Gyrokinetic analysis of tearing modes in a collisionless plasma

The linear and nonlinear dynamics of tearing instabilities in a collisionless plasma are investigated analytically using a gyrokinetic description. The effects of Landau and $\rm \nabla B$ resonances on the linear characteristics of $\Delta^{\prime} $-driven tearing modes are discussed by including short wavelength variations across the confining magnetic field and long wavelength variations along the field. For a simple case when electrons are adiabatic and ions are fluid-like, the solutions of dispersion relations are obtained for modewidths, $\rm x_w$, lying between electron and ion excursion lengths, namely, $\rm X_{e,i}$, where $\rm X_{e,i} = \omega L_s/(k_{y} v_ {e,i})$, $\rm k_y$ is the wavenumber, $\rm L_s$ is the magnetic shear length, and $\rm v_{e,i}$ represent electron and ion thermal speeds. It is shown that electron Landau damping effect can drive the tearing mode unstable with growth rate proportional to $(\Delta^{\prime})^{1/2}$. For this mode, it is further shown that the effects of compressional mode coupling and finite Larmor radius can combine to have a slightly stabilizing effect. In another physical situation, it is demonstrated that the electron $\rm \nabla B$ resonance effect can significantly destabilize the gyrokinetic tearing mode with growth rates varying as fractional powers of $\Delta^{\prime}$ and $\rm k_y$. The nonlinear implications of these effects are investigated by deriving an appropriate Rutherford equation for the magnetic island evolution.