  We validate a new law for the aspect ratio $\alpha = H/L$ of vortices in a
rotating, stratified flow, where $H$ and $L$ are the vertical half-height and
horizontal length scale of the vortices. The aspect ratio depends not only on
the Coriolis parameter f and buoyancy (or Brunt-Vaisala) frequency $\bar{N}$ of
the background flow, but also on the buoyancy frequency $N_c$ within the vortex
and on the Rossby number $Ro$ of the vortex such that $\alpha = f \sqrt{[Ro (1
+ Ro)/(N_c^2- \bar{N}^2)]}$. This law for $\alpha$ is obeyed precisely by the
exact equilibrium solution of the inviscid Boussinesq equations that we show to
be a useful model of our laboratory vortices. The law is valid for both
cyclones and anticyclones. Our anticyclones are generated by injecting fluid
into a rotating tank filled with linearly-stratified salt water. The vortices
are far from the top and bottom boundaries of the tank, so there is no Ekman
circulation. In one set of experiments, the vortices viscously decay, but as
they do, they continue to obey our law for $\alpha$, which decreases over time.
In a second set of experiments, the vortices are sustained by a slow continuous
injection after they form, so they evolve more slowly and have larger |Ro|, but
they also obey our law for $\alpha$. The law for $\alpha$ is not only validated
by our experiments, but is also shown to be consistent with observations of the
aspect ratios of Atlantic meddies and Jupiter's Great Red Spot and Oval BA. The
relationship for $\alpha$ is derived and examined numerically in a companion
paper by Hassanzadeh et al. (2012).
