PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka PALM – model equations.

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka

PALM – model equations

Palm-SeminarZingst July 2004

Micha Gryschka

Institut für Meteorologie und KlimatologieUniversität Hannover

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka

Structure

• Basic equations

• Boussinesq-approximation and filtering

• poisson equation for pressure

• Prandtl-layer

• how Cloud physics is imbedded

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka

Symbols

T

zyx

,(ix

wvu

,(iu

i

i

,,

)3,21

,,

)3,21

QQ

f

gz

p

ijk

i

,

,

velocity components

spatial coordinates

potential temperature

passive scalar

actual temperature

pressure

density

geopotential height

Coriolis parameter

alternating symbol

molecular diffusivity

sources or sinks

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka

Basic equations

k

k

x

u

t

k

k

ik

i

ikjijk

ik

ik

i

x

u

xx

u

xuf

x

p

x

uu

t

u

3

112

2

1. Navier-stokes equations

3. continuity equation

Qxx

ut k

hk

k

2

2

2. First principle of thermodynamics and equation for any passive scalar ψ

Qxx

ut kk

k

2

2

• Variety of solutions

• some solutions are not importent for meteorological questions

• some solutions cost a lot computer power (f.e. sonic waves decrease the timestep)

meteorological meaningfull simplifications

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka

Boussinesq-approximation (I)

000

00

0

0

0

1

, 1

RTp

gz

pfu

y

pfv

x

pgg

0**

0

*0

; t)z,y,(x,)(),,,(

t)z,y,(x,),,(),,,(

ztzyx

pzyxptzyxp

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka

• in the horizontal components: terms with * are negligible

in the vertikal component: term g */ 0 is not neglibible

• replacing

Boussinesq-approximation (II)

2

2

30

**

033

1

k

ii

ikkikjijk

k

ik

i

x

ug

x

pufuf

x

uu

t

ugeo

0

*

0

*

in case of shallow convection!

incrompressible (divergence free) flow (no solution for acustic waves)

0

0

0

0

0

k

k

x

u

k

k

x

u

x

u

t

k

k

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka

Are the equations directly solveable?

• Numerical solving of the equations implies discrete solving on a grid

• If the grid is small enough, the equations could be discretized directly

• In LES the grid is not small enough

• Equations have to be filtered:

– large structures, resolved from the grid (and timestep)

– small structures, unresolved from the grid (and timestep)

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka

Filtering the equations (I)

ψψψψψ

uuuuu iiiii

;

;

• Splitting the variables into mean part ( ¯ ) and deviation ( )’

• By filtering, a turbulent diffusion term comes into being

k

iki

ikkikjijk

k

ik

i

xg

x

pufuf

x

uu

t

ugeo

03

0

**

033

11

jiij uuτ '' subgrid-scale (SGS) stress tensor

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka

Filtering the equations (II)

k

rik

ii

kkikjijkk

ik

i

kk*

ijkkijrik

rikijkkij

xg

xufεufε

x

uu

t

u

τpπ

δττττδττ

geo

03

0

*

033

11

3

13

1

3

1

• The SGS stress tensor is splitted into an isotropic and an anisotropic

part:

rij anisotropic SGS stress tensor

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka

Qx

u

xu

t k

k

kk

Qx

u

xu

t k

k

kk

k

rik

ii

kkikjijkk

ik

i

xg

xufuf

x

uu

t

ugeo

03

0

*

033

11

The filtered equations

0

k

k

x

u

1. Boussinesq-approximated Reynolds equations for incompressible flows

3. continuity equation for incompressible flows

2. First principle of thermodynamics and equation for any passive scalar

: has to be parametrized

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka

1. Calculating a preliminary velocity field without considering the pressure term

2. Solving the poisson equation

3. Correcting the velocitiy field with considering the pressure term

The Poisson-equation (I)

• This strategy connects the continuity equation with the motion equations,

so it's guaranteed that the flow is divergence free.

• Solving the poisson equation is one of the most costs of computer power!

ii x

u

txi

pre

02

2

After filtering and parametrizing, only the pressure is unknown.

Solution: considering the continuity equation

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka

The Poisson-equation (II)origin of the poisson equation:

ii x

ut

ut i

0pre

1

ix

2

2

0

pre 1

iii

i

xx

u

tx

u

ti

0

ii x

u

txi

pre

02

2

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka

The Prandtl-layer (I)

h

m

zz

z

u

z

u

*

*

*

0*

00*

u

w

uwu

Φm and Φh: Dyer-Businger functions

friction velocity

characteristic temperature in the Prandtl-layer

between ground and first grid layer

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka

The Prandtl-layer (II)

zu

uw

θwθ

g

0~Rif Richardson flux number

Dyer-Businger

functions for

momentum

and heat

2/1

4/1

Rif 16-1

1

Rif 51

Rif 16-1

1

Rif 51

h

m

stable stratification

neutral stratification

unstable stratification

stable stratification

neutral stratification

unstable stratification

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka

how Cloud Physics is imbedded prognostic equation for ui

connection of dynamics and cloud physics via temperature

v ( qv , ql , l )

prognostic equation for l

sources and sinks:

(t l)rad , (t l)prec

prognostic equation for q

sources and sinks:

(t q)prec

qv = q - ql

cloud physics model