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  • View in gallery

    Snapshots of the true (reference) h field for the periodic case at (a) 30, (b) 36, (c) 42, (d) 48, (e) 54, and (f) 60 mo. Solid curves stand for h ≥ 500 m and dashed curves for h < 500 m; the contour interval (CI) is 5 m.

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    Time evolution of h anomalies along x = 200 km for 72 mo: upper panel for true h anomalies (periodic); middle panel for noise-corrupted anomalies; and lower panel for assimilation of altimetric data, starting at 18 mo (CI = 7 m). The zero-anomaly contour has been omitted for clarity.

  • View in gallery

    As in Fig. 2 but for an aperiodic case extended to 120 mo (CI = 10 m.)

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    Snapshots of the true (reference) h field for the aperiodic case at (a) 30, (b) 39, (c) 73, (d) 81, (e) 90, and (f) 106 mo. All curves as in Fig. 1 but with CI = 10 m.

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    Time evolution of spatially averaged total energy (unit: 1012Jm2) for the aperiodic case: (a) Simulation results with no data—thick line for true case, thin line for noise-perturbed case; (b) assimilation results for altimetry data being processed starting at 18 mo.

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    As in Fig. 1 but for the error-contaminated simulation.

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    The evolution of rms errors for h component averaged over the whole domain: (a) periodic and (b) aperiodic cases. The rms errors are normalized by rms errors at the 72nd (periodic case) and 120th (aperiodic case) month (cf. ho value in the inset), respectively. “Mod”indicates the model’s rms error growth, with no observations being assimilated, while “IOI” shows the rms error reduction when the model is updated by altimeter data on a daily basis, starting at 18 mo. The arrow labeled “Obs” marks the observational error level in h.

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    As in Fig. 4 but for the error-contaminated simulation.

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    The h-simulation error (hmodelhtrue) at the end of each experiment: (a) and (c) for periodic case, (b) and (d) for aperiodic case; (a) and (b) without assimilating observations, (c) and (d) with data assimilation. CIs are 10 m for (a,b) and 1 m for (c,d); the zero contour is omitted for clarity.

  • View in gallery

    Idealized altimetry networks used in the data assimilation experiments; the repeat periods and track separations correspond to (a) Geosat and (b) TOPEX/POSEIDON. Model-simulated sea surface height variability is sampled along ascending (solid lines) and descending (dotted lines) groundtracks. The daily sequence in which satellite tracks cross the domain is numbered outside the frame (only ascending tracks are numbered). The background contours represent two typical persistent patterns for the aperiodic case: (a) jet and (b) eddy-meander patterns.

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    Ten normalized eigenmodes of the correlation matrix C [Eq. (2.4)], corresponding to the ten largest eigenvalues, in a one-dimensional case with 50 grid points.

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    As in Fig. 1 (periodic case) but for data-updated h fields.

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    As in Fig. 4 (aperiodic case) but for data-updated h fields.

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    Snapshots of h field updated for 6 months by altimetry data with (a) combined ascending and descending Geosat tracks (GEO), (b) combined ascending and descending TOPEX/POSEIDON tracks (TPX), (c) descending Geosat tracks only (GEO-D), and (d) ascending TOPEX/POSEIDON tracks only (TPX-A). All curves as in Fig. 4 (aperiodic case).

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    Simulation error fields corresponding to Fig. 14. All curves as in Fig. 9 but with CI = 2 m.

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    The 6-mo evolution of the global rms errors between the updated and the true h fields for assimilation experiments that differ in temporal and spatial coverage: heavy solid for combined ascending and descending Geosat tracks, light solid for combined ascending and descending TOPEX/POSEIDON tracks, heavy dashed for descending Geosat tracks only, and light dashed for ascending TOPEX/POSEIDON tracks only. All cases are normalized by the observation error, which equals 7.7 m in this case.

