Double tearing instability with shear flow

Abstract

The linear evolution of the double tearing mode with parallel to the magnetic field equilibrium shear flow and viscosity is investigated numerically. Numerically obtained growth rates are found to agree with the solutions of the double tearing dispersion relation in the parameter range of validity. Solutions of the incompressible, time-dependent, linearized, viscoresistive magnetohydrodynamic equations for the double tearing mode with parallel flow are found for wide relevant parameter ranges. Large (weakly coupled) and small (strongly coupled) rational surface separation ys are investigated. The magnetic Reynolds number S is varied up to 108, and ambient flow velocities up to 0.57 of the Alfvén speed VA far from the tearing layer are considered. The normalized wave number α is 0.05 (long wavelength) and 0.5 (short wavelength). Spatial variations of the perturbed magnetic field ψ and flow W indicate the ‘‘nonconstant-ψ’’ effects for small ys. Shear flow decouples the rational surfaces, reduces the growth rate, and transforms the instability to the standard tearing mode. Overstable modes are found from the solutions of the dispersion relation and in the numerical computations, and their frequencies are not affected by the value of viscosity. The temporal oscillations of the solutions increase with the flow at the resonant surfaces at a rate slower than that of the Doppler shift. For viscous Reynolds number Sv comparable to or larger than the magnetic Reynolds number a stabilizing effect was found, and in the presence of large flow the real growth rate γR scaling approaches the standard tearing mode scaling γR∼Sv1/6.