Nonlinear Dynamics of Non-uniform Current-Vortex Sheets in Magnetohydrodynamic Flows (original) (raw)
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Nonlinear motion of non-uniform current-vortex sheets in magnetohydrodynamic flows
Fluid Dynamics Research, 2014
Nonlinear motion of vortex sheets with non-uniform current is investigated using the vortex blob method. We show that the current induced on a vortex sheet leads to a strong amplification of the magnetic field when the Lorentz force term in the governing equation is sufficiently small. When the Lorentz force term is large, an oscillation due to the Alfvén wave appears and the nonlinear growth is suppressed. We present various interfacial profiles depending on the magnitude of the Atwood number (density ratio) and Lorentz force, and discuss the complicated motion of non-uniform current-vortex sheets.
Nonlinear interfacial motion in magnetohydrodynamic flows
High Energy Density Physics, 2019
Nonlinear motion of vortex sheets with non-uniform current is investigated taking the magnetohydrodynamic Richtmyer-Meshkov instability (MHD RMI) and the magnetohydrodynamic Kelvin-Helmholtz instability (MHD KHI) as the examples. As the ratio of the magnetic force to the convective force increases, Alfvén oscillations appear and the nonlinear growth of the interface as a vortex sheet is suppressed. We show that the turbulent energy possessing the interface flows into the magnetic energy, which causes the strong magnetic field amplification for both instabilities. We also discuss the difference of the temporal evolution of the interface between MHD RMI and MHD KHI.
Nonlinear Stability of the Current-Vortex Sheet to the Incompressible MHD Equations
Communications on Pure and Applied Mathematics, 2017
In this paper, we solve a long-standing open problem: nonlinear stability of current-vortex sheet in the ideal incompressible Magneto-Hydrodynamics under the linear stability condition. This result gives a first rigorous confirmation of the stabilizing effect of the magnetic field on Kelvin-Helmholtz instability.
Archive for Rational Mechanics and Analysis, 2008
Compressible vortex sheets are fundamental waves, along with shock and rarefaction waves, in entropy solutions to the multidimensional hyperbolic systems of conservation laws; and understanding the behavior of compressible vortex sheets is an important step towards our full understanding of fluid motions and the behavior of entropy solutions. For the Euler equations in two-dimensional gas dynamics, the classical linearized stability analysis on compressible vortex sheets predicts stability when the Mach number M > √ 2 and instability when M < √ 2; and Artola-Majda's analysis reveals that the nonlinear instability may occur if planar vortex sheets are perturbed by highly oscillatory waves even when M > √ 2. For the Euler equations in three-dimensions, every compressible vortex sheet is violently unstable and this violent instability is the analogue of the Kelvin-Helmholtz instability for incompressible fluids. The purpose of this paper is to understand whether compressible vortex sheets in three dimensions, which are unstable in the regime of pure gas dynamics, become stable under the magnetic effect in three-dimensional magnetohydrodynamics (MHD). One of the main features is that the stability problem is equivalent to a free boundary problem whose free boundary is a characteristic surface, which is more delicate than noncharacteristic free boundary problems. Another feature is that the linearized problem for current-vortex sheets in MHD does not meet the uniform Kreiss-Lopatinskii condition. These features cause additional analytical difficulties and especially prevent a direct use of the standard Picard iteration to the nonlinear problem. In this paper, we develop a nonlinear approach to deal with these difficulties in three-dimensional MHD. We first carefully formulate the linearized problem for the current-vortex sheets to show rigorously that the magnetic effect makes the problem weakly stable and establish energy estimates, especially high-order energy estimates, in terms of the nonhomogeneous terms and variable coefficients without loss of the order. Then we exploit these results to develop a suitable iteration scheme of Nash-Moser-Hörmander type and establish its convergence, which leads to the existence and stability of compressible current-vortex sheets, locally in time, in the three-dimensional MHD.
Journal of Computational Physics, 2001
Five model flows of increasing complexity belonging to the class of stationary two-dimensional planar field-aligned magnetohydrodynamic (MHD) flows are presented which are well suited to the quantitative evaluation of MHD codes. The physical properties of these five flows are investigated using characteristic theory. Grid convergence criteria for flows belonging to this class are derived from characteristic theory, and grid convergence is demonstrated for the numerical simulation of the five model flows with a standard high-resolution finite volume numerical MHD code on structured body-fitted grids. In addition, one model flow is presented which is not field-aligned, and it is discussed how grid convergence can be studied for this flow. By formal grid convergence studies of magnetic flux conservation and other flow quantities, it is investigated whether the Powell source term approach to controlling the ∇ · B constraint leads to correct results for the class of flows under consideration.
