Interaction between mean flow and turbulence in two dimensions

Falkovich, Gregory

doi:10.1098/rspa.2016.0287

Cited by 18 publications

(23 citation statements)

References 46 publications

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“…The zeroth mode, m = 0, deserves a separate consideration (see also 10,33 ). As was noted before, and can be deduced from the (φ, φ) component of the steady state equation…”

Section: Fluctuationsmentioning

confidence: 99%

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The culmination of an inverse cascade: Mean flow and fluctuations

Frishman

2017

Physics of Fluids

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Two dimensional turbulence has a remarkable tendency to self-organize into large, coherent structures, forming a mean flow. The purpose of this paper is to elucidate how these structures are sustained, and what determines them and the fluctuations around them. A recent theory for the mean flow will be reviewed. The theory assumes turbulence is excited by a forcing supported on small scales, and uses a linear shear model to relate the turbulent momentum flux to the mean shear rate. Extending the theory, it will be shown here that the relation between the momentum flux and mean shear is valid, and the momentum flux is non-zero, for both an isotropic and an anisotropic forcing, independent of the dissipation mechanism at small scales. This conclusion requires taking into account that the linear shear model is an approximation to the real system. The proportionality between the momentum flux and the inverse of the shear can then be inferred most simply on dimensional grounds. Moreover, for a homogeneous pumping, the proportionality constant can be determined by symmetry considerations, recovering the result of the original theory. The regime of applicability of the theory, its compatibility with observations from simulations, a formula for the momentum flux for an inhomogeneous pumping, and results for the statistics of fluctuations, will also be discussed.

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“…The zeroth mode, m = 0, deserves a separate consideration (see also 10,33 ). As was noted before, and can be deduced from the (φ, φ) component of the steady state equation…”

Section: Fluctuationsmentioning

confidence: 99%

“…Section III B concerns the vortex mean flow, describing the analytic solution for its profile, found recently 9,10 . The derivation relies on a particular closure of the energy balance, which connects the Reynolds stress (turbulent momentum flux uv ) to the mean shear rate, U ′ .…”

Section: Introductionmentioning

confidence: 99%

The culmination of an inverse cascade: Mean flow and fluctuations

Frishman

2017

Physics of Fluids

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“…On the contrary, large-scale motions are usually less dissipative, so that an inverse cascade can proceed unimpeded, either producing larger and larger scales or reaching the box size and creating a coherent mode of growing amplitude. That process is now actively studied in 2D incompressible turbulence [3,4,[9][10][11][12][13], including in a curved space, where vortex rings rather than vortices are created [14]. The energy of an incompressible flow in an unbounded domain grows unlimited when the friction factors go to zero at a finite energy input rate.…”

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confidence: 99%

How vortices and shocks provide for a flux loop in two-dimensional compressible turbulence

Falkovich

Kritsuk

2017

Phys. Rev. Fluids

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Large-scale turbulence in fluid layers and other quasi-two-dimensional compressible systems consists of planar vortices and waves. Separately, wave turbulence usually produces a direct energy cascade, while solenoidal planar turbulence transports energy to large scales by an inverse cascade. Here, we consider turbulence at finite Mach numbers when the interaction between acoustic waves and vortices is substantial. We employ solenoidal pumping at intermediate scales and show how both direct and inverse energy cascades are formed starting from the pumping scale. We show that there is an inverse cascade of kinetic energy up to a scale ℓ, where a typical velocity reaches the speed of sound; this creates shock waves, which provide for a compensating direct cascade. When the system size is less than ℓ, the steady state contains a system-size pair of long-living condensate vortices connected by a system of shocks. Thus turbulence in fluid layers processes energy via a loop: Most energy first goes to large scales via vortices and is then transported by waves to small-scale dissipation.Inverse cascade is a counterintuitive process of selforganization of turbulence. Predicted almost simultaneously for two-dimensional (2D) incompressible flows [1] and sea wave turbulence [2] and established in many cases of turbulence in plasma, optics, etc. [3][4][5][6][7][8], it is predicated on the existence of two quadratic conserved quantities having different wave-number dependencies. Excitation at some intermediate wave number then leads to two cascades: a direct one to small scales and an inverse one to large scales. There is always a strong dissipation at small scales which acts as a sink for the direct cascade. On the contrary, large-scale motions are usually less dissipative, so that an inverse cascade can proceed unimpeded, either producing larger and larger scales or reaching the box size and creating a coherent mode of growing amplitude. That process is now actively studied in 2D incompressible turbulence [3,4,[9][10][11][12][13], including in a curved space, where vortex rings rather than vortices are created [14]. The energy of an incompressible flow in an unbounded domain grows unlimited when the friction factors go to zero at a finite energy input rate. The same happens to the action of wave turbulence [15], if long waves of large amplitude are stable. For example, optical turbulence in media with defocusing nonlinearity produces a growing condensate [7,8,16]. On the contrary, in the focusing case, condensate instability results in wave collapses which provide for a loop of inverse cascade to the small-scale dissipation so that there is a steady state with only small-scale dissipation [8].Here, we consider compressible two-dimensional turbulence which is of significant importance for numerous geophysical, astrophysical and industrial applications. We show that it realizes a third possibility of a steady state with only small-scale dissipation: On the one hand, an inverse cascade is able to produce long-living stable vorti...

