  Planetary turbulent flows are observed to self-organize into large scale
structures such as zonal jets and coherent vortices. One of the simplest models
of planetary turbulence is obtained by considering a barotropic flow on a
beta-plane channel with turbulence sustained by random stirring. Non-linear
integrations of this model show that as the energy input rate of the forcing is
increased, the homogeneity of the flow is broken with the emergence of
non-zonal, coherent, westward propagating structures and at larger energy input
rates by the emergence of zonal jets. We study the emergence of non-zonal
coherent structures using a non-equilibrium statistical theory, Stochastic
Structural Stability Theory (S3T, previously referred to as SSST). S3T directly
models a second order approximation to the statistical mean turbulent state and
allows identification of statistical turbulent equilibria and study of their
stability. Using S3T, the bifurcation properties of the homogeneous state in
barotropic beta-plane turbulence are determined. Analytic expressions for the
zonal and non-zonal large scale coherent flows that emerge as a result of
structural instability are obtained. Through numerical integrations of the S3T
dynamical system, it is found that the unstable structures equilibrate at
finite amplitude. Numerical simulations of the nonlinear equations confirm the
characteristics (scale, amplitude and phase speed) of the structures predicted
by S3T.
