Giant radio relics are the arc-shaped diffuse radio emission regions observed in the outskirts of some merging galaxy clusters. They are believed to trace shock-waves in the intra-cluster medium. Recent observations demonstrated that some prominent radio relics exhibit a steepening above 2 GHz in their radio spectrum. This challenges standard theoretical models because shock acceleration is expected to accelerate electrons to very high energies with a power-law distribution in momentum. In this work we attempt to reconcile these data with the shock-acceleration scenario. We propose that the spectral steepening may be caused by the highest energy electrons emitting preferentially in lower magnetic fields than the bulk of synchrotron bright electrons in relics. Here, we focus on a model with an increasing mag- netic field behind the shock front, which quickly saturates and then declines. We derive the time-evolution of cosmic-ray electron spectra in time variable magnetic fields and an expanding medium. We then apply the formalism on the large radio relic in the cluster CIZA J2242.8+5301 (the Sausage relic). We show that under favourable circumstances of magnetic field amplification downstream, our model can explain the observed radio spectrum, the brightness profile and the spectral index profile of the relic. A possible interpretation for the required amplification of the magnetic field downstream is a dynamo acting behind the shock with an injection scale of magnetic turbulence of about 10 kpc. Our models require injection efficiencies of CRe - which are in tension with simple diffusive shock acceleration from the thermal pool. We show that this problem can likely be alleviated considering pre-existing CRe.
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