# WCNA-96

### (last updated May 7, 1996)

Organizers: Paul G. Schmidt and A. J. Meir
Department of Mathematics, Parker Hall
Auburn University, AL 36849-5310, USA
 Email: pgs@math.auburn.edu, ajm@math.auburn.edu Phone: (334)844-6579, (334)844-6580 Fax: (334)844-6555

### Summary:

Magnetohydrodynamics (or MHD) is the theory of the macroscopic interaction of electrically conducting fluids with a magnetic field. It is of importance in connection with many engineering problems as well as in geophysics and astronomy. Much research has been devoted to the physical modeling and computational simulation of MHD-dominated processes, but there is still a shortage of rigorous analytical and numerical methods. Future progress will likely depend on an intensified interdisciplinary dialogue between analysts, computationalists, physicists, and engineers. The proposed special session will attempt to contribute to this dialogue, if only on a small scale, by bringing together researchers from different disciplines concerned with mathematical and computational questions of MHD. Speakers will be encouraged to emphasize interdisciplinary aspects of their research, so that the session should appeal to a broad audience.

## Speakers, Titles, and Abstracts:

### Part I

#### 1. Peter M. Laurence

Dipartimento di Matematica, Universita di Roma "La Sapienza"
Piazzale Aldo Moro 5, 00185 Roma, Italy
laurence@mat.uniroma1.it

Title of Presentation: Variational problems with topological constraints in magnetohydrodynamics

Abstract: We give an overview of joint work with Edward Stredulinsky on variational problems arising in laboratory and in astrophysical plasmas, which involve topological constraints, due to the frozen-in invariants associated to ideal plasmas. We will try to cover two aspects : 1) The aspect related to weak closures of classes of diffeomorphisms and the associated question of an intrinsic characterization of topological constraints when dealing with Sobolev functions. Our results here apply to gradient fields, i.e., to level sets of a Sobolev function. Although the results are n-dimensional, the physically relevant case to which the theory is pertinent is the case where an ignorable coordinate is present and the magnetic field is representable as a rotated gradient. Less complete results are also available for certain classes of $L^2$ divergence-free fields in three dimensions. 2) The construction of a functional having an integral representation and analogous to the helicity integral, which describes higher order linking, as for instance in dealing with the Borremean rings and other Brunnian links, for which the pairwise linking integrals vanish but the third-order linking integrals due to Massey are non-zero.

#### 2. Jesus Ildefonso Diaz

jidiaz@mat.ucm.es

Phone: 34-1-3944452, Fax: 34-1-3944607

Title of Presentation: Mathematical treatment of the magnetic confinement of a plasma in Stellarator devices

Abstract: We consider the existence and some qualitative properties of the solutions to a two-dimensional free boundary problem modeling the magnetic confinement of a plasma in a Stellarator configuration. Such a model can be stated as an inverse problem since several nonlinear terms of the elliptic equation are not a priori known. A nonlocal formulation is obtained by using the notion of relative rearrangement.

#### 3. Bernard Saramito

Laboratoire de Mathematiques Appliquees
Universite Blaise Pascal (Clermont-Ferrand 2)
F-63177 Aubiere cedex, France
saramito@ucfma.univ-bpclermont.fr

Title of Presentation: Problems of MHD stability in thermonuclear fusion plasmas

Abstract: Different types of instabilities of the equilibrium state of a plasma can be described by MHD equations, such as tearing, kink, or interchange instabilities, which are really observed in tokamak experiments. First, we give a mathematical formulation of the problem in terms of bifurcation theory, taking into account the nonlinearity of dissipative coefficients and the compressibility of the plasma. Then, with a new 3D code, for which we give some results of convergence, we numerically obtain branches of nonlinear solutions, for instance for double tearing instability.

