Numerical simulation of quasi-geostrophic turbulence

Abstract

The development of turbulence is studied numerically through the use of a simplified two-level quasi-geostrophic model. The domain occupied by the fluid has 64 times 64 grid points in the horizontal and obeys cyclic boundary conditions in the x - and y -directions. The turbulence is generated by a heating function composed of the 2-dimensional fourier components having the larger wave number, m = 8. Three cases are investigated for two values of H , the amplitude of the heating function and two values of v , the coefficient of eddy viscosity. Quasistationary spectra are obtained in each case. The slopes of the spectra for 8 ? m ? 16 on a log-log graph vary from ?1.7 to ?3.4 depending on v and H . The nonlinear cascade of available potential energy is found to be toward larger wavenumbers. The nonlinear cascade of kinetic energy is from components for which 8 ? m ? 14, mainly toward 1 ? m ? 7, the larger scales. The gains for m > 14 resulting from the cascade is relatively small. These transfers are in qualitative agreement with atmospheric observations. Gains of enstrophy are observed to be fairly constant for 14 ? m ? 28. In an inertial subrange they should be zero. Relatively small gains are recorded for m < 7 while the spectral region 8 ? m ? 14 acts as source. Except for the wavenumber of transition from negative to positive values the distribution of losses and gains of enstrophy due to the inertial cascade is similar to that resulting from calculations based on atmospheric observations. It is concluded that the two dimensional turbulence theories enunciated by Fjortoft, Kraichnan and Leith pertain to the model in this study and the atmosphere (so far as our model resembles the atmosphere) in the prediction of the dominant direction of the nonlinear cascades of kinetic energy and enstrophy in wavenumber space. Inertial subranges in which the cascades result in zero changes are neither observed in the numerical experiments of this study nor in calculations based on atmospheric observations in the spectral region 8 ? m ? 16. The experiments described in this paper were performed prior to the theory proposed by Charney and so were not designed to be a test of it. However, some assumptions and conclusions are discussed in the light of that theory. DOI: 10.1111/j.2153-3490.1973.tb00607.x