MHD Equations

The MHD equations take the form:

$$\frac{\partial n}{\partial t} + \nabla\cdot(n\,{\bf V}_e) =0,$$ (4.205)

$$m_e \,n\,\frac{\partial {\bf V}_e}{\partial t} + m_e \,n\,({\bf V}_e\cdot\nabla){\bf V}_e+ \nabla p_e$$

$$+ [\delta^{\,-1}]\,e\, n\, ({\bf E} + {\bf V}_e\times {\bf B}) = [\zeta]\,{\bf F}_U + {\bf F}_T,$$ (4.206)

$$\frac{3}{2}\frac{\partial p_e}{\partial t} + \frac{3}{2} \,({\bf V}_e\cdot\nabla)\, p_e + \frac{5}{2}\,p_e\,\nabla\cdot{\bf V}_e = -[\delta^{\,-1}\,\zeta\,\mu^2]\,W_i,$$ (4.207)

and

$$\frac{\partial n}{\partial t} + \nabla\cdot(n\,{\bf V}_i) =0,$$ (4.208)

$$m_i \,n\,\frac{\partial {\bf V}_i}{\partial t} + m_i\, n\,({\bf V}_i\cdot\nabla) {\bf V}_i + \nabla p_i$$

$$- [\delta^{\,-1}]\, e\, n\, ({\bf E} + {\bf V}_i\times {\bf B}) =- [\zeta]\,{\bf F}_U -{\bf F}_T,$$ (4.209)

$$\frac{3}{2}\frac{\partial p_i}{\partial t} + \frac{3}{2} \,({\bf V}_i\cdot\nabla) \,p_i + \frac{5}{2}\,p_i\,\nabla\cdot{\bf V}_i =[\delta^{\,-1}\,\zeta\,\mu^2]\, W_i.$$ (4.210)

These equations can also be simplified by making use of the smallness of the mass ratio $m_e/m_i$ . Now, according to the ordering adopted in Section 4.11, $U \sim \delta\,V_e\sim \delta\,V_i$ in the MHD limit. It follows from Equations (4.200) and (4.201) that

$${\bf V}_i \simeq {\bf V} + {\cal O}\left(\frac{m_e}{m_i}\right),$$ (4.211)

and

$${\bf V}_e \simeq {\bf V} - [\delta]\,\frac{{\bf j}}{n\,e} + {\cal O}\left(\frac{m_e} {m_i}\right).$$ (4.212)

The main point, here, is that in the MHD limit the velocity difference between the electron and ion fluids is relatively small.

Equations (4.205) and (4.208) yield the continuity equation:

$$\frac{dn}{dt} + n\,\nabla\cdot{\bf V} = 0,$$ (4.213)

where $d/dt \equiv \partial/\partial t + {\bf V} \cdot\nabla$ .

Equations (4.206) and (4.209) can be summed to give the equation of motion:

$$m_i \,n\,\frac{d{\bf V}}{dt} + \nabla p - {\bf j}\times{\bf B} \simeq 0.$$ (4.214)

Here, $p=p_e+p_i$ is the total pressure. Note that all terms in the previous equation are the same order in $\delta$ .

The ${\cal O}(\delta^{\,-1})$ components of Equations (4.206) and (4.209) yield the Ohm's law:

$${\bf E} + {\bf V}\times {\bf B} \simeq {\bf0}.$$ (4.215)

This is sometimes called the perfect conductivity equation, because it is identical to the Ohm's law in a perfectly conducting liquid.

Equations (4.207) and (4.210) can be summed to give the energy evolution equation:

$$\frac{3}{2} \frac{dp}{dt} + \frac{5}{2}\, p \,\nabla\cdot{\bf V} \simeq 0.$$ (4.216)

Equations (4.213) and (4.216) can be combined to give the more familiar adiabatic equation of state:

$$\frac{d}{dt}\!\left(\frac{p}{n^{5/3}}\right) \simeq 0.$$ (4.217)

Finally, the ${\cal O}(\delta^{\,-1})$ components of Equations (4.207) and (4.210) yield

$$W_i \simeq 0,$$ (4.218)

or $T_e\simeq T_i$ [see Equation (4.101)]. Thus, we expect equipartition of the thermal energy between electrons and ions in the MHD limit.