Emergence and Equilibration of Jets in Beta-Plane Turbulence: Applications of Stochastic Structural Stability Theory

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

doi:10.1175/jas-d-13-076.1

Cited by 65 publications

(111 citation statements)

References 51 publications

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“…In geophysical fluid dynamics, zonal jets or zonal flows occur when the velocity field is aligned with latitude circles, and they depend only on the longitude coordinate y, i.e., = U y v e ( ) x . Furthermore, multiple zonal jet configurations were observed as dynamical attractors in the quasi-geostrophic model for the same set of parameters [36]. These examples provide the necessary motivation to try to understand rare transitions between two attractors for geophysical fluid flows.…”

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confidence: 76%

“…However, it must be stated that this is only an approximate measure because the structure of the noise correlation will also contribute. Moreover, many cases of multiple steady-state solutions of the barotropic equation with differing numbers of jets have been observed for the same sets of parameters [36]. These multiple states are assumed to be linearly stable for the unforced (with or without dissipation) dynamics.…”

Section: Dynamics Of the Statistically Steady Statementioning

confidence: 99%

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Computation of rare transitions in the barotropic quasi-geostrophic equations

Laurie

Bouchet

2015

New J. Phys.

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We investigate the theoretical and numerical computation of rare transitions in simple geophysical turbulent models. We consider the barotropic quasi-geostrophic and two-dimensional Navier-Stokes equations in regimes where bistability between two coexisting large-scale attractors exist. By means of large deviations and instanton theory with the use of an Onsager-Machlup path integral formalism for the transition probability, we show how one can directly compute the most probable transition path between two coexisting attractors analytically in an equilibrium (Langevin) framework and numerically otherwise. We adapt a class of numerical optimization algorithms known as minimum action methods to simple geophysical turbulent models. We show that by numerically minimizing an appropriate action functional in a large deviation limit, one can predict the most likely transition path for a rare transition between two states. By considering examples where theoretical predictions can be made, we show that the minimum action method successfully predicts the most likely transition path. Finally, we discuss the application and extension of such numerical optimization schemes to the computation of rare transitions observed in direct numerical simulations and experiments and to other, more complex, turbulent systems. models. Stochastic resonance, however, remains a very interesting possibility. To address such issues one should study the attractors and the dynamics of the rare transition between attractors in a hierarchy of models from the simplest to more complex ones used by the climate community. For these complex climate models, which genuinely reproduce the turbulent nature of the Earthʼs atmosphere and ocean dynamics, such a task is currently inconceivable and seems unreachable in the foreseeable future using direct numerical simulations. The reasons are the rarity of the transitions and the computational complexity of these models. The main aim of this paper is to make a step in the direction of this challenge by studying bistability and the associated transitions in turbulent dynamics using tools that will allow one to compute transitions in more complex systems in the near future.These rare transitions are essential phenomena because they correspond to drastic changes in complex system behavior. Moreover, they cannot be studied using conventional tools. They contain dynamics occurring on multiple and extremely different timescales, usually with no spectral gap. This prevents the use of classical tools from dynamical system theory. The theoretical understanding of these transitions is an extremely difficult problem due to the complexity, the large number of degrees of freedom, and the non-equilibrium nature of many of these flows. Up to now, there have been an extremely limited number of theoretical results, where analysis has been limited to analogies with models of very few degrees of freedom [27] or to specific classes of systems that can be directly related to equilibrium Langevin dynamics [28]. For this reason, the u...

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mentioning

confidence: 76%

Section: Dynamics Of the Statistically Steady Statementioning

confidence: 99%

Computation of rare transitions in the barotropic quasi-geostrophic equations

Laurie

Bouchet

2015

New J. Phys.

