Cosmic Ray Self-Confinement, Escape and Transport

Propagation of cosmic rays (CR) in a self-confinement regime is discussed. A self-similar solution for a CR-cloud expansion along the magnetic field strongly deviates from test-particle results. The normalized CR partial pressure is close to $\mathcal{P}\left(p,z,t\right)=2\left[\left|z\right|^{5/3}+z_{{\rm dif}}^{5/3}\left(p,t\right)\right]^{-3/5}\exp\left[-z^{2}/4D_{B}\left(p\right)t\right]$, where $p$ is the momentum of CR and $z$ is directed along the field. The core of the cloud expands as $z_{dif}\propto\sqrt{D_{ NL}\left(p\right)t}$ and decays in time as $\mathcal{P}\propto2z_{dif}^{-1}\left(t\right)$. The diffusion coefficient $D_{NL}$ is strongly suppressed compared to its background value $D_{{\rm B}}$: $D_{{\rm NL}}\sim D_{{\rm B}}\exp\left(-\Pi\right)\ll D_{{\rm B}}$ for sufficiently high field-line-integrated CR partial pressure, $\Pi$. When $\Pi\gg1$, the CRs drive Alfven waves efficiently enough to build a $\it{transport~barrier}$ ($\mathcal{P}\approx2/\left|z\right|$ -``pedestal'') that strongly reduces the leakage. The solution has a spectral break in momentum spectrum at $p=p_{{\rm br}}$, where $p_{{\rm br}}$ satisfies the following equation $D_{{\rm NL}}\left(p_{{\rm br}}\right)\simeq z^{2}/t$. Magnetic focusing effects in CR transport are briefly discussed.