Extended Kalman filtering for vortex systems. Part II: Rankine vortices and observing-system design

Point-vortex systems with both regular and chaotic motion can be tracked by a combination of Lagrangian observations of vortex position and of Eulerian observations of fluid velocity at a few fixed points. If only Eulerian observations are available, singularities in the tracking can develop due to nonlinearity in the observing functions on two occasions: when point vortices pass very close to an observing station, and when the observed speed is small. In Part II of this two-part paper, the vorticity concentrations are approximated by Rankine vortices with finite core in solid-body rotation, while a cut-off criterion is imposed on the Eulerian observations; this removes the singularities and hence results in satisfactory tracking when the observations are judiciously chosen. The main result is that a number of observations comparable to that of the distinct vorticity concentrations suffices for very accurate tracking. The key elements for successful tracking of Rankine-vortex systems using nonlinear, Eulerian observations are: distribution of available stations, threshold criteria for the selection of stations used in each update, and the observing frequency. Simple analysis provides the optimal estimate for these key elements, together with the optimal size of the estimated Rankine vortices. © 1997 Elsevier Science B.V. 1. I n t r o d u c t i o n Geophysical fluid dynamicists have used data assimilation to describe the motion of the atmosphere and oceans from incomplete and noisy data (Ghil, 1989, Daley, 1991, * Corresponding author. 0377-0265/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S0377-0265(97)000171 334 K. lde. M. Ghil / Dynamics ~)['Atmospheres and Oceans 27 (1997) 333-350 Panel on Model-Assimilated Data Sets for Atmospheric and Oceanic Research, 1991, Ghil and Malanotte-Rizzoli, 1991). Kalman filtering and its nonlinear extension, extended Kalman filtering (EKF), are advanced data-assimilation methods rooted in sequential estimation theory (Kalman, 1960: Kalman and Bucy, 1961; Jazwinski, 1970; Ghil et al., 1981). They allow us to forecast and estimate the variables of interest by combining dynamical and statistical aspects of the system. In meteorology and oceanography, data assimilation has been applied to help understand the fundamental physics and dynamics of the flow or for the practical purposes of numerical prediction (Bengtsson et al., 1981; Wunsch, 1988; Robinson et al., 1989; Ghil and Malanotte-Rizzoli, 1991). To describe the dynamical aspects of geophysical fluids, most studies so tar used an Eulerian framework, independent of the goals and techniques of the data-assimilation process. In this framework, fluid quantities such as velocity and density are computed on a spatial or spectral grid according to a finite-dimensional dynamical system of ordinary differential equations obtained by discretizing an infinite-dimensional system of partial differential equations, such as the primitive equations or the quasi-geostrophic equations which describe the flow fields of interest. However, the atmosphere (McWilliams, 1980) as well as the ocean (Robinson, 1983) are known to have much of their energy and vorticity concentrated in approximately two-dimensional localized coherent structures, rather than having a spatially near-homogeneous distribution of fluid quantities (McWilliams, t991). In Part I of this study (Ide and Ghil, 1997a), we saw that, using a Lagrangian representation of the flow field in terms of point vortices, a number of observations comparable to that of vortices suffices, in general, to estimate well the entire flow field. Besides the dynamics of the flow, that may be represented better by a Lagrangian description, the observations used are another important aspect of data assimilation. Observations can be Lagrangian, e.g. buoy trajectories (Carter, 1989) or sea-surface height contours based on remote sensing (Mariano, 1990), as well as Eulerian, such as temperature or current measurements at a fixed station. Therefore geophysical data assimilation may need to deal with Lagrangian dynamics combined with either Lagrangian or Eulerian data sets, or with some combination of both. Based on knowledge about the underlying flow dynamics and possible observations on it, one can design an observing system that provides efficient and successful estimation. Observing-system optimization (Ghil and Ide, 1994) is critical for the oceans where each data point counts, due to the limited number of observations. Barth and Wunsch (1990) and Barth (1992) considered specific optimization problems for acoustic tomography data in linear ocean models (steady-state and time-dependent, respectively) using simulated annealing. Bennett (1992) also considered, in fairly general terms, the problem of observing-system, or 'antenna' design for ocean models. In order to estimate the large-scale flow patterns of geophysical or planetary flows, we used the EKF on point-vortex system in Part I. The EKF consists of the sequential application of two steps (Gelb, 