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“…Stochastically forced models were also utilized in the context of stochastic structural stability theory to study jet formation and equilibration in barotropic beta-plane turbulence (Farrell & Ioannou 2003Bakas & Ioannou 2011;Constantinou et al 2014a;Bakas & Ioannou 2014). Recently, it was demonstrated that a feedback interconnection of the streamwise-constant NS equations with a stochastically-driven streamwisevarying linearized model can generate self-sustained turbulence in Couette and Poiseuille flows (Farrell & Ioannou 2012;Thomas et al 2014;Constantinou et al 2014b).…”

Section: Stochastic Forcing and Flow Statisticsmentioning

confidence: 99%

Colour of turbulence

2017

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In this paper, we address the problem of how to account for second-order statistics of turbulent flows using low-complexity stochastic dynamical models based on the linearized Navier-Stokes equations. The complexity is quantified by the number of degrees of freedom in the linearized evolution model that are directly influenced by stochastic excitation sources. For the case where only a subset of velocity correlations are known, we develop a framework to complete unavailable second-order statistics in a way that is consistent with linearization around turbulent mean velocity. In general, white-in-time stochastic forcing is not sufficient to explain turbulent flow statistics. We develop models for colored-intime forcing using a maximum entropy formulation together with a regularization that serves as a proxy for rank minimization. We show that colored-in-time excitation of the Navier-Stokes equations can also be interpreted as a low-rank modification to the generator of the linearized dynamics. Our method provides a data-driven refinement of models that originate from first principles and captures complex dynamics of turbulent flows in a way that is tractable for analysis, optimization, and control design.

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“…While there are several studies which show the utility of the quasi-linear approximation (e.g. Srinivasan & Young 2012;Constantinou, Farrell & Iouannou 2014) and while this approximation would certainly be valid in some parameter regimes (as per our scalings in ε early in the paper), we recognise that the approach may principally be useful as a qualitative rather than quantitative guide to behaviour in full nonlinear simulations. We have also assumed scale separation between waves and mean flows, and this can be relaxed by using the correlation function formulation developed by Srinivasan & Young (2012) and discussed by Bakas & Iouannou (2013).…”

Section: Discussionmentioning

confidence: 86%

“…What is neglected is the interactions of waves giving smaller-scale waves, in other words the beginning of a turbulent cascade. In the context of zonostrophic instability, these ideas and applications are developed in the stochastic structural stability theory of Farrell & Iouannou (2003, Bakas & Iouannou (2011 and Constantinou, Farrell & Iouannou (2014). Here coupled equations for mean flow and waves, with additional stochastic forcing, are solved numerically in a variety of systems, showing robust jet formation, and the quasi-linear theory validated against direct simulations.…”

Section: Introductionmentioning

confidence: 99%

Transport and instability in driven two-dimensional magnetohydrodynamic flows

Durston

Gilbert

2016

J. Fluid Mech.

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This paper concerns the generation of large scale flows in forced two-dimensional systems. A Kolmogorov flow with a sinusoidal profile in one direction (driven by a body force) is known to become unstable to a large scale flow in the perpendicular direction at a critical Reynolds number. This can occur in the presence of a beta-effect and has important implications for flows observed in geophysical and astrophysical systems. It has recently been termed 'zonostrophic instability' and studied in a variety of settings, both numerically and analytically.The goal of the present paper is to determine the effect of magnetic field on such instabilities using the quasi-linear approximation, in which the full fluid system is decoupled into a mean flow and waves of one scale. The waves are driven externally by a given, random body force and move on a fast time scale, while their stress on the mean flow causes this to evolve on a slow time scale. Spatial scale separation between waves and mean flow is also assumed, to allow analytical progress.The paper first discusses purely hydrodynamic transport of vorticity including zonostrophic instability, the effect of uniform background shear, and calculation of equilibrium profiles in which the effective viscosity varies spatially, through the mean flow. After brief consideration of passive scalar transport or equivalently kinematic magnetic field evolution, the paper then proceeds to study the full MHD system and to determine effective diffusivities and other transport coefficients using a mixture of analytical and numerical methods. This leads to results on the effect of magnetic field, background shear and beta-effect on zonostrophic instability and magnetically driven instabilities.

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Emergence and Equilibration of Jets in Beta-Plane Turbulence: Applications of Stochastic Structural Stability Theory

Cited by 65 publications

References 51 publications

Computation of rare transitions in the barotropic quasi-geostrophic equations

Computation of rare transitions in the barotropic quasi-geostrophic equations

Colour of turbulence

Transport and instability in driven two-dimensional magnetohydrodynamic flows

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