Tangential oscillations of a differentially rotating spherical fluid layer

The natural harmonic oscillations of a differentially rotating fluid layer under the action of a potential force are considered. The rise in the layer level is assumed to be negligible. The oscillations satisfy an ordinary second-order differential equation with singular coefficients that depend on the spatial coordinate. This equation is solved by the method of local separation of singularities based on the use of the properties of the Fuchs series for a bounded solution. Various laws for the latitude dependence of the angular rate of ocean rotation and the effect of these laws on the problem spectrum are considered. An equation is obtained for the streamlines of the oscillations investigated. Two cases in which the latitude dependence of the base flow velocity coincides with the real dependence for a celestial body are considered and the corresponding modes are found.     

Non-linear oscillations of a spherical particle in a rotating fluid

The motion of a spherical particle contained within a rotating fluid is analysed using equations of classical mechanics. Conditions are established under which the second order differential system simplifies to the well known Svedberg equation which describes the radial motion in centrifugation theory. The complete particle trajectories are obtained by solving the second order system numerically and approximations to these are derived using perturbation methods.     

Self-gravitating stability of a rotating fluid layer

The self-gravitating instability of the present model is discussed by using a simple linear theory. The problem is formulated for a rotating fluid layer, and a dispersion relation valid for all kinds of perturbations is derived and discussed. The self-gravitating force is found to be a destabilizing factor for a small range of wavenumbers, while it is stabilizing in other ranges, depending on the density ratio of the fluids. For high values of the angular velocity, the rotational force produces a stabilizing effect and can suppress the self-gravitating instability. In the absence of the self-gravitating force, the model of a rotating fluid layer is marginally stable.     

Axisymmetric oscillations of two spherical shells in a compressible fluid

To find the interaction between spherical shells at the frequency of their free oscillations in a fluid, we examine the problem of axisymmetric oscillations of two identical spherical shells under the assumption that the shell centers of curvature do not coincide. The solution is found for the cases of a compressible and an incompressible fluid by the series method with reduction to an infinite system of linear equations. A mathematical justification of the method used is presented.     

Vortex motion in a rotating barotropic fluid layer

To study vortex motion and the mechanisms of geostrophic adjustment (i.e. the equilibrium between pressure gradient and Coriolis force, which leads to the weakening of inertio-gravity waves) in large scale geophysical flows, we simulate the dynamics of a shallow-water layer in uniform rotation, without any forcing other than the initial injection of energy and potential enstrophy. Such a flow generates inertio-gravity waves which interact with the rotational eddies. We found that both inertio-gravity waves and rotation reduce the non-linear interactions between vortices, namely the condensation of the vorticity field into isolated coherent vortices, corresponding to the inverse rotational energy cascade, and the associated production of vorticity filaments, due to the direct potential enstrophy cascade. Rotation also inhibits the direct inertio-gravitational energy cascade for scales larger than the Rossby deformation radius. Therefore, if inertio-gravity waves are initially excited at large enough scales, they will remain trapped there due to rotation and there will be no geostrophic adjustment. On the contrary, if inertio-gravity waves are only present at scales smaller than the Rossby deformation radius, which are insensitive to the effect of rotation, they will non-linearly interact and cascade towards the dissipative scales, leaving the flow in geostrophic equilibrium.     

Stability of small radial oscillations of a spherical layer of liquid

An isothermal spherical layer of a viscoelastic liquid described by the one-parameter Maxwell model is considered. When the model parameter is taken equal to zero, a model of a purely viscous Newtonian fluid is obtained. The stability of the spherical layer of liquid with respect to small radial perturbations of the velocity and pressure is investigated for both types of liquids. Leningrad. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 170–171, September–October, 1988.     

Instability of fluid motion in a thin spherical layer

We study flow stability in a thin spherical layer with respect to small disturbances. It is shown that for each given layer thickness there is a sequence of critical Reynolds numbers above which the motion is unstable. In its form, the critical disturbance is reminiscent of the secondary flow which develops upon loss of stability of the basic fluid flow between rotating cylinders (Taylor problem).The author wishes to thank M. I. Shliomis for his continued interest in this study.     

Nonlinear surface waves on a ferromagnetic fluid of variable depth

A nonlinear ray method is used to study surface waves on a ferromagnetic fluid of variable depth subject to a horizontal magnetic field, and an equation of the KdV type with variable coefficients is derived. An approximate solution of the equation representing a three-dimensional soliton with varying amplitude and phase is constructed and numerical results are presented.     

Large-scale turbulence in a thin layer of nonisothermal rotating fluid

The dynamics of large-scale nonisothermal turbulence in a thin rotating layer of fluid are investigated. An hierarchical model, obtained by averaging the initial Boussinesq equations with respect to the vertical coordinates and subsequently projecting the two-dimensional equations onto a basis consisting of a system of axisymmetric spiral vortices of progressively decreasing scale, is proposed. It is shown that the presence of horizontal temperature inhomogeneities leads to a considerable increase in the turbulence decay time.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 48–55, July–August, 1988.     

Natural frequencies of longitudinal oscillations for a nonuniform variable cross-section rod

The natural frequencies of longitudinal oscillations of a rod such that its Young’s modulus, the density, and the cross-sectional area are functions of the longitudinal coordinate are analyzed. For solving the corresponding problem, an integral formula is used to represent the general solution to the original Helmholtz equation with variable coefficients in terms of the general solution to the accompanying equation with constant coefficients. Frequency equations are derived in the form of rapidly converging Leibniz series for three types of boundary conditions. For these cases the frequency zerothapproximation equations are given to quickly find the lowest natural frequencies with an adequate accuracy.     

Convective heat transfer in a rotating horizontal cylindrical fluid layer

This paper describes the thermal convection and heat transfer in a cylindrical fluid layer rotating around a horizontal axis, with various constant temperatures set at the layer boundaries. The influence of the rotational speed of the cylindrical fluid layer on the convective heat transfer in this layer is studied. The study results are presented as functions of dimensionless parameters that characterize the action of two convective mechanisms: centrifugal and thermal-oscillatory. It is shown that, with low rotational speed, the heat transfer is determined by quasistationary gravitational convection.     

Convective stability of a horizontal rotating fluid layer with helical turbulence

The problem of the corrective stability of a horizontal layer of turbulent fluid rotating about a vertical axis with a fixed heat flow at the boundaries is investigated in the case in which the intensity of the helical background does not depend on the rate of rotation and the degree of heating.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 33–39, January–February, 1992.The author is grateful to S. S. Moiseev for proposing the subject and to G. Z. Gershuni and D. V. Lyubimov for useful discussions.