Electromagnetic Torques

The flux-surface integrated poloidal and toroidal electromagnetic torque densities acting on the plasma can be written

$$T_\theta(r,t) = \oint\oint \,(\delta {\bf j}\times \delta{\bf B})_\theta\,{\cal J}\,d\theta\,d\varphi,$$ (14.61)

$$T_\varphi(r,t) = \oint\oint \,(\delta {\bf j}\times \delta{\bf B})_\varphi\,{\cal J}\,d\theta\,d\varphi,$$ (14.62)

respectively. Equations (14.3), (14.8), (14.9), (14.44), and (14.58)–(14.60) yield

$$T_\theta = - \frac{2\pi^2\,R_0}{\mu_0}\sum_{k=1,K}{\rm Im}\left([\chi_k]_{r_{k-}}^{r_{k+}}\,\psi_k^{\,\ast}\right)\,\delta(r-r_k),$$ (14.63)

$$T_\varphi = \frac{2\pi^2\,R_0}{\mu_0}\sum_{k=1,K}{\rm Im}\left(\frac{n}{m_k}\,[\chi_k]_{r_{k-}}^{r_{k+}}\,\psi_k^{\,\ast}\right)\,\delta(r-r_k).$$ (14.64)

Let

$${\mit\Psi}_k(t) =\frac{\psi_k(r_k,t)}{m_k},$$ (14.65)

$${\mit\Delta\Psi}_k(t) = [\chi_k]_{r_{k-}}^{r_{k+}}.$$ (14.66)

Note that these quantities are, in general, complex. It follows that [11,14,16]

$$T_\theta(r,t) = \sum_{k=1,K} T_{\theta\, k}(t)\,\delta(r-r_k),$$ (14.67)

$$T_\varphi(r,t) =\sum_{k=1,K} T_{\varphi \,k}(t)\,\delta(r-r_k),$$ (14.68)

where

$$T_{\theta \,k} = - \frac{2\pi^2\,R_0\,m_k}{\mu_0}\,{\rm Im}({\mit\Delta\Psi}_k\,{\mit\Psi}_k^{\,\ast}),$$ (14.69)

$$T_{\varphi\,k} = \frac{2\pi^2\,R_0\,n}{\mu_0}\,{\rm Im}({\mit\Delta\Psi}_k\,{\mit\Psi}_k^{\,\ast}).$$ (14.70)

It can be seen, by analogy with the analysis of Sections 3.3, 3.8, and 3.13, that ${\mit\Psi}_k$ is the reconnected magnetic flux at the $k$th rational surface, whereas ${\mit\Delta\Psi}_k$ parameterizes the amplitude and phase of the current sheet flowing (parallel to the equilibrium magnetic field) at the same surface.