Evolution of solitary vortices in a rotating fluid

The evolution of cyclones and anticyclones with a characteristic scale substantially exceeding the radius of deformation is investigated numerically. An equation obtained by an asymptotic method for small values of the Kibel'-Rossby number is employed. The model ensures conservation of the potential vorticity in the fluid particles for motions of finite amplitude (when the thickness of the layer H deviates considerably from the undisturbed value H₀). It is shown that anticyclones of a certain type adapt themselves to a steady shape and travel in a direction opposite to the direction of rotation of the sphere (westwards). It is found that the existence of a steadily migrating anticyclone requires the presence of a region of closed isolines of the potential vorticity (trapping region), in which the fluid is transported together with the anticyclone. Cyclones drift westwards more slowly than anticyclones and migrate towards the poles, losing energy by emitting Rossby waves. It is found that in cyclones the trapping region retains an almost circular shape, the decay of the eddy, associated with wave emission, showing as its initial intensity increases.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 52–59, July–August, 1986.     

Numerical analysis of the rotating viscous flow approaching a solid sphere

A numerical simulation is performed to investigate the flow induced by a sphere moving along the axis of a rotating cylindrical container filled with the viscous fluid. Three‐dimensional incompressible Navier–Stokes equations are solved using a finite element method. The objective of this study is to examine the feature of waves generated by the Coriolis force at moderate Rossby numbers and that to what extent the Taylor–Proudman theorem is valid for the viscous rotating flow at small Rossby number and large Reynolds number. Calculations have been undertaken at the Rossby numbers (R_o) of 1 and 0.02 and the Reynolds numbers (R_e) of 200 and 500. When R_o=O(1), inertia waves are exhibited in the rotating flow past a sphere. The effects of the Reynolds number and the ratio of the radius of the sphere and that of the rotating cylinder on the flow structure are examined. When R_o ? 1, as predicted by the Taylor–Proudman theorem for inviscid flow, the so‐called ‘Taylor column’ is also generated in the viscous fluid flow after an evolutionary course of vortical flow structures. The initial evolution and final formation of the ‘Taylor column’ are exhibited. According to the present calculation, it has been verified that major theoretical statement about the rotating flow of the inviscid fluid may still approximately predict the rotating flow structure of the viscous fluid in a certain regime of the Reynolds number. Copyright © 2004 John Wiley & Sons, Ltd.     

Calculation of autooscillations of the type of azimuthal waves occurring at loss of stability of flow of a viscous fluid between concentric cylinders rotating in opposite directions

Starting with the Navier-Stokes equation we use the Lyapunov-Schmidt method to investigate the nature of the loss of stability of Couette flow between cylinders as the Reynolds number passes through its critical value. We consider the rotation of the cylinders in opposite directions with the ratio of the angular velocities such that the role of the most dangerous disturbances passes over from rotationally symmetric to nonrotationally symmetric disturbances. Branching nonstationary secondary flows (autooscillations) are found in the form of azimuthal waves; the longitudinal wave number and the azimuthal wave number m are assumed given. The amplitude of autooscillations and the wave velocity are calculated for m = 1, and it is shown that depending on the value of both weak excitation of stable and strong excitation of unstable autooscillations are possible and the wave number for which the critical Reynolds number is a minimum corresponds to a stable wave regime in the supercritical region. The linear problem of the stability of the circular flow of a viscous fluid with respect to nonrotationally symmetric disturbances is discussed in [1–3]. Di Prima [1] solved the problem numerically by the Galerkin method when the gap is small and the cylinders rotate in the same direction. Di Prima's analysis is extended in [2] to cylinders rotating in opposite directions, and in [3] it is extended to gaps which are not small. The nonlinear stability problem is treated in [4], where for fixed = 3 and cylinders rotating in opposite directions the axisymmetric stationary secondary flow the Taylor vortex is calculated. The formation of azimuthal waves in the fluid between the cylinders was studied experimentally in detail by Coles [5].Translated from Zhurnal Prikladnoi Mekhanika i Tekhnicheskoi Fiziki, No. 2, pp. 68–75, March–April, 1976.     