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    Fig. A1. Fields of normally distributed random numbers: (a) σh(tk), no spatial correlation; (b) ηh(tk)(= Lσh(tk)), with the spatial correlation given by Eq. (2.4) and the correlation scale LD = 100 km; (c) u; and (d) v components of ηv(tk), geostrophically projected from ηh(tk). CIs are 1.0 for (a), 0.5 for (b), 0.3 for (c), and (d).

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Tracking Nonlinear Solutions with Simulated Altimetric Data in a Shallow-Water Model

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  • 1 Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts
  • | 2 Department of Atmospheric Sciences and Institute of Geophysics and Planetary Physics, University of California, Los Angeles,Los Angeles, California
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Abstract

Low-frequency variability of western boundary currents (WBCs) is pervasive in both observations and numerical models of the oceans. Because advection is of the essence in WBCs, nonlinearities are thought to be important in causing their variability. In numerical models, this variability can be distorted by our incomplete knowledge of the system’s dynamics, manifested in model errors. A reduced-gravity shallow-water model is used to study the interaction of model error with nonlinearity. Here our focus is on a purely periodic solution and a weakly aperiodic one.

For the periodic case, the noise-corrupted system loses its periodicity due to nonlinear processes. For the aperiodic case, the intermittent occurrences of two relatively persistent states—a straight jet with high total energy and a meandering one with low total energy—in the perturbed model are almost out of phase with the unperturbed one. For both cases, the simulation errors are trapped in the WBC region, where the nonlinear dynamics is most vigorous.

Satellite altimeters measure sea surface height globally in space and almost synoptically in time. They provide an opportunity to track WBC variability through its pronounced sea surface signature. By assimilating simulated Geosat data into the stochastically perturbed model with the improved optimal interpolation method, the authors can faithfully track the periodic behavior that had been lost and capture the correct occurrences of two relatively persistent patterns for the aperiodic case. The simulation errors accumulating in the WBC region are suppressed, thus improving the system’s predictability. The domain-averaged rms errors reach a statistical equilibrium below the observational error level.

Comparison experiments using simulated Geosat and TOPEX/POSEIDON tracks show that spatially dense sampling yields lower rms errors than temporally frequent sampling for the present model. A criterion defining spatial oversampling—that is, diminishing returns—is also addressed.

Corresponding author address: Dr. Shi Jiang, Department of Earth, Atmospheric, and Planetary Science, Massachusetts Institute of Technology, Room 54-1422, Cambridge, MA 02139.

Abstract

Low-frequency variability of western boundary currents (WBCs) is pervasive in both observations and numerical models of the oceans. Because advection is of the essence in WBCs, nonlinearities are thought to be important in causing their variability. In numerical models, this variability can be distorted by our incomplete knowledge of the system’s dynamics, manifested in model errors. A reduced-gravity shallow-water model is used to study the interaction of model error with nonlinearity. Here our focus is on a purely periodic solution and a weakly aperiodic one.

For the periodic case, the noise-corrupted system loses its periodicity due to nonlinear processes. For the aperiodic case, the intermittent occurrences of two relatively persistent states—a straight jet with high total energy and a meandering one with low total energy—in the perturbed model are almost out of phase with the unperturbed one. For both cases, the simulation errors are trapped in the WBC region, where the nonlinear dynamics is most vigorous.

Satellite altimeters measure sea surface height globally in space and almost synoptically in time. They provide an opportunity to track WBC variability through its pronounced sea surface signature. By assimilating simulated Geosat data into the stochastically perturbed model with the improved optimal interpolation method, the authors can faithfully track the periodic behavior that had been lost and capture the correct occurrences of two relatively persistent patterns for the aperiodic case. The simulation errors accumulating in the WBC region are suppressed, thus improving the system’s predictability. The domain-averaged rms errors reach a statistical equilibrium below the observational error level.

Comparison experiments using simulated Geosat and TOPEX/POSEIDON tracks show that spatially dense sampling yields lower rms errors than temporally frequent sampling for the present model. A criterion defining spatial oversampling—that is, diminishing returns—is also addressed.

