Kink-like mode of a double gradient instability in a compressible plasma current sheet

Abstract

A linear MHD instability of the electric current sheet, characterized by a small normal magnetic field component, varying along the sheet, is investigated. The tangential magnetic field component is modeled by a hyperbolic function, describing Harris-like variations of the field across the sheet. For this problem, which is formulated in a 3D domain, the conventional compressible ideal MHD equations are applied. By assuming Fourier harmonics along the electric current, the linearized 3D equations are reduced to 2D ones. A finite difference numerical scheme is applied to examine the time evolution of small initial perturbations of the plasma parameters. This work is an extended numerical study of the so called “double gradient instability”, – a possible candidate for the explanation of flapping oscillations in the magnetotail current sheet, which has been analyzed previously in the framework of a simplified analytical approach for an incompressible plasma. The dispersion curve is obtained for the kink-like mode of the instability. It is shown that this curve demonstrates a quantitative agreement with the previous analytical result. The development of the instability is investigated also for various enhanced values of the normal magnetic field component. It is found that the characteristic values of the growth rate of the instability shows a linear dependence on the square root of the parameter, which scales uniformly the normal component of the magnetic field in the current sheet.

Numerical linearized MHD model of flapping oscillations

“…Essentially, dispersion curves of oscillations, obtained here, and the corresponding instability, studied in Ref. 44, are close to the same analytical prediction (1) and (2). This supports the wave generation mechanism suggested in Ref.…”

“…The model was tested by means of two-dimensional linearized MHD and three-dimensional full MHD simulations of the unstable kink mode, developing in a Harris-like current sheet 44 and at the Pritchett and Coroniti 41 background. 34,35 It is found out that numerical growth rates are close to the analytical prediction with x f ¼ max½x f ðxÞ for 2D simulations and x f ¼ hx f ðxÞi for 3D simulations, where angle brackets mean averaging.…”

Kinetic ballooning/interchange instability in a bent plasma sheet

“…[34] The above continuous observation of flapping signatures could agree with drift-kink [Lapenta and Brackbill, 1997;Daughton, 1999;Karimabadi et al, 2003aKarimabadi et al, , 2003bZelenyi et al, 2009] and double-gradient instability [e.g., Erkaev et al, 2008;Korovinskiy et al, 2011] theories. The drift-kink instability does not, however, explain the complementary, significant in-phase oscillations of the Z GSM component of the magnetic field shown in the third panel of Figure 10.…”

“…Theory predicts that a plasma sheet configuration with negative ∂ B Z /∂ X may be unstable for both a double‐gradient instability [ Erkaev et al , 2008; Korovinskiy et al , 2011] and a ballooning/interchange instability [ Pritchett and Coroniti , 2010]. While the former generates plasma sheet flapping oscillations, the latter is responsible for the formation of sausage‐like fingers.…”

Stability of Current Sheets in the Solar Corona

“…This inhomogeneity corresponds to larger spatial and time scales and can support relatively slow long-wavelength oscillations of current sheets. However, direct application of the theory of current sheet oscillations Biernat, 2007, 2010;Korovinskiy et al, 2011) to descriptions of possible dynamical properties of current sheets in the solar corona is impossible due to the absence of the magnetic field shear component (along the current direction) in this theory. The presence of significantly sheared magnetic field components in current sheets of solar flares (the magnetic field along the current direction can be ten times larger than the transverse component) is evidenced in numerous observations of both magnetic fields at the level of the photosphere (e.g.…”