dynamics and the l a rge s i z e eddies would p lay an important r o l e i n the tu rbu len t f l ow o f i c e f l oes . Hence the s t a t i s t i c a l features o f t u rbu len t f l ow o f pack i c e would be d i f f e r e n t from both the wind tunnel turbulence and the atmospheric turbulence.

There are many o ther fac to rs which are not important i n the study o f wind tunnel turbulence and/or atmospheric turbulence bu t which p lay important ro les i n the i c e dynamics turbulence, p a r t i c u l a r l y those fac to rs such as ( i ) the exchange processes on the top and the bottom surfaces o f the i c e f i e l d , ( i i ) source terms i n i c e dynamics due t o f reez ing and me l t i ng o f ice, (iii) the i n t e r a c t i o n and c o l l i s i o n o f i c e f l o e s and ( i v ) the d i f f e rence o f v e l o c i t i e s of i c e and the surrounding water and many o the r fac to rs . We s h a l l b r i e f l y discuss some o f these e f f e c t s i n the next few sect ions.

3 . Euler ian and Lagrangian descr ip t ions o f turbulence

I n most o f t he t h e o r e t i c a l and the experimental i nves t i ga t i ons o f wind tunnel turbulence and atmospheric turbulence, we use the Eu le r ian descr ip- t i o n . In such a desc r ip t i on , we consider the t u r b u l e n t v e l o c i t i e s a t f i x e d po in ts i n space? a t var ious times, i .e. , u(?, t). The c o r r e l a t i o n

func t ion such as u l (x l , t l )u ' (x2 , t2 ) and the corresponding spectrum func t ions

--- _I_

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are measured and analyzed. However, most o f the measurements i n i c e dynamics are based on Lagrangian d e s c r i p t i o n i n which we f o l l o w the movement o f a s i n g l e i c e f l o e t i o n funct ions has been w e l l developed bu t the s t a t i s t i c a l theory o f the form o f Lagrangian c o r r e l a t i o n has n o t been w e l l developed ye t . Furthermore, the r e l a t i o n between the Euler ian and the Lagrangian c o r r e l a t i o n c o e f f i c i e n t s i s not known i n general, even though some simple cases have been discussed i n reference 3 . For instance, i n some simple cases, i t was found t h a t the Euler ian and the Lagrangian c o r r e l a t i o n c o e f f i c i e n t s have approximately s i m i l a r shape and t h a t the Lagrangian c o r r e l a t i o n has a l a r g e r t ime scale. From the measurement o f the t r a c k of an i c e f l o e , we may o b t a i n t h e Lagrangian c o r r e l a t i o n c o e f f i c i e n t . o f the i c e f low depends on the Eu ler ian c o r r e l a t i o n c o e f f i c i e n t . f u r t h e r study o f the r e l a t i o n between the Eu ler ian and the Lagrangian corre- l a t i o n s i s o f spec ia l importance i n the study of i c e dynamics.

u A ( t ) o r a few i c e f l o e s . The theory o f Eu le r ian c o r r e l a -

I n the mathematical model ,lo the t u r b u l e n t s t ress Hence a

4.

The f i r s t successful theory o f t u r b u l e n t d i f f u s i o n i s T a y l o r ' s theory5 o f 1921, i n which he considered the random walk of a drunkard having some memory: be def ined as

Tay lo r ' s , Richardson's and --.- Go lds te in 's theor ies o f t u r b u l e n t d i f f u s i o n

I n T a y l o r ' s the0ry,~,5 a t u r b u l e n t d i f f u s i o n c o e f f i c i e n t D may

where y ( t ) i s the p o s i t i o n o f the p a r t i c l e a t t ime t. d i f f u s i o n equat ion f o r a d i f f u s i b l e phys ica l q u a n t i t y s , e.g., the concen- t r a t i o n o f p o l l u t i o n p a r t i c l e s , i s

The corresponding

/

The ex is tence o f the t u r b u l e n t d i f f u s i o n c o e f f i c i e n t o f equat ion ( 1 ) impl ies t h a t (i) the p a r t i c l e s move w i t h an i n f i n i t e v e l o c i t y and ( i i ) there i s no c o r r e l a t i o n o f the d i r e c t i o n s o f motion. Both of these f a c t s are not good assumptions f o r the random mot ion o f i c e f l o e s .

For the t r a c k o f i c e f l o e s , we are in te res ted i n the d ispers ion o f the coord inate o f a d i f f u s i n g p a r t i c l e , i .e., 2 o f equat ion (2). we may in t roduce a Lagrangian c o r r e l a t i o n RL( t ) such t h a t

To determine 2,

- RL(t) - - v ( t )V(t+T) = v2 R L ( ' r ) (4)

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where v = d y ( t ) / d t . I f we consider s t a t i o n a r y turbulence, such t h a t

v2 = coins t a n t , we have

T a y l o r ' s theory i s e s s e n t i a l l y f o r small sca le tu rbu len t d i f f u s i o n S O t h a t the main c h a r a c t e r i s t i c s of t u rbu len t d i f f u s i o n i s s i m i l a r t o those of molecular d i f f u s i o n , When the scale of turbulence z i s no t small i n com- par ison w i t h the dimension of the flow f i e l d we are i n te res ted i n , the laws o f t u r b u l e n t d i f f u s i o n are e s s e n t i a l l y d i f f e r e n t from those o f molecular d i f f u s i o n . For instance, i n cont ras t t o molecular d i f f u s i o n , the r a t e o f change o f the d is tance L between two d i f f u s i n g p a r t i c l e s depends on the d is tance i t s e l f ; the r a t e o f separat ion, on the average, i s no t la rge when L remains small but i t grows la rge when L becomes la rge . Thus the smal l -sca le motions on ly s l i g h t l y change t h i s d is tance L w h i l e the la rge-sca le motions (>>L) simultaneously t r a n s f e r two p a r t i c l e s w i thout e s s e n t i a l l y changing the d is tance between them. As a r e s u l t , the increase o f s i z e of the p a r t i c l e c loud leads t o the increase o f the e f f e c t i v e d i f f u s i o n c o e f f i c i e n t .

