Observational evidence of strongly amplified magnetic fields at interstellar supernova remnant (SNR) shocks has been proposed to result from downstream small scale dynamo; however, quantitative comparison with observations of X-ray time variability has been not extensively explored. We present an interpretation of the time variability of the X-ray flux reported from a 15 years multi-epoch observational campaign of the supernova remnant Cassiopeia A by Chandra. The increase of the [4.2-6] keV non-thermal flux up to 50% is shown quantitatively to trace the growth of the magnetic field due to vortical amplification mechanism downstream of inward shocks colliding with inner overdensities. The fast synchrotron cooling as compared with shock-acceleration time scale is consistent with the subsequent flux decrease.