Solution of Plasma Angular Equations of Motion
In many situations of interest, the perturbed angular velocity profiles,
and
, are
localized in the vicinity of the rational surface [4]. Hence, it is reasonable to express the perturbed angular
equations of motion, (3.161) and (3.165), in the simplified forms
where
,
, and
.
Let us write [3,5]
where
Here,
is a Bessel function, and
denotes its
th zero [1]. Note that
Equations (3.170)–(3.173) automatically satisfy the boundary conditions (3.166) and (3.167).
It is easily demonstrated that [13]
and
Equations (3.139), (3.140), and (3.168)–(3.177) yield
Here,
![$\displaystyle \tau_{M} = \frac{a^{\,2}}{{\mit\Xi}_{\perp\,s}},$](img1758.png) |
(3.180) |
is the momentum confinement time,
![$\displaystyle \tau_A = \left(\frac{\mu_0\,\rho_s\,a^{\,2}}{B_z^{\,2}}\right)^{1/2}$](img1759.png) |
(3.181) |
is the Alfvén time,
![$\displaystyle \epsilon_s=\frac{r_s}{R_0}\ll 1$](img1760.png) |
(3.182) |
is the inverse aspect-ratio of the rational surface, and