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.

Streamwise rolls and accompanying streamwise streaks are ubiquitous in wall-bounded shear flows, both in natural settings, such as the atmospheric boundary layer, as well as in controlled settings, such as laboratory experiments and numerical simulations. The streamwise roll and streak structure has been associated with both transition from the laminar to the turbulent state and with maintenance of the turbulent state. This close association of the streamwise roll and streak structure with the transition to and maintenance of turbulence in wall-bounded shear flow has engendered intense theoretical interest in the dynamics of this structure. In this work, stochastic structural stability theory (SSST) is applied to the problem of understanding the dynamics of the streamwise roll and streak structure. The method of analysis used in SSST comprises a stochastic turbulence model (STM) for the dynamics of perturbations from the streamwise-averaged flow coupled to the associated streamwise-averaged flow dynamics. The result is an autonomous, deterministic, nonlinear dynamical system for evolving a second-order statistical mean approximation of the turbulent state. SSST analysis reveals a robust interaction between streamwise roll and streak structures and turbulent perturbations in which the perturbations are systematically organized through their interaction with the streak to produce Reynolds stresses that coherently force the associated streamwise roll structure. If a critical value of perturbation turbulence intensity is exceeded, this feedback results in modal instability of the combined streamwise roll/streak and associated turbulence complex in the SSST system. In this instability, the perturbations producing the destabilizing Reynolds stresses are predicted by the STM to take the form of oblique structures, which is consistent with observations. In the SSST system this instability exists together with the transient growth process. These processes cooperate in determining the structure of growing streamwise roll and streak. For this reason, comparison of SSST predictions with experiments requires accounting for both the amplitude and structure of initial perturbations as well as the influence of the SSST instability. Over a range of supercritical turbulence intensities in Couette flow, this instability equilibrates to form finite amplitude time-independent streamwise roll and streak structures. At sufficiently high levels of forcing of the perturbation field, equilibration of the streamwise roll and streak structure does not occur and the flow transitions to a time-dependent state. This time-dependent state is self-sustaining in the sense that it persists when the forcing is removed. Moreover, this self-sustaining state rapidly evolves toward a minimal representation of wall-bounded shear flow turbulence in which the dynamics is limited to interaction of the streamwise-averaged flow with a perturbation structure at one streamwise wavenumber. In this minimal realization of the self-sustaining process, the time-dependent streamwise roll and streak structure is maintained by perturbation Reynolds stresses, just as is the case of the time-independent streamwise roll and streak equilibria. However, the perturbation field is maintained not by exogenously forced turbulence, but rather by an endogenous and essentially non-modal parametric growth process that is inherent to time-dependent dynamical systems.

Coherent jets, such as the Jovian banded winds, are  prominent feature of rotating turbulence. Shallow water turbulence models capture the essential mechanism of jet formation, which is systematic eddy momentum flux directed up the mean velocity gradient. Understanding how the systematic eddy flux convergence is maintained and how the mean zonal flow and the eddy field mutually adjust to produce the observed jet structure constitutes a fundamental theoretical problem. In this work shallow-water equatorial beta plane model implementation of stochastic structural stability theory (SSST) is used to study the mechanism of zonal jet formation. In SSST a stochastic model for the ensemble-mean turbulent eddy fluxes is coupled with an equation for the mean jet dynamics to produce a nonlinear model of the mutual adjustment between the field of turbulent eddies and the zonal jets. In a weak turbulence, and for parameters appropriate to Jupiter, both prograde and retrograde equatorial jets are found to be stable solutions of the SST system, but only the prograde equatorial jet remains stable strong turbulence. In addition to the equatorial jet, multiple midlatitude zonal jets are also maintained in these stable equilibria. These midlatitude jets have structure and spacing in agreement with observed zonal jets and exhibit the observed robust reverals in sign of both absolute and potential vorticity gradient.

Understanding the physical mechanism maintaining fluid turbulence remains a fundamental theoretical problem. The two-layer model is an analytically and computationally simple system in which the dynamics of turbulence can be conveniently studied; in this work, a maximally simplified model of the statistically steady turbulent state in this system is constructed to isolate and identify the essential mechanism of turbulence. In this minimally complex turbulence model the effects of nonlinearity are parameterized using an energetically consistent stochastic process that is white in both space and time, turbulent fluxes are obtained using a stochastic turbulence model (STM), and statistically steady turbulent states are identified using stochastic structural stability theory (SSST). These turbulent states are the fixed-point equilibria of the nonlinear SSST system. For parameter values typical of the midlatitude atmosphere, these equilibria predict the emergence of marginally stable eddy-driven baroclinic jets. The eddy variances and fluxes associated with these jets and the power-law scaling of eddy variances and fluxes are consistent with observations and simulations of baroclinic turbulence. This optimally simple model isolates the essential physics of baroclinic turbulence: maintenance of variance by transient perturbation growth, replenishment of the transiently growing subspace by nonlinear energetically conservative eddy-eddy scattering, and equilibration to a statistically steady state of marginal stability by a combination of nonlinear eddy-induced mean jet modification and eddy dissipation. These statistical equilibrium states provide a theory for the general circulation of baroclinically turbulent planetary atmospheres.

