This study investigates the impact of magnetic turbulence on cosmic ray (CR) electrons through Fermi-II acceleration behind merger-driven shocks in the intracluster medium and examines how the ensuing synchrotron radio emission is influenced by the decay of magnetic energy through dissipation in the postshock region. We adopt simplified models for the momentum diffusion coefficient, specifically considering transit-time-damping resonance with fast-mode waves and gyroresonance with Alfvén waves. Utilizing analytic solutions derived from diffusive shock acceleration theory, at the shock location, we introduce a CR spectrum that is either shock-injected or shock-reaccelerated. We then track its temporal evolution along the Lagrangian fluid element in the time domain. The resulting CR spectra are mapped onto a spherical shell configuration to estimate the surface brightness profile of the model radio relics. Turbulent acceleration proves to be a significant factor in delaying the aging of postshock CR electrons, while decaying magnetic fields have marginal impacts due to the dominance of inverse Compton cooling over synchrotron cooling. However, the decay of magnetic fields substantially reduces synchrotron radiation. Consequently, the spatial distribution of the postshock magnetic fields affects the volume-integrated radio spectrum and its spectral index. We demonstrate that the Mach numbers estimated from the integrated spectral index tend to be higher than the actual shock Mach numbers, highlighting the necessity for accurate modeling of postshock magnetic turbulence in interpreting observations of radio relics.

Introduction
Physical Models and Numerical Method 
    Postshock Magnetic Turbulence
    DSA CR Spectrum at the Shock Position
    Model Parameters
     Evolution of CR Spectrum in the Postshock Flow
Results
    Postshock Cooling and TA of CR Electrons
    Volume Integrated Radio Emission
    Surface Brightness Profile of Model Radio Relics
Summary 
Acknowledgments
References

• Hyesung Kang(Department of Earth Sciences, Pusan National University, Busan 46241, Republic of Korea) Corresponding author