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tokens: a stronger field provides more magnetic energy to damp. We fit the slopes of this “big trend” for all the curves by appropriately removing a few points far away from this “big trend” and taking the average of these slopes. Interestingly, both viscous and magnetic dissipations obey an identical scaling law:The dashed lines in Figures 5 (b) and (c) denote this scaling law. This scaling law for the relation of the two dissipations and magnetic fields reveals that there exists self-similarity for the magnetic effect on dynamical tide. Figure 5. (a) Kinetic energy, (b) viscous dissipation, and (c) magnetic dissipation vs. S 2 at various ω. Pm = 1.0. The two dashed lines denote the scaling law ∝ S 2 . 3.2. Investigation of Pm In this subsection we investigate Pm . Similar to Figure 2 , Figure 6 shows viscous dissipation, magnetic dissipation, total dissipation, and the ratio of magnetic to viscous dissipations versus tidal frequency at various Pm with S 2 fixed to be 0.1, namely in a strong field regime. The first three subfigures show that for all the tidal frequencies, a larger Pm leads to higher viscous dissipation and lower magnetic dissipation but total dissipation is almost independent of Pm . This is reasonable because Pm is the ratio of viscosity to magnetic diffusivity (see Equation ( 9 )) but it cannot influence the total diffusivity. More interestingly, Figure 6 (d) shows that magnetic dissipation wins out over viscous dissipation for and this ratio is approximately equal to Pm −1 . In the astrophysical situation Pm is very small, and it can be inferred that magnetic dissipation dominates over viscous dissipation with a moderate or stronger magnetic field. Note again that with a very weak field, e.g., S ≤ 10 −4 , viscous dissipation wins out over magnetic dissipation even at a small Pm , e.g., (Lin Ogilvie 2018 ). Figure 6. (a) Viscous dissipation, (b) magnetic dissipation, (c) total dissipation, and (d) ratio of magnetic to viscous dissipations vs. tidal frequency at various Pm . . To find the scaling laws for Pm , we plot the two dissipations versus Pm as shown in Figure 7 . The dependence on Pm looks more regular than the dependence on S in Figure 5 because the field strength influences the resonant frequencies, whereas diffusivities cannot. By taking the average of the slopes for different tidal frequencies, we obtain the scaling lawsThe dashed lines in Figures 7 (a) and (b) denote these two scaling laws. The ratio of magnetic to viscous dissipations is indeed close to Pm −1 , as shown in Figure 6 (d). Note that our parameter regime is not the real regime, e.g., E and Pm are not too small, so these scaling laws might not be realistic. Lacking knowledge of the physical properties in stars and planets, we do not know the real parameters, and even estimating the order of magnitude is not convincing. These scaling laws obtained in the moderate parameter regime at least give us some qualitative evidence that a moderate or stronger