Ideal Incompressible Magnetohydrodynamics Research Articles

Local well-posedness of the free-boundary incompressible magnetohydrodynamics with surface tension

We prove the local well-posedness of the 3D free-boundary incompressible ideal magnetohydrodynamics (MHD) equations with surface tension, which describe the motion of a perfect conducting fluid in an electromagnetic field. We adapt the ideas developed in the remarkable paper [11] by Coutand and Shkoller to generate an approximate problem with artificial viscosity indexed by κ>0 whose solution converges to that of the MHD equations as κ→0. However, the local well-posedness of the MHD equations is no easy consequence of Euler equations thanks to the strong coupling between the velocity and magnetic fields. This paper is the continuation of the second and third authors' previous work [38] in which the a priori energy estimate for incompressible free-boundary MHD with surface tension is established. But the existence is not a trivial consequence of the a priori estimate as it cannot be adapted directly to the approximate problem due to the loss of the symmetric structure.

Inverse scattering of Alfvén waves in three dimensional ideal magnetohydrodynamics

The purpose of this paper is to solve the inverse scattering problem of nonlinear Alfvén waves governed by the three dimensional ideal incompressible magnetohydrodynamics (MHD) system. By bridging together geometric methods and weighted energy estimates, we establish a set of scattering isomorphisms to substantially strengthen our previous rigidity results. This answer is consistent with the physical intuition that Alfvén waves behave exactly in the same manner as their scattering fields that are detected by faraway observers. The novelty of the present work is twofold: for one thing, the relationship between Alfvén waves emanating from the plasma and their scattering fields at infinities is explored to the best; for another thing, the null structure inherent in MHD equations is thoroughly examined, especially when we estimate the pressure term.

Well-posedness of the free boundary problem in incompressible MHD with surface tension

In this paper, we study the two phase flow problem with surface tension in the ideal incompressible magnetohydrodynamics. First, we prove the local well-posedness of the two phase flow problem with surface tension. Second, for the initial data satisfying a Syrovatskij type stability condition, we prove that as surface tension tends to zero, the solution of the two phase flow problem with surface tension converges to the solution of the two phase flow problem without surface tension.

Weakly nonlinear surface waves on the plasma–vacuum interface

We consider the free boundary problem for a plasma–vacuum interface in ideal incompressible magnetohydrodynamics. Unlike the classical statement, where the vacuum magnetic field obeys the div-curl system of pre-Maxwell dynamics, we do not neglect the displacement current in the vacuum region and consider the Maxwell equations for electric and magnetic fields. Our aim is to construct highly oscillating surface wave solutions in weakly nonlinear regime to this plasma–vacuum interface problem.Under a necessary and sufficient stability condition for a piecewise constant background state, we construct approximate solutions at any arbitrarily large order of accuracy to the free boundary problem in three space dimensions when the initial discontinuity displays high frequency oscillations. Moreover, such approximate surface waves have nontrivial residual non-oscillatory components.

Regularity and Energy Conservation of the Nonhomogeneous Incompressible Ideal Magnetohydrodynamics Equations

In this paper, we study the regularity and local energy equation of the weak solutions for nonhomogeneous incompressible ideal magnetohydrodynamics system. The conditions given on the regularity of solutions guarantee the energy to be conserved. The main method we have employed relies on the commutator estimates.

Dyadic models for ideal MHD

We study two dyadic models for incompressible ideal magnetohydrodynamics, one with a uni-directional energy cascade and the other one with both forward and backward energy cascades. Global existence of weak solutions and local well-posedness are established for both models. In addition, solutions to the model with uni-directional energy cascade associated with positive initial data are shown to develop blow-up at a finite time. Moreover, a set of fixed points is found for each model. Linear instability about some particular fixed points is proved.

Well-posedness for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equations

We study the well-posedness theory for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equations in a bounded domain. We express the magnetic field in terms of the velocity field and the deformation tensors in the Lagrangian coordinates, and substitute the magnetic field into the momentum equation to get an equation of the velocity in which the initial magnetic field serves only as a parameter. Then, we linearize this equation with respect to the position vector field whose time derivative is the velocity, and obtain the local-in-time well-posedness of the solution by using energy estimates of the tangential derivatives and the curl with the help of Lie derivatives and the smooth-out approximation.

Global exact controllability of ideal incompressible magnetohydrodynamic flows through a planar duct

This article is concerned with the global exact controllability for ideal incompressible magnetohydrodynamics in a rectangular domain where the controls are situated in both vertical walls. First, global exact controllability via boundary controls is established for a related Elsässer type system by applying the return method, introduced in Coron [Math. Control Signals Syst. 5 (1992) 295–312]. Similar results are then inferred for the original magnetohydrodynamics system with the help of a special pressure-like corrector in the induction equation. Overall, the main difficulties stem from the nonlinear coupling between the fluid velocity and the magnetic field in combination with the aim of exactly controlling the system. In order to overcome some of the obstacles, we introduce ad-hoc constructions, such as suitable initial data extensions outside of the physical part of the domain and a certain weighted space.

