Continuous Spectrum of Shear Alfvén Waves within Magnetic Islands

The radial structure of the shear Alfvén wave continuous spectrum is calculated inside the separatrix of a magnetic island. We find that, within a magnetic island, there is a continuous spectrum very similar to that of tokamak plasmas, where a generalized safety factor $q$ can be defined and a wide frequency gap is formed, analogous to the ellipticity induced Alfvén eigenmode gap in tokamaks. This is due to the strong eccentricity of the island cross section. In this gap, a magnetic-island induced Alfvén eigenmode (MIAE) can exist as a bound state, essentially free of continuum damping, which can be resonantly excited by energetic particles and interact nonlinearly with the magnetic island. Because of the frequency dependence of the shear Alfvén wave continuum on the magnetic-island size, the possibility of utilizing MIAE frequency scalings as a novel magnetic-island diagnostic is also discussed.

Figure 1

The safety factor inside a magnetic island, normalized to $q(0)=1/(\sqrt{M}{n}_{\mathrm{isl}})=\gamma /(|s|\sqrt{1-e})$. For typical magnetic islands, $1-e<{10}^{-2}$ and we have $q(0)>10$.

Figure 2

Continuous spectrum $\Omega (x)$ for the small eccentricity case, $e\ll 1$, corresponding to $M\simeq 1$, plotted versus the radial position inside the island. Typical values of the equilibrium parameters have been chosen and $n_{\mathrm{isl}}=1$. The $O$ point is at $x=0$ and the separatrix at $x=1$. Modes with $n=10$ are considered. The MIAE gap is found at frequencies $\Omega \sim \sqrt{M}{n}_{\mathrm{isl}}$.

Figure 3

Continuous spectrum $\Omega (x)$ for $n=1$ and a typical size magnetic island: $M={10}^{-2}$, corresponding to $e\simeq 0.99$. In this case, the structure of the continuous spectrum is the same as for $M=1$, but the MIAE gap is much wider. This implies that a wide range of frequencies exists, where modes contained within the magnetic island are not affected by continuum damping.