According to the equations of marginally-stable ideal-MHD, the electron number density continuity equation
takes the form
![$\displaystyle \frac{\partial n_e}{\partial t} + \nabla\cdot\left(n_e\,{\bf V} + \frac{{\bf b} \times \nabla p_i}{e\,B_z}\right)= 0,$](img1501.png) |
(3.45) |
where use has been made of Equations (2.139), (2.380), and (3.26). Here,
. It follows that
and
.
Thus,
![$\displaystyle \nabla\cdot\left( \frac{{\bf b} \times \nabla p_i}{e\,B_z}\right)\simeq 0,$](img1505.png) |
(3.46) |
where use has been made of Equation (3.29).
Now, the equilibrium plasma flow is written
![$\displaystyle {\bf V} = V_\theta(r)\,{\bf e}_\theta + V_z(r)\,{\bf e}_z.$](img1506.png) |
(3.47) |
It follows that
![$\displaystyle \nabla\cdot {\bf V} = 0.$](img1507.png) |
(3.48) |
The linearized form of Equation (3.45) is
![$\displaystyle \frac{\partial\delta n_e}{\partial t} + {\bf V}\cdot\nabla\delta n_e+ \delta {\bf V}\cdot\nabla n_e+ n_e\,\nabla\cdot\delta{\bf V}=0,$](img1508.png) |
(3.49) |
where
is the perturbed plasma velocity, and use has been made of Equations (3.46) and (3.48). Let us write
![$\displaystyle \delta {\bf V} =\nabla\phi\times {\bf e}_z + \delta v\,{\bf B}.$](img1510.png) |
(3.50) |
It follows that
![$\displaystyle \nabla\cdot \delta {\bf V} = {\bf B}\cdot \nabla \delta v = {\rm i}\,F\,\delta v,$](img1511.png) |
(3.51) |
where use has been made of Equations (3.8) and (3.19).
Hence,
writing
Equation (3.49) reduces to
![$\displaystyle \delta v = -\frac{m}{r\,F}\left[\phi + \left(\frac{\omega-{\mit\Omega}}{F}\right)\delta\psi\right]\frac{n_e'}{n_e},$](img1516.png) |
(3.54) |
where use has been made of Equation (3.42).