Lermusiaux, P.F.J. and F. Lekien, 2005. *Dynamics and Lagrangian Coherent Structures in the Ocean and their Uncertainty.* Extended Abstract in report of the "Dynamical System Methods in Fluid Dynamics" Oberwolfach Workshop. Jerrold E. Marsden and Jurgen Scheurle (Eds.), Mathematisches Forschungsinstitut Oberwolfach, July 31st - August 6th, 2005, Germany. 2pp.

The observation, computation and study of “Lagrangian Coherent Structures”
(LCS) in turbulent geophysical
flows have been active areas of research in
fluid
mechanics for the last 30 years. Growing evidence for the existence of LCSs in
geophysical
flows (e.g., eddies, oscillating jets, chaotic mixing) and other
fluid
flows
(e.g., separation prole at the surface of an airfoil, entrainment and detrainment
by a vortex) generates an increasing interest for the extraction and understanding
of these structures as well as their properties.
In parallel, realistic ocean modeling with dense data assimilation has developed
in the past decades and is now able to provide accurate nowcasts and predictions
of ocean
flow fields to study coherent structures. Robust numerical methods
and sufficiently fast hardware are now available to compute real-time forecasts of
oceanographic states and render associated coherent structures. It is therefore
natural to expect the direct predictions of LCSs based on these advanced models.
The impact of uncertainties on the coherent structures is becoming an increasingly
important question for practical applications. The transfer of these uncertainties
from the ocean state to the LCSs is an unexplored but intriguing scientific
problem. These two questions are the motivation and focus of this presentation.
Using the classic formalism of continuous-discrete estimation [1], the spatially
discretized dynamics of the ocean state vector x and observations are described
by
(1a) dx =M(x; t) + d
yok
(1b) = H(xk; tk) + k
where M and H are the model and measurement model operator, respectively.
The stochastic forcings d and k are Wiener/Brownian motion processes,
N(0;Q(t)), and white Gaussian sequences, k N(0;Rk), respectively. In other
words, Efd(t)d
T
(t)g
:=
Q(t) dt. The initial conditions are also uncertain and
x(t0) is random with a prior PDF, p(x(t0)), i.e. x(t0) = bx0 + n(0) with n(0)
random. Of course, vectors and operators in Eqs. (1a-b) are multivariate which
impacts the PDFs: e.g. their moments are also multivariate.
The estimation problem at time t consists of combining all available information
on x(t), the dynamics and data (Eqs. 1a-b), their prior distributions and the initial
conditions p(x(t0)). Defining the set of all observations prior to time t by yt