Magnetohydrodynamic shallow water equations with the alpha effect: Rossby-dynamo waves in solar–stellar tachoclines

¹ Institute of Physics, University of Graz, Universitätsplatz 5, 8010 Graz, Austria
² High Altitude Observatory, NCAR, 3080 Center Green Drive, Boulder, CO 80301, USA
³ Department of Astronomy and Astrophysics at Space Research Center, Ilia State University, Kakutsa Cholokashvili Ave 3/5, 0179 Tbilisi, Georgia
⁴ Evgeni Kharadze Georgian National Astrophysical Observatory, Mount Kanobili, 0301 Abastumani, Georgia

^⋆ Corresponding author: teimuraz.zaqarashvili@uni-graz.at

Received: 26 June 2025
Accepted: 28 August 2025

Abstract

Context. The activity of Sun-like stars is governed by the magnetic field, which is believed to be generated in a thin layer between convective and radiative envelopes. The dynamo layer, also called the tachocline, permits the existence of Rossby waves (r-modes) described by magnetohydrodynamic shallow water models, which may lead to short-term cycles in stellar activity.

Aims. Convective cells penetrate into the layer creating an overshoot upper part, where they transport an additional energy for vigorous activity. The aim of this paper is to study the influence of overshooting convection on the dynamics of Rossby waves in the tachoclines of Sun-like stars.

Methods. Here we write the magnetohydrodynamic shallow water equations with the effect of the penetrative convection and study the dynamics of wave modes in the layer.

Results. The formalism leads to the excitation of new oscillation modes connected with the dynamo coefficient, α, causing periodic modulations of all parameters in the tachocline. The modes are coupled with the Rossby waves resulting mutual exchange of convective and rotation energies. The timescales of Rossby-dynamo waves, for certain parameters, correspond to Schwabe (∼11 years) and Rieger (∼150–170 days) cycles as observed in solar activity.

Conclusions. The waves provide a new paradigm for internal magnetism and may drive the dynamos of Sun-like stars. Theoretical properties of the waves and observations can be used for magneto-seismological sounding of stellar interiors.