When assimilating Lagrangian data into a regional primitive equation model, 
Lagrangian coordinates offer two distinct advantages over Eulerian coordinates. 
First, the positions of the floats are the natural dependent variables of the 
model. Second, the coordinates facilitate modeling in Lagrangian open domains 
which, unlike Eulerian open domains, lead to well posed forward and backward 
problems. These are essential for the construction of efficient inversion 
algorithms.

Working with the shallow water model, we have determined that forward 
Lagrangian integrations accurately describe a domain that moves with uniformly 
rotating flow. For time and space scales of North Atlantic mesoscale 
variability, the solutions from inviscid Lagrangian integrations are within a 
few percent of those from inviscid Eulerian integrations for approximately 20 to 
30 days. By this time, the turbulent cascade of entrophy to small scales renders 
the solution meaningless.

Including viscosity in double periodic domains on the flat earth extends the 
time limit to over a hundred days, for grid Reynolds numbers as large as 5 but 
not 10. This time limit is entirely satisfactory and flat earth doubly periodic 
domains will be used in first tests of assimilations of float data over a few 
months.

Anticipating the need to work eventually in nonperiodic open domains on the 
spherical earth, it must be conceded that viscosity resolves the ill posedness 
in principle, but not in practice as discontinuities are replaced with spurious 
boundary layers. Thus the second attraction of Lagrangian coordinates, namely 
that they facilitate computation of well posed Lagrangian open boundary 
problems, remains.