Richardson6 was the f i r s t one t o pay a t t e n t i o n t o t h i s phenomenon i n 1926. He suggested t h a t t h i s phenomenon may be descr ibed by a d is tance neighbor func t i on g(L , t ) which i s the p r o b a b i l i t y dens i ty f o r d is tance L between two d i f f u s i n g p a r t i c l e s . Richardson suggested l a t e r t h a t the change o f the func- t i o n g(L , t ) should be descr ibed by a pa rabo l i c d i f f u s i o n equat ion w i t h a d i f f u s i o n c o e f f i c i e n t 0 depending on L. With the he lp o f empi r i ca l data, Richardson found t h a t D(L) i s p ropor t i ona l t o I. 4/3 This law i s v a l i d f o r phenomena o f d i f f e r e n t scales from d i f f u s i o n i n the sur face l aye r o f the atmosphere up t o ho r i zon ta l mix ing on the scale of the general c i r c u l a t i o n of the atmosphere. Richardson's d i f f u s i o n equat ion has been ex tens i ve l y used f o r atmospheric turbulence.

Both Tay lo r ' s theory and Richardson's theory i nd i ca te t h a t the tu rbu len t d i f f u s i o n equations are o f pa rabo l i c type which means t h a t the v e l o c i t y o f propagat ion o f smal l d is turbance i n the tu rbu len t d i f f u s i o n f i e l d i s i n f i n i t e . Ac tua l l y , the v e l o c i t y o f propagat ion o f the d is turbance should be f i n i t e . Hence the pa rabo l i c equat ion would g i ve some essen t ia l e r ro rs , p a r t i c u l a r l y when we consider a la rge f i e l d such as the i c e f i e l d i n the A r c t i c Ocean and i t s surrounding areas. The c l a s s i c a l theory o f Tay lor has been genera l ized by Goldstein,8 who found tha t the tu rbu len t d i f f u s i o n can be represented by a hyperblol ic equat ion so t h a t the d is turbance o f t u rbu len t d i f f u s i o n would propagate a t f i n i t e speed. Goldste in 's equat ion i s as fo l lows:

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where s may be considered as the number dens i ty o f i c e f loes , AV$ = D i s the t u r b u l e n t d i f f u s i o n c o e f f i c i e n t , vd i s the speed of propagat ion o f d isturbance, B i s a constant, C i s the source f u n c t i o n o f i c e f l o e s , y i s the s p a t i a l coord inate (here we consider the one dimensional f l a w ) and t i s t h e t i m e . I f we assume t h a t vd i s i n f i n i t e but AV2 = D i s f i n i t e , Eq.6(6) i s reduced t o t h e c l a s s i c a l equat ion o f t u r b u l e n t i f f u s i o n o f Tay lor , where the source term C i s assumed t o be zero. For the o ther l i m i t i n g case, i f we take, A +a, B +a, vd i s f i n i t e and C = 0, Eq. (6) becomes

Eq. (7 ) i s t h e well-known wave equat ion f o r tu rbu len t d i f f u s i o n . I f we take B -f 03, but both A and vd f i n i t e and C = 0, Eq. (6) becomes

Eq. (8) i s the te legraph equat ion of t u r b u l e n t d i f f u s i o n proposed by Goldstein, which i s o f hyperbo l i c type w i t h f i n i t e v e l o c i t y vd.

From experimental data, we may determine the t u r b u l e n t d i f f u s i o n c h a r a c t e r i s t i c constants A, B and vd. measurements o f the random motion o f a number o f i c e f l o e s w i t h t ime and space distance y. We have the i n i t i a l number densi ty d i s t r i b u t i o n o f the i c e f l o e s so. We may f i n d the s o l u t i o n o f Eq. (6) f o r the number dens i ty s ( y , t ) and check i t w i t h the experimental data so t h a t the c h a r a c t e r i s t i c constants A, b and vd are determined. t o p r e d i c t the i c e f l o e movement i n the A r c t i c Ocean and check the t h e o r e t i c a l r e s u l t s w i t h the experimental data obtained by f i e l d measurements o r by remote sensing from a i r c r a f t o r sate1 1 i t e .

I t i s i n t e r e s t i n g t o n o t i c e t h a t the est imate o f the d i f f u s i b l e q u a n t i t y s by T a y l o r ' s o r Richardson's equat ion i s not s u f f i c i e n t except i n the case when the motion o f the d i f f u s i n g p a r t i c l e s a r e independent o f each other . Hence f o r i c e dynamics, Goldste in 's equat ion i s b e t t e r . For f u r t h e r improve- ment o f the d i f f u s i o n equation f o r i c e dynamics, we may use the general approach o f k i n e t i c theory i n which a six-dimensional space: th ree coordinates and three v e l o c i t i e s are used. I n a way, t h i s general approach i s s i m i l a r t o the theory of Brownian motion o r the k i n e t i c theory o f gases. '0bukhov3 has suggested such an approach. A lso L ink and others8 proposed t o use the Langevin equat ion t o study the t u r b u l e n t d i f f u s i o n i n s t a t i o n a r y turbulence which may be considered as a f i r s t approximation f o r the theory s i m i l a r t o Brownian motion. L i n ' s theory gives T a y l o r ' s and Richardson's r e s u l t s as specia l cases but i t does no t g i v e Go lds te in 's r e s u l t nor those w i t h the

For example, one o f the experiments w i l l be the

We may then use these c h a r a c t e r i s t i c constants

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e f f e c t o f c o l l i s i o n between p a r t i c l e s , such as should reexamine L i n ' s approach so t h a t Goldste as a spec ia l case f o r i c e dynamics.

(Pai and L i )

ce f l o e s , i n genera1.9 We n ' s r e s u l t may be considered

5 . -- One-par t ic le d ispers ion I_.- and t w o - p a r t i c l e .

