Nonlinear Rayleigh-Taylor instability

The Rayleigh-Taylor instability in hydromagnetics is investigated with the use of the method of multiple scales. It is shown that when the wavenumberK is equal to the critical wavenumberK _c , the amplitude modulation of a standing wave can be described by the nonlinear Schrödinger equation from which the nonlinear cutoff wavenumber is subsequently derived.     

Rayleigh-Taylor instability as a mechanism for corona formation on Venus

In this study we explore the idea that coronae have formed on Venus as a result of gravitational (Rayleigh-Taylor) instability of the lithosphere. The lithosphere is represented by a system of stratified homogeneous viscous layers (low-density crust over high density mantle, over lower density layer beneath the lithosphere). A small harmonic perturbation imposed on the base of the lithosphere is observed to result in gravitational instability under the constraint of assumed axisymmetry. Topography develops with time under the influence of dynamic stress associated with downwelling or upwelling, and spatially variable crustal thickening or thinning. Topography may therefore be elevated or depressed above a mantle downwelling, but the computed gravity anomaly is always negative above a mantle downwelling in a homogeneous asthenosphere. The ratio of peak gravity to topography anomaly depends primarily on the ratio of crust to lithospheric viscosity. Average observed ratios are well resolved for two groups of coronae (∼40 mgal km⁻¹), consistent with models in which the crust is perhaps 5 times stronger than the lithosphere. Group 3a (rim surrounding elevated central region) coronae are inferred to arise from a central upwelling model, whereas Group 8 (depression) coronae are inferred to arise from central downwelling. Observed average coronae radii are consistent with a lithospheric thickness of only 50 km. An upper low-density crustal layer is 10-20 km thick, as inferred from the amplitude of gravity and topography anomalies.     

Finite larmor-radius effects on Rayleigh-Taylor instability of a dusty magnetized plasma

This paper discusses the Rayleigh-Taylor (RT) instability of an infinitely conducting medium having an exponential density distribution which includes the effects of finite ion Larmor-radius (FLR) corrections and suspended particles in the presence of a uniform horizontal magnetic field. The relevant equations of the problem are linearized and from the linearized perturbation equations a dispersion relation is obtained, using appropriate boundary conditions. It has been found that the criterion for the stable density stratification remains uninfluenced by the simultaneous inclusion of the FLR corrections and suspended particles. The stability of the medium has been proved for the case of stable stratification when the FLR corrections are not considered in the analysis. The growth rate of unstable RT modes with increasing relaxation frequency of the suspended particles is evaluated analytically. It has been shown that the presence of suspended particles in the medium suppresses the growth rate of the unstable RT modes, thereby implying a stabilizing influence of the particles on the considered configuration.     

The Earth's core formation due to the Rayleigh-Taylor instability

The recent theories of planetary formation lead to a gravitationally unstable structure of the proto-Earth in the accretion stage, which is composed of three layers: an innermost undifferentiated solid core, an intermediate metal-melt layer, and an outermost silicate-melt layer. Taking this configuration as an initial state, we investigate the Earth's core formation due to the Rayleigh-Taylor instability by using the quantitative results on the instability in a self-gravitating fluid sphere obtained from another paper (S. Ida, Y. Nakagawa, and K. Nakagawa, submitted). We find that the instability occurs through the translational mode on a time scale of about 10 hr if the thickness of the metal-melt layer ⪆1 km. This leads to the conclusion that the Earth's core began to form through the translation of the innermost undifferentiated solid core as soon as the outer layer was melted and differentiated in the late accretion stage. In addition, we examine the rotational effects of the instability; the translation occurs most often along the rotational axis. But this preference is weak, since the rotational energy is small compared to the gravitational one.     

Rayleigh-Taylor instability of a Hall plasma with arbitrary density gradient

The Rayleigh-Taylor instability of a plasma layer in the presence of a horizontal magnetic field is investigated, taking into account the effects of Hall-currents and an arbitrarily large density gradient. It is shown for the general case that if the density decreases vertically upward, the system is thoroughly stable.For a plasma layer with exponentially varying density an approximate dispersion relation is obtained using the Galerkin's method. An analysis of the roots of the dispersion relation reveals that the Hall-currents loosen the stabilizing influence of the magnetic field and impart instability to the system. For sufficiently large values of the density gradient and the Hall currents the system is throughly destabilized for all perturbations.     

