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).