Analytical model for fast reconnection in large guide field plasma configurations

Significant progress in understanding magnetic reconnection {\it without a guide field} was made recently by deriving quantitatively accurate analytical models for reconnection in electron [1] and Hall [2] MHD. However, no such analytical model is available for reconnection {\it with a guide field}. Here, we derive such an analytical model for the large-guide-field, low-$\beta$, cold-ion fluid model [3] with electron inertia, ion viscosity $\mu$, and resistivity $\eta$. We find that the reconnection is Sweet-Parker-like when the Sweet-Parker layer thickness $\delta_{SP} > (\rho_s^4 + d_e^4)^{1/4}$, with $\rho_s$ and $d_e$ the sound Larmor radius and electron inertial length. However, reconnection changes character otherwise, resulting in reconnection rates $E_z/B_x^2 \approx \sqrt{2 \eta/\mu} (\rho_s^2 + d_e^2)/(\rho_s w)$ with $B_x$ the upstream magnetic field and $w$ the diffusion region length. Unlike the zero-guide-field case, $\mu$ plays crucial role in manifesting fast reconnection rates. If it represents the perpendicular viscosity [3], $\sqrt{\eta/\mu} \sim \beta^{-1} \sqrt{(m_e/m_i)(T_i/T_e)}$ and $E_z$ becomes dissipation independent and therefore potentially fast.\\[0pt] [1] L. Chac\'{o}n, A. N. Simakov, and A. Zocco, {\it PRL} {\bf 99}, 235001 (2007).\\[0pt] [2] A. N. Simakov and L. Chac\'on, {\it PRL} {\bf 101}, 105003 (2008).\\[0pt] [3] D. Biskamp, {\it Magnetic reconnection in plasmas}, Cambridge University Press, 2000.