Mini Symposium on Lagrangian Structure, Lagrangian Data
Organisers: Colin Cotter, Sophia Olhede and Jonathan Lilly
Wednesday 1 July 2009

Speakers

Simon Cotter (Warwick) Lagrangian Data Assimilation for a Viscous Incompressible Fluid (Collaboration with Masoumeh Dashti, James Robinson and Andrew Stuart)

We study the inverse problem of determining the initial state, and possibly the forcing, of a viscous incompressible fluid observed indirectly through the positional data from passive Lagrangian tracers,
over a period of time. We formulate this as a Bayesian inverse problem, giving rise to a probability measure on function space for the initial vector field, and the forcing (or model error). We will describe a well-posed mathematical setting for this problem and describe the results of effective MCMC methods that allow us to sample from such a posterior distribution.

Shane Elipot (Proudman Oceanographic Laboratory) The transfer function for wind-driven oceanic currents

The international oceanographic community maintains an "array" of approximately 1200 freely-drifting buoys at the surface of the World Ocean. These Lagrangian surface "drifters" are drogued at 15 m depth so that their displacements with time are representative of near-surface currents. A large component of the variance of these currents is ascribable to direct forcing by the atmospheric wind stress.

The drifter velocity data, combined with wind data, are used to infer the transfer function in the spectral domain from the atmospheric wind stress to the wind-driven current. In order to interpret physically the observed response to atmospheric forcing, these observations are compared to theoretical transfer functions arising from a suite of theoretical models for the oceanic top boundary layer.

Some general mathematical properties of these transfer functions are discussed and it is shown in particular that the boundary conditions of the theoretical models are crucial in setting their limiting behavior around the resonant inertial frequency.

Emilio Hernandez-Garcia IFISC (CSIC-UIB) Stretching fields and lines in ocean transport dynamics

Developments of the Lagrangian description of fluid motion have allowed the introduction of dynamical systems tools into the characterization of transport dynamics and mixing in fluid flows. In this talk I will describe applications, mainly based in the use of the finite-size Lyapunov exponent, of these ideas to understanding transport in ocean flows. The techniques are flexible enough to deal with velocity fields obtained by simulation and by satellite altimetry observations. Plankton and marine birds are shown to be affected by the physical ocean structures revealed by the Lyapunov analysis.

Jonathan Lilly (Earth and Space Research) • Joint work with Sofia Olhede, UCL • Inferring Oceanic Vortex Properties from Lagrangian Data

The large amounts of data from freely – drifting, or "Lagrangian", subsurface floats have opened a unique window into the structure and variability of the deep ocean currents. Plots of the trajectories from these floats are typically called "spaghetti diagrams" because of their chaotic and frequently looping character. We have developed a method to use this data to infer the properties of oceanic vortices, also known as "coherent eddies". Such vortices are known to be important for transport and mixing of heat and salt, but are difficult to parameterize in models on account of their longevity and ability to translate long distances.

Our method is based on an extension of the ideas of "instantaneous moments" for the analysis of non-stationary (i.e., inhomogeneous in time) univariate signals to the bivariate case. Estimation of the time-varying instantaneous moments is accomplished via a generalization of wavelet ridge analysis to multivariate signals. Random fluctuations – due to instrument noise and the red background flow---lead to stochastic distributions of the estimated instantaneous moments, which we solve for.

Due to a direct relationship between eddy properties and the instantaneous moments of a bivariate oscillation, important aspects of the eddy structure such as its time-varying vorticity can then be inferred, and bounded by error bars. This formalizes an automated and objective solution for the extraction of vortex structures, a task which has hitherto been accomplished by more subjective means.

See also:
Mathematics Research Centre
Mathematical Interdisciplinary Research at Warwick (MIR@W)
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