A brief history of ELM

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.

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