Rayleigh-Taylor instability of a viscous compressible plasma of variable density

A study has been made of the problem of the Rayleigh-Taylor instability of a hydromagnetic plasma of varying density to investigate the influence of the simultaneous presence of the effects of compressibility and viscosity. The solution is shown to be characterized by a variational principle. Based on the variational principle proper solutions have been obtained for a semi-infinite plasma, in which the density has a one-dimensional gradient along the direction of a uniform vertical magnetic field, confined between two planes. Both the viscosity and magnetic field are found to have a stabilizing influence. The effect of compressibility is found to be destabilizing.     

Rayleigh-Taylor instability of two viscous superposed rotating and conducting fluids

Classical R-M and synthetic W-D analysis of V758 Centauri are presented. Two solutions (semi-detached and contact) were found from differential corrections approach. The semi-detached model is physically acceptable since the system is thermally decoupled. The solution for this case and the photometric data are consistent with a B9 primary and A9 secondary components having parameters close to Main-Sequence values. It is suggested that V758 Centauri is a B-type W UMa system at the brokencontact phase predicted by the Thermal Relaxation Oscillations theory.     

Rayleigh-Taylor instability of ionized viscous fluids with FLR-corrections and surface-tension

The problem of Rayleigh-Taylor instability of superposed viscous magnetized fluids through porous medium is investigated in a partially-ionized medium. The fluid has ionized and neutralized particle components interacting with collisions. The effect of surface tension on R-T instability is also included in the present problem. The magnetohydrodynamic equations are modified for finite-Larmor radius corrections which is in the form of tensor. The equations of problem are linearized and using appropriate boundary condition, general dispersion relation is derived for two superposed fluids separated by horizontal boundary. The first part of the dispersion relation gives stable mode and condition is investigated using Hurwitz conditions. The second part of the dispersion relation shows that the growth rate of unstable system is reduced due to FLR corrections, viscosity, and collisional frequency of the neutrals. The role of surface tension on the system is also discussed.     

Rayleigh-Taylor instability of a plasma with finite ion Larmor radius effects

The stability of an infinitely conducting plasma of variable density has been investigated taking into account the finiteness of the ion Larmor radius. The perturbations propagating along the ambient magnetic field are considered. It is established that, in general,n ² is necessarily real, wheren is the growth rate of disturbance, thus ruling out the possibility of overstability or damped oscillations. The solution is shown to be characterized by a variational principle, which provides the basis for obtaining an approximate solution of the problem. Two density distributions are considered: (i) a continuously stratified plasma layer and (ii) two semi-infinitely extending plasmas of constant densities separated by a horizontal interface. In both cases it has been shown that for the said disturbances the stability criterion remains unaffected by the inclusion of finite Larmor radius effects, though the amplified motion is strongly inhibited due to their inclusion.     

Finite Larmor radius effects on the Rayleigh-Taylor instability of a rotating plasma of variable density

The Rayleigh-Taylor instability in a rotating plasma of variable density has been investigated to include simultaneously the effects of viscosity and the finiteness of the ion Larmor radius. It is shown that, for a plasma in which the density is stratified along the vertical, the solution is characterized by a variational principle. Making use of this, proper solutions have been otained for a semiinfinite plasma in which the density varies exponentially. The dispersion relation has been solved numerically and it is found that the influence of the effects of both FLR and viscosity is stabilizing. The Coriolis forces are found to have a dual role, stabilizing for small wave numbers and destabilizing for large wave numbers. The range of the small wave numbers, over which the Coriolis forces have a stabilizing influence, is found to increase with Coriolis forces.     

Rayleigh-Taylor instability of rotating fluid in the presence of a vertical magnetic field

The Rayleigh-Taylor instability of two rotating superposed fluids in the presence of a vertical magnetic field has been investigated. It is shown thatn ² is purely real, wheren is the growth rate of a perturbation. In the basis of this fact it is shown that a unique dispersion relation exists if the lighter fluid lies beneath the heavier one. However, if the heavier fluid lies beneath the lighter fluid, then no unique dispersion relation exists. The effect of rotation is to slow down the rate at which potentially unstable stratification departs from the equilibrium position.     

Nonlinear Rayleigh-Taylor instability in the presence of magnetic field and mass and heat transfer

The nonlinear Rayleigh-Taylor instability in the presence of magnetic field and mass and heat transfer is studied using a simplified formulation. The method of multiple scale expansion is employed for the investigation. It is found that the nonlinear effects of magnetic field and mass and heat transfer stabilize the classically unstable system. A simple non-dimensional parameter is also found to characterize the stability of the system.     