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“…While this work gave theoretical ground to the approach, explicit formula for the Reynolds stress cannot be expected in general. However, in a recent work [13], an expression for the Reynolds stress has been derived from the momentum and energy balance equations by neglecting the perturbation cubic terms in the energy balance (this follows from the quasilinear approach justification [12]), but also neglecting pressure terms (not justified so far, see [14]). This approach surprisingly predicts a constant velocity profile for the outer region of a large scale vortex in two dimensions that does not depend on the detailed characteristics of the stochastic forcing but only on the total energy injection rate expressed in m 2 s −3 .…”

mentioning

confidence: 99%

“…The small scale forcing limit K 1 is the most common framework for turbulence studies (see for ex. [14]) and relevant for Jupiter troposphere. Also, computing the pressure from the Navier-Stokes equations involves inverting a Laplacian.…”

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confidence: 99%

Theoretical prediction of Reynolds stresses and velocity profiles for barotropic turbulent jets

Woillez¹,

Bouchet²

2017

EPL

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It is extremely uncommon to be able to predict the velocity profile of a turbulent flow. In twodimensional flows, atmosphere dynamics, and plasma physics, large scale coherent jets are created through inverse energy transfers from small scales to the largest scales of the flow. We prove that in the limits of vanishing energy injection, vanishing friction, and small scale forcing, the velocity profile of a jet obeys an equation independent of the details of the forcing. We find another general relation for the maximal curvature of a jet and we give strong arguments to support the existence of an hydrodynamic instability at the point with minimal jet velocity. Those results are the first computations of Reynolds stresses and self consistent velocity profiles from the turbulent dynamics, and the first consistent analytic theory of zonal jets in barotropic turbulence.Theoretical prediction of velocity profiles of inhomogeneous turbulent flows is a long standing challenge, since the nineteenth century. It involves closing hierarchy for the velocity moments, and for instance obtaining a relation between the Reynolds stress and the velocity profile. Since Boussinesq in the nineteenth century, most of the approaches so far have been either empirical or phenomenological. Even for the simple case of a three dimensional turbulent boundary layer, plausible but so far unjustified similarity arguments may be used to derive von Kármán logarithmic law for the turbulent boundary layer (see for instance [1]), but the related von Kármán constant [2] has never been computed theoretically. Still this problem is a crucial one and has some implications in most of scientific fields, in physics, astrophysics, climate dynamics, and engineering. Equations (6-7), (9), and (11) are probably the first prediction of the velocity profile for turbulent flows, and relevant for barotropic flows.In this paper we find a way to close the hierarchy of the velocity moments, for the equation of barotropic flows with or without effect of the Coriolis force. This two dimensional model is relevant for laboratory experiments of fluid turbulence [3], liquid metals [4], plasma [5], and is a key toy model for understanding planetary jet formation [6] and basics aspects of plasma dynamics on Tokamaks in relation with drift waves and zonal flow formation [7]. It is also a relevant model for Jupiter troposphere organization [8]. Moreover, our approach should have future implications for more complex turbulent boundary layers, which are crucial in climate dynamics in order to quantify momentum and energy transfers between the atmosphere and the ocean.It has been realized since the sixties and seventies in the atmosphere dynamics and plasma communities that in some regimes two dimensional turbulent flows are * Eric.Woillez@ens-lyon.fr † Freddy.Bouchet@ens-lyon.fr strongly dominated by large scale coherent structures. Jets and large vortices are often observed in numerical simulations or in experiments, but the general mechanism leading to such an organization of t...

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Interaction between mean flow and turbulence in two dimensions

Cited by 18 publications

References 46 publications

The culmination of an inverse cascade: Mean flow and fluctuations

The culmination of an inverse cascade: Mean flow and fluctuations

How vortices and shocks provide for a flux loop in two-dimensional compressible turbulence

Theoretical prediction of Reynolds stresses and velocity profiles for barotropic turbulent jets

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