A brief study of instabilities in the context of space magnetohydrodynamic simulations

Revista Brasileira de Ensino de Fı́sica, vol. 38, nº 1, 1309 (2016) www.scielo.br/rbef DOI: http://dx.doi.org/10.1590/S1806-11173812098 Artigos Gerais cbnd Licença Creative Commons A brief study of instabilities in the context of space magnetohydrodynamic simulations Um breve estudo de instabilidades no contexto de simulações de magneto-hidrodinâmica espacial Edgard de F.D. Evangelista∗, Margarete O. Domingues, Odim Mendes, Oswaldo D. Miranda Instituto Nacional de Pesquisas Espaciais, São José dos Campos, SP, Brasil. Recebido em 17 de agosto de 2015. Aceito em 8 de setembro de 2015 The study of the hydrodynamic instabilities in the context of the magnetohydrodynamics (MHD) is very important in many branches of physics. Particularly, we can mention geophysical and astrophysics, where we have several processes involving hydrodynamic effects, such as shock waves, plasma flows and the propagation of waves. In these scenarios it is frequent the onset of instabilities. For example, let a system be formed by two phases with different densities and relative velocities. Besides, consider these phases are in contact with each other by means of a tangential surface, that is, an interface where there is no transference of matter and where there are only relative tangential velocities. In this case, under certain circumstances, we will have a particular type of phenomenon, the so-called Kelvin-Helmholtz (KH) instability. In this paper we will address to the basic theory of such instabilities, explaining how they arise from the hydrodynamic equations and showing the numerical simulation of a particular case. Besides, we show examples of other MHD instabilities which are usually found in astrophysical processes. Keywords: magnetohydrodynamics, instabilities, FLASH Code. O estudo das instabilidades hidrodinâmicas no contexto da magneto-hidrodinâmica (MHD) é muito importante para várias áreas da fı́sica. Particularmente, podemos mencionar a geofı́sica e a astrofı́sica, em que temos diversos processos envolvendo efeitos hidrodinâmicos, tais como ondas de choque, fluxos de plasma a propagação de ondas. Nestes cenários é frequente o surgimento de instabilidades. Por exemplo, seja um sistema formado por duas fases com diferentes densidades e velocidades relativas. Além disso, considere que estas fases estão em contato entre si por meio de uma superfı́cie tangencial, isto é, uma interface onde não há transferência de matéria e onde há somente velocidades relativas tangenciais. Nesse caso, sob certas circunstâncias, teremos um tipo particular de fenômeno, conhecido como instabilidade de Kelvin-Helmholtz (KH). Nesse artigo abordaremos a teoria básica de tais instabilidades, explicando como elas surgem das equações hidrodinâmicas e mostrando a simulação numérica de um caso particular. Além disso, são mostrados exemplos de outras instabilidades em MHD, as quais são geralmente encontradas em processos astrofı́sicos. Palavras-chave: magnetohidrodinâmica, instabilidades, código FLASH. 1. Introduction Magnetohydrodynamics consists in the study of the fluids which are compressible and conductor of electricity under the influence of magnetic fields. Roughly speaking, the equations governing the behavior of these fluids under such conditions are obtained through the combination of the Euler equa∗ Endereço de correspondência: . Copyright by Sociedade Brasileira de Fı́sica. Printed in Brazil. tions of the fluid mechanics with the Maxwell equations of the electromagnetism. The formalism of the MHD is of great interest for several branches of physics, such as space geophysics, astrophysics and engineering. For example, the MHD is applicable in many scenarios in astrophysics and cosmology, once most of the baryonic matter in the universe is formed by plasma, including stars and interplanetary, interstellar and inter- 1309-2 A brief study of instabilities in the context of space magnetohydrodynamic simulations galactic media. Besides, many astrophysical systems are not in local thermodynamic equilibrium, which requires an extra kinematic treatment for the complete description of the phenomena. Particularly, solar winds and blasts are understood under the framework of the MHD. Concerning the instabilities, let a system be initially at a stationary state, that is, the variables which define its configuration do not depend on time. If that system undergo small perturbations, which are gradually smoothed such that there are not appreciable deviation from the stationary state, we can say such a system is stable [1]. On the other hand, if we imagine the system undergo small deviation in a given region of its domain, in such a manner the acting forces tend to increase more and more the deformations, we have an unstable configuration. Such an unstable behavior can occur in several forms, that is, there are many types of instabilities, having particular characteristics. According to [2], there is a diversity of dynamic instabilities which are characteristic of fluids. For example, a static fluid in a gravitational field can undergo convective inversions when its inferior portion is heated or its superior portion is cooled. Generally speaking, a vertical stratification in the density profile can be caused by a temperature gradient, yielding the so-called Rayleigh-Taylor (RT) instabilities [3]. Further, the presence of a radiation field with spatially variable opacity can induce unstable temperature and density distributions. Of particular interest is the case where the fluid has two adjacent phases with different densities and which have a tangential movement relative to each other. In this case, at the interface between the phases, under certain circumstances, we can have the KH instabilities. Such instabilities came from the combined effect of the pressure, the gravity and the Reynolds strain. In this paper we address to the MHD and to its instabilities, particularly the KH type. The basic formalism of the MHD is treated in Section 2; in Section 3 we discuss some examples in the context of space physics where the phenomena related to the MHD play important roles; in Section 4 we focus on the KH instabilities and we show the simulation of a particular case using FLASH Code; next, we present our conclusions. Besides, in Annex A there is a brief discussion on FLASH Code. Revista Brasileira de Ensino de Fı́sica, vol. 38, nº 1, 1309, 2016 2. Basic formalism of the MHD The ideal MHD describes the interplay between a magnetic field and a compressible fluid, with no viscosity and which is a perfect conductor of electricity (hence the term “ideal”) [4]. Besides, we consider the fluid has a non-relativistic behavior, that is, at any point of the domain and at any instant, the velocities are small when compared to the speed of light in vacuum. In general, the model which describes the phenomena related to that interaction is built through the combination of the equations (...truncated)