We examine the critical merging distance between two equal-volume, equal-potential-
vorticity quasi-geostrophic vortices. We focus on how this distance depends on the
vertical offset between the two vortices, each having a unit mean height-to-width
aspect ratio. The vertical direction is special in the quasi-geostrophic model (used to
capture the leading-order dynamical features of stably stratified and rapidly rotating
geophysical flows) since vertical advection is absent. Nevertheless vortex merger may
still occur by horizontal advection.
In this paper, we first investigate the equilibrium states for the two vortices as a
function of their vertical and horizontal separation. We examine their basic properties
together with their linear stability. These findings are next compared to numerical
simulations of the nonlinear evolution of two spheres of potential vorticity. Three
different regimes of interaction are identified, depending on the vertical offset. For
a small offset, the interaction differs little from the case when the two vortices are
horizontally aligned. On the other hand, when the vertical offset is comparable to
the mean vortex radius, strong interaction occurs for greater horizontal gaps than
in the horizontally aligned case, and therefore at significantly greater full separation
distances. This perhaps surprising result is consistent with the linear stability analysis
and appears to be a consequence of the anisotropy of the quasi-geostrophic equations.
Finally, for large vertical offsets, vortex merger results in the formation of a metastable tilted dumbbell vortex.