| An investigation is made on the self-similar spectral evolution of quasi-geotstrophic (QG) turbulence on an f-plane affected by horizontal divergence, which is proportional to the squared inverse of the radius of deformation. To purify the effect of horizontal divergence we adopt the Charney-Hasegawa-Mima equation as the governing equation. First, freely decaying turbulence is examined as a first step. In case of vanishing horizontal divergence, Batchelor provided a well-known theory for self-similar spectral evolution of two-dimensional turbulence. One more parameter of horizontal divergence invalidates such a purely dimensional argument as by Batchelor. A similarity form is proposed here including the effect of horizontal divergence. The formula is reduced to that of Batchelor when there is no horizontal divergence. On the other hand it turns out to agree with the spectral evolution discussed later for strong horizontal divergence. Next, spectral forms in inertial ranges are studied based on the balance of terms in the vorticity equation. In this argument the power of the stream function or kinetic energy is preferred to that of total energy. In reality the kinetic energy spectrum in the inertial subrange is shown to be the same irrespective of the strength of horizontal divergence: k-5/3 for energy cascade and k-3 for enstrophy cascade with k being the wavenumber, just as in the ordinary two-dimensional turbulence without horizontal divergence. Finally self-similar spectral evolution is discussed in terms of the ordinary differential equations (ODEs) for the three variables of peak wavenumber, total energy and total enstrophy. The dynamics based on the ODEs gives quite a comprehensive and systematic explanation to the self-similar evolution of QG turbulence on an f-plane with and without horizontal divergence in various situations including the classical case of Batchelor. The method can be applied to both freely decaying and forced turbulence, which may suffer finite dissipation near the truncation wavenumber in numerical experiment. In order to examine the validity of the argument a numerical experiment was carried out, where the kinetic energy (not total energy) was kept constant by a rather artificial numerical adjustment for the case of strong horizontal divergence. Numerical experiment showed a good agreement with the prediction based on the dynamics that the spectral peak scale increases as a third power of time. |
|
|