Large scale motions in the atmosphere and in the oceans are well expressed using quasi-geostrophic systems. Simple one-layer quasi-geostrophic potential vorticity equation (also called Charney-Hasegawa-Mima equation as an equivalent) often describes the essential dynamics of such large scale motions. In turbulent motions governed by this quasi-geostrophic equation, the energy is transported upward to the small wavenumber region and thus coherent patterns are formed. The energy upward cascade is also observed in two-dimensional turbulence, but there are some quantitative differences such as the speed of the upward cascade. Dynamic scaling laws for quasi-geostrophic turbulence, which describes quantitatively this energy upward cascade, is developed by Watanabe et al. (1998) theoretically and by numerical simulations. However, in their theory, they evaluated a parameter necessary for the dynamic scaling law from the result of numerical calculation: a parameter which shows the total energy dissipation rate.
In the quasi-geostrophic turbulence, most of the energy is transported to the small wavenumber region, but a small amount of energy is transported to the large wavenumber region owing to the small but finite dissipation. We estimated the energy dissipation rate based on the assumption that the total energy dissipation in this system is equal to the energy transported to the large wavenumber region. The shift of the energy spectrum to the small wavenumber region must be balanced with the energy transport to the large wavenumber region, and this relation enables us to evaluate the energy transported to the large wavenumber region from the wavenumber of the energy spectrum peak. This theoretical estimation complements the dynamic scaling laws of Watanabe et al. (1998): the parameter expressing the total energy dissipation which is left unknown and empirically determined in their study is theoretically derived.
In numerical calculations with finite cut-off wavenumber, we use artificial hyperviscosity coefficients in order to avoid the pile-up of the cascaded energy near the cut-off wavenumber region. This estimation also suggests the means to determine the appropriate hyperviscosity coefficient used in numerical simulations. Considering that the artificial dissipation near the cut-off wavenumber region is a substitute of the energy which should have been transported beyond the cut-off wavenumber, we can determine the appropriate value of the hyperviscosity from the condition that the energy dissipated by the hyperviscosity should be the same as the energy cascaded to the large wavenumber region. In the numerical calculation by Watanabe et al. (1998), the energy spectrum has a too steep gradient in the large wavenumber region. The numerical calculation using the value suggested by the consideration here successfully derived the theoretically expected energy spectrum. |
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