Data dependence of approximate current-vortex sheets near the onset of instability
Journal of Hyperbolic Differential Equations
The paper is concerned with the free boundary problem for two-dimensional current-vortex sheets in ideal incompressible magneto-hydrodynamics near the transition point between the linearized stability and instability. In order to study the dynamics of the discontinuity near the onset of the instability, Hunter and Thoo [On the weakly nonlinear Kelvin–Helmholtz instability of tangential discontinuities in MHD, J. Hyperbolic Differ. Equations 8(4) (2011) 691–726] have introduced an asymptotic quadratically nonlinear integro-differential equation for the amplitude of small perturbations of the planar discontinuity. The local-in-time existence of smooth solutions to the Cauchy problem for the amplitude equation was shown in [Approximate current-vortex sheets near the onset of instability, J. Math. Pures Appl. 105(4) (2016) 490–536; Existence of approximate current-vortex sheets near the onset of instability, J. Hyperbolic Differ. Equations]. In the present paper, we prove the continuous...
Incompressible models of magnetohydrodynamic Richtmyer-Meshkov instability in cylindrical geometry
Physical Review Fluids, 2019
The Richtmyer-Meshkov instability (RMI) occurs when a shock impulsively accelerates an interface between two different fluids, and it is important in many technological applications such as inertial confinement fusion (ICF) and astrophysical phenomena such as supernova. Here, we present incompressible models of an impulsively accelerated interface separating conducting fluids of different densities in cylindrical geometry. The present study complements earlier investigations on linear and nonlinear simulations of RMI. We investigate the influence of a normal or an azimuthal magnetic field on the growth rate of the interface. This is accomplished by solving the linearized initial value problem using numerical inverse Laplace transform. For a finite normal magnetic field, although the initial growth rate of the interface is unaffected by the presence of the magnetic field, at late-time the growth rate of the interface decays. This occurs by transporting the vorticity by two Alfvén fronts which propagate away from the interface. For the azimuthal magnetic field configuration, the suppression mechanism is associated with the interference of two waves propagating parallel and antiparallel to the interface that transport vorticity and cause the growth rate to oscillate in time with nearly a zero mean value. Comparing the results of the incompressible models with linear compressible MHD simulations show reasonable agreement at early time of simulations.
Journal of Fluids Engineering, 2017
We investigate the linear stability of both positive and negative Atwood ratio interfaces accelerated either by a fast magnetosonic or hydrodynamic shock in cylindrical geometry. For the magnetohydrodynamic (MHD) case, we examine the role of an initial seed azimuthal magnetic field on the growth rate of the perturbation. In the absence of a magnetic field, the Richtmyer–Meshkov growth is followed by an exponentially increasing growth associated with the Rayleigh–Taylor instability (RTI). In the MHD case, the growth rate of the instability reduces in proportion to the strength of the applied magnetic field. The suppression mechanism is associated with the interference of two waves running parallel and antiparallel to the interface that transport vorticity and cause the growth rate to oscillate in time with nearly a zero mean value.
Transition from two-dimensional to three-dimensional magnetohydrodynamic turbulence
Journal of Fluid Mechanics, 2007
We report a theoretical investigation of the robustness of two-dimensional inviscid magnetohydrodynamic (MHD) flows at low magnetic Reynolds numbers with respect to three-dimensional perturbations. We use a combination of linear stability analysis and direct numerical simulations to analyse three problems, namely the flow in the interior of a triaxial ellipsoid, and two unbounded flows: a vortex with elliptical streamlines and a vortex sheet parallel to the magnetic field. The flow in a triaxial ellipsoid is found to present an exact analytical model which demonstrates both the existence of inviscid unstable three-dimensional modes and the stabilizing role of the magnetic field. The nonlinear evolution of the flow is characterized by intermittency typical of other MHD flows with long periods of nearly two-dimensional behaviour interrupted by violent three-dimensional transients triggered by the instability. We demonstrate, using the second model, that motion with elliptical streamlines perpendicular to the magnetic field becomes unstable with respect to the elliptical instability once the magnetic interaction parameter falls below a critical magnitude whose value tends to infinity as the eccentricity of the streamlines increases. Furthermore, the third model indicates that vortex sheets parallel to the magnetic field, which are unstable for any velocity and any magnetic field, emit eddies with vorticity perpendicular to the magnetic field. Whether the investigated instabilities persist in the presence of small but finite viscosity, in which case two-dimensional turbulence would represent a singular state of MHD flows, remains an open question.
Short-wavelength analysis of magnetorotational instability of resistive MHD flows
2016
Local stability analysis is made of axisymmetric rotating flows of a perfectly conducting fluid and resistive flows with viscosity, subjected to external azimuthal magnetic field to non-axisymmetric as well as axisymmetric perturbations. For perfectly conducting fluid (ideal MHD), we use the Hain-Lust equation, capable of dealing with perturbations over a wide range of the axial wavenumber k to take short wavelength approximation. When the magnetic field is sufficiently weak, the maximum growth rate is given by the Oort A-value. As the magnetic field is increased, the Keplerian flow becomes unstable to waves of short axial wavelength. We also incorporate the effect of the viscosity and the electric resistivity and apply the WKB method in the same way as we do to the perfectly conducting fluid. In the inductionless limit, i.e. when the magnetic diffusivity is much larger than the viscosity, Keplerian-rotation flow of arbitrary distributions of the magnetic field, including the Liu li...