Thermal stratification, organized motion, and the onset of counter-gradient flows within canopies

Thermal stratification, organized motion, and the onset of counter-gradient flows within canopies

Daniela Cava1, Gabriel Katul2, Antonio Scrimieri1,3, Davide Poggi2,4, Alessandro Cescatti5, and

Umberto Giostra6

1CNR - Institute of Atmosphere Sciences and Climate section of Lecce, Lecce, Italy.

2Nicholas School of the Environment and Earth Sciences, Duke University, Durham, N.C., U.S.A.

3Material Science Department, University of Lecce, Lecce, Italy.

4Dipartimento di Idraulica, Trasporti ed Infrastrutture Civili, Politecnico di Torino, Torino, Italy.

5Centro di Ecologia Alpina, 38040 Viote del Monte Bondone (Trento), Italy

6Environmental Science Department, University of Urbino, Urbino, Italy

Update on Daniela

Out of the hospital And back to Lecce

Historical PerspectiveIn 1988, Raupach1 concluded his review by

noting that the reasons for the failure of gradient-diffusion theory inside uniform canopy on flat terrain are now understood….

“looking further ahead, several thorny problems await systematic investigation…., the problem of buoyancy effects …..”

1Raupach, M., 1988, Canopy Transport Processes, in Flow and Transport in the Natural Environment: Advances and Applications Ed. W.L. Steffen and O.T. Denmead, Springer Verlag, pp.95-127.

A decade laterIn 1998, Mahrt2 ended his review on the Stable

Boundary Layer by noting that: “formulation of turbulence in the very stable

boundary layer is uncertain and the stable boundary layer contains a number of physical influences not present in any of the existing models….”

“…even small future advances justify more work”.

2Mahrt, L., 1998, Stratified Atmospheric Boundary Layers and Breakdown of Models, Theoret. Comput. Fluid Dynamics, 11: 263–279.

IntroductionFailure of gradient-diffusion theory inside

canopies is often linked to three inter-related factors:

Variable scalar source distribution within the canopy strongly impacts the apparent diffusivity (near-field effects).

Lack of local balance between turbulent production and dissipation.

Vertical transport occurs by organized eddy motion whose size is comparable to the canopy height.

Objective Investigate the interplay between

ejections and sweeps (often used as signatures of large-scale organized motion) and local thermal stability in the onset of zero- or counter-gradient flows inside canopies.

A necessary first step towards a ‘small advance’.

25 m

33 m

17.5 m11 m

4 m

Mixed hardwood forest Lavarone, Italy

Aerial View of the Site

Tower

Lavarone Experiments

Unstable

Near-neutral

Stable

Theory – 1:Budget Equations

Mean Continuity Equation:

Heat Flux – Budget Equation:

TSz

Tw

t

T

0

22 10 VT

T

g

z

pT

z

Tww

z

Tw

t

Tw

Theory – 2:Closure ModelsFlux-Gradient Closure for Triple Moment: (e.g. Donaldson, 19731)

Scalar-Pressure Interaction (Andre et al., 19792)

ST

5

w Tw w T C w w

z

24

1

3

p w T gT C T

z T

1Donaldson, C., 1973, Construction of a dynamic model for the production of atmospheric turbulence and the dispersal of atmospheric pollutants, in Workshop on Micrometeorology, American Meteorological Society, 313-392. 2Andre, J.C., G. De Moor, P. Lacarrere, G. Therry, and R. du Vachat, 1979, The clipping approximation and inhomogeneous turbulence simulations, Turbulent Shear Flows – I, Springer Verlag, 307-318.

Result:Second-Order Closure Model Result

‘Near field’Effects from canopy heat source vertical variations

Buoyancyeffects (+ve)

ProductionTerm

Model is not explicit in terms of ejection-sweep cycle

2 2 25

4

4

3T V

T gw T w C w S T

C z z T

w w T

Organized Motion

The interest in ejections and sweeps dates back to early experiments by Kline et al. (1967) who demonstrated via flow visualization that fluid motion near a wall is “far from being completely chaotic in nature” revealing a definite “sequence of ordered motion”.

