Slow magnetohydrodynamic waves in the solar atmosphere

There is increasingly strong observational evidence that slow magnetoacoustic modes arise in the solar atmosphere, either as propagating or standing waves. Sunspots, coronal plumes and coronal loops all appear to support slow modes. Here we examine theoretically how the slow mode may be extracted from the magnetohydrodynamic equations, considering the special case of a vertical magnetic field in a stratified medium: the slow mode is described by the Klein-Gordon equation. We consider its application to recent observations of slow waves in coronal loops.     

Damping of magnetohydrodynamic waves by resonant absorption in the solar atmosphere

In the last decade we have been overwhelmed by an avalanche of discoveries of magnetohydrodynamic (MHD) waves by the Solar and Heliospheric Observatory and Transition Region and Coronal Explorer observatories. Both standing and propagating versions of fast magnetoacoustic and slow magnetoacoustic MHD waves have been detected. Information on the damping times and damping distances of these waves is less detailed and less accurate than that on periods and amplitudes. Nevertheless, observations show the damping times and damping lengths are often short. Also, different types of MHD waves in different types of magnetic structures likely require different damping mechanisms. The phenomenon of fast damping is well documented for the standing fast magnetosonic kink waves in coronal loops. This paper concentrates on standing fast magnetosonic waves. It reports on results on periods and damping times due to resonant absorption in one-dimensional and two-dimensional models of coronal loops. Special attention is given to multiple modes.     

Nonlinear waves in the Pitaevskii-Gross equation

Nonlinear waves, solitary and periodic, are studied exactly in the Pitaevskii-Gross equation for the wave function of the condensate of a superfluid. We also study the relationship between these two waves and Bogoliubov's phonon, and the energies associated with these waves. The creation energy of a solitary wave with amplitudeA is proportional toA ^3/2. Solitary waves show interesting behavior on their collision due to their localized character. The effect of collision on solitary waves can be described by the phase shift. We give a formula of the phase shift on a collision of two solitary waves. We further discuss the decay of an arbitrary initial disturbance into solitary waves.On leave from Department of Physics, Kyushu University, Fukuoka, Japan.     

Nonlinear water waves

This introduction to the issue provides a review of some recent developments in the study of water waves. The content and contributions of the papers that make up this Theme Issue are also discussed.     

Nonlinear surface waves in magnetohydrodynamics

Summary The magnetohydrodynamic effects on the nonlinear gravity waves and capillary waves on the surface of an infinitely-conducting liquid confined by a vacuum magnetic field aligned with the unperturbed surface are studied. For the nonlinear problem, the method of strained parameters is used to obtain solutions as perturbations about the linear oscillations. The cases with wavenumbers away from the second-harmonic quasi-resonant value and wavenumbers near the latter are treated separately. Several distinctive features associated with magnetohydrodynamic nonlinear surface wave motions are pointed out.With 1 Figure     

Numerical modelling of MHD waves in the solar chromosphere

Acoustic waves are generated by the convective motions in the solar convection zone. When propagating upwards into the chromosphere they reach the height where the sound speed equals the Alfvén speed and they undergo mode conversion, refraction and reflection. We use numerical simulations to study these processes in realistic configurations where the wavelength of the waves is similar to the length scales of the magnetic field. Even though this regime is outside the validity of previous analytic studies or studies using ray-tracing theory, we show that some of their basic results remain valid: the critical quantity for mode conversion is the angle between the magnetic field and the k-vector: the attack angle. At angles smaller than 30 degrees much of the acoustic, fast mode from the photosphere is transmitted as an acoustic, slow mode propagating along the field lines. At larger angles, most of the energy is refracted/reflected and returns as a fast mode creating an interference pattern between the upward and downward propagating waves. In three-dimensions, this interference between waves at small angles creates patterns with large horizontal phase speeds, especially close to magnetic field concentrations. When damping from shock dissipation and radiation is taken into account, the waves in the low-mid chromosphere have mostly the character of upward propagating acoustic waves and it is only close to the reflecting layer we get similar amplitudes for the upward propagating and refracted/reflected waves. The oscillatory power is suppressed in magnetic field concentrations and enhanced in ring-formed patterns around them. The complex interference patterns caused by mode-conversion, refraction and reflection, even with simple incident waves and in simple magnetic field geometries, make direct inversion of observables exceedingly difficult. In a dynamic chromosphere it is doubtful if the determination of mean quantities is even meaningful.     

