Convective Oscillations of Cosmic Rays

Turbulent motion of the background plasma can introduce fluctuations in the spatial distribution of cosmic rays (CRs). For an ensemble-averaged Gaussian CR distribution with a width $\mathrm{\Lambda }(t,p)$ at time $t$ and momentum $p$, it is found that $P_{\text{s}}(t,k,p)$ and $P_{\text{d}}(t,k,p)$, the fluctuation power spectra induced by incompressible and acoustic turbulent convection, respectively, have power-law asymptotics in wavenumber $k$: $P_{\text{s}}(t,k,p)\propto k^4$ and $P_{\text{d}}(t,k,p)\propto k^2$ for $k\ll 1\mathrm{/}\mathrm{\Lambda }(t,p)$; $P_{\text{s}}(t,k,p)\propto k^{{\gamma }_{\text{s}}-4}$ for $k\gg 1\mathrm{/}{L}_{\text{s}}$, and $P_{\text{d}}(t,k,p)\propto k^{{\gamma }_{\text{d}}-2}$ for $k\gg 1\mathrm{/}{L}_{\text{d}}$, where ${\gamma }_{\text{s}}$ and ${\gamma }_{\text{d}}$ are the spectral indices of the velocity power spectra of the incompressible and acoustic turbulence below their outer scales $L_{\text{s}}$ and $L_{\text{d}}$, respectively. These results have implications for probing interstellar turbulence via future high-resolution images of pulsar gamma-ray halos.