arXiv subjects

Petros J. Ioannou

Publications and source records attributed to Petros J. Ioannou.

At least 19 recordsLinked to original sources

arXiv2026

Statistical State Dynamics Eigenmodes and Equilibria support Couette Turbulence

Wide-channel Couette (WCC) turbulence consists primarily of quasi-steady roll-streak structures (RSS) maintained by a self-sustaining process (SSP), yet the Navier-Stokes equations in velocity variables admit no linear RSS instability or stable equilibrium, leaving the WCC turbulent state's analytic basis obscure. We show that, in the statistical state dynamics (SSD) of a second-order closure of the Navier-Stokes equations, Couette turbulence arises from a modal RSS instability and equilibrates as a stable fixed point at spanwise wavenumber 3. This fixed point comprises a streamwise-mean flow and a pair of counterpropagating neutral eigenmodes, regularized by roll advection rather than viscosity. This stable equilibrium identifies both a canonical turbulence solution and its self-sustaining process (SSP), and characterizes their analytic structure. WCC turbulence arises as a spanwise tiling by this stable fixed-point RSS unit cell. Predictions of this SSD theory are corroborated by comparison with direct numerical simulation.

physics.flu-dyn↗

arXiv2026

Statistical state dynamics modes and equilibria underlie the structure and mechanism of wide channel Couette turbulence

Wide channel Couette (WCC) turbulence is striking in being dominated by a large-scale spanwise periodic structure composed of streamwise streaks and associated roll superstructures. This apparent equilibrium is shown in this work to correspond to a fixed point solution of the Navier-Stokes equations expressed in the statistical state dynamics (SSD) framework. Moreover, this fixed point solution is found to be rank-three, consisting of one analytically identified roll-streak structure (RSS) constituting the first cumulant of the SSD, and two analytically determined eigenmodes supporting the second cumulant. This minimal representation captures both the structure and the dynamics of WCC turbulence, while the remaining spectral components contribute negligibly to the equilibrium dynamics. Turbulent Couette flows other than WCC can be understood to be limit cycle and chaotic extensions supported by the same underlying mechanism as the fixed point WCC turbulence except for the two eigenmodes supporting WCC turbulence being replaced by two Floquet modes and two Lyapunov vectors, respectively. These results provide an analytic solution for turbulence in Couette flow.

physics.flu-dyn↗

arXiv2024

Fluctuation covariance-based study of roll-streak dynamics in Poiseuille flow turbulence

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.

physics.flu-dyn↗

arXiv2023

Statistical state dynamics-based study of the stability of the mean statistical state of wall-bounded turbulence

Turbulence in wall-bounded flows is characterized by stable statistics. Although, in many 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 is on a transient chaotic attractor in the S3T statistical state space and the time-mean statistical state is neither a stable fixed point of this SSD nor, if it is maintained as an equilibrium, is it stable. Nevertheless, sufficiently small perturbations from the ensemble/time-mean state 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 stable dynamics. While simplifying insight into turbulent flows has 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 the 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.

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arXiv2023

POD-based study of turbulent plane Poiseuille flow: comparing structure and dynamics between quasi-linear simulations and DNS

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.

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arXiv2021

Synchronization of Low Reynolds Number Plane Couette Turbulence

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.

physics.flu-dyn↗

arXiv2018

Statistical state dynamics of weak jets in barotropic beta-plane turbulence

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.

physics.ao-ph↗

arXiv2018

Parametric mechanism maintaining Couette flow turbulence verified in DNS implies novel control strategies

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 $\approx1000$ by making available to perturbations the kinetic energy of the externally forced flow. Understanding the physical mechanism underlying this energy transfer 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 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.

physics.flu-dyn↗

arXiv2018

Statistical State Dynamics Based Study of the Role of Nonlinearity in the Maintenance of Turbulence in Couette Flow

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 are the deviations from the streamwise mean and two mechanisms potentially contributing to maintaining these 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 of 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.

physics.flu-dyn↗

arXiv2018

Is spontaneous generation of coherent baroclinic flows possible?

Geophysical turbulence is observed to self-organize into large-scale flows such as zonal jets and coherent vortices. Previous studies on barotropic beta-plane turbulence have shown that coherent flows emerge out of a background of homogeneous turbulence as a bifurcation when the turbulence intensity increases and 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 with no imposed mean north-south thermal gradient and turbulence supported by an external random stirring. We use a second order closure of the statistical state dynamics (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 of zonal jets and/or large scale waves when the energy input rate of the excitation passes a critical threshold is analytically studied. The theory predicts that when the flow transitions to a statistical state with large-scale structures, these states are barotropic if the scale of excitation is larger than the deformation radius. 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 non-significant for higher energy input rates. The results of the S3T theory are compared to nonlinear simulations. The theory is found to accurately predict both the critical transition arameters and the scales of the emergent structures but underestimates their amplitude.

physics.ao-ph↗

arXiv2018

The mechanism by which nonlinearity sustains turbulence in plane Couette flow

Turbulence in wall-bounded shear flow results from a synergistic interaction between linear non-normality and nonlinearity in which non-normal growth of a subset of perturbations configured to transfer energy from the externally forced component of the turbulent state to the perturbation component maintains the perturbation energy, while the subset of energy-transferring perturbations is replenished by nonlinearity. Although it is accepted that both linear non-normality mediated energy transfer from the forced component of the mean flow and nonlinear interactions among perturbations are required to maintain the turbulent state, the detailed physical mechanism by which these processes interact in maintaining turbulence has not been determined. In this work a statistical state dynamics based analysis is performed on turbulent Couette flow at $R=600$ and a comparison to DNS is used to demonstrate that the perturbation component in Couette flow turbulence is replenished by a non-normality mediated parametric growth process in which the fluctuating streamwise mean flow has been adjusted to marginal Lyapunov stability. It is further shown that the alternative mechanism in which the subspace of non-normally growing perturbations is maintained directly by perturbation-perturbation nonlinearity does not contribute to maintaining the turbulent state. This work identifies parametric interaction between the fluctuating streamwise mean flow and the streamwise varying perturbations to be the mechanism of the nonlinear interaction maintaining the perturbation component of the turbulent state, and identifies the associated Lyapunov vectors with positive energetics as the structures of the perturbation subspace supporting the turbulence.