Corresponding author address: Dr. Shi Jiang, Department of Earth, Atmospheric, and Planetary Science, Massachusetts Institute of Technology, Room 54-1422, Cambridge, MA 02139.

1. Introduction and motivation

Remotely sensed altimetric data show that the largest variability of the ocean’s circulation is associated with swift western boundary currents (WBCs), off the east coasts of the continents (Nerem et al. 1994; Wunsch and Stammer 1994). The low-frequency variability and, to some extent, bimodality of various WBCs are also well documented by either conventional or satellite data [e.g., Qiu and Joyce (1992) for the Kuroshio; Olson et al. (1988) for the Brazil Current; Roemmich and Cornuelle (1990) for the East Australia Current, and Brown and Evans (1987) for the Gulf Stream]. Because advection is of the essence in WBCs, nonlinearities are thought to be important in causing their variability.

In a nonlinear system, multiple equilibria may exist, and the system can undergo abrupt transitions from one equilibrium to another as parameters change. Such bifurcation phenomena have been demonstrated in various atmospheric and climate models (Ghil and Childress 1987) and, more recently, in moderately well resolved numerical models of the wind-driven circulation (Chao 1984; Cessi and Ierley 1995; Jiang et al. 1995, JJG hereafter; Ierley and Sheremet 1995). As the nonlinearity becomes sufficiently strong, transition to irregular, chaotic behavior of certain deterministic flows can occur (Lorenz 1963). Chaotic wind-driven circulations have been obtained in both a low-order barotropic system (Veronis 1963) and a two-layer quasigeostrophic (QG) model (McCalpin and Haidvogel 1996). The latter exhibits strong low-frequency variability of a double-gyre system between high- and low-energy states. During the high- energy regime, an extended eastward jet is present and quasi-stable, while the low-energy states are characterized by vigorous meandering and eddy activity. Transitions from equilibria to limit cycles and on to chaotic solutions were studied systematically in a double-gyre, shallow-water (SW) model (JJG; Speich et al. 1995).

The presence of large WBC variability in both the observed data and numerical models poses a challenging problem to data assimilation. There are two steps to be considered in tracking WBC variability in detail: 1) driving model states toward the observations’ attractor (fixed point, limit cycle, or strange attractor) and 2) matching the amplitude and phase of (regular or irregular) oscillations on the attractor. The successful completion of both steps depends on many factors, such as the accuracy of model and data, spatial and temporal coverage of measurements, and assimilation method.

When a nonlinear system is not perfectly described, the model error can distort solutions in any regime: steady, periodic, or chaotic. As a step in studying the interaction of model error with nonlinearity, we examine here how periodic and aperiodic solutions of a deterministic nonlinear SW model are disturbed by stochastic noise. The noise is attributed to the incomplete knowledge of the system’s dynamics, mainly to subgrid-scale errors. As demonstrated later, the system’s noise can lead to loss of periodicity or to improper timing of persistent patterns in the simulation. These results motivate us to estimate the true variability from observed information in a noise-perturbed version of the SW model.

Satellite altimetry offers near-global coverage of surface height and, therewith, derived near-surface currents on synoptic timescales and in all weather. A variety of altimetry missions have been or will be launched (e.g., Arnault and Perigaud 1992). Among these missions, the Geodesy Satellite (Geosat) and the Ocean Topography Experiment (TOPEX/POSEIDON, Fu et al. 1994) bring increasingly high accuracy to bear upon the challenge of ocean topographic mapping. Therefore, Geosat and TOPEX/POSEIDON observations will be simulated in this study’s tracking experiments. In the absence of a sufficiently long stream of real altimetric data for the purpose of our decade-long simulations, we perform twin-type experiments.

