Logarithmic discretization and systematic derivation of shell models in two-dimensional turbulence

Gürcan, Ö. D.; Morel, Pierre; Kobayashi, Sumire; Singh, Rameswar; Xu, Shi; Diamond, P. H.

doi:10.1103/physreve.94.033106

Cited by 5 publications

(6 citation statements)

References 46 publications

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“…An example is visually represented in figure 2 and more detailed pictures of allowed scale interactions can be found in [90]. We see that fork p the scale q can be at most k 2 and at least 0.…”

Section: Interaction Conditions For Scalesmentioning

confidence: 94%

Gyrokinetic turbulence: between idealized estimates and a detailed analysis of nonlinear energy transfers

2017

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Using large resolution numerical simulations of gyrokinetic (GK) turbulence, spanning an interval ranging from the end of the fluid scales to the electron gyroradius, we study the energy transfers in the perpendicular direction for a proton-electron plasma in a slab equilibrium magnetic geometry. The plasma parameters employed here are relevant to kinetic Alfvén wave turbulence in solar wind conditions. In addition, we use an idealized test representation for the energy transfers between two scales, to aid our understanding of the diagnostics applicable to the nonlinear cascade in an infinite inertial range. For GK turbulence, a detailed analysis of nonlinear energy transfers that account for the separation of energy exchanging scales is performed. Starting from the study of the energy cascade and the scale locality problem, we show that the general nonlocal nature of GK turbulence, captured via locality functions, contains a subset of interactions that are deemed local, are scale invariant (i.e. a sign of asymptotic locality) and possess a locality exponent that can be recovered directly from measurements on the energy cascade. It is the first time that GK turbulence is shown to possess an asymptotic local component, even if the overall locality of interactions is nonlocal. The results presented here and their implications are discussed from the perspective of previous findings reported in the literature and the idea of universality of GK turbulence.

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Section: Interaction Conditions For Scalesmentioning

confidence: 94%

Gyrokinetic turbulence: between idealized estimates and a detailed analysis of nonlinear energy transfers

2017

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“…The resulting model is a stiff set of ordinary differential equations (ODEs) on an exponantially coarse grid, somewhat similar to the 2D model discussed in Ref. [11].…”

Section: Numerical Resultsmentioning

confidence: 99%

“…This is important, as we will see consecutively, since it provides a "derivation" of shell models and their generalizations without a detailed knowledge of the conservation laws. A similar effort was discussed earliler for 2D turbulence [11].…”

Section: B Conservation Lawsmentioning

confidence: 90%

“…This means that now we need to solve 4 × 10 equations for each scale or, noting that half of these points are simply reflections of the rest of the points, 4 × 5 = 20 equations for each shell. The fourfold degeneracy which appears upon going from 3D to 2D is remarkable, since in the standard 2D formulation one would consider only vorticity (not helicities of both signs) and only one kind of polygon as a representation of the 2D k space [11]. The effort discussed in this work for going to a 2D formulation may be worthwhile, however, due to the issue of unphysical shell equipartition overwhelming the inverse cascade in 2D logarithmic discretization.…”

Section: E Transition To Two Dimensionsmentioning

confidence: 99%

“…[10]), are also simplified models that try to exploit this particular aspect of the geometry of the turbulent cascade. Logarithmically discretized models (LDMs) for two dimensional turbulence, which can be derived from a systematic self-similar reduction of the Fourier space [11], can also be counted among these models.…”

Section: Introductionmentioning

confidence: 99%

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Nested polyhedra model of turbulence

Gürcan

2017

Phys. Rev. E

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A discretization of the wave-number space is proposed, using nested polyhedra, in the form of alternating dodecahedra and icosahedra that are self-similarly scaled. This particular choice allows the possibility of forming triangles using only discretized wave vectors when the scaling between two consecutive dodecahedra is equal to the golden ratio and the icosahedron between the two dodecahedra is the dual of the inner dodecahedron. Alternatively, the same discretization can be described as a logarithmically spaced (with a scaling equal to the golden ratio), nested dodecahedron-icosahedron compounds. A wave vector which points from the origin to a vertex of such a mesh, can always find two other discretized wave vectors that are also on the vertices of the mesh (which is not true for an arbitrary mesh). Thus, the nested polyhedra grid can be thought of as a reduction (or decimation) of the Fourier space using a particular set of self-similar triads arranged approximately in a spherical form. For each vertex (i.e., discretized wave vector) in this space, there are either 9 or 15 pairs of vertices (i.e., wave vectors) with which the initial vertex can interact to form a triangle. This allows the reduction of the convolution integral in the Navier-Stokes equation to a sum over 9 or 15 interaction pairs, transforming the equation in Fourier space to a network of "interacting" nodes that can be constructed as a numerical model, which evolves each component of the velocity vector on each node of the network. This model gives the usual Kolmogorov spectrum of k −5/3 . Since the scaling is logarithmic, and the number of nodes for each scale is constant, a very large inertial range (i.e., a very high Reynolds number) with a much lower number of degrees of freedom can be considered. Incidentally, by assuming isotropy and a certain relation between the phases, the model can be used to systematically derive shell models.

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A turbulent cascade model of bounce averaged gyrokinetics

Morel

Gürcan

2018

Physics of Plasmas

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A shell model is derived for the description of nonlinear bounce averaged gyrokinetics, which is one of the simplest kinetic descriptions in magnetized plasmas. In order to validate the numerical implementation, detailed linear evolution of the system is compared with a linear benchmark based on solving the linear dispersion relation numerically. The resulting wave number spectrum, which extends over 3–4 decades, is shown to have a robust general structure to model parameters, such as dissipation or the ratio of linear energy injection to nonlinear transfer. In a range of wave numbers where the nonlinear transfer term is dominant, a power law spectrum, roughly of the form k−4, is observed for the spectral electrostatic potential energy density. The model, being fully kinetic, can be used to obtain the free energy spectra for ion and electron distribution functions as functions of E. This model constitutes the first numerical implementation of a kinetic shell model.

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Logarithmic discretization and systematic derivation of shell models in two-dimensional turbulence

Cited by 5 publications

References 46 publications

Gyrokinetic turbulence: between idealized estimates and a detailed analysis of nonlinear energy transfers

Gyrokinetic turbulence: between idealized estimates and a detailed analysis of nonlinear energy transfers

Nested polyhedra model of turbulence

A turbulent cascade model of bounce averaged gyrokinetics

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