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

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie Sonja Weinbrecht

Structure

• Basic equations

• Boussinesq-approximation and filtering

• SGS-parameterization

• Prandtl-layer

• Cloud physics

• Boundary conditions

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie Sonja Weinbrecht

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

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie Sonja Weinbrecht

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 Sonja Weinbrecht

• 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

**

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie Sonja Weinbrecht

• 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

*

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie Sonja Weinbrecht

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

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie Sonja Weinbrecht

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

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie Sonja Weinbrecht

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

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie Sonja Weinbrecht

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

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie Sonja Weinbrecht

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

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie Sonja Weinbrecht

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

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie Sonja Weinbrecht

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

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie Sonja Weinbrecht

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 Sonja Weinbrecht

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

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie Sonja Weinbrecht

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

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie Sonja Weinbrecht

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

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie Sonja Weinbrecht

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:

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie Sonja Weinbrecht

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:

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie Sonja Weinbrecht

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

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie Sonja Weinbrecht

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:

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie Sonja Weinbrecht

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

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie Sonja Weinbrecht

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)

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie Sonja Weinbrecht

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

PALM model equations Universität Hannover

Institut für Meteorologie und Klimatologie Sonja Weinbrecht

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