A remarkable phenomenon in turbulent flows is the spontaneous emergence of coherent large spatial scale zonal jets. In this work a comprehensive theory for the interaction of jets with turbulence, stochastic structural stability theory, is applied to the problem of understanding the formation and maintenance of the zonal jets that are crucial for enhancing plasma confinement in fusion devices.

Turbulent fluids are frequently observed to spontaneously self-organize into large spatial-scale jets; geophysical examples of this phenomenon include the Jovian banded winds and the earth’s polar-front jet. These relatively steady large-scale jets arise from and are maintained by the smaller spatial- and temporal- scale turbulence with which they coexist. Frequently these jets are found to be adjusted into marginally stable states that support large transient growth. In this work, a comprehensive theory for the interaction of jets with turbulence, stochastic structural stability theory (SSST), is applied to the two-layer baroclinic model with the object of elucidating the physical mechanism producing and maintaining baroclinic jets, understanding how jet amplitude, structure, and spacing is controlled, understanding the role of parameters such as the temperature gradient and static stability in determining jet structure, understanding the phe- nomenon of abrupt reorganization of jet structure as a function of parameter change, and understanding the general mechanism by which turbulent jets adjust to marginally stable states supporting large transient growth. When the mean thermal forcing is weak so that the mean jet is stable in the absence of turbulence, jets emerge as an instability of the coupled system consisting of the mean jet dynamics and the ensemble mean eddy dynamics. Destabilization of this SSST coupled system occurs as a critical turbulence level is exceeded. At supercritical turbulence levels the unstable jet grows, at first exponentially, but eventually equilibrates nonlinearly into stable states of mutual adjustment between the mean flow and turbulence. The jet structure, amplitude, and spacing can be inferred from these equilibria.

With weak mean thermal forcing and weak but supercritical turbulence levels, the equilibrium jet structure is nearly barotropic. Under strong mean thermal forcing, so that the mean jet is unstable in the absence of turbulence, marginally stable highly nonnormal equilibria emerge that support high transient growth and produce power-law relations between, for example, heat flux and temperature gradient. The origin of this power-law behavior can be traced to the nonnormality of the adjusted states.

As the stochastic excitation, mean baroclinic forcing, or the static stability are changed, meridionally confined jets that are in equilibrium at a given meridional wavenumber abruptly reorganize to another meridional wavenumber at critical values of these parameters.

The equilibrium jets obtained with this theory are in remarkable agreement with equilibrium jets ob- tained in simulations of baroclinic turbulence, and the phenomenon of discontinuous reorganization of confined jets has important implications for storm-track reorganization and abrupt climate change.

Theoretical understanding of the growth of wind-driven surface water waves has been based on two distinct mechanisms: growth due to random atmospheric pressure fluctuations unrelated to wave amplitude and growth due to wave coherent atmospheric pressure fluctuations proportional to wave amplitude. Wave-independent random pressure forcing produces wave growth linear in time, while coherent forcing proportional to wave amplitude produces exponential growth. While observed wave development can be parameterized to fit these functional forms and despite broad agreement on the underlying physical process of momentum transfer from the atmospheric boundary layer shear flow to the water waves by atmospheric pressure fluctuations, quantitative agreement between theory and field observations of wave growth has proved elusive. Notably, wave growth rates are observed to exceed laminar instability predictions under gusty conditions. In this work, a mechanism is described that produces the observed enhancement of growth rates in gusty conditions while reducing to laminar instability growth rates as gustiness vanishes. This stochastic parametric instability mechanism is an example of the universal process of destabilization of nearly all time-dependent flows.

This work continues the generalized stability theory (GST) analysis of baroclinic shear flow in the primitive equations (PE), focusing on the regime in which the mean baroclinic shear and the stratification are of the same order. The Eady model basic state is used and solutions obtained using the PE are compared to quasigeostrophic (QG) solutions.

