Vertical modes of quasi-geostrophic flows in an ocean with bottom topography : evolution equation and energetics

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Quasi-geostrophic current is expanded in terms of vertical modes such as barotropic and baroclinic ones. Then the evolution of quasi-geostrophic motion is understood from the behavior of each vertical mode. There are some subtle issues, however, as regards vertical modes: boundary conditions, difference between a level model and a layer model, and so on. A comprehensive formulation is given of the expansion of the quasi-geostrophic flows in terms of vertical modes both for a level model and for a layer model. Vertical modes are defined in almost the same manner for a level model and a layer model. Evolution equation and energetics of each mode are derived and argued. In addition to the ordinary nonlinear advective three-mode interaction, a gentle relief of bottom topography allows inter-modal energy transfer via bottom topography; in particular the so-called JEBAR effect and topographic torque is classified into one of such two-mode interaction (coupling). Under the rigid lid surface condition, the barotropic mode does not make a triplet with two different baroclinic modes, in the three-mode nonlinear advective interaction. Detailed balance turns out to hold for inter-modal energy transfer with respect to three-mode nonlinear advective interaction and two-mode coupling via topography. Potential enstrophy transfer among vertical modes is formulated also. It is shown that detailed balance is assured for inter-modal enstrophy transfer as well, if potential enstrophy is chosen properly. On the other hand detailed balance is not expected for inter-modal enstrophy transfer by two-mode coupling via bottom topography.

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