**Titre/Title :**

Exact solutions to ideal magnetohydrodynamics equations

**Contact :**

Jan-Bert Flor

**Résumé/Abstract :**

Analytical modeling of plasma physics is based on the exact solutions

of the MHD equations. Due to the nonlinearity and complexity of the

equations, it is impossible to construct a general solution. However,

particular solutions of MHD equations that are based on Lie symmetry

analysis, allow for a describtion of the main features.

We present results of such a Lie symmetry analysis for the

construction and investigation of exact partially invariant solutions

to ideal MHD equations.

We discuss mainly two classes of solutions: partially invariant

solutions with respect to the group of rotations in 3D space, and

solutions with constant total pressure of stationary incompressible

MHD equations.

The first class of solutions describe a vortex motion with planar

trajectories and magnetic field lines driven by a spherical source.

Each magnetic line circumscribes the same planar curve in 3D space,

however, position and orientation of each curve is unique and is

prescribed by a certain directional field on a sphere.

We discuss singularities and characteristics of the flow governed by

the solution.

in the second part we observe stationary flows of an infinitely

conducting incompressible fluid. It is shown that contact magnetic

surfaces of such flows are translational surfaces, i.e. are swept out

by translating one curve rigidly along another curve. Explicit

examples of solutions possessing a significant functional

arbitrariness are discussed. Further we discuss possibilities for

application of Lie symmetry analysis to equations of rotating shallow

water equations, gas dynamics, and Navier-Stokes equations.