Stability analysis of convection in the intracluster medium

Abstract

We use the machinery usually employed for studying the onset of Rayleigh-Bénard convection in hydro-and magnetohydrodynamic settings to address the onset of convection induced by the magnetothermal instability and the heat-flux-buoyancy-driveninstability in the weakly-collisional magnetized plasma permeating the intracluster medium. Since most of the related numerical simulations consider the plasma being bounded between two 'plates' on which boundary conditions are specified, our strategy provides a framework that could enable a more direct connection between analytical and numerical studies. We derive the conditions for the onset of these instabilities considering the effects of induced magnetic tension resulting from a finite plasma beta. We provide expressions for the Rayleigh number in terms of the wave vector associated with a given mode, which allow us to characterize the modes that are first to become unstable. For both the heat-flux-buoyancy-driven-instability and the magnetothermal instability, oscillatory marginal stable states are possible.PACS numbers 98.65. Hb, 44.25.+f

On the helium fingers in the intracluster medium

“…As far as the governing equations of motion of this system is concerned, they are the following well-known set of equations (see also Pessah & Chakraborty (2013); Gupta et al (2016)):…”

“…Henceforth an infinitesimal perturbation of a physical quantity, say f , about its equilibrium state value has been denoted as δ f . Since the sound speed is much faster than the speed at which the unstable convective modes grows, it is justified to employ the Boussinesq approximation (Balbus 2000(Balbus , 2001Quataert 2008;Pessah & Chakraborty 2013;Gupta et al 2016), i.e., ∇•δu = 0. Also one has, due to the solenoidal character of the magnetic field, ∇•δB = 0.…”

Effect of composition gradient on magnetothermal instability modified by shear and rotation

“…After the linear stability analyses of the composition gradient modified HBI and MTI have been done by Pessah & Chakraborty (2013) using the standard WKB-type approximations, Sadhukhan, Gupta & Chakraborty (2017)-following Gupta et al (2016)have shown how the problem can also be studied by employing the technique (Chandrashekhar 1981) traditionally used to study the convection problem in a Rayleigh-Bénard set-up. The latter technique has some added advantages, e.g., it connects with the boundary conditions usually adopted in the numerical simulations, it helps one to find critical Rayleigh number corresponding to the HBI and the MTI, it facilitates easier analytical estimation of the contribution from the magnetic tension, and it may help us to come up with low dimensional models for convection, like the Lorenz model (Lorenz 1963).…”

“…Out of many possible boundary conditions, we adopt the following boundary conditions (Gupta et al 2016) at the bounding plates: δu z = 0, ∂ 2 z δu z = 0, ∂ z δw z = 0, δB z = 0, ∂ z j z = 0, δθ = 0, and δc = 0. These conditions physically imply that the normal component of the velocity must be zero on the boundary surfaces, surfaces are stress-free, boundaries are perfectly conducting, the boundary surfaces are at a constant temperature, and the surfaces are at constant concentration as well.…”

“…Further studies on this interplay between the temperature gradient and the concentration gradient has been well explored both numerically and analytically (Berlok & Pessah 2015, 2016aSadhukhan, Gupta & Chakraborty 2017). In a parallel investigation, researchers have drawn an analogy between the convection in the ICM to the paradigmatic Rayleigh-Bénard convection (Gupta et al 2016) and have found the critical value of the temperature gradient at which the MTI and the HBI are triggered. The analogy has been further extended to study the effect of the composition gradient on the critical value of temperature gradient required for the onset of the HBI (Sadhukhan, Gupta & Chakraborty 2017).…”