palm model equations universität hannover institut für meteorologie und klimatologie sonja...
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![Page 1: PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut](https://reader035.vdocument.in/reader035/viewer/2022070305/5514eefa55034693478b5db7/html5/thumbnails/1.jpg)
PALM model equations Universität Hannover
Institut für Meteorologie und Klimatologie Sonja Weinbrecht
PALM – model equations
Sonja Weinbrecht
Institut für Meteorologie und KlimatologieUniversität Hannover
![Page 2: PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut](https://reader035.vdocument.in/reader035/viewer/2022070305/5514eefa55034693478b5db7/html5/thumbnails/2.jpg)
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
![Page 3: PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut](https://reader035.vdocument.in/reader035/viewer/2022070305/5514eefa55034693478b5db7/html5/thumbnails/3.jpg)
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
![Page 4: PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut](https://reader035.vdocument.in/reader035/viewer/2022070305/5514eefa55034693478b5db7/html5/thumbnails/4.jpg)
PALM model equations Universität Hannover
Institut für Meteorologie und Klimatologie Sonja Weinbrecht
Symbols
T
zyx
,(ix
wvu
,(iu
i
i
,,
)3,21
,,
)3,21
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
![Page 5: PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut](https://reader035.vdocument.in/reader035/viewer/2022070305/5514eefa55034693478b5db7/html5/thumbnails/5.jpg)
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
**
![Page 6: PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut](https://reader035.vdocument.in/reader035/viewer/2022070305/5514eefa55034693478b5db7/html5/thumbnails/6.jpg)
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
*
![Page 7: PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut](https://reader035.vdocument.in/reader035/viewer/2022070305/5514eefa55034693478b5db7/html5/thumbnails/7.jpg)
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
![Page 8: PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut](https://reader035.vdocument.in/reader035/viewer/2022070305/5514eefa55034693478b5db7/html5/thumbnails/8.jpg)
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
![Page 9: PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut](https://reader035.vdocument.in/reader035/viewer/2022070305/5514eefa55034693478b5db7/html5/thumbnails/9.jpg)
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
![Page 10: PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut](https://reader035.vdocument.in/reader035/viewer/2022070305/5514eefa55034693478b5db7/html5/thumbnails/10.jpg)
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
![Page 11: PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut](https://reader035.vdocument.in/reader035/viewer/2022070305/5514eefa55034693478b5db7/html5/thumbnails/11.jpg)
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
![Page 12: PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut](https://reader035.vdocument.in/reader035/viewer/2022070305/5514eefa55034693478b5db7/html5/thumbnails/12.jpg)
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
![Page 13: PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut](https://reader035.vdocument.in/reader035/viewer/2022070305/5514eefa55034693478b5db7/html5/thumbnails/13.jpg)
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
![Page 14: PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut](https://reader035.vdocument.in/reader035/viewer/2022070305/5514eefa55034693478b5db7/html5/thumbnails/14.jpg)
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
![Page 15: PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut](https://reader035.vdocument.in/reader035/viewer/2022070305/5514eefa55034693478b5db7/html5/thumbnails/15.jpg)
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
![Page 16: PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut](https://reader035.vdocument.in/reader035/viewer/2022070305/5514eefa55034693478b5db7/html5/thumbnails/16.jpg)
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
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
![Page 17: PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut](https://reader035.vdocument.in/reader035/viewer/2022070305/5514eefa55034693478b5db7/html5/thumbnails/17.jpg)
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
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
![Page 18: PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut](https://reader035.vdocument.in/reader035/viewer/2022070305/5514eefa55034693478b5db7/html5/thumbnails/18.jpg)
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:
![Page 19: PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut](https://reader035.vdocument.in/reader035/viewer/2022070305/5514eefa55034693478b5db7/html5/thumbnails/19.jpg)
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:
![Page 20: PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut](https://reader035.vdocument.in/reader035/viewer/2022070305/5514eefa55034693478b5db7/html5/thumbnails/20.jpg)
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
q
K
622.01
622.0116.11
61.01
1
![Page 21: PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut](https://reader035.vdocument.in/reader035/viewer/2022070305/5514eefa55034693478b5db7/html5/thumbnails/21.jpg)
PALM model equations Universität Hannover
Institut für Meteorologie und Klimatologie Sonja Weinbrecht
Cloud Physics (IV)
)(
61.01
0 zp
p
T
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:
![Page 22: PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut](https://reader035.vdocument.in/reader035/viewer/2022070305/5514eefa55034693478b5db7/html5/thumbnails/22.jpg)
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
![Page 23: PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut](https://reader035.vdocument.in/reader035/viewer/2022070305/5514eefa55034693478b5db7/html5/thumbnails/23.jpg)
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)
![Page 24: PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut](https://reader035.vdocument.in/reader035/viewer/2022070305/5514eefa55034693478b5db7/html5/thumbnails/24.jpg)
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
qqqq
t
q
![Page 25: PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut](https://reader035.vdocument.in/reader035/viewer/2022070305/5514eefa55034693478b5db7/html5/thumbnails/25.jpg)
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