Tearing Mode Evolution Equations

Equations (3.90), (3.102), (3.108), (3.122)–(3.128), (3.178), (3.179), and (3.186) can be combined to give the following complete set of equations that determine the time evolution of the reconnected magnetic flux at the rational surface in the presence of a resistive vacuum vessel and an external magnetic field-coil:

$${\mit\Delta\hat{\Psi}}_s\equiv \tau_s\left(\frac{d\hat{\mit\Psi}_s}{dt} + {\rm i}\,\omega\,\hat{\mit\Psi}_s\right) = E_{ss}\,\hat{\mit\Psi}_s + E_{sw}\,\hat{\mit\Psi}_w,$$ (3.187)

$${\mit\Delta\hat{\Psi}}_w \equiv \tau_w\,\frac{d\hat{\mit\Psi}_w}{dt} = E_{ws}\,\hat{\mit\Psi}_s +\tilde{E}_{ww}\,\hat{\mit\Psi}_w +\hat{I}_c,$$ (3.188)

$$\omega(t) = \omega_{0} - \sum_{p=1,\infty}\left[\alpha_p(t)+\beta_p(t)\right],$$ (3.189)

$$\frac{d\alpha_p}{dt} + \left(\frac{1}{\tau_{\theta}}+\frac{j_{1p}^{\,2}}{\tau_{M}}\right)\alpha_p = \frac{m^{2}\,[J_1(j_{1p}\,r_s/a)]^{\,2}}{\tau_A^{\,2}\,\epsilon... ..._{1p})]^{\,2}} \,{\rm Im}({\mit\Delta\hat{\Psi}}_s\,\hat{\mit\Psi}_s^{\,\ast}),$$ (3.190)

$$\frac{d\beta_p}{dt} + \frac{j_{0p}^{\,2}}{\tau_{M}}\,\beta_p = \frac{n^{2}\,[J_0(j_{0p}\,r_s/a)]^{\,2}}{\tau_A^{\,2}\,[J_1(j_{0p})]^{\,2}}\, {\rm Im}({\mit\Delta\hat{\Psi}}_s\,\hat{\mit\Psi}_s^{\,\ast}).$$ (3.191)

Here,

$$\hat{\mit\Psi}_w = \frac{{\mit\Psi}_w}{R_0\,B_z},$$ (3.192)

$${\mit\Delta\hat{\Psi}}_w =\frac{{\mit\Delta\Psi}_w}{R_0\,B_z},$$ (3.193)

$$\hat{I}_c(t) = \left(\frac{r_w}{r_c}\right)^m\frac{\mu_0\,I_c(t)}{R_0\,B_z},$$ (3.194)

$$\tilde{E}_{ww} =E_{ww} - \frac{E_{wc}\,E_{cw}}{E_{cc}}= E_{ww} + \frac{2m\,(r_w/r_c)^{2m}}{1-(r_w/r_c)^{2m}}.$$ (3.195)