Most o f the analyses o f t u rbu len t motion o f p a r t i c l e s a re based on the movement o f a s i n g l e pa r t i c l e5 ,7 which may apply t o the experimental r e s u l t s o f the movement ( o f a s i n g l e i c e f l o e . We may determine the one-pa r t i c l e Lagrangian v e l o c i t y c o r r e l a t i o n func t i on R L ( T and the corresponding t ime scale T(T). For a s t a t i o n a r y turbulence w i t h 3 constant, t h i s Lagrangian c o r r e l a t i o n func t i on RL(T) i s very c lose t o the s imple exponent ia l exp(-BT) where a = BTf and T f i s the t ime sca le o f the fo rce f on the p a r t i c l e according t o Langevin equat ion* and 6.r = O ( 1 ) o r g rea ter . The Lagrangian t ime microscale TL i s

and the Lagrangian turbulence sca le i s

For Richardson's theory and i n some measurements o f the t racks o f i c e f l o e s , we are i n te res ted i n the r e l a t i v e movement o f two d i f f e r e n t p a r t i c l e s . Hence i t would be o f i n t e r e s t t o study the two-pa r t i c l e d ispers ion. I n the study o f the two-pa r t i c l e d ispers ion r e l a t i o n , one o f the new fac to rs i s the i n i t i a l separat ion d is tance o f the two p a r t i c l e s considered. I t has been f i r s t found by L in4 t h a t the i n i t i a l growth r a t e o f the mean square p a r t i c l e separat ion fo l lows a t3-1aw, i .e.,

whereB1 i s a constant. I n L i n ' s theory, i t was assumed tha t the i n i t i a l r e l a t i v e v e l o c i t y i s very small and t h a t the i n i t i a l separat ion d is tance i s much smal le r than the microscale A but l a r g e r than the Kolmogoroff sca le nlC. L i n ' s forrniula (11) i s e s s e n t i a l l y Richardson's law o f d ispers ion bu t here the c o e f f i c i e n t o f t 3 i s given i n terms of the i n i t i a l separat ion d is tance between the p a r t i c l e s . t h a t i n the two-pa r t i c l e problem, the r e l a t i v e acce le ra t i on of the p a r t i c l e s i s a non-stat ionary random f u n c t i o n of d i f f u s i o n time. Only i n the case t h a t t h e i n i t i a l separat ion L i s a t i s f i e s the i n e q u a l i t y Qlc<<Li<<X, the non-stat ionary e f f e c t s a re n e g l i g i b l e .

Krasnof f and Peskin8 extended L i n ' s ana lys is and found

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The above L i n ' s r e s u l t s may not h o l d t r u e f o r i c e dynamics because the i n i t i a l separat ion d is tance between two i c e f l o e s I n most cases i s large. Hence f u r t h e r study on the t w o - p a r t i c l e d ispers ion f o r the i c e dynamics should be made.

6. One-, two- and three-dimensional motions of t u r b u l e n t d i f f u s i o n

For both wind tunnel turbulence and atmospheric turbulence, the t u r b u l e n t f l u c t u a t i o n s are e s s e n t i a l l y three-dimensional f low. However, i n both of these cases, there i s one main d i r e c t i o n o f d i f f u s i o n which i s of spec ia l i n t e r e s t , For wind tunnel turbulence, the d i r e c t i o n o f the mean f l o w x i s o f spec ia l i n t e r e s t . We would l i k e t o know, f o r instance, the d ispers ion o f p a r t i c l e s as a f u n c t i o n o f the s p a t i a l coordinate x. O r d i n a r i l y , Tay lo r ' s assumption has been used such t h a t the t ime spectrum o f turbulence i s assumed t o be equal t o the one-dimensional l o n g i t u d i n a l spectrum o f space. Hence the one- dimensional c o r r e l a t i o n e u a t i o n o f turbulence, p a r t i c u l a r l y f o r homogeneous and i s o t r o p i c turbulence,? has been ex tens ive ly invest igated. Since f o r i c e dynamics, Goldste in 's equat ion i s more s u i t a b l e than Tay lo r ' s equation, we should reexamine T a y l o r ' s assumption f o r i c e dynamics from both Go lds tk in 's equat ion and experimental r e s u l t s i n the case when we consider some one- d i mens i onal app rox i mat ion o f i ce dynami cs . For atmospheric turbulence, the v e r t i c a l d is tance plays a specia l r o l e because o f the g r a v i t a t i o n a l e f f e c t s . Hence the v a r i a t i o n o f t u r b u l e n t d i f f u s i o n w i t h a l t i t u d e i s one o f the most important problems i n atmospheric turbulence. For the v a r i a t i o n o f t u r b u l e n t d i f f u s i o n i n atmosphere w i t h a l t i t u d e , some specia l features should be considered such as t h e terminal v e l o c i t y o f the p a r t i c l e s . I n i c e dynamics, the terminal v e l o c i t y does not p l a y an important r o l e because the densi ty o f i c e i s less than t h a t o f the surrounding sea water and the i c e i s f l o a t i n g on the water surface ra ther than s i n k i n g down. From t h i s p o i n t o f view, the features o f i c e dynamics turbulence would be g r e a t l y d i f f e r e n t from those o f atmospheric turbulence.

For the motion o f i c e f l o e s , the v e l o c i t y f i e l d i s e s s e n t i a l l y two-dimensional-- t h a t i s , the motions o f i c e f l o e s are e s s e n t i a l l y on the sur face o f the ocean and the v e r t i c a l motions are very small and r e s t r i c t e d i n a t h i n l a y e r o f the sur face o f the ocean whose thickness i s much smal ler than the two dimensions o f the surface o f the ocean. The tu rbu len t mot ion o f i c e f l o e s i n a two-dimensional space has not been w e l l s tud ied. Since we know t h a t there i s a great d i f f e r e n c e between the two-dimensional and the three-dimensional random walk, i t would be o f great i n t e r e s t t o f i n d ou t the main features o f the two-dimensional random walk process o f i c e f l o e s i n the ocean.

7. Special f a c t o r s o f random motion o f i c e f l o e s

For the random motion o f i c e f l o e s , there are a number o f spec ia l important problems which may not be o f importance f o r wind tunnel turbulence and/or atmospher c turbulence and which have no t been c a r e f u l l y invest igated. Specia a t t e n t i o n s h o u l d be given t o these problems when we study the random motion o f i c e f l o e s i n the A r c t i c Ocean and i t s surrounding areas.

