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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

Classical stability theory is extended to include transient growth processes. The central role of the nonnormality of the linearized dynamical system in the stability problem is emphasized, and a generalized stability theory is constructed that is applicable to the transient as well as the asymptotic stability of time-independent flows. Simple dynamical systems are used as examples including an illustrative nonnormal two-dimensional operator, the Eady model of baroclinic instability, and a model of convective instability in baroclinic flow.

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Brian F. Farrell and Petros J. Ioannou

Abstract

An extension of classical stability theory to address the stability of perturbations to time-dependent systems is described. Nonnormality is found to play a central role in determining the stability of systems governed by nonautonomous operators associated with time-dependent systems. This pivotal role of nonnormality provides a conceptual bridge by which the generalized stability theory developed for analysis of autonomous operators can be extended naturally to nonautonomous operators. It has been shown that nonnormality leads to transient growth in autonomous systems, and this result can be extended to show further that time-dependent nonnormality of nonautonomous operators is capable of sustaining this transient growth leading to asymptotic instability. This general destabilizing effect associated with the time dependence of the operator is explored by analysing parametric instability in periodic and aperiodic time-dependent operators. Simple dynamical systems are used as examples including the parametrically destabilized harmonic oscillator, growth of errors in the Lorenz system, and the asymptotic destabilization of the quasigeostrophic three-layer model by stochastic vacillation of the zonal wind.

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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

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.

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Navid C. Constantinou, Brian F. Farrell, and Petros J. Ioannou

Abstract

Jets coexist with planetary-scale waves in the turbulence of planetary atmospheres. The coherent component of these structures arises from cooperative interaction between the coherent structures and the incoherent small-scale turbulence in which they are embedded. It follows that theoretical understanding of the dynamics of jets and planetary-scale waves requires adopting the perspective of statistical state dynamics (SSD), which comprises the dynamics of the interaction between coherent and incoherent components in the turbulent state. In this work, the stochastic structural stability theory (S3T) implementation of SSD for barotropic beta-plane turbulence is used to develop a theory for the jet–wave coexistence regime by separating the coherent motions consisting of the zonal jets together with a selection of large-scale waves from the smaller-scale motions that constitute the incoherent component. It is found that mean flow–turbulence interaction gives rise to jets that coexist with large-scale coherent waves in a synergistic manner. Large-scale waves that would exist only as damped modes in the laminar jet are found to be transformed into exponentially growing waves by interaction with the incoherent small-scale turbulence, which results in a change in the mode structure, allowing the mode to tap the energy of the mean jet. This mechanism of destabilization differs fundamentally and serves to augment the more familiar S3T instabilities in which jets and waves arise from homogeneous turbulence with the energy source exclusively from the incoherent eddy field and provides further insight into the cooperative dynamics of the jet–wave coexistence regime in planetary turbulence.

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Nikolaos A. Bakas, Navid C. Constantinou, and Petros J. Ioannou

Abstract

Zonal jets in a barotropic setup emerge out of homogeneous turbulence through a flow-forming instability of the homogeneous turbulent state (zonostrophic instability), which occurs as the turbulence intensity increases. This has been demonstrated using the statistical state dynamics (SSD) framework with a closure at second order. Furthermore, it was shown that for small supercriticality the flow-forming instability follows Ginzburg–Landau (G–L) dynamics. Here, the SSD framework is used to study the equilibration of this flow-forming instability for small supercriticality. First, we compare the predictions of the weakly nonlinear G–L dynamics to the fully nonlinear SSD dynamics closed at second order for a wide range of parameters. A new branch of jet equilibria is revealed that is not contiguously connected with the G–L branch. This new branch at weak supercriticalities involves jets with larger amplitude compared to the ones of the G–L branch. Furthermore, this new branch continues even for subcritical values with respect to the linear flow-forming instability. Thus, a new nonlinear flow-forming instability out of homogeneous turbulence is revealed. Second, we investigate how both the linear flow-forming instability and the novel nonlinear flow-forming instability are equilibrated. We identify the physical processes underlying the jet equilibration as well as the types of eddies that contribute in each process. Third, we propose a modification of the diffusion coefficient of the G–L dynamics that is able to capture the evolution of weak jets at scales other than the marginal scale (side-band instabilities) for the linear flow-forming instability.

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Nikolaos A. Bakas, Navid C. Constantinou, and Petros J. Ioannou

Abstract

Zonal jets and nonzonal large-scale flows are often present in forced–dissipative barotropic turbulence on a beta plane. The dynamics underlying the formation of both zonal and nonzonal coherent structures is investigated in this work within the statistical framework of stochastic structural stability theory (S3T). Previous S3T studies have shown that the homogeneous turbulent state undergoes a bifurcation at a critical parameter and becomes inhomogeneous with the emergence of zonal and/or large-scale nonzonal flows and that these statistical predictions of S3T are reflected in direct numerical simulations. In this paper, the dynamics underlying the S3T statistical instability of the homogeneous state as a function of parameters is studied. It is shown that, for weak planetary vorticity gradient β, both zonal jets and nonzonal large-scale structures form from upgradient momentum fluxes due to shearing of the eddies by the emerging infinitesimal flow. For large β, the dynamics of the S3T instability differs for zonal and nonzonal flows but in both the destabilizing vorticity fluxes decrease with increasing β. Shearing of the eddies by the mean flow continues to be the mechanism for the emergence of zonal jets while nonzonal large-scale flows emerge from resonant and near-resonant triad interactions between the large-scale flow and the stochastically forced eddies. The relation between the formation of large-scale structure through modulational instability and the S3T instability of the homogeneous state is also investigated and it is shown that the modulational instability results are subsumed by the S3T results.

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Nikolaos A. Bakas, Petros J. Ioannou, and George E. Kefaliakos

Abstract

Three-dimensional perturbations producing optimal energy growth in stratified, unbounded constant shear flow are determined. The optimal perturbations are intrinsically three-dimensional in structure. Streamwise rolls emerge as the optimally growing perturbations at long times, but their energy growth factor is limited by stratification to E = O(1/Ri), where Ri is the Richardson number. The perturbations that attain the greatest energy growth in the flow are combinations of Orr solutions and roll solutions that maximize their energy growth in typically O(10) advective time units. These optimal perturbations are localized in the high-shear regions of the boundary layer and are associated with strong updrafts and downdrafts that evolve into streamwise velocity streaky structures in the form of hairpin vortices in agreement with observations.

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Navid C. Constantinou, Brian F. Farrell, and Petros J. Ioannou

Abstract

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 between 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 structures. 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.

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