Publications

In wall-bounded shear flow the primary coherent structure is the streamwise roll and streak (R-S). Absent of an associated instability the R-S has been ascribed to non-normality mediated interaction between the mean flow and perturbations. This interaction may occur either directly due to excitation of a transiently growing perturbation or indirectly due to destabilization of the R-S by turbulent Reynolds stresses. A fundamental distinction between the direct and the indirect mechanisms, which is central to understanding the physics of turbulence, is that in the direct mechanism the R-S is itself the growing structure while in the indirect mechanism the R-S emerges as a self-organized structure. In the emergent R-S theory the fundamental mechanism is organization by the streak of Reynolds stresses configured to support its associated roll by the lift-up process. This requires that a streak organizes turbulent perturbations such as to produce Reynolds stresses configured to reinforce the streak. In this paper we provide detailed analysis explaining physically why this positive feedback occurs and is a universal property of turbulence in shear flow. DNS data from the same turbulent flow as that used in the theoretical study (Poisseullle flow at $R=1650$) is also analyzed verifying that this mechanism operates in DNS.

While linear non-normality underlies the mechanism of energy transfer from the externally driven flow to the perturbation field that sustains turbulence, nonlinearity is also known to play an essential role. The goal of this study is to better understand the role of nonlinearity in sustaining turbulence. The method used in this study is implementation in Couette flow of a statistical state dynamics (SSD) closure at second order in a cumulant expansion of the Navier-Stokes equations in which the averaging operator is the streamwise mean. The perturbations in this SSD are the deviations from the streamwise mean and two mechanisms potentially contributing to maintaining these second cumulant perturbations are identified. These are parametric perturbation growth arising from interaction of the perturbations with the fluctuating mean flow and transient growth of perturbations arising from nonlinear interaction between components of the perturbation field. By the method of comparing the turbulence maintained in the SSD and in the associated direct numerical simulation (DNS) in which these mechanisms have been selectively included and excluded, parametric growth is found to maintain the perturbation field of the turbulence while the more commonly invoked mechanism associated with transient growth of perturbations arising from scattering by nonlinear interaction is found to suppress perturbation growth. In addition to verifying that the parametric mechanism maintains the perturbations in DNS it is also verified that the Lyapunov vectors are the structures that dominate the perturbation energy and energetics in DNS. It is further verified that these vectors are responsible for maintaining the roll circulation that underlies the self-sustaining process (SSP) and in particular the maintenance of the fluctuating streak that supports the parametric perturbation growth.

The no-slip boundary condition results in a velocity shear forming in fluid flow near a solid surface. This shear flow supports the turbulence characteristic of fluid flow near boundaries at Reynolds numbers above \(\approx\)1000 by making available to perturbations the kinetic energy of the externally forced flow. Understanding the physical mechanism underlying transfer of energy from the forced mean flow to the turbulent perturbation field that is required to maintain turbulence poses a fundamental question. Although qualitative understanding that this transfer involves nonlinear destabilization of the roll-streak coherent structure has been established, identification of this instability has resisted analysis. The reason this instability has resisted comprehensive analysis is that its analytic expression lies in the Navier–Stokes equations (NS) expressed using statistical rather than state variables. Expressing NS as a statistical state dynamics (SSD) at second order in a cumulant expansion suffices to allow analytical identification of the nonlinear roll-streak instability underlying turbulence in wall-bounded shear flow. In this nonlinear instability the turbulent perturbation field is identified by the SSD with the Lyapunov vectors of the linear operator governing perturbation evolution about the time-dependent streamwise mean flow. In this work the implications of the predictions of SSD analysis that this parametric instability underlies the dynamics of turbulence in Couette flow and that the perturbation structures are the associated Lyapunov vectors are interpreted to imply new conceptual approaches to controlling turbulence. It is shown that the perturbation component of turbulence is supported on the streamwise mean flow, which implies optimal control should be formulated to suppress perturbations from the streamwise mean. It is also shown that suppressing only the top few Lyapunov vectors on the streamwise mean vectors results in laminarization. These results are verified using DNS.

Although the roll-streak (R-S) is fundamentally involved in the dynamics of wall-turbulence, the physical mechanism responsible for its formation and maintenance remains controversial. In this work we investigate the dynamics maintaining the R-S in turbulent Poiseuille flow at R=1650. Spanwise collocation is used to remove spanwise displacement of the streaks and associated flow components, which isolates the streamwise-mean flow R-S component and the second-order statistics of the streamwise-varying fluctuations that are collocated with the R-S. This streamwise-mean/fluctuation partition of the dynamics facilitates exploiting insights gained from the analytic characterization of turbulence in the second-order statistical state dynamics (SSD), referred to as S3T, and its closely associated restricted nonlinear dynamics (RNL) approximation. Symmetry of the statistics about the streak centerline permits separation of the fluctuations into sinuous and varicose components. The Reynolds stress forcing induced by the sinuous and varicose fluctuations acting on the R-S is shown to reinforce low- and high-speed streaks respectively. This targeted reinforcement of streaks by the Reynolds stresses occurs continuously as the fluctuation field is strained by the streamwise-mean streak and not intermittently as would be associated with streak-breakdown events. The Reynolds stresses maintaining the streamwise-mean roll arise primarily from the dominant POD modes of the fluctuations, which can be identified with the time average structure of optimal perturbations growing on the streak. These results are consistent with a universal process of R-S growth and maintenance in turbulent shear flow arising from roll forcing generated by straining turbulent fluctuations, which was identified using the S3T SSD.

Turbulence in wall-bounded flows is characterized by stable statistics for the mean flow and the fluctuations both for the case of the ensemble and the time mean. Although, in a substantial set of turbulent systems, this stable statistical state corresponds to a stable fixed point of an associated statistical state dynamics (SSD) closed at second order, referred to as S3T, this is not the case for wall-turbulence. In wall-turbulence the trajectory of the statistical state evolves on a transient chaotic attractor in the S3T statistical state phase space and the time-mean statistical state is neither a stable fixed point of this SSD nor, if the time-mean statistical state is maintained as an equilibrium state, is it stable. Nevertheless, sufficiently small perturbations from the ensemble/time-mean state of wall-turbulence are expected to relax back to the mean statistical state following an effective linear dynamics. In this work the dynamics of spanwise uniform perturbations to the time-mean flow are studied using a linear inverse model (LIM) to identify the linear operator governing the ensemble stability of the ensemble/time-mean state by obtaining the time mean stability properties over the transient attractor of the turbulence identified by the S3T SSD. The ensemble/time-mean stability of an unstable equilibrium can be understood by noting that even when every member of an ensemble is unstable the ensemble mean may be stable with perturbations following an identifiable stable dynamics. While simplifying insight into turbulent flows has commonly been obtained by identifying and studying ensemble mean statistical states, less attention has been accorded to identifying and studying the ensemble mean dynamics. We show that in the case of wall turbulence, even though stable fixed point SSD equilibria are not available to allow application of traditional perturbation analysis methods to identify the perturbation stability of the mean state, an effective linear stability analysis can be obtained to identify the perturbation dynamics of the ensemble/time-mean statistical state.

Turbulence in the restricted nonlinear (RNL) dynamics is analyzed and compared with DNS of Poiseuille turbulence at $R=1650$. The structures are obtained by POD analysis of the two components of the flow partition used in RNL dynamics: the streamwise-mean flow and fluctuations. POD analysis of the streamwise-mean flow indicates that the dominant POD modes, in both DNS and RNL, are roll-streaks harmonic in the spanwise. However, we conclude that these POD modes do not occur in isolation but rather are Fourier components of a coherent roll-streak structure. POD analysis of the fluctuations in DNS and RNL reveals similar complex structures consisting in part of oblique waves collocated with the streak. The origin of these structures is identified by their correspondence to POD modes predicted using a stochastic turbulence model (STM). These predicted POD modes are dominated by the optimally growing structures on the streak, which the STM predicts correctly to be of sinuous oblique wave structure. This close correspondence between the roll-streak structure and the associated fluctuations in DNS, RNL and the STM implies that the self-sustaining mechanism operating in DNS is essentially the same as that in RNL, which has been previously associated with optimal perturbation growth on the streak.