Propagation of quasi-simple waves in a compressible rotating atmosphere

Summary A class of self-propagating linear and nonlinear travelling wave solutions for compressible rotating fluid is studied using both numerical and analytical techiques. It is shown that, in general, a three dimensional linear wave is not periodic. However, for some range of wave numbers depending on rotation, horizontally propagating waves are periodic. When the rotation is equal to , all horizontal waves are periodic. Here, is the ratio of specific heats. The analytical study is based on phase space analysis. It reveals that the quasi-simple waves are periodic only in some plane, even when the propagation is horizontal, in contrast to the case of non-rotating flows for which there is a single parameter family of periodic solutions provided the waves propagate horizontally. A classification of the singular points of the governing differential equations for quasi-simple waves is also appended.     

Nonlinear waves in an active-dissipative disperse medium

Nonlinear waves in a medium involving dissipation, dispersion, and enhancement described by the generalized Kuramoto-Sivashinsky equation are discussed. Analytical solutions of the equation are obtained in the form of solitary waves. For numerical modeling of the nonlinear waves a difference scheme is suggested. Interaction of nonlinear waves described by the Kuramoto-Sivashinsky model is considered. It is shown that for specified values of the problem parameters there is one solitary wave described by the initial model. The dependences of the velocity and amplitude of this wave on the problem parameters are determined. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 71, No. 1, pp. 149–154, January–February, 1998.     

MHD waves and oscillations in the solar plasma. Introduction

The Sun's magnetic field is responsible for many spectacularly dynamic and intricate phenomena, such as the 11 year solar activity cycle, the hot and tenuous outer atmosphere called the solar corona, and the continuously expanding stream of solar particles known as the solar wind.Recently, there has been an enormous increase in our understanding of the role of solar magnetism in producing the observed complex atmosphere of the Sun. One such advance has occurred in the detection, by several different high-resolution space instruments on-board the Solar and Heliospheric Observatory and Transition Region and Coronal Explorer satellites, of magnetic waves and oscillations in the solar corona.The new subjects of solar atmospheric and coronal seismology are undergoing rapid development. The aim of this Scientific Discussion Meeting was to address the progress made through observational, theoretical and numerical studies of wave phenomena in the magnetic solar plasma. Major theoretical and observational advances were reported by a wide range of international scientists and pioneers in this field, followed by lively discussions and poster sessions on the many intriguing questions raised by the new results. Theoretical and observational aspects of magnetohydrodynamic waves and oscillations in general, and how these wave phenomena differ in various regions of the Sun, including sunspots, the transient lower atmosphere and the corona (in magnetic loops, plumes and prominences), were addressed through invited review papers and selected poster presentations. The results of these deliberations are collected together in this volume.     

Nonlinear axisymmetric waves in compressible hyperelastic rods: long finite amplitude waves

Summary This paper investigates nonlinear axisymmetric waves in compressible hyperelastic circular cylindrical rods. We consider first a compressible Mooney-Rivlin material to obtain exact governing equations. To further study the problem, we introduce the notion of long finite amplitude waves and derive the corresponding simplified model equations, which gives the framework for studying problems like wave-interactions arising through collision or reflection. The asymptotically valid far-field equation is consequently deduced from the simplified model equations. Then, using a strained-coordinate method, we obtain the second-order solitary wave solution. The result is not only of interest itself, but also provides a suitable initial condition for wave interaction problems. Finally, the results for a general hyperelastic rod are presented.     

Nonlinear thermomagnetic waves of finite amplitude in superconductors

A theoretical analysis of the dynamics of thermal and electromagnetic perturbations in a superconductor shows that nonlinear thermal and electric dissipative wave structures may form under certain conditions on the sample surface. The structures possess a finite-amplitude and propagate at a constant velocity. The appearance of these structures is qualitatively described and the wave propagation velocity is estimated.     

