In this paper, we revisit the governing equations for linear magnetohydrodynamic (MHD) waves and instabilities existing within a magnetized, plane-parallel, self-gravitating slab. Our approach allows for fully non-uniformly magnetized slabs, which deviate from isothermal conditions, such that the well-known Alfvén and slow continuous spectra enter the page_content. We generalize modern MHD textbook treatments, by showing how self-gravity enters the MHD wave equation, beyond the frequently adopted Cowling approximation. This clarifies how Jeans’ instability generalizes from hydro to magnetohydrodynamic conditions without assuming the usual Jeans’ swindle approach. Our main contribution lies in reformulating the completely general governing wave equations in a number of mathematically equivalent forms, ranging from a coupled Sturm-Liouville formulation, to a Hamiltonian formulation linked to coupled harmonic oscillators, up to a convenient matrix differential form. The latter allows us to derive analytically the eigenfunctions of a magnetized, self-gravitating thin slab. In addition, as an example we give the exact closed form dispersion relations for the hydrodynamical p- and Jeans-unstable modes, with the latter demonstrating how the Cowling approximation modifies due to a proper treatment of self-gravity. The various reformulations of the MHD wave equation open up new avenues for future MHD spectral studies of instabilities as relevant for cosmic filament formation, which can e.g. use modern formal solution strategies tailored to solve coupled Sturm-Liouville or harmonic oscillator problems.

keywords:
magnetic fields - MHD - methods: analytical.
††pagerange: An MHD spectral theory approach to Jeans’ magnetized gravitational instability-An MHD spectral theory approach to Jeans’ magnetized gravitational instability††pubyear: 2020

1.1 Motivations from Astrophysics and Cosmology

Sheet-like and filamentary structures of matter are ubiquitous in the Universe. For example, they are routinely observed in the interstellar medium of our Galaxy, in which giant molecular clouds are shaped by the combined action of gravity, supernovae explosions, thermal instability, cloud-cloud collisions, turbulence and magnetic fields Schneider & Elmegreen (1979); Bally et al. (1987); Mizuno et al. (1995); Hartmann (2002); Myers (2009); Pudritz & Kevlahan (2013); André (2015). Similarly, at cosmological scales, gravity organizes matter into a cosmic web of voids delineated by cosmological walls and filaments, as demonstrated by numerical simulations (Klypin & Shandarin, 1983; Klar & Mücket, 2010). At the nodes of this web lie galaxy clusters, which are supplied with matter, baryonic and dark, flowing along the filaments that interconnect them. Part of this accretion occurs intermittently (Dekel et al., 2009a, b; Kereš et al., 2009; Sánchez Almeida et al., 2014), suggesting that denser clumps of matter might form not only within galaxy clusters, but also either in the voids, the walls or the filaments of the cosmic web. While a fraction of these clumps may in fact be numerical artifacts, most of the clumps are believed to have a true physical origin (Springel, 2010; Hobbs et al., 2013; Nelson et al., 2013; Hobbs et al., 2016). Do baryons in cosmological walls and filaments fragment due to their own gravitational instability, or are these (numerically) observed gas clumps exclusively the product of the growth of primordial overdensities, as baryons fall into the gravitational potential of collapsed dark matter halos that they are embedded into? The general motivation of this paper is to contribute to the study of the fragmentation of magnetized and unmagnetized self-gravitating gas structures, in order to predict the size and growth rates of formation of clumps from astrophysical to cosmological scales. Such predictions are essential to better understand star and galaxy formation.

Many different instabilities may in principle give rise to this fragmentation. The thermal, Rayleigh-Taylor or Kelvin-Helmholtz instabilities surely play a role, for example in the cosmological context, in the denser environments of massive haloes (e.g. Kereš & Hernquist, 2009). But another well identified universal actor at play, which is the main focus of the present work, is Jeans’ gravitational instability, including magnetic fields, given their important dynamical role (Parker, 1979; Cox, 2005). In the literature, ‘gravitational instability’ may refer to the convective instability or the Rayleigh-Taylor instability, but here we deliberately choose a closure relation which switches-off convection, because our focus is on Jeans’ gravitational instability.