Both linear growth processes associated with non-normality of the mean flow and nonlinear interaction transferring energy among fluctuations contribute to maintaining turbulence. However, a detailed understanding of the mechanism by which they cooperate in sustaining the turbulent state is lacking. In this report, we examine the role of fluctuation-fluctuation nonlinearity by varying the magnitude of the associated term in the dynamics of Couette flow turbulence to determine how this nonlinear component helps maintain and determine the structure of the turbulent state, and particularly whether this mechanism is parametric or regenerative. Having determined that the mechanism supporting the fluctuation field in Navier-Stokes turbulence is parametric, we then study the mechanism by which the fluctuation component of turbulence is maintained by parametric growth in a time-dependent mean flow by examining the parametric growth mechanism in the frequency domain using analysis of the time-dependent resolvent.

We demonstrate that a separation of the velocity field in large and small scales according to a streamwise Fourier decomposition identifies subspaces with stable Lyapunov exponents and allows the dynamics to exhibit properties of an inertial manifold, such as the synchronization of the small scales in simulations sharing the same large scales or equivalently the decay of all small scale flow states to the state uniquely determined from the large scale flow. This behaviour occurs for deviations with streamwise wavelength smaller than 130 wall units which was shown in earlier studies to correspond to the streamwise spectral peak of the cross-flow velocity components of the top Lyapunov vector of the turbulent flow.

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 ranges 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 asymmetric evolution for weak jets at scales other than the marginal scale (side-band instabilities) for the linear flow-forming instability.

Traditionally, single realizations of the turbulent state have been the object of study in shear flow turbulence. When a statistical quantity was needed it was obtained from a spatial, temporal or ensemble average of sample realizations of the turbulence. However, there are important advantages to studying the dynamics of the statistical state (the SSD) directly. In highly chaotic systems statistical quantities are often the most useful and the advantage of obtaining these statistics directly from a state variable is obvious. Moreover, quantities such as the probability density function (pdf) are often difficult to obtain accurately by sampling state trajectories even if the pdf is stationary. In the event that the pdf is time dependent, solving directly for the pdf as a state variable is the only alternative. However, perhaps the greatest advantage of the SSD approach is conceptual: adopting this perspective reveals directly the essential cooperative mechanisms among the disparate spatial and temporal scales that underly the turbulent state. While these cooperative mechanisms have distinct manifestation in the dynamics of realizations of turbulence both these cooperative mechanisms and the phenomena associated with them are not amenable to analysis directly through study of realizations as they are through the study of the associated SSD. In this review a selection of example problems in the turbulence of planetary and laboratory flows is examined using recently developed SSD analysis methods in order to illustrate the utility of this approach to the study of turbulence in shear flow.

Planetary turbulence is observed to self-organize into large-scale structures such as zonal jets and coherent vortices. One of the simplest models that retains the relevant dynamics of turbulent self-organization is a barotropic flow in a beta-plane channel with turbulence sustained by random stirring. Non-linear integrations of this model show that as the energy input rate of the forcing is increased, the homogeneity of the flow is first broken by the emergence of non-zonal, coherent, westward propagating structures and at larger energy input rates by the emergence of zonal jets. The emergence of both non-zonal coherent structures and zonal jets is studied using a statistical theory, Stochastic Structural Stability Theory (S3T). S3T directly models a second-order approximation to the statistical mean turbulent state and allows the identification of statistical turbulent equilibria and study of their stability. Using S3T, the bifurcation properties of the homogeneous state in barotropic beta-plane turbulence are determined. Analytic expressions for the zonal and non-zonal large-scale coherent flows that emerge as a result of structural instability are obtained and the equilibration of the incipient instabilities is studied through numerical integrations of the S3T dynamical system. The dynamics underlying the formation of zonal jets are also investigated. It is shown that zonal jets form from the upgradient momentum fluxes that result from the shearing of the eddies by the emerging infinitesimal large-scale flow. Finally, numerical simulations of the nonlinear equations confirm the characteristics (scale, amplitude and phase speed) of the structures predicted by S3T, even in highly non-linear parameter regimes such as the regime of zonostrophic turbulence.

Geophysical turbulence is observed to self-organize into large-scale flows such as zonal jets and coherent vortices. Previous studies of barotropic beta-plane turbulence have shown that coherent flows emerge from a background of homogeneous turbulence as a bifurcation when the turbulence intensity increases. The emergence of large-scale flows has been attributed to a new type of collective, symmetry-breaking instability of the statistical state dynamics of the turbulent flow. In this work, we extend the analysis to stratified flows and investigate turbulent self-organization in a two-layer fluid without any imposed mean north–south thermal gradient and with turbulence supported by an external random stirring. We use a second-order closure of the statistical state dynamics, that is termed S3T, with an appropriate averaging ansatz that allows the identification of statistical turbulent equilibria and their structural stability. The bifurcation of the statistically homogeneous equilibrium state to inhomogeneous equilibrium states comprising zonal jets and/or large-scale waves when the energy input rate of the excitation passes a critical threshold is analytically studied. Our theory predicts that there is a large bias towards the emergence of barotropic flows. If the scale of excitation is of the order of (or larger than) the deformation radius, the large-scale structures are barotropic. Mixed barotropic–baroclinic states with jets and/or waves arise when the excitation is at scales shorter than the deformation radius with the baroclinic component of the flow being subdominant for low energy input rates and insignificant for higher energy input rates. The predictions of the S3T theory are compared with nonlinear simulations. The theory is found to accurately predict both the critical transition parameters and the scales of the emergent structures but underestimates their amplitude.

The no-slip boundary condition results in a velocity shear forming in fluid flow near a solid surface. This shear flow supports the turbulence characteristic of fluid flow near boundaries at Reynolds numbers above  1000 by making available to perturbations the kinetic energy of the externally forced flow. Understanding the physical mechanism underlying transfer of energy from the forced mean flow to the turbulent perturbation field that is required to maintain turbulence poses a fundamental question. Although a qualitative understanding that this transfer involves nonlinear destabilization of the roll-streak coherent structure has been established, identication of this instability has resisted analysis. This instability has resisted comprehensive analysis because its analytic expression lies in the Navier-=Stokes equations (NS) expressed with statistical rather than state variables. Expressing NS as a statistical state dynamics (SSD) at second-order in a cumulant expansion suffices to allow analytical identication of the nonlinear roll-streak instability underlying turbulence in wall-bounded shear flow. In this nonlinear instability the turbulent perturbationfield is identied by the SSD, with the Lyapunov vectors of the linear operator governing perturbation evolution about the time-dependent streamwise mean flow. In this work the implications of the predictions of SSD analysis, that this parametric instability underlies the dynamics of turbulence in Couette flow and that the perturbation structures are the associated Lyapunov vectors, are shown to imply new conceptual approaches to controlling turbulence. It is shown that the perturbation component of turbulence is supported by the parametric instability of the streamwise mean flow, which implies optimal control should be formulated to suppress perturbations from the streamwise mean. It is also shown that suppressing only the top few Lyapunov vectors on the streamwise mean vectors results in laminarization. These results are veried by DNS.

Turbulence in wall-bounded shear flow results from a synergistic interaction between linear non-normality, by which a subset of perturbations configured to transfer energy from the mean component of the turbulent state to the perturbation component maintain the perturbation field energy, and nonlinearity by which the subset of energy-transferring perturbations is replenished and maintained in the statistically steady state. Although it is accepted that both linear non-normality mediated energy transfer and nonlinear interactions among perturbations are required to maintain the turbulent state, the detailed physical mechanism by which these processes operate to maintain the turbulent state has not been determined. In this work a statistical state dynamics based analysis is performed demonstrating that the perturbation component in Couette flow turbulence is maintained by a parametric growth process associated with marginal Lyapunov stability of the streamwise mean flow, and that interaction among streamwise varying components of the perturbation field does not contribute positively to the maintenance of the turbulent state. This work identifies the parametric interaction between the fluctuating streamwise mean and the streamwise varying perturbations to be the mechanism of the nonlinear interaction maintaining the turbulent state, and identifies the associated Lyapunov vectors with positive energetics as the energetically active perturbation subspace.

