Turbulent flows are often observed to be organized into large-spatial-scale jets such as the familiar zonal 
jets in the upper levels of the Jovian atmosphere. These relatively steady large-scale jets are not forced 
coherently but are maintained by the much smaller spatial- and temporal-scale turbulence with which they 
coexist. The turbulence maintaining the jets may arise from exogenous sources such as small-scale convec-
tion or from endogenous sources such as eddy generation associated with baroclinic development processes 
within the jet itself. Recently a comprehensive theory for the interaction of jets with turbulence has been 
developed called stochastic structural stability theory (SSST). In this work SSST is used to study the 
formation of multiple jets in barotropic turbulence in order to understand the physical mechanism produc-
ing and maintaining these jets and, specifically, to predict the jet amplitude, structure, and spacing. These 
jets are shown to be maintained by the continuous spectrum of shear waves and to be organized into stable 
attracting states in the mutually adjusted mean flow and turbulence fields. The jet structure, amplitude, and 
spacing and the turbulence level required for emergence of jets can be inferred from these equilibria. For 
weak but supercritical turbulence levels the jet scale is determined by the most unstable mode of the SSST 
system and the amplitude of the jets at equilibrium is determined by the balance between eddy forcing and 
mean flow dissipation. At stronger turbulence levels the jet amplitude saturates with jet spacing and 
amplitude satisfying the Rayleigh-Kuo stability condition that implies the Rhines scale. Equilibrium jets 
obtained with the SSST system are in remarkable agreement with equilibrium jets obtained in simulations 
of fully developed -plane turbulence.

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.

Turbulence in fluids is commonly observed to coexist with relatively large spatial and temporal scale coherent jets. These jets may be steady, vacillate with a definite period, or be irregular. A comprehensive theory for this phenomenon is presented based on the mutual interaction between the coherent jet and the turbulent eddies. When a sufficient number of statistically independent realizations of the eddy field participate in organizing the jet a simplified asymptotic dynamics emerges with progression, as an order parameter such as the eddy forcing is increased, from a stable fixed point associated with a steady symmetric zonal jet through a pitchfork bifurcation to a stable asymmetric jet followed by a Hopf bifurcation to a stable limit cycle associated with a regularly vacillating jet and finally a transition to chaos. This underlying asymptotic dynamics emerges when a sufficient number of ensemble members is retained in the stochastic forcing of the jet but a qualitative different mean jet dynamics is found when a small number of ensemble members is retained as is appropriate for many physical systems. Example applications of this theory are presented including a model of midlatitude jet vacillation, emergence and maintenance of multiple jets in turbulent flow, a model of rapid reorganization of storm tracks as a threshold in radiative forcing is passed, and a model of the quasi-biennial oscillation. Because the statistically coupled wave-mean flow system discussed is generally globally stable this system also forms the basis for a comprehensive theory for equilibration of unstable jets in turbulent shear flow.

The strong mean shear in the vicinity of the boundaries in turbulent boundary layer flows prefer- entially amplifies a particular class of perturbations resulting in the appearance of coherent structures and in characteristic associated spatial and temporal velocity spectra. This enhanced response to certain perturba- tions can be traced to the nonnormality of the linearized dynamical operator through which transient growth arising in dynamical systems with asymptotically stable operators is expressed. This dynamical amplification process can be comprehensively probed by forcing the linearized operator associated with the boundary layer flow stochastically to obtain the statistically stationary response.

In this work the spatial wave-number/temporal frequency spectra obtained by stochastically forcing the linearized model boundary layer operator associated with wall-bounded shear flow at large Reynolds number are compared with observations of boundary layer turbulence. The verisimilitude of the stochastically excited synthetic turbulence supports the identification of the underlying dynamics maintaining the turbulence with nonnormal perturbation growth

The level of variance maintained in a stochastically forced asympttically stable linear dynamical system with a non-normal dynamical operator cannot be fully characterized by the decay rate of its normal modes, unlike normal dynamical systems. The nonorthogonality of modes may lead to transient growth which supports variance far in excess of that anticipated from the decay rate given by the eigenvalues of the operator. As an example, the variance maintained by stochastic forcing in a canonical shear slow is found to increase with a power of the reynolds number between 1.5 and 3. This great amplification of variance suggests a fundamentally linear mechanism underlying shear flow turbulence.

Transient amplification of a particular set of favorably configured forcing functions in the stochastically driven Navier-Stokes equations linearized about a mean shear flow is shown to produce high levels of variance concentrated in a distinct set of response functions. The dominant forcing functions are found as solutions of a Lyapunov equation and the response functions are found as the distinct solutions of a related Lyapunov equation. Neither the forcing nor the response functions can be identified with the normal modes of the linearized dynamical operator. High variance levels are sustained in these systems under stochastic forcing, largely by transfer of energy from the mean flow to the perturbation field, despite the exponential stability of all normal modes of the system. From the perspective of modal analysis the explanation for this amplification of variance can be traced to the non-normality of the linearized dynamical operator. The great amplification of perturbation variance found for Couette and Poiseuille flow implies a mechanism for producing and sustaining high levels of variance in shear flows from relatively small intrinsic or extrinsic forcing disturbances.
I

The problem of growth of small perturbations in fluid flow and the related problem of maintenance of perturbation variance has traditionally been studies by appeal to exponential modal instability of the flow. in the event that a flow supports and exponentially growing modal solution, the initially unbounded growth of the mode is taken as more of less compelling evidence for eventual flow breakdown. However, atmospheric flows are characterized by large thermally forced background rates of strain and are subject to perturbations that are not infinitesimal in amplitude. Under these circumstances there is an alternative mechanism for growth and maintenance of perturbation variance: amplification is straining flow of stochastically forced perturbations in the absence of exponential instabilities. From this viewpoint the flow is regarded as a driven amplifier rather than as an unstable oscillator. We explore this mechanism using as examples unbounded constant shear and pure deformation flow for which closed-form solutions are available and neither of which supports a nonsingular mode. With diffusive dissipation we find that amplification of isotropic band-limited stochastic driving is unlinear velocity profile. A phenomenological model of the contribution of linear and nonlinear damped modes to the maintenance of variance results in variance levels increasing linearly with shear. We conclude that amplification of stochastic forcing in straining field can maintain a variance field substantially more energetic than that resulting from the same forcing in the absence of a background straining flow. our results further indicate that existence of linear and non linear damped modes is important in maintaining high levels of variance byt he mechanism of stochastic excitation.