150

( P a i and L

(i) Meteoro log ica l cond i t ions . One o f the major f ac to rs t h a t a f f e c t s the motion o f i c e f l o e s i s the meteoro log ica l cond i t ions , p a r t i c u l a r l y wind, - --

temperature, barometr ic pressure and the c loud cond i t ions over the i c e f i e which a f f e c t the s o l a r r a d i a t i o n . We w i l l i nc lude the meteoro log ica l con- d i t i o n s i n the study of random mot ion o f i c e f l o e s so t h a t an improved form o f t u rbu len t d i f f u s i o n equat ion over Go lds te in 's equat ion may be obtained. Since the re are many fac to rs due t o meteoro log ica l cond i t ions , i t might not be poss ib le t o examine a l l o f these fac to rs i n t h i s sho r t paper, bu t we w i l l l i s t m o s t o f the important f ac to rs i n the f o l l o w i n g sect ions and t r y t o p o i n t out t he i r importance.

( i i ) in te rac t i lon between the i c e and the atmosphere p lays a very important r o l e .

Boundary e f f e c t s and exchange processes. On the upper sur face, the - -__II_

The shear ing s t ress due t o the wind i s a d r i v i n g fo rce o f the motion o f the i ce f loes.10 On the lower sur face, the i n t e r a c t i o n between the i c e and the ocean cu r ren t i s important and the shear ing s t ress due t o the ocean cur ren t i s another d r i v i n g fo rce . We w i l l reexamine the random walk o f i c e f l oes under the i n f l uence o f ex te rna l random forces due t o the wind and the ocean cur ren t5 (and improve Go lds te in 's equat ion (61 f o r the i c e f l oes .

When we study the motion o f i c e near the edge o f a la rge i c e f i e l d , the boundary e f f e c t o f the tu rbu len t motion o f i c e f l o e s becomes important. I n Go lds te in 's equat ion (6), no r e s t r i c t i o n on the boundary i s made f o r the ran- dom movement o f i c e f l o e s . Near the boundary,3 such a r e s t r i c t i o n should be made and we w i l l have some mod i f i ca t ions on the Go lds te in 's equat ion (6) due t o t h i s boundary e f f e c t . The tu rbu len t c h a r a c t e r i s t i c s o f the motion o f i c e f l oes near the edge o f a l a rge i c e f i e l d w i l l change accord ing ly . Since the motion o f i c e near the edge o f a la rge i c e f i e l d impacts f i s h e r y a c t i v i - t i es , t ranspor ta t i on , and o f f sho re m i n i ng opera t ion i n the Arc t i c reg i on, i t i s essen t ia l t h a t a thorough i n v e s t i g a t i o n o f the i n t e r a c t i o n o f the i c e and the water near the edge o f a l a rge i c e f i e l d be s tud ied.

( i i i ) I n t e r a c t i o n and c o l l i s i o n o f i c e f l o e s . I n the c l a s s i c a l theor ies o f Tay lo r , Richardson and Goldstein. the i n t e r a c t i o n and c o l l i s i o n p lay

---- VI-

. . ra the r important ro les i n the random motion o f i c e f l oes . One way t o inc lude the e f f e c t o f co l1 i s i o n and i n t e r a c t i o n i s t o develop a general ized Bo1 tzmann equat ion f o r the random d i s t r i b u t i o n func t i on o f i c e f l o e s . Such a study requi res a longer per iod o f i n v e s t i g a t i o n before any p r a c t i c a l r e s u l t s may be obtained. Another approach i s t o in t roduce proper r e l a t i o n f o r the d i f f u s i o n v e l o c i t y s o tha t some e f f e c t s o f the c o l l i s i o n may be included. A simple model should be developed f o r t h i s s i m p l i f i e d approach.

( i v ) Other fac to rs . random walk theory o f i c e f l o e s . These are (a) source terms due t o me l t i ng

There a re o ther fac to rs which may be considered i n the --I---

- and f reez ing , and (b) the d i f f e r e n c e o f v e l o c i t i e s o f . ice f l oes from t h a t o f t he surrounding medium.

I n conclusion, the f i r s t s tep t o study the tu rbu len t motion o f i c e f l o e s may be the appl l icat ion of Goldste in 's equat ion o f tu rbu len t d i f f u s i o n ( 6 ) and the determinat lon o f tu rbu len t c h a r a c t e r i s t i c s based on Golds te in 's equat ion ( 6 ) . The next s tep w i l l be the m o d i f i c a t i o n of Go lds te in 's equat ion so t h a t a b e t t e r c o r r e l a t i o n between the exper imental data and t h e o r e t i c a l ana lys i s may be obtained.

151

(Pai and L i )

8. References

1. Ketchum, R. D. J r , , and Wittmann, W. 1 . Recent remote sensing s tud ies o f the East Greenland pack ice. Proceedings o f In te rna t i ona l Sea Conference, Reykjavik, Iceland, May 10-13, 1971, ed. T. Karlsson.

2. Pai, S. 1 . Viscous Flow Theory I I . Turbulent Flow, D. Van Nostrand Co., N .Y . 1957.

3. F renk ie l , F. N., and Sheppard, P. A. (ed.). Atmospheric D i f f u s i o n and A i r P o l l u t i o n , vo l . 6 o f Advances i n Geophysics, Academic Press, N.Y. 1959.

4. L in , C. C., and Reid, W. H. Turbulent Flow, Theoret ica l Aspects. Handbuch der Physik, V I I1/2, Springer Verlag, B e r l i n . 1963.

5 . Tay lor , G. I . D i f f u s i o n by continuous movement. Proceedings London Math. SOC. 2, XX, p. 169, 1971.

6. Richardson, L. F. Atmospheric d i f f u s i o n shown on a distance neighbor graph. Proc. Roy. SOC. A-135, p. 709, 1926.

7. Goldstein, S . On d i f f u s i o n by discontinuous movements and on the Quar t . Jour. o f Math. and Appl. Mech., Lo l . 4, p. 129, telegraph equation.

1951.

8. Krasnoff, E., and Peskin, R. L. The Langevin mo-el f o r t u rbu len t d i f f us ion . Geophysical F l u i d Dynamics, vo l . 2, pp. 123-146, Gordon & Breach Science Pub1 ishers, 1971.

9. Timokhov, L. A. One-dimensional s tochas t ic i c e d r i f t . Trudy A A N l l , Leningrad, vo l . 281, pp. 121-129, 1967, a l so t rans la ted i n AIDJEX B u l l e t i n No. 3, pp. 80-93, November 1970.

