ABSTRACT
Regional assimilation of Lagrangian data faces two challenges: the forward and backward boundary - initial value problems for the primitive equations in fixed open regions are ill-posed, (Oliger and Sandstrom,1978; Bennett and Chua,1994) and Lagrangian data are characterized in Eulerian coordinates by nonlinear measurement functionals.
The first challenge makes impossible the transformation of the tangent-linear estimation problem from a state space search into data- space search. Thus weak-constraint estimation of three-dimensional, unsteady circulation is intractable.
The second greatly complicates the iterative application of tangent-linear estimation to nonlinear circulation dynamics.
The first challenge is resolved by adopting a boundary that moves, at each depth, with the local particle speed. In this 'comoving' reference frame, all internal gravity waves are subcritical. The boundary condition count is therefore unambiguous and so the foward and backward problems are well posed. Hence the transforamtion to a data space search is feasible, and may be effected by working with the Euler-Lagrange equations. The latter equations comprise a well-posed two-point boundary value problem in time regardless of the ill-posedness of the forward and backward problems (Bennett,1992), but are not readily solved without recourse to backward and forward integrations.
The comoving domain is naturally expressed in Lagrangian coordinates. Moreover, Lagrangian measurement functionals are linear in such coordinates.
We shall discuss steps towards regional assimilation of Lagrangian data in light of these theoretical considerations.