Linearized Reduced Drift-MHD Equations

Let us now take the tearing mode perturbation into account. In accordance with Equations (5.22)–(5.25), and (5.30)–(5.31), we can write

$$\psi(\hat{x},\zeta,\hat{t}) = \frac{\hat{x}^{\,2} }{2\,\hat{L}_s}+ \tilde{\psi}(\hat{x})\,{\rm e}^{\,{\rm i}\,(\zeta-\hat{\omega}\,\hat{t})},$$ (5.33)

$$N(\hat{x},\zeta,\hat{t}) = -\hat{V}_\ast\,\hat{x} +\left(\frac{1+\tau}{\tau}\right)\tilde{N}(\hat{x})\,{\rm e}^{\,{\rm i}\,(\zeta-\hat{\omega}\,\hat{t})},$$ (5.34)

$$\phi(\hat{x},\zeta,\hat{t}) =-\hat{V}_E\,\hat{x}+ \skew{3}\tilde{\phi}(\hat{x})\,{\rm e}^{\,{\rm i}\,(\zeta-\hat{\omega}\,\hat{t})},$$ (5.35)

$$V(\hat{x},\zeta,\hat{t}) = \hat{V}_\parallel + \left(\frac{1+\tau}{\tau}\right)\tilde{V}(\hat{x})\,{\rm e}^{\,{\rm i}\,(\zeta-\hat{\omega}\,\hat{t})},$$ (5.36)

$$J(\hat{x},\zeta,\hat{t}) =-\left(\frac{2}{s_s}-1\right)\frac{1}{\hat{L}_s}+ \hat{\nabla}^2\tilde{\psi}(\hat{x})\,{\rm e}^{\,{\rm i}\,(\zeta-\hat{\omega}\,\hat{t})},$$ (5.37)

$$U(\hat{x},\zeta,\hat{t}) = \hat{\nabla}^2\skew{3}\tilde{\phi}(\hat{x})\,{\rm e}^{\,{\rm i}\,(\zeta-\hat{\omega}\,\hat{t})},$$ (5.38)

where $\hat{\omega}=r_s\,\omega/V_A$, and $\omega$ is the rotation frequency of the tearing perturbation in the laboratory frame. Here, $\tilde{A}$ denotes a perturbed quantity. Substituting Equations (5.33)–(5.38) into the reduced drift-MHD model, (5.8)–(5.13), and only retaining terms that are first order in perturbed quantities, we obtain the following set of linear equations:

$$-{\rm i}\,(\omega-\omega_E -\omega_{\ast\,e})\,\tau_H\,\tilde{\psi} = -{\rm i} \,\hat{x}\,(\skew{3}\tilde{\phi}-\tilde{N}) + S^{-1}\,\hat{\nabla}^2\tilde{\psi},$$ (5.39)

$$-{\rm i}\,(\omega-\omega_E)\,\tau_H\,\tilde{N} = -{\rm i}\,\omega_{\ast\,e}\,\tau_H\,\skew{3}\tilde{\phi}-{\rm i... ...ac{\tau}{1+\tau}\right)\hat{d}_\beta^{\,2}\,\hat{x}\,\hat{\nabla}^2\tilde{\psi}$$ (5.40)

$$\phantom{=} +S^{-1}\,P_\parallel\left(\omega_{\ast\,e}\,\tau_H\,\... ...-\hat{x}^2\,\tilde{N}\right) + S^{-1}\,P_\perp\,\hat{\nabla}_\perp^2 \tilde{N},$$

$$-{\rm i}\,(\omega-\omega_E-\omega_{\ast\,i})\,\tau_H\,\hat{\nabla}^2\skew{3}\tilde{\phi} = -{\rm i}\,\hat{x}\,\hat{\nabla}^2\tilde{\psi}+S^{-1}\,P_\varphi\,\hat{\nabla}^4\!\left(\skew{3}\tilde{\phi} + \frac{\tilde{N}}{\tau}\right),$$ (5.41)

$$-{\rm i}\,(\omega-\omega_E)\,\tau_H\,\tilde{V} = {\rm i}\,\omega_{\ast\,e}\,\tau_H\,\tilde{\psi} - {\rm i}\,\hat{x}\,\tilde{N} + S^{-1}\,P_\varphi\,\hat{\nabla}^2\tilde{V}.$$ (5.42)

Here,

$$\tau_H = \frac{L_s}{m\,V_A}$$ (5.43)

is the hydromagnetic time,

$$\omega_E = \frac{m}{r_s}\,V_E(r_s)$$ (5.44)

the E-cross-B frequency,

$$\omega_{\ast\,e} = \left(\frac{\tau}{1+\tau}\right)\omega_\ast,$$ (5.45)

the electron diamagnetic frequency,

$$\omega_{\ast\,i} = -\left(\frac{1}{1+\tau}\right)\omega_\ast,$$ (5.46)

the ion diamagnetic frequency,

$$\omega_\ast =-\frac{m}{r_s}\,V_\ast(r_s)= -\frac{m}{r_s}\frac{1}{e\,n_0\,B_z}\left.\frac{dp}{dr}\right\vert _{r_s},$$ (5.47)

the (total) diamagnetic frequency,

$$S=\frac{\tau_R}{\tau_H}$$ (5.48)

the Lundquist number [note that this is a slightly different definition to that given in Equation (1.84)],

$$\tau_R = \mu_0\,r_s^{\,2}\,\sigma_\parallel(r_s)$$ (5.49)

the resistive diffusion time [note that this is a slightly different definition to that given in Equation (1.83)],

$$\tau_\varphi = \frac{r_s^{\,2}}{{\mit\Xi}_{\perp\,i}(r_s)}$$ (5.50)

the toroidal momentum confinement time,

$$\tau_\parallel = r_s^{\,2}\left/\left\{\frac{2}{3}\,(1-c_\beta^2)... ...ft(\frac{\eta_i}{1+\eta_i}\right)\chi_{\parallel\,i}(r_s)\right]\right\}\right.$$ (5.51)

the effective parallel energy equilibration time, and

$$\tau_\perp =r_s^{\,2}\left/\left\{\frac{2}{3}\,(1-c_\beta^2)\left... ...)\left(\frac{\eta_i}{1+\eta_i}\right)\chi_{\perp\,i}(r_s)\right]\right\}\right.$$ (5.52)

the effective energy confinement time. Furthermore,

$$P_\varphi = \frac{\tau_R}{\tau_\varphi},$$ (5.53)

$$P_\perp = \frac{\tau_R}{\tau_\perp},$$ (5.54)

$$P_\parallel =(n\,\epsilon_s\,s_s)^2\frac{\tau_R}{\tau_\parallel}.$$ (5.55)