Motion of a current-vortex sheet in the magnetic Kelvin-Helmholtz instability.

In this paper, we consider the Kelvin-Helmholtz instability in the magnetohydrodynamic flow. The motion of the interface is described by a current-vortex sheet. We examine the linear stability of the current-vortex sheet model and determine the growth rate of the interface. The interface is linearly stable for M_{A}<2 where M_{A} represents the Alfvén Mach number. It is found that the interface is linearly unstable in the limit of the critical Alfvén Mach number M_{A}=2, due to resonance of eigenvalues. We perform numerical simulations for the current-vortex sheet for both regimes of M_{A}<2 and M_{A}>2. The numerical results show the stabilizing effects of the magnetic field on the evolution of the current-vortex sheet when the magnetic field is sufficiently large. For the regime M_{A}<2, the sheet oscillates both longitudinally and transversely and the transverse surface wave is pronounced for a large M_{A}. Remarkably, the interface is nonlinearly unstable for M_{A}≈2, for M_{A}<2, which may be due to the propagation of surface waves. For the regime M_{A}>2, the roll-up of the spiral is weakened and the spiral is more pinched and stretched for smaller M_{A}. A comparison of the unstable evolutions of large and small values of M_{A} shows significant differences of the magnetic field and vortex sheet strength.

Stability of barotropic vortex strip on a rotating sphere.

We study the stability of a barotropic vortex strip on a rotating sphere, as a simple model of jet streams. The flow is approximated by a piecewise-continuous vorticity distribution by zonal bands of uniform vorticity. The linear stability analysis shows that the vortex strip becomes stable as the strip widens or the rotation speed increases. When the vorticity constants in the upper and the lower regions of the vortex strip have the same positive value, the inner flow region of the vortex strip becomes the most unstable. However, when the upper and the lower vorticity constants in the polar regions have different signs, a complex pattern of instability is found, depending on the wavenumber of perturbations, and interestingly, a boundary far away from the vortex strip can be unstable. We also compute the nonlinear evolution of the vortex strip on the rotating sphere and compare with the linear stability analysis. When the width of the vortex strip is small, we observe a good agreement in the growth rate of perturbation at an early time, and the eigenvector corresponding to the unstable eigenvalue coincides with the most unstable part of the flow. We demonstrate that a large structure of rolling-up vortex cores appears in the vortex strip after a long-time evolution. Furthermore, the geophysical relevance of the model to jet streams of Jupiter, Saturn and Earth is examined.

Long-time simulations of the Kelvin-Helmholtz instability using an adaptive vortex method.

The nonlinear evolution of an interface subject to a parallel shear flow is studied by the vortex sheet model. We perform long-time computations for the vortex sheet in density-stratified fluids by using the point vortex method and investigate late-time dynamics of the Kelvin-Helmholtz instability. We apply an adaptive point insertion procedure and a high-order shock-capturing scheme to the vortex method to handle the nonuniform distribution of point vortices and enhance the resolution. Our adaptive vortex method successfully simulates chaotically distorted interfaces of the Kelvin-Helmholtz instability with fine resolutions. The numerical results show that the Kelvin-Helmholtz instability evolves a secondary instability at a late time, distorting the internal rollup, and eventually develops to a disordered structure.

Effects of surface tension and viscosity on the growth rates of Rayleigh-Taylor and Richtmyer-Meshkov instabilities.

We present an analytical model for unstable interfaces with surface tension in fluids of arbitrary viscosity. Linear and nonlinear asymptotic solutions are obtained for growth rates of Rayleigh-Taylor and Richtmyer-Meshkov instabilities. In Rayleigh-Taylor instability, both surface tension and viscosity decrease the asymptotic bubble velocity. For Richtmyer-Meshkov instability, the analysis of the model suggests a dependence of the decaying rate of the bubble velocity on the relative importance of viscosity and surface tension. Results of numerical simulations are also given, and comparisons of the solutions of the model with numerical results are in good agreement.

Quantitative modeling of bubble competition in Richtmyer-Meshkov instability.

We present a quantitative model for the evolution of single and multiple bubbles in the Richtmyer-Meshkov (RM) instability. The higher-order solutions for a single-mode bubble are obtained, and distinctions between RM and Rayleigh-Taylor bubbles are investigated. The results for multiple-bubble competition from the model shows that the higher-order correction to the solution of the bubble curvature has a large influence on the growth rate of the RM bubble front. The model predicts that the bubble front of RM mixing grows as h approximately ttheta with theta approximately (0.3-0.35)+/-0.02 .

Bubble interaction model for hydrodynamic unstable mixing.

The analytic model for the evolution of single and multiple bubbles in Rayleigh-Taylor mixing is presented for the system of arbitrary density ratio. The model is the extension of Zufiria's potential theory, which is based on the velocity potential with point sources. We present solutions for a single bubble, at various stages, from the model and show that the solutions for the bubble velocity and curvature are in good agreement with numerical results. We demonstrate the evolution of multiple bubbles for finite density contrast and investigate dynamics of bubble competition, whereby leading bubbles grow in size at the expense of neighbors. The model shows that the growth coefficient alpha for the scaling law of the bubble front depends on the Atwood number and increases logarithmically with the initial perturbation amplitude. It is also found that the aspect ratio of the bubble size to the bubble height exhibits a self-similar behavior in the bubble competition process, and its values are insensitive to the Atwood number. The predictions of the model for the similarity parameters are in accordance with experimental and numerical results.

Density dependence of a Zufiria-type model for Rayleigh-Taylor bubble fronts.

We present the analytic model for the evolution of bubbles of arbitrary density ratio in Rayleigh-Taylor and Richtmyer-Meshkov instabilities. The model is the generalization of Zufiria's potential theory, which is based on the velocity potential with a point source and previously applied only for the interface of infinite density ratio. The analytic expressions for asymptotic solutions of bubbles are obtained. The predictions from the Zufiria model agree well with the numerical results not only for the bubble velocity, but also for the bubble curvature. It is found that the asymptotic curvature of a Richtmyer-Meshkov bubble is smaller than that of a Rayleigh-Taylor bubble for all Atwood numbers.

Vortex model and simulations for Rayleigh-Taylor and Richtmyer-Meshkov instabilities.

The vortex method is applied to simulations of Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) instabilities. The numerical results from the vortex method agree well with analytic solutions and other numerical results. The bubble velocity in the RT instability converges to a constant limit, and in the RM instability, the bubble and spike have decaying growth rates, except for the spike of infinite density ratio. For both RT and RM instabilities, bubbles attain constant asymptotic curvatures. It is found that, for the same density ratio, the RT bubble has slightly larger asymptotic curvature than the RM bubble. The vortex sheet strength of the RM interface has different behavior than that of the RT interface. We also examine the validity of theoretical models by comparing the numerical results with theoretical predictions.

Simple potential-flow model of Rayleigh-Taylor and Richtmyer-Meshkov instabilities for all density ratios.

We generalize the Layzer-type model for unstable interfaces to the system of arbitrary density ratio. The predictions from the generalized model for bubble growth rates of Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) instabilities are in good agreement with numerical results. We present the theoretical prediction for asymptotic growth rates for RT and RM bubbles for finite density ratios in two and three dimensions.