Cylindrical Tearing Mode Equation

The magnetic structure of a tearing perturbation is determined by the so-called cylindrical tearing mode equation, (3.35), which can be written in the form [4,14]

$$\frac{\partial^2\delta\psi}{\partial r^2} + \frac{1}{r}\,\frac{\p... ...tial r}-\frac{m^2}{r^2}\,\delta\psi - \frac{J_z'\,\delta\psi}{r\,(1/q-n/m)}= 0,$$ (3.60)

where

$$J_z(r)= \frac{R_0\,\mu_0\,j_z(r)}{B_z}$$ (3.61)

is a dimensionless measure of the toroidal current density profile. Note that Equation (3.60) is singular at the so-called rational magnetic flux-surface, radius $r=r_s$, at which

$$q(r_s) = \frac{m}{n}.$$ (3.62)

At the rational surface, ${\bf k}\cdot{\bf B}=0$, where ${\bf B}$ is the equilibrium magnetic field, and ${\bf k} = (k_r,\,m/r,\,-n/R_0)$ the wavevector of the tearing perturbation.