  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 (viz. temperature).
