The intracluster medium (ICM) is expected to experience on average about three passages of weak shocks with low sonic Mach numbers, M . 3, during the formation of galaxy clusters. Both protons and electrons could be accelerated to become high energy cosmic rays (CRs) at such ICM shocks via diusive shock acceleration (DSA). We examine the eects of DSA by multiple shocks on the spectrum of accelerated CRs by including in situ injection/acceleration at each shock, followed by repeated re- acceleration at successive shocks in the test-particle regime. For simplicity, the accelerated particles are assumed to undergo adiabatic decompression without energy loss and escape from the system, before they encounter subsequent shocks. We show that in general the CR spectrum is attened by multiple shock passages, compared to a single episode of DSA, and that the acceleration eciency increases with successive shock passages. However, the decompression due to the expansion of shocks into the cluster outskirts may reduce the amplication and attening of the CR spectrum by multiple shock passages. The nal CR spectrum behind the last shock is determined by the accumulated eects of repeated re-acceleration by all previous shocks, but it is relatively insensitive to the ordering of the shock Mach numbers. Thus multiple passages of shocks may cause the slope of the CR spectrum to deviate from the canonical DSA power-law slope of the current shock.

Abstract
1. INTRODUCTION
2. DSA SPECTRUM BY MULTIPLE SHOCKS
    2.1. Prerequisites
    2.2. Injected Spectrum at Each Shock
    2.3. Re-acceleration by Subsequent Shocks
3. RESULTS
    3.1. DSA by Multiple Identical Shocks
    3.2. DSA by Multiple Shocks with Different M’s
    3.3. DSA Efficiency
4. SUMMARY
REFERENCES

• Hyesung Kang(Department of Earth Sciences, Pusan National University) Corresponding author