Toroidal vortices over isolated topography in geophysical flows

This work deals with a model of a topographically trapped vortex appearing over isolated topography in a geophysical flow. The main feature of the study is that we pay special attention to the vertical structure of a topographically trapped vortex. The model considered allows one to study the vertical motion which is known not to be negligible in many cases. Given topography in the form of an isolated cylinder, and radial symmetry and stationarity of a uniform flow, in the linear approximation, we formulate a boundary value problem that determines all the components of the velocity field through a six-order differential operator, and nonincreasing boundary conditions at the center of the topography, and at infinity. The eigenvalues of the boundary value problem correspond to bifurcation points, in which the flow becomes unstable, hence non-negligible vertical velocities occur. We formulate a condition for the boundary value problem to have a discrete spectrum of these bifurcation points, and hence to be solvable. Conducting a series of test calculations, we show that the resulting vortex lies in the vicinity of topography, and can attain the distance up to half of the topography characteristic radius.