The classical identical-twin experiment lacks realism, since both model and observations are assumed to be perfect. Two approaches improve upon this unrealistic type of assimilation study: 1) Assimilation of data from one dynamical model into a different one, for example, multilayer SW and multilevel primitive equation (PE) or QG and PE models; this allows one to test whether the scheme is able to correct for systematic errors. 2) Perturbing both model and observation fields with random noise when using a single model; this permits us to check whether random errors can be corrected by the assimilation scheme. These two approaches could be called dizygotous-twin or, more simply, sibling experiments. In the present work, we adopt the latter approach: Data are sampled from the unperturbed simulation, with observation errors added to account for geoid, tide, orbit, and atmospheric transmissivity uncertainties. A major reason for resorting to this type of experiment is to focus on a highly idealized model in which the variability is caused essentially by the system’s nonlinearity.

Ghil and Malanotte-Rizzoli (1991) have reviewed a wide variety of data assimilation methods for both atmospheric and oceanic flows. Daley (1991), Bennett (1992), and Wunsch (1996) provide further details on modern data assimilation for the atmosphere and oceans, respectively. The present paper represents another step in studying the performance of sequential estimation methods for highly unstable, strongly nonlinear flows. Ghil and Ide (1994) used the extended Kalman filter (EKF) in nonlinear models of vorticity dynamics to study the observability problem for such flows. Miller et al. (1994) used both the EKF and variational methods to track solutions in low-order flow models. Optimization of the updating interval in a nonlinear SW model has been discussed by Jiang (1994). Todling and Ghil (1994) used the Kalman filter in a linear SW model to track barotropic instabilities with very few observations. However, the computational burden of the EKF prevents this method from being directly applied into the present model with n = 15000 gridpoint variables [see discussion of current EKF implementations at the end of Jiang and Ghil (1993)]. As a compromise between optimality and feasibility, we choose a suboptimal scheme, called improved optimal interpolation (IOI: Daley 1992; Todling and Cohn 1994; Jiang 1994), to perform our data assimilation experiments. The initial analysis error covariance is estimated empirically, as in conventional optimal interpolation (OI: e.g., Hao and Ghil 1994), but with a more appropriate inclusion of model error, as in the EKF.

The paper consists of two parts. First, we give a brief introduction to the SW model and its behavior. The model is used to produce true (unperturbed) periodic and aperiodic solutions in section 2a. The procedure for perturbing these solutions with random noise is outlined and the perturbed solutions are discussed in section 2b, emphasizing the loss of periodicity in the periodic regime and phase shifting of persistent jet and meander patterns in the aperiodic regime.

The second part presents the tracking experiments. In section 3, two altimetry networks, Geosat and TOPEX/POSEIDON, are outlined; the observation errors are specified in terms of the corresponding accuracies. The IOI method and its eigenstructure analysis are described in section 4. Section 5 discusses the tracking experiments results, comparing Geosat and TOPEX/POSEIDON performance. In section 6, we summarize the present results and discuss possible directions for future work.

2. The stochastically perturbed nonlinear model

The present model is a higher-resolution, nonlinear version of Jiang and Ghil’s (1993) SW model; it is described in greater detail by JJG. The fluid is assumed to consist of a single active layer of constant density ρand variable thickness h(x,y,t), overlying a deep and motionless layer of density ρ + Δρ. The model domain is confined to a rectangular basin given by 0 ≤ xLand 0 ≤ yD, with x eastward and y northward Cartesian coordinates. The governing equations are
i1520-0485-27-1-72-e2-1a
i1520-0485-27-1-72-e2-1b
V = hv = h(ui + υj) is the horizontal transport, with iand j pointing eastward and nothward, respectively; f = fo + β(y − 0.5D) is the Coriolis parameter, with k pointing upward; and g′ = (Δρ/ρ)g is the reduced gravity. The model ocean is driven by a zonal wind stress modeled as a body force τ̃ = τxi; we choose τx = −τocos(2πy/D), where τo is the amplitude. The bottom friction and the lateral viscosity are scaled by R and A, respectively. No- slip boundary conditions are applied, and the geometric term in the β-plane approximation of the pressure gradient [Eq. (6.3.13a) in Pedlosky (1987)] is neglected. JJG used systematic changes in two nondimensional parameters, αAand ατ, to explore the model’s range of behavior. The discretization by finite differences is described in Jiang (1994), and the model is marginally eddy resolving, with Δx = Δy= 20 km, as in JJG. The experimental parameters are listed in Table 1.

