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**Subgrid-Scale Models – an Overview**

Sonja Weinbrecht Institut für Meteorologie und Klimatologie Universität Hannover

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

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**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

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**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

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**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

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**The Smagorinsky-model (III)**

Modification by Deardorff (1980) – implemented in PALM: Turbulent kinetic energy Characteristic grid spacing Wall adjustment factor

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**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

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**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

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**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)

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**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

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**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

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**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

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**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.

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**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).

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