  In geophysical and plasma contexts, zonal flows are well known to arise out
of turbulence. We elucidate the transition from statistically homogeneous
turbulence without zonal flows to statistically inhomogeneous turbulence with
steady zonal flows. Starting from the Hasegawa--Mima equation, we employ both
the quasilinear approximation and a statistical average, which retains a great
deal of the qualitative behavior of the full system. Within the resulting
framework known as CE2, we extend recent understanding of the symmetry-breaking
`zonostrophic instability'. Zonostrophic instability can be understood in a
very general way as the instability of some turbulent background spectrum to a
zonally symmetric coherent mode. As a special case, the background spectrum can
consist of only a single mode. We find that in this case the dispersion
relation of zonostrophic instability from the CE2 formalism reduces exactly to
that of the 4-mode truncation of generalized modulational instability. We then
show that zonal flows constitute pattern formation amid a turbulent bath.
Zonostrophic instability is an example of a Type I$_s$ instability of
pattern-forming systems. The broken symmetry is statistical homogeneity. Near
the bifurcation point, the slow dynamics of CE2 are governed by a well-known
amplitude equation, the real Ginzburg-Landau equation. The important features
of this amplitude equation, and therefore of the CE2 system, are multiple.
First, the zonal flow wavelength is not unique. In an idealized, infinite
system, there is a continuous band of zonal flow wavelengths that allow a
nonlinear equilibrium. Second, of these wavelengths, only those within a
smaller subband are stable. Unstable wavelengths must evolve to reach a stable
wavelength; this process manifests as merging jets. These behaviors ...
