Application of the Forward Sensitivity Method to Data Assimilation of the Lagrangian Tracer Dynamics

Abstract

The analysis of the dynamics of a tracer/drifter/buoy floating on the free surface of the water waves in the open ocean whose motion is described by the shallow water model equations is of great interest in Lagrangian data assimilation. A special case of the low/reduced order version of the linearized shallow water model equations gives rise to a class of tracer dynamics given a system of two first order, nonlinear, time varying systems of ordinary differential equations whose flow field is the sum of a time invariant geostrophic mode that depends on a parameter u0 ∈ R and a time varying inertial-gravity mode that depends on a set of three parameters, αˆ = (u1(0), v1(0), h1(0))T ∈ R3. In this thesis we provide a complete characterization of the properties of the equilibria of the tracer dynamics along with with bifurcation as the four parameters in α = (u0,αˆ)T ∈ R4 are varied. It is shown that the impact of the four parameters can be effectively captured by to systems of intersecting hyperbolas in two dimensions. We then apply the Forward Sensitivity Method (FSM) to assimilate data in the twin experiments following the dynamics of the Lagrangian tracers in the shallow water model. In these experiments, we assume that the error results from the incorrect estimation of the control vector α = (u0, u1(0), v1(0), h1(0))T ∈ R4. We also analyze the sensitivity of the model to changes in the elements of the control vector α in order to improve placement of the observations. We have found that sensitivity, together with the condition number of the matrix constructed with sensitivity values, gives a good prognostications about success of the data assimilation.