Turbulence on Hyperbolic Plane: The Fate of Inverse Cascade

We describe ideal incompressible hydrodynamics on the hyperbolic plane which is an infinite surface of constant negative curvature. We derive equations of motion, general symmetries and conservation laws, and then consider turbulence with the energy density linearly increasing with time due to action of small-scale forcing. In a flat space, such energy growth is due to an inverse cascade, which builds a constant part of the velocity autocorrelation function proportional to time and expanding in scales, while the moments of the velocity difference saturate during a time depending on the distance. For the curved space, we analyze the long-time long-distance scaling limit, that lives in a degenerate conical geometry, and find that the energy-containing mode linearly growing with time is not constant in space. The shape of the velocity correlation function indicates that the energy builds up in vortical rings of arbitrary diameter but of width comparable to the curvature radius of the hyperbolic plane. The energy current across scales does not increase linearly with the scale, as in a flat space, but reaches a maximum around the curvature radius. That means that the energy flux through scales decreases at larger scales so that the energy is transferred in a non-cascade way, that is the inverse cascade spills over to all larger scales where the energy pumped into the system is cumulated in the rings. The time-saturated part of the spectral density of velocity fluctuations contains a finite energy per unit area, unlike in the flat space where the time-saturated spectrum behaves as k^-5/3.