On the dynamics of the fluid balancer

This paper is concerned with the dynamics of a so-called fluid balancer; a hula hoop ring-like structure containing a small amount of liquid which, during rotation, is spun out to form a thin liquid layer on the outermost inner surface of the ring. The liquid is able to counteract unbalanced mass in an elastically mounted rotor. The paper derives the equations of motion for the coupled fluid–structure system, with the fluid equations based on shallow water theory. An approximate analytical solution is obtained via the method of multiple scales. For a rotor with an unbalance mass, and without fluid, it is well known that the unbalance mass is in the direction of the rotor deflection at sub-critical rotation speeds, and opposite to the direction of the rotor deflection at super-critical rotation speeds (when seen from a rotating coordinate system, attached to the rotor). The perturbation analysis of the problem involving fluid shows that the mass center of the fluid layer is in the direction of the rotor deflection for any rotation speed. In this way a surface wave on the fluid layer can counterbalance an unbalanced mass.     

分层流体中内孤立波在台阶上的反射和透射

基于匹配渐近展开和格林函数的方法，研究了两层流体系统中内孤立波在台阶地形上透射、 反射及其分裂的演化特征. 通过保角变换和求解奇异Fredholm积分方程，获得了反映地形 效应对Boussinesq方程影响的约化边界条件，藉此建立了KdV演化方程的``初值'问题， 根据散射反演理论获得了反射波和透射波的解析表达式. 分析结果表明：上下流体层的厚度 比、密度比以及台阶高度对于反射和透射波振幅及其分裂具有显著的影响. 尤其当上层流体 厚度小于下层厚度时，由于存在临界点，在其附近反射波的幅值随台阶高度的演化由单调增 变为单调减，透射波的幅值由单调减变为单调增；上台阶的反射波与入射波反相，其最大幅 值可达到入射波的数倍；此外，下台阶反射波也可发展为单支孤立波，它区别于单层流体中 反射波仅为衰减的振荡波列.     

Frontal collision of internal solitary waves of first mode

The dynamics and energetics of a frontal collision of internal solitary waves (ISW) of first mode in a fluid with two homogeneous layers separated by a thin interfacial layer are studied numerically within the framework of the Navier–Stokes equations for stratified fluid. It was shown that the head-on collision of internal solitary waves of small and moderate amplitude results in a small phase shift and in the generation of dispersive wave train travelling behind the transmitted solitary wave. The phase shift grows as amplitudes of the interacting waves increase. The maximum run-up amplitude during the wave collision reaches a value larger than the sum of the amplitudes of the incident solitary waves. The excess of the maximum run-up amplitude over the sum of the amplitudes of the colliding waves grows with the increasing amplitude of interacting waves of small and moderate amplitudes whereas it decreases for colliding waves of large amplitude. Unlike the waves of small and moderate amplitudes collision of ISWs of large amplitude was accompanied by shear instability and the formation of Kelvin–Helmholtz (KH) vortices in the interface layer, however, subsequently waves again become stable. The loss of energy due to the KH instability does not exceed 5%–6%. An interaction of large amplitude ISW with even small amplitude ISW can trigger instability of larger wave and development of KH billows in larger wave. When smaller wave amplitude increases the wave interaction was accompanied by KH instability of both waves.     

Flow structures,nonlinear inertial waves and energy transfer in rotating spheres

We investigate flow structures, nonlinear inertial waves and energy transfer in a rotating fluid sphere, using a Galerkin spectral method based on helical-wave decomposition (HWD). Numerical simulations of flows in a sphere are performed with different system rotation rates, where a large-scale forcing is employed. For the case without system rotation, the intense vortex structures are tube-like. When a weak rotation is introduced, small-scale structures are reduced and vortex tubes tend to align with the rotation axis. As the rotation rate increases, a large-scale anticyclonic vortex structure is formed near the rotation axis. The structure is shown to be led by certain geostrophic modes. When the rotation rate further increases, a cyclone and an anticyclone emerge from the top and bottom of the boundary, respectively, where two quasi-geostrophic equatorially symmetric inertial waves dominate the flow. Based on HWD, effects of spherical confinement on rotating turbulence are systematically studied. It is found that the forward cascade becomes weaker as the rotation increases. When the rotation rate becomes larger than some critical value, dual energy cascades emerge, with an inverse cascade at large scales and a forward cascade at small scales. Finally, the flow behavior near the boundary is studied, where the average boundary layer thickness gets smaller when system rotation increases. The flow behavior in the boundary layer is closely related to the interior flow structures, which create significant mass flux between the boundary layer and the interior fluid through Ekman pumping.     

