Abstract
We study the thermodynamic equilibrium spectra of the Charney–Hasegawa–Mima (CHM) equation in its weakly nonlinear limit. In this limit, the equation has three adiabatic invariants, in contrast to the two invariants of the 2D Euler or Gross–Pitaevskii equations, which are examples for comparison. We explore how the third invariant considerably enriches the variety of equilibrium spectra that the CHM system can access. In particular we characterise the singular limits of these spectra in which condensates occur, i.e. a single Fourier mode (or pair of modes) accumulate(s) a macroscopic fraction of the total invariants. We show that these equilibrium condensates provide a simple explanation for the characteristic structures observed in CHM systems of finite size: highly anisotropic zonal flows, large-scale isotropic vortices, and vortices at small scale. We show how these condensates are associated with combinations of negative thermodynamic potentials (e.g. temperature).
Highlights
It is well known that the turbulent flow of statistically isotropic ideal fluids in two dimensions (2D) conserves the total kinetic energy E and the total mean square vorticity, or enstrophy Ω
Kraichnan [1, 6] considered continuous vorticity fields by examining the 2D Euler equation with Fourier truncations at the small and large scale. He established that their statistical equilibria predicted the large-scale accumulation of energy, with the equilibrium being parameterised by a negative temperature, see review [7]
In the present paper we offer an alternative explanation of the tendency for zonal modes to condense energy, based on the statistical equilibria of quasi2D flows with a β effect in the weakly nonlinear limit, and show that these too predict the condensation of energy into zonal flows
Summary
It is well known that the turbulent flow of statistically isotropic ideal fluids in two dimensions (2D) conserves the total kinetic energy E and the total mean square vorticity, or enstrophy Ω. One of the earliest formulations is the simple and robust argument of Fjørtoft [2], whose conclusion can be expressed as a principle involving the sign-definite global invariants of the flow: each such invariant is pushed by all the others towards the sector of Fourier space where its spectral weight is greatest [3, 4] This statement, regarding the dynamics of 2D flow, is reflected in the equilibrium statistical mechanics of 2D turbulence. As is common in WT theory, including in [22], we work here in the microcanonical ensemble, considering an isolated system evolving from an initial condition and finding a unique equilibrium through ergodic dynamics This is in contrast to Kraichnan’s analysis of 2D turbulence which took place in the grand canonical ensemble, see [23] for a recent microcanonical treatment of the Fourier-truncated 2D Euler equation. In the appendix we recapitulate results on infinite-sized GPE systems in 3D and 2D in slightly more detail, as well as apply the results of this work to the 2D GPE in finite-sized systems
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