Nonlinear surface waves with allowance for the saturation effect

It is shown that for a medium with the dielectric function given by the expression ε=ε ₁+(ε ₂−ε ₁)tanh(E ²/E _s ² ), where ε ₁ and ε ₂ are constants and E _s is the saturation field, there is a threshold value of the power density at which a nonlinear surface wave can be excited. T. G. Shevchenko Pridnestrov State-Corporate University Pis’ma Zh. Tekh. Fiz. 23, 36–39 (December 12, 1997)     

Nonlinear surface waves at ferrite-metamaterial waveguide structure

A new ferrite slab made of a metamaterial (MTM), surrounded by a nonlinear cover cladding and a ferrite substrate, was shown to support unusual types of electromagnetic surface waves. We impose the boundary conditions to derive the dispersion relation and others necessary to formulate the proposed structure. We analyse the dispersion properties of the nonlinear surface waves and we calculate the associated propagation index and the film–cover interface nonlinearity. In the calculation, several sets of the permeability of the MTM are considered. Results show that the waves behaviour depends on the values of the permeability of the MTM, the thickness of the waveguide and the film–cover interface nonlinearity. It is also shown that the use of the singular solutions to the electric field equation allows to identify several new properties of surface waves which do not exist in conventional waveguide.     

Nonlinear capillary-gravity waves under an edge condition

The objective of this paper is to investigate the forced motion of nonlinear capillary-gravity waves in a waterfilled circular basin by a harmonic vibration applied to its side wall under Evans's or Hocking's edge condition at a contact line. The forcing frequency is near a resonance frequency under the classical edge condition of the basin. Two complex-amplitude equations for the excited eigenmode at the resonance frequency corresponding to these two edge conditions are derived. The solutions to these equations display quite different behavior and an edge condition indeed has a great influence on the excited surface waves.     

Nonlinear surface waves on an elastic solid

The effect of quadratic elastic nonlinearity on the propagation of surface Rayleigh waves on an isotropic elastic solid is examined. Using the method of multiple scales an approximate solution is obtained which is uniformly valid in both spatial directions as well as in time. An arbitrary wave profile is considered and an integro-differential equation is derived for the Fourier transform of the displacement on the boundary. In the case of a quasi-monochromatic wave explicit expressions are derived for the variations of the amplitudes of the fundamental and second and third harmonics along the boundary.     

Nonlinear surface acoustic waves on an isotropic solid

A systematic derivation of the approximate coupled amplitude equations governing the propagation of a quasi-monochromatic Rayleigh surface wave on an isotropic solid is presented, starting from the non-linear governing differential equations and the non-linear free-surface boundary conditions, using the method of mulitple scales. An explicit solution of these equations for a signalling problem is obtained in terms of hyperbolic functions. In the case of monochromatic excitation, it is shown that the second harmonic amplitude grows initially at the expense of the fundamental and that the amplitudes of the fundamental and second harmonic remain bounded for all time.     

Nonlinear electrocapillary waves on a charged surface of the ideal liquid

An analytical expression for the profile of a plane wave traveling over a charged surface of the ideal incompressible liquid under the action of surface tension and gravity is obtained in the second order of smallness with respect to the wave motion amplitude. An analysis of this expression shows that, for a certain fixed value of the surface charge density (subcritical from the standpoint of realization of the Tonks-Frenkel instability), the wave profiles are qualitatively different from those of nonlinear capillary-gravitational waves existing on the uncharged liquid surface (for this reason, the obtained wave branch can be called electrocapillary).     

Nonlinear reflection of surface acoustic waves from contact lines

We present the results of an experimental investigation of nonlinear processes accompanying the interaction of surface acoustic waves (SAWs) with real cracking defects and contact lines. The dynamic characteristics of the nonlinear reflection of higher harmonics are presented. It is shown that the third harmonic generation has a pronounced threshold character, the dynamic characteristics exhibit hysteresis, and the efficiency of nonlinear response depends on the duration of SAW action on the defect. The results lead to a conclusion that the nonlinear reflection of SAWs from a contact line provides an adequate model of reflection from a real cracking defect.