Conditional Sampling, Ejections- Sweeps & Quadrant Analysis

W’

U’

Ejections

Sweeps

Quadrant 2

Quadrant 4

Frenkiel and Klebanoff (1967), Lu and Willmarth (1973), Antonia (1981) – Conditionalsampling and quadrant analysisRelative contribution

of ejections and sweeps toMomentum flux:

Nakagawa and Nezu (1977)& Raupach (1981):

24o

u w u wS

u w

Connecting the Ejection-Sweep Cycle with the Flux-Transport

Using 3rd order cumulant expansion methods (CEM) in Nakagawa and Nezu (1977) and Raupach (1981), and a sensitivity analysis in Katul et al. (1997), an ‘Incomplete CEM’ (or ICEM) was proposed by Poggi et al. (2004).

21 12

1

2 2o

uw

S M MR

Validation of ICEM

From Katul, G.G., D. Poggi, D. Cava, and and J.J. Finnigan, 2005, BLM, in Press

Further Simplification

*Value from a Pine Forest experiment

*21 12| | ; 0.6M C M C

From Cava et al. (2006)

Sweeps dominate heat flux0TS

w T

0TS

w T

Ejections dominate heat flux

Equate Gradient-Diffusion Model and ICEM representation of w w T

5(1 ) 1

2 2o w T

C CS S

w T

z/hc

2

chTw

Tw

1

00

DaytimeNightime

0;0

dz

TwdTw

0;0

dz

TwdTw

Ejections

Sweeps

All Sweeps

0;0

dz

TwdTw

Canonical Profiles From Kaimal and Finnigan (1994)

Prognostic Equations

From Poggi et al. (2006) – Flume experiments

From Katul et al. (2001) – IREX [rice]and Siqueria and Katul (2002) - pine

From triple moments

22 24 4

( )3

tt w T

K Cw T w T T gK w T z

z z z z z T

25 ( )

wtK C z ( )c

w

h

z

0.1

4 52.9; 2.5C C

Three Variants on the Model

K-theory [Dummy model]

No buoyancy

Full Model

24 ( )w

C Tw T z

z

224 ( )t

t w

K Cw T w T TK w T z

z z z z z

22 24 4

( )3

tt w T

K Cw T w T T gK w T z

z z z z z T

Lavarone Experiments

Unstable

Neutral

Stable

DataK-theory

g=0

Fullmodel

Uh

(m

s-1

)012345

-1 .5-1

-0 .50

0 .51

1 .5

w' (

m s

-1)

T' (

K)

-1 .2-0 .8-0 .400 .40 .81 .2

-80

-40

0

40

80

CO

2' (

mg

m-3

)

q' (

mg

m-3

)

- 6- 4- 20246 x 1 0 -4

T im e (m in u tes)

-50-40-30-20-10

0

RN

(W

m-2

)

0 4 8 12 16 20 24 28

a )

b )

c )

d )

e )

f)

Thorny issue1:

Very Stable Conditions1) Flow is not high

Reynolds # (and Peclet #..)

2) Flow is not stationary

3) Standard turbulence closure scheme that assume independence of Reynolds number need not apply.

4) Flow may not be independent of transients in the upper boundary conditions

1From Cava et al. (2004)

DUKE FOREST

Conclusions Two analytical expressions relating

For neutral to slightly stable flows, neglecting the buoyancy contribution is preferred given that the temperature variance is always finite.

, , , ,p T T w

Tw T f S

z

Prognostic

, , , ,d o T w

Tw T f S

z

Diagnostic

Publications

•Cava et al. , 2006, Buoyancy and the sensible heat flux budget within dense canopies, Boundary Layer Meteorology, to appear.

•Poggi, et al. 2006, Scalar Dispersion within a Model Canopy: Measurements and Three-Dimensional Lagrangian Models, Advances in Water Resources , to appear

•Katul, G.G., D. Poggi, D. Cava, and J.J. Finnigan, 2006, The relative importance of ejections and sweeps to momentum transfer in the atmospheric boundary layer. Boundary Layer Meteorology, to appear.

•Cava et al., 2004, Organized motion and radiative perturbations in the nocturnal canopy sublayer above an even-aged pine forest, Boundary-Layer Meteorology, 112, 129-15