Instability of the magnetohydrodynamics system at small but finite Reynolds number

The aim of this paper is to give a result concerning the stability properties of the solutions of magnetohydrodynamics equations at small but finite Reynolds numbers. These solutions are found using the alpha-effect: this method gives us solutions which are highly oscillating spatially on the scale of the underlying flow but are growing on a larger scale depending on a parameter epsilon. We prove nonlinear stability and instability results for a dense subset of initial velocity field of the flow at given Reynolds number.Comment: 14 page

Toward an asymptotic behaviour of the ABC dynamo

The ABC flow was originally introduced by Arnol'd to investigate Lagrangian chaos. It soon became the prototype example to illustrate magnetic-field amplification via fast dynamo action, i.e. dynamo action exhibiting magnetic-field amplification on a typical timescale independent of the electrical resistivity of the medium. Even though this flow is the most classical example for this important class of dynamos (with application to large-scale astrophysical objects), it was recently pointed out (Bouya Isma\"el and Dormy Emmanuel, Phys. Fluids, 25 (2013) 037103) that the fast dynamo nature of this flow was unclear, as the growth rate still depended on the magnetic Reynolds number at the largest values available so far $(\text{Rm} = 25000)$ . Using state-of-the-art high-performance computing, we present high-resolution simulations (up to 40963) and extend the value of $\text{Rm}$ up to $5\cdot10^5$ . Interestingly, even at these huge values, the growth rate of the leading eigenmode still depends on the controlling parameter and an asymptotic regime is not reached yet. We show that the maximum growth rate is a decreasing function of $\text{Rm}$ for the largest values of $\text{Rm}$ we could achieve (as anticipated in the above-mentioned paper). Slowly damped oscillations might indicate either a new mode crossing or that the system is approaching the limit of an essential spectrum

Revisiting the ABC flow dynamo

The ABC flow is a prototype for fast dynamo action, essential to the origin of magnetic field in large astrophysical objects. Probably the most studied configuration is the classical 1:1:1 flow. We investigate its dynamo properties varying the magnetic Reynolds number Rm. We identify two kinks in the growth rate, which correspond respectively to an eigenvalue crossing and to an eigenvalue coalescence. The dominant eigenvalue becomes purely real for a finite value of the control parameter. Finally we show that even for Rm = 25000, the dominant eigenvalue has not yet reached an asymptotic behaviour. Its still varies very significantly with the controlling parameter. Even at these very large values of Rm the fast dynamo property of this flow cannot yet be established

Instabilités en magnétohydrodynamique

The magnetohydrodynamics, or dynamo effect, involves the generation of electrical energy from mechanical energy. More specifically, it consists in studying the evolution of a magnetic field generated by a fluid conductor. This phenomenon is found in planets, stars, or even galaxies, where the magnetic field comes from the inner movement. In this thesis, we focus on magnetohydrodynamic instabilities: starting from a conductive fluid with no magnetic field, is a slight disturbance of the flow and magnetic fields (for example, a residue of the magnetic field coming from another system) sufficient to give an amplification of the magnetic field, thus creating a dynamo? The second question consists in the time required to achieve such an amplification of the magnetic field. This thesis is the study of these two questions, and gives two theoretical results and two numerical results.La magnétohydrodynamique, ou effet dynamo, consiste en la génération d'énergie électrique à partir d'énergie mécanique. Plus précisément, on étudie l'évolution d'un champ magnétique généré par un fluide conducteur. Ce phénomène se retrouve dans les planètes, les étoiles, ou même les galaxies, où le champ magnétique provient du mouvement interne. Dans cette thèse, nous nous intéressons plus précisément aux instabilités en magnétohydrodynamique : partant d'un fluide conducteur sans champs magnétique, est-ce qu'une perturbation légère de l'écoulement et du champ magnétique (par exemple, un résidu de champs magnétique arrivant d'un autre système) peut engendrer une amplification de ce champ magnétique, créant ainsi une dynamo ? La deuxième interrogation consiste en le temps nécessaire pour obtenir une telle amplification du champ magnétique. Cette thèse consiste donc en l'étude de ces deux questions, et donne deux résultats d'ordre théorique et deux résultats d'ordre numérique

Instabilités en magnétohydrodynamique

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