1.2 Brief summary on Jeans’ instability analysis

The analysis of Jeans’ instability in astrophysics has a very long history. References of historical importance studying the equilibrium states of self-gravitating structures include Ledoux (1951) for planar structures, and Ostriker (1964a) for cylinders, and more recently an extremely detailed study of polytropes has been performed by Horedt (2004). As for the stability of these equilibria, a plethora of studies could be quoted, such that the following list is by no means exhaustive. Historically, the investigation of gravitational instability was triggered by the works of Jeans (e.g. Jeans, 1928). Later the stability of sheet-like structures has been explored by Ledoux (1951) and extended by Simon (1965b) numerically. Effects of deviations from isothermality can be found for instance in Goldreich & Lynden-Bell (1965), in which stability criteria for pressure bounded, uniformly rotating polytropic sheets are obtained. Gradually, more and more ingredients relevant to describing astrophysical and cosmological environments were taken into account. The effects of an external pressure (e.g. Elmegreen & Elmegreen, 1978; Miyama et al., 1987a, b; Narita et al., 1988), of uniform and differential rotation (Safronov, 1960; Simon, 1965a; Narita et al., 1988; Papaloizou & Savonije, 1991; Burkert & Hartmann, 2004), of flow (Lacey, 1989), of the background expansion of the Universe and the dark matter component (Umemura, 1993; Anninos et al., 1995; Hosokawa et al., 2000), of the possible advent of convective instability (e.g. Mamatsashvili & Rice, 2010; Breysse et al., 2014), of the local expansion (or collapse) of the structure (Inutsuka & Miyama, 1992; Iwasaki et al., 2011), of curvature (Chandrasekhar & Fermi, 1953; Ostriker, 1964b; Sadhukhan et al., 2016), and, last but not least, of magnetic fields (Strittmatter, 1966; Kellman, 1972, 1973; Langer, 1978; Nakano & Nakamura, 1978; Tomisaka & Ikeuchi, 1983; Nakano, 1988; Hosseinirad et al., 2017). Despite these numerous works, there is still no rigorous and complete study of the magnetized Jeans’ instability, i.e. a derivation yielding analytically, without simplifying assumptions, the explicit expressions for eigenvalues and eigenfunctions of the corresponding eigenvalue problem. The two major difficulties to do so are the following.

The first difficulty is that systems including gravity are necessarily stratified. Considering a homogeneous background violates the static equilibrium Poisson equation, and doing so is known in the literature as Jeans’ swindle. The Universe being statistically homogeneous and isotropic at its largest scales and is not static, this simplification yields good results in the cosmological context, aside from studies of the cosmic web, which is obviously a stratified medium. Now, the study of waves and instabilities in stratified media is a very complex topic, which is still active ongoing research by hydrodynamicists and plasma physicists. The most celebrated example of an important subtlety arising from inhomogeneity is Landau damping, due to inhomogeneity in velocity space (kinetic page_content), but a similar damping arises in the fluid page_content of magnetohydrodynamics (MHD) in spatially inhomogeneous systems. Mathematically speaking, taking rigorously into account stratification involves considerations on continuous spectra and generalized eigenfunctions (i.e. distributions) in the framework of spectral theory. The spectral theory of linear operators has a wide field of applications in physics. It is at the foundation of quantum mechanics (von Neumann, 1955) as well as of MHD (Lifschitz, 1989; Goedbloed et al., 2019), it is useful in astrophysics (Adam, 1986a, b; Winfield, 2016) and in Earth seismology (Margerin, 2009; Margerin et al., 2009), to page_seo_title but a few examples.

