Propagation of vortices within a rotating,fluid shell

We consider a spherical, solid planet surrounded by a thin layer of an incompressible, inviscid fluid. The planet rotates with constant angular velocityWe study the vortex motion within this rotating ocean. For this purpose, we obtain a linearized version of the Navier-Stokes equation and adopt it as our ocean model; next, we prove analytically that a certain function of vorticity is an invariant of motion.Using this ocean model and this invariant property of vorticity, we are able to establish a general equation governing the motion of vortices within a fluid shell: it is a nonlinear partial differential equation of the third order for the stream function of motion.We finally examine some particular solutions of this vorticity equation that represent solitary waves of permanent form and decay within a finite distance. These solutions have been represented in terms of quadratic, exponential, and hyperbolic functions.The question whether these vortices that propagate as solitary waves could be solitons depends on their behavior when they collide with each other; this has not yet been resolved.Retired, U.S. Naval Research Laboratory, Washington, D.C., U.S.A.     

The motion of vortices within a rotating,fluid shell

We consider a spherical, solid planet surrounded by a thin layer of an incompressible, inviscid fluid. The planet rotates with constant angular velocity.Within the constraints of the geostrophic approximation of hydrodynamics, we determine the equation that governs the motion of a vortex tube within this rotating ocean. This vorticity equation turns out to be a nonlinear partial differential equation of the third order for the stream function of the motion.We next examine the existence of particular solutions to the vorticity equation that represent travelling waves of permanent form but decaying at infinity. A particular solution is obtained in terms of I ₁ and k ₁, the modified Bessel functions of order one.The question whether these localized vortices that move like solitary waves could even be solitons depends on their behavior during and after collision with each other and has not yet been resolved.Retired, U.S. Naval Research Laboratory, Washington, D.C., U.S.A.     

Solitary waves of vortices

We study the propagation of solitary waves of vortices within a spherical shell which constitutes the uppermost layer of a solid planet. This solid-liquid configuration rotates with constant angular velocity about an axis which is fixed with respect to the solid surface. The fluid within the shell is inviscid, incompressible, and of constant density. The motion imparted by the planetary rotation upon this fluid mass is governed by the Laplace tidal equation from which the potential of the extraplanetary forces has been deleted. Consistent with this ocean model, we establish that the stream function of a solitary wave of vortices must satisfy a third-order partial differential equation. We obtain solutions to this wave equation by imposing the condition that the vertical component of vorticity be functionally related to the stream function. We find that this dependence must necessarily be of the exponential type and that the solution to the wave equation then reduces to a quadrature depending on some arbitrary parameters. We prove that we can always choose the values of these parameters in order to approximate the integral in question by means of an analytic function: we reach a representation of the stream function of a solitary wave of vortices in terms of hyperbolic functions of time and position.This paper is dedicated to the memory of Professor Zdenek Kopal.     

Convection currents within a rotating Earth

We establish the equations of motion for a fluid spherical shell which constitutes the upper-most layer of a rotating body and which is heated from below because of radiogenic decay occurring in the lower layers. We solve these equations in terms of series of products of Bessel functions and Gegenbauer polynomials. By the step-wise solution of an infinite-order determinant equation we determine those values for the Taylor and Rayleigh numbers of the flow, up to the 10¹⁶ order, that will give rise to convective cells in the liquid layer.Using these eigenvalues, we solve linear systems of equations to determine the coefficients of the series solutions.     

Density distortion within a rotating body

This paper ascertains the distortion of the density distribution within a self-gravitating body in hydrostatic equilibrium under the influence of rotation.For this purpose, the Poisson equation has been solved by using the undistorted density profile _o(a) within the Laplacian to obtain the distorted density (a, ). The Laplacian has been expressed in terms of a system of curvilinear coordinates for which the equipotential surfaces constitute a family of fundamental surfaces.In performing the requisite algebraic manipulations, the Clairaut and Radau equations developed in a previous paper (Lanzano, 1974) were utilized to eliminate the derivatives of the elements pertaining to the equipotential surfaces.The density distortion has been obtained up to third-order terms in a small rotational parameter.     

Self-organization of electromagnetic waves into vortices in a magnetized electron-positron plasma

This paper presents a new class of well localized dipolar vortex solutions to the newly derived set of coupled nonlinear equations governing the dynamics of low-frequency electromagnetic waves in a strongly magnetized electron-positron plasma.     

Electrostatic drift vortices in a hot rotating strongly magnetized electron-positron pulsar plasma

Electrostatic drift wave in a hot rotating and strongly magnetized electron-positron pulsar plasma is considered. Using relativistic two fluid equations a pair of coupled nonlinear equations is derived. It is shown that the wave can propagate in the form of two-dimensional dipolar vortices at ultrarelativistic temperature (Tmc ²) of the plasma. The latter may affect the energy transport in the hot plasma, which can lead to a new turbulent state in the pulsar magnetosphere.     

