  An object's wake in a plasma with small Debye length that drifts
\emph{across} the magnetic field is subject to electrostatic electron
instabilities. Such situations include, for example, the moon in the solar wind
wake and probes in magnetized laboratory plasmas. The instability drive
mechanism can equivalently be considered drift down the potential-energy
gradient or drift up the density-gradient. The gradients arise because the
plasma wake has a region of depressed density and electrostatic potential into
which ions are attracted along the field. The non-linear consequences of the
instability are analysed in this paper. At physical ratios of electron to ion
mass, neither linear nor quasilinear treatment can explain the observation of
large-amplitude perturbations that disrupt the ion streams well before they
become ion-ion unstable. We show here, however, that electron holes, once
formed, continue to grow, driven by the drift mechanism, and if they remain in
the wake may reach a maximum non-linearly stable size, beyond which their
uncontrolled growth disrupts the ions. The hole growth calculations provide a
quantitative prediction of hole profile and size evolution. Hole growth appears
to explain the observations of recent particle-in-cell simulations.
