Instabilities in a neutron star can generate Alfven 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 Alfven 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.
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