  At the poles of Jupiter, cyclonic vortices are clustered together in patterns
made up of equilateral triangles called vortex crystals. Such patterns are seen
in laboratory flows but never before in a planetary atmosphere, where the
planet's rotation and gravity add new physics. Here we use the shallow water
(SW) equations at the pole of a rotating planet to study the emergence and
evolution of vortices starting from an initial random pattern of small-scale
turbulence. The flow is in a single layer with a free surface whose slope
produces the horizontal pressure gradient force. We explored three parameters
in the problem: the mean kinetic energy of the initial turbulence, the
horizontal scale of the initial turbulence, and the radius of deformation of
the undisturbed fluid layer. We find that some regions of this parameter space
lead to vortex crystals and others lead to chaotic behavior and mergers. Our
results identified that the relative change of the layer thickness is the key
quantity that determines whether the vortex crystal or chaotic patterns emerge.
