Fluid Continuity

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

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

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

where use has been made of Equation (3.29).

Now, the equilibrium plasma flow is written

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

It follows that

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

The linearized form of Equation (3.45) is

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

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

It follows that

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

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

$$\frac{\partial}{\partial t} = -{\rm i}\,\omega,$$ (3.52)

$${\mit\Omega}(r) = \frac{m}{r}\,V_\theta - \frac{n}{R_0}\,V_\phi,$$ (3.53)

Equation (3.49) reduces to

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

where use has been made of Equation (3.42).