1. Subgrid-Scale Models – an Overview
Sonja Weinbrecht
Institut für Meteorologie und Klimatologie Universität Hannover

2. Structure
What has to be parameterized ?
Eddy diffusion models
Dynamic models
Mixed models
Backscatter models

3. What has to be parameterized ?
Leonard-stresses
cross-stresses
Reynolds-stresses
LES-Probleme in Bodennähe – SGS-Modelle
Quantitativer Vergleich sowohl zur Evaluierung von LES-Modell prinzipiell, als auch (im zweiten Schritt) zur Bewertung verschiedener SGS-Modelle

4. Eddy-diffusion models – The Smagorinsky-model
Filtered strain rate tensor
Characteristic filtered rate of strain
eddy viscosity or turbulent viscosity
Smagorinsky coefficient
Productionterm of kinetic energy

5. The Smagorinsky-model (II)
Problems/Disadvantages:
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

6. The Smagorinsky-model (III)
Modification by Deardorff (1980) – implemented in PALM:
Turbulent kinetic energy
Characteristic grid spacing
Wall adjustment factor

7. The Smagorinsky-model (IV)- Deardorff’s modification
Prognostic equation for the turbulent kinetic energy has to be solved:
C-e-definition is no longer the original Deardorff-model but was modified by Moeng and Wyngaard

8. The Smagorinsky-model (V)
Modification by Sullivan et al (1994) – tested in PALM:
The so-called two-part eddy viscosity model:
Isotropy factor
eddy coefficient for inhomogeneous turbulence
denotes average over homogeneous directions

9. 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)

10. Dynamic Models (II) Smagorinsky-coefficient Csn is no longer constant
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

11. 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

12. Backscatter Models (I)
E.g. Mason and Thomson (1992), Schumann (1995)
Energy transfer from smaller to larger scales is explicitly modeled
Stochastic stress tensor
Random number
Characteristic correlation time

13. 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.

14. Comparison of two SGS-models in PALM
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 simulation results, dotted line: simulation results with the model of Moeng (1984).