#### 4. Zensho Yoshida

Faculty of Engineering, University of Tokyo Hongo, Tokyo 113, Japan
yoshida@plasma.gen.u-tokyo.ac.jp

Phone: 81-3-5800-6859, Fax: 81-3-5800-6859

Title of Presentation: Application of Beltrami functions in plasma physics

Abstract: Beltrami functions represent a special class of steady states in fluids and plasmas. They are also used to span the phase space of electromagnetic fields with an invariant measure of nonlinear dynamics. We discuss mathematical backgrounds and applications in the physics of plasmas.

### Part II

#### 5. Richard K. Jordan

Department of Mathematics and Center for Nonlinear Analysis
Carnegie Mellon University, Pittsburgh, PA 15213-3890, USA
rjordan@andrew.cmu.edu

Title of Presentation: Organized states in two-dimensional magnetohydrodynamics: A continuum statistical equilibrium model

Abstract: We propose a statistical equilibrium model to characterize large-scale organized states, or coherent structures, in two-dimensional magneto- hydrodynamic turbulence. In this continuum model, the magnetic field and velocity field pair provide a microscopic description of the magnetofluid system, while a macroscopic description is furnished by a local joint probability distribution, or Young measure, on the values of the magnetic field and velocity field at each point in the spatial domain. This macroscopic description captures the long-time behavior of the system, in which the field and the flow develop fluctuations on increasingly fine spatial scales. The most probable macrostate is found by maximizing an appropriate entropy functional subject to constraints dictated by the conservation of energy, flux and cross-helicity under ideal magnetohydrodynamics. This statistical equilibrium state is, for each point in the spatial domain, a Gaussian probability distribution on the values of the magnetic field and the velocity field. The associated local mean is an exact stationary solution of the ideal MHD equations. Thus, the model predicts that the system will evolve into a coherent stationary mean field-flow coupled with turbulent Gaussian local fluctuations. Furthermore, the predictions of the continuum statistical model are in excellent qualitative and quantitative agreement with recent high-resolution numerical simulations of two-dimensional magnetofluid turbulence.

#### 6. Bruce E. Turkington

Department of Mathematics & Statistics
University of Massachusetts, Amherst, MA 01003-0113, USA
turk@math.umass.edu

Title of Presentation: Lattice models of ideal magnetofluid turbulence

Abstract: We construct statistical equilibrium models of MHD turbulence. These models describe the turbulent relaxation of a nondissipative magnetofluid into a coherent state, consisting of a large-scale, steady mean field and flow together with small-scale, local fluctuations. We build the models by approximating the Gibbs measure for the ideal MHD invariants (energy, cross-helicity, and flux or magnetic helicity) on a discrete grid. This analysis yields a Gaussian distribution whose mean state solves a nonlinear eigenvalue problem. The corresponding continuum models are obtained in the limit as the grid spacing tends to zero. In two dimensions the theory is now well understood, and the characteristics of the relaxed state can be predicted from the given values of the global invariants. In three dimensions, a similar theory appears to be feasible.

#### 7. Olivier Coulaud

IECN and INRIA Lorraine, 615 Rue du Jardin Botanique, B.P. 101
F-54602 Villers les Nancy cedex, France
Olivier.Coulaud@loria.fr

Phone: +33 83 59 30 32 or +33 83 91 26 69, Fax: +33 83 27 83 19

Title of Presentation: Multiple time scales and perturbation methods for high-frequency electromagnetic-hydrodynamic coupling in the treatment of liquid metals

