ISSN 1028-3358, Doklady Physics, 2013, Vol. 58, No. 5, pp. 186–190. © Pleiades Publishing, Ltd., 2013. Original Russian Text © V.N. Zyryanov, E.A. Ryzhov, K.V. Koshel’, 2013, published in Doklady Akademii Nauk, 2013, Vol. 450, No. 2, pp. 171–175. 186 In this article, we present a model that describes one of the mechanisms of diapycnal mixing in topo- graphic eddies above seamounts in the Ocean [1, 2]. It is known that the topographic eddies are described in the quasi-geostrophic approximation [1–4]; i.e., ver- tical velocities are considered to be negligibly small (of about the Rossby number). However, the data of observations indicate the presence of a well-mixed fluid column in the vicinity of seamounts [5]. Descrip- tion of such vertical mixing is possible only by taking into account the vertical velocity component. A mathematical theory that describes the appear- ance of finite vertical velocities was suggested recently [6, 7]. The theory was constructed for the specific con- figuration of the seamount in the form of two coaxial cylinders of different diameters, in which the smaller diameter cylinder was arranged above the larger diam- eter cylinder. It was shown that the bifurcation of the topographic vortex occurs in a ring region over the ter- race under definite conditions, as a result of which, secondary toroidal vortices appear. Such a configura- tion of the underwater perturbation was specified in order to use the analogy with the Taylor–Couette flow between two rotating cylinders as applied to a ring region over the terrace but already on the rotating- f-plane. Two circular Stewartson layers on liquid con- tinuations of cylinder boundaries play the role of ana- logs of solid side boundaries for velocity perturbations. Thus, the obtained toroidal vortices were arranged between two Stewartson layers in a ring region “cut” in the fluid by continuations of the side boundaries of cylinders. However, the impermeability of the bound- aries for perturbations of the flow velocity is a rather strong assumption since the Stewartson turbulent layer can be rather wide and, in addition, it is not an imper- meable barrier for liquid particles in reality. In this report, we state a more rigorous statement of the problem free of the mentioned disadvantage. In the context of this statement, it turned out to be possi- ble to consider the generation process of toroidal vor- tices for the case of configuration of the seamount in the form of one isolated cylinder. STATEMENT OF THE PROBLEM Let us initially consider the problem for two coaxial cylinders from [6]. The starting spectral problem for- mulated in [6] has the form MECHANICS Vortex Tori above Bottom Perturbations in a Rotating Fluid V. N. Zyryanov a, b , E. A. Ryzhov c , and K. V. Koshel’ c, d Presented by Academician V.P. Dymnikov November 14, 2012 Received December 23, 2012 Abstract—In this study, a model that describes one of the mechanisms of diapycnal mixing in topographic vortices above seamounts in the Ocean is presented. It is known that topographic eddies are described in the quasi-geostrophic approximation; i.e., vertical velocities are considered to be negligibly small (of about the Rossby number). However, the data of observations indicate the presence of a well-mixed fluid column in the vicinity of seamounts. Description of such vertical mixing is possible only due to account of the vertical veloc- ity component. The spectral problem for the stability of a topographic vortex over the perturbation of the bot- tom relief in the form of a simple cylinder and for a more complex configuration consisting of two coaxial cyl- inders is solved. It is shown that the eigenvalues of the spectrum are the bifurcation points of the flow, due to which secondary toroidal vortices are generated. These toroidal vortices perform intense vertical mixing of waters above seamounts in the Ocean. The method of invariant imbedding is used to solve the spectral problem. DOI: 10.1134/S1028335813050121 a Water Problems Institute, Russian Academy of Sciences, Gubkin str., 3, Moscow, 119333 Russia b Moscow State University, Moscow, 119899 Russia c Pacific Institute of Oceanology, Far East Branch, Russian Academy of Sciences, Baltiiskaya ul. 43, Vladivostok, 690041 Russia d Far East Federal University, ul. Sukhanova 8, Vladivostok, 690600 Russia