The second difficulty is that formally the study of gravitational instability is an eigenvalue problem involving an integro-differential operator (cf. section 4 below). Physically, this stems from the fact that gravity is a long range force, without negative masses (mass being the gravitational equivalent of charge in electromagnetism). In Newtonian gravity there is no screening mechanism, no gravitational equivalent of a Debye sphere, which could reduce the interaction to a local one effectively. Consequently, the system of equations governing this eigenvalue problem is of fourth order. In some fields, notably in asteroseismology and in laboratory MHD, the perturbation of the gravitational potential induces only small effects, and it is common practice to neglect it. This is called the Cowling approximation (Cox, 1980; Unno et al., 1989), and doing so reduces this fourth-order problem to a second order one, enabling an approximate analytic treatment. However, this approximation is not relevant for our purpose, since it precisely discards the term responsible for the Jeans instability.

1.3 Overview of related analytical approaches

Efforts to understand analytically the evolution of perturbations in self-gravitating structures without the Cowling approximation are ongoing. So far, the analytic dispersion relations for Jean’s instability were derived in special cases only, notably in the incompressible case (Goldreich & Lynden-Bell, 1965; Tassoul, 1967), in the thin sheet limit (Tomisaka & Ikeuchi, 1985; Wünsch et al., 2010), focusing on marginal stability only (Oganesyan, 1960; Goldreich & Lynden-Bell, 1965), or working under simplifying assumptions about the scale of perturbations (Lubow & Pringle, 1993; Clarke, 1999). Variational approaches such as in Chandrasekhar (1961); Lynden-Bell & Ostriker (1967); Raoult & Pellat (1978) provide general stability criteria but do not give explicit expressions for the eigenvalues. An upper bound on the perturbed self-gravitational energy associated with the Lagrangian displacement was derived by Keppens & Demaerel (2016); Demaerel & Keppens (2016), and Durrive & Langer (2019) decomposed, in the planar hydrostatic case, the fourth-order eigenvalue problem into a sequence of second-order problems that can be solved separately. In addition, the eigenvalue problem related to Jeans’ instability being formally very similar to the one relevant for stellar oscillations, the following analytical studies are also noteworthy. Since the Cowling approximation is not accurate for long wavelength oscillations (Cox, 1980), it poorly describes the dipolar f-mode in stellar oscillations, and taking advantage of the fact that dipolar oscillations have the specific property of yielding a first integral from momentum conservation, Takata (2005, 2006) reduced his fourth-order system of equations into a second-order one, and was able to analyze adiabatic dipolar oscillations of stars without making the Cowling approximation. This analysis is restricted to a specific mode, does not consider magnetic fields, and assumes a hydrostatic equilibrium. Other mathematically-oriented stellar physics studies include Beyer (1995b, a); Beyer & Schmidt (1995) in the framework of operator theory, and Takata (2012) who suggests a way to give a complete mathematical justification to the conventional classification of stellar eigenmodes into p-modes, g-modes, and f-modes, adopting an approach from the field of geoseismology based on wedge products. Finally, Poedts et al. 1985) use both the technique from Goedbloed (1975). That from Pao (1975) to derive a reduced eigenvalue problem focused on continuum modes. They conclude that the perturbation of the gravitational potential has no effect on the continuous spectrum. However, their study excludes, by construction, discrete modes and in particular Jeans’ instability.

1.4 Aim of this MHD spectral approach

In the present paper, we aim at addressing the two above difficulties and contributing to the challenge of understanding rigorously the magnetized Jeans’ gravitational instability as follows. Our goal is to reformulate the problem in order to exhibit the fundamental singularities underlying this problem, given that singularities of differential equations are key to understand dynamics (e.g. Adam, 1986c). More precisely, we will extend the approach of Goedbloed et al. (2019): Based on the work of Goedbloed (1971), the modern textbook treatment in Goedbloed et al. (2019) (section 7.3) exhibits the spectrum of a non-uniformly magnetized plasma slab embedded in a uniform gravitational field, making the Cowling approximation. To do so, they write in Sturm-Liouville form the equation satisfied by the component of the displacement vector in the direction of the stratification. From this equation, the spectrum may be read: the zeros of the numerator of the coefficient of the highest order term correspond to the slow and Alfvén genuine singularities (continuous spectra), while the zeros of its denominator correspond to the slow and fast magneto-acoustic apparent singularities. In addition, the theorem derived by Goedbloed & Sakanaka (1974), which extends the classical Sturm-Liouville oscillation theorem (relevant for a linear eigenvalue problem) to this non-linear eigenvalue problem, indicates the monoticity of the discrete parts of the spectrum lying between these continuous ranges of singularities. In the present work, we complement this study by deriving the MHD wave equation of a self-gravitating slab, taking into account both the equilibrium and the perturbed Poisson equations.