Solitary waves and conjugate flows in a three-layer fluid

Interfacial symmetric solitary waves propagating horizontally in a three-layer fluid with constant density of each layer are investigated. A fully nonlinear numerical scheme based on integral equations is presented. The method allows for steep and overhanging waves. Equations for three-layer conjugate flows and integral properties like mass, momentum and kinetic energy are derived in parallel. In three-layer fluids the wave amplitude becomes larger than in corresponding two-layer fluids where the thickness of a pycnocline is neglected, while the opposite is true for the propagation velocity. Waves of limiting form are particularly investigated. Extreme overhanging solitary waves of elevation are found in three-layer fluids with large density differences and a thick upper layer. Surprisingly we find that the limiting waves of depression are always broad and flat, satisfying the conjugate flow equations. Mode-two waves, obtained with a periodic version of the numerical method, are accompanied by a train of small mode-one waves. Large amplitude mode-two waves, obtained with the full method, are close to one of the conjugate flow solutions.     

On permanent nonlinear waves in a rotating fluid

The governing equation for long nonlinear gravity waves in a rotating fluid changes with the value of the Coriolis parameter f. (1) When f is large, i.e. in the strong rotation case, in an infinite ocean, there are only Sverdrup waves; in a semi-infinite ocean or in a channel, there are either solitary Kelvin waves, for which the governing equation is a KdV equation, or Poincaré waves, which can be obtained by superposition of two Sverdrup waves. (2) When f is small, i.e. in the weak rotation case, in an infinite ocean there are solitary or cnoidal waves governed by the Ostrovskiy equation, and we provide an explicit solution for both solitary and cnoidal Ostrovskiy progressive waves; and in a semi-infinite ocean or a channel, there are Sverdrup waves, which are governed either by Ostrovskiy equations or by the Grimshaw-Melville equation. (3) When f is very small, i.e. in the very weak rotation case, in an infinite ocean, or in a channel, there are solitary waves with a horizontal crest, but with a velocity component or a pressure gradient, which are governed by KdV equations as in the non-rotating case. Physically, that means that the most determining factor is the ratio of the Rossby radius of deformation over a characteristic length of the wave.     

Adjustment processes and radiating solitary waves in a regularised ostrovsky equation

The Ostrovsky equation is an adaptation of the Korteweg-de Vries equation widely used to describe the effect of rotation on surface and internal solitary waves. It has been shown that the effect of rotation is to destroy such solitary waves in finite time due to the emission of trailing radiation. Here this issue is re-examined for a regularised Ostrovsky equation. The regularisation is necessary to remove an anomaly in the Ostrovsky equation whereby there is a discontinuity in the mass field at the initial moment. It is demonstrated that in the regularised Ostrovsky equation there is a rapid adjustment of the mass which is transported a large distance in the opposite direction to that in which the solitary wave propagates.     

Finite-amplitude solitary waves in a two-layer fluid

Second-mode nonlinear internal waves at a thin interface between homogeneous layers of immiscible fluids of different densities have been studied theoretically and experimentally. A mathematical model is proposed to describe the generation, interaction, and decay of solitary internal waves which arise during intrusion of a fluid with intermediate density into the interlayer. An exact solution which specifies the shape of solitary waves symmetric about the unperturbed interface is constructed, and the limiting transition for finite-amplitude waves at the interlayer thickness vanishing is substantiated. The fine structure of the flow in the vicinity of a solitary wave and its effect on horizontal mass transfer during propagation of short intrusions have been studied experimentally. It is shown that, with friction at the interfaces taken into account, the mathematical model adequately describes the variation in the phase and amplitude characteristics of solitary waves during their propagation.     

Effect of a latitudinal temperature gradient on convective motions arising in a rotating spherical layer

The hydrodynamics of planetary atmospheres and Interiors are frequently directly or indirectly connected with convective motions taking place in rotating liquid spherical layers in the field of a central force. Convective stability in a spherical layer at rest, in a central gravity field, was first discussed in [1, 2]. It was shown that the critical Rayleigh number Rao at which convective instability sets in and the wave number of the critical perturbations depend essentially on the thickness of the layer. As in the plane case, the problem of the convective stability of a spherical layer is found to be degenerate, and the form of the critical perturbations cannot be determined from the linear problem. In actuality, minimization of the Rayleigh number permits establishing only the wave numberl for the spherical harmonic Y _l ^m (θ, ?), realized at the limit of stability; the parameter m remains indeterminate and thus 2l+1 independent convective modes correspond to Rao. In [3] a study was made of the convective stability of a liquid in a slowly rotating thin spherical layer. It was shown that the presence of rotation eliminates the degeneracy; at the limit of stability there arise motions corresponding to the Y _l l (θ, ?) -harmonic with a degenerate maximum at the equator, and propagating in a wave manner toward the side opposite to the rotation. In the present work a study is made of the convective stability of a flow of liquid, arising in a rotating spherical layer due to a nonuniform distribution of the temperatures at one of the boundaries of the layer. In such a statement of the problem it is possible to model large-scale motions in the atmospheres of large planets having internal sources of heat and absorbing solar radiation near the cloud cover of the atmosphere. It is established that, depending on the relationships between the parameters imparting the rotation and the inhomogeneous distribution of the temperature, there is either stabilization or destabilization of the layer in comparison with a fixed layer of the same thickness and with the same, but uniformly distributed heat flux supplied to the layer. A study is made of the form of the corresponding critical perturbations.     

