Real-Space Manifestations of Bottlenecks in Turbulence Spectra

Frisch, U.; Ray, Samriddhi Sankar; Sahoo, Ganapati; Banerjee, Debarghya; Pandit, Rahul

doi:10.1103/physrevlett.110.064501

Cited by 23 publications

(29 citation statements)

References 31 publications

(23 reference statements)

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“…The initial condition IC2 is like IC1 but, in addition, it has a finite initial condensate population N i 0 =|ψ(k = 0, t) | 2 ( k) 2 at time t = 0. Note the study of [41] uses a hyper-viscosity term ν(−∇ 2 ) n ψ, which is absent in our study; such hyperviscosity terms can modify energy spectra in important ways, as has been discussed in the context of turbulence in the Navier-Stokes equation in [50,63].…”

Section: Numerical Methods and Initial Conditionsmentioning

confidence: 99%

Turbulence in the two-dimensional Fourier-truncated Gross–Pitaevskii equation

2013

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We undertake a systematic, direct numerical simulation of the twodimensional, Fourier-truncated, Gross-Pitaevskii equation to study the turbulent evolutions of its solutions for a variety of initial conditions and a wide range of parameters. We find that the time evolution of this system can be classified into four regimes with qualitatively different statistical properties. Firstly, there are transients that depend on the initial conditions. In the second regime, powerlaw scaling regions, in the energy and the occupation-number spectra, appear and start to develop; the exponents of these power laws and the extents of the scaling regions change with time and depend on the initial condition. In the third regime, the spectra drop rapidly for modes with wave numbers k > k c and partial thermalization takes place for modes with k < k c ; the self-truncation wave number k c (t) depends on the initial conditions and it grows either as a power of t or as log t. Finally, in the fourth regime, complete thermalization is achieved and, if we account for finite-size effects carefully, correlation functions and spectra are consistent with their nontrivial Berezinskii-Kosterlitz-Thouless

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Section: Numerical Methods and Initial Conditionsmentioning

confidence: 99%

Turbulence in the two-dimensional Fourier-truncated Gross–Pitaevskii equation

2013

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“…Specifically, equipartition solutions to the Galerkin-truncated Gross-Pitaevskii [19], magnetohydrodynamic [20], Burgers [5,17], and Euler equations [6,21] have been studied extensively in recent years. Alongside the very important theoretical under-pinnings of such studies, thermalised or partially thermalised states have been shown [22][23][24] to be a possible explanation of the ubiquitous bottleneck [25] in the energy spectrum of turbulent flows.…”

Section: Introductionmentioning

confidence: 99%

The onset of thermalization in finite-dimensional equations of hydrodynamics: insights from the Burgers equation

Venkataraman

Ray

2017

Proc. R. Soc. A.

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Solutions to finite-dimensional (all spatial Fourier modes set to zero beyond a finite wavenumber KG), inviscid equations of hydrodynamics at long times are known to be at variance with those obtained for the original infinite dimensional partial differential equations or their viscous counterparts. Surprisingly, the solutions to such Galerkin-truncated equations develop sharp localised structures, called tygers [Ray, et al., Phys. Rev. E 84, 016301 (2011)], which eventually lead to completely thermalised states associated with an equipartition energy spectrum. We now obtain, by using the analytically tractable Burgers equation, precise estimates, theoretically and via direct numerical simulations, of the time τc at which thermalisation is triggered and show that τc ∼ KG ξ , with ξ = −4/9. Our results have several implications including for the analyticity strip method to numerically obtain evidence for or against blow-ups of the three-dimensional incompressible Euler equations.

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“…When this expression is converted to the spectral domain, it recovers the spectral bottleneck reported at the crossover from inertial to dissipation (Katul et al, ). This bottleneck is commonly identified by a bump when the compensated spectrum k 5/3 E w w ( k ) is plotted against k in the vicinity of k η ≈0.1 and has been the subject of active research (Davidson, ; Dobler et al, ; Donzis & Sreenivasan, ; Frisch et al, , ; Herring et al, ; Hill, ; Katul et al, ; Meyers & Meneveau, ). The assumed spectrum by Lamont and Scott () exhibits an inertial subrange adjusted with a power‐law viscous cutoff based on the Kovasznay spectrum.…”

Section: Theorymentioning

confidence: 99%

A Structure Function Model Recovers the Many Formulations for Air‐Water Gas Transfer Velocity

Katul

Mammarella

Grönholm

et al. 2018

Water Resources Research

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Two ideas regarding the structure of turbulence near a clear air‐water interface are used to derive a waterside gas transfer velocity kL for sparingly and slightly soluble gases. The first is that kL is proportional to the turnover velocity described by the vertical velocity structure function Dww(r), where r is separation distance between two points. The second is that the scalar exchange between the air‐water interface and the waterside turbulence can be suitably described by a length scale proportional to the Batchelor scale lB=ηSc−1/2, where Sc is the molecular Schmidt number and η is the Kolmogorov microscale defining the smallest scale of turbulent eddies impacted by fluid viscosity. Using an approximate solution to the von Kármán‐Howarth equation predicting Dww(r) in the inertial and viscous regimes, prior formulations for kL are recovered including (i) kL=2false/15Sc−1false/2vK, vK is the Kolmogorov velocity defined by the Reynolds number vKη/ν = 1 and ν is the kinematic viscosity of water; (ii) surface divergence formulations; (iii) kL∝Sc−1false/2u∗, where u∗ is the waterside friction velocity; (iv) kL∝Sc−1false/2gνfalse/u∗ for Keulegan numbers exceeding a threshold needed for long‐wave generation, where the proportionality constant varies with wave age, g is the gravitational acceleration; and (v) kL=2false/15Sc−1false/2false(νgβoqofalse)1false/4 in free convection, where qo is the surface heat flux and βo is the thermal expansion of water. The work demonstrates that the aforementioned kL formulations can be recovered from a single structure function model derived for locally homogeneous and isotropic turbulence.

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Real-Space Manifestations of Bottlenecks in Turbulence Spectra

Cited by 23 publications

References 31 publications

Turbulence in the two-dimensional Fourier-truncated Gross–Pitaevskii equation

Turbulence in the two-dimensional Fourier-truncated Gross–Pitaevskii equation

The onset of thermalization in finite-dimensional equations of hydrodynamics: insights from the Burgers equation

A Structure Function Model Recovers the Many Formulations for Air‐Water Gas Transfer Velocity

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