The Scaling Law of Quasi-Geostrophic Turbulence with Weak Energy Dissipation

Abstract

When a problem without viscosity is considered as an ideal case, a small dissipation term is necessary in many numerical calculations. Quasi-geostrophic turbulence, which is described by the barotropic version of quasi-geostrophic potential vorticity equation, (also called Charney-Hasegawa-Mima equation as an equivalent) is one of these situations: a dissipation term must be added in order to prevent the energy from piling up in the large wavenumber region. Consequently, the total energy decreases gradually, which should be conserved in an ideal situation. In this article, the total energy dissipation rate in quasi-geostrophic turbulence is estimated, based on the assumption that the energy dissipated in this system is equal to the energy transported to the large wavenumber region. This estimation complements the dynamic scaling laws for the quasi-geostrophic turbulence developed by Watanabe et al. (1998): the parameter expressing the total energy dissipation, which was determined from the result of the numerical calculation in their study, is derived from the parameters of the setting of the numerical calculation. This estimation also suggests the means to determine the appropriate artificial hyperviscosity coefficient used in numerical simulations.