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MHD Equations

The MHD equations take the form:

$\displaystyle \frac{\partial n}{\partial t} + \nabla\cdot(n\,{\bf V}_e)$ $\displaystyle =0,$ (4.205)
$\displaystyle m_e \,n\,\frac{\partial {\bf V}_e}{\partial t} + m_e \,n\,({\bf V}_e\cdot\nabla){\bf V}_e+ \nabla p_e$    
$\displaystyle + [\delta^{\,-1}]\,e\, n\, ({\bf E} + {\bf V}_e\times {\bf B})$ $\displaystyle = [\zeta]\,{\bf F}_U + {\bf F}_T,$ (4.206)
$\displaystyle \frac{3}{2}\frac{\partial p_e}{\partial t} + \frac{3}{2} \,({\bf V}_e\cdot\nabla)\, p_e + \frac{5}{2}\,p_e\,\nabla\cdot{\bf V}_e$ $\displaystyle = -[\delta^{\,-1}\,\zeta\,\mu^2]\,W_i,$ (4.207)

and

$\displaystyle \frac{\partial n}{\partial t} + \nabla\cdot(n\,{\bf V}_i)$ $\displaystyle =0,$ (4.208)
$\displaystyle m_i \,n\,\frac{\partial {\bf V}_i}{\partial t} + m_i\, n\,({\bf V}_i\cdot\nabla) {\bf V}_i + \nabla p_i$    
$\displaystyle - [\delta^{\,-1}]\, e\, n\, ({\bf E} + {\bf V}_i\times {\bf B})$ $\displaystyle =- [\zeta]\,{\bf F}_U -{\bf F}_T,$ (4.209)
$\displaystyle \frac{3}{2}\frac{\partial p_i}{\partial t} + \frac{3}{2} \,({\bf V}_i\cdot\nabla) \,p_i + \frac{5}{2}\,p_i\,\nabla\cdot{\bf V}_i$ $\displaystyle =[\delta^{\,-1}\,\zeta\,\mu^2]\, W_i.$ (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

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

and

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

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

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

$\displaystyle {\bf E} + {\bf V}\times {\bf B} \simeq {\bf0}.$ (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:

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

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

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

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

$\displaystyle W_i \simeq 0,$ (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.


next up previous
Next: Drift Equations Up: Plasma Fluid Theory Previous: Cold-Plasma Equations
Richard Fitzpatrick 2016-01-23