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Petros J. Ioannou

Abstract

Recently, a new theoretical and conceptual model of quasigeostrophic turbulence has been advanced in which eddy variance is regarded as being maintained by transient growth of perturbations arising from sources including the nonlinear interactions among the eddies, but crucially without a direct contribution of unstable modal growth to the maintenance of variance. This theory is based on the finding that stochastic forcing of the subcritical atmospheric flow supports variance arising from induced transfer of energy from the background flow to the disturbance field that substantially exceeds the variance expected from the decay rate of the associated normal modes in an equivalent normal system. Herein the authors prove that such amplification of variance is a general property of the stochastic dynamics of systems governed by nonnormal evolution operators and that consequently the response of the atmosphere to unbiased forcing is always underestimated when consideration is limited to the response of the system's individual normal modes to stochastic excitation.

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Eli Tziperman and Petros J. Ioannou

Abstract

The physical mechanisms of transient amplification of initial perturbations to the thermohaline circulation (THC), and of the optimal stochastic forcing of THC variability, are discussed using a simple meridional box model. Two distinct mechanisms of transient amplification are found. One such mechanism, with a transient amplification timescale of a couple of years, involves an interaction between the THC induced by rapidly decaying sea surface temperature anomalies and the THC induced by the slower-decaying salinity mode. The second mechanism of transient amplification involves an interaction between different slowly decaying salinity modes and has a typical growth timescale of decades. The optimal stochastic atmospheric forcing of heat and freshwater fluxes are calculated as well. It is shown that the optimal forcing induces low-frequency THC variability by exciting the salinity-dominated variability modes of the THC.

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Brian F. Farrell and Petros J. Ioannou

Abstract

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.

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Brian F. Farrell and Petros J. Ioannou

Abstract

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 convection 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 producing 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 β-plane turbulence.

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Brian F. Farrell and Petros J. Ioannou

Abstract

The problem of growth of small perturbations in fluid flow and the related problem of maintenance of perturbation variance has traditionally been studied by appeal to exponential modal instability of the flow. In the event that a flow supports an exponentially growing modal solution, the initially unbounded growth of the mode is taken as more or 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 in a 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 unbounded for the case of pure deformation and bounded by a threefold increase at large shear for the case of a linear 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 a 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 nonlinear damped modes is important in maintaining high levels of variance by the mechanism of stochastic excitation.

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Brian F. Farrell and Petros J. Ioannou

Abstract

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.

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Brian F. Farrell and Petros J. Ioannou

Abstract

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.

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Brian F. Farrell and Petros J. Ioannou

Abstract

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. Remarkably, 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.

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Brian F. Farrell and Petros J. Ioannou

Abstract

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 is a central goal of dynamic meteorology and climate studies. Recently, stochastic forcing of highly nonnormal dynamical systems, such as arise from analysis of the equations governing perturbations to the midlatitude westerly jet, has 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 stochastic forcing is described 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 agrees with observed seasonal mean energetics.

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Brian F. Farrell and Petros J. Ioannou

Abstract

The maintenance of variance and attendant heat flux in linear, forced, dissipative baroclinic shear flows 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 to arise in response to transient amplification of a subset of forcing functions that obtain energy from the mean flow and project this energy on a distinct subset of response functions (E0Fs) that are in turn distinct 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.

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