Perturbed Current Density

Let $\delta {\bf j}$ be the perturbed current density associated with the tearing mode. We can write

$\displaystyle {\cal J}\,\mu_0\,\delta j^{\,r} = \frac{\partial\,\delta B_\varphi}{\partial\theta} -\frac{\partial \,\delta B_\theta}{\partial\varphi},$ (14.55)
$\displaystyle {\cal J}\,\mu_0\,\delta j^{\,\theta} = \frac{\partial\,\delta B_r}{\partial\varphi} -\frac{\partial \,\delta B_\varphi}{\partial r},$ (14.56)
$\displaystyle {\cal J}\,\mu_0\,\delta j^{\,\varphi} = \frac{\partial\,\delta B_\theta}{\partial r} -\frac{\partial \,\delta B_r}{\partial\theta}.$ (14.57)

Normally, according to our previous assumptions, all three contravariant components of $\delta {\bf j}$ are zero. Consider, however, the behavior in the vicinity of the $k$th rational surface, $r=r_k$, at which $n\,q(r_k)=m_k$. (See Section 3.7.) In general, $\psi_k$, $\psi_{j\neq k}$, and $\chi_{j\neq k}$ are continuous across the surface, whereas $\chi_k$ is discontinuous [11,14]. Hence, we deduce that

$\displaystyle {\cal J}\,\mu_0\,\delta j^{\,r}(r,\theta,\varphi,t)$ $\displaystyle =0,$ (14.58)
$\displaystyle {\cal J}\,\mu_0\,\delta j^{\,\theta}(r,\theta,\varphi,t)$ $\displaystyle =-\sum_{k=1,K}\frac{n}{m_k}\,[\chi_k]_{r_{k-}}^{r_{k+}}\,\delta(r-r_k)\,{\rm e}^{\,{\rm i}\,(m_k\,\theta-n\,\varphi)},$ (14.59)
$\displaystyle {\cal J}\,\mu_0\,\delta j^{\,\varphi}(r,\theta,\varphi,t)$ $\displaystyle =-\sum_{k=1,K}[\chi_k]_{r_{k-}}^{r_{k+}}\,\delta(r-r_k)\,{\rm e}^{\,{\rm i}\,(m_k\,\theta-n\,\varphi)},$ (14.60)

where use has been made of Equations (14.45) and (14.54). Here, it is assumed that there are $K$ rational surfaces in the plasma, numbered sequentially from 1 to $K$, in the order of the innermost to the outermost. It is easily demonstrated from Equations (14.7)–(14.9), (14.19)–(14.22), and (14.58)–(14.60) that $\delta{\bf j}\times {\bf B}=0$ at a given rational surface. Thus, we conclude that a current sheet forms at each rational surface in the plasma. Moreover, each sheet is made up of current filaments that flow parallel to the local equilibrium magnetic field.