# 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.