Three-wave coupling observed between a shear Alfv\'en wave and a kink-unstable magnetic flux rope*

Results from a laboratory experiment are presented in which,

a shear Alfv\'{e}n wave is launched using an antenna in a current-carrying plasma

that is either stable or unstable to the kink oscillation. 

As the plasma is driven kink unstable, the power spectrum of the Alfv\'en wave evolves from a single peak to a peak with multiple sidebands separated by integer multiples of the kink frequency.

The main sidebands (one on either side of the launched wave peak in the power spectrum) are analyzed using azimuthal wavenumber matching; perpendicular;  parallel wavenumber decomposition; and bispectral analysis.

The dispersion relation and three-wave matching conditions are satisfied given that each sideband is a propagating Alfv\'en wave, resulting from the interaction of the pump Alfv\'en wave and the co-propagating component of a half-wavelength, standing kink mode. The interaction is shown to generate smaller perpendicular wavelength Alfv\'en waves that drive energy transport to scales that will approach the dissipation scale of $k_{\perp}\rho_{s}=1$, with $k_{\perp}$ the perpendicular wavenumber and $\rho_{s}$ the ion gyroradius at the electron temperature.