10. Campbell, W. J. The wind dr iven c i r c u l a t i o n o f i c e and water i n a p o l a r ocean. 15, 1965.

Jour. Geophys. Res., vo l . 70, no. 14, pp. 3279-3301, Ju l y

152

DATA MANAGEMENT REPORT ON 1972 A I D J E X P I L O T STUDY

The t a b l e which appears on t h e next four pages is an up-to-daqe

summary of d a t a received by t h e AIDJEX Data Bank from t h e 2 1 p r i n c i p a l

i n v e s t i g a t o r s who p a r t i c i p a t e d i n t h e 1972 p i l o t study.

37 p r o j e c t s y i e lded an independent set of d a t a , and t h e Data Bank has

sets from 2 1 of t hese p r o j e c t s , a v a i l a b l e as d i g i t a l d a t a , c h a r t s ,

photographs, and f i n a l r epor t s . For a more d e t a i l e d d e s c r i p t i o n of

t h e p r o j e c t s , t h e reader is r e f e r r e d t o t h e AIDJEX operat ions Manual

(January 1972) and AIDJEIGBulletin No. 14 ( Ju ly 1972).

each of t h e

The p r o j e c t s l i s t e d i n the t a b l e include experiments which were

performeld bu t not s p e c i f i c a l l y mentioned i n t h e Operations Manual.

Because they r e s v l t e d i n no usable d a t a , t h e d i r e c t shea r stress p r o j e c t

(Businger) and t h e accelerometer p r o j e c t (Evans) are omitted. P r o j e c t

t i t l es and remarks r e f l e c t t h e v a r i a b l e s observed r a t h e r than u l t ima te

use of da t a . The d a t a management summary r e f e r s t o t h a t which t h e Data

Manager requested of t he p r i n c i p a l i n v e s t i g a t o r s on 31 May 1972.

ALL comments and i n q u i r i e s concerning the AIDJEX Data Bank should

be addressed t o Murray J . Stateman, AIDJEX Data Manager, 4059 Roosevelt

Way N . E . , S e a t t l e , Washington 98105.

153

DATA FILES FROM AIDJEX PILOT STUDY 1972

Data M g m t . P r i n c i p a l Summary Data

P r o j e c t I n v e s t i g a t o r s I n s t i t u t i o n Received Received Remarks

Delco buoys

Trans i t pos i t i on ing

Acoustic bottom

ABR depth

re f e ren ce

Mesoscale s t r a i n

Azimuth P fc cn Ice morphology

Remot e-$ens ing over f 1 i gh ts

Microscale s t r a i n

A i r , ice tempera-

S t r e s s i n i c e

tu re

Nansen casts

I n t e r i o r flow

Buck/Mar t i n

Martin/ Thorn dike

Martin/

Martin/

Weeks

Tho m dike

Tho m d i k e

Thorndike

Weeks

Weeks

Tab ata

Tabata

Tabata

Coachman

Coachman

Delc o/AIDJEX

AIDJEX 6/ 12/ 72

A I D j E X

AIDJEX 6/12/72

CRREL 6/14/72,

AIDJEX 6/12/72

CRREL 6/14/ 72

CRREL

Ho kka i do Japan

Ho kka i do J ap an

Hokkaido J ap an

U. of Wash.

U. of Wash.

10/10/72 4 b w y s - a i r p r e s s u r e

7/26/72 Navigational satell i te

11/21/72 7 r e fe rence transponders

12/1/72 Transit t i m e

Laser: angle, d i s t a n c e measurements

7/15/72 Published i n AIDJEX Bu l l . 814

3/29/73 Ice and snow thickness , s a l i n i t y . Data list. Analysis i n AIDJEX B u l l . 1/19.

7/17/72(2) 6 of 7 mosaics of 10x10 km 3/18/73(4) a t main camp

Laser - 1 km r a d i u s

Also wind v e l o c i t y

P res su re and s t ra in gauges i n ic'e

11/30/ 72 Hydrographic

Water v e l o c i t y , 5 depths, 3 s t a t i o n s

DATA FILES FROM A I D J E X PILOT STUDY 1972 (continued)

Data Xgmt. P r i n c i p a1 Sumary Data

P r o j e c t Inves t iga to r s I n s t i t u t i o n Received Received Remarks

~ CTD

0 ce an b o un da r y l a y e r .

Unde r-i ce p r o f i l e

Water stress

P cn cn

Current p r o f i l e

Deep s t a t i o n STD

CTD

Atmospheric shear stress

Ice su r face s k i n drag

Coachman / Smi t h

J . D . Smith

J . D . Smith

Hunkins

Hunkins

Amos /Hunkins

Pounder

Goddard

S.D. Smith/ Banke

U. of Wash.

U. of Wash.

U. of Wash.

Lamon t-Doher t y Columbia Univ.

Lamon t-Doh e r t y Columbia Univ.

Lamon t -Doh er t y Columbia Univ.

P CSP / M c G i 11

U. o f C a l i f . , Davis

Bedford I n s t .

6/16/72

6/16/72

10/3/72

Gu i ld l ine f i s h ; Vidar d a t a logge r

3- dimensional c u r r e n t meter a r r a y s

1/15/73 SCUBA d i v e r ' s r e p o r t Published i n AIDJEX Bul l . 1/18

4/3/73 2-dimensional cu r ren t meters on r i g i d masts. March 28- Apr i l 26, 1972.

3/19/73 2-dimensional cu r ren t m e t e r casts t o 170 m

Casts to ocean f l o o r

11/8/72 Resu l t s contained i n g raph ica l r e p o r t

Micrometeorology ; wind speed and temperature

12/21/ 72 Resu l t s contained i n graphical r e p o r t

DATA FILES FROM AIDJEX PILOT STUDY 1972 (continued) Data M g m t .

