# MHD Equations¶

## Introduction¶

The magnetohydrodynamics (MHD) equations are:

(1) (2) (3) (4) assuming is constant. See the next section for a derivation. We can now apply the following identities (we use the fact that ): So the MHD equations can alternatively be written as:

(5) (6) (7) (8) One can also introduce a new variable , that simplifies (6) a bit.

## Derivation¶

The above equations can easily be derived. We have the continuity equation: Navier-Stokes equations (momentum equation) with the Lorentz force on the right-hand side: where the current density is given by the Maxwell equation (we neglect the displacement current ): and the Lorentz force: from which we eliminate : and put it into the Maxwell equation: so we get: assuming the magnetic diffusivity is constant, we get: where we used the Maxwell equation: ## Finite Element Formulation¶

We solve the following ideal MHD equations (we use , but we drop the star):

(9) (10) (11) (12) If the equation (12) is satisfied initially, then it is satisfied all the time, as can be easily proved by applying a divergence to the Maxwell equation (or the equation (10), resp. (3)) and we get , so is constant, independent of time. As a consequence, we are essentially only solving equations (9), (10) and (11), which consist of 5 equations for 5 unknowns (components of , and ).

We discretize in time by introducing a small time step and we also linearize the convective terms:

(13) (14) (15) Testing (13) by the test functions , (14) by the functions and (15) by the test function , we obtain the following weak formulation:

(16) (17) (18) To better understand the structure of these equations, we write it using bilinear and linear forms, as well as take into account the symmetries of the forms. Then we get a particularly simple structure: where: E.g. there are only 4 distinct bilinear forms. Schematically we can visualize the structure by:

 A -X -B A -Y -B X Y -B A -B A

In order to solve it with Hermes, we first need to write it in the block form: comparing to the above, we get the following nonzero forms: where: and , ..., are the same as above.