On the nature of the magnetic Rayleigh–Taylor instability in astrophysical plasma: the case of uniform magnetic field strength

Abstract The magnetic Rayleigh–Taylor instability has been shown to play a key role in many astrophysical systems. The equation for the growth rate of this instability in the incompressible limit, and the most-unstable mode that can be derived from it, are often used to estimate the strength of the magnetic field that is associated with the observed dynamics. However, there are some issues with the interpretations given. Here, we show that the class of most unstable modes ku for a given θ, the class of modes often used to estimate the strength of the magnetic field from observations, for the system leads to the instability growing as σ2 = 1/2Agku, a growth rate which is independent of the strength of the magnetic field and which highlights that small scales are preferred by the system, but not does not give the fastest growing mode for that given k. We also highlight that outside of the interchange (k ⋅ B = 0) and undular (k parallel to B) modes, all the other modes have a perturbation pair of the same wavenumber and growth rate that when excited in the linear regime can result in an interference pattern that gives field aligned filamentary structure often seen in 3D simulations. The analysis was extended to a sheared magnetic field, where it was found that it was possible to extend the results for a non-sheared field to this case. We suggest that without magnetic shear it is too simplistic to be used to infer magnetic field strengths in astrophysical systems. instabilities, magnetic fields, MHD 1 THE RAYLEIGH–TAYLOR INSTABILITY IN ASTROPHYSICAL PLASMA The Rayleigh–Taylor instability, first proposed by Rayleigh (1900) and Taylor (1950), is a fundamental process in many space and astrophysical systems. For a contact discontinuity that is formed where a heavy fluid is supported above a light fluid against gravity, this boundary is unstable to perturbations that grow by converting gravitational potential energy into kinetic energy creating rising and falling fingers. The growth rate (σ) for this instability in the absence of magnetic field or viscous effects is give as \begin{equation} \sigma =\sqrt{A k g}, \end{equation} (1) where k is the wavenumber, g is the acceleration due to gravity and A is the Atwood number given as (ρu − ρl)/(ρu + ρl) for where ρu is the upper density and ρl is the lower density (Chandrasekhar 1961). Daly (1967) described how the evolution of the Rayleigh–Taylor instability changes from the symmetry of the low Atwood number Rayleigh–Taylor instability to be replaced by the formation of rising bubbles and sharp falling spikes in the high Atwood number limit. For a review of the hydrodynamic Rayleigh–Taylor instability see, for example, Sharp (1984). The inclusion of a horizontal magnetic field to the Rayleigh–Taylor instability adds a directionality to the system (Kruskal & Schwarzschild 1954). The interchange mode, where the wavevector |${\boldsymbol {k}}$| is perpendicular to the magnetic field and so magnetic tension does not have an effect, reduces the problem to one analogous to the hydrodynamic situation where a total pressure replaces the role of the gas pressure. The undular mode, where the wavevector |${\boldsymbol {k}}$| is parallel to the magnetic field and as such drives distortion of the magnetic field, creates a magnetic tension force that works to suppress high wavenumber perturbations along the magnetic field. The growth rate of the magnetic Rayleigh–Taylor instability for a mixed mode perturbation where the magnetic field is only in the y-direction is given as \begin{equation} \sigma ^2=kg\left[A- \frac{B^2k_y^2}{2 \pi (\rho _{\rm u} + \rho _{\rm l})gk } \right], \end{equation} (2) where B is the magnetic field strength in the y-direction (Chandrasekhar 1961). This implies that the system is always unstable providing a perturbation with sufficiently small kx is given. For equation (2), a critical wavelength (λc) that gives a growth rate of σ = 0 can be defined as λc = 2π/kc = B2cos2θ/(ρu − ρl)g, where θ is the angle between the k vector and the magnetic field. The Rayleigh–Taylor instability drives many observed features in astrophysical systems. Hachisu et al. (1992) described how the instability can lead to element mixing in supernova explosions. Hester et al. (1996) compared the observational characteristics of the Crab nebula with simulations of the magnetic Rayleigh–Taylor instability performed by Jun, Norman & Stone (1995), finding that the magnetic Rayleigh–Taylor instability could explain the observed filamentary structure. Recent axisymmetric simulations of the Crab nebula by Porth, Komissarov & Keppens (2014), using adaptive mesh refinement to provide high resolution, found that the magnetic field is insufficient to suppress the growth of the instability. In the Earth's ionosphere, the rise of regions of depleted plasma against the gravitational field during the Equatorial Spread-F phenomenon has been interpreted as the occurrence of the magnetic Rayleigh–Taylor instability in a low-beta magnetic plasma environment (Kelley et al. 1976; Takahashi et al. 2009). Gratton, Farrugia & Cowley (1996) described how variations in the solar wind can lead to expansion and contraction of the magnetopause pointing out that in the expansion phase there is an effective gravity that drives the magnetic Rayleigh–Taylor instability, a similar physical process can also happen at the heliopause as a result of changes due to the solar cycle (Borovikov & Pogorelov 2014). Observations by Berger et al. (2008, 2010) show this instability occurring in quiescent prominences. This instability has also been found in plasma jet experiments (Moser & Bellan 2012). Numerical investigations in the magnetic Rayleigh–Taylor instability in 3D have revealed a great deal about its evolution. One of the key aspects that these simulations have revealed is that in the 3D magnetic Rayleigh–Taylor simulations is that in 3D the instability results in the formation of structures that are elongated in the direction of the magnetic field (Stone & Gardiner 2007). This feature has been of particular importance in understanding the formation of plumes in solar prominences (Hillier et al. 2011, 2012; Keppens, Xia & Porth 2015; Terradas et al. 2015) or in the formation of filamentary structure associated with emerging magnetic flux (Isobe et al. 2006). To make the linear theory for the magnetic Rayleigh–Taylor instability more applicable to the astrophysical settings described above, there has been development of the linear theory beyond the simple model used by Chandrasekhar (1961). Ruderman, Terradas & Ballester (2014) investigated the role of magnetic shear on the instability both for the single discontinuity and for a dense slab embedded in a tenuous atmosphere. This can be treated as an extension of case of a plasma interacting with a vacuum where there is shear in the magnetic field as elucidated in Chapter 6 of Goedbloed & Poedts (2004), th (...truncated)