Fluid flow in an arbitrary cascade on an axisymmetric stream surface in a variable thickness layer

A solution is given for the problem of flow past a cascade on an axisymmetric stream surface in a layer of variable thickness, which is a component part of the approximate solution of the three-dimensional problem for a three-dimensional cascade. Generalized analytic functions are used to obtain the integral equation for the potential function, which is solved via iteration method by reduction to a system of linear algebraic equations. An algorithm and a program for the Minsk-2 computer are formulated. The precision of the algorithm is evaluated and results are presented of the calculation of an example cascade.In the formulation of [1, 3] the problem of flow past a three-dimensional turbomachine cascade is reduced approximately to the joint solution of two-dimensional problems of the averaged axisymmetric flow and the flow on an axisymmetric stream surface in an elementary layer of variable thickness.In the following we solve the second problem for an arbitrary cascade with finite thickness rotating with constant angular velocity in ideal fluid flow: the solution is carried out on a Minsk-2 computer.Many studies have been devoted to this problem. A method for solving the direct problem for a cascade of flat plates in a hyperbolic layer was presented in [2]. Methods were developed in [1, 3] for constructing the flow for the case of a channel with variable thickness; these methods are approximately applicable for dense cascades but yield considerable error for small-load turbomachine cascades. The solution developed in [4], somewhat reminiscent of that of [2], is applicable for thin, slightly curved profiles in a layer with monotonically varying thickness. A solution has been given for a circular cascade for layers varying logarithmically [5] and linearly [6]. Approximate methods for slightly curved profiles in a monotonically varying layer with account for layer variability only in the discharge component were examined in [7–9]. A solution is given in [10] for an arbitrary layer by means of the relaxation method, which yields a roughly approximate flow pattern. The general solution of the problem by means of potential theory and the method of singularities presented in [11] is in error because of neglect of the crossflow through the skeletal line. The computer solution of [12] contains an unassessed error for the calculations in an arbitrary layer. The finite difference method is used in [13] to solve the differential equation of flow, which is illustrated by numerical examples for monotonie layers of axial turbomachines. The numerical solution of [13] is very complex.The solution presented below is found in the general formulation with respect to the geometric parameters of the cascade and the axisymmetric surface and also in terms of the layer thickness variation law.The numerical solution requires about 15 minutes of machine time on the Minsk-2 computer.     

Some properties of multi-fluid stokes flows

The solution of Stokes' equations for a rotating axisymmetric body which possesses reflection symmetry about a planar interface between two infinite immiscible quiescent viscous fluids is shown to be independent of the viscosities of the fluids and identical with the solution when the fluids have the same viscosity. The result is generalized to a rotating axisymmetric system of bodies which possesses reflection symmetry about each interface of a plane stratified system of fluids. An analogous result for two-fluid systems with a nonplanar static interface is also derived. The effect on torque reduction produced by the presence of a second fluid layer adjacent to a rotating axisymmetric body is considered and explicit calculations are given for the case of a sphere. A proof of uniqueness for unbounded multi-fluid Stokes' flow is given and the asymptotic far field structure of the velocity field is determined for axisymmetric flow caused by the rotation of axisymmetric bodies.     

Internal solitary waves moving over the low slopes of topographies

Internal solitary waves moving over uneven bottoms are analyzed based on the reductive perturbation method, in which the amplitude, slope and horizontal lengthscale of a topography on the bottom are of the orders of , ^5/2 and ^−3/2, respectively, where the small parameter is also a measure of the wave amplitude. A free surface condition is adopted at the top of the fluid layer. That condition contains two parameters, δ and Δ, the first of which concerns the discontinuity of the basic density between the outer layer and the inner one; the second concerns the discontinuity of the mean density between them. An amplitude equation for the disturbance of order decomposes into a Korteweg-de Vries (KdV) equation and a system of algebraic equations for a stationary disturbance around a topography on the bottom. Solitary waves moving over a localized hill are studied in a simple case where both the basic flow speed and the Brunt-Vaisalla frequency are constant over the fluid layer. For this case, the expression for the amplitude of the stationary disturbance contains singular points with respect to basic flow speed. These singularities correspond to the resonant conditions modified by the free surface condition. The advancing speeds of solitary waves are changed by the influence of bottom topography, in a case where the long internal waves propagate in the direction opposite to the basic flow, but their waveforms remain almost unchanged.     

