Hall MHD

Generalized Ohm's law	\frac{m_e}{n e^2} \frac{\partial J}{\partial t} - \frac{1}{ne}\nabla \cdot \mathbf{P_e} = \mathbf{E} + \frac{1}{c} \mathbf{v} \times \mathbf{B} - \frac{1}{nec} \mathbf{J} \times \mathbf{B} - \frac{\mathbf{J}}{\sigma}
Ideal MHD	0 = \mathbf{E} + \frac{1}{c} \mathbf{v}\times \mathbf{B}
Resistive MHD	0 = \mathbf{E} + \frac{1}{c} \mathbf{v} \times \mathbf{B} - \frac{\mathbf{J}}{\sigma}
Hall MHD	0 = \mathbf{E} + \frac{1}{c} \mathbf{v} \times \mathbf{B} - \frac{1}{nec} \mathbf{J} \times \mathbf{B}

Combining these with the induction equation

\frac{\partial \mathbf{B}}{\partial t} = -c \nabla \times \mathbf{E}

we arrive at

Ideal MHD	\frac{\partial \mathbf{B}}{\partial t} = \nabla \times \left [ \mathbf{v}\times \mathbf{B} \right ]
Resistive MHD	\frac{\partial \mathbf{B}}{\partial t} = \nabla \times \left [ \mathbf{v} \times \mathbf{B} - c \frac{\mathbf{J}}{\sigma} \right ]
Hall MHD	\frac{\partial \mathbf{B}}{\partial t} = \nabla \times \left [ \mathbf{v} \times \mathbf{B} - \frac{1}{ne} \mathbf{J} \times \mathbf{B} \right ]

Now for Hall MHD, we can write the induction equation as

\frac{\partial \mathbf{B}}{\partial t} = \nabla \times \left [ \mathbf{v} \times \mathbf{B} - \mathbf{v}_\mbox{H} \times \mathbf{B} \right ]

where

\mathbf{v}_\mbox{H} = \frac{\mathbf{J}}{ne}

Note the Hall velocity is always parallel to the current and therefore perpendicular to the field (using Ampere's law without Maxwell's addition ignoring time varying electric fields) and as a result it does no work on the magnetic field and does not need any additional energy terms.

Expanding just the Hall term further, we arrive at

\frac{\partial \mathbf{B}}{\partial t} = -\nabla \times \left [ \frac{1}{ne} \left ( \nabla \times \mathbf{B} \right ) \times \mathbf{B} \right ]

which we can compare to the resistive mhd term

\frac{\partial \mathbf{B}}{\partial t} = -\nabla \times \left [ \left ( \eta \nabla \times \mathbf{B} \right ) \right ]

Discretization

For resistive MHD, we calculate \xi = \nabla \times \mathbf{B} at cell edges for which each component can be calculated by differencing the 4 cell faces that share the edge.

For Hall MHD we need to calculate \xi = \left (\nabla \times \mathbf{B} \right) \times \mathbf{B} along with n at cell edges.

\xi_i = B_k d_l B_m \epsilon_{ijk} \epsilon_{jlm} = B_k d_l B_m \left(\delta_{kl} \delta_{im} - \delta_{km} \delta_{il} \right )

So we have

\xi_z = B_x \left ( d_x B_z - d_z B_x \right) + B_y \left ( d_y B_z - d_z B_y \right)

So we need

• edge centered estimates of the perpendicular fields (which can be found by averaging the two adjacent faces)
• transverse gradients of the parallel field (which can be found by differencing adjacent face centered averages of the parallel field (found by averaging the adjacent cell centered fields)
• parallel gradients of the transverse field

So steps are

\xi^z_{i+1/2,j+1/2,k} = \frac{B^x_{i+1/2,j,k} + B^x_{i+1/2,j+1,k}}{2} \left ( \frac{b^z_{i+1,j,k} + b^z_{i+1,j+1,k} - b^z_{i,j,k}-b^z_{i,j+1,k}}{2 \Delta x} - \frac{B^x_{i+1/2, j, k+1} + B^x_{i+1/2,j+1,k+1}-B^x_{i+1/2,j,k-1}-B^x_{i+1/2,j+1,k-1}}{4 \Delta z} \right )

+\frac{B^y_{i,j+1/2,k} + B^y_{i+1,j+1/2,k}}{2} \left ( \frac{b^z_{i,j+1,k} + b^z_{i+1,j+1,k} - b^z_{i,j,k}-b^z_{i+1,j,k}}{2 \Delta y} - \frac{B^y_{i, j+1/2, k+1} + B^y_{i+1,j+1/2,k+1}-B^y_{i,j+1/2,k-1}-B^y_{i+1,j+1/2,k-1}}{4 \Delta z} \right )

Now if we represent edge centered perpendicular B fields \beta^x_{i+1/2,j+1/2,k} = \frac{B^x_{i+1/2,j,k}+B^x_{i+1/2,j+1,k}}{2}

and face centered parallel fields

\alpha^z_{i,j+1/2,k} = \frac{b^z_{i,j,k}+b^z_{i,j+1,k}}{2}

this simplifies to

\xi^z_{i+1/2,j+1/2,k} = \beta^x_{i+1/2,j+1/2,k} \left ( \frac{\alpha^z_{i+1,j+1/2,k} - \alpha^z_{i,j+1/2,k}}{\Delta x} - \frac{\beta^x_{i+1/2, j+1/2, k+1} -\beta^x_{i+1/2,j+1/2,k-1}}{2 \Delta z} \right )

+\beta^y_{i+1/2,j+1/2,k} \left ( \frac{\alpha^z_{i+1/2,j+1,k} - \alpha^z_{i+1/2,j,k}}{\Delta x} - \frac{\beta^y_{i+1/2, j+1/2, k+1} -\beta^y_{i+1/2,j+1/2,k-1}}{2 \Delta z} \right )

So to calculate 'z' edges we need adjacent face centered z fields and edge centered transverse fields

These fields won't be used for any of the other directional updates - so can be discarded after each dimension