a. Unperturbed solutions

By setting ατ = 0.8 and αA = 1.0, we obtain a purely periodic solution, which has been discussed by JJG in detail. Snapshots of the h field are shown in Fig. 1 at 6-month intervals for a total of 30 months (slightly shorter than the basic 34-month period); they clearly exhibit a periodic strengthening and weakening of the subtropical recirculation, accompanied by stronger or weaker meandering of the eastward jet. The upper panel of Fig. 2 shows the time-continuous h anomalies along a meridional section at x = 200 km, where the eastward jet is strongest after the confluence of the two WBCs near the maximum wind stress line. The h anomalies are largest near the jet axis, with alternating positive and negative features, and correspond to a perfectly periodic oscillation of the eastward jet.

Starting from the periodic solution, aperiodic solutions can be achieved by either increasing ατ or decreasing αA. Here we focus on an aperiodic case with ατ = 0.8 and αA = 0.75. Shown in the upper panel of Fig. 3 is the 120-month (10-year) evolution of h anomalies along the same meridian as in Fig. 2: A 48-year run of this aperiodic solution has been carried out by JJG; the present case is chosen from years 36 to 46 of that run. The eastward jet now oscillates aperiodically with shorter 30-month episodes alternating with longer 70-month ones (see Fig. 10b of JJG). The positive anomalies during the shorter episode in Fig. 3 here move southward (from month 6 to 36) faster than during the longer episode (from month 36 to 109). Each rapid transition between two episodes is accompanied by vigorous eddy activity in both the subtropical and the subpolar gyres.

The ratio of the shorter versus longer episodes’ duration in Fig. 3 is approximately 1:2. Given the irregular length of the episodes in the full 48-year run, it is not clear whether this solution lies along a period doubling (Feigenbaum 1978; Kadanoff 1983) or an intermittency (Pomeau and Manneville 1980) route to chaos (e.g., Seydel 1988). Both types of transition have recently been found in a hierarchy of coupled ocean–atmosphere models for the tropical Pacific (Münnich et al. 1991; Chang et al. 1994; Jin et al. 1994, 1996; Tziperman et al. 1994). Routes to chaos in the wind-driven double- gyre problem are currently the subject of further investigation (H. Dijkstra, K. Ide, and Z. Pan 1996, personal communication). Figure 4 shows instantaneous h fields at 6 selected months, corresponding to transition stages between the shorter and longer episodes (month 39 and 106), high- and low-energy states (month 39 and 81, cf. Fig. 5 below), or stages when the root-mean-square (rms) error of the stochastically perturbed model is near maximum or minimum (month 73 and 90, cf. Fig. 7bbelow) as discussed next.

The total energy for the aperiodic case is calculated to gain more insight into the double gyre’s variability. As shown in Fig. 5a, the total energy (heavy solid curve) also oscillates aperiodically with shorter (from month 6 to 36) and longer (from month 36 to 109) episodes. The high-energy state persists longer than the low-energy one; this behavior is sustained throughout the 48-year experiment (see Fig. 9c in JJG). The h fields corresponding to low- and high-energy states are easy to distinguish from each other. In the low-energy state (e.g., 39 months), eddies and meanders are vigorous and the existence of a free jet is not obvious (see Fig. 4b). By contrast, the high-energy state (e.g., 81 months) is characterized by a strong eastward jet penetrating into the interior of the domain, so that the meander occurs farther downstream and is much weaker (see Fig. 4d). These features of the high- and low-energy states are robust throughout the 48-year experiment (not shown).