In the process, we recast the problem into various compact, classical forms, suited to analyze the spectrum and make the solutions explicit. In particular, we manage to factorize this MHD wave equation. Previous authors did not take advantage of the wave equation formulation. For instance Ledoux (1950) and Ledoux & Walraven (1958) say that Pekeris (1938) has been the first to carry out completely the necessary eliminations and derived a fourth order equation ‘which is too complicated to be reproduced [in their paper]’. Similarly, it is written in Goldreich & Lynden-Bell (1965) that they derived this equation but they ‘did not find the result very enlightening so [they] shall not repeat it [in their paper]’. One example in which such an equation is made fully explicit is Elmegreen & Elmegreen (1978). However, it is limited to the hydrodynamical and isothermal case, and only the equation on the gravitational potential is derived, while we will formulate the equation on the Lagrangian displacement vector, which is the most fundamental variable since all the other linearized quantities may be directly deduced from it. In addition, the equation in Elmegreen & Elmegreen (1978) is left unfactorized. Similarly, often this eigenvalue problem is written as a set of two coupled second order differential equations (e.g. equations (22)-(33) of Nagai et al., 1998) or as a 4×4444\times 44 × 4 matrix differential equation (e.g. equations (15)-(27) of Nakamura et al., 1991), but without any particular form, such that it is impossible to tell from the coefficients which frequencies are the genuine singularities at the heart of the dynamics.

1.5 Paper organization

The paper is organized as follows. First, we present the equilibrium state under consideration (section 2), and then the MHD equations linearized about this equilibrium (section 3). In section 4 we present the eigenvalue problem, which we then transform into various forms, from which we discuss its spectral properties and its solutions: (i) a coupled Sturm-Liouville form (obtained through sections 5, 6, 7), (ii) a coupled harmonic oscillator form, including its Hamiltonian form (section 8), (iii) a first order matrix differential equation (section 9), and (iv) a scalar wave equation (section 10). In section 11, we give the expression of the displacement vector and perturbation of the gravitational potential in terms of the solution of the above matrix differential equation. In that sense, we reduced the challenge of solving the initial problem to solving a much simpler problem (which we do fully solve in a certain limit). In section 12 we illustrate through a simple example how we may obtain explicitly the dispersion relation thanks to the above reformulation, and in particular we give the analytic expression for the mode corresponding to Jeans’ instability. Finally, we conclude in section 13, presenting some of the prospects of this work.

2 Equilibrium relations

Let us consider a magnetized, polytropic, self-gravitating, planar medium in static equilibrium, governed by the following relations. Using Cartesian coordinates x?¥xitalic_x, y?¦yitalic_y and z?§zitalic_z, we choose x?¥xitalic_x as the direction of stratification. Thus, all equilibrium quantities, denoted with subscripts 00, are functions of x?¥xitalic_x only. The slab contains a magnetic field ?©0subscript?©0\boldsymbolB_0bold_italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT confined to plane layers perpendicular to the stratified direction x?¥xitalic_x, but whose components vary along the stratification, page_seo_titlely