MHD waves in coronal loops with a shell

We consider a model of a coronal loop in the form of a cord surrounded by a coaxial shell. Two slow magnetosonic waves longitudinally propagate within a thin flux tube on the m=0 cylindrical mode with velocities close to the tube velocities in the cord and the shell. One wave propagates inside the cord, while the other propagates inside the shell. A peculiar feature of the second wave is that the plasma in the cord and the shell oscillates with opposite phases. There are two fast magnetosonic waves on each of the cylindrical modes with m>0. If the plasma density in the shell is lower than that in the surrounding corona, then one of the waves is radiated into the corona, which causes the loop oscillations to be damped, while the other wave is trapped by the cord, but can also be radiated out under certain conditions. If the plasma density in the shell is higher than that in the cord, then one of the waves is trapped by the shell, while the other wave can also be trapped by the shell under certain conditions. In the wave trapped by the shell and the wave radiated by the tube, the plasma in the cord and the shell oscillates with opposite phases.     

Magneto-gravitational instability of a rotating and finitely-conducting fluid

The gravitational instability of an infinite homogeneous finitely conducting viscid fluid through porous medium is studied in the presence of a uniform vertical magnetic field and finite ion Larmor radius (FLR) effects. The medium is considered uniformly rotating along and perpendicular to the direction of the prevalent magnetic field. A general dispersion relation is obtained from the relevant linearized perturbation equations of the problem. Furthermore, the wave propagation along and perpendicular to the direction of existing magnetic field has been discussed for each direction of the rotation. It is found that the simultaneous presence of viscosity finite conductivity, rotation, medium porosity, and FLR corrections does not essentially change the Jeans's instability condition. The stabilizing influence of FLR in the case of transverse propagation is reasserted for a non-rotating and inviscid porous medium. It is shown that the finite conductivity has destabilizing influence on the transverse wave propagation whereas for longitudinal propagation finite conductivity does not affect the Jean's criterion.     

Unsteady magnetohydrodynamic flows in a rotating elasto-viscous fluid

The unsteady flow of an incompressible electrically-conducting and elasto-viscous fluid (Walter's liquidB), filling the semi-infinite space, in contact with an infinite non-conducting plate, in a rotating medium and in the presence of a transverse magnetic field is investigated. An arbitrary time-dependent forcing effect on the motion of the plate is considered and the plate and fluid rotate uniformly as a rigid body. The solution of the problem is obtained with the help of the Laplace transform technique and the analytical expressions for the velocity field as well as for the skin-friction are given.     

Unsteady magnetohydrodynamic flows in a rotating elastico-viscous fluid

This paper points out the errors in the solutions of a research work by N. Nanousis under the same title published in this journal, volume 199, 1993. The correct solutions of the problem for the velocity field and the drag on the plate, by the Laplace transform technique, are presented. The results are discussed for two cases of an arbitrary time-dependent forcing effect. It is shown that the viscoelastic parameterk > 0 influences the velocity and introduces reverse flow. For a suddenly accelerated plate,k > 0 increases the velocity forz < and decreases it forz > . In the case of the ramp-type boundary condition,k > 0 tends to decrease the velocity.     

Unsteady magnetohydrodynamic free-convection flows in a rotating fluid

The three-dimensional free-convection flow near an infinite vertical plate moving in a rotating fluid in the presence of a transverse magnetic field is studied in the case when the plate temperature undergoes a thermal transient. An exact solution has been obtained by defining a complex velocity with the help of the Laplace-transform technique, when the plate is moving with a velocity which is an arbitrary function of time. Three special cases of physical interest are also discussed.     

Strong cylindrical magnetogasdynamic shock waves in a rotating interplanetary medium

Strong cylindrical magnetogasdynamic shock waves in rotating interplanetary medium has been studied and an analytic solution for their propagation has been obtained. Using characteristic method and considering the effect of Coriolis force, we have shown that magnetic field has significant effect on the velocity of the shock wave.     

Magnetic reconnection via tearing instability in a rotating viscous fluid

The resistive tearing instability of a sheet pinch, first investigated by Kuang & Roberts (1990) for the case of a rapidly rotating inviscid fluid, is studied for arbitrary rotation rate in a visco‐resistive fluid. Altogether there are three regimes of the resistive tearing instability which correspond to the particular parameter domain in the (Ω, P_m) plane. Here Ω is the angular velocity of the medium which is normalized to the Alfvén time and P_m is the magnetic Prandtl number. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Tidal dissipation in rotating fluid bodies: a simplified model

We study the tidal forcing, propagation and dissipation of linear inertial waves in a rotating fluid body. The intentionally simplified model involves a perfectly rigid core surrounded by a deep ocean consisting of a homogeneous incompressible fluid. Centrifugal effects are neglected, but the Coriolis force is considered in full, and dissipation occurs through viscous or frictional forces. The dissipation rate exhibits a complicated dependence on the tidal frequency and generally increases with the size of the core. In certain intervals of frequency, efficient dissipation is found to occur even for very small values of the coefficient of viscosity or friction. We discuss the results with reference to wave attractors, critical latitudes and other features of the propagation of inertial waves within the fluid, and comment on their relevance for tidal dissipation in planets and stars.     

Hall effects on hydromagnetic convection flow in a rotating fluid

An analysis of Hall effects on the hydromagnetic free convection resulting from the combined effects of thermal and mass diffusion of an electrically-conducting liquid passed an infinite vertical porous plate in a rotating frame of reference is carried out when a strong magnetic field is imposed in a plane which makes an angle with the normal to the plate. The expressions for the mean velocity, mean temperature in the boundary layer, and the mean skin-friction, the mean rate of heat transfer on the plate are derived. The influence of Hall currents on the flow is studied for various values of .