This paper describes a study of the self-sustaining process in wall-turbulence. The study is based on a second order statistical state dynamics model of Couette flow in which the state variables are the streamwise mean flow (first cumulant) and perturbation covariance (second cumulant). This statistical state dynamics model is closed by either setting the third cumulant to zero or by replacing it with a stochastic parameterization. Statistical state dynamics models with this form are referred to as S3T models. S3T models have been shown to self-sustain turbulence with a mean flow and second order perturbation structure similar to that obtained by direct numerical simulation of the equations of motion. The use of a statistical state dynamics model to study the physical mechanisms underlying turbulence has important advantages over the traditional approach of studying the dynamics of individual realizations of turbulence. One advantage is that the analytical structure of S3T statistical state dynamics models isolates the interaction between the mean flow and the perturbation components of the turbulence. Isolation of the interaction between these components reveals how this interaction underlies both the maintenance of the turbulence variance by transfer of energy from the externally driven flow to the perturbation components as well as the enforcement of the observed statistical mean turbulent state by feedback regulation between the mean and perturbation fields. Another advantage of studying turbulence using statistical state dynamics models of S3T form is that the analytical structure of S3T turbulence can be completely characterized. For example, the perturbation component of turbulence in the S3T system is demonstrably maintained by {\color{blue} a parametric perturbation growth mechanism in which fluctuation of the mean flow maintains the perturbation field which in turn maintains the mean flow fluctuations} in a synergistic interaction. Furthermore, the equilibrium statistical state of S3T turbulence can be demonstrated to be enforced by feedback regulation in which transient growth of the perturbations episodically suppresses streak growth preventing runaway parametric growth of the perturbation component. Using S3T to isolate the parametric growth and feedback regulation mechanisms allows a detailed characterization of the dynamics of the self-sustaining process in S3T turbulence with compelling implications for advancing understanding of wall-turbulence.

Coherent jets containing most of the kinetic energy of the flow are a common feature in observations of atmospheric turbulence at the planetary scale. In the gaseous planets these jets are embedded in a field of incoherent turbulence on scales small relative to the jet scale. Large-scale coherent waves are sometimes observed to coexist with the coherent jets and the incoherent turbulence with a prominent example of this phenomenon being the distortion of Saturn's north polar jet (NPJ) into a distinct hexagonal form. Observations of this large-scale jet-wave-turbulence coexistence regime raise the question of identifying the mechanism responsible for forming and maintaining this turbulent state. The coherent planetary scale jet component of the turbulence arises and is maintained by interaction with the incoherent small-scale turbulence component. It follows that theoretical understanding of the dynamics of the jet-wave-turbulence coexistence regime can be facilitated by employing a statistical state dynamics (SSD) model in which the interaction between coherent and incoherent components is explicitly represented. In this work a two-layer \(\beta\)-plane SSD model closed at second order is used to develop a theory that accounts for the structure and dynamics of the NPJ. An asymptotic analysis is performed of the SSD equilibrium in the weak jet damping limit that predicts a universal jet structure in agreement with observations of the NPJ. This asymptotic theory also predicts the wave number of the prominent jet perturbation. Analysis of the jet-wave-turbulence regime dynamics using this SSD model reveals that jet formation is controlled by the effective value of \( \beta \) and the required value of this parameter for correspondence with observation is obtained. As this is a robust prediction it is taken as an indirect observation of a deep poleward sloping stable layer beneath the NPJ. The slope required is obtained from observations of the magnitude of the zonal wind component of the NPJ. The amplitude of the wave-6 perturbation then allows identification of the effective turbulence excitation maintaining this combined structure. The observed jet structure is then predicted by the theory as is the wave-6 disturbance. The wave-6 perturbation, which is identified as the least stable mode of the equilibrated jet, is shown to be primarily responsible for equilibrating the jet with the observed structure and amplitude.

Although the roll-streak structure is ubiquitous in both observations and simulations of pretransitional wall-bounded shear flow, this structure is linearly stable if the idealization of laminar flow is made. Lacking an instability, the large transient growth of the roll-streak structure has been invoked to explain its appearance as resulting from chance occurrence in the background turbulence of perturbations configured to optimally excite it. However, there is an alternative interpretation for the role of the background turbulence in the genesis of the roll-streak structure which is that the background turbulence interacts with the roll-streak structure to destabilize it. Statistical state dynamics (SSD) provides analysis methods for studying instabilities of this type which arise from interaction between the coherent and incoherent components of turbulence. Stochastic structural stability theory (S3T), which implements SSD in the form of a closure at second order, is used in this work to analyze the SSD modes arising from interaction between the coherent streamwise invariant component and the incoherent background turbulence. In pre-transitional Couette flow a manifold of stable modes with roll-streak form is found to exist in the presence of small amplitude background turbulence. The least stable mode of this manifold is destabilized at a critical value of a parameter controlling background turbulence intensity and a stable finite amplitude roll-streak structure arises from this instability through a bifurcation in this parameter. Although this bifurcation has analytical expression only in SSD, it is closely reflected in both the dynamically similar quasi-linear system, referred to as the restricted non-linear (RNL) system, and in the associated nonlinear system (NL). This correspondence is verified using ensemble implementations of the RNL and NL systems. S3T also predicts a second bifurcation at a higher value of the turbulent excitation parameter that results in destabilization of the finite amplitude roll-streak equilibria. This second bifurcation is shown to lead first to time dependence of the roll-streak in the S3T system and then to chaotic fluctuation corresponding to minimal channel turbulence. This transition scenario is verified in simulations of the RNL and NL systems. This bifurcation from a finite amplitude roll-streak equilibrium provides a direct route to the turbulent state through the S3T roll-streak instability.

This paper reviews results obtained using statistical state dynamics (SSD) that demonstrate the benefits of adopting this perspective for understanding turbulence in wall-bounded shear flows. The SSD approach used in this work employs a second-order closure that retains only the interaction between the streamwise mean flow and the streamwise mean perturbation covariance. This closure restricts nonlinearity in the SSD to that explicitly retained in the streamwise constant mean flow together with nonlinear interactions between the mean flow and the perturbation covariance. This dynamical restriction, in which explicit perturbation–perturbation nonlinearity is removed from the perturbation equation, results in a simplified dynamics referred to as the restricted nonlinear (RNL) dynamics. RNL systems, in which a finite ensemble of realizations of the perturbation equation share the same mean flow, provide tractable approximations to the SSD which is equivalent to an infinite ensemble RNL system. This infinite ensemble system, referred to as the stochastic structural stability theory system, introduces new analysis tools for studying turbulence. RNL systems provide computationally efficient means to approximate the SSD, producing self-sustaining turbulence exhibiting qualitative features similar to those observed in direct numerical simulations despite its greatly simplified dynamics. Finally, we show that RNL turbulence can be supported by as few as a single streamwise varying component interacting with the streamwise constant mean flow and that judicious selection of this truncated support or 'band-limiting' can be used to improve quantitative accuracy of RNL turbulence. The results suggest that the SSD approach provides new analytical and computational tools allowing new insights into wall turbulence.

Οταν ο Voyager προσέγγισε τον πλανήτη Κρόνο αποκάλυψε στα υψηλά γεωγραφικά πλάτη ένα εξαγωνικό αεροχείμαρρο μέσης ταχυτήτας της τάξης των 100 m/s. Παρότι ο αεροχείμαρρος αυτός βρίσκεται σε περιβάλλον ισχυρής τύρβης είναι σταθερός και σχεδόν αμετάβλητος. Στην εργασία αυτή μελετούμε την στατιστική δυναμική της τυρβώδους ροής στο περιβάλλον του Κρόνου και δείχνουμε ότι οι ροές στους εξωτερικούς πλανήτες λαμβάνουν ασυμπτωτικά καθολική μορφή συμβατή με τις παρατηρήσεις και προτείνουμε εξήγηση για τη σταθερότητά τους.

When Voyager approached Saturn it was realized that near the North pole of the planet at latitude 74\(^\textrm{o}\) a powerful zonal jet in the shape of a hexagon was located. Despite the powerful turbulence surrounding the jet, the jet appeared to be a steady feature of the planet. In this paper we present a statistical dynamical theory appropriate for turbulent conditions at high latitudes at Saturn and show that the zonal jets that are supported by the ambient turbulent field assume a universal structure that is hydrodynamically stable and agrees with observations.

