palm model equations universität hannover institut für meteorologie und klimatologie sonja...

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie Sonja Weinbrecht

PALM – model equations

Sonja Weinbrecht

Institut für Meteorologie und KlimatologieUniversität Hannover



Structure

• Basic equations

• Boussinesq-approximation and filtering

• SGS-parameterization

• Prandtl-layer

• Cloud physics

• Boundary conditions



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



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



• Pressure p, density ρ, and

temperature T are split into a basic

part ()0, which only depends on height

(except from p), and a deviation from

it ()*, which is small compared with

the basic part

• The basic pressure p0 fulfills the

equations shown on the right.

Boussinesq-approximation (I)

00

0

0

0

0

1

1

gz

p

fuy

p

fvx

p

g

g

00

0**

0

0**

0

;

;

;

TTT TT

ppppp

**



• Neglecting the density variations in all terms except from the

buoyancy term

• Density variations are replaced by potential temperature variations

Boussinesq-approximation (II)

0

1

0

0

0

0

2

2

30

**

033

k

k

x

u

k

k

k

ii

ikkikjijk

k

ik

i

x

u

x

u

t

x

ug

x

pufuf

x

uu

t

u

k

k

geo

0

*

0

*



Filtering the equations (I)

ψψψψψ

uuuuu iiiii

;

;

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

• By filtering, a turbulent diffusion term comes into being

• compared with the turbulent diffusion term the molecular diffusion term

can be neglected

k

iki

ikkikjijk

k

ik

i

xg

x

pufuf

x

uu

t

ugeo

3

0

**

033

1

jijiij uuuuτ 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

30

*

033

1

3

1

3

1

3

1

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

part:

rij anisotropic SGS stress tensor



The filtered equations

Qx

u

xu

t k

k

kk

k

rik

ii

kkikjijkk

ik

i

xg

xufuf

x

uu

t

ugeo

3

0

*

033

1

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

Qx

u

xu

t k

k

kk



The parameterization model (I) (Deardorff, 1980)

),(),(

2

txelCtx

S

mm

ijmrij

1.0.

2

2

1

constC

uue

x

u

x

uS

m

ii

i

j

j

iij

strain rate tensor

turbulent kinetic energy

anisotropic SGS stress tensor

eddy viscosity for momentum



The parameterization model (II) (Deardorff, 1980)

ms

h

ihi

ltx

xu

21),(

8.1.

else: ,min

stratifiedstably : 76.0,,min

3/1

2/1

0

constF

zyx

Fz

z

geFz

l

s

s

s

Mixing length

Characteristic grid spacing

Wall adjustment factor

s

F

l

Eddy viscosity for heat



The parameterization model (III)

• Prognostic equation for the turbulent kinetic energy has to be solved:

00

peu

xw

g

x

u

x

eu

t

ej

jj

iij

jj

see

ej

ej

jj

lc

l

ec

Kx

eK

x

peu

x

74.019.0 ;

2 ;

2/3

0



The Prandtl-layer (I)

h

m

zz

z

u

z

u

*

*

*

0*

00*

u

w

uwu

velocity- and temperature gradients in the Prandl-layer

Φm and Φh are the Dyer-Businger functions for

momentum and heat

friction velocity

characteristic temperature in the Prandtl-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



Cloud Physics (I)

• Suppositions: liquid water content and water vapor are in thermodynamic equilibrium

• All thermodynamic processes are reversible

• Potential liquid water temperature θl and total water content q as prognostic variables

• For moist adiabatic processes, θl and q are conserved.

