# Global magneto-convection models of stars with varying rotation rate

by Mariangela Viviani

Date of Examination:2020-01-24

Date of issue:2020-06-12

Advisor:Prof. Dr. Maarit Käpylä

Referee:Prof. Dr. Maarit Käpylä

Referee:Prof. Dr. Andreas Tilgner

Referee:Prof. Dr. Paul Charbonneau

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## Abstract

### English

The solar magnetic field is mostly axisymmetric at the largest scale. Latitudinal dynamo waves propagate from mid-latitudes towards the equator, as can be seen, for example, from sunspot observations. Reproducing this pattern is one of the main demands that a numerical model of the solar dynamo has to satisfy. One of the most common theories to explain the solar cycle is the $\alpha \Omega$ dynamo mechanism, that depicts dynamo action in the Sun as an interplay of differential rotation ($\Omega$ effect) and helical turbulence ($\alpha$ effect). As the rotation rate increases, the turbulence effects increase, hence the $\alpha-$effect increases. As a consequence, the dynamo moves from the $\alpha \Omega-$ regime to the $\alpha^2-$regime, where the inductive effects from turbulence are the dominant source for the toroidal and poloidal components of the magnetic field. This is postulated to lead to the emergence of non-axisymmetric solutions, but the details on how this transition occurs are not yet clear. The aim of this project is to quantitatively study how and why the types of the dynamo solutions and their cyclicity change as a function of rotation rate, together with some other important governing system parameters. In Section 5.1, we first compare semi-global models at increasing rotation rates: models covering $1/4$ of the full longitudinal extent and models covering the full longitude. We find that only when the longitudinal wedge assumption is relaxed, non-axisymmetric modes can develop and produce azimuthal dynamo waves. However, due to numerical constraints, our models lack the polar caps. We therefore expect that full-sphere models will show features more close to stellar observations, such as more dipolar magnetic fields at higher rotation, that cannot develop now because of the missing poles. We also find out that high enough resolution is necessary to reach the non-axisymmetric regime. In the axisymmetric regime, we find two different kind of solutions: one oscillatory in the presence of a solar-like differential rotation, where the equator rotates faster than the poles, and the other one still oscillatory, but in combination with an anti-solar differential rotation profile, with the equator slower than the poles. The main difference between the two solutions is the migration direction of the magnetic field, equatorward in the first case and poleward in the second one. The solar-like solutions were already analyzed with the test-field method, hence, here we concentrate on the oscillating solution in the anti-solar differential rotation regime. In Section 5.2 we study the turbulent transport coefficients in the axisymmetric models with the test-field module and analyze the dynamo operating in this regime. An important aspect of the model is the description of heat conduction in the convection zone, and the aim in this study was to make it as realistic as possible. The common approach is to prescribe it according to mixing-length theory, so that the depth of the convection zone is fixed. Recent local models have shown that, in a more realistic non-linear setup, where heat conduction is a function of temperature and density, the depth of the convection zone can considerably vary, and the bottom of the domain can become sub-adiabatic. Thus, the actual extent of the convective layer cannot be determined a priori, but is an output result of the simulations. We therefore also investigate the nature of dynamo solutions using such a non-linear model in Section~ 5.3 by adopting Kramers opacity law and compare with the results of Section 5.1. The different form of heat transport affects the azimuthal dynamo wave propagation direction. Also, the transition point to the non-axisymmetric regime is shifted towards a higher rotation rate, moving the simulations transition closer to the observed one.**Keywords:**Magnetohydrodynamics; Dynamo; Convection; Turbulence; Solar and stellar dynamos