Η μελέτη της εξέλιξης των μη γραμμικών βέλτιστων διαταραχών (ΜΓΒΔ) μιας διατμητικής ροής προσδιορίζει την αλληλουχία των μηχανισμών που επιφέρουν τη μετάβαση στην τυρβώδη κατάσταση σε ένα ιδεατό περιβάλλον άνευ θορύβου. Η ύπαρξη θορύβου κατά τη μετάβαση είναι όμως ένας σημαντικός παράγοντας που πρέπει να λάβουμε υπόψιν μας. Σε αυτή την εργασία υπολογίζουμε τις ΜΓΒΔ που αντιστοιχούν στο μικρότερο χωρίο που υποστηρίζει τυρβώδη δυναμική σε μία ροή Couette και προσδιορίζουμε τη συνεισφορά των ΜΓΒΔ στην διαδικασία μετάβασης σε περιβάλλον θορύβου.

The study of the evolution of the nonlinear optimal perturbations (NLOP) of wall-bounded flows unveil the mechanisms that lead to transition to turbulence under ideal conditions and the absence of free-stream turbulence. However, free-stream turbulence may play an important role in the transition process. In this work we determine the NLOP in a minimal channel in a Couette flow and investigate the contribution of the NLOPs in the transition process in the presence of free-stream turbulence.

Planetary turbulent flows are observed to self-organize into large scale structures such as zonal jets and coherent vortices. Recently, it was shown that a comprehensive understanding of the properties of these large scale structures and of the dynamics underlying their emergence and maintenance is gained through the study of the dynamics of the statistical state of the flow. Previous studies addressed the emergence of the coherent structures in barotropic turbulence and showed the zonal jets emerge as an instability of the Statistical State Dynamics (SSD). In this work, the equilibration of the incipient instabilities and the stability of the equilibrated jets near onset is investigated. It is shown through a weakly nonlinear analysis of the SSD that the amplitude of the jet evolves according to a Ginzburg—Landau equation. The equilibrated jets were found to have a harmonic structure and an amplitude that is an increasing function of the planetary vorticity gradient. It is also shown that most of the equilibrated jets are unstable and will evolve through jet merging and branching to the stable jets that have a scale close to the most unstable emerging jet.

Planetary turbulent flows are observed to self-organize into large scale structures such as zonal jets and coherent vortices. In this work, the eddy—mean flow dynamics underlying the formation of both zonal and nonzonal coherent structures in a barotropic turbulent flow is investigated within the statistical framework of stochastic structural stability theory (S3T). Previous studies have shown that the coherent structures emerge due to the instability of the homogeneous turbulent flow in the statistical dynamical S3T system and that the statistical predictions of S3T are reflected in direct numerical simulations. In this work, the dynamics underlying the structure forming S3T instability are studied. It is shown that, for weak planetary vorticity gradient beta, both zonal jets and non-zonal large-scale structures form from upgradient momentum fluxes due to shearing of the eddies by the emerging flow. For large beta, the dynamics of the S3T instability differs for zonal and non-zonal flows. Shearing of the eddies by the mean flow continues to be the mechanism for the emergence of zonal jets while non-zonal large-scale flows emerge from resonant and near-resonant triad interactions between the large-scale flow and the stochastically forced eddies.

The structure of turbulence in a reduced model of turbulence (RNL) is analyzed by means of a Proper Orthogonal Decomposition (POD modes). POD analysis was carried out on two different components of the flow, the roll/streak and the perturbation structure. The POD structure in both RNL and direct numerical simulations (DNS) is similar and this correspondence suggests that the dynamics retained in RNL are the essential dynamical ingredients underlying the self-sustaining mechanism of the turbulent state.

The perspective of statistical state dynamics (SSD) has recently been applied to the study of mechanisms underlying turbulence in a variety of physical systems. An SSD is a dynamical system that evolves a representation of the statistical state of the system. An example of an SSD is the second order cumulant closure referred to as stochastic structural stability theory (S3T), which has provided insight into the dynamics of wall turbulence and specifically the emergence and maintenance of the roll/streak structure. S3T comprises a coupled set of equations for the streamwise mean and perturbation covariance, in which nonlinear interactions among the perturbations has been removed, restricting nonlinearity in the dynamics to that of the mean equation and the interaction between the mean and perturbation covariance. In this work, this quasi-linear restriction of the dynamics is used to study the structure and dynamics of turbulence in plane Poiseuille flow at moderately high Reynolds numbers in a closely related dynamical system, referred to as the restricted non-linear (RNL) system. Simulations using this RNL system reveal that the essential features of wall-turbulence dynamics are retained. Consistent with previous analyses based on the S3T version of SSD, the RNL system spontaneously limits the support of its turbulence to a small set of streamwise Fourier components giving rise to a naturally minimal representation of its turbulence dynamics. Although greatly simplified, this RNL turbulence exhibits natural-looking structures and statistics albeit with quantitative differences from those in direct numerical simulations (DNS) of the full equations. Surprisingly, even when further truncation of the perturbation support to a single streamwise component is imposed the RNL system continues to self-sustain turbulence with qualitatively realistic structure and dynamic properties. RNL turbulence at the Reynolds numbers studied is dominated by the roll/streak structure in the buffer layer and similar very-large-scale structure (VLSM) in the outer layer. In this work diagnostics of the structure, spectrum and energetics of RNL and DNS turbulence are used to demonstrate that the roll/streak dynamics supporting the turbulence in the buffer and logarithmic layer is essentially similar in RNL and DNS.

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 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 which 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 energy source exclusively from the incoherent eddy field and provides further insight into the cooperative dynamics of the jet—waves coexistence regime in planetary turbulence.

In this work, we examine the turbulence maintained in a Restricted Nonlinear (RNL) model of plane Couette flow. This model is a computationally efficient approximation of the second order statistical state dynamics obtained by partitioning the flow into a streamwise averaged mean flow and perturbations about that mean, a closure referred to herein as the RNL\(_{\infty}\) model. The RNL model investigated here employs a single member of the infinite ensemble that comprises the covariance of the RNL\(_{\infty}\) dynamics. The RNL system has previously been shown to support self-sustaining turbulence with a mean flow and structural features that are consistent with direct numerical simulations (DNS). Regardless of the number of streamwise Fourier components used in the simulation, the RNL system’s self-sustaining turbulent state is supported by a small number of streamwise varying modes. Remarkably, further truncation of the RNL system’s support to as few as one streamwise varying mode can suffice to sustain the turbulent state. The close correspondence between RNL simulations and DNS that has been previously demonstrated along with the results presented here suggest that the fundamental mechanisms underlying wall-turbulence can be analyzed using these highly simplified RNL systems.

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 b, 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 \(\beta\), the dynamics of the S3T instability differs for zonal and nonzonal flows but in both the destabilizing vorticity fluxes decrease with increasing \(\beta\). 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.

Planetary turbulence is observed to self-organize into large scale structures such as zonal jets and coherent vortices. One of the simplest models that retains the relevant dynamics of turbulent self-organization is a barotropic flow in a beta-plane channel with turbulence sustained by random stirring. Non-linear integrations of this model show that as the energy input rate of the forcing is increased, the homogeneity of the flow is first broken by the emergence of non-zonal, coherent, westward propagating structures and at larger energy input rates by the emergence of zonal jets. The emergence of both non-zonal coherent structures and zonal jets is studied using a statistical theory, Stochastic Structural Stability Theory (S3T). S3T models a second order approximation to the statistical mean state and allows identification of statistical equilibria and study of their stability. It is found that when the homogeneous turbulent state becomes S3T unstable, coherent structures emerge (non-zonal large scale structures and zonal jets). Analytic expressions for their characteristics (scale, amplitude and phase speed) are obtained and their non-linear equilibration is studied numerically. Direct Numerical Simulations of the nonlinear equations show that the structures predicted by S3T dominate the turulent flow.