Control of nonlinear vibrations of vibrating ring microgyroscope

The influence of the basement rotation on the variations in the spectrum of vibration frequencies of thin elastic shells and rings was known already at the end of the 19th century [1]. The physical phenomenon of inertness of elastic waves occurring free vibrations of an axisymmetric body, first explained in [2], were practically used in developing new types of gyros [2–6]. The foundations of the theory of wave gyros were laid in [2, 4], and the errors of such gyroscopes for various shapes of the vibrating resonator were studied in [2, 4, 7, 8]. It was shown that the error of the resonator manufacturing (the variable density, thickness, anisotropy of the material elastic properties, etc.) [2, 8] and the geometric nonlinearity of the resonator flexural vibrations studied in [2, 7] lead to splitting of the natural frequency of flexural vibrations [2], which is reflected in the wave picture of the resonator vibrations and characterizes the gyroscope precision.In the present paper, we study the errors of the vibrating microgyroscope which arise because of nonlinear elastic properties of the ring resonator material. We construct a control of the potential on the electrodes which allows one to maintain the prescribed amplitude of the normal resonator deflection and compensate for the gyroscope errors arising because of the nonlinear elastic properties of the material.     

On the origin of super-rotating layers in magnetohydrodynamic flows

An asymptotic analysis has been performed for the magnetohydrodynamic flow between perfectly conducting concentric cylindrical shells. The flow in the model geometry exhibits all the features which had been discovered in the past for the case of differentially rotating spherical shells considered in the context of geophysical analyses. For strong magnetic fields, the flow domain splits into distinct subregions and exhibits two different types of cores which are separated from each other by a tangent shear layer. The fluid in the inner core flows similar to a solid-body rotation and the outer core is entirely stagnant. With increasing magnetic fields the shear layer becomes thinner and, since the flow rate carried by the layer asymptotes to a finite value, the velocity in the layer increases as the layer thickness decreases. Moreover, the flux carried by the layer rotates in opposite direction compared with the rotation of the body. It is shown that the rotating jet is driven by the electric potential difference between the edges of the inner and the outer core.     

Secondary regimes in couette flow

We consider the asymptotic solutions of secondary steady flows in a fluid contained between cylinders rotating in the same direction for large Reynolds numbers.The existence of secondary axisymmetric steady flows in a fluid contained between cylinders rotating in the same direction was shown in [1, 2]. In the following we present the asymptotic behavior of such solutions for the case of large Reynolds numbers. The construction follows the scheme suggested in [3].     

Forced dissipative Boussinesq equation for solitary waves excited by unstable topography

In the paper, the effects of topographic forcing and dissipation on solitary Rossby waves are studied. Special attention is given to solitary Rossby waves excited by unstable topography. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a forced dissipative Boussinesq equation. By using the modified Jacobi elliptic function expansion method and the pseudo-spectral method, the solutions of homogeneous and inhomogeneous dissipative Boussinesq equation are obtained, respectively. With the help of these solutions, the evolutional character of Rossby waves under the influence of dissipation and unstable topography is discussed.     

Motion of a fluid between rotating cones

A study is made of the steady axisymmetric flow of a viscous fluid between two cones rotating in opposite ways round a common axis. It is shown that as in the case of the flow of fluid swirled by plane disks rotating at different speeds [1], there can be two regimes of motion in the system: a Batchelor regime with quasirigid rotation of the fluid outside the boundary layers [2] and a Stewartson regime in which the azimuthal flow is concentrated only in the boundary layers [3]. In the Stewartson regime, a boundary layer analogous to that in the single disk problem (see, for example, [4–6]) forms in the region of each cone far from the apex. For the flows outside the boundary layers, simple expressions are found which make it possible to obtain a conception of the circulation of the fluid as a whole. With minor alterations, the results can be applied to the case of the rotation of other curved surfaces.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 58–64, March–April, 1985.The author thanks A. M. Obukhov for suggesting the subject and for his interest in the work, and A. V. Danilov and S. V. Nesterov for useful discussions.