The results for our aperiodic case agree with what McCalpin and Haidvogel (1996) found in a wind-driven QG model, although they define low- and high-energy states by constant thresholds, while our low- and high- energy states are located around the trough and the crest of each episode, respectively. Distinct low- and high- energy states are also present in a non-wind-driven isopycnal model due to the interaction between the deep WBC and the Gulf Stream (Spall 1996). Observational evidence of straight-jet and meandering states is found for the Kuroshio with a period of several years (e.g., Taft 1972) and for the Gulf Stream with a 9-month period (e.g., Kelly et al. 1994; Lee and Cornillon 1995).

b. Noise-perturbed solutions

Miller et al. (1994) studied separately the interplay between stochastic perturbations and nonlinearity and between the latter and vigorous instabilities in data assimilation, while Ghil and Ide (1994) studied both deterministically chaotic and stochastically perturbed vortex systems. We investigate here, therefore, nonlinear response to model noise η(tk), k being the time step index and η accounting for our imperfect knowledge of true dynamics. The model noise η(tk) is commonly assumed to be white in time and correlated in space; it contains the velocity components ηv(tk) and mass component ηh(tk). The velocity noise ηv(tk) is derived from ηh(tk) through the geostrophic relations. We further assume that ηh(tk) is Gaussian and white in time, with zero mean and covariance matrix Q(tk),
i1520-0485-27-1-72-e2-2a
i1520-0485-27-1-72-e2-2b
where the operator E denotes the ensemble expectation and δkl is the Kronecker delta.
The true covariance matrix Q(tk) can be approximated by the product
QtkDttk½CDttk½
here the superscript t denotes “true,” Dt(tk) is a diagonal variance matrix Dt(tk) = diagQ(tk) that is allowed to vary in time, and C is a correlation matrix carrying the information on the spatial structure of ηh(tk), as done for the forecast-error covariance matrix in OI (Cohn et al. 1981; Marshall 1985), and is kept fixed in time. In view of the close dynamical similarity between the midlatitude atmosphere and oceans, we adopt the form of the height–height correlation function commonly used in meteorological applications—the Gaussian exponential function:
i1520-0485-27-1-72-e2-4
where Ci,j is the element (i,j) in C and ri,j is the distance between grid points i and j. The decorrelation scale LDdepends on the scale of the features in h (Balgovind et al. 1983); for the WBC systems here LD is of the order of the internal radius of deformation LR. The value of LR ≈ 56 km in Table 1 is fairly realistic for the North Atlantic (see, e.g., Feliks and Ghil 1996a,b), and we take, for simplicity, LD = 100 km. The correlation matrix C and variance matrix Dt(tk) are further assumed to be homogeneous and isotropic in space.
Given Q(tk), the model noise η(tk) is generated as described in the appendix and is then fed into the model at every time step, that is,
i1520-0485-27-1-72-e2-5
for each k where superscripts p and f stand for the “perturbed” and “forecast” fields and G is the geostrophic projector [see also Eq. (4.1b) and discussion there]. In this study, we choose the variance of model noise to be 0.5 m2 day−1, corresponding to a standard deviation of about 0.019 m per time step; this choice is based on Eq. (4.2) in Jiang and Ghil (1993) and arguments given there. Note that the only source of model error is ηh(tk) in the continuity equation (2.1b). This approach differs somewhat from Miller and Cane’s (1989) and Hao and Ghil’s (1994), who both assumed that the dominant source of model error (in a tropical basin) is error in the wind stress. The two main reasons for including random model noise here in the continuity equation are 1) the importance of heat—or, equivalently in our simple model, mass—flux in the mid- and high-latitude ocean circulation and its poor representation in various ocean models with multiple layers (Hogan et al. 1992) or levels (Holland and Bryan 1993; Yu and Malanotte-Rizzoli 1996) and 2) our emphasis on surface-height measurements from altimetry. Net accumulation of mass is avoided by ensuring that η(tk) have spatial zero mean (see appendix) and geostrophic correlation with the model error in the v component according to Eq. (2.5)above.

