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



Structure

• Basic equations

• Boussinesq-approximation and filtering

• poisson equation for pressure

• Prandtl-layer

• how Cloud physics is imbedded



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



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



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



• 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



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)



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



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



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



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



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



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



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



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

Documents

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