P r i n c i p a l Summary Data P ro jec t I n v e s t i g a t o r s I n s t i t u t i o n Received Received Remarks

Pounder A tmosphe r i c

p r o f i l e 6/19/72 8/15/72 Micrometeorology ; wind speed and temperature

PCSP/McGill

Goddard/ McBe t h

Goddard

UC Davis/ NCAR

U. of Ca l i f . , Davis

U. of Alaska

Boundary p r o f i l e

Energy t r a n s f e r

system Balloon sonde

Air /i ce b ounda r y

Resu l t s contained i n f i n a l p r o j e c t r e p o r t

6/19/72 31231 73

3/28/73

Radiation f luxes Weller

Campbell/ Gloersen

USGS/ NASA

Remote-sensing ove r f 1 igh t s cn P m

8 microwave mosaics 6 v i s u a l mosaics 70 mm nega t ives

RemQte-sensing o v e r f l i g h t s 1012172 2700 9x9 photos

3 mosaics

Microwave senso r - h e l i c o p t e r base

Ice s a l i n i t y , dens i ty , temperature, t e x t u r e

Microwave senso r - mobile tower; f ina l r e p o r t

W i t tmann NAVO CEANO

Barring t on /

Ramseier

Adey CRC Canada

Environment Canada

Aero j e t Gener a1

Ice thickness

6 1141 72

8/11 72

Ice thickness

Ice thickness Edgert on

B iax ia l

Ocean tilt

tilt meter Lamon t-Doherty Columbia Univ. 6/16/72

DEMR, Canada

Hmkins

Weber

Ice tilt

Sea tilt and ice t i l t

DATA FILES FROM AIDJEX PILOT STUDY 1972 (continued)

Data M g m t . P r i n c i p a l Summary Data

Project I n v e s t i g a t o r s I n s t i t u t i o n Received Received Remarks

I U S buoy P o s i t i o n d a t a from p o s i t i o n Haugen APL, U. of W. 6/16/72 1/ 231 73 May-November 19 72

Gravity

Ce le s t i a l fixes

Web e r DEMR, Canada

G i l l / Hunkins Lamon t -Dohe r t y Columbia Univ.

6 1 30 / 72 Main camp; publ ished i n A I D J E X Bu l l . 814

Ground weather AID JEX 6/30/72 A l l t h r e e camps; publ i shed i n AIDJEX B u l l . #14

ABSTRACTS OF INTEREST

Gerd Wendler. 1973. Sea i c e observat ion by means of sa te l l i te . Journal of GeophysicaZ Research 78(9): 142791448.

For a s m a l l area (about 120 km X150 lan) of t h e Arctic Ocean o f f t h e shore of n o r t h Alaska t h e ice condi t ions were examined i n d e t a i l by s a t e l l i t e imagery. Instead of using 64 s t e p s of gray, 15 b r igh tness s t e p s were d i s t ingu i shed by computer f o r each d a t a po in t . b r igh tness composites were used. d r a f t e d i n g r e a t d e t a i l f o r 5-day per iods f o r t h e summer of 1969, t h i s f a c t showing the value of sa te l l i tes as a unique t o o l f o r sea ice reconnaissance, The ice condi t ions were found t o depend s t r o n g l y on t h e wind d i r e c t i o n . t h e ice o u t , and wind toward t h e shore brought i t back. A good c o r r e l a t i o n w a s found with t h e wind taken from the weather maps, although t h e d a t a from a ground-based c l ima to log ica l s t a t i o n were no t always i n good agreement wi th t h e i ce movement, probably because of orographic e f f e c t s . A comparison with i ce c h a r t s mapped by more conventional methods showed good greement i n most cases, The sa te l l i t e p i c t u r e , however, gave much g r e a t e r d e t a i l . Monthly mean albedo maps w e r e constructed from t h e b r igh tness composites f o r t h e 4 months from mid-May t o mid- September and compared with previous measurements.

To suppress t h e t r a n s i e n t c loudiness , minimum The i ce condi t ions could be

Offshore wind moved

158

CONTENTS OF PAST AIDJEX BULLETINS

No. 1 (Sept. 1970): S t a tus Report - no Zonger avaiZabZe T i m e schedule Program elements and working groups

Analysis and s imula t ion A i r and w a t e r stress Ice mechanics and morphology Radiat ion and h e a t budget Automatic d a t a buoy Rem0 t e s ens i n g I c e roughness measurement

1971 p i l o t s t u d i e s I n t e r n a t i o n a l p a r t i c i p a t i o n Budget outlook Mailing l i s t

No. 2 (Oct. '1970, rep. Oct. 1971): Theoretical Discussions The kinematics and mechanical behavior of pack i c e : t h e s ta te

of t he s u b j e c t (Rothrock) La tes t experiments with ice rheology (Campbell and Rasmussen) Notes om a p o s s i b l e c o n s t i t u t i v e l a w f o r a rc t ic sea ice (Evans) Thought:; on a viscous model f o r sea ice (Glen) The p res su re term i n t h e c o n s t i t u t i v e l a w of an i c e pack (Rothrock) A s tudy of i ce dynamics r e l evan t t o AIDJEX (Solomon) Techniques f o r measuring s t r a i n rate (Thorndike) Bibliography Power spectrum a n a l y s i s of i ce r idges (Hibler and LeSchack)

No. 3 (Nov. '1970): Selected Soviet Research - no longer available Resu l t s of expedi t ion i n v e s t i g a t i o n s of t h e d r i f t and dynamics of

t h e ice cover of t h e Arctic Basin during t h e sp r ing of 1961 (Bushuyev, Volkov, Gudkovich, and Loshchilov)

(Do r o n i n)

t h e in f luence of wind during t h e navigat ion period (Nikiforov, Gudkovich , Yef imov , and Romanov)

On a method of c a l c u l a t i n g t h e compactness and d r i f t of i ce f l o e s

P r i n c i p l e s of a method f o r c a l c u l a t i n g t h e i c e r e d i s t r i b u t i o n under

Dynamics of t h e i ce cover and changes i n i t s compactness (Timokhov) One-dimensional s t o c h a s t i c ice d r i f t (Timokhov) Dynamics and kinematics of i ce f l o e s (Timokhov) On r e l i e f forms of f l o a t i n g i c e (Yakovlev)

159

No. 4 (Jan. 1971): Water Stress Studies - no longer available AIDJ'EX oceanographic i n v e s t i g a t i o n s (Smith) A r e p o r t on t h e 1970 AIDJEX p i l o t s tudy (Coachman and Smith) An a r c t i c under-ice d iv ing experiment (Martin) 1971 AIDJEX water stress p i l o t s t u d i e s

- - Introduct ion (Smith) --Lamont measurements of water stress and ocean cu r ren t s (Hunkins) --University of Washington water stress s t u d i e s (C0achma.n and Smith)

No. 5 (Feb. 1971):, Remote Sensing and Ice Morphology - y10 longer available Remote sens ing , i c e mechanics and morphology: an o u t l i n e of t h e