Abstract: Many processes in the treatment of molten metals are based on the use of electromagnetic fields. One of the advantages of the magnetic induction is that it can act at distance and without contact. Its effects can be thermal (Joule's effect) or mechanical (brazing's effect). There are a lot of applications, for example electromagnetic shaping or casting, levitation of melting metals, stirring, surfacing, surface stabilization, heating, etc... Here, we are only interested in the mechanical effects. The complete modelization of these phenomena requires the coupling between electromagnetic aspects and hydrodynamic ones. Moreover the air-metal interface is a free boundary, which adds to the difficulty. Finally the industrial configurations are three-dimensional. One important phenomenon in this kind of problems is the so-called "skin effect": the eddy currents are concentrated near the surface of the melting metals. The thickness of this layer is proportional to the inverse of the square root of the frequency $\omega$ of the imposed alternating current. For very high frequencies like $10^4$Hz or more, due to the fact that the magnetic field does not penetrate into the liquid, we can consider that the eddy currents are only on the surface. Also, the motion of the liquid can be neglected. The model obtained is then magnetostatic. In this limit model, we first compute the shape taken by the metal and then, if we need, the motion inside the metal. Our goal is to justify this limit model when the frequency is large and to build efficient numerical schemes. We consider the complete set of equations (Maxwell and Navier-Stokes). An analysis shows that we have three important small parameters, the screen parameter, the magnetic Reynolds number, and the Reynolds number. By using the theory of perturbations and multiple scales, we obtain classical models used in high-frequency MHD problems such as for example the above-mentioned magnetostatic model. We have a singular perturbation problem in parameters $\epsilon$ and Reynolds; in fact the skin effect is like a free boundary layer. We analyze it. Moreover we are able to construct the magnetic field and the velocity in the layer.

#### 8. Rachid Touzani

Laboratoire de Mathematiques Appliquees
Universite Blaise Pascal (Clermont-Ferrand 2)
F-63177 Aubiere cedex, France
touzani@ucfma.univ-bpclermont.fr

Phone: (33) 73 40 77 06, Fax: (33) 73 40 70 64

Title of Presentation: Mathematical and numerical problems in incompressible MHD

Abstract: We consider a rigorous derivation of incompressible MHD equations for melting flows arising in industrial applications like electromagnetic casting, forming, or stirring. The model treats the time-harmonic case. Existence results for the resulting coupled equations are given as well as uniqueness in the case where the injected current is small enough. Numerical analysis is also considered using a coupled Boundary Element / Finite Element method, and convergence results are outlined.

### Part III

#### 9. Kumbakonam R. Rajagopal

Department of Mechanical Engineering
University of Pittsburgh, Pittsburgh, PA 15260, USA
raj@vms.cis.pitt.edu

Title of Presentation: Mathematical modelling of field-dependant materials

Abstract: In this talk I shall first present constitutive relations for both field-dependant fluids (electrorheological fluids) and solids (electro- active elastomers) within the frame-work of continuum mechanics. Using the above constitutive relations, I shall discuss the solution of several initial-boundary value problems.

#### 10. Gerhard O. Strohmer

Department of Mathematics, University of Iowa
Iowa City, IA 52241, USA
strohmer@math.uiowa.edu

Title of Presentation: Partially regular weak solutions of some abstract equations, with applications to magneto-hydrodynamics

Abstract: First an existence proof for partially regular weak solutions of the abstract evolution equation U'(t)+AU(t)=F(U(t),t) is discussed, where A is a self-adjoint positive operator with a compact inverse, and F is a nonlinearity fulfilling certain conditions. Then we consider some applications to magneto-hydrodynamics.

#### 11. Paul G. Schmidt

Department of Mathematics, Parker Hall Auburn University, AL 36849-5310, USA
pgs@math.auburn.edu

Phone: (334)844-6579, Fax: (334)844-6555

Title of Presentation: Analytical and numerical methods for viscous incompressible MHD

Abstract: We will present a new approach to viscous incompressible MHD that allows us to efficiently deal with a variety of coupled nonlinear flow problems where a conducting fluid, confined to a bounded region of space, interacts with the outside world via the magnetic field and through currents flowing in external conductors. Analytical results regarding the well-posedness of such problems will be discussed in time-dependent as well as stationary situations. We will also describe a finite-element solver for the numerical approximation of steady-state solutions and report on the results of some computational experiments.

MHD MHD Publications

ajm@math.auburn.edu