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.

Synoptic-scale eddy variance and fluxes of heat and momentum in midlatitude jets are sensitive to small changes in mean jet velocity, dissipation, and static stability. In this work the change in the jet producing the greatest increase in variance or flux is determined. Remarkably, a single jet structure change completely characterizes the sensitivity of a chosen quadratic statistical quantity to modification of the mean jet in the sense that an arbitrary change in the jet influences a chosen statistical quantity in proportion to the projection of the change on this single optimal structure. The method used extends previous work in which storm track statistics were obtained using a stochastic model of jet turbulence.

A theory for quasigeostrophic turbulence in barclonic jets is examined in which interaction between the mean flow and the perturbations is explicitly modeled by the nonnormal operator obtained by linearization about the mean flow, while the eddy--eddy interactions are parameterized by a combination of stochastic excitation and effective dissipation. The quasi-linear equilibrium is the stationary state in dynamical balance between the mean flow forcing and eddy forcing produced by the linear stochastic model. The turbulence model depends on two parameters that specify the magnitude of the effective dissipation an stochastic excitation. The quasi-linear model produces heat fluxes (upgradient), momentum fluxes, and mean zonal winds, which are remarkably consistent with those produced by the nonlinear model over a wide range of parameter values despite energy and edstrophy imbalances associated with the parameterization for eddy-eddy interactions. The quasi-linear equilibrium also appears consistent with most aspects of the energy cycle, and with the negative correlation between transient eddy transport and other transports observed in the atmosphere. The model overestimated the equilibrium eddy kinetic energy in cases in which it achieves correct eddy fluxes and energy balance. Understanding the role of forcing orthogonal functions rationalized this behavior and provides the basis for addressing the role of transient eddies in climate.

The innate tendency of the background straining field of the midlatitude atmospheric jet to preferentially amplify a subset of disturbances produces a characteristic response to stochastic perturbation whether the perturbations are internally generated by non linear processes or externally imposed. This physical property of enhances response to a subset of perturbations is expressed analytically through the nonnormality of t he linearized dynamical operator, which can be studied to determine the transient growth of particular disturbances over time through solution of the initial value problem or, alternatively, to determine the stationary response to continual excitation through solution of the related stochastic problem. Making use of the fact that the background flow dominates the strain rate field, a theory for the turbulent state can be constructed based on the nonormality of the dynamical operator linearized about the background flow. While the initial value problem provides an explanation for the individual cyclogenesis events, solution of the stochastic problem provides a theory for the statistics of the ensemble of all cyclones including structure, frequency, intensity, and resulting fluxes of heat and momentum, which together constitute the synoptic-scale influence on midlatitude climate. Moreover, the observed climate can be identified with the background thermal and velocity structure that is in self-consistent equilibrium with both its own induced fluxes and imposed large-scale thermal forcing. In order to approach the problem of determining the self-consistent statistical equilibrium of the midlatitude jet it is first necessary to solve the stochastic problem for the mixed baroclinic/barotropic jet because fluxes of both heat and momentum are involved in this balance.

In this work the response to stochastic forcing of a linearized nonseparable quasigeostrophic model of the midlatitude jet is solved. The observed distribution of transient eddy variance with frequency and wavenumber, the observed vertical structures, and the observed heat and momentum flux distributions are obtained. Associated energetics and implications for maintenance of the climatological jet are discussed.

The authors explore the hypothesis that nonlinear eddy interactions in quasigeosptrophic turbulence can be parameterized as a stochastic excitation plus an augmented dissipation in a statistically stationary equilibrium. The focus primarily on models sufficiently simple to be solves analytically a In particular, closed form solutions are obtained for the linear response to stochastic excitation of horizontally uniform barotropic and two-layer baroclinic flow. The response of the barotrophoic model is very simple to understand because the governing equations are mathematically normal. In contrast, the two-layer model is non-normal in the presence of vertical shear and/or vertically asymmetric dissipation and yields rather complicated results. The space-time spectra of the streamfunction and the hear fluxes derived from the two lyer model are in quantative agreement  with the corresponding observed quantities at 50N. The velocity variance predicted from the parameterication is a weaker function of the temperature gradient than indicated by observations. For strong thermal forcing, the parameterized fluxes vary inversely with the difference between a critical temperature gradient and the ambient gradient. This parameterization yields behavior suggestive of baroclinic adjustment  but operates by mechanisms fundamentally different from those conventionally associated with instability theory.

Obtaining a physically based understanding of the variations with spatial scale of the amplitude and dispersive properties of midlatitude transient baroclinic waves and the heat flux associated with these waves central goal of dynamic meteorology and climate studies. Recently, stochastic forcing of highly nonnormal dynamical systems, such as arise from analysis of these equations governing perturbations to the midlatitude westerly jet, had been shown to induce large transfers of energy from the mean to the perturbation scale. In the case of a baroclinic atmospheric jet, this energy transfer to the synoptic scale produces dispersive properties, distributions of wave energy with wavenumber, and heat fluxes that are intrinsically associated with the nonnormal dynamics underlying baroclinic wave development.

In this work a method for calculating the spectrum and heat flux arising from the stochastic forcing is describes and predictions of this theory for a model atmosphere are compared with observations. The calculated energy spectrum is found to be in remarkable agreement with observations, in contrast with the predictions of modal instability theory. The calculated heat flux exhibits a realistic distribution with height and its associated energetic cycle with observed seasonal mean energetics.

The maintenance of variance and attendant heat flux in linear, forced dissipatative baroclinic shear flow subject to stochastic excitation is examined. The baroclinic problem is intrinsically nonnormal and its stochastic dynamics is found to differ significantly from the more familiar stochastic dynamics of normal systems. When the shear is sufficiently great in comparison to dissipative effects, stochastic excitation supports highly enhanced variance levels in these nonnormal systems compared to variance levels supported by the same forcing and dissipation in related normal systems. The eddy variance and associated heat flux are found from the mean flow and project this energy on a distinct subset of response functions (EOFs) that are in turn from the set of normal modes of the system. A method for obtaining the dominant forcing and response functions as well as the distribution of heat flux for a given flow is described.