Abstract:
Difficulties in the assimilation of Lagrangian data arise
because the state of the prognostic model is generally described in
terms of Eulerian variables computed on a fixed grid in space, as a
result there is no direct connection between the model variables and
Lagrangian observations that carry time-integrated information. I will
present a method for assimilating Lagrangian tracer positions, observed
at discrete times, directly into the model. The idea is to augment the
model with tracer advection equations and to track the correlations
between the flow and the tracers via the extended Kalman filter. The
method is applied to point vortex flows with a Gaussian noise term that
is added to simulate unresolved processes. Numerical experiments
demonstrate successful tracking for
systems of two and four (in regular and chaotic regime) vortices
provided the observations are reasonably frequent and accurate and the
system noise level is not too high. Taking into account Lagrangian
nature of drifter observations allows to extract maximal information
about the flow, as a result the
scheme performs better than an alternative indirect approach that
assimilates flow velocity estimated from consecutive tracer positions.
When the noise levels are increased the scheme is prone to abrupt
divergence
caused by the passage of tracers through the neighborhoods of
saddle-points of the velocity field.
Analysis of the filter malfunction shows that it is caused by the
breakdown of the tangent linear approximation (that is used to predicted
the model error covariance matrix) due to the fast exponential error
growth in the vicinity of a saddle.