Διαταραχές που εμφανίζουν μεταβατική αύξηση αποτελούν έναν αποτελεσματικό τρόπο για να μεταφερθεί ενέργεια από μια μέση ροή στις διαταραχές και για να έχουμε μετάβαση στην τυρβώδη κατάσταση. Εξετάζουμε στην περίπτωση μιας ροής Poiseuille 2D σε αριθμό \(Re=4000\) εάν οι μη γραμμικές βέλτιστες διαταραχές αποτελούν τις καταλληλότερες αρχικές συνθήκες για να καταλήξουν στη ζώνη ημιισορροπίας και συν τω χρόνω στην χρονοεξαρτώμενη ευσταθή λύση της ροής στισ δύο διαστάσεις. Ο προσδιορισμός των μη γραμμικών διαταραχών γίνεται με μεθόδους συζυγούς βελτιστοποίησης. Συμπεραίνουμε ότι σε 2 διαστάσεις οι μη γραμμικές βέλτιστες διαταραχές δεν αποφέρουν σημαντική βελτίωση της απόδοσης και δεν ακολουθούν μια διαφορετική διαδικασία διέγερσης του άνω κλάδου της δευτερογενούς λύσης.

Transiently growing disturbances are an efficient way to gain energy from a mean velocity profile and trigger transition. In the case of a 2D Poiseuille flow at \(Re = 4000\) we are interested in determining whether the nonlinear optimal disturbances are the most efficient initial conditions for reaching the zone of quasi-equilibria and eventually exciting the upper branch solution of the flow. We conclude that in two dimensions nonlinear optimals do not result in a significant improvement of energy growth and do not produce an alternative path for the excitation of the upper-branch solution.

Παρουσιάζουμε αριθμητικές προσομοιώσεις στις οποίες συντηρείται τυρβώδης κατάσταση στα \(Re_\tau = 970\) με μία μόνο μη μηδενική αρμονική στη διεύθυνση της ροής και χωρίς καμία άλλη παρέμβαση στις εξισώσεις Navier-Stokes. Η τυρβώδης κατάσταση λειτουργεί με τον αναγεννητικό κύκλο (self-sustaining process – SSP) που συντηρεί την τυρβώδη κατάσταση σε διατμητικές τυρβώδεις ροές. Η εύρεση της απλοποιημένης αυτής τυρβώδους κατάστασης μπορεί να οδηγήσει στη κατανόηση του αναγεννητικού κύκλου SSP και στην εξεύρεση μεθόδων ελέγχου της τυρβώδους ροής πέραν της γραμμικής θεωρίας.

We present numerical simulations which show that a realistic and self-sustaining turbulent state at \(Re_\tau = 970\) can be maintained with a single nonzero Fourier streamwise component without any other modification of the Navier-Stokes equations. The turbulent state is operating with the characteristic self-sustaining process (SSP) that operates in the inner wall region. This simplified turbulent state can lead to understanding of the dynamics of the SSP and also serve as the platform for designing control strategies of the turbulent state that go beyond the already existing linear strategies.

This paper demonstrates the maintenance of self-sustaining turbulence in a restricted nonlinear (RNL) model of plane Couette flow. The RNL system is derived directly from the Navier-Stokes equations and permits higher resolution studies of the dynamical system associated with the stochastic structural stability theory (S3T) model, which is a second order approximation of the statistical state dynamics of the flow. The RNL model shares the dynamical restrictions of the S3T model but can be easily implemented by reducing a DNS code so that it retains only the RNL dynamics. Comparisons of turbulence arising from DNS and RNL simulations demonstrate that the RNL system supports self-sustaining turbulence with a mean flow as well as structural and dynamical features that are consistent with DNS. These results demonstrate that the simplified RNL system captures fundamental aspects of fully developed turbulence in wall-bounded shear flows and motivate use of the RNL/S3T framework for further study of wall-turbulence.

Atmospheric turbulence is observed to self-organize into large scale structures such as zonal jets and coherent vortices. One of the simplest models that retains the relevant dynamics is a barotropic flow in a beta-plane channel with turbulence sustained by random stirring. Non- linear integrations of this model show that as the energy input rate of the forcing is increased, the homogeneity of the flow is first broken by the emergence of non-zonal, coherent, westward propagating structures and at larger energy input rates by the emergence of zonal jets. We study the emergence of non-zonal coherent structures using a statistical theory, Stochastic Structural Stability Theory (S3T). S3T directly models a second order approximation to the statistical mean turbulent state and allows identification of statistical turbulent equilibria and study of their stability. We find that when the homogeneous turbulent state becomes S3T unstable, non-zonal large scale structures emerge and we obtain analytic expressions for their characteristics (scale, amplitude and phase speed). Numerical simulations of the non-linear equations are found to reproduce the characteristics of the structures predicted by S3T.

Turbulent fluids often appear to self-organize forming large-scale zonal structures. Examples from meteorology are the midlatidute polar jet in the Earth’s atmosphere and the zonal winds in the atmosphere of Jupiter. These large-scale zonal structures are formed and also maintained by the small-scale baroclinic or barotropic turbulence with which they coexist. We present a new theory, named S3T, that explains the emergence and equilibration at finite amplitude of large-scale zonal flows in planetary turbulence. We apply this theory to make predictions for the emergence of zonal flows from a background of homogeneous turbulence as a function of parameters, in a barotropic fluid on a beta-plane. We show that the transition of a homogeneous turbulent state to an inhomogeneous state, dominated by large-scale zonal flows, occurs as a bifurcation phenomenon. We also show the accuracy of the theory by comparing its predictions to non-linear numerical simulations of the turbulent fluid. This theory provides a vehicle for studying the structural stability of large-scale atmospheric flows and can be used to determine climate sensitivity.

S3T (Stochastic Structural Stability Theory) employs a closure at second order to obtain the dynamics of the statistical mean turbulent state. When S3T is implemented as a coupled set of equations for the streamwise mean and perturbation states, nonlinearity in the dynamics is restricted to interaction between the mean and perturbations. The S3T statistical mean state dynamics can be approximately implemented by similarly restricting the dynamics used in a direct numerical simulation (DNS) of the full Navier-Stokes equations (referred to as the NS system). Although this restricted nonlinear system (referred to as the RNL system) is greatly simplified in its dynamics in comparison to the associated NS, it nevertheless self-sustains a turbulent state in wall-bounded shear flow with structures and dynamics comparable to that in observed turbulence. Moreover, RNL turbulence can be analyzed effectively using theoretical methods developed to study the closely related S3T system. In order to better understand RNL turbulence and its relation to NS turbulence, an extensive comparison is made of diagnostics of structure and dynamics in these systems. Although quantitative differences are found, the results show that turbulence in the RNL system closely parallels that in NS and suggest that the S3T/RNL system provides a promising reduced complexity model for studying turbulence in wall-bounded shear flows.

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.

Planetary turbulent flows are observed to self-organize into large scale structures such as zonal jets and coherent vortices. One of the simplest models of planetary turbulence is obtained by considering a barotropic flow on a beta-plane channel with turbulence sustained by random stirring. Non-linear integrations of this model show that as the energy input rate of the forcing is increased, the homogeneity of the flow is broken with the emergence of non-zonal, coherent, westward propagating structures and at larger energy input rates by the emergence of zonal jets. We study the emergence of non-zonal coherent structures using a non-equilibrium statistical theory, Stochastic Structural Stability Theory (S3T, previously referred to as SSST). S3T directly models a second order approximation to the statistical mean turbulent state and allows identification of statistical turbulent equilibria and study of their stability. Using S3T, the bifurcation properties of the homogeneous state in barotropic beta-plane turbulence are determined. Analytic expressions for the zonal and non-zonal large scale coherent flows that emerge as a result of structural instability are obtained. Through numerical integrations of the S3T dynamical system, it is found that the unstable structures equilibrate at finite amplitude. Numerical simulations of the nonlinear equations confirm the characteristics (scale, amplitude and phase speed) of the structures predicted by S3T.

In this Letter, we use a nonequilibrium statistical theory, the stochastic structural stability theory (SSST), to show that an extended version of this theory can make predictions for the formation of nonzonal as well as zonal structures (lattice and stripe patterns) in forced homogeneous turbulence on a barotropic \(\beta\) plane. Comparison of the theory with nonlinear simulations demonstrates that SSST predicts the parameter values for the emergence of coherent structures and their characteristics (scale, amplitude, phase speed) as they emerge and at finite amplitude. It is shown that nonzonal structures (lattice states or zonons) emerge at lower energy input rates of the stirring compared to zonal flows (stripe states) and their emergence affects the dynamics of jet formation.