Well-posedness of the two-phase flow problem in incompressible MHD

<p style='text-indent:20px;'>In this paper, we study the two phase flow problem in the ideal incompressible magnetohydrodynamics. We propose a Syrovatskij type stability condition, and prove the local well-posedness of the two phase flow problem with initial data satisfies such condition. This result shows that the magnetic field has a stabilizing effect on Kelvin-Helmholtz instability even the fluids on each side of the free interface have different densities.</p>

On well-posedness of the plasma-vacuum interface problem with displacement current in vacuum

We consider the plasma-vacuum interface problem in ideal incompressible magneto-hydrodynamics. Unlike the classical statement when the vacuum magnetic field obeys the div-curl system of pre-Maxwell dynamics, to better understand the influence of the electric field in vacuum we do not neglect the displacement current in the vacuum region and consider the Maxwell equations for electric and magnetic fields. Under the necessary and sufficient stability condition for a planar interface found earlier by Trakhinin, we prove an energy a priori estimate for the linearized constant coefficient problem. The process of derivation of this estimate is based on various methods, including a secondary symmetrization of the vacuum Maxwell equations, the derivation of a hyperbolic evolutionary equation for the interface function and the construction of a degenerate Kreiss-type symmetrizer for an elliptic-hyperbolic problem for the total pressure.

Elasticity of tangled magnetic fields

The fundamental difference between incompressible ideal magnetohydrodynamics and the dynamics of a non-conducting fluid is that magnetic fields exert a tension force that opposes their bending; magnetic fields behave like elastic strings threading the fluid. It is natural, therefore, to expect that a magnetic field tangled at small length scales should resist a large-scale shear in an elastic way, much as a ball of tangled elastic strings responds elastically to an impulse. Furthermore, a tangled field should support the propagation of ‘magnetoelastic waves’, the isotropic analogue of Alfvén waves on a straight magnetic field. Here, we study magnetoelasticity in the idealised context of an equilibrium tangled field configuration. In contrast to previous treatments, we explicitly account for intermittency of the Maxwell stress, and show that this intermittency necessarily decreases the frequency of magnetoelastic waves in a stable field configuration. We develop a mean-field formalism to describe magnetoelastic behaviour, retaining leading-order corrections due to the coupling of large- and small-scale motions, and solve the initial-value problem for viscous fluids subjected to a large-scale shear, showing that the development of small-scale motions results in anomalous viscous damping of large-scale waves. Finally, we test these analytic predictions using numerical simulations of standing waves on tangled, linear force-free magnetic-field equilibria.

Stability of an incompressible plasma–vacuum interface with displacement current in vacuum

We study the free boundary problem for a plasma–vacuum interface in ideal incompressible magnetohydrodynamics. Unlike the classical statement when the vacuum magnetic field obeys the div‐curl system of pre‐Maxwell dynamics, to better understand the influence of the electric field in vacuum, we do not neglect the displacement current in the vacuum region and consider the Maxwell equations for electric and magnetic fields. Under the necessary and sufficient stability condition for a planar interface found earlier by Trakhinin, we prove an energy a priori estimate for the linearized constant coefficient problem. The process of derivation of this estimate is based on various methods, including a secondary symmetrization of the vacuum Maxwell equations, the derivation of a hyperbolic evolutionary equation for the interface function, and the construction of a degenerate Kreiss‐type symmetrizer for an elliptic‐hyperbolic problem for the total pressure.

Lagrangian and Dirac constraints for the ideal incompressible fluid and magnetohydrodynamics

The incompressibility constraint for fluid flow was imposed by Lagrange in the so-called Lagrangian variable description using his method of multipliers in the Lagrangian (variational) formulation. An alternative is the imposition of incompressibility in the Eulerian variable description by a generalization of Dirac’s constraint method using noncanonical Poisson brackets. Here it is shown how to impose the incompressibility constraint using Dirac’s method in terms of both the canonical Poisson brackets in the Lagrangian variable description and the noncanonical Poisson brackets in the Eulerian description, allowing for the advection of density. Both cases give the dynamics of infinite-dimensional geodesic flow on the group of volume preserving diffeomorphisms and explicit expressions for this dynamics in terms of the constraints and original variables is given. Because Lagrangian and Eulerian conservation laws are not identical, comparison of the various methods is made.

