Extended Kalman filtering for vortex systems. Part 1: Methodology and point vortices

Abstract Planetary flows-atmospheric and oceanic-are approximately two-dimensional and dominated by coherent concentrations of vorticity. Data assimilation attempts to determine optimally the current state of a fluid system from a limited number of current and past observations. In this two-part paper, an advanced method of data assimilation, the extended Kalman filter, is applied to the Lagrangian representation of a two-dimensional flow in terms of vortex systems. Smaller scales of motion are approximated here by stochastic forcing of the vortices. In Part I, the systems studied have either two point vortices, leading to regular motion or four point vortices and chaotic motion, in the absence of stochastic forcing. Numerical experiments are performed in the presence or absence of stochastic forcing. Point-vortex systems with both regular and chaotic motion can be tracked by a combination of Lagrangian observations of vortex positions and of Eulerian observations of fluid velocity at a few fixed points. Dynamically, the usual extended Kalman filter tends to yield insufficient gain if stochastic forcing is absent, whether the underlying system is regular or chaotic. Statistically, the type and accuracy of observations are the key factors in achieving a sufficiently accurate flow description. A simple analysis of the update mechanism supports the numerical results and also provides geometrical insight into them. In Part II, tracking of Rankine vortices with a finite core area is investigated and the results are used for observing-system design.