  The manner in which the rate of magnetic reconnection scales with the
Lundquist number in realistic three-dimensional (3D) geometries is still an
unsolved problem. It has been demonstrated that in 2D rapid non-linear tearing
allows the reconnection rate to become almost independent of the Lundquist
number (the `plasmoid instability'). Here we present the first study of an
analogous instability in a fully 3D geometry, defined by a magnetic null point.
The 3D null current layer is found to be susceptible to an analogous
instability, but is marginally more stable than an equivalent 2D
Sweet-Parker-like layer. Tearing of the sheet creates a thin boundary layer
around the separatrix surface, contained within a flux envelope with a
hyperbolic structure that mimics a spine-fan topology. Efficient mixing of flux
between the two topological domains occurs as the flux rope structures created
during the tearing process evolve within this envelope. This leads to a
substantial increase in the rate of reconnection between the two domains.
