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Most of the observed features of radio {\it gischt} relics such as spectralsteepening across the relic width and power-law-like integrated spectrum can beadequately explained by diffusive shock acceleration (DSA) model, in whichrelativistic electrons are (re-)accelerated at shock waves induced in theintracluster medium. However, the steep spectral curvature in the integratedspectrum above $\sim 2$ GHz detected in some radio relics such as the Sausagerelic in cluster CIZA J2242.8+5301 may not be interpreted by simple radiativecooling of postshock electrons. In order to understand such steepening, we hereconsider a model in which a spherical shock sweeps through and then exits outof a finite-size cloud with fossil relativistic electrons. The ensuingintegrated radio spectrum is expected to steepen much more than predicted foraging postshock electrons, since the re-acceleration stops after thecloud-crossing time. Using DSA simulations that are intended to reproduce radioobservations of the Sausage relic, we show that both the integrated radiospectrum and the surface brightness profile can be fitted reasonably well, if ashock of speed, $u_s \sim 2.5-2.8\times 10^3 \kms$, and sonic Mach number, $M_s\sim 2.7-3.0$, traverses a fossil cloud for $\sim 45$ Myr and the postshockelectrons cool further for another $\sim 10$ Myr. This attempt illustrates thatsteep curved spectra of some radio gischt relics could be modeled by adjustingthe shape of the fossil electron spectrum and adopting the specificconfiguration of the fossil cloud.

Re-acceleration Model for Radio Relics with Spectral Curvature