physics.flu-dyn↗

arXiv2018

Statistical State Dynamics: a new perspective on turbulence in shear flow

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.

physics.flu-dyn↗

arXiv2017

Statistical state dynamics-based analysis of the physical mechanisms sustaining and regulating turbulence in Couette flow

This paper describes a study of the self-sustaining process in wall-turbulence based on a second order statistical state dynamics (SSD) model of Couette flow. SSD models with this form are referred to as S3T models and self-sustain turbulence with a mean flow and second order perturbation structure similar to that obtained by DNS. The use of a SSD model to study the physical mechanisms underlying turbulence has advantages over the traditional approach of studying the dynamics of individual realizations of turbulence. One advantage is that the analytical structure of SSD isolates and directly expresses the interaction between the coherent mean flow and the incoherent 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 SSD models is that the analytical structure of S3T turbulence can be completely characterized. For example, turbulence in the S3T system is maintained by a parametric growth mechanism. Furthermore, the equilibrium statistical state of the turbulence can be demonstrated to be enforced by feedback regulation in which transient growth of the incoherent perturbations episodically suppresses coherent streak growth preventing runaway parametric growth of the incoherent turbulent component. Using S3T to isolate these 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 understanding the mechanism of wall-turbulence.

physics.flu-dyn↗

arXiv2017

Statistical state dynamics based theory for the formation and equilibration of Saturn's north polar jet

Coherent jets with most of the kinetic energy of the flow are common in atmospheric turbulence. In the gaseous planets these jets are maintained by incoherent turbulence excited by small-scale convection. Large-scale coherent waves are sometimes observed to coexist with the jets; a prominent example is Saturn's hexagonal North polar jet (NPJ). The mechanism responsible for forming and maintaining such a turbulent state remains elusive. The coherent planetary-scale component of the turbulence arises and is maintained by interaction with the incoherent small-scale turbulence component. Theoretical understanding of the dynamics of the jet/wave/turbulence coexistence regime is gained by employing a statistical state dynamics (SSD) model. Here, a second-order closure implementation of a two-layer beta-plane SSD is used to develop a theory that accounts for the structure and dynamics of the NPJ. Asymptotic analysis of the SSD equilibrium in the weak jet damping limit predicts a universal jet structure in agreement with NPJ observations. This asymptotic theory also predicts the wavenumber (six) of the prominent jet perturbation. Analysis with this model of the jet/wave/turbulence regime dynamics reveals that jet formation is controlled by the effective value of $β$; 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 NPJ structure as is the small-scale turbulence excitation required to maintain the jet. The observed jet structure is then predicted by the theory as is the wave-six disturbance. This wave, which is identified with the least stable mode of the equilibrated jet, is shown to be primarily responsible for equilibrating the jet with the observed structure and amplitude.

physics.ao-ph↗

arXiv2017

Instability of the roll/streak structure induced by free-stream turbulence in pre-transitional Couette flow

Although the roll/streak structure is ubiquitous in pre-transitional 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 free-stream turbulence (FST) of perturbations configured to optimally excite it. However, there is an alternative interpretation which is that FST 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 to analyze the SSD modes arising from interaction between the coherent streamwise invariant component and the incoherent FST component of turbulence. The least stable S3T mode is destabilized at a critical value of a parameter controlling FST intensity and a 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 DNS. S3T also predicts a second bifurcation at a higher value of the turbulent excitation parameter. This second bifurcation is shown to lead to transition to turbulence. Bifurcation from a finite amplitude roll/streak equilibrium provides a direct route to the turbulent state through the S3T roll/streak instability.

physics.flu-dyn↗

arXiv2016

A statistical state dynamics approach to wall-turbulence

This paper reviews results from the study of wall-bounded turbulent flows 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 which isolates the interaction between the streamwise mean and the equivalent of the perturbation covariance. This closure restricts nonlinearity in the SSD to that explicitly retained in the streamwise constant mean together with nonlinear interactions between the mean 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 an ensemble of a finite number of realizations of the perturbation equation share the same mean flow provide tractable approximations to the equivalently infinite ensemble RNL system. The infinite ensemble system, referred to as the S3T, introduces new analysis tools for studying turbulence. The RNL with a single ensemble member can be alternatively viewed as a realization of RNL dynamics. 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 (DNS) 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.

physics.flu-dyn↗

arXiv2016

A statistical state dynamics-based study of the structure and mechanism of large-scale motions in plane Poiseuille flow

The perspective of statistical state dynamics (SSD) has recently been applied to the study of mechanisms underlying turbulence in various physical systems. An example implementation of SSD is the second order 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. This closure eliminates 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. Here, 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 nonlinear (RNL) system. RNL simulations reveal that the essential features of wall-turbulence dynamics are retained. Remarkably, 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. Surprisingly, even when further truncation of the perturbation support to a single streamwise component is imposed the RNL system continues to produce self-sustaining turbulent structure and dynamics. 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. 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.

physics.flu-dyn↗

arXiv2016

Statistical state dynamics of jet/wave coexistence in barotropic beta-plane turbulence

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 only exist 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.

physics.ao-ph↗