Plasma Heating by Alfvén-Wave Phase Mixing

We propose the heating of collisionless plasma by utilizing a spatial phase mixing by shear Alfvén wave resonance and discuss the application to toroidal plasma. The phase mixing takes place as a result of the nonuniform Alfvén speed. The approximate heating rate per cycle of the wave frequency is given by $\frac{\left(\frac{b_0^{}{}_{}{}^2}{{\mu }_0}\right)\kappa }{k_{\perp }}$ where $\kappa$ is the measure of the nonuniformity, $k_{\perp }$ is the wave number perpendicular to the direction of the magnetic field and the nonuniformity, and $b_0$ is the flux density of the applied-wave magnetic field.