Magnetic Rayleigh–Taylor instability at a contact discontinuity with an oblique magnetic field

Aims.We investigate the nature of the magnetic Rayleigh–Taylor instability at a density interface that is permeated by an oblique homogeneous magnetic field in an incompressible limit.Methods.Using the system of linearised ideal incompressible magnetohydrodynamics equations, we derive the dispersion relation for perturbations of the contact discontinuity by imposing the necessary continuity conditions at the interface. The imaginary part of the frequency describes the growth rate of waves due to instability. The growth rate of waves is studied by numerically solving the dispersion relation.Results.The critical wavenumber at which waves become unstable, which is present for a parallel magnetic field, disappears because the magnetic field is inclined. Instead, waves are shown to be unstable for all wavenumbers. Theoretical results are applied to diagnose the structure of the magnetic field in prominence threads. When we apply our theoretical results to observed waves in prominence plumes, we obtain a wide range of field inclination angles, from 0.5° up to 30°. These results highlight the diagnostic possibilities that our study offers.

Weakly nonlinear surface waves in magnetohydrodynamics. I

This work is devoted to the construction of weakly nonlinear, highly oscillating, current vortex sheet solutions to the system of ideal incompressible magnetohydrodynamics. Current vortex sheets are piecewise smooth solutions that satisfy suitable jump conditions on the (free) discontinuity surface. In this article, we complete an earlier work by Alì and Hunter (Quart. Appl. Math. 61(3) (2003) 451–474) and construct approximate solutions at any arbitrarily large order of accuracy to the three-dimensional free boundary problem when the initial discontinuity displays high frequency oscillations. As evidenced in earlier works, high frequency oscillations of the current vortex sheet give rise to ‘surface waves’ on either side of the sheet. Such waves decay exponentially in the normal direction to the current vortex sheet and, in the weakly nonlinear regime which we consider here, their leading amplitude is governed by a nonlocal Hamilton–Jacobi type equation known as the ‘HIZ equation’ (standing for Hamilton–Il’insky–Zabolotskaya (J. Acoust. Soc. Amer. 97(2) (1995) 891–897)) in the context of Rayleigh waves in elastodynamics. The main achievement of our work is to develop a systematic approach for constructing arbitrarily many correctors to the leading amplitude. We exhibit necessary and sufficient solvability conditions for the corrector equations that need to be solved iteratively. The verification of these solvability conditions is based on mere algebra and arguments of combinatorial analysis, namely a Leibniz type formula which we have not been able to find in the literature. The construction of arbitrarily many correctors enables us to produce infinitely accurate approximate solutions to the current vortex sheet equations. Eventually, we show that the rectification phenomenon exhibited by Marcou in the context of Rayleigh waves (C. R. Math. Acad. Sci. Paris 349(23–24) (2011) 1239–1244) does not arise in the same way for the current vortex sheet problem.

Ill-Posedness of Free Boundary Problem of the Incompressible Ideal MHD

In the present paper, we show the ill-posedness of the free boundary problem of the incompressible ideal magnetohydrodynamics (MHD) equations in two spatial dimensions for any positive vacuum permeability $\mu_0$, in Sobolev spaces. The analysis is uniform for any $\mu_0>0$.

Small Alfvén number limit for incompressible magneto-hydrodynamics in a domain with boundaries

for any fixed Alfven number, the local well-posedness is proved for the equations of three-dimensional ideal incompressible magneto-hydrodynamics in a domain with boundaries. Under appropriate conditions, a smooth solution is shown to exist in a time interval independent of the Alfven number, and the solutions of the original system tend to the solutions of a two-dimensional Euler flow coupled with a linear transport equation as the Alfven number goes to zero.

Well-Posedness of the Plasma–Vacuum Interface Problem for Ideal Incompressible MHD

In this paper, we prove the local well-posedness of the plasma–vacuum interface problem for ideal incompressible magnetohydrodynamics under the stability condition: the magnetic field $${\mathbf h}$$ and the vacuum magnetic field $${\hat{{\mathbf h}}}$$ are non-collinear on the interface (i.e., $$|{\mathbf h}\times {\hat{{\mathbf h}}}|>0$$ ) which was introduced by Trakhinin as a stability condition for the compressible plasma–vacuum interface problem.

Decoupled Cascades of Kinetic and Magnetic Energy in Magnetohydrodynamic Turbulence.

Magnetic energy (ME) and kinetic energy (KE) in ideal incompressible magnetohydrodynamics are not global invariants and, therefore, it has been justified to discuss only the cascade of their sum total energy. We provide a physical argument based on scale locality, along with compelling evidence that ME and KE budgets statistically decouple beyond a transitional "conversion" range. This arises because magnetic field-line stretching is a large-scale process which vanishes on average at intermediate and small scales within the inertial-inductive range, thereby allowing each of the mean ME and KE to cascade conservatively and at an equal rate, yielding a turbulent magnetic Prandtl number of unity over these scales.

On the Gevrey regularity of solutions to the 3D ideal MHD equations

In this paper, similar to the incompressible Euler equation, we prove the propagation of the Gevrey regularity of solutions to the three-dimensional incompressible ideal magnetohydrodynamics (MHD) equations. We also obtain an uniform estimate of Gevery radius for the solution of MHD equation.