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Stochastic Structural Stability Theory (S3T) provides analytical methods forunderstanding the emergence and equilibration of jets from the turbulence inplanetary atmospheres based on the dynamics of the statistical mean state ofthe turbulence closed at second order. Predictions for formation andequilibration of turbulent jets made using S3T are critically compared withresults of simulations made using the associated quasi-linear and nonlinearmodels. S3T predicts the observed bifurcation behavior associated with theemergence of jets, their equilibration and their breakdown as a function ofparameters. Quantitative differences in bifurcation parameter values betweenpredictions of S3T and results of nonlinear simulations are traced tomodification of the eddy spectrum which results from two processes: nonlineareddy-eddy interactions and formation of discrete non-zonal structures.Remarkably, these non-zonal structures, which substantially modify theturbulence spectrum, are found to arise from S3T instability. Formation aslinear instabilities and equilibration at finite amplitude of multipleequilibria for identical parameter values in the form of jets with distinctmeridional wavenumbers is verified as is the existence at equilibrium of finiteamplitude non-zonal structures in the form of nonlinearly modified Rossbywaves. When zonal jets and nonlinearly modified Rossby waves coexist at finiteamplitude the jet structure is generally found to dominate even if it islinearly less unstable. The physical reality of the manifold of S3T jets andnon-zonal structures is underscored by the existence in nonlinear simulationsof jet structure at subcritical S3T parameter values which are identified withstable S3T jet modes excited by turbulent fluctuations.

Emergence and equilibration of jets in beta-plane turbulence: applications of Stochastic Structural Stability Theory