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Magnetic Field Flux in 1D

Consider the induction equation for a MHD fluid:

\frac{\partial \mathbf{B}}{\partial t} = \mathbf{\nabla} \times \left( \mathbf{v} \times \mathbf{B} \right) - \mathbf{\nabla} \times \left[(\eta+\alpha_2) ( \mathbf{\nabla} \times \mathbf{B}) \right] \\

For the x axis, v_y, \ v_z, \ B_x, \ B_z \rightarrow 0 . Hence, only the y component of the above equation is important.

\begin{align} \frac{\partial B_y}{\partial t} = - & \left[ \frac{\partial }{\partial x} (v_x B_y) - \frac{\partial }{\partial x} \left( \eta \frac{\partial B_y}{\partial x} + \alpha_2 \frac{\partial B_y}{\partial x} - \eta \frac{\partial B_x}{\partial y} - \alpha_2 \frac{\partial B_x}{\partial y} \right) \right. \\ & \left. - \frac{\partial }{\partial z} \left( \eta \frac{\partial B_y}{\partial z} + \alpha_2 \frac{\partial B_y}{\partial z} - \eta \frac{\partial B_z}{\partial y} - \alpha_2 \frac{\partial B_z}{\partial y} \right) + B_y \frac{\partial v_z}{\partial z} \right] \nonumber \end{align}

This gives us 10 terms to plot on the RHS, but before doing that we take integral wrt x on both sides from x = -R to x (the asteroid surface is taken to be at 0). The resulting plot is shown below for the 3 cases. The plots are labelled in the order they appear in the above equation.

astrobear

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Magnetic Field Flux in 1D

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