Remote sensing and ground t r u t h i n v e s t i g a t i o n s (Campbell) CRREL-USGS i c e mechanics and morphology program (Weeks and Kovacs) NASA f l i g h t program (Petersen) U.S. Coast Guard research r e l evan t t o AIDJEX (Johnson, McIntosh, and

NAVOCEANO p a r t i c i p a t i o n i n AIDJEX (Ketchum) N O M s a t e l l i t e resources a v a i l a b l e t o AIDJEX (Fleming) 1971 p i l o t study summaries AIDJEX spokesmen (Fle tcher ) Log i s t i c support f o r AIDJEX-71 p i l o t s t u d i e s (Bjorner t ) 1 9 7 1 p i l o t study t i m e schedule Numerical modeling group s e s s i o n

problem (Langleben and Pounder)

B r es lau)

No. 6 (March 1971): Ice Dynamics - no longer available Measurement of tilt of a f rozen sea (Weber and L i l l e s t r a n d ) I c e balance i n the Arc t i c Ocean (Koerner) On the r e l a t i o n between turbulen t and averaged i ce - f loe motion (Timokhov) Theory of t he d r i f t i n i c e f i e l d s i n a ho r i zon ta l ly inhomogeneous

Abs t rac ts Progress r e p o r t , 1971 AIDJEX p i l o t s tudy AIDJEX mail ing l i s t

wind f i e l d (Yegorov)

No. 7 (Am41 1971 : Arctic Data BUOY and Posi t i o n i n q Smtems - no longer The Soviet DARMS (Olenicoff) avai Zable Sono-Drift-Buoys (Defence Research Establishment P a c i f i c - Canada) Experimental a r c t i c da t a buoy (Haugen) A remote automatic multipurpose station--RAMS (Buck, Brumbach, and Brown) The navigat ion of F le t che r ' s I c e I s l and (T-3) (Hunkins and Hal l ) AIDJEX mesoscale measurements (Campbell) A review of r ad io pos i t i on ing f o r AIDJEX (Martin)

No. 8 (May 1971): 1971 Pilot S tudv Narratives - no longer available CRREL-USGS program a t Camp 200 (Weeks and Campbell) Report of NASA Convair 990 remote-sensing missions over AIDJSX s i te ,

Lamont ocean cu r ren t program (Hunkins) Universi ty of Washington oceanographic s t u d i e s (Coachman and Smith) 1971 a r c t i c f l o e s t u d i e s , A r c t i c Submarine Lab (Brown) Log i s t i c s r e p o r t of t he 1971 AIDJEX p i l o t s tudy (Bjorner t )

March 9-16, 1971 (Noen)

160

No. 9 (Aus 1971): AIDJEX Planning Conference Preface (Unters te iner and F le t che r ) In t roduc t ion AIDJEX planning conference minutes A I D J E X o rgan iza t ion Mathematical model of two-phase flow f o r t h e dynamics of pack ice

i n t h e Arctic Ocean (Pa i and L i )

No. 10 (Sepl:. 1971 ) : Trudy, AANII , Vol . 296 Role of t h e po la r regions i n g loba l i nves t iga t ions of t h e oceanic and

atmospheric c i r c u l a t i o n (Borisenkov and Treshnikov) I n t e r a c t i o n between t h e atmosphere and the ocean, b a s i c problems

( Dor onin) I n v e s t i g a t i o n of t he i n t e r a c t i o n between the atmosphere and t h e ocean

i n t h e Arctic by means of hydrodynamical models (Doronin) I n t e g r a t i o n of t h e f u l l equations of hydrothermodynamics i n t h e

numerical f o r e c a s t i n g of meteorological f i e l d s (Nagurnyi) A problem i n the f o r e c a s t i n g of t he geopo ten t i a l and wind f i e l d s with

t h e a i d of energy models (Borisenkov) The r o l e of cloudiness i n v a r i a t i o n s of t h e thermobaric f i e l d i n t h e

atmosphere (Doronin and Semenova) Analogous energy levels i n the atmosphere and t h e ocean (Borisenkov

and Nagurnyi) Large-scale i n t e r a c t i o n between the ocean and the atmosphere: l i t e r a t u r e

review (Khrol) Some phys ica l and s t a t i s t i c a l c h a r a c t e r i s t i c s of t h e w a t e r - and air-

temperature f i e l d s i n the North A t l a n t i c (Vladimirov and Nikolaev) Detect ion of a hidden p e r i o d i c i t y i n a set of multidimensional q u a n t i t i e s

( N i ko laev) Semi-annual p e r i o d i c i t y i n the seasonal changes of some hydrometeoro-

l o g i c a l q u a n t i t i e s i n t h e North A t l a n t i c (Vladimirov and Smirnov) Solar a c t i v i t y and f l u c t u a t i o n s of the b a r i c f i e l d s i n high l a t i t u d e s of

t h e Northern Hemisphere (Vladimirov, Nikolaev, and Smirnov) I n v e s t i g a t i o n of t h e turbulence of t h e lower atmospheric l a y e r over

t h e ocean (Ordanovich, Nagurnyi, and Andreev) Ef fec t of measurement e r r o r s upon the accuracy of determinations of t he

ver t ical temperature p r o f i l e from outgoing microwave r a d i a t i o n (Bazlova)

No. 11 ~ ~ o Y , , , . 1971) --- The Polar Experiment (Borisenkov and Treshnikov) The American “Arctic I c e Dynamics J o i n t Experiment” P r o j e c t (Treshnikov

Trans l a t ions now under c o n t r a c t S c i e n t i f i c operat ions of t he 1972 AIDJEX p i l o t s tudy L o g i s t i c s p l an f o r t h e 1972 AIDJEX p i l o t study (Bjornert and Heiberg) Sa fe ty precaut ions on t h e ice (Unters te iner)

e t a l . )

161

No. 12 (Feb. 1972) Progress r e p o r t on 25 cm r ada r observat ions of t h e 1971 AIDJEX s t u d i e s

No.