Similar optimal growth is obtained in the PE and QG frameworks for eddies with horizontal scale equal 
to or larger than the Rossby radius, although PE growth rates always exceed QG growth rates. The primary 
energy growth mechanism is the conventional baroclinic conversion of mean available potential energy to 
perturbation energy mediated by the eddy meridional heat flux. However, for eddies substantially smaller 
than the Rossby radius, optimal growth rates in the PE greatly exceed those found in the QG. This 
enhanced growth rate in the PE is dominated by conversion of mean kinetic energy to perturbation kinetic 
energy mediated by the vertical component of zonal eddy momentum flux. This growth mechanism is 
filtered in QG. In the intermediate Richardson number regime mixed Rossby-gravity modes are nonor-
thogonal in energy, and these participate in the process of energy transfer from the barotropic source in the 
mean shear to predominantly baroclinic waves during the transient growth process.

The response of shear flow in the intermediate Richardson number regime to spatially and temporally uncorrelated stochastic forcing is also investigated. It is found that a comparable amount of shear turbulent variance is maintained in the rotational and mixed Rossby-gravity modes by such unbiased forcing suggesting that any observed dominance of rotational mode energy arises from restrictions on the effective forcing and damping.

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.

A generalized stability theory (GST) analysis of baroclinic shear flow is performed using primitive equations (PEs), and typical synoptic-scale midlatitudinal values of vertical shear and stratification. GST is a comprehensive linear stability theory that subsumes modal stability theory and extends it to account for nonmodal interactions such as may occur among the approximately geostrophically balanced modes and the nearly divergent gravity wave modes supported by the PE. Unbounded constant shear flow and the Eady model are taken as examples and energy is used as the reference norm. Comparison is made with results obtained using quasigeostrophic (QG) analysis. While the PE and the QG dynamics give similar results for timescales of a few days, the PE initial growth rate during the first few hours exceeds the QG growth at all wavelengths and can attain values as much as two orders of magnitude greater as the wavelength decreases. This PE growth is due both to the direct kinetic energy growth mechanism, which is filtered out by QG, and to the interaction between the QG modes and the gravity waves.

An important application of GST is to study shear turbulence using a method of analysis based on stochastically forcing the mean state. The PE response of the Eady model to spatially and temporally uncorrelated forcing reveals the rotational and the divergent modes support a comparable amount of variance. The observed spectrum and physical mechanisms influencing the spectrum are discussed.

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.

In studies of perturbation dynamics in physical systems, certain specification of the gov-
erning perturbation dynamical system is generally lacking, either because the perturba-
tion system is imperfectly known or because its specification is intrinsically uncertain, 
while a statistical characterization of the perturbation dynamical system is often avail-
able. In this report exact and asymptotically valid equations are derived for the ensemble 
mean and moment dynamics of uncertain systems. These results are used to extend the 
concept of optimal deterministic perturbation of certain systems to uncertain systems. 
Remarkably, the optimal perturbation problem has a simple solution: In uncertain sys-
tems there is a sure initial condition producing the greatest expected second moment 
perturbation growth.

Perturbation growth in uncertain systems associated with fluid flow is examined concentrating on deriving, solving, and interpreting equations governing the ensemble mean covariance. Covariance evolution equations are obtained for fluctuating operators and illustrative physical examples are solved. Stability boundaries are obtained constructively in terms of the amplitude and structure of operator fluctuation required for existence of bounded second-moment statistics in an uncertain system. The forced stable uncertain system is identified as a primary physical realization of second-moment dynamics by using an ergodic assumption to make the physical connection between ensemble statistics of stable stochastically excited systems and observations of time mean quantities. Optimal excitation analysis plays a central role in generalized stability theory and concepts of optimal deterministic and stochastic excitation of certain systems are extended in this work to uncertain systems. Re- markably, the optimal excitation problem has a simple solution in uncertain systems: there is a pure structure producing the greatest expected ensemble perturbation growth when this structure is used as an initial condition, and a pure structure that is most effective in exciting variance when this structure is used to stochastically force the system distributed in time.

Optimal excitation analysis leads to an interpretation of the EOF structure of the covariance both for the case of optimal initial excitation and for the optimal stochastic excitation distributed in time that maintains the statistically steady state. Concepts of pure and mixed states are introduced for interpreting covariances and these ideas are used to illustrate fundamental limitations on inverting covariances for structure in stochastic systems in the event that only the covariance is known.

