Linear waves in a slowly rotating, compressible and perfectly conducting gas embedded in magnetic and gravitational fields

We discuss the dispersion relation of local linear waves in a compressible and perfectly conducting gas possessing magnetic and gravitational fields in a slowly rotating frame of reference. Instead of the full energy equation and a gas law, a not necessarily adiabatic equation of state p=p(ϱ) is used to close the system of equations, — an arguably flexible way of treatment when we are not clear about the contributions by radiation and conductivity to the energy transport.

We give a general dimensionless dispersion relation, (8). This reduces to (9) if the magnetic field B is zero; to (10) if, further, rotation φ is zero; to the relation for accoustic waves (11), if further the gravitational field G is zero. When B is not zero, we consider various cases with the propagation vector K always perpendicular to B: the relation now reduces to (13) if K is not perpendicular to φ; to (14) if, further, K is parallel to G; to (15) if φ=0; to the relation for last magneto-accoustic waves (16) if G=0. It reduces to (17) if K is perpendicular to G; to the fast magneto-accoustic waves (18), if, further, φ=0. It reduces to (19) if K is perpendicular to φ and to (20), if, further, K is parallel to G.

Our study shows that, in general, there are no pure modes, only hybrids. In particular, a rotation gives rise to modes that are dependent on the latitude, which we call “physico-geometrical” waves.

The present study is preliminary, and we may expect even more interesting results when we take into consideration the energy equation and the effects of radiation.