*Vortical structures are commonly found in the Earth's atmosphere
and oceans. Vortices -or coherent structures of potential vorticity
(PV)- play an important dynamical role in these environments. When
both the Froude and Rossby number are small, the atmosphere and oceans
exhibit a quasi-`balanced' state in which the full motion can approximately
be derived from the PV distribution alone.*

*The quasi-geostrophic (QG) fluid model is the simplest `balanced'
model containing the dominant features of stratification and rotation operating
in the atmosphere and oceans. It combines geostrophic and hydrostatic
balance. Hence, while the PV distribution is three dimensional,
the motion is constrained to remain parallel to stratification surfaces,
i.e. is layerwise two-dimensional. Moreover, in absence of dissipative
and diabatic effects, which are often weak, PV is materially conserved.*

*Despite the simplicity of the QG model, accurate direct numerical
simulations still require long time to be performed. For that reason mostly,
even the problem of two-vortex interaction is little known. For uniform-PV
volumes, vortex interactions still depend on 6 parameters: the volume
ratio between the two vortices, the PV ratio, the height-to-width aspect
ratios, the vertical offset and the horizontal offset. The comprehensive
study of such a large parameter space using direct numerical simulation
is arguably impossible. One needs to use a simpler, approximate
approach.*

*This has been made possible with a new model, called the `quasi-geostrophic
ellipsoidal model' (ELM) which represents vortices as uniform-PV ellipsoids
while filtering high-order non-ellipsoidal deformations. This approach
greatly simplifies the problem since a vortex is fully described
by its centroid position and a 3*3 `shape'-matrix in which only 5 variable
coefficients are independent. ELM is a finite Hmailtonian system that reads*

*where X is the centroid position of a given ellipsoid, B is a 3*3
matrix in terms of which the equation of the ellipsoid reads*

*H is the Hamiltonian of the system (i.e. its energy devided by 4*pi)
, S is the "flow matrix" and kappa is the strength of the vortex (volume
intergral of its PV scaled by 4*pi).*

*To read more about the derivation see the reference document linked
from the ELM main page.*
*To see vortices interacting, see the galleries linked from the ELM
main page.*