  We examine the effect of an external guide field and current sheet thickness
on the growth rates and nature of three dimensional unstable modes of an
electron current sheet driven by electron shear flow. The growth rate of the
fastest growing mode drops rapidly with current sheet thickness but increases
slowly with the strength of the guide field. The fastest growing mode is
tearing type only for thin current sheets (half thickness $\approx d_e$, where
$d_e=c/\omega_{pe}$ is electron inertial length) and zero guide field. For
finite guide field or thicker current sheets, fastest growing mode is
non-tearing type. However growth rates of the fastest 2-D tearing mode and 3-D
non-tearing mode are comparable for thin current sheets ($d_e < $half thickness
$ < 2\,d_e$) and small guide field (of the order of the asymptotic value of the
component of magnetic field supporting electron current sheet). It is shown
that the general mode resonance conditions for electron-magnetohydrodynamic
(EMHD) and magnetohydrodynamic (MHD) tearing modes depend on the effective
dissipation mechanism (electron inertia and resistivity in cases of EMHD and
MHD, respectively). The usual tearing mode resonance condition
($\mathbf{k}.\mathbf{B}_0=0$, $\mathbf{k}$ is the wave vector and
$\mathbf{B}_0$ is equilibrium magnetic field) can be recovered from the general
resonance conditions in the limit of weak dissipation. Necessary conditions
(relating current sheet thickness, strength of the guide field and wave
numbers) for the existence of tearing mode are obtained from the general mode
resonance conditions.

Plasma Physics