Perturbation growth in uncertain systems is examined and related to previous work in which linear stability 
concepts were generalized from a perspective based on the nonnormality of the underlying linear operator. In 
this previous work the linear operator, subject to an initial perturbation or a stochastic forcing distributed in 
time, was either fixed or time varying, but in either case the operator was certain. However, in forecast and 
climate studies, complete knowledge of the dynamical system being perturbed is generally lacking; nevertheless, 
it is often the case that statistical properties characterizing the variability of the dynamical system are known. 
In the present work generalized stability theory is extended to such uncertain systems. The limits in which 
fluctuations about the mean of the operator are correlated over time intervals, short and long, compared to the 
timescale of the mean operator are examined and compared with the physically important transitional case of 
operator fluctuation on timescales comparable to the timescales of the mean operator. Exact and asymptotically 
valid equations for transient ensemble mean and moment growth in uncertain systems are derived and solved. 
In addition, exact and asymptotically valid equations for the ensemble mean response of a stable uncertain 
system to deterministic forcing are derived and solved. The ensemble mean response of the forced stable uncertain 
system obtained from this analysis is interpreted under the ergodic assumption as equal to the time mean of the 
state of the uncertain system as recorded by an averaging instrument. Optimal perturbations are obtained for 
the ensemble mean of an uncertain system in the case of harmonic forcing. Finally, it is shown that the remarkable 
systematic increase in asymptotic growth rate with moment in uncertain systems occurs only in the context of 
the ensemble.

Asymptotic linear stability of time-dependent flows is examined by extending to nonautonomous systems 
methods of nonnormal analysis that were recently developed for studying the stability of autonomous systems. 
In the case of either an autonomous or a nonautonomous operator, singular value decomposition (SVD) analysis 
of the propagator leads to identification of a complete set of optimal perturbations ordered according to the 
extent of growth over a chosen time interval as measured in a chosen inner product generated norm. The long-
time asymptotic structure in the case of an autonomous operator is the norm-independent, most rapidly growing 
normal mode while in the case of the nonautonomous operator it is the first Lyapunov vector that grows at the 
norm independent mean rate of the first Lyapunov exponent. While information about the first normal mode 
such as its structure, energetics, vorticity budget, and growth rate are easily accessible through eigenanalysis of 
the dynamical operator, analogous information about the first Lyapunov vector is less easily obtained. In this 
work the stability of time-dependent deterministic and stochastic dynamical operators is examined in order to 
obtain a better understanding of the asymptotic stability of time-dependent systems and the nature of the first 
Lyapunov vector. Among the results are a mechanistic physical understanding of the time-dependent instability 
process, necessary conditions on the time dependence of an operator in order for destabilization to occur, 
understanding of why the Rayleigh theorem does not constrain the stability of time-dependent flows, the de-
pendence of the first Lyapunov exponent on quantities characterizing the dynamical system, and identification 
of dynamical processes determining the time-dependent structure of the first Lyapunov vector.

The structure and dynamics of axisymmetric tornado-like vortices are explored with a numerical model of 
axisymmetric incompressible flow based on recently developed numerical methods. The model is first shown 
to compare favorably with previous results and is then used to study the effects of varying the major parameters 
controlling the vortex: the strength of the convective forcing, the strength of the rotational forcing, and the 
magnitude of the model eddy viscosity. Dimensional analysis of the model problem indicates that the results 
must depend on only two dimensionless parameters. The natural choices for these two parameters are a convective 
Reynolds number (based on the velocity scale associated with the convective forcing) and a parameter analogous 
to the swirl ratio in laboratory models. However, by examining sets of simulations with different model parameters 
it is found that a dimensionless parameter known as the vortex Reynolds number, which is the ratio of the far-
field circulation to the eddy viscosity, is more effective than the conventional swirl ratio for predicting the 
structure of the vortex.

As the value of the vortex Reynolds number is increased, it is observed that the tornado-like vortex transitions from a smooth, steady flow to one with quasiperiodic oscillations. These oscillations, when present, are caused by axisymmetric disturbances propagating down toward the surface from the upper part of the domain. Attempts to identify these oscillations with linear waves associated with the shears of the mean azimuthal and vertical winds give mixed results.

The parameter space defined by the choices for model parameters is further explored with large sets of numerical simulations. For much of this parameter space it is confirmed that the vortex structure and time-dependent behavior depend strongly on the vortex Reynolds number and only weakly on the convective Reynolds number. The authors also find that for higher convective Reynolds numbers, the maximum possible wind speed increases, and the rotational forcing necessary to achieve that wind speed decreases. Physical reasoning is used to explain this behavior, and implications for tornado dynamics are discussed.

