  Stochastic Structural Stability Theory (S3T) provides analytical methods for
understanding the emergence and equilibration of jets from the turbulence in
planetary atmospheres based on the dynamics of the statistical mean state of
the turbulence closed at second order. Predictions for formation and
equilibration of turbulent jets made using S3T are critically compared with
results of simulations made using the associated quasi-linear and nonlinear
models. S3T predicts the observed bifurcation behavior associated with the
emergence of jets, their equilibration and their breakdown as a function of
parameters. Quantitative differences in bifurcation parameter values between
predictions of S3T and results of nonlinear simulations are traced to
modification of the eddy spectrum which results from two processes: nonlinear
eddy-eddy interactions and formation of discrete non-zonal structures.
Remarkably, these non-zonal structures, which substantially modify the
turbulence spectrum, are found to arise from S3T instability. Formation as
linear instabilities and equilibration at finite amplitude of multiple
equilibria for identical parameter values in the form of jets with distinct
meridional wavenumbers is verified as is the existence at equilibrium of finite
amplitude non-zonal structures in the form of nonlinearly modified Rossby
waves. When zonal jets and nonlinearly modified Rossby waves coexist at finite
amplitude the jet structure is generally found to dominate even if it is
linearly less unstable. The physical reality of the manifold of S3T jets and
non-zonal structures is underscored by the existence in nonlinear simulations
of jet structure at subcritical S3T parameter values which are identified with
stable S3T jet modes excited by turbulent fluctuations.
