  A 2D linear theory of the instability of Sweet-Parker (SP) current sheets is
developed in the framework of Reduced MHD. A local analysis is performed taking
into account the dependence of a generic equilibrium profile on the outflow
coordinate. The plasmoid instability [Loureiro et al, Phys. Plasmas {\bf 14},
100703 (2007)] is recovered, i.e., current sheets are unstable to the formation
of a large-wave-number chain of plasmoids ($k_{\rm max}\Lsheet \sim S^{3/8}$,
where $k_{\rm max}$ is the wave-number of fastest growing mode, $S=\Lsheet
V_A/\eta$ is the Lundquist number, $\Lsheet$ is the length of the sheet, $V_A$
is the Alfv\'en speed and $\eta$ is the plasma resistivity), which grows
super-Alfv\'enically fast ($\gmax\tau_A\sim S^{1/4}$, where $\gmax$ is the
maximum growth rate, and $\tau_A=\Lsheet/V_A$). For typical background
profiles, the growth rate and the wave-number are found to {\it increase} in
the outflow direction. This is due to the presence of another mode, the
Kelvin-Helmholtz (KH) instability, which is triggered at the periphery of the
layer, where the outflow velocity exceeds the Alfv\'en speed associated with
the upstream magnetic field. The KH instability grows even faster than the
plasmoid instability, $\gmax \tau_A \sim k_{\rm max} \Lsheet\sim S^{1/2}$. The
effect of viscosity ($\nu$) on the plasmoid instability is also addressed. In
the limit of large magnetic Prandtl numbers, $Pm=\nu/\eta$, it is found that
$\gmax\sim S^{1/4}Pm^{-5/8}$ and $k_{\rm max} \Lsheet\sim S^{3/8}Pm^{-3/16}$,
leading to the prediction that the critical Lundquist number for plasmoid
instability in the $Pm\gg1$ regime is $\Scrit\sim 10^4Pm^{1/2}$. These results
are verified via direct numerical simulation of the linearized equations, using
a new, analytical 2D SP equilibrium solution.
