Solitary waves 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 about a fixed axis. The motion imparted by this planetary rotation upon the fluid particles of the ocean has been assumed to be governed by a linear version of the Navier-Stokes equation.We study the vortex motion within this rotating ocean and establish that the propagation of vortices depends on a third-order partial differential equation for the stream function. We prove that, in the most general case, this vorticity equation cannot generate any solitary waves; however, should the vertical component of vorticity satisfy a certain functional relationship, then we have obtained a family of solitary waves of permanent form.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.     

Non-linear dynamics of the corotation torque

The excitation of spiral waves by an external perturbation in a disc deposits angular momentum in the vicinity of the corotation resonance (the radius where the speed of a rotating pattern matches the local rotation rate). We use matched asymptotic expansions to derive a reduced model that captures non-linear dynamics of the resulting torque and fluid motions. The model is similar to that derived for forced Rossby wave critical layers in geophysical fluid dynamics. Using the model we explore the saturation of the corotation torque, which occurs when the background potential (specific) vorticity is redistributed by the disturbance. We also consider the effects of dissipation. If there is a radial transport of potential vorticity, the corotation torque does not saturate. The main application is to the creation, growth and migration of protoplanets within discs like the primordial solar nebula. The disturbance also nucleates vortices in the vicinity of corotation, which may spark further epochs of planet formation.     

Optical layouts of the WSO-UV spectrographs

Electron acoustic blow up solitary waves and periodic waves are studied in a classical unmagnetized plasma containing cold electron fluid, kappa distributed hot electrons and stationary ions. We obtain Korteweg-de Vries (KdV) equation for electron acoustic waves (EAWs) using the reductive perturbation technique (RPT). Applying bifurcation theory of planar dynamical systems to the obtained KdV equation, we prove the existence of electron acoustic blowup solitary and periodic wave solutions. Depending on different physical parameters, two types of exact explicit solutions of the mentioned waves are derived. Our model may be applied to explain blow up solitary and periodic wave features that may occur in the planetary magnetosphere and the plasma sheet boundary layer.     

On structure formation in the universe

The formation of structures in the universe is one of the most challenging problems of cosmology. In this paper, an attempt to explain the formation of galaxies through the generation of vortices (with dissipation) in an uniformly expanding perfect fluid is made. The equation governing the mean square vorticity for a turbulent (isotropic and homogeneous) fluid is derived. It is shown that the mechanism of stretching vortices could enhance the mean square vorticity as a function of time. However, ultimately expansion and dissipation dominate and the solution for the mean square vorticity reaches the prediction by linear theory.     

Bifurcations of ion acoustic solitary waves and periodic waves in an unmagnetized plasma with kappa distributed multi-temperature electrons

Ion acoustic solitary waves and periodic waves in an unmagnetized plasma with superthermal (kappa distributed) cool and hot electrons have been investigated using non-perturbative approach. We have transformed basic model equations to an ordinary differential equation involving electrostatic potential. Then we have applied the bifurcation theory of planar dynamical systems to the obtained equation and we have proved the existence of solitary wave solutions and periodic wave solutions. We have derived two exact solutions of solitary and periodic waves depending on the parameters. From the solitary wave solution and periodic wave solution, we have shown the effects of density ratio p of cool electrons and ions, spectral index κ, and temperature ratio σ of cool electrons and hot electrons on characteristics of ion acoustic solitary and periodic waves.     

Bifurcations of electron acoustic traveling waves in an unmagnetized quantum plasma with cold and hot electrons

Bifurcations of nonlinear electron acoustic solitary waves and periodic waves in an unmagnetized quantum plasma with cold and hot electrons and ions has been investigated. The one dimensional quantum hydrodynamic model is used to study electron acoustic waves (EAWs) in quantum plasma. Applying the well known reductive perturbation technique (RPT), we have derived a Korteweg-de Vries (KdV) equation for EAWs in an unmagnetized quantum plasma. By using the bifurcation theory and methods of planar dynamical systems to this KdV equation, we have presented the existence of two types of traveling wave solutions which are solitary wave solutions and periodic traveling wave solutions. Under different parametric conditions, some exact explicit solutions of the above waves are obtained.     

