Religion: The Conservation Law

Hall MHD models solve the following governing equations of ion mass density \(\rho\), magnetic field \(\mathbf{B}\), ion bulk velocity \(\mathbf{u}\), total thermal pressure \(p_t=p_i+p_e\):

\[ \begin{gather} \frac{\partial \rho}{\partial t} + \nabla\!\cdot(\rho\,\mathbf{u}) = 0 \\[6pt] \rho\,\frac{\partial \mathbf{u}}{\partial t} + \rho\,(\mathbf{u}\!\cdot\!\nabla)\mathbf{u} = -\nabla p_t + \mathbf{G} + \mathbf{J}\times\mathbf{B} \\[6pt] \frac{\partial p_t}{\partial t} + \mathbf{u}\!\cdot\!\nabla p_t + \gamma\,p_t\,\nabla\!\cdot\!\mathbf{u} = (\gamma-1) (\mathbf{G}\!\cdot\!\mathbf{u} + \eta\,J^2) \\[6pt] \frac{\partial \mathbf{B}}{\partial t} = -\,\nabla\times\mathbf{E} \\[6pt] \end{gather} \]

Hall term (\(\mathbf{J}\times\mathbf{B}\)) does no local work in this form. On the other hand, instead of the original form shown above, the BATSRUS code actrually solves the conservative form of these equations, by default.

Definitions

\[ \begin{aligned} \mathbf{E} &= \eta\,\mathbf{J} - \mathbf{u}\times\mathbf{B} + \frac{\mathbf{J}\times\mathbf{B}}{e\,n_e}, \\ n_e &= n = \frac{\rho}{m}, \\ \mathbf{J} &= \frac{1}{\mu_0}\,\nabla\times\mathbf{B}, \\ \mathbf{u}_e &= \mathbf{u} - \frac{\mathbf{J}}{e\,n_e} = \mathbf{u} + \mathbf{u}_H. \end{aligned} \]

Notation: \(\mathbf{I}\) is the identity tensor, \(B^2=\mathbf{B}\cdot\mathbf{B}\), \(\mu_0\) is the vacuum permeability, \(e\) is the elementary charge, \(\eta\) is the resistivity, and \(\gamma\) is the adiabatic index. The \(p_e\) term is neglected in the \(\mathbf{E}\) equation.

Most ion inertial length \(d_i\) formulas are written in cgs, including wiki. I write it in SI here: \(\lambda_i=\sqrt{ \frac{m_i}{\mu_0 e^2 n_i} }\)[Toth et al. 2017].

Fancy Plasma parameter Calculators

Ying-Dong Jia's Recent Research Results

Solar wind interacting with Earth dipole. These figures show how parameters change at the shock and the pile up boundary. We show a large domain to exhibit the oblique shock regions.



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At the dawn-dusk terminator, flow shear is still only ~100km/s.

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