Zonal jets are commonly observed to spontaneously emerge in a \(\beta\)-plane channel from a background of turbulence that is sustained in a statistical steady state by homogeneous stochastic excitation and dissipation of vorticity. The mechanism for jet formation is examined in this work within the statistical wave–mean flow interaction framework of stochastic structural stability theory (SSST) that makes predictions for the emergence of zonal jets in \(\beta\)-plane turbulence. Using the coupled dynamical SSST system that governs the joint evolution of the second-order statistics and the mean flow, the structural stability of the spatially homogeneous statistical equilibrium with no mean zonal jets is studied. It is shown that close to the structural stability boundary, the eddy–mean flow dynamics can be split into two competing processes: The first, which is shearing of the eddies by the local shear described by Orr dynamics in a b plane, is shown in the limit of infinitesimal shear to lead to the formation of jets. The second, which is momentum flux divergence resulting from lateral wave propagation on the nonuniform local mean vorticity gradient, is shown to oppose jet formation. The upgradient momentum fluxes due to shearing of the eddies are shown to act exactly as negative viscosity for an anisotropic forcing and as negative hyperviscosity for isotropic forcing. The downgradient fluxes due to wave flux divergence are shown to act hyperdiffusively.

Tropical cyclones are among the most life threatening and destructive natural phenomena on Earth. A dynamical mechanism for cyclone intensification that has been proposed is based on the idea that patches of high vorticity associated with individual convective systems are quickly axisymmetrized, feeding their energy into the circular vortex. In this work, Stochastic Structural Stability Theory (SSST) is used to achieve a comprehensive understanding of this physical mecha- nism. According to SSST, the distribution of momentum fluxes arising from the field of asymmetric eddies associated with a given mean vortex structure, is obtained using a linear model of stochastic turbulence. The resulting momentum flux distribution is then coupled with the equation governing the evolution of the mean vortex to produce a closed set of eddy/mean vortex equations. We apply the SSST tools to a two dimensional, non-divergent model of stochastically forced asymmetric eddies. We show that the process intensifying a weak vortex is shearing of asymmetric eddies with small azimuthal scale that produces upgradient fluxes. For stochastic forcing with amplitude larger than a certain threshold, these upgradient fluxes lead to a structural instability of the eddy/mean vortex system and to an exponentially growing vortex.

The prominence of streamwise elongated structures in wall-bounded shear flow turbu- lence previously motivated turbulence investigations using streamwise constant (2D/3C) and streamwise averaged (SSST) models. Results obtained using these models imply that the statistical mean turbulent state is in large part determined by streamwise constant structures, particularly the well studied roll and streak. In this work the role of stream- wise structures in transition and turbulence is further examined by comparing theoretical predictions of roll/streak dynamics made using 2D/3C and SSST models with DNS data. The results confirm that the 2D/3C model accurately obtains the turbulent mean veloc- ity profile despite the fact that it only includes one-way coupling from the cross-stream perturbations to the mean flow. The SSST system augments the 2D/3C model through the addition of feedback from this streamwise constant mean flow to the dynamics of streamwise varying perturbations. With this additional feedback, the SSST system sup- ports a perturbation/mean flow interaction instability leading to a bifurcation from the laminar mean flow to a self-sustaining turbulent state. Once in this self-sustaining state the SSST collapses to a minimal representation of turbulence in which a single streamwise perturbation interacts with the mean flow. Comparisons of DNS data with simulations of this self-sustaining state demonstrate that this minimal representation of turbulence produces accurate statistics for both the mean flow and the perturbations. These results suggest that SSST captures fundamental aspects of the mechanisms underlying transi- tion to and maintenance of turbulence in wall-bounded shear flows.

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.

Highly stratified shear layers are rendered unstable even at high stratifications by Holmboe instabilities when the density stratification is concentrated in a small region of the shear layer. These instabilities may cause mixing in highly stratified environments. However, these instabilities occur in limited bands in the parameter space. We perform Generalized Stability analysis of the two dimensional perturbation dynamics of an inviscid Boussinesq stratified shear layer and show that Holmboe instabilities at high Richardson numbers can be excited by their adjoints at amplitudes that are orders of magnitude larger than by introducing initially the unstable mode itself. We also determine the optimal growth that is obtained for parameters for which there is no instability. We find that there is potential for large transient growth regardless of whether the background flow is exponentially stable or not and that the characteristic structure of the Holmboe instability asymptotically emerges as a persistent quasi-mode for parameter values for which the flow is stable.

Large-scale mean flows often emerge in turbulent fluids. In this work, we formulate a stability theory, the stochastic structural stability theory (SSST), for the emergence of jets under external random excitation. We analytically investigate the structural stability of a two-dimensional homogeneous fluid enclosed in a channel and subjected to homogeneous random forcing. We show that two generic competing mechanisms control the instability that gives rise to the emergence of an infinitesimal jet: advection of the eddy vorticity by the mean flow that is shown to be jet forming and advection of the vorticity gradient of the jet by the eddies that is shown to hinder the formation of the mean flow. We show that stochastic forcing with small streamwise coherence and an amplitude larger than a certain threshold leads to the emergence of jets in the channel through a bifurcation of the non-linear SSST system.

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.

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.

Coherent jets, such as the Jovian banded winds, are a prominent feature of rotating turbulence. Shallow- water turbulence models capture the essential mechanism of jet formation, which is systematic eddy mo- mentum flux directed up the mean velocity gradient. Understanding how this systematic eddy flux conver- gence 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 a 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 weak turbulence, and for parameters appropriate to Jupiter, both prograde and retrograde equatorial jets are found to be stable solutions of the SSST system, but only the prograde equatorial jet remains stable in strong turbulence. In addition to the equatorial jet, multiple mid- latitude zonal jets are also maintained in these stable SSST equilibria. These midlatitude jets have structure and spacing in agreement with observed zonal jets and exhibit the observed robust reversals in sign of both absolute and potential vorticity gradient.

Shear flows with a free surface possess diverse branches of modal instabilities. By approximating the mean flow with a piecewise linear profile, an understanding and classification of the instabilities can be achieved by studying the interaction of the edge waves that arise at the density discontinuity at the surface and the vorticity waves that are supported at the mean vorticity gradient discontinuities in the interior. The various branches of instability are identified and their physical origin is clarified. The edge waves giving rise to the modal instabilities can also lead to a modest transient growth that extends into the regions of neutrality of the flow. However, when the continuous spectrum is excited substantial transient growth can arise and the optimal perturbations attain greater energy when compared with the energy of the fastest modal growing perturbation. These optimal perturbations utilize the continuous spectrum to excite at large amplitude the neutral or amplifying modes of the system.

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 struc- ture, 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 obtained 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.

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 \(\beta\)-plane turbulence.

In this paper, the emission of internal gravity waves from a local westerly shear layer is studied. Thermal and/or vorticity forcing of the shear layer with a wide range of frequencies and scales can lead to strong emission of gravity waves in the region exterior to the shear layer. The shear flow not only passively filters and refracts the emitted wave spectrum, but also actively participates in the gravity wave emission in conjunction with the distributed forcing. This interaction leads to enhanced radiated momentum fluxes but more importantly to enhanced gravity wave energy fluxes. This enhanced emission power can be traced to the nonnormal growth of the perturbations in the shear region, that is, to the transfer of the kinetic energy of the mean shear flow to the emitted gravity waves. The emitted wave energy flux increases with shear and can become as large as 30 times greater than the corresponding flux emitted in the absence of a localized shear region.
Waves that have horizontal wavelengths larger than the depth of the shear layer radiate easterly momentum away, whereas the shorter waves are trapped in the shear region and deposit their momentum at their critical levels. The observed spectrum, as well as the physical mechanisms influencing the spectrum such as wave interference and Doppler shifting effects, is discussed. While for large Richardson numbers there is equipartition of momentum among a wide range of frequencies, most of the energy is found to be carried by waves having vertical wavelengths in a narrow band around the value of twice the depth of the region. It is shown that the waves that are emitted from the shear region have vertical wavelengths of the size of the shear region.