The dynamics of both transient and exponentially growing disturbances in two-dimensional vortices that are 
maintained by the radial inflow of a fixed cylindrical deformation field are investigated. Such deformation fields 
are chosen so that both one-celled and two-celled vortices may be studied. The linearized evolution of asymmetric 
perturbations is expressed in the form of a linear dynamical system dx/dt 5 Ax. The shear of the mean flow 
results in a nonnormal dynamical operator A, allowing for the transient growth of perturbations even when all 
the modes of the operator are decaying. It is found that one-celled vortices are stable to asymmetric perturbations 
of all azimuthal wavenumbers, whereas two-celled vortices can have low-wavenumber instabilities. In all cases, 
generalized stability analysis of the dynamical operator identifies the perturbations that grow the fastest, both 
instantaneously and over a finite period of time. While the unstable modal perturbations necessarily convert 
mean-flow vorticity to perturbation vorticity, the perturbations with the fastest instantaneous growth rate use 
the deformation of the mean flow to rearrange their vorticity fields into configurations with higher kinetic energy. 
Also found are perturbations that use a hybrid of these two mechanisms to achieve substantial energy growth 
over finite time periods.

Inclusion of the dynamical effects of radial inflow—vorticity advection and vorticity stretching—is found to be extremely important in assessing the potential for transient growth and instability in these vortices. In the two-celled vortex, neglecting these terms destabilizes the vortex for azimuthal wavenumbers one and two. In the one-celled vortex, neglect of the radial inflow terms results in an overestimation of transient growth for all wavenumbers, and it is also found that for high wavenumbers the maximum transient growth decreases as the strength of the radial inflow increases.

The effects of these perturbations through eddy flux divergences on the mean flow are also examined. In the one-celled vortex it is found that for all wavenumbers greater than one the net effect of most perturbations, regardless of their initial configuration, is to increase the kinetic energy of the mean flow. As these perturbations are sheared over they cause upgradient eddy momentum fluxes, thereby transferring their kinetic energy to the mean flow and intensifying the vortex. However, for wavenumber one in the one-celled vortex, and all wavenumbers in the two-celled vortex, it was found that nearly all perturbations have the net effect of decreasing the kinetic energy of the mean flow. In these cases, the kinetic energy of the perturbations accumulates in nearly neutral or unstable modal structures, so that energy acquired from the mean flow is not returned to the mean flow but instead is lost through dissipation.

The effects of stochastically excited asymmetric disturbances on two-dimensional vortices are investigated. 
These vortices are maintained by the radial inflow of fixed cylindrical deformation fields, which are chosen so 
that both one-celled and two-celled vortices may be studied. The linearized perturbation equations are reduced 
to the form of a linear dynamical system with stochastic forcing, that is, dx/dt 5 Ax 1 Fj, where the columns 
of F are forcing functions and the elements of j are Gaussian white-noise processes. Through this formulation 
the stochastically maintained variance of the perturbations, the structures that dominate the response (the empirical 
orthogonal functions), and the forcing functions that contribute most to this response (the stochastic optimals) 
can be directly calculated.

For all cases the structures that most effectively induce the transfer of energy from the mean flow to the perturbation field are close approximations to the global optimals (i.e., the initial perturbations with the maximum growth in energy in finite time), and that the structures that account for most of the variance are close approximations to the global optimals evolved forward in time to when they reach their maximum energy. For azimuthal wavenumbers in each vortex where nearly neutral modes are present (k 5 1 for the one-celled vortex and 1 # k # 4 for the two-celled vortex), the variance sustained by the stochastic forcing is large, and in these cases the variance may be greatly overestimated if the radial inflow that sustains the mean vortex is neglected in the dynamics of the perturbations.

Through a modification of this technique the ensemble average eddy momentum flux divergences associated with the stochastically maintained perturbation fields can be computed, and this information is used to determine the perturbation-induced mean flow tendency in the linear limit. Examination of these results shows that the net effect of the low wavenumber perturbations is to cause downgradient eddy fluxes in both vortex types, while high wavenumber perturbations cause upgradient eddy fluxes. However, to determine how these eddy fluxes actually change the mean flow, the local accelerations caused by the eddy flux divergences must be incorporated into the equation for the steady-state azimuthal velocity. From calculations of this type, it is found that the effect of the radial inflow can be crucial in determining whether or not the vortex is intensified or weakened by the perturbations: though the net eddy fluxes are most often downgradient, the radial inflow carries the transported angular momentum back into the vortex core, resulting in an increase in the maximum wind speed. In most cases for the vortex flows studied, the net effect of stochastically forced asymmetric perturbations is to intensify the mean vortex. Applications of the same analysis techniques to vortices with azimuthal velocity profiles more like those used in previous studies give similar results.