Rutherford Island Width Evolution Equation

Suppose that the rigid wall surrounding the plasma is perfectly conducting (i.e., $\tau_w\rightarrow\infty$, where $\tau_w$ is the time constant of the wall—see Section 3.10), and that there is no current flowing through the external magnetic field-coil [i.e., $\hat{I}_c=0$, where $\hat{I}_c$ is the normalized coil current—see Equation (3.194)]. In this case, Equations (3.187), (3.188), and (7.5), yield

$$\frac{{\mit\Delta}\hat{\mit\Psi}_s}{\hat{\mit\Psi}_s} ={\mit\Delta}_{pw},$$ (9.2)

where ${\mit\Delta}_{pw}=E_{ss}$ is the (real dimensionless) perfect-wall tearing stability index, $\hat{\mit\Psi}_s$ the normalized reconnected helical magnetic flux at the rational surface [see Equation (3.184)], and ${\mit\Delta}\hat{\mit\Psi}_s$ the normalized helical sheet current density at the rational surface [see Equation (3.183)]. It is clear from the previous equation that

$${\rm Im}\!\left(\frac{{\mit\Delta}\hat{\mit\Psi}_s}{\hat{\mit\Psi}_s}\right) = 0,$$ (9.3)

which implies that zero electromagnetic torque is exerted at the rational surface. (See Section 3.13.)

Equation (9.2) yields

$${\rm Re}\!\left(\frac{{\mit\Delta}\hat{\mit\Psi}_s}{\hat{\mit\Psi}_s}\right) = {\mit\Delta}_{pw},$$ (9.4)

which can be combined with Equations (8.1), (8.108), and (9.3) to give the so-called Rutherford island width evolution equation [13]:

$$I_1\,\tau_R\,\frac{d}{dt}\!\left(\frac{W}{r_s}\right)= {\mit\Delta}_{pw}.$$ (9.5)

Here, $W$ is the full radial width of the magnetic island chain that develops at the rational surface, $r_s$ the minor radius of the surface, and $I_1=0.8227$ [see Equation (8.124)]. We conclude that, in the nonlinear regime, the width of the magnetic island chain grows algebraically in time on the resistive diffusion time, $\tau _R$. Obviously, this is a much slower time evolution than that predicted in the linear regime.