Turbulence on hyperbolic plane: the fate of inverse cascade
type: Item
instance of: scholarly article
title: Turbulence on hyperbolic plane: the fate of inverse cascade
zbMATH Open document ID: 1307.35192
author: Gregory Falkovich
author: Krzysztof Gawȩdzki
published in: Journal of Statistical Physics
publication date: 2014-10-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/1402.1797
review text: The authors consider an inverse cascade scenario for the 2D turbulence in the hyperbolic plane \(\mathbb{H}_R\) with the parametrization by the complex coordinate \(z\), \(| z|<1\) \[ X^1+iX^2=\frac{2Rz}{1-| z|^2},\quad X^3=\frac{R(1+| z|^2)}{1-| z|^2}. \] In this coordinates, the Riemannian metric is \[ g_R=R^2\frac{4\,dz\,d\bar{z}}{(1-| z|^2)^2}. \] Here, \(R\) is the scale of the metric with the dimension of length. At the same time \(R\) is the constant negative curvature of \(\mathbb{H}_R\). The turbulent flow is described by the Navier-Stokes equations with a random Gaussian white-in-time forcing operating on scales much shorter then the parameter \(R\). The long-distance behavior of the energy is investigated in the paper. It is found that the energy is cumulated in ring-like vortices of arbitrary diameter but of width of order \(R\). The theory of the Euler and Navier-Stokes equations on Riemannian manifolds is recalled in the paper. The long-time long-distance scaling limit of the theory on the hyperbolic plane is studied too. There are six appendices collected a supplementary and technical material.
reviewed by: Il'ya Sh. Mogilevskii
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35Q31
Mathematics Subject Classification ID: 76F20
Mathematics Subject Classification ID: 37D40
zbMATH DE Number: 6352059
zbMATH Keywords: turbulence
zbMATH Keywords: inverse cascade
zbMATH Keywords: hyperbolic plane
MaRDI profile type: Publication
OpenAlex ID: W3102506862
arXiv ID: 1402.1797
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DOI: 10.1007/S10955-014-0995-X
description: scientific article; zbMATH DE number 6352059