Natural Velocity of Magnetic Islands

The phase velocity of magnetic islands is calculated in the semicollisional regime with cold ions. Two solution branches arise, corresponding to islands propagating with the ions and with the electrons. For the ion branch the phase velocity and the polarization current are small. For the electron branch, the phase velocity depends on the ratio of $W$, the half-width of the island, and ${\rho }_s$, the Larmor radius calculated with the electron temperature. For $W\gg {\rho }_s$ the phase velocity is larger than the electron drift velocity and the polarization current is destabilizing. For $W\ll {\rho }_s$, the situation is reversed provided that the density and temperature gradients point in the same direction.

Figure 1

Average along the streamlines of the azimuthal plasma velocity ${\partial }_x\phi$ for various values of the applied torque. The parameters are $w=0.5$, ${\eta }_e=2$, $D=\mu =0.6{\kappa }_{\perp }$, and $C=1.65{\kappa }_{\perp }$ resulting in a natural propagation velocity $v_{\mathrm{phase}}=-0.78$. Maintaining steady state requires ${\widehat{\Delta }}^{\prime }=0.21$, indicating that the polarization current is stabilizing.

Figure 2

Offset in the velocity as a function of the flow forcing for $D=\mu =0.6{\kappa }_{\perp }$. $C=1.65{\kappa }_{\perp }$ for the points with $w\le 1$, while $C\ll {\kappa }_{\perp }$ for the points with $w>1$. The solid and dashed lines represent the thin and intermediate width asymptotic solutions, and the dotted lines are quadratic fits.