Manuel Núñez

A theorem of existence for the equations of magnetohydrodynamics of partially ionized plasmas

The equations of magnetohydrodynamics of partially ionized plasmas have been known for a long time, but rarely studied. Instead, several simplifications have been applied to different physical models, ranging from magnetic reconnection to ambipolar drift. The original system relates the electric field to kinetic magnitudes by means of a non-local law, so that the equations describing it involve partial differentials as well as functional operators. We prove a theorem of existence and uniqueness of solutions for a finite time by means of a fixed point argument in an appropriate functional setting.

Spectral analysis of viscous static compressible fluid equilibria

It is generally assumed that the study of the spectrum of the linearized Navier-Stokes equations around a static state will provide information about the stability of the equilibrium. This is obvious for inviscid barotropic compressible fluids by the self-adjoint character of the relevant operator, and rather easy for viscous incompressible fluids by the compact character of the resolvent. The viscous compressible linearized system, both for periodic and homogeneous Dirichlet boundary problems, satisfies neither condition, but it does turn out to be the generator of an immediately continuous, almost stable semigroup, which justifies the analysis of the spectrum as predictive of the initial behaviour of the flow. As for the spectrum itself, except for a unique negative finite accumulation point, it is formed by eigenvalues with negative real part, and nonreal eigenvalues are confined to a certain bounded subset of complex numbers.

The limit states of magnetic relaxation

It is generally believed that a viscous, non-resistive plasma will eventually decay to a magnetostatic state, probably possessing contact discontinuities. We prove that even in the presence of a decaying forcing, the kinetic energy of the system tends to zero, which justifies the belief that the limit state is static. Regarding the magnetic field, the fact that the magnetic energy remains bounded proves the existence of weak sequential limits of the field as the time goes to infinity, but this does not imply that the magnetic field tends to a single state: we present an example where there is no limit, even in a weak sense. One additional condition upon the velocity, however, is enough to guarantee existence of a single limit magnetic configuration.

Estimates on hyperdiffusive magnetohydrodynamics

Abstract A plasma is hyperdiffusive when the dissipation of the small scales is more efficient than the viscous-resistive dissipation. Mathematically it is represented by the addition of a fourth-order diffusive operator to the standard magnetohydrodynamic (MHD) equations. Since hyperdiffusivity has been widely used in astrophysical numerical models, its consequences upon the behavior of the MHD system must be studied. It is found that, unlike what is known in the classical case, solutions are bounded in the H1-norm for all time; for kinematic dynamos, it provides estimates upon the exponential growth of the magnetic field improving considerably the ones of standard MHD. All this points out that hyperdiffusive MHD evolution may differ substantially from the usual MHD one.

Ion--neutral friction as a model for magnetic relaxation

It is generally admitted that a plasma in the absence of forcing will relax to a minimum energy state compatible with appropriate constraints. Usually this is a force-free state, which, in two dimensions, implies a potential magnetic field except by the possible presence of current sheets. The precise mechanism of this relaxation, and in particular the plasma velocity, are generally ignored. There exists, however, a physically well-defined process that should produce magnetic relaxation: ion--neutral (or ambipolar) friction. While there is no guarantee of the existence of a limit of this process when t → ∞, there exists a family of sequential limits for whom the Lorentz force tends to zero. To analyze the configuration of these limit states, we study the evolution of several moments of the magnetic energy. We prove that for as long as the enstrophy remains bounded, the current density energy also remains bounded in two dimensions: this excludes all classical configurations of current sheets across which the magnetic field reverses direction. Hence, these sheets cannot be the limit of ion--neutral diffusion unless the flow becomes increasingly irregular over time.

Dynamic effects on the stretching of the magnetic field by a plasma flow

A key mechanism in the growth of magnetic energy in kinematic dynamos is the stretching of the magnetic field vector by making it point in an unstable direction of the strain matrix. Our objective is to study whether this feature may be maintained in an ideal plasma when also considering the back reaction of the magnetic field upon the flow through the Lorentz force. Several effects occur: in addition to the nonlocal ones exerted by the total pressure, a complex geometry of magnetic field lines decreases the rate of growth of magnetic energy, rotation of the flow enhances it and above all the rate of growth decreases with minus the square of the eigenvalue associated with the magnetic field direction. Thus local dynamics tend to rapidly quench the stretching of the field.

Dissipation of Kinetic Energy in a Magnetic Dynamo

In a dynamo the magnetic field grows exponentially for some time. If there is no energy injected into the system, this growth must be done at the expense of the plasma kinetic energy. It is proved that for a certain growth rate, the kinetic energy decreases at least by a fixed amount independent of how large the magnetic field is; hence velocity fields allowing dynamos posses a minimum rate of dissipation. The key of the proof is a new inequality on the induction equation.

On the regularity of the magnetic field in a diffusive plasma

It is known that magnetic fields in ideal chaotic plasmas tend to become extremely irregular and to concentrate in a fractal set, and it is assumed that the presence of a positive resistivity will have a smoothing effect. Here we try to quantify this effect by proving new inequalities which, on the one hand, relate the local and global size of velocity and magnetic field with the gradient of this field, and on the other provide a bound of the area of generalized level surfaces.

Spectral analysis in magnetohydrodynamic equilibria

It has been universally assumed that the spectrum of the magnetohydrodynamics equations, linearized around an equilibrium state, provides enough information on the short-term evolution of the plasma to study certain stability properties. We show that this is true if one takes into account viscous and resistive effects and the equilibrium satisfies certain regularity conditions.

On a class of partial differential equations with singular coefficients

We study the equation div(fΔu)=0 for functions f which may become infinite at some subset of the domain of definition. After defining adequate Sobolev-like spaces of functions, we make a careful study of the behaviour of admissible solutions in the neighbourhood of singular points. This enables us to show that the Dirichlet problem possesses a space of solutions whose dimension is determined by the number and characteristics of the connected components of the set f=∞, a similar conclusion holds for the Neumann problem.