Generalized Ohm's law

Referece [2] gives a clear derivation of the generalized Ohm's law, which takes the following form

$$\mathbf{E}+\mathbf{u} \times \mathbf{B}- \eta \mathbf{J}= \frac{1... ...\partial t} + \nabla \cdot (\mathbf{J}\mathbf{u}+\mathbf{u}\mathbf{J}) \right],$$ (12)

where the first term on the right-hand side is called the ``Hall term'', the second term is the electron pressure term, and the third term is called the ``electron inertia term'' since it is proportional to the mass of electrons.

Note that both $\mathbf{u}$ and $\mathbf{J}$ are the first-order moments, with $\mathbf{u}$ being the (weighted) sum of the first-order moment of electrons and ions while $\mathbf{J}$ being the difference between them. The generalized Ohm's law is actually the difference between the electrons and ions first-order moment equations. The generalized Ohm's law is an equation that governs the time evolution of $\mathbf{J}$. Also note that Ampere's law, with the displacement current retained, is an equation governing the time evolution of $\mathbf{E}$. However, in the approximation of the resistive MHD, the time derivative terms $\partial \mathbf{E}/ \partial t$ and $\partial \mathbf{J}/ \partial t$ are ignored in Ampere's law and Ohm's law, respectively. In this approximation, Ohm's law is directly solved to determine $\mathbf{E}$ and Ampere's law is directly solved to determine $\mathbf{J}$.