Fluid Continuity

According to the equations of marginally-stable ideal-MHD, the electron number density continuity equation takes the form

$\displaystyle \frac{\partial n_e}{\partial t} + \nabla\cdot\left(n_e\,{\bf V} + \frac{{\bf b} \times \nabla p_i}{e\,B_z}\right)= 0,$ (3.45)

where use has been made of Equations (2.139), (2.380), and (3.26). Here, ${\bf b} \equiv {\bf B}/B\simeq (\epsilon/q)\,{\bf e}_\theta + {\bf e}_z$. It follows that $\nabla\cdot{\bf b}=0$ and $r\,\nabla\times {\bf b}\sim{\cal O}(\epsilon/q)$. Thus,

$\displaystyle \nabla\cdot\left( \frac{{\bf b} \times \nabla p_i}{e\,B_z}\right)\simeq 0,$ (3.46)

where use has been made of Equation (3.29).

Now, the equilibrium plasma flow is written

$\displaystyle {\bf V} = V_\theta(r)\,{\bf e}_\theta + V_z(r)\,{\bf e}_z.$ (3.47)

It follows that

$\displaystyle \nabla\cdot {\bf V} = 0.$ (3.48)

The linearized form of Equation (3.45) is

$\displaystyle \frac{\partial\delta n_e}{\partial t} + {\bf V}\cdot\nabla\delta n_e+ \delta {\bf V}\cdot\nabla n_e+ n_e\,\nabla\cdot\delta{\bf V}=0,$ (3.49)

where $\delta{\bf V}$ is the perturbed plasma velocity, and use has been made of Equations (3.46) and (3.48). Let us write

$\displaystyle \delta {\bf V} =\nabla\phi\times {\bf e}_z + \delta v\,{\bf B}.$ (3.50)

It follows that

$\displaystyle \nabla\cdot \delta {\bf V} = {\bf B}\cdot \nabla \delta v = {\rm i}\,F\,\delta v,$ (3.51)

where use has been made of Equations (3.8) and (3.19). Hence, writing

$\displaystyle \frac{\partial}{\partial t}$ $\displaystyle = -{\rm i}\,\omega,$ (3.52)
$\displaystyle {\mit\Omega}(r)$ $\displaystyle = \frac{m}{r}\,V_\theta - \frac{n}{R_0}\,V_\phi,$ (3.53)

Equation (3.49) reduces to

$\displaystyle \delta v = -\frac{m}{r\,F}\left[\phi + \left(\frac{\omega-{\mit\Omega}}{F}\right)\delta\psi\right]\frac{n_e'}{n_e},$ (3.54)

where use has been made of Equation (3.42).