1974, Ghil, 1989, Miller et al., 1994): in the first step, the state variables and their expected error-covariance matrix are forecast from one observing time to the next, according to the underlying dynamics of the system; in the second step, an update of model variables at observing time combines forecast and K. lde, M. Ghil / Dynamics of Atmospheres and Oceans 27 (1997) 333-350 335 observations so as to minimize the estimated error in a least-square sense. In this two-part paper, we focus on observing-system design, and hence emphasize the second step. Ide and Ghil (1997b) discuss a number of issues concerning the first step. Our numerical and analytical results in Part I showed that, despite the possible complex motion of point-vortex systems (Aref, 1984), the EKF works well when all the vortex positions are observed, albeit with relatively large errors. EKF tracking may, however, fail when the number of vortex-position data is insufficient and they are replaced entirely or partially by velocity observations at given locations. There are two main causes of unsuccessful EKF performance when station-velocity observations are used: (1) when at least one of the vortices is very close to the station, so that the measured velocity is very large; (2) when the magnitude of the measured velocity is almost zero, either near a stagnation point where induced velocities due to all the vortices present cancel out or when all the vortices are far away from the station. There are simple, but crude remedies for both problems. In order to avoid the second one, a cut-off criterion for the minimum velocity can be imposed. To avoid the velocity singularity due to the close approach of the vortices to the stations, a maximum velocity cut-off criterion could be imposed as well. At each update time, stations for which either the observed velocity or the one reconstructed from the estimated vortex positions do not meet either cut-off criterion can be ignored in the update process. These criteria were implemented for the point-vortex systems of Part I (not shown there); they do help avoid poor EKF performance in a variety of situations. But the latter type of cut-off can also have deleterious effects, since velocity updates perform best when the vortices are close to the station used in the update--as shown analytically in Part I. In order to overcome the difficulties encountered with point-vortex systems at a more fundamental level, we approximate in Part II the flow field using Rankine vortices. A Rankine vortex is a finite-core circular vortex with top-hat vorticity distribution. The velocity inside the core is a linear function of distance from the center of the vortex, while the outside velocity field decays like that of a point vortex (e.g. Ghil and Solan, 1973). As the core radius goes to zero while keeping the total circulation constant, a Rankine vortex becomes a point vortex. As long as all vortices are well separated, the centers of the Rankine vortices move as if they were point vortices with the same circulation, while the singularities associated with the latter area removed from the flow field. Although this is a crude approximation, it is realistic in the sense that it captures the main interaction between localized coherent vorticity structures, as well as representing the local velocity field around each vortex more realistically than a point vortex does (e.g. Olson, 1980, Feliks and Ghil, 1996). When the minimum cut-off criterion is properly imposed on the Rankine-vortex system, numerical and analytical results show that tracking using only nonlinear Eulerian observation can be successful as well as reasonably efficient. Throughout this study, the total circulation and core radius of the Rankine vortices are assumed to be known. These values can be also estimated, along with the evolution of the vortices, by treating the vortex-system tracking as a parameter-identification problem (Ide and Ghil, 1995). Part II of this paper is organized as follows: in Section 2 we introduce the Rankine-vortex model and the corresponding EKF formulation. In Section 3 numerical 336 K. Ide, M. Ghil / Dynamics of Atmospheres and Oceans 27 (1997) 333-350 results using the nonlinear Eulerian data only are shown; in Section 4 we give a brief analysis for the optimization of the observing system and point out the key elements in successful tracking for the Rankine-vortex system. Analytical and numerical results are summarized and discussed in Section 5. The notation used here follows Ide et al. (1997). 2. Model formulation In the absence of other vortices, a steady Rankine vortex (Fig. 1) is an exact, axisymmetric solution of the Euler equations for two-dimensional flow. A Rankine vortex whose core area is 7to-2 and total circulation is F undergoes solid-body rotation of angular velocity F/27r02 without deforming nor changing its position; it induces the same outside flow field as a point vortex of circulation F located at the center of the Rankine vortex. The velocity field of the Rankine vortex whose center is located at the origin of the coordinate system is m ~ m dz* i2 0 -2 Z for [zl < 0-, ( 1 )