# Statistical Dynamical Closures and Subgrid Modeling for Inhomogeneous QG and 3D Turbulence

## Abstract

**:**

## 1. Introduction

## 2. Flow over Topography in a Baroclinic Quasigeostrophic Model

## 3. Navier Stokes Equations for Three-Dimensional Inhomogeneous Turbulence

## 4. Quasi-Diagonal DIA Closure Equations

## 5. Langevin Equation for QDIA Closure

## 6. Subgrid-Scale Parameterizations

_{T}to C

_{R}< C

_{T}, where C

_{R}is the resolution of the resolved scales. In the previous sections, the summations over

**p**and

**q**are such that $p\le {C}_{T},q\le {C}_{T}$ or $(p,q)\in \mathcal{T}$ where the set:

**p**and

**q**are subgrid:

**p**and

**q**, can then be split into resolved scale terms for which $(p,q)\in \mathcal{R}$and subgrid scale terms for which $(p,q)\in \mathcal{S}$. For $(p,q)\in \mathcal{S}$we define ${\eta}_{k}^{\mathbf{K}ab}(t,s),{f}_{H}^{\mathbf{K}ab}(k,t),{\chi}_{k}^{\mathbf{K}ab}(t,s)$ by right hand side of Equations (18a), (18b), (18c), ${S}_{k}^{\mathbf{K}ab}(t,s),{\pi}_{k}^{\mathbf{K}ab}(t,s),{P}_{k}^{\mathbf{K}ab}(t,s)$ by (23b), (23c), (23d), and ${f}_{S}^{\mathbf{K}ab}(k,t)and{f}_{P}^{\mathbf{K}ab}(k,t)$by Equations (27a) and (27b) respectively. Similar expressions with superscript $\mathcal{R}$ may be defined for$(p,q)\in \mathcal{R}$.

#### 6.1. Mean Field

#### 6.2. Fluctuating Field

#### 6.3. Response Function and Covariance

#### 6.4. Langevin Equation for QDIA with Subgrid-Scale Parameterizations

## 7. Effective Dissipation and Viscosity Parameterizations

**k**. This is the case, in particular, near canonical equilibrium.

## 8. Discussion and Conclusions

#### 8.1. Quasi-Diagonal LET and SCFT Inhomogeneous Closures

#### 8.2. Regularization and Non-Gaussian initial Conditions

#### 8.3. Concluding Comments

## Acknowledgements

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## Appendix A: Perturbation Theory

## Appendix B: Quasigeostrophic Model with Continuous Vertical Variations

## Appendix C: Quasigeostrophic Model in Terms of Barotropic and Baroclinic Components

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Frederiksen, J.S.
Statistical Dynamical Closures and Subgrid Modeling for Inhomogeneous QG and 3D Turbulence. *Entropy* **2012**, *14*, 32-57.
https://doi.org/10.3390/e14010032

**AMA Style**

Frederiksen JS.
Statistical Dynamical Closures and Subgrid Modeling for Inhomogeneous QG and 3D Turbulence. *Entropy*. 2012; 14(1):32-57.
https://doi.org/10.3390/e14010032

**Chicago/Turabian Style**

Frederiksen, Jorgen S.
2012. "Statistical Dynamical Closures and Subgrid Modeling for Inhomogeneous QG and 3D Turbulence" *Entropy* 14, no. 1: 32-57.
https://doi.org/10.3390/e14010032