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.     

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.     

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.     

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.     

Ion acoustic solitary waves in a weakly relativistic plasma with <Emphasis Type="Italic">q</Emphasis>-nonextensive electrons and thermal positrons

Nonlinear ion acoustic solitary waves (IASWs) are addressed in a weakly relativistic plasma consisting of cold ion fluid, q-nonextensive electron velocity distribution and Boltzmann distributed positron. The Korteweg-de Vries- (KdV) equation is derived by reductive perturbation method. We investigate the effect of nonextensive electrons on solitary waves in this medium. It is found that only compressive solitons can be appeared in the existence of nonextensive electrons. It is shown that the structure of soliton depend sensitively on the q-nonextensive parameter.     

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.     

Arbitrary amplitude ion-acoustic solitary waves in superthermal electron-positron-ion magnetoplasma

Arbitrary amplitude ion-acoustic solitary waves propagating in a magnetized plasma composed of positive ions, superthermal electrons and positrons are investigated. For this purpose, the ions are represented by the hydrodynamical fluid equations while the non-Maxwellian electrons and positrons densities are assumed to follow kappa (κ) distribution. The basic equations are reduced to a pseudoenergy-balance equation. Existence conditions for large amplitude solitary waves are presented. The analytical and numerical analysis of the latter show that the ion-acoustic solitary wave can propagate only in the subsonic region in our plasma system and it is significantly influenced by the plasma parameters. The present analysis could be helpful for understanding the nonlinear ion-acoustic solitary waves propagating in interstellar medium and pulsar wind, which contain an excess of superthermal particles.     

On dust ion acoustic solitary waves in collisional dusty plasmas with ionization effect

The propagation of solitary waves in an unmagnetized collisional dusty plasma consisting of a negatively charged dust fluid, positively charged ions, isothermal electrons, and background neutral particles is studied. The ionization, ion loss, ion–neutral, ion–dust, and dust–neutral collisions are considered. Applying a reductive perturbation theory, a damped Korteweg–de Vries (DKdV) equation is derived. On the other hand, at a critical phase velocity, the dynamics of solitary waves is governed by a damped modified Korteweg–de Vries (DMKdV) equation. The nonlinear properties of solitary waves in the two cases are discussed.     

Propagation of solitary waves in non uniform dusty plasmas

Interaction of dust acoustic solitary waves in plasmas consisting of medium disorders is investigated. Disorders and inhomogeneities of the medium are added to the equation of motion as perturbative terms through the medium parameters. The effects of these perturbations on the behaviour of solitary waves are studied with numerical simulations and the results are compared with theoretical predictions in a uniform media.     

Detection of ionospheric perturbation due to a soft gamma ray repeater SGR J1550-5418 by very low frequency radio waves

The properties of cylindrical and spherical dust acoustic (DA) solitary and shock waves in an unmagnetized electron depleted dusty plasma consisting of inertial dust fluid and ions featuring Tsallis statistics are investigated by employing the reductive perturbation technique. A Korteweg-de Vries Burgers (KdVB) equation is derived and its numerical solution is obtained. The effects of ion nonextensivity and dust kinematic viscosity on the basic features of DA solitary and shock waves are discussed in nonplanar geometry. It is found that nonextensive nonplanar DA waves behave quite differently from their one-dimensional planar counterpart.     

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.     

Compressive and Rarefactive Dust-acoustic Solitary Structures in a Magnetized Two-ion-temperature Dusty Plasma

A rigorous theoretical investigation has been made of obliquely propagating dust-acoustic solitary structures in a cold magnetized two-ion-temperature dusty plasma consisting of a negatively charged, extremely massive, cold dust fluid and ions of two different temperatures. The reductive perturbation method has been employed to derive the Korteweg-de Vries (K-dV) equation which admits a solitary wave solution for small but finite amplitude limit. It has been shown that the presence of second component of ions modifies the nature of dust-acoustic solitary structures and may allow rarefactive dust-acoustic solitary waves (solitary waves with density dip) to exist in such a dusty plasma system. The effects of obliqueness and external magnetic field on the properties of these dust-acoustic solitary structures are also briefly discussed.     

Time-fractional KdV equation for electron-acoustic waves in plasma of cold electron and two different temperature isothermal ions

The time fractional KdV equation is derived for small but finite amplitude electron-acoustic solitary waves in plasma of cold electron fluid with two different temperature isothermal ions. The effects of the time fractional parameter on the electrostatic solitary structures are presented. It is shown that the effect of time fractional parameter can be used to modify the amplitude of the electrostatic waves (viz. the amplitude, width and electric field) of the electron-acoustic solitary waves. The model may provide a possible explanation for the low-frequency component of the broadband electrostatic noise in the plasma sheet boundary layer of the Earth’s magnetotail where the electron beams are not present.     

Ion-acoustic waves in the solar atmosphere

A model for ion-acoustic waves in the solar atmosphere is presented. In the limit of strongly magnetized plasma this model leads to the Zakharov-Kuznetsov equation which possesses a flat solitary wave solution. An initial-value problem for this equation is solved numerically to show a transition of the flat solitary waves into spherical solitary waves. The paper suggests further developments of an ion-acoustic wave theory that may improve our knowledge of ion-acoustic waves and lead to the possibility of their being detected in the solar atmosphere.     

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.     

Arbitrary amplitude electron acoustic waves in a magnetized nonextensive plasma

Arbitrary amplitude electron acoustic (EA) solitary waves in a magnetized nonextensive plasma comprising of cool fluid electrons, hot nonextensive electrons, and immobile ions are investigated. The linear dispersion properties of EA waves are discussed. We find that the electron nonextensivity reduces the phase velocities of both modes in the linear regime: similarly the nonextensive electron population leads to decrease of the EA wave frequency. The Sagdeev pseudopotential analysis shows that an energy-like equation describes the nonlinear evolution of EA solitary waves in the present model. The effects of the obliqueness, electron nonextensivity, hot electron temperature, and electron population are incorporated in the study of the existence domain of solitary waves and the soliton characteristics. It is shown that the boundary values of the permitted Mach number decreases with the nonextensive electron population, as well as with the electron nonextensivity index, q. It is also found that an increase in the electron nonextensivity index results in an increase of the soliton amplitude. A comparison with the Vikong Satellite observations in the dayside auroral zone is also taken into account.     

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.