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Subgrid-Scale Models Universität Hannover
Institut für Meteorologie und Klimatologie Sonja Weinbrecht
Subgrid-Scale Models – an Overview
Sonja Weinbrecht
Institut für Meteorologie und KlimatologieUniversität Hannover
Subgrid-Scale Models Universität Hannover
Institut für Meteorologie und Klimatologie Sonja Weinbrecht
Structure
• What has to be parameterized ?
• Eddy diffusion models
• Dynamic models
• Mixed models
• Backscatter models
Subgrid-Scale Models Universität Hannover
Institut für Meteorologie und Klimatologie Sonja Weinbrecht
What has to be parameterized ?
jiij
jijiij
jijiij
ijijijjijiij
uuR
uuuuC
uuuuL
RCLuuuu
Leonard-stresses
cross-stresses
Reynolds-stresses
ijkk
ijkkijrij
p
3
13
1
*
j
rij
ikjijkij
iji
xguf
xx
uu
t
u
3
0
*1
Subgrid-Scale Models Universität Hannover
Institut für Meteorologie und Klimatologie Sonja Weinbrecht
Filtered strain rate tensor
Characteristic filtered rate of strain
eddy viscosity or turbulent viscosity
Smagorinsky coefficient
Productionterm of kinetic energy
Eddy-diffusion models – The Smagorinsky-model
2
22
2
2
2
1
2
SSSSP
SCSl
SSS
x
u
x
uS
S
ijijijrij
s
ijij
i
j
j
iij
ijrij
ijS
S
sC
P
Subgrid-Scale Models Universität Hannover
Institut für Meteorologie und Klimatologie Sonja Weinbrecht
The Smagorinsky-model (II)
• Cs is a constant here but actually varies for different types of flow
• The Smagorinsky-model is very dissipative
• Backscatter of energy from smaller to larger structures can not be
considered
• The model is only valid for isotropic turbulence
• The model overestimates the wind shear near the ground
Problems/Disadvantages:
Subgrid-Scale Models Universität Hannover
Institut für Meteorologie und Klimatologie Sonja Weinbrecht
8.1.
1.0.
: ,min
: 76.0,,min
2
),(),(
3/1
2/1
0
constF
constC
zyx
Fz
z
geFz
l
uue
txelCtx
m
s
s
s
ii
m
cases other
stratifiedstably
The Smagorinsky-model (III)
Modification by Deardorff (1980) – implemented in PALM:
Turbulent kinetic energy
Characteristic grid spacing
Wall adjustment factor
e
s
F
Subgrid-Scale Models Universität Hannover
Institut für Meteorologie und Klimatologie Sonja Weinbrecht
The Smagorinsky-model (IV)- Deardorff’s modification
• Prognostic equation for the turbulent kinetic energy has to be solved:
03
0
peu
xu
g
x
u
x
eu
t
ej
jj
iij
jj
se
e
e
je
jj
j
Δ
l..c
l
ec
K
x
eK
x
peu
x
740190
2
2/3
0
Subgrid-Scale Models Universität Hannover
Institut für Meteorologie und Klimatologie Sonja Weinbrecht
The Smagorinsky-model (V)
2)(
)(
)()(
2
22
*
*
22
*
ijijpm
ppTpT
pm
p
pm
ppT
ijijijij
ijTijij
SSzu
zzvzz
zu
wvwuz
z
uzz
SSSSP
SS
Modification by Sullivan et al (1994) – tested in PALM:The so-called two-part eddy viscosity model:
ijij
ijijijij
SSS
SSSSS
SS
S
2
2
Isotropy factor
eddy coefficient for inhomogeneous turbulence
denotes average over homogeneous directions
T
Subgrid-Scale Models Universität Hannover
Institut für Meteorologie und Klimatologie Sonja Weinbrecht
Dynamic Models (I)
• As prototype: model of Germano
et al. (1991)
• Needs filtering twice (grid filter and
test filter)
• u can be split into a resolved part
(I), a subgrid-scale part (III), and a
part on a scale between and
(I)
• Three stress tensors are defined
as shown (Lij can be directly
computed from the filtered velocity
components)
scales filter
filter test
filter grid
:~
:~
,,),(~
:,,),(
rdrGtrxutxu
rdrGtrxutxu
IIIIII
uuuuu )~
(~
jijiijijij
jijiij
jijiij
uuuuTL
uuuuT
uuuu
~~~
~~
~
Subgrid-Scale Models Universität Hannover
Institut für Meteorologie und Klimatologie Sonja Weinbrecht
Dynamic Models (II)
Advantages:
• Smagorinsky-coefficient
Csn is no longer
constant
• Csn can take negative
values, which could be
interpreted as
backscatter – but which
could also cause
problems with
numerical stability
klkl
ijij
sn
ijijij
ijsnijijsnijkkijrij
ijsnijkkijr
ij
ssn
ijsnijrij
MM
LMC
SSSSM
MCSSSSCLLL
SSCTTT
CC
SSCS
~~~2
~~~2
3
1
~~~2
3
1
22
22
22
2
2
2
Subgrid-Scale Models Universität Hannover
Institut für Meteorologie und Klimatologie Sonja Weinbrecht
Mixed Models
• E.g. Bardina et al. (1980)
• Assumption: the Smagorinsky-parameterization is only made
for Cij+ Rij
• The amount of Lij is explicitly added
ijsnijkkijrij SSCLL 2
modelBardina
23
1
Subgrid-Scale Models Universität Hannover
Institut für Meteorologie und Klimatologie Sonja Weinbrecht
Backscatter Models (I)• E.g. Mason and Thomson (1992), Schumann (1995)
• Energy transfer from smaller to larger scales is explicitly modeled
; 3
2
0 ; 3
2
2
ii
stijijjim
stij
stijij
rij
gev
RevvR
RS
1
exp
0
e
τ
tt)xxδ(δ)t,x,t)g(xg(
g
v
vij
i
Stochastic stress tensor
Random number
Characteristic correlation time
stijR
ig
v
Subgrid-Scale Models Universität Hannover
Institut für Meteorologie und Klimatologie Sonja Weinbrecht
Backscatter Models (II)
• γm is a parameter to describe the portion of random stress
• [kc,nkc] is the wavelength interval, where interaction takes place
• m is the spectrum slope
• For m=-5/3 and n = 2, γ = 0.9.
m
k
m
nk
k
m
m n
dkk
dkk
c
c
c 21
2
2
2 1
Subgrid-Scale Models Universität Hannover
Institut für Meteorologie und Klimatologie Sonja Weinbrecht
Comparison of two SGS-models in PALM
Dimensionless wind shear: on the left: SGS-model of Deardorff (1980); on the right: Dimensionless wind shear: on the left: SGS-model of Deardorff (1980); on the right: SGS-model of Sullivan et al (1994) – dashed line: theoretical solution, solid line: PALM SGS-model of Sullivan et al (1994) – dashed line: theoretical solution, solid line: PALM simulation results, dotted line: simulation results with the model of Moeng (1984).simulation results, dotted line: simulation results with the model of Moeng (1984).