Instabilities in a neutron star can generate Alfv\'en waves in its magnetosphere. Propagation along the curved magnetic field lines strongly shears the wave, boosting its electric current $j_{ \rm A}$. We derive an analytic expression for the evolution of the wave vector $\boldsymbol{ k}$ and the growth of $j_{ \rm A}$. In the strongly sheared regime, $j_{ \rm A}$ may exceed the maximum current $j_{ 0}$ that can be supported by the background $e^{ \pm}$ plasma. We investigate these "charge-starved" waves, first using a simplified two-fluid analytic model, then with first-principles kinetic simulations. We find that the Alfv\'en wave continues to propagate successfully even when $\kappa \equiv j_{ \rm A}/j_{ 0} \gg 1$. It sustains $j_{ \rm A}$ by compressing and advecting the plasma along the magnetic field lines with particle Lorentz factors $\sim \kappa^{ 1/2}$. The simulations show how plasma instabilities lead to gradual dissipation of the wave energy, giving a dissipation power $L_{ \rm diss}\sim 10^{ 35}(\kappa/100)^{ 1/2} (B_w/10^{ 11}\,{ \rm G})\,\mathrm{ erg/s}$, where $B_w$ is the wave amplitude. Our results imply that dissipation due to charge starvation is not sufficient to power observed fast radio bursts (FRBs), in contrast to recent proposals.
Categories: astro-ph.HE
