On some inverse problems in MHD

Usually, MHD is understood as the forward problem of determiningvelocity and magnetic fields in the sense of solving the coupled system of Navier-Stokes equations and induction equation if some boundary conditions or types of forcing are given. For most laboratory and technical applications with small magnetic Reynolds number Rm the magnetic field is disturbed only slightly by the flow whereas it can modify the latter significantly via Lorentz forces. For higher Rm, however, the fluid flow can change the magnetic field drastically or can even lead to self-excitation of a magnetic field via the dynamo effect.

In the inverse problems approach one strives to get information about the distribution of certain material parameters or temperature and velocity fields inside the fluid from measurements of appropriate quantities at the fluid boundary and/or outside the fluid. We restrict our interest on the spatial reconstruction of the velocity field (or of related mean-field quantities) purely from magnetic fields and electric potentials which can be measured outside the fluid body and at its boundary, respectively.

For the case of small Rm, which is most interesting for a number of technological applications like metal casting and crystal growth, it is necessary to apply external magnetic fields and to measure the flow induced magnetic fields and electric potentials. Analytical results concerning the uniqueness problem of velocity reconstruction /1/ as well as numerical illustrations /2/ are presented. The need of regularization of the inverse problem is discussed.

Some aspects of generalizing the method to high Rm, i.e. to an inverse dynamo theory, are also discussed. In particaluar, considering a modified Krause-Steenbeck dynamo model with radially varying alpha we show how this radial dependence can be inferred from the growth rates of several magnetic field modes. The connection of this problem with similar problems in quantum mechanics, like isospectral potential deformations, is worked out.