Dissipative structures in the F-region of the equatorial ionosphere generated by Rayleigh-Taylor instability

The non-linear regime of electrostatic perturbations of the equatorial ionospheric F-region generated by Rayleigh-Taylor instability has been discussed, taking into account conductivity along magnetic field lines. A closed non-linear equation has been derived in the stationary limit for the polarization electric field potential. It coincides with the Karman equation of an ideal liquid. To solve the equation, the averaged variational Whitham method has been proposed. Some solutions localized along and across the geomagnetic field, B, as well as quasi-periodic solutions in the transverse direction, have been investigated. Non-linear longitudinal localization of perturbations has been shown to be due to electron-ion collisions.     

Rayleigh-Taylor instability of magnetized partially-ionized superposed fluids with rotation and surface tension in porous medium

The Rayleigh-Taylor instability of the plane interface separating the two partially-ionized superposed fluids through porous medium is analysed. The effect of variable horizontal magnetic field, surface tension and rotation along the vertical axis are also incorporated. The relevant linearized perturbation equations are taken and using normal mode analysis the general relation is obtained from which the dispersion relation for two superposed fluids of different densities is derived. It is found that the surface tension and horizontal magnetic field have the stabilizing effect on the R-T-instability. The condition of instability remains unaffected by the permeability of porous medium, presence of neutral particles in the fluids and rotation.It is concluded that the system is unstable only for those positive wave numbers which are less than certain critical value in case of an adverse density gradient.     

Note on the Helmholtz instability in stratified flows with magnetic fields

The nature of the neutral curves for the stability of a Helmholtz velocity profile in a stratified, Boussinesq fluid in the presence of a uniform magnetic field for the cases (1) an infinite fluid (2) a semi-infinite fluid with a rigid boundary is discussed.     

Rayleigh-Taylor instability of a two-fluid layer under a general rotation field and a horizontal magnetic field

In this paper the Rayleigh-Taylor instability (RTI) of a two-fluid layer system under the simultaneous action of a general rotation field and a horizontal magnetic field is presented. An approximate and an exact solution of the eigenvalue equation are calculated. These solutions are important not only to understand more deeply the physical problem but also to determine the correct numerical solutions. Numerical calculations are done for an unstable density stratification in the cases of horizontal magnetic field parallel and perpendicular to the horizontal component of the angular velocity. For an adverse density stratification, it is shown that in comparison to previous works, the horizontal magnetic field creates new angular areas (of the angle of propagation of the perturbation) at which the perturbation is stable and propagates as traveling waves. It is also shown that the vertical component of the angular velocity has a destabilizing effect because it works to eliminate the stable angular areas.     

Dissipative-acoustic instability in accretion disks at a nonlinear stage

Nonstationary hydrodynamic models of a viscous accretion disk around a central compact object were constructed. Two different numerical methods (TVD and SPH) are used to study the dynamics of dissipatively unstable acoustic perturbations at the nonlinear stage in terms of the standard α-disk model. The standard disk accretion in the Shakura-Sunyaev model is unstable against acoustic waves for various parameters of the system. If the α parameter, which specifies the level of turbulent viscosity, exceeds α?0.03, then a complex nonstationary system of small-scale weak shock waves is formed. The growth rate of the perturbations is higher in the central disk region. For α?0.2, the relative shock amplitude can exceed 50% of the equilibrium disk parameters. The reflection of waves from the disk boundaries and their nonlinear interaction are important factors that can produce unsteady accretion. The luminosity of such a disk undergoes quasi-periodic oscillations at a level of several percent (?5%) of the equilibrium level.     

Secondary tearingmode in the nonlinear evolution of magnetorotational instability

Numerical investigation of the two‐dimensional magnetic reconnection is given in the context of the nonlinear evolution of the Magneto‐Rotational Instability (MRI). With a careful comparison to various theories using both one‐ and twodimensional analysis, it is found that a new stabilizing effect of the centrifugal force on tearing instability must be present in the specific geometry of the MRI.Magnetic reconnection might play a key role to the formation of the nonaxisymmetric structures observed in MRI experiments. The results may also be useful for the estimate of the accretion rate in various astrophysical objects. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)