For the periodic case, Fig. 6 shows the h field produced by the noise-driven model at 6-month intervals. The divergence from the purely deterministic model simulations in the corresponding Fig. 1 is obvious. The shape, strength, and position of the recirculating gyres (e.g., at 30 months), eastward jet (e.g., at 36 months), interior eddies (e.g., at 30 months), and downstream meanders (e.g., at 42 months) are all severely distorted when compared to their counterparts in Fig. 1. The difference of time-evolving h anomalies between the true and noise-contaminated solutions (see upper and middle panels of Fig. 2) is even more striking. The contaminated h anomalies are irregularly distributed along the cross section at x = 200 km, although the fluctuation is still strongest near the jet axis. Two strong positive anomalies are apparent but, in comparison with the true patterns in the upper panel, occur later by about 12 months.

The true periodicity of the solutions is lost after about 10 months. Around the 18th month, negative anomalies occur near the middle of the cross section, where maximum positive anomalies are supposed to occur in the unperturbed solution. Shown in Fig. 7a is the 72-month evolution of domain-averaged rms error in h (the curve labeled “Mod”), for this case in which the unperturbed model behaves periodically; the rms error δh is defined by
i1520-0485-27-1-72-e2-6
where N stands for the total number of grid points. The error growth is parabolic at first, as in linear noise-driven models. It saturates in about 3 months, after which it fluctuates in time due to nonlinear effects.

For the aperiodic case, the comparison between upper and middle panels in Fig. 3 indicates that the stochastically perturbed model exchanges the relative positions in time of the longer (from 9 to 83 months) and shorter (from 83 to 111 months) episodes. Large anomalies are still concentrated near the strongly nonlinear region. The total energy of model solution (light solid line in Fig. 5a) is approximately in phase opposition with the true solution (heavy solid line). In other words, the perturbed model persists in a high-energy state when the true solution is in a low-energy state, and vice versa. This finding is further confirmed by examining snapshots of h in Fig. 8 that correspond in timing to those in Fig. 4. For example, the perturbed model produces a well-penetrating eastward jet at 39 months (Fig. 8b) when an eddy-rich field should obtain (Fig. 4b); the opposite occurs at 81 months (see Figs. 8d and 4d). Accordingly, in Figs. 8a and 8e, the model starts to transit from a low- to a high-energy state, while the true transition should be in the opposite direction, as indicated in Figs. 4a and 4e. Figure 7b shows the rms error of h fluctuating much more vigorously than in the periodic case (Fig. 7a), due to the solution’s stronger nonlinearity.

Interestingly, three major rms error peaks at 30, 73, and 100 months occur during the transitions between low- and high-energy states (or vice versa), instead of during intervals when the energy difference between the perturbed model and unperturbed solutions is maximum. This is in excellent agreement with Miller et al.’s (1994)results on both the stochastically perturbed double well and the Lorenz (1963) model and with Kimoto et al.’s (1992) in a numerical weather prediction model: the largest errors in strongly nonlinear, unstable systems occur at regime transitions. In the present model, the total energy is the sum of the available potential and the kinetic energies; the latter is quadratic in the velocity components, the former in the upper-layer thickness anomalies. The relative size of the two terms seems to change at transition times, which might help explain the discrepancy in behavior of rms errors and total energy differences.

In Fig. 9, we show the simulation error in h, hmodelhtrue, for a periodic case at 72 months (panel a) and an aperiodic case at 120 months (panel b). In both cases, the simulation errors are largest near the strongly nonlinear region, where the separation and confluence of WBCs occur, and are small in the nearly linear Sverdrup region. This is generally true for other epochs, although the amplitude and patterns of the errors might differ. Remember that the net accumulation of model errors at any point is zero, according to Eq. (2.2a). However, when feeding ηh(tk) into the model at time step k, it will be advected and reorganized in time by the model dynamics. Figures 9a and 9b indicate that the nonlinear dynamics plays a key role in propagating simulation errors into the highly advective zone and maintaining them there. This trapping of the largest errors in a relatively limited area should help the design of future observing systems (Bennett 1992; Ghil and Ide 1994) for midlatitude ocean basins.

