The culmination of an inverse cascade: Mean flow and fluctuations

Frishman, Anna

doi:10.1063/1.4985998

Cited by 21 publications

(22 citation statements)

References 37 publications

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“…Indeed, the effective correlation time τ = l/V of the pumping diminishes as the large-scale velocity V grows and therefore the energy production ∼ f 2 τ diminishes. Similar conclusions are presented in the recent paper [24].…”

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confidence: 92%

Profile of coherent vortices in two-dimensional turbulence

Kolokolov

Лебедев

2015

Jetp Lett.

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We examine the velocity profile of coherent vortices appearing as a consequence of the inverse cascade of two-dimensional turbulence in a finite box in the case of static pumping. We demonstrate that in the passive regime the flat velocity profile is realized, as in the case of pumping short correlated in time. However, in the static case the energy flux to large scales is dependent on the system parameters. We demonstrate that it is proportional to f 4/3 where f is the characteristic force exciting turbulence.Turbulence is a chaotic state of fluid motions realized at large Reynolds numbers Re, see, e.g., [1]. In a number of cases turbulence appears to be effectively two-dimensional [2]. Already first theoretical works [3][4][5] devoted to two-dimensional turbulence reveal its principal difference from three-dimensional one. The difference is related to existence of two quadratic quantities (energy and enstrophy) conserved by two-dimensional Euler equation. That leads to two different cascades produced by the non-linear interaction: enstrophy is carried from the pumping scale to smaller scales (direct cascade) whereas energy is carried to larger scales (inverse cascade). The enstrophy is dissipated due to viscosity at scales smaller than the pumping length and the energy is dissipated due to bottom friction at scales larger than the pumping length.Statistical properties of the velocity fluctuations in the inverse cascade were investigated both experimentally [6] and numerically [7]. Results of the works are in a good agreement with the analytical theory developed for an unbound system [8]. Particularly, the normal Kolmogorov scaling is observed in the inverse cascade [2]. The normal scaling in the inverse cascade of twodimensional turbulence is in contrast with the anomalous scaling observed in three-dimensional turbulence [9]. Some theoretical arguments in favor of the normal scaling in the inverse cascade were presented in the work [10], where the feature was related to a leading role of the converging Lagrangian trajectories in the inverse cascade.In an unbound two-dimensional system the inverse cascade is terminated at a scale L α , which is determined by a balance between the energy flux ǫ (per unit mass) toward large scales and the bottom friction. Using the Kolmogorov estimate (ǫL α ) 1/3 for the velocity at the scale L α , one obtains from the balance L α = ǫ 1/2 α −3/2 , where α is the bottom friction coefficient. If the box size L is smaller than L α , then the energy carried by the inverse cascade to scales of the order of the box size L starts to accumulate there. The scenario is realized if the pumping scale l is much less than the box size L, l ≪ L, otherwise there is no space for the inverse cascade carrying the energy to the scale L.The accumulation of the energy at the scale L leads to appearing an intensive large-scale motion including big coherent vortices. The tendency towards the formation of the vortices was indicated already in the first works devoted to two-dimensional turbulence, both exp...

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confidence: 92%

Profile of coherent vortices in two-dimensional turbulence

Kolokolov

Лебедев

2015

Jetp Lett.

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“…As shown previously [10,11,14], within the quasilinear approximation -justified for δ 1 -and once (2) is observed to hold.…”

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confidence: 76%

“…This is called the quasi-linear approach; it was used extensively to study dynamics in numerical simulations [6], often under different names such as Direct Statistical Simulation (DSS), Cumulant Expansion (CE) [7] or Stochastic Structural Stability Theory (SSST) [8], and justified theoretically using adiabatic reduction [9]. Actually, if turbulence is excited at asymptotically small scales, the perturbative treatment allows to analytically derive an explicit formula for the mean-flow and momentum flux profiles, as demonstrated for the vortex condensate [10,11], and discussed for jets [12,14] and on the sphere [3]. Until now, the only part of these predictions which was quantitatively checked against data from Direct Numerical Simulation (DNS) is the profile of the mean-flow [10].…”

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confidence: 99%

Turbulence Statistics in a Two-Dimensional Vortex Condensate

Frishman

Herbert

2018

Phys. Rev. Lett.

