Calculating rotating hydrodynamic and magneto-hydrodynamic waves to understand magnetic effects on dynamical tides

TL;DR

This paper analytically investigates how magnetic fields influence rotating hydrodynamic and magneto-hydrodynamic waves, revealing magnetic splitting of resonant frequencies and effects on tidal dissipation relevant to astrophysical objects.

Contribution

It provides an analytical framework for understanding magnetic effects on dynamical tides, including frequency splitting and dissipation characteristics in rotating MHD flows.

Findings

Magnetic field splits the resonant frequency into two, positive and negative.

Dissipations are inversely proportional to the Ekman number and square of wavenumber.

Frequency-averaged dissipation remains constant across scales.

Abstract

For understanding magnetic effects on dynamical tides, we study the rotating magneto-hydrodynamic (MHD) flow driven by harmonic forcing. The linear responses are analytically derived in a periodic box under the local WKB approximation. Both the kinetic and Ohmic dissipations at the resonant frequencies are calculated and the various parameters are investigated. Although magnetic pressure may be negligible compared to thermal pressure, magnetic field can be important for the first-order perturbation, e.g. dynamical tides. It is found that magnetic field splits the resonant frequency, namely the rotating hydrodynamic flow has only one resonant frequency but the rotating MHD flow has two, one positive and the other negative. In the weak field regime the dissipations are asymmetric around the two resonant frequencies and this asymmetry is more striking with a weaker magnetic field. It is also found that both the kinetic and Ohmic dissipations at the resonant frequencies are inversely proportional to the Ekman number and the square of wavenumber. The dissipation at the resonant frequency on small scales is almost equal to the dissipation at the non-resonant frequencies, namely the resonance takes its effect on the dissipation at intermediate length scales. Moreover, the waves with phase propagation perpendicular to magnetic field are much more damped. It is also interesting to find that the frequency-averaged dissipation is constant. This result suggests that in compact objects magnetic effects on tidal dissipation should be considered.