Penetrative magneto-convection of a rotating Boussinesq flow in f-planes

In this study, we conducted linear instability analysis of penetrative magneto-convection in rapidly rotating Boussinesq flows within tilted f-planes, under the influence of a uniform background magnetic field. We integrated wave theory and convection theory to elucidate the penetration dynamics in rotating magneto-convection. Our findings suggest that efficient penetration in rapidly rotating flows with weakly stratified stable layers at low latitudes can be attributed to the resonance of wave transmission near the interface between unstable and stable layers. In the context of strongly stratified flows, we derived the scaling relationships of penetrative distances Delta with the stability parameter delta. Our calculation shows that, for both rotation-dominated and magnetism-dominated flows, Delta obeys a scaling of Delta similar to O(delta(-1/2)). In rotation-dominated flows, we noted a general decrease in penetrative distance with an increased rotational effect, and a minor decrease in a penetrative distance with an increased latitude. When a background magnetic field is introduced, we observed a significant shift in the penetrative distance as the Elsasser number Lambda approaches one. The penetrative distance tends to decrease when Lambda << 1 and increase when Lambda >> 1 with the rotational effect, indicating a transition from rotation-dominated to magnetism-dominated flow. We have further investigated the impact of the background magnetic field when it is not aligned with the rotational axis. This presents a notable contrast to the case where the magnetic field is parallel to the rotational axis.