• Condensation and evaporation do not have to be explicitly described

• Only totally saturated or totally unsaturated grid cells are allowed in the model

lv

lp

l

qqq

qTc

L



Cloud Physics - Symbols

s

l

r

v

l

sv

l

E

T

.κ

p

L

q

qq

q

2860

hPa 1000

,

potential liquid water temperature

total water content

specific humidity, s.h. for saturated air

liquid water content

latent heat / vaporization enthalpy

virtual potential temperature

pressure (reference value)

adiabatic coefficient

actual liquid water temperature

saturation vapour pressure



Cloud Physics (II)

q

iq

i

i

i

l

i

lil

Qx

q

x

qu

t

q

Qxx

u

t ll

2

2

2

2

prec

precrad

t

qQ

ttQ

q

l

i

ll

Filtered equations of θl and q (molecular diffusion term neglected)

Equations of θl and q:

prec

precrad

t

q

x

H

x

qu

t

q

ttx

W

x

u

t

i

i

i

i

l

i

l

i

i

i

lil

ququH

uuW

iii

lilii



Cloud Physics (III)

• only totally saturated or totally unsaturated grid cells are allowed in the model

ref

)(

622.0

p

zpT

Tc

L

TR

L

ll

lpl

ls

lsls

lslss

TEzp

TETq

Tq

qTqq

377.0)(622.0

1

1

86.35

16.273269.17exp78.610

l

lls T

TTE

0

if ; TqqTqqq

ss

l

• the specific humidity for saturated air qs is computed as follows:



Cloud Physics – SGS-Parameterization

• The buoyancy-term in the TKE-equation is modified (the potential

temperature θ is replaced by the virtual potential temperature θv):

iHi

i

lWi

x

qvH

xvW

Hw

mH vs

lv

21

03,

0

peu

xH

g

x

u

x

eu

t

ej

jv

vj

iij

jj

• Parameterisation of Wi and Hi:



Cloud Physics – SGS-Parameterization (II)

1

61.0

1

2K

Tc

LK

p

• Hv,3: subgrid-scale vertical flux

of virtual potential temperature

(buoyancy flux)

• K1, K2: Coefficients for

• unsaturated moist air

• and saturated moist air

respectively

vv wWKHKH 32313,

sp

s

qTc

LRTL

RTL

qq

q

K

622.01

622.0116.11

61.01

1



Cloud Physics (IV)

)(

61.01

0 zp

p

T

qq

r

lvv

llp

lv qqqTc

L61.161.01

• Prognosticating θl and q, the virtual potential temperature θv and the quotient of potential and actual temperature θ/T have to be computed as follows:



Cloud Physics – Radiation Model

)()(),()()(

)0()()0,()0()(

1

toptoptop

0rad

zFzBzzzFzF

BzBzBzF

FFF

zFzFzcTt p

l

• based on the work of Cox (1976)

• vertical gradients of radiant flux as sources of energy

• the downward radiation at the top of the model is prescribed



Cloud Physics – Radiation Model - Symbols

1212

021

2121

2121

kgm 158 ;kgm 130

LWP

LWPexp1

LWPexp1,

,

2

1

ba

qdzzz

,zzb,zzε

,zzazzε

zB

zFzF

z

z l

upward and downward radiant fluxes

black body radiation

cloud emissivities between z1 and z2

Liquid Water Path

mass exchange coefficient (empirical data)



Cloud Physics – Precipitation Model

• Kessler scheme (Kessler, 1969)

• only autoconversion (production of rain by coalescence) is considered

• precipitation starts when a threshold value qlt is exceeded

• τ is a retarding time constant

1

precprec

prec

gkg 05.0

; 0

;

l

p

l

ltl

ltlltl

q

t

q

Tc

L

t

qq

qqqq

t

q



Boundary conditions

000

0,

1,0,00,

max

kwzw

nkvu

kvukvuzvu

0max0

max

max

0

s ;0

0 ; 10

10

init

prescribed is re temperatusurface if ;0

fluxheat constant if ; 10

snkz

sks

nkpkpkp

keke

znk

z

k

kk

z

palm model equations universität hannover institut für meteorologie und klimatologie sonja...

Documents

parameterization model

momentum slide

heat slide

filtered equations flux

passive scalar slide

prandtllayer slide

continuity equation

prognostic equation