D. K. Callebaut

EHD Envelope Solitons of Capillary-Gravity Waves in Fluids of Finite Depth

We consider the nonlinear electrohydrodynamic stability of wave packets of capillary-gravity waves in fluids of any depth; they travel predominantly in one direction, however the wave amplitudes are modulated slowly in both horizontal directions. The method of multiple time scales is used to obtain a nonlinear Schrodinger equation describing the behaviour of the perturbed system. The envelope solutions of steady form are obtained in terms of the Jacobian elliptic functions. It follows that various types of envelope solutions of the modulated Stokes waves may exist depending on the relative signs of terms representing dispersive and nonlinear effects; the solitary and periodic envelope solutions for the general case of any liquid depth are described. It is also shown that the evolution of the envelope is governed by two coupled partial differential equations with cubic nonlinearity. The stability of solitons with respect to transverse perturbations is investigated. It is found that such wave packets are stable to short waves, and unstable to long disturbances, and the envelope solitons and the waveguides are always unstable, and the stability of the system depends on the values of the dielectric constant ratio, the electric field, the wavenumber, and the depth of the fluid.

General soliton solutions for nonlinear dispersive waves in convective type instabilities

The two-dimensional Ginzburg–Landau equation (GLE) is obtained from basic equations by a linear stability analysis. This equation governs the evolution of slowly varying envelopes of periodic spatio-temporal patterns related to Rayleigh–Benard convective instabilities. In addition, the phase instabilities of the complex GLE (CGLE) with quintic and space-dependent cubic terms modelling the Eckhaus and zigzag convective instabilities are reported. We find soliton solution classes to the elliptic and hyperbolic CGLE, by applying the function transformation method. The two-dimensional CGLE is transformed to a sine-Gordon equation, a sinh-Gordon equation and other equations, which depend only on one function χ. The general solution of the equation in χ leads to a general soliton solution of the two-dimensional CGLE. The obtained solutions contain some interesting specific solutions such as plane solitons, N multiple solitons and propagating breathers. We also discuss the soliton stability of the CGLE.

Heating of plasma by Alfvén waves envelope

Alfven waves are investigated including dissipation (resistivity, viscosity) and the Hall effect (dispersion), searching for the dominant nonlinear effects. First a modulational instability arises, which heats the plasma and causes a solition envelope to develop. Both the stationary and non-stationary soliton envelopes were studied. For certain conditions shock waves are generated. Finally the emission of ion-sound waves by the accelerated solition envelope in inhomogeneous plasma is discussed.

Linear and nonlinear interaction of electromagnetic waves with weakly ionized magnetized plasmas

We derive the basic equations for magnetoacoustic hydrodynamics (MAD). A new branch of electromagnetic waves having small damping is obtained in the range of the electron cyclotron frequency in a strong collisional plasma. Filamentational and modulational instabilities are elaborated. The effect of helicon waves (whistlers) on the plasma is considered and instabilities are studied. The formation of solitons and shock waves is investigated.

Large-scale unipolar regions generated from undeep magnetic fields

We explain the generation of the large-scale unipolar magnetic field regions (global magnetic regions) by the same dynamo action as for the generation of the sunspots and the polar faculae butterfly diagrams as given by Callebaut (2006). The previous global magnetic regions through meridional circulation now serve as the main seed fields (flux-transport dynamo for the global field regions), possibly supplemented by leftovers from the sunspots and some weak fields generated at the tachocline.

Kelvin-Helmholtz instability in magnetohydrodynamic flows

The Rayleigh-Taylor instability (RTI) of a continuously stratified fluid has implica- tions on the stability of solar and planetary interiors. A nonlinear stage of the two-dimensional RTI is studied by including various effects. By using the multiple scale method, we derived a non- linear Schrodinger equation (NLSE) in 2+1 dimensions. We show the general soliton solutions of the NLSE and this allows to discuss their stability.

On general transformations and variational principles of three-dimensional incompressible gravitating flows in ideal MHD

In this paper, we apply the general theory of Arnold (1965, 1966) and Moffatt et al. (1997). We search sufficient conditions for the linear stability of steady three-dimensional incompressible gravitating flows in ideal magnetohydrodynamics (MHD). The results suggest that the solar and the stellar convection zones must be sensitive to the density stratification.