?©0=By(x)??y+Bz(x)??z,subscript?©0subscript?µ?¦?¥subscript???¦subscript?µ?§?¥subscript???§\boldsymbolB_0=B_y(x)\boldsymbole_y+B_z(x)\boldsymbole_z,bold_italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( italic_x ) bold_italic_e start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT + italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_x ) bold_italic_e start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT , (1)
where we denote by ??xsubscript???¥\boldsymbole_xbold_italic_e start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT, ??ysubscript???¦\boldsymbole_ybold_italic_e start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT and ??zsubscript???§\boldsymbole_zbold_italic_e start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT the unit vectors in the x?¥xitalic_x, y?¦yitalic_y and z?§zitalic_z directions. As in Goedbloed et al. (2019), throughout this paper we make use of units where vacuum permeability ?0subscript??0\mu_0italic_? start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is unity. To restore mks units one should make the substitutions ?©??©/?0??©?©subscript??0\boldsymbolB\rightarrow\boldsymbolB/\sqrt\mu_0bold_italic_B ? bold_italic_B / square-root start_ARG italic_? start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG and ????0?????subscript??0??\boldsymbolj\rightarrow\sqrt\mu_0\boldsymboljbold_italic_j ? square-root start_ARG italic_? start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG bold_italic_j in the formulae. Thus, the equilibrium currents are given by ??0=?×?©0=-Bz???y+By???zsubscript??0bold-?subscript?©0subscriptsuperscript?µ??§subscript???¦subscriptsuperscript?µ??¦subscript???§\boldsymbolj_0=\boldsymbol\nabla\times\boldsymbolB_0=-B^\prime_z% \boldsymbole_y+B^\prime_y\boldsymbole_zbold_italic_j start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = bold_? × bold_italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = - italic_B start_POSTSUPERSCRIPT ? end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT bold_italic_e start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT + italic_B start_POSTSUPERSCRIPT ? end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT bold_italic_e start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT. These currents remain along magnetic surfaces (the (y,z)?¦?§(y,z)( italic_y , italic_z ) planes at a given height x?¥xitalic_x), and since they are in general not aligned with ?©0subscript?©0\boldsymbolB_0bold_italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, there exists a non-vanishing Lorentz force

??0×?©0=-?(12B02),subscript??0subscript?©0bold-?12superscriptsubscript?µ02\boldsymbolj_0\times\boldsymbolB_0=-\boldsymbol\nabla\left(\tfrac1% 2B_0^2\right),bold_italic_j start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT × bold_italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = - bold_? ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) , (2)
where B02=By2+Bz2superscriptsubscript?µ02superscriptsubscript?µ?¦2superscriptsubscript?µ?§2B_0^2=B_y^2+B_z^2italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. However, there is no magnetic curvature term in this configuration. The slab is self-gravitating, meaning that the equilibrium gravitational acceleration ??0subscript??0\boldsymbolg_0bold_italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT satisfies the Poisson equation

????0=-4?G?0,?bold-?subscript??04???ºsubscript??0\boldsymbol\nabla\cdot\boldsymbolg_0=-4\pi G\rho_0,bold_? ? bold_italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = - 4 italic_? italic_G italic_? start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , (3)
where ?0subscript??0\rho_0italic_? start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the equilibrium density field and G?ºGitalic_G is Newton’s gravitational constant. This equation introduces the parameter

?02?4?G?0(x),superscriptsubscript??024???ºsubscript??0?¥\omega_0^2\equiv 4\pi G\rho_0(x),italic_? start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ? 4 italic_? italic_G italic_? start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x ) , (4)
which corresponds physically to the local (due to its x?¥xitalic_x-dependence) free-fall timescale. This timescale is the fundamental new ingredient as compared to the textbook treatment in Goedbloed et al. (2019). The Lorentz and gravitational forces are in competition with gradients of the pressure p0subscript?0p_0italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, such that the equilibrium force balance reads

-?p0+??0×?©0+?0??0=??.bold-?subscript?0subscript??0subscript?©0subscript??0subscript??00-\boldsymbol\nablap_0+\boldsymbolj_0\times\boldsymbolB_0+\rho_0% \boldsymbolg_0=\boldsymbol0.- bold_? italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + bold_italic_j start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT × bold_italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_? start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT bold_italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = bold_0 . (5)
Now, given the planar geometry we may write

