7.2. MHD Equations

7.2.1. Introduction

The magnetohydrodynamics (MHD) equations are:

(7.2.1.1)\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho {\bf v}) = 0

(7.2.1.2)\rho\left(\frac{\partial {\bf v}}{\partial t} + ({\bf v} \cdot \nabla) {\bf v} \right) = -\nabla p + {1\over\mu}(\nabla\times{\bf B}) \times {\bf B} + \rho {\bf g}

(7.2.1.3){\partial {\bf B}\over\partial t} = \nabla\times({\bf v}\times{\bf B}) + \eta\nabla^2{\bf B}

(7.2.1.4)\nabla\cdot{\bf B} = 0

assuming \eta is constant. See the next section for a derivation. We can now apply the following identities (we use the fact that \nabla\cdot{\bf B}=0):

\left[(\nabla\times{\bf B}) \times {\bf B}\right]_i = \varepsilon_{ijk}(\nabla\times{\bf B})_j B_k = \varepsilon_{ijk}\varepsilon_{jlm}(\partial_l B_m)B_k = (\delta_{kl}\delta_{im}-\delta_{km}\delta_{il})(\partial_l B_m)B_k = =(\partial_k B_i)B_k - (\partial_i B_k)B_k =\left[({\bf B}\cdot\nabla){\bf B} - {1\over2}\nabla|{\bf B}|^2\right]_i (\nabla\times{\bf B}) \times {\bf B} &= ({\bf B}\cdot\nabla){\bf B} - {1\over2}\nabla|{\bf B}|^2= ({\bf B}\cdot\nabla){\bf B} + {\bf B}(\nabla\cdot{\bf B}) - {1\over2}\nabla|{\bf B}|^2 =\nabla\cdot({\bf B}{\bf B}^T) - {1\over2}\nabla|{\bf B}|^2\\ \nabla\times({\bf v} \times {\bf B}) &= ({\bf B}\cdot\nabla){\bf v} - {\bf B}(\nabla\cdot{\bf v}) +{\bf v}(\nabla\cdot {\bf B}) - ({\bf v}\cdot\nabla) {\bf B} = \nabla\cdot({\bf B}{\bf v}^T - {\bf v}{\bf B}^T)\\ \nabla\cdot(\rho{\bf v}{\bf v}^T) &= \left(\nabla\cdot(\rho{\bf v})\right){\bf v} + \rho({\bf v}\cdot\nabla){\bf v}= -{\bf v}\frac{\partial \rho}{\partial t} + \rho({\bf v}\cdot\nabla){\bf v}

So the MHD equations can alternatively be written as:

(7.2.1.5)\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho {\bf v}) = 0

(7.2.1.6)\frac{\partial \rho{\bf v}}{\partial t} + \nabla\cdot(\rho{\bf v}{\bf v}^T) = -\nabla p + {1\over\mu}\left(\nabla\cdot({\bf B}{\bf B}^T) - {1\over2}\nabla|{\bf B}|^2\right) + \rho {\bf g}

(7.2.1.7){\partial {\bf B}\over\partial t} = \nabla\cdot({\bf B}{\bf v}^T - {\bf v}{\bf B}^T) + \eta\nabla^2{\bf B}

(7.2.1.8)\nabla\cdot{\bf B} = 0

One can also introduce a new variable p^* = p + {1\over2}\nabla|{\bf B}|^2, that simplifies (7.2.1.6) a bit.

7.2.2. Derivation

The above equations can easily be derived. We have the continuity equation:

\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho {\bf v}) = 0

Navier-Stokes equations (momentum equation) with the Lorentz force on the right-hand side:

\rho\left(\frac{\partial {\bf v}}{\partial t} + ({\bf v} \cdot \nabla) {\bf v} \right) = -\nabla p + {\bf j} \times {\bf B} + \rho {\bf g}

where the current density {\bf j} is given by the Maxwell equation (we neglect the displacement current {\partial{\bf E}\over\partial t}):

{\bf j} = {1\over\mu}\nabla\times{\bf B}

and the Lorentz force:

{1\over\sigma}{\bf j} = {\bf E} + {\bf v}\times{\bf B}

from which we eliminate {\bf E}:

{\bf E} = - {\bf v}\times{\bf B} + {1\over\sigma}{\bf j} = - {\bf v}\times{\bf B} + {1\over\sigma\mu}\nabla\times{\bf B}

and put it into the Maxwell equation:

{\partial {\bf B}\over\partial t} = -\nabla\times{\bf E}

so we get:

{\partial {\bf B}\over\partial t} = \nabla\times({\bf v}\times{\bf B}) - \nabla\times\left({1\over\sigma\mu}\nabla\times{\bf B}\right)

assuming the magnetic diffusivity \eta={1\over\sigma\mu} is constant, we get:

{\partial {\bf B}\over\partial t} = \nabla\times({\bf v}\times{\bf B}) - \eta\nabla\times\left(\nabla\times{\bf B}\right) = \nabla\times({\bf v}\times{\bf B}) + \eta\left(\nabla^2{\bf B}-\nabla(\nabla\cdot{\bf B})\right) = \nabla\times({\bf v}\times{\bf B}) + \eta\nabla^2{\bf B}

where we used the Maxwell equation:

\nabla\cdot{\bf B} = 0

7.2.3. Finite Element Formulation

We solve the following ideal MHD equations (we use p^* = p + {1\over2}\nabla|{\bf B}|^2, but we drop the star):

(7.2.3.1)\frac{\partial {\bf u}}{\partial t} + ({\bf u} \cdot \nabla) {\bf u} - ({\bf B}\cdot\nabla){\bf B} + \nabla p = 0

(7.2.3.2){\partial {\bf B}\over\partial t} + ({\bf u}\cdot\nabla){\bf B} - ({\bf B}\cdot\nabla){\bf u} = 0

(7.2.3.3)\nabla\cdot{\bf u} = 0

(7.2.3.4)\nabla\cdot{\bf B} = 0

If the equation (7.2.3.4) is satisfied initially, then it is satisfied all the time, as can be easily proved by applying a divergence to the Maxwell equation {\partial {\bf B}\over\partial t} = -\nabla\times{\bf E} (or the equation (7.2.3.2), resp. (7.2.1.3)) and we get {\partial \over\partial t}(\nabla\cdot{\bf B}) = 0, so \nabla\cdot{\bf B} is constant, independent of time. As a consequence, we are essentially only solving equations (7.2.3.1), (7.2.3.2) and (7.2.3.3), which consist of 5 equations for 5 unknowns (components of {\bf u}, p and {\bf B}).

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

(7.2.3.5)\frac{{\bf u}^n-{\bf u}^{n-1}}{\tau} + ({\bf u}^{n-1} \cdot \nabla) {\bf u}^n - ({\bf B}^{n-1}\cdot\nabla){\bf B}^n + \nabla p = 0

(7.2.3.6){{\bf B}^n-{\bf B}^{n-1}\over\tau} + ({\bf u}^{n-1}\cdot\nabla){\bf B}^n - ({\bf B}^{n-1}\cdot\nabla){\bf u}^n = 0

(7.2.3.7)\nabla\cdot{\bf u}^n = 0

Testing (7.2.3.5) by the test functions (v_1, v_2), (7.2.3.6) by the functions (C_1, C_2) and (7.2.3.7) by the test function q, we obtain the following weak formulation:

(7.2.3.8)\int_\Omega {u_1 v_1\over\tau} + ({\bf u}^{n-1}\cdot\nabla)u_1 v_1 - ({\bf B}^{n-1}\cdot\nabla)B_1 v_1 -p {\partial v_1\over\partial x} \,{\rm d}{\bf x} = \int_\Omega {u_1^{n-1} v_1\over\tau}\,{\rm d}{\bf x} \int_\Omega {u_2 v_2\over\tau} + ({\bf u}^{n-1}\cdot\nabla)u_2 v_2 - ({\bf B}^{n-1}\cdot\nabla)B_2 v_2 -p {\partial v_2\over\partial y} \,{\rm d}{\bf x} = \int_\Omega {u_2^{n-1} v_2\over\tau}\,{\rm d}{\bf x}

(7.2.3.9)\int_\Omega {B_1 C_1\over\tau} + ({\bf u}^{n-1}\cdot\nabla)B_1 C_1 - ({\bf B}^{n-1}\cdot\nabla)u_1 C_1 \,{\rm d}{\bf x} = \int_\Omega {B_1^{n-1} C_1\over\tau}\,{\rm d}{\bf x} \int_\Omega {B_2 C_2\over\tau} + ({\bf u}^{n-1}\cdot\nabla)B_2 C_2 - ({\bf B}^{n-1}\cdot\nabla)u_2 C_2 \,{\rm d}{\bf x} = \int_\Omega {B_2^{n-1} C_2\over\tau}\,{\rm d}{\bf x}

(7.2.3.10)\int_\Omega {\partial u_1\over\partial x}q + {\partial u_2\over\partial y}q \,{\rm d}{\bf x} = 0

