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.

Lattice Boltzmann model for collisionless electrostatic drift wave turbulence obeying Charney–Hasegawa–Mima dynamics

“…As a result the transformed signals do not alter the overall power law coefficient of the k-spectrum. The obtained power law coefficients resemble the theoretically [43] and numerically [37,39,44] predicted strong turbulence laws…”

“…The deviation at t = 0 of the initial variables is based in the differing resolutions underlying the Fourier transforms of the k-spectra, and in the initialisation of the dynamical variables itself. The fitted power law coefficients are of same magnitude as the estimates of U ∝ t −0.5 and E ∝ t −0.05 [39]. …”

“…The occurring shear in the x-direction reduces with the drift parameter κ n and persists even if the approximated velocity shifts (eqs. (38) and (39)) are replaced by the exact expressions. Hence the shear effect is intrinsically related to the compressible limit of the resolved ion continuity and momentum equations.…”

“…The initial energy spectrum E δφ (k) ∝ k 30 (k + k 0 ) −60 is determined in k-space by a random phase factor and a narrow peak around a wavenumber k 0 [39,40], which is here is fixed to k 0 = 0.5. Hence around 32 field modes are initialised into a physical box size of 64ρ s .…”

Two possible mechanisms for vortex self-organization

“…where ψ is the geostrophic stream function, λ the characteristic wave number, and J the Jacobian opera- [20,21] . As far as the dissipative term, v = 0 in some works [22] .…”