Bifurcations of dust ion acoustic travelling waves in a magnetized quantum dusty plasma

Bifurcation behavior of nonlinear dust ion acoustic travelling waves in a magnetized quantum dusty plasma has been studied. Applying the reductive perturbation technique (RPT), we have derived a Kadomtsev-Petviashili (KP) equation for dust ion acoustic waves (DIAWs) in a magnetized quantum dusty plasma. By using the bifurcation theory of planar dynamical systems to the KP equation, we have proved that our model has solitary wave solutions and periodic travelling wave solutions. We have derived two exact explicit solutions of the above travelling waves depending on different parameters.     

Covariant formulation of the dynamical equations of quantum vortices in type II superconductors

We have derived the closed system of covariant equations which describe the motion of quantum vortices regarded as a two-2dimensional polarized liquid. We have obtained the covariant expressions of the forces acting on the vortices; from the equilibrium condition of these forces we have deduced the equation satisfied by the velocity field of the fluid. It is shown that this velocity field depends on the friction coefficient, the density of vortices and the superconducting current. From this closed system of equations we derived the relaxation equation when a variable magnetic field is applied. Published in Astrofizika, Vol. 50, No. 3, pp. 381–391 (August 2007).     

Bifurcations of dust acoustic solitary waves and periodic waves in an unmagnetized plasma with nonextensive ions

Bifurcations of dust acoustic solitary waves and periodic waves in an unmagnetized plasma with q-nonextensive velocity distributed ions are studied through non-perturbative approach. Basic equations are reduced to an ordinary differential equation involving electrostatic potential. After that by applying the bifurcation theory of planar dynamical systems to this equation, we have proved the existence of solitary wave solutions and periodic wave solutions. Two exact solutions of the above waves are derived depending on the parameters. From the solitary wave solution and periodic wave solution, the effect of the parameter (q) is studied on characteristics of dust acoustic solitary waves and periodic waves. The parameter (q) significantly influence the characteristics of dust acoustic solitary and periodic structures.     

Force-free electromagnetic waves

The time-dependent Force-Free Electromagnetic Field (FFEMF) is studied. In contrast to the case of Force-Free Magnetic Field (FFMF), it is shown that the FFEMF can occur in the form of waves. The FFEMF wave equation is solved in the case of one spatial dimension. Besides a periodical linear FFEMF wave solutions, the existence of solitary wave solutions is demonstrated. The possible application of FFEMF solutions to solar flares is discussed.Work done at the Space Environment Laboratory, NOAA/ERL/SEL, Boulder, CO 80303, U.S.A.     

New exact solutions for nonlinear solitary waves in Thomas-Fermi plasmas with (G′/G)-expansion method

The nonlinear propagation of ion acoustic waves in an ideal plasmas containing degenerate electrons is investigated. The Korteweg-de-Vries (K-dV) equation is derived for ion acoustic waves by using reductive perturbation method. The analytical traveling wave solutions of the K-dV equation investigated, through the (G′/G)-expansion method. These traveling wave solutions are expressed by hyperbolic function, trigonometric functions are rational functions. When the parameters are taken special values, the solitary waves are derived from the traveling waves. Also, numerically the effect different parameters on these solitary waves investigated and it is seen that exist only the compressive solitary waves in Thomas-Fermi plasmas.     

Nonlinear periodic and algebraic ion-acoustic solitary waves in a relativistic plasma

We derive a mixed modified Korteweg-de Vries (MK-dV) equation from a semi-relativistic ion acoustic wave with hot ions by the fluid approximation. The positive cubic nonlinearity of the mixed MK-dV equation give rise to the periodic progressive waves and the algebraic solitary waves. The periodic wave bears a series of solitary pulses, and the algebraic solitary wave reduces the rarefactive solitary wave in the limit of the particular boundary condition. These nonlinear wave modes explain, respectively, the periodic pulse of the potential and the rarefactive solitary wave of the fine structure observed in space.     

Alternative method to solve soliton envelope equation

In studying the nonlinear electrohydrodynamic stability of solitary wave packets of capillary-gravity waves, in (2+1) dimensions, for dielectric fluids, we found that the complex amplitude of the surface elevation can be described by a nonlinear Schrödinger equation which can be written in the form of a soliton envelope equation. Using the tanh method we get in a very simple way the solitary wave solutions of this equation which we had obtained before by using the Jacobian elliptic functions.     

