The outskirts of galaxy clusters are continuously disturbed by mergers and gas infall along filaments , which in turn induce turbulent flow motions and shock waves .
We examine the properties of shocks that form within r _ { 200 } in sample galaxy clusters from structure formation simulations .
While most of these shocks are weak and inefficient accelerators of cosmic rays ( CRs ) , there are a number of strong , energetic shocks which can produce large amounts of CR protons via diffusive shock acceleration .
We show that the energetic shocks reside mostly in the outskirts and a substantial fraction of them are induced by infall of the warm-hot intergalactic medium from filaments .
As a result , the radial profile of the CR pressure in the intracluster medium is expected to be broad , dropping off more slowly than that of the gas pressure , and might be even temporarily inverted , peaking in the outskirts .
The volume-integrated momentum spectrum of CR protons inside r _ { 200 } has the power-law slope of 4.25 - 4.5 , indicating that the average Mach number of the shocks of main CR production is in the range of \left < M _ { s } \right > _ { CR } \approx 3 - 4 .
We suggest that some radio relics with relatively flat radio spectrum could be explained by primary electrons accelerated by energetic infall shocks with M _ { s } \gtrsim 3 induced in the cluster outskirts .