??0=g0(x)??x,subscript??0subscript??0?¥subscript???¥\boldsymbolg_0=g_0(x)\boldsymbole_x,bold_italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x ) bold_italic_e start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT , (6)
so that Poisson’s equation (3) reduces to

g0?=-?02,superscriptsubscript??0?superscriptsubscript??02\displaystyle g_0^\prime=-\omega_0^2,italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ? end_POSTSUPERSCRIPT = - italic_? start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , (7)
and using (2), the force balance (5) reduces to

(p0+12B02)?=?0g0.superscriptsubscript?012superscriptsubscript?µ02?subscript??0subscript??0\left(p_0+\tfrac12B_0^2\right)^\prime=\rho_0g_0.( italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT ? end_POSTSUPERSCRIPT = italic_? start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT . (8)
Throughout this work we consider a polytropic equation of state

p0=??0?,subscript?0??superscriptsubscript??0?¾p_0=\kappa\rho_0^\gamma,italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_? italic_? start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_? end_POSTSUPERSCRIPT , (9)
where ??¾\gammaitalic_? is called the polytropic exponent and ???\kappaitalic_? is a constant that depends on the specific entropy. The isothermal equation of state corresponds to ?=1?¾1\gamma=1italic_? = 1, in which case ???\kappaitalic_? reduces to the speed of sound squared. Finally, two local (due to their x?¥xitalic_x-dependence) speeds appear in this problem: the speed of sound

c(x)??p0?0,??¥?¾subscript?0subscript??0c(x)\equiv\sqrt\gamma\fracp_0\rho_0,italic_c ( italic_x ) ? square-root start_ARG italic_? divide start_ARG italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG italic_? start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG end_ARG , (10)
and the Alfvén speed

b(x)?B0?0,??¥subscript?µ0subscript??0b(x)\equiv\fracB_0\sqrt\rho_0,italic_b ( italic_x ) ? divide start_ARG italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG square-root start_ARG italic_? start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG end_ARG , (11)
associated with the propagation of purely magnetic waves called Alfvén waves, which are in essence vectorial since ?©0subscript?©0\boldsymbolB_0bold_italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is a vector, but given the planar stratification considered, only the above scalar Alfvén speed appears here.

3 Perturbation equations

The ideal MHD equations describing the dynamics of a self-gravitating magnetized fluid are

?t?+??(???)=0,?(?t??+??????)=-?p+??×?©+???,??=?×?©,?t?©=?×(??×?©),????=-4?G?,subscript?¡???bold-?????0??subscript?¡?????bold-???bold-?????©??????bold-??©subscript?¡?©bold-????©?bold-???4???º??\beginarray[]l\partial_t\rho+\boldsymbol\nabla\cdot(\rho\boldsymbolv% )=0,\\ \rho(\partial_t\boldsymbolv+\boldsymbolv\cdot\boldsymbol\nabla% \boldsymbolv)=-\boldsymbol\nablap+\boldsymbolj\times\boldsymbolB+\rho% \boldsymbolg,\\ \boldsymbolj=\boldsymbol\nabla\times\boldsymbolB,\\ \partial_t\boldsymbolB=\boldsymbol\nabla\times(\boldsymbolv\times% \boldsymbolB),\\ \boldsymbol\nabla\cdot\boldsymbolg=-4\pi G\rho,\endarraystart_ARRAY start_ROW start_CELL ? start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_? + bold_? ? ( italic_? bold_italic_v ) = 0 , end_CELL end_ROW start_ROW start_CELL italic_? ( ? start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT bold_italic_v + bold_italic_v ? bold_? bold_italic_v ) = - bold_? italic_p + bold_italic_j × bold_italic_B + italic_? bold_italic_g , end_CELL end_ROW start_ROW start_CELL bold_italic_j = bold_? × bold_italic_B , end_CELL end_ROW start_ROW start_CELL ? start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT bold_italic_B = bold_? × ( bold_italic_v × bold_italic_B ) , end_CELL end_ROW start_ROW start_CELL bold_? ? bold_italic_g = - 4 italic_? italic_G italic_? , end_CELL end_ROW end_ARRAY (12)
corresponding respectively to mass conservation, the momentum equation (with pressure gradients, Lorentz’s force and the gravitational force), Ampère’s law, the induction equation (Faraday’s law with Ohm’s law in the infinite electric conductivity limit) and Poisson’s equation. To analyze waves and instabilities in self-gravitating magnetized fluids, we linearize these equations around the equilibrium state detailed in section 2. Following the usual procedure, for each quantity Q=(?,??,?©,??,??)???????©????Q=(\rho,\boldsymbolv,\boldsymbolB,\boldsymbolj,\boldsymbolg)italic_Q = ( italic_? , bold_italic_v , bold_italic_B , bold_italic_j , bold_italic_g ) we write Q=Q0+Q1??subscript??0subscript??1Q=Q_0+Q_1italic_Q = italic_Q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT where subscripts 00 and 1111 indicate equilibrium and perturbed quantities respectively, assuming |Q1|?|Q0|much-less-thansubscript??1subscript??0|Q_1|\ll|Q_0|| italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | ? | italic_Q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT |. Doing so, mass conservation and the momentum equation read