(Thompson, Bishop, and Brown)

Weeks, Ackley, and Hib le r ) A s tudy of a mul t iyea r p re s su re r i d g e i n t h e Beaufort Sea, 1971 (Kovacs,

Turbulence measurements over i ce a t Camp 200, 1971 (Banke and Smith) Water stress and ocean c u r r e n t measurements a t Camp 200, 1971 (Hunkins) Water and ice motion i n t h e Beaufort Sea, s p r i n g 1970 (Coachman and Newton) S p a t i a l a spec t s of p re s su re r i d g e statist ics (Mock, Hartwell, and Hibler) S t a t i s t i c a l a s p e c t s of sea-ice r i d g e d i s t r i b u t i o n s (Hibler , Weeks, and Mock)

1 3 (May 1972) Airphoto a n a l y s i s of ice deformation i n t h e Beaufort Sea, March 1971

(Hartwell) Mesoscale s t r a i n measurements on the Beaufort Sea pack ice , AIDJEX 1971

(Hibler , Weeks, Ackley, Kovacs, and Campbell) Top and bottom roughness of a mult iyear i ce f l o e (Hibler , Ackley, Weeks,

and Kovacs) The measurement of conduct ivi ty and temperature i n t h e sea f o r s a l i n i t y

determinat ion (Lewis and Sudar) S a l i n i t y c a l c u l a t i o n s from i n s i t u measurements (Perkin and Walker)

No. 14 ( Ju ly 1972): 1972 AIDJEX P i l o t Study , ----*r*l

Camp operat ions Pro j e c t summaries Preliminary d a t a

-.--- No. 15 (Aug. 1972): AIDJEX S c i e n t i f i c Plan Abstract In t roduc t ion S c i e n t i f i c o b j e c t i v e s Experiment design Analysis and modeling P r a c t i c a l a p p l i c a t i o n s of AIDJEX Appendix References

No. 16 (Oct. 1972): Trudy, AANII, Vol. 303 ( F i r s t Half) Cyclic v a r i a t i o n of t h e ice-cover c o e f f i c i e n t of t h e arctic seas

Test ing a numerical model of spring-summer r e d i s t r i b u t i o n of sea ice

Many-year v a r i a t i o n s of t h e ice coverage o f t h e Greenland Sea and

Manifestat ion of atmospheric cyc le s i n ice-cover c o e f f i c i e n t (Zakharov) Application of discr iminant ana lys i s t o long-range ice f o r e c a s t i n g f o r

The e f f e c t of long-period t i d e s on ice condi t ions i n t h e arctic seas

Results of t h e s tudy of nonuniform ice d r i f t i n t h e Arctic Basin

(Volkov and Sleptsov-Shevlevich)

(Doronin, Zhukovskaya , and Smetannikova)

methods of f o r e c a s t i n g it ( K i r i l l o v and Khromtsova)

the a r c t i c seas (Nikolaev and Kovalev)

(Vorob 'ev and Gudkovich)

(Volkov, Gudkovich, and Uglev)

162

No. 16 (Qct. 1972): Trudy, AANII, Vol. 383 (First Half - continued) Exc i t a t ion of compressive stresses i n i c e during t h e hydrodynamic s t a g e

of compact ice d r i f t (Kheisin) The e f f e c t of long-period t i d e s on ice d r i f t i n the Arctic Basin

(Gudkovich and Evdokimov) Ice d r i f t i n an inhomogeneous p re s su re f i e l d (Egorov) Wave d r i f t of an i s o l a t e d f l o e (Arikainen)

No. 17 (Dec. 1972): Trudy, AANII, Vol. 303 (Second Half) Seasonal f e a t u r e s of t h e "polar t i de" p re s su re wave over t h e Arctic

The r e l a t i o n of t he r e s u l t a n t monthly average wind t o the p res su re

Experimental determination of t h e wind drag on an i ce s h e e t (Karelin

S ta t i s t i . ca1 c h a r a c t e r i s t i c s of some ice cover parameters i n t h e Arctic

Estimating the lateral melt ing of d r i f t ice (Nazintsev) Snow acc:umulation of Kara Sea ice (Nazintsev) Shear measurements of n a t u r a l ice with o p t i c a l t h e o d o l i t e s (Legen 'kov,

0bservat:ions of ice motion w i t h o p t i c a l t heodo l i t e s on t h e SEVERNYI

(Gudkovich and Santsevich)

gradient (Nikolaev)

and Ti-mokhov)

(Buzuev and Dubovtsev)

Uglev, and Blinov)

POLYUS-17 d r i f t i n g s t a t i o n (methodology and accuracy) (Legen'kov and Uglev)

Application of computers f o r determining age c h a r a c t e r i s t i c s of a r c t i c ice (Movikov)

No. 18 (Feb. 1973) AIDJEX f i e l d operat ions, f a l l 1972 (Heiberg) Observations of i c e motion and i n t e r i o r flow f i e l d during 1971 AIDJEX

Diving r e p o r t , 1972 AIDJEX p i l o t s tudy (Welch, Par tch, Lee, and Smith) Some seasonal v a r i a t i o n s of t h e ice cover in t h e Beaufort Sea: evidence

Mapping t h e underside of a rc t ic sea ice by backsca t t e r ing sound (Berkson,

C i rcu la t ion of an incompressible ice cover (Rothrock) Crack propagation i n sea ice: a f i n i t e element approach (Mukherji) AIDJEX and the AGU Gymposium

p i l o t s tudy (Newton and Coachman)

of mac.roscale ice dynamics phenomena (DeRycke)

Clay, and Kan)

Some impressions of A I D J E X (Rooth) Abstracts

Determining t h e s t r e n g t h of sea ice s h e e t s (Mohaghegh)

No. 19 (Mar. 1973) S a l i n i t y v a r i a t i o n s i n sea ice (Cox and Weeks) 1972 AIDJEX i n t e r i o r flow f i e l d study:

Air f r i c t i o n and form drag on a r c t i c sea ice (Arya) Mechanical models of r i dg ing i n t h e arct ic sea ice cover (Parmerter and Coon) S p a t i a l v a r i a b i l i t y of topside and bottomside ice roughness and i t s

Abstracts AIDJEX Data Bank d i g i t a l d a t a i n d i c e s and prel iminary u s e r ' s guide

prel iminary r e p o r t and comparison wi th previous r e s u l t s (Nexton and Coachman)

relevance t o underside a c o u s t i c r e f l e c t i o n l o s s (Kozo and Diachok)

16 3