Abstract

We propose a parameterization of 2D geophysical turbulence in the form of a relaxation equation similar to a generalized Fokker–Planck equation [P.H. Chavanis, Phys. Rev. E 68 (2003) 036108]. This equation conserves circulation and energy and increases a generalized entropy functional determined by a prior vorticity distribution fixed by small-scale forcing [R. Ellis, K. Haven, B. Turkington, Nonlinearity 15 (2002) 239]. We discuss applications of this formalism to jovian atmosphere and Jupiter’s great red spot. We show that, in the limit of small Rossby radius where the interaction becomes short-range, our relaxation equation becomes similar to the Cahn–Hilliard equation describing phase ordering kinetics. This strengthens the analogy between the jet structure of the great red spot and a “domain wall”. Our relaxation equation can also serve as a numerical algorithm to construct arbitrary nonlinearly dynamically stable stationary solutions of the 2D Euler equation. These solutions can represent jets and vortices that emerge in 2D turbulent flows as a result of violent relaxation. Due to incomplete relaxation, the statistical prediction may fail and the system can settle on a stationary solution of the 2D Euler equation which is not the most mixed state. In that case, it can be useful to construct more general nonlinearly dynamically stable stationary solutions of the 2D Euler equation in an attempt to reproduce observed phenomena.

Keywords: Long-range interactions; 2D turbulence; Vortex dynamics

PACS: 05.90.+m; 05.70.− a; 47.10.+g; 47.32.− y

Article Outline

1. Introduction

2. Summary of the statistical theories

2.1. The quasi-geostrophic equations

2.2. The statistical equilibrium state

2.3. Prior vorticity distribution and generalized entropy

2.4. Refined condition of nonlinear dynamical stability

3. A parameterization of geophysical flows

4. The limit of small Rossby radius

5. Conclusion

Appendix A. Relative entropy

Appendix B. Diffusion coefficient in the Q.G. model

Appendix C. Non-equilibrium distributions

References