Perturbed Magnetic Field

Let $\delta{\bf B}$ be the perturbed magnetic field associated with a tearing mode to which the plasma is subject. Now, $\nabla \cdot \delta{\bf B}=0$, so we obtain

$$\frac{\partial ({\cal J}\,\delta B^{\,r})}{\partial r}+ \frac{\pa... ...al\theta} +\frac{\partial({\cal J}\,\delta B^{\,\varphi})}{\partial\varphi} =0,$$ (14.34)

where use has been made of Equation (14.13).

It is easily demonstrated from Equations (14.1)–(14.5) that

$$\delta B_r = \left(\frac{1}{\vert\nabla r\vert^{2}}\right)\delta B^{\,r}-\le... ...\frac{\nabla r\cdot\nabla\theta}{\vert\nabla r\vert^{2}}\right)\delta B_\theta,$$ (14.35)

$$\delta B^{\,\theta} = \left(\frac{\nabla r\cdot\nabla\theta}{\vert\nabla r\vert^{2}}\... ...t(\frac{R_0^{\,2}}{r^{2}\,R^{2}\,\vert\nabla r\vert^{2}}\right)\delta B_\theta.$$ (14.36)

Suppose, for the moment, that the tearing perturbation has $m$ periods in the poloidal direction and $n$ periods in the toroidal direction. Let us adopt the simplifying approximation that the perturbed current density, $\delta {\bf j}$, is negligible in the regions lying between the various rational surfaces in the plasma [16]. Given that $\mu_0\,\delta{\bf j} = \nabla\times \delta{\bf B}$, it follows from Equations (14.13)–(14.15) that

$$\frac{\partial\,\delta B_\varphi}{\partial\theta} \simeq \frac{\partial \,\delta B_\theta}{\partial\varphi},$$ (14.37)

$$\frac{\partial\,\delta B_r}{\partial\varphi} \simeq\frac{\partial \,\delta B_\varphi}{\partial r},$$ (14.38)

$$\frac{\partial\,\delta B_\theta}{\partial r} \simeq \frac{\partial \,\delta B_r}{\partial \theta}.$$ (14.39)

Assuming that $\partial/\partial r\sim m/r$, the previous three equations imply that

$$r\,\delta B_r\sim \delta B_\theta\sim \frac{m}{n}\,\delta B_\varphi.$$ (14.40)

Hence, we deduce that

$$r\,\delta B^{\,r}\sim r^2\,\delta B^{\,\theta} \sim \frac{m}{n}\,R^2\,\delta B^{\,\varphi}.$$ (14.41)

Consequently, the final term on the left-hand side of Equation (14.34) is of order $(n/m)^2\,(r/R)^2$ smaller than the other two terms. Let us assume that this final term is negligible, as would be the case in a large aspect-ratio (i.e., $R_0\gg r_{100}$) torus. It follows that

$$r\,\frac{\partial}{\partial r}\left(\frac{r\,R^{2}\,\delta B^{\,r... ...}}\right) +\left(\frac{1}{\vert\nabla r\vert^{2}}\right)\delta B_\theta\right],$$ (14.42)

where use has been made of Equation (14.3), (14.35), and (14.36). Finally, Equations (14.35) and (14.39) yield

$$r\,\frac{\partial\,\delta B_\theta}{\partial r} \simeq \frac{\par... ...nabla r\cdot\nabla\theta}{\vert\nabla r\vert^{2}}\right)\delta B_\theta\right].$$ (14.43)

Let

$$\frac{r\,R^{2}\,\delta B^{\,r}(r,\theta,\varphi,t)}{R_0^{\,2}} ={\rm i} \sum_j \psi_j(r,t)\,{\rm e}^{\,{\rm i}\,(m_j\,\theta-n\,\varphi)},$$ (14.44)

$$\delta B_\theta (r,\theta,\varphi,t) =-\sum_j \chi_j(r,t)\,{\rm e}^{\,{\rm i}\,(m_j\,\theta-n\,\varphi)},$$ (14.45)

where the sum is over all relevant poloidal harmonics of the perturbed magnetic field. Here, we are now taking account of the fact that a tearing mode in a toroidal tokamak plasma possesses a unique toroidal mode number, but consists of many coupled poloidal harmonics with different poloidal mode numbers [5,6,11,14,22,31]. Operating on Equations (14.42) and (14.43) with $\oint(\cdots)\, {\rm e}^{-{\rm i}\,m_j\,\theta}\,d\theta/(2\pi)$, we obtain

$$r\,\frac{\partial\psi_j}{\partial r} \simeq m_j\sum_{j'}\left(-c_{jj'}\,\psi_{j'}+a_{jj'}\,\chi_{j'}\right),$$ (14.46)

$$r\,\frac{\partial\chi_j}{\partial r} \simeq m_j\sum_{j'}\left(-c_{jj'}\,\chi_{j'} +b_{jj'}\,\psi_{j'}\right),$$ (14.47)

where

$$a_{jj'}(r) = \oint \frac{1}{\vert\nabla r\vert^{2}}\, {\rm e}^{-{\rm i}\,(m_j-m_{j'})\,\theta} \,\frac{d\theta}{2\pi},$$ (14.48)

$$b_{jj'}(r) = \oint \frac{R_0^{\,2}}{R^{2}\,\vert\nabla r\vert^{2}}\, {\rm e}^{-{\rm i}\,(m_j-m_{j'})\,\theta} \,\frac{d\theta}{2\pi},$$ (14.49)

$$c_{jj'}(r) = \oint \frac{{\rm i}\,r\,\nabla r\cdot\nabla \theta}{\vert\nabla r\vert^{2}}\, {\rm e}^{-{\rm i}\,(m_j-m_{j'})\,\theta} \,\frac{d\theta}{2\pi}.$$ (14.50)

Incidentally, we can recover the approximate relations (14.46) and (14.47) from the completely general analysis of Reference [11] by neglecting the equilibrium plasma current, as well as by taking the limit that $(n/m)^2\,(r/R_0)^2\ll 1$. This procedure is roughly equivalent to neglecting the term involving $J_z'$ in the cylindrical tearing mode equation, (3.60). Hence, by analogy with this equation, we would expect our toroidal tearing mode to be classically stable (given that the drive for the classical tearing instability in the cylindrical tearing mode equation derives from the term involving $J_z'$). However, this does not preclude the possibility that our toroidal tearing mode could be unstable as a neoclassical tearing mode. (See Chapter 12.)

Finally, it is readily demonstrated that

$$\frac{r\,R^2\,\delta B^{\,\theta}(r,\theta,\varphi,t)}{R_0^{\,2}} = - \sum_j\frac{1}{m_j}\,\frac{\partial\psi_j}{\partial r}\,{\rm e}^{\,{\rm i}\,(m_j\,\theta-n\,\varphi)},$$ (14.51)

$$R^2\,\delta B^{\,\varphi}(r,\theta,\varphi,t) =n\sum_j\frac{\chi_j}{m_j}\,{\rm e}^{\,{\rm i}\,(m_j\,\theta-n\,\varphi)},$$ (14.52)

$$\delta B_r (r,\theta,\varphi,t) ={\rm i}\sum_j\frac{1}{m_j}\,\frac{\partial \chi_j}{\partial r}\,{\rm e}^{\,{\rm i}\,(m_j\,\theta-n\,\varphi)},$$ (14.53)

$$\delta B_\varphi(r,\theta,\varphi,t) =n\sum_j\frac{\chi_j}{m_j}\,{\rm e}^{\,{\rm i}\,(m_j\,\theta-n\,\varphi)}.$$ (14.54)