Solution of Plasma Angular Equations of Motion

In many situations of interest, the perturbed angular velocity profiles, ${\mit\Delta\Omega}_\theta(r,t)$ and ${\mit\Delta\Omega}_z(r,t)$, 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

$$4\pi^2\,R_0\left[\rho_s\,r^3\,\frac{\partial {\mit\Delta\Omega}_\... ...\!\left(r^3\,\frac{\partial{\mit\Delta\Omega}_\theta}{\partial r}\right)\right] =T_{\theta\,s}\,\delta(r-r_s),$$ (3.168)

$$4\pi^2\,R_0^{\,3}\left[\rho_s\,r\,\frac{\partial {\mit\Delta\Omeg... ...artial{\mit\Delta\Omega}_z}{\partial r}\right)\right] =T_{z\,s}\,\delta(r-r_s),$$ (3.169)

where $\rho_s=\rho(r_s)$, $\tau_{\theta}=\tau_{\theta\,i}(r_s)$, and ${\mit\Xi}_{\perp\,s}={\mit\Xi}_{\perp\,i}(r_s)$.

Let us write [3,5]

$${\mit\Delta\Omega}_\theta(r,t) = - \frac{1}{m}\sum_{p=1,\infty} \alpha_p(t)\,\frac{y_p(r)}{y_p(r_s)},$$ (3.170)

$${\mit\Delta\Omega}_z(r,t) = \frac{1}{n}\sum_{p=1,\infty} \beta_p(t)\,\frac{z_p(r)}{z_p(r_s)},$$ (3.171)

where

$$y_p(r) = \frac{J_1(j_{1p}\,r/a)}{r/a},$$ (3.172)

$$z_p(r) = J_0(j_{0p}\,r/a).$$ (3.173)

Here, $J_m(z)$ is a Bessel function, and $j_{mp}$ denotes its $p$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]

$$\frac{d}{dr}\!\left(r^{3}\,\frac{dy_p}{dr}\right) = -\frac{j_{1p}^{\,2}\,r^{3}\,y_p}{a^{2}},$$ (3.174)

$$\frac{d}{dr}\!\left(r\,\frac{dz_p}{dr}\right) = -\frac{j_{0p}^{\,2}\,r\,z_p}{a^{2}},$$ (3.175)

and

$$\int_0^a r^{3}\,y_p(r)\,y_q(r)\,dr = \frac{a^{4}}{2}\,[J_2(j_{1p})]^{\,2}\,\delta_{pq},$$ (3.176)

$$\int_0^a r\,z_p(r)\,z_q(r)\,dr = \frac{a^{2}}{2}\,[J_1(j_{0p})]^{\,2}\,\delta_{pq}.$$ (3.177)

Equations (3.139), (3.140), and (3.168)–(3.177) yield

$$\frac{d\alpha_p}{dt} + \left(\frac{1}{\tau_{\theta}}+\frac{j_{1p}^{\,2}}{\tau_{M}}\right)\alpha_p = \frac{m^{2}\,[J_1(j_{1p}\,r_s/a)]^{\,2}}{\tau_A^{\,2}\,\epsilon... ..._{1p})]^{\,2}}\, {\rm Im}({\mit\Delta\hat{\Psi}_s}\,\hat{\mit\Psi}_s^{\,\ast}),$$ (3.178)

$$\frac{d\beta_p}{dt} + \frac{j_{0p}^{\,2}}{\tau_{M}}\,\beta_p = \frac{n^{2}\,[J_0(j_{0p}\,r_s/a)]^{\,2}}{\tau_A^{\,2}\,[J_1(j_{0p})]^{\,2}}\, {\rm Im}({\mit\Delta\hat{\Psi}_s}\,\hat{\mit\Psi}_s^{\,\ast}).$$ (3.179)

Here,

$$\tau_{M} = \frac{a^{\,2}}{{\mit\Xi}_{\perp\,s}},$$ (3.180)

is the momentum confinement time,

$$\tau_A = \left(\frac{\mu_0\,\rho_s\,a^{\,2}}{B_z^{\,2}}\right)^{1/2}$$ (3.181)

is the Alfvén time,

$$\epsilon_s=\frac{r_s}{R_0}\ll 1$$ (3.182)

is the inverse aspect-ratio of the rational surface, and

$${\mit\Delta\hat{\Psi}}_s =\frac{{\mit\Delta\Psi}_s}{R_0\,B_z},$$ (3.183)

$$\hat{\mit\Psi}_s =\frac{{\mit\Psi}_s}{R_0\,B_z}.$$ (3.184)