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Disentangling the evolution of a coherent mean-flow and turbulent fluctuations, interacting through the nonlinearity of the Navier-Stokes equations, is a central issue in fluid mechanics. It affects a wide range of flows, such as planetary atmospheres, plasmas, or wall-bounded flows, and hampers turbulence models. We consider the special case of a two-dimensional flow in a periodic box, for which the mean flow, a pair of box-size vortices called "condensate," emerges from turbulence. As was recently shown, a perturbative closure describes correctly the condensate when turbulence is excited at small scales. In this context, we obtain explicit results for the statistics of turbulence, encoded in the Reynolds stress tensor. We demonstrate that the two components of the Reynolds stress, the momentum flux and the turbulent energy, are determined by different mechanisms. It was suggested previously that the momentum flux is fixed by a balance between forcing and mean-flow advection: using unprecedently long numerical simulations, we provide the first direct evidence supporting this prediction. By contrast, combining analytical computations with numerical simulations, we show that the turbulent energy is determined only by mean-flow advection and obtain for the first time a formula describing its profile in the vortex.

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“…Finally, for a 2‐D neutral fluid, one can show that, in the presence of a mean flow (such as vortices or jets) and turbulent fluctuations, the mean momentum stress is proportional to the mean shear (Frishman, ). The mean flow energy is obtained from a balance between the large‐scale friction and the turbulent dissipation at small scale (Frishman & Herbert, ).…”

Section: Coupling To a Magnetic Fieldmentioning

confidence: 99%

“…It is becoming increasingly clear that the role of large‐scale shear is central in many turbulent flows (Pumir, ), as in the structure of late‐time coherent eddies in two dimensional turbulence (Frishman, ; Frishman & Herbert, ), in the destabilization of stratified flows as discussed above (see Figure , top), or when it leads to the formation of fronts between metastable and stable states or between quiet and turbulent regions (Pomeau, ; Waleffe, ). The connection, in the context of the dynamics between large‐scale predator (the shear flow) and prey (the turbulent eddies), and in which the details of the small‐scale turbulent eddies are rather irrelevant, allows for the classification of turbulent flows by analogy with directed percolation (Barkley, ; Pomeau, ), as supported by laboratory experiments (Sano & Tamai, ).…”

Section: The Bidirectional or Dual Cascades In Turbulencementioning

confidence: 99%

Helicity Dynamics, Inverse, and Bidirectional Cascades in Fluid and Magnetohydrodynamic Turbulence: A Brief Review

Pouquet

Rosenberg

Stawarz

et al. 2019

Earth and Space Science

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We briefly review helicity dynamics, inverse and bidirectional cascades in fluid and magnetohydrodynamic (MHD) turbulence, with an emphasis on the latter. The energy of a turbulent system, an invariant in the nondissipative case, is transferred to small scales through nonlinear mode coupling. Fifty years ago, it was realized that, for a two‐dimensional fluid, energy cascades instead to larger scales and so does magnetic excitation in MHD. However, evidence obtained recently indicates that, in fact, for a range of governing parameters, there are systems for which their ideal invariants can be transferred, with constant fluxes, to both the large scales and the small scales, as for MHD or rotating stratified flows, in the latter case including quasi‐geostrophic forcing. Such bidirectional, split, cascades directly affect the rate at which mixing and dissipation occur in these flows in which nonlinear eddies interact with fast waves with anisotropic dispersion laws, due, for example, to imposed rotation, stratification, or uniform magnetic fields. The directions of cascades can be obtained in some cases through the use of phenomenological arguments, one of which we derive here following classical lines in the case of the inverse magnetic helicity cascade in electron MHD. With more highly resolved data sets stemming from large laboratory experiments, high‐performance computing, and in situ satellite observations, machine learning tools are bringing novel perspectives to turbulence research. Such algorithms help devise new explicit subgrid‐scale parameterizations, which in turn may lead to enhanced physical insight, including in the future in the case of these new bidirectional cascades.

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The culmination of an inverse cascade: Mean flow and fluctuations

Cited by 21 publications

References 37 publications

Profile of coherent vortices in two-dimensional turbulence

Profile of coherent vortices in two-dimensional turbulence

Turbulence Statistics in a Two-Dimensional Vortex Condensate

Helicity Dynamics, Inverse, and Bidirectional Cascades in Fluid and Magnetohydrodynamic Turbulence: A Brief Review

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