Energy spectra and coherent structures in forced two-dimensional and beta-plane turbulence

Results from a wide range of direct numerical simulations of forced-dissipative, differentially rotating two-dimensional turbulence are presented, in order to delineate the broad dependence of flow type on forcing parameters. For most parameter values the energy spectra of simulations forced at low wavenumbers are markedly steeper than the classical k −3 enstrophy inertial-range prediction, and although k −3 spectra can be produced under certain circumstances, the regime is not robust, and the Kolmogorov constant is not universal unless a slight generalization is made in the phenomenology. Long-lived, coherent vortices form in many cases, accompanied by steep energy spectra and a higher than Gaussian vorticity kurtosis. With the addition of differential rotation (the β-effect), a small number of fairly distinct flow regimes are observed. Coherent vortices weaken and finally disappear as the strength of the β-effect increases, concurrent with increased anisotropy and decreased kurtosis. Even in the absence of coherent vortices and with a Gaussian value of the kurtosis, the spectra remain relatively steep, although not usually as steep as for the non-rotating cases. If anisotropy is introduced at low wavenumbers, the anisotropy is transferred to all wavenumbers in the inertial range, where the dynamics are isotropic. For those simulations that are forced at relatively high wavenumbers, a well resolved and very robust k −5/3 energy inertial range is observed, and the Kolmogorov constant appears universal. The low-wavenumber extent of the reverse energy cascade is essentially limited by the β-effect, which produces an effective barrier in wavenumber space at which energy accumulates, and by frictional effects which must be introduced to achieve equilibrium. Anisotropy introduced at large scales remains largely confined to the low wavenumbers, rather than being cascaded to small scales. When there is forcing at both large and small scales (which is of relevance to the Earth's atmosphere), energy and enstrophy inertial ranges coexist, with an upscale energy transfer and downscale enstrophy transfer in the same wavenumber interval, without the need for any dissipation mechanism between forcing scales.