Turbulence on hyperbolic plane: the fate of inverse cascade
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.
scientific article; zbMATH DE number 6352059
scientific article