Correlation between changes in the polytropic index and changes in incompressible (left) and compressible (right) turbulence cascade rates across shocks. The scatter points show the downstream-to-upstream ratios of the polytropic index and cascade rates for shocks in Group 1 (black), Group 2 (red), and neither group (gray). The dashed lines indicate linear fits, with the corresponding Pearson correlation coefficients shown in the figure.