Interaction of ion beam with dust grains produces dust-acoustic solitary waves in Herbig-Haro objects

Properties of dust-acoustic solitary waves in a warm dusty plasma are analyzed by using the hydrodynamic model for massive dust grains, electrons, ions, and streaming ion beam. For this purpose, Korteweg-de Vries (KdV) equation for the first-order perturbed potential and linear inhomogeneous KdV-type equation for the second-order perturbed potential have been derived and their analytical solutions are presented. In order to show the characteristics of the dust-acoustic solitary waves are influenced by the plasma parameters, the relevant numerical analysis of the KdV and linear inhomogeneous KdV-type equations are obtained. The dust-acoustic solitary waves, as predicted here, may be associated with the nonlinear structures caused by the interaction of polar jets with the interstellar medium, which is known as Herbig-Haro objects.     

Nonlinear dispersive waves on the surface of a self-gravitating fluid layer

The evolution of two dimensional wave packets on the surface of a self-gravitating fluid layer is investigated and shown to be governed by a nonlinear Schrödinger equation. The wave train of finite amplitude is modulationally unstable. Obtained also are the dynamical equations for the second harmonic resonance. The analysis reveals that the general motion consists of both amplitude and phase modulated waves of which the pure phase and amplitude modulated waves, solitary waves, and phase jump are just the special cases.     

Jupiter's cyclones and anticyclones vorticity from Voyager and Galileo images

We have measured the internal velocity field in jovian synoptic-scale cyclones and anticyclones by tracking cloud elements in very high spatial resolution images obtained by the Voyager 1 and 2 (in 1979) and Galileo (1996-2000) spacecrafts. In total we have studied 24 different closed vortices (6 cyclones, 18 anticyclones) spanning a latitude range from ∼60° N to 60° S and with East-West sizes larger than ∼2000 km. The tangential component of the velocity as a function of the distance to the vortex center and position angle is used to retrieve the vorticity field. We find that the velocity increases in all the vortices from a nearly quiescent center to a maximum at the vortex periphery, with a record of about 180 m s⁻¹ for the GRS. The vorticity of cyclones and anticyclones increases in general toward their periphery with absolute values in the range from ∼2-14×¹⁰⁻⁵ s⁻¹. There is a marked tendency to increase the vortices vorticity with their latitude location. However the vorticity does not depend on the vortex size, circulation sense, or ambient background meridional wind shear. The vortex Rossby number ranges from ∼0.2 to 0.5. A study of the interaction between the Great Red Spot with other vortices show that the GRS does not change its vorticity upon their absorption. The two White Ovals mergers showed contradictory results, with greater vorticity in the case of BE, but lower vorticity in the case of BA, although data are poorer for this last case. We present the case of a short lived but large coherent cyclone at −59° that was embedded in a weakly anticyclone wind shear domain. We show that jovian vortices do not follow the simple Kida vortex relationship between vorticity and aspect ratio as it has been previously suggested.     

On the speed and shape of electron acoustic solitary waves

Electron acoustic solitary waves in a collisionless plasma consisting of a cold electron fluid and non-thermal hot electrons are investigated by a direct analysis of the field equations. The Sagdeev potential is obtained in terms of electron acoustic speed by simply solving an algebraic equation. It is found that the amplitude and width of the electron acoustic solitary waves as well as the parametric regime where the solitons can exist are very sensitive to the population of energetic non-thermal hot electrons. The soliton and double layer solutions are obtained as a small-amplitude approximation. The effect of non-thermal hot electrons is found to significantly change the properties of the electron acoustic solitary waves (EAWs). A comparison with the Viking Satellite observations in the day side auroral zone is also discussed.     

Solitons and double-layers of electron-acoustic waves in magnetized plasma; an application to auroral zone plasma

A theoretical investigation is carried out for understanding the properties of electron-acoustic potential structures (i.e., solitary waves and double-layers) in a magnetized plasma whose constituents are a cold magnetized electron fluid, hot electrons obeying a nonthermal distribution, and stationary ions. For this purpose, the hydrodynamic equations for the cold magnetized electron fluid, nonthermal electron density distribution, and the Poisson equation are used to derive the corresponding nonlinear evolution equation; modified Zakharov–Kuznetsov (MZK) equation, in the small amplitude regime. The MZK equation is analyzed to examine the existence regions of the solitary pulses and double-layers. It is found that rarefactive electron-acoustic solitary waves and double-layers strongly depend on the density and temperature ratios of the hot-to-cold electron species as well as the nonthermal electron parameter.