Nonlinear Shock Acceleration beyond the Bohm Limit

We suggest a physical mechanism whereby the acceleration time of cosmic raysby shock waves can be significantly reduced. This creates the possibility ofparticle acceleration beyond the knee energy at ~10^15eV. The accelerationresults from a nonlinear modification of the flow ahead of the shock supportedby particles already accelerated to the knee momentum at p ~ p_*. The particlesgain energy by bouncing off converging magnetic irregularities frozen into theflow in the shock precursor and not so much by re-crossing the shock itself.The acceleration rate is thus determined by the gradient of the flow velocityand turns out to be formally independent of the particle mean free path(m.f.p.). The velocity gradient is, in turn, set by the knee-particles at p ~p_* as having the dominant contribution to the CR pressure. Since it isindependent of the m.f.p., the acceleration rate of particles above the kneedoes not decrease with energy, unlike in the linear acceleration regime. Thereason for the knee formation at p ~ p_* is that particles with $p > p_*$ areeffectively confined to the shock precursor only while they are within limiteddomains in the momentum space, while other particles fall into``loss-islands'', similar to the ``loss-cone'' of magnetic traps. Thisstructure of the momentum space is due to the character of the scatteringmagnetic irregularities. They are formed by a train of shock waves thatnaturally emerge from unstably growing and steepening magnetosonic waves or asa result of acoustic instability of the CR precursor. These losses steepen thespectrum above the knee, which also prevents the shock width from increasingwith the maximum particle energy.Comment: aastex, 13 eps figure