Heat spots as structural elements of synoptic turbulence model

Abstract

The large-scale dynamics of heat spots in a thin layer of incompressible rotating fluid under the action of Coriolis and gravity is considered to obtain a simple model of synoptic turbulence. The derivation of equations describing the evolution of the spots is based on the use of the variational principle, perturbation theory, and the assumption of geostrophic balance. Exact solutions with piecewise constant contour curvature are found. The simplest of them look like circular spots that move with constant velocity depending on their radius. More complex solutions, such as ``loop solitons'', are shown to have fractal structure and are constructed by the gluing method. Heat spots can act as structural elements of turbulence. In particular, we show that the spectral energy density for the velocity field of a random ensemble of heat spots has the power asymptotics $E\propto k^{\alpha}$ with exponents $\alpha=2$ at $k\bar{R}<1$, and $\alpha=-1$ at $k\bar{R}>1$, where $\bar{R}$ is the average (over ensemble) radius of spot.