  In this paper, we examine the dimensionality of a single electrically driven
vortex bounded by two no-slip and perfectly insulating horizontal walls distant
by $h$. The study was performed in the weakly inertial limit by means of an
asymptotic expansion, which is valid for any Hartmann number. We show that the
dimensionality of the leading order can be fully described using the single
parameter $l_z^\nu / h$, where $l_z^\nu$ represents the distance over which the
Lorentz force is able to act before being balanced by viscous dissipation. The
base flow happens to introduce inertial recirculations in the meridional plane
at the first order, which are shown to follow two radically different
mechanisms: inverse Ekman pumping driven by a vertical pressure gradient along
the axis of the vortex, or direct Ekman pumping driven by a radial pressure
gradient in the Hartman boundary layers. We demonstrate that when the base flow
is quasi-2D, the relative importance of direct and inverse pumping is solely
determined by the aspect ratio $\eta / h$, where $\eta$ refers to the width of
the vortex. Of both mechanisms, only inverse pumping appears to act as a
significant source of helicity.
