Cold-Plasma Equations

Previously, we used the smallness of the magnetization parameter, $\delta$, to derive the cold-plasma equations:

$$\frac{\partial n}{\partial t} + \nabla\cdot(n\,{\bf V}_e) =0,$$ (4.175)

$$m_e \,n\,\frac{\partial {\bf V}_e}{\partial t} + m_e\, n\, ({\bf V}_e\cdot \nabla ){\bf V}_e + e \,n\, ({\bf E} + {\bf V}_e\times {\bf B}) = [\zeta]\,{\bf F}_U,$$ (4.176)

and

$$\frac{\partial n}{\partial t} + \nabla\cdot(n\,{\bf V}_i) =0,$$ (4.177)

$$m_i\, n\,\frac{\partial {\bf V}_i}{\partial t} +m_i \,n\,({\bf V}_i\cdot \nabla){\bf V}_i - e \,n\, ({\bf E} + {\bf V}_i\times {\bf B}) =- [\zeta]\,{\bf F}_U.$$ (4.178)

Let us now use the smallness of the mass ratio $m_e/m_i$ to further simplify these equations. In particular, we would like to write the electron and ion fluid velocities in terms of the center-of-mass velocity,

$${\bf V} = \frac{m_i\,{\bf V}_i + m_e\,{\bf V}_e}{m_i+ m_e},$$ (4.179)

and the plasma current

$${\bf j} = -n\,e\,{\bf U},$$ (4.180)

where ${\bf U} = {\bf V}_e-{\bf V_i}$. According to the ordering scheme adopted in the previous section, $U\sim V_e\sim V_i$ in the cold-plasma limit. We shall continue to regard the mean-free-path parameter $\zeta$ as ${\cal O}(1)$.

It follows from Equations (4.179) and (4.180) that

$${\bf V}_i \simeq {\bf V} + {\cal O}\left(\frac{m_e}{m_i}\right),$$ (4.181)

and

$${\bf V}_e \simeq {\bf V} - \frac{{\bf j}}{ne} + {\cal O}\!\left(\frac{m_e}{m_i}\right).$$ (4.182)

Equations (4.175), (4.177), (4.181), and (4.182) yield the continuity equation:

$$\frac{dn}{dt} + n\,\nabla\cdot{\bf V} = 0,$$ (4.183)

where $d/dt \equiv \partial/\partial t + {\bf V} \cdot\nabla$. Here, use has been made of the fact that $\nabla\cdot{\bf j} =0$ in a quasi-neutral plasma.

Equations (4.176) and (4.178) can be summed to give the equation of motion:

$$m_i \,n\,\frac{d{\bf V}}{dt} -{\bf j}\times {\bf B} \simeq {\bf0}.$$ (4.184)

Finally, Equations (4.176), (4.181), and (4.182) can be combined to give a modified Ohm's law:

$${\bf E} + {\bf V}\times {\bf B} \simeq \frac{{\bf F}_U}{n\,e} + \frac{{\bf j}\times{\bf B}} {n\,e... ...}\cdot\nabla)\,{\bf V} - \frac{m_e}{n^2 \,e^3}\, ({\bf j}\cdot\nabla)\,{\bf j}.$$ (4.185)

The first term on the right-hand side of the previous equation corresponds to resistivity, the second corresponds to the Hall effect, the third corresponds to the effect of electron inertia, and the remaining terms are usually negligible.