?t?1+??(?0??1)=0,subscript?¡subscript??1?bold-?subscript??0subscript??10\partial_t\rho_1+\boldsymbol\nabla\cdot\left(\rho_0\boldsymbolv_1% \right)=0,? start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_? start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + bold_? ? ( italic_? start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT bold_italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = 0 , (13)
and

?0?t??1=-?p1+??1×?©0+??0×?©1+?1??0+??0??1,subscript??0subscript?¡subscript??1bold-?subscript?1subscript??1subscript?©0subscript??0subscript?©1subscript??1subscript??0??subscript??0subscript??1\rho_0\partial_t\boldsymbolv_1=-\boldsymbol\nablap_1+\boldsymbolj% _1\times\boldsymbolB_0+\boldsymbolj_0\times\boldsymbolB_1+\rho_% 1\boldsymbolg_0+\eta\rho_0\boldsymbolg_1,italic_? start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ? start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT bold_italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = - bold_? italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + bold_italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × bold_italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + bold_italic_j start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT × bold_italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_? start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT bold_italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_? italic_? start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT bold_italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , (14)
where ?1,??1,??1,?©1subscript??1subscript??1subscript??1subscript?©1\rho_1,\boldsymbolv_1,\boldsymbolj_1,\boldsymbolB_1italic_? start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and ??1subscript??1\boldsymbolg_1bold_italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT are respectively the perturbations of mass density, velocity, current density, magnetic field, and gravitational acceleration. Physically, the terms in the right hand side of (14) correspond to the forces applied on volume elements, and are modeled as follows. Note that we linearize about a static equilibrium where ??0=??subscript??00\boldsymbolv_0=\boldsymbol0bold_italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = bold_0, such that the linearization of (12) involves (13). POSTSUBSCRIPT. Let us consider that the timescales of the perturbations - the oscillation period if stable. Growth timescale if unstable - are sufficiently short so that no heat is exchanged between neighboring fluid elements. Let us consider that the timescales of the perturbations - the oscillation period if stable. Growth timescale if unstable - are sufficiently short so that no heat is exchanged between neighboring fluid elements. Then the evolution of the perturbations may be considered as adiabatic and, from thermodynamical considerations, it can be shown (cf. Thompson (2006) for example) that the equation expressing the absence of heat exchange ?Q=0?¿??0\delta Q=0italic_? italic_Q = 0 becomes the following relation between the Lagrangian variation of pressure ?p?¿?\delta pitalic_? italic_p and the Lagrangian variation of density ???¿??\delta\rhoitalic_? italic_?:

?pp0=?ad???0,?¿?subscript?0subscri

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