Given the importance of this result, it is worth noting that Speich et al. (1995) have specifically shown that potential vorticity anomalies (i.e., the field of differences from a mean field) for a periodic solution of the same JJG model evolve outward from the model’s nonlinear, confluence-and-recirculation area in roughly concentric circles (their Fig. 8b). A fairly detailed explanation of the oscillatory instability that gives rise, through Hopf bifurcation, to the finite-amplitude periodic solutions is also given there, in terms of initial Rossby-like wave growth, local steepening, and multipole (di-, tri-, and quadrupole, at various points along the limit cycle arising from the bifurcation) vortex interactions. By carrying out also numerical experiments in a rectangular basin with a size similar to the North Atlantic, 6400 km × 4400 km (their Fig. 16), these authors showed that the concentration of kinetic energy in the same limited area apparent here and in JJG persists, with decay away from it into a fairly regular, quiescent, and very large Sverdrup region.

We next attempt to suppress the accumulated simulation errors in the highly nonlinear region and track both periodic and aperiodic solutions using observed data. It is of particular interest to see whether we are able to capture the eddy persistence (low energy) or the jet persistence (high energy) states through assimilation of surface height information.

3. Space–time sampling strategy

A satellite altimeter measures sea level along a subsatellite track, its orbit being determined by independent ground tracking. The measured sea surface height contains two components: the geoid (the sea level when the ocean is at rest) and a small deviation from it. This small deviation (∼ 1 m) represents many ocean dynamical processes, such as wind-driven gyres and their variability discussed here. Due to inhomogeneities in earth’s density distribution, the geoid undulates around the reference ellipsoid by about 100 m. Accurate determination of the geoid is crucially important in deriving the small deviation from it, induced by oceanic flows. Marshall (1985) showed how to improve geoid estimation through data assimilation, and Thompson (1986)showed how to cope with geoid uncertainty when assimilating altimetry data into a mesoscale ocean model. Future satellite gravity missions, such as the Geopotential Research Mission, could help solve this problem, which we do not treat here.

Two altimetry missions, Geosat and TOPEX/POSEIDON, provide more accurate altimetry information than other past or present ones. By design, these two platforms sample the sea surface height along different repeating orbits. For a repeating satellite orbit, the track- separation distance Ds, in degrees of longitude, is inversely related to the repeat period Pr,
i1520-0485-27-1-72-e3-1a
Here Pr is an integer number of days between exact repeats, and is nondimensional, while Sf is called the fundamental interval and is the longitudinal distance between one track and the parallel one that follows it along the satellite orbit, in the same direction of motion, ascending or descending; Sf is equal to the nodal period Pn times the difference between the rotation rate of earth ωE = 2π rad day−1 and the precession rate of the line of nodes ωn = 2π/365 rad day−1 (Parke et al. 1987),
SfPnωEωn

Note that Ds is the distance between two adjacent (i.e., geometrically closest) tracks, after the complete repeating pattern has been laid out, while Sf is the distance between two subsequent (i.e., temporally closest) tracks. Relation (3.1) limits orbital designs to a choice between high resolutions in either time or space and leads to a sampling dilemma: spatially high sampling may cause poor temporal sampling, and vice versa.

The original Geosat mission lasted from March 1985 to September 1989 and did not have an exact repeat orbit; the forthcoming Geosat Follow-On (GFO) satellite with exactly repeating tracks is currently scheduled to be launched in mid-1997 [(Barry et al. 1995), as updated on the GFO web site at this time, http://GFO.bmpcoe.org]. Geosat’s approximate repeat period was of about 17 days and the track separation at the equator of about 130 km, while TOPEX/POSEIDON flies a recurring pass more frequently (∼ 10 days) with a sparser coverage (∼ 315 km).

Using an eddy-resolving QG model, Holland and Malanotte-Rizzoli (1