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:

$$\begin{array}{lclclclclcl} +A(u_1, v_1) && && -X(p, v_1) && -B(B_1, v_1) && &=& l_1(v_1)\\ && +A(u_2, v_2) && -Y(p, v_2) && && -B(B_2, v_2) &=& l_2(v_2)\\ +X(q, u_1) && +Y(q, u_2) && && && &=& 0\\ -B(u_1, C_1) && && && +A(B_1, C_1) && &=& l_4(C_1)\\ && -B(u_2, C_2) && && && +A(B_2, C_2) &=& l_5(C_2) \end{array}$$

where:

A(u, v) &= \int_\Omega {u v\over\tau} + ({\bf u}^{n-1}\cdot\nabla)u v\,{\rm d}{\bf x}\\ B(u, v) &= \int_\Omega ({\bf B}^{n-1}\cdot\nabla)uv\,{\rm d}{\bf x}\\ X(u, v) &= \int_\Omega u {\partial v\over\partial x} \,{\rm d}{\bf x}\\ Y(u, v) &= \int_\Omega u {\partial v\over\partial y} \,{\rm d}{\bf x}\\ l_1(v) &= \int_\Omega {u_1^{n-1} v\over\tau} \,{\rm d}{\bf x}\\ l_2(v) &= \int_\Omega {u_2^{n-1} v\over\tau} \,{\rm d}{\bf x}\\ l_4(v) &= \int_\Omega {B_1^{n-1} v\over\tau} \,{\rm d}{\bf x}\\ l_5(v) &= \int_\Omega {B_2^{n-1} v\over\tau} \,{\rm d}{\bf x}

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:

$$\begin{array}{lclclclclcl} a_{11}(u_1, v_1) &+& a_{12}(u_2, v_1) &+& a_{13}(p, v_1) &+& a_{14}(B_1, v_1) &+& a_{15}(B_2, v_1) &=& l_1(v_1)\\ a_{21}(u_1, v_2) &+& a_{22}(u_2, v_2) &+& a_{23}(p, v_2) &+& a_{24}(B_1, v_2) &+& a_{25}(B_2, v_2) &=& l_2(v_2)\\ a_{31}(u_1, q) &+& a_{32}(u_2, q) &+& a_{33}(p, q) &+& a_{34}(B_1, q) &+& a_{35}(B_2, q) &=& l_3(q)\\ a_{41}(u_1, C_1) &+& a_{42}(u_2, C_1) &+& a_{43}(p, C_1) &+& a_{44}(B_1, C_1) &+& a_{45}(B_2, C_1) &=& l_4(C_1)\\ a_{51}(u_1, C_2) &+& a_{52}(u_2, C_2) &+& a_{53}(p, C_2) &+& a_{54}(B_1, C_2) &+& a_{55}(B_2, C_2) &=& l_5(C_2) \end{array}$$

comparing to the above, we get the following nonzero forms:

$$\begin{array}{lclclclclcl} a_{11}(u_1, v_1) &+& 0 &+& a_{13}(p, v_1) &+& a_{14}(B_1, v_1) &+& 0 &=& l_1(v_1)\\ 0 &+& a_{22}(u_2, v_2) &+& a_{23}(p, v_2) &+& 0 &+& a_{25}(B_2, v_2) &=& l_2(v_2)\\ a_{31}(u_1, q) &+& a_{32}(u_2, q) &+& 0 &+& 0 &+& 0 &=& 0\\ a_{41}(u_1, C_1) &+& 0 &+& 0 &+& a_{44}(B_1, C_1) &+& 0 &=& l_4(C_1)\\ 0 &+& a_{52}(u_2, C_2) &+& 0 &+& 0 &+& a_{55}(B_2, C_2) &=& l_5(C_2) \end{array}$$

where:

a_{11}(u_1, v_1) &= A(u_1, v_1)\\ a_{22}(u_2, v_2) &= A(u_2, v_2)\\ a_{44}(B_1, C_1) &= A(B_1, C_1)\\ a_{55}(B_2, C_1) &= A(B_2, C_2)\\ a_{13}(p, v_1) &= -X(p, v_1)\\ a_{31}(u_1, q) &= X(q, u_1)\\ a_{23}(p, v_2) &= -Y(p, v_2)\\ a_{32}(u_2, q) &= Y(q, u_2)\\ a_{14}(B_1, v_1) &= -B(B_1, v_1)\\ a_{41}(u_1, C_1) &= -B(u_1, C_1)\\ a_{25}(B_2, v_2) &= -B(B_2, v_2)\\ a_{52}(u_2, C_2) &= -B(u_2, C_2)

and l1, …, l5 are the same as above.