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 be- tween 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 nonzonal struc- tures. Remarkably, these nonzonal 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 nonzonal 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 nonzonal structures is underscored by the existence in nonlinear simulations of jet structure at subcritical S3T parameter values that are identified with stable S3T jet modes excited by turbulent fluctuations.

Two-dimensional perturbations configured for maximum energy growth in laminar viscous shear flow are shown to develop into quasisteady finite amplitude structures, provided that the initial perturbation has sufficient energy and a nearby nonlinear mode exists. For Poiseuille flow, which supports finite amplitude equilibria for Reynolds numbers above -2900, an optimal perturbation with initial energy density equal to or greater than 0.1% of the mean flow energy density closely approaches the quasiequilibrium state within 10 advective time units. For Couette flow, which has no finite amplitude solution, the optimal perturbations decay rapidly after reaching maximum amplitude unless the configuration is sufficiently close to a linear mode with slow exponential decay rate. While the quasiequilibrium structure for Poiseuille flow is locally infiectional, it supports only weak instabilities with scales larger than the local region. 

We obtain an exact non linear stationary solution for barotropic wares in a b-plane channel and show that it can be excited under a range of initial conditions. Results show that a finite-amplitude wave in a constant shear flow, given an initial phase tilt against the shear and a sufficient initial amplitude, interacts with the mean flow to produce a nearly steady state close to the exact stationary solution. This equilibration process involves nonlinear transients; in particular, as the flow equilibrates, the emergence of critical levels is accompanied by the neutralization of local mean vorticity gradients at these levels, thus allowing the solution to attain a nonsingular modal structure.