Minimizing forecast error requires accurately specifying the initial state from which the forecast is made by optimally using available observing resources to obtain the most accurate possible analysis. The Kalman filter accomplishes this for linear systems and experience shows that the extended Kalman filter also performs well in nonlinear systems. Unfortunately, the Kalman filter and the extended Kalman filter require computation of the time dependent error covariance matrix which presents a daunting computational burden. However, the dynamically relevant dimension of the forecast error system is generally far smaller than the full state dimension of the forecast model which suggests the use of reduced order error models to obtain near optimal state estimators. A method is described and illustrated for implementing a Kalman filter on a reduced order approximation of the forecast error system. This reduced order system is obtained by balanced truncation of the Hankel operator representation of the full error system. As an example application a reduced order Kalman filter is constructed for a time-dependent quasi-geostrophic storm track model. The accuracy of the state identification by the reduced order Kalman filter is assessed and comparison made to the state estimate obtained by the full Kalman filter and to the estimate obtained using an approximation to 4D-Var. The accuracy assessment is facilitated by formulating the state estimation methods as observer systems. A practical approximation to the reduced order Kalman filter that utilizes 4D-Var algorithms is examined.

Understanding of the stability of deterministic and stochastic dynamical systems has evolved recently from a traditional grounding in the system’s normal modes to a more comprehensive foundation in the system’s propagator and especially in an appreciation for the role of non-normality of the dynamical operator in determining the system’s stability as revealed through the propagator. This set of ideas which approach stability analysis from a non-modal perspective will be referred to as Generalized Stability Theory (GST). Some applications of GST to deterministic and statistical forecast are discussed in this review. Perhaps the most familiar of these applications is identifying initial perturbations resulting in greatest error in deterministic error systems which is in use for ensemble and targeting applications. But of increasing importance is elucidating the role of temporally distributed forcing along the forecast trajectory and obtaining a more comprehensive understanding of the prediction of statistical quantities beyond the horizon of deterministic prediction. The optimal growth concept can be extended to address error growth in nonautonomous systems in which the fundamental mechanism producing error growth can be identified with the necessary non-normality of the system. The influence of model error in both the forcing and the system is examined using the methods of stochastic dynamical systems theory. In this review deterministic and statistical prediction, that is forecast and climate prediction, are separately discussed.

Temporally distributed deterministic and stochastic excitation of the tangent linear forecast system governing forecast error growth and the tangent linear observer system governing assimilation error growth is examined. The method used is to determine the optimal set of distributed deterministic and stochastic forcings of the forecast and observer systems over a chosen time interval. Distributed forcing of an unstable system addresses the effect of model error on forecast error in the presumably unstable forecast error system. Distributed forcing of a stable system addresses the effect on the assimilation of model error in the presumably stable data assimilation system viewed as a stable observer. In this study, model error refers both to extrinsic physical error forcing, such as that which arises from unresolved cumulus activity, and to intrinsic error sources arising from imperfections in the numerical model and in the physical parameterizations.

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 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.

In studies of perturbation dynamics in physical systems, certain specification of the governing perturbation dynamical system is generally lacking, either because the perturbation system is imperfectly known or because its specification is intrinsically uncertain, while a statistical characterization of the perturbation dynamical system is often available. 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 systems 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. 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.

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.

Minimizing forecast error requires accurately specifying the initial state from which the forecast is made by optimally using available observing resources to obtain the most accurate possible analysis. The Kalman filter accomplishes this for a wide class of linear systems, and experience shows that the extended Kalman filter also performs well in nonlinear systems. Unfortunately, the Kalman filter and the extended Kalman filter require computation of the time-dependent error covariance matrix, which presents a daunting computational burden. However, the dynamically relevant dimension of the forecast error system is generally far smaller than the full state dimension of the forecast model, which suggests the use of reduced-order error models to obtain near- optimal state estimators. A method is described and illustrated for implementing a Kalman filter on a reduced- order approximation of the forecast error system. This reduced-order system is obtained by balanced truncation of the Hankel operator representation of the full error system and is used to construct a reduced-order Kalman filter for the purpose of state identification in a time-dependent quasigeostrophic storm track model. The accuracy of the state identification by the reduced-order Kalman filter is assessed by comparison to the true state, to the state estimate obtained by the full Kalman filter, and to the state estimate obtained by direct insertion.

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 \(E = O(1/\textrm{Ri})\), where \(\textrm{Ri}\) 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.

Methods for approximating a stable linear autonomous dynamical system by a system of lower order are examined. Reducing the order of a dynamical system is useful theoretically in identifying the irreducible dimension of the dynamics and in isolating the dominant spatial structures supporting the dynamics, and practically in providing tractable lower-dimension statistical models for climate studies and error covariance models for forecast analysis and initialization. Optimal solution of the model order reduction problem requires simultaneous representation of both the growing structures in the system and the structures into which these evolve. For autonomous operators associated with fluid flows a nearly optimal solution of the model order reduction problem with prescribed error bounds is obtained by truncating the dynamics in its Hankel operator representation. Simple model examples including a reduced-order model of Couette flow are used to illustrate the theory. Practical methods for obtaining approximations to the optimal order reduction problem based on finite-time singular vector analysis of the propagator are discussed and the accuracy of the resulting reduced models evaluated.

In this work we study the growth of perturbations in Keplerian disks. Despite the asymptotic stability of the disk, a subset of optimal perturbations are found to exhibit large transient growth. The transient growth is due to the nonnormality of the underlying operator which governs the perturbation dynamics. It is shown that the amplifying perturbations produce positive momentum fluxes and a tendency of outward angular momentum expulsion during amplification. We calculate the statistical steady state that emerges under white forcing in space and time. The perturbation structure is found to be organized in coherent structures that invariably export angular momentum outward. The radial structure of the resulting angular momentum flux is in agreement with the predictions of the equilibrium theory of accretion disks. The e†ect of spatial localization and temporal band limiting of the forcing on the maintained momentum Ñuxes is investigated. We Ðnd that if the forcing is broadband and adequately distributed, accretion to the main body can be maintained by stochastic forcing.

A comprehensive assessment is made of transient and asymptotic two-dimensional perturbation growth in compressible shear flow using unbounded constant shear and the Couette problem as examples. The unbounded shear flow example captures the essential dynamics of the rapid transient growth processes at high Mach numbers, while excitation by nonmodal mechanisms of nearly neutral modes supported by boundaries in the Couette problem is found to be important in sustaining high perturbation amplitude at long times. The optimal growth of two-dimensional perturbations in viscous high Mach number flows in both unbounded shear flow and the Couette problem is shown to greatly exceed the optimal growth obtained in incompressible flows at the same Reynolds number.

Current understanding of how chemical sources and sinks in the atmosphere interact with the physical processes of advection and diffusion to produce local and global distributions of constituents is based primarily on analysis of chemical models. One example of an application of chemical models which has important implications for global change is to the problem of determining sensitivity of chemical equilibria to changes in natural and anthropogenic sources. This sensitivity to perturbation is often summarized by quantities such as a mean lifetime of a chemical species estimated from reservoir turnover time or the decay rate of the least damped normal mode of the species obtained from eigenanalysis of the linear perturbation equations. However, the decay rate of the least damped normal mode or a mean lifetime does not comprehensively reveal the response of a system to perturbation. In this work, sensitivity to perturbations of chemical equilibria is assessed in a comprehensive manner through analysis of the system propagator. When chemical perturbations are measured using the proper linear norms, it is found that the greatest disturbance to chemical equilibrium is achieved by introducing a single chemical species at a single location, and that this optimal perturbation can be easily found by a single integration of the transpose of the dynamical system. Among other results are determination of species distributions produced by impulsive, constant, and stochastic forcing; release sites producing the greatest and least perturbation in a chosenconstituent at another chosen site; and a critical assessment of chemical lifetime measures. These results are general and apply to any perturbation chemical model, including three-dimensional global models, provided the perturbations are sufficiently small that the perturbation dynamics are linear.

An energy transfer analysis of turbulent plane Couette flow is performed. It is found that nonlinear interaction between the \( [0, \pm 1]\) modes is principally responsible for maintaining the mean streamwise turbulent velocity profile. The \([0,\pm 1]\) modes extract energy from the laminar flow by linear non-modal growth mechanisms and transfer it directly to the mean flow mode. The connection of this work to linear/nonlinear models of transition is discussed.

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 dependence 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 large-scale magnetic fields of stellar and galactic bodies are generally understood to be organized and amplified by motions in the conducting fluid media of these bodies. This article examines a mechanism by which continual excitation of the conducting fluid by small-scale fields results in production of large-scale fields. The excitation of the induction equation by small-scale fields is parameterized as stochastic forcing, and the crucial role of the nonnormality of the induction operator in determining the spatial and temporal structure of variation in the large-scale fields is emphasized. A cylindrically symmetric helical flow is used to provide illustrative examples.

The mechanism by which large-scale magnetic Ðelds in stars and galaxies arise remains uncertain, but it is believed that initially small internally generated or primordial seed fields are amplified and organized by motions in the conducting fluid interiors of these bodies. Methods for analyzing this process in the weak field limit are based on the induction equation and fall into two classes: those involving advection of the magnetic field as a passive tracer, and those involving calculation of exponential instabilities. The former is a nonmodal stability analysis, while the latter is essentially modal. In this work these two methods of analysis are synthesized, making use of recent advances in the theory of nonnormal system dynamics. An application of this generalized stability analysis to the helical dynamo model of Lortz is described in which the maximum field growth over prescribed time intervals and the perturbation structures producing this growth are identified.

The strong mean shear in the vicinity of the boundaries in turbulent boundary layer flows preferentially 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 perturbations 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.

It has recently been recognized that the non-normality of the dynamical operator obtained by the linearization of the equations of motion about the strongly sheared background flow plays a central role in the dynamics of both fully developed turbulence and laminar/turbulent transition. This advance has led to the development of a deterministic theory for the role of coherent structures in shear turbulence as well as a stochastic theory for the maintenance of the turbulent state. In this work the theory of stochastically forced non-normal dynamical systems is extended to explore the possibility of controlling the transition process and of suppressing fully developed shear turbulence. Modeling turbulence as a stochastically forced non-normal dynamical system allows a great variety of control strategies to be explored and their physical mechanism understood. Two distinct active control mechanisms have been found to produce suppression of turbulent energy by up to 70%. A physical explanation of these effective control mechanisms is given and possible applications are discussed.

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 analyzing 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.

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.

Recently, a new theoretical and conceptual model of quasigeostrophic turbulence has been advanced in whιch 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 cons1deratton ts limited to the response of the system's individual normal modes to stochastic excitation.

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 nonlinear processes or externally imposed. This physical property of enhanced response to a subset of perturbations is expressed analytically through the nonnonnality of the 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 nonnormality of the dynamical operator linearized about the background flow. While the initial value problem provides an explanation for 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 the imposed large-scale thermal forcing. lo 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 level of variance maintained in a stochastically forced asymptotically 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 flow 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.

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 cen.tral goal of dynamic meteorology and climate studies. Recently, stochastic forcing of highly nonnorrnal 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 nonnorrnal 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.

The gravitational tidal response at the visible cloud level of Jupiter is obtained as a function of static stability in the planetary interior. It is suggested that confirmation of the presence of static stability in the planetary interior could be achieved by observing tidal fields at cloud level. We also calculate the mean How accelerations induced by tidal fields and suggest that, if the interior is even marginally statically stable, the tides may provide the momentum source maintaining the alternating zonal jets observed at the cloud level of the planet.

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.

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 non normal 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 (EΟFs) 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.

Disturbance structures that achieve maximum growth over a specified interval of time have recently been obtained for unbounded constant shear flow making use of closed-form solutions. Optimal perturbations have also been obtained for the canonical bounded shear flows, the Couette, and plane Poiseuille flows, using numerical solution of the linearized Navier-Stokes equations. In this note it is shown that these optimal perturbations have similar spectra and structure indicating an underlying universality of traditional methods.

The three-dimensional perturbations to viscous constant shear flow that increase maximally in energy over a chosen time interval are obtained by optimizing over the complete set of analytic solutions. These optimal perturbations are intrinsically three dimensional, of restricted morphology, and exhibit large energy growth on the advective time scale, despite the absence of exponential normal modal instability in constant shear flow. The optimal structures can be interpreted as combinations of two fundamental types of motion associated with two distinguishable growth mechanisms: streamwise vortices growing by ‘advection of mean streamwise velocity to form streamwise streaks, and upstream tilting waves growing by the down gradient Reynolds stress mechanism of two-dimensional shear instability. The optimal excitation over a chosen interval of time comprises a combination of these two mechanisms, characteristically giving rise to tilted roll vortices with greatly amplified perturbation energy. It is suggested that these disturbances provide the initial growth leading to transition to turbulence, in addition to providing an explanation for coherent structures in a wide variety of turbulent shear flows.

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 oflinear 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 oflinear and nonlinear damped modes is important in maintaining high levels of variance by the mechanism of stochastic excitation.

Transient development of perturbations in inviscid stratified shear flow is investigated. Use is made of closed form analytic solutions that allow concise identification of optimally growing plane-wave solutions for the case of an unbounded flow with constant shear and stratification. For the case of channel flow, variational techniques are employed to determine the optimally growing disturbances.
The maximum energy growth attained over a specific time interval decreases continuously with increasing stratification, and no special significance attaches to \(\textrm{Ri} = 0.25\). Indeed, transient growth can be substantial even for \(\textrm{Ri} = O( 1 )\). A general lower bound on the energy growth attained by an optimal perturbation in a stratified flow over a given time interval is the square root of the growth attained by the corresponding perturbation in unstratified flow. Enhanced perturbation persistence is found for mean-flow stratification lying in the range \(0.1 < \textrm{Ri} < 0.3\). Small but finite perturbations in mean flow with \(\textrm{Ri} < 0.4\) produce regions with locally negative total density gradient, which are expected to overturn. Although the perturbations are of wave form, buoyancy fluxes mediate transfer between perturbation kinetic and potential energy during transient development, thus implying that buoyancy flux is not a determinative diagnostic for distinguishing between waves and turbulence in stratified flows.

The theory of excitation of tidal oscillations in a fluid planetary body is formulated, and separable equations are derived that extend the results of the classical theory of tides to the nonhydrostatic interiors of planets. The theory is applied to the example of the gravitational tidal response of Jupiter to forcing by Io. The tidal response is found to crucially depend on the static stability in the interior of the planet, the response of the planet being as much as two to three orders of magnitude greater than the response with a neutral interior. The tidal dissipation factor \(Q\) is calculated for Jupiter and found to agree with the values required by the astronomical arguments only if the interior has finite though small static stability. We are led to the conclusion that the interior of Jupiter must have regions which are stably stratified.

Classical tidal theory is applied to the gravitational excitation of the atmospheres of the gaseous planets. The only departure made from classical theory is the retention of the effects of nonhydrostaticity which are important in the deeper atmosphere or wherever one expects extremely small static stability. The meridional structure of the tidal response is shown to depend only on the ratio of the period of gravitational forcing to the period of rotation of the planet. Forcing by the low-inclination orbits of the satellites of Jupiter, Saturn, and Uranus excites primarily symmetric Hough modes. Consideration of the vertical structure equation shows that although the gravitational tidal forcing is proportional to the first symmetric spherical harmonic with zonal wavenumber 2, the tidal response will be concentrated in higher order meridional structures confined equatorward of 50° N on Jupiter, 76° N on Saturn, and 45° N on Uranus. The meridional structure of these modes resembles the visible banding on these planets. The excitation of the tides depends on the distribution of static stability in the interior. Estimates are made showing that observation of the tidal response of the planets at the visible cloud level may be within reach of current observational capability. Detection of this signal is shown to provide information about the thermodynamic structure of the interior.
A primary purpose of the present paper, in addition to the above, is the presentation of computational results concerning the eigenvalues and eigenfunctions relevant to gravitational tides in the outer planets.

The baroclinic instability of a frontal mean state is investigated using the WKBJ approximation. The results are compared with numerical calculations performed on the same mean state. Excellent agreement (within 5%) is found for jets whose half-width is as small as a Rossby radius of deformation. For jets 20% broader, the agreement is almost perfect.

A formalism is developed for the calculation ofbaroclinic instability for barotropically stable jets. The formalism is applied to jet versions of both the Eady and Charney problems. It is found that jets act to confine instabilities meridionally, thus internally determining meridional wave scales. Once this internally determined meridional scale is taken into account, results correspond plausibly to classical results without a jet.
Consideration of the effect of such instabilities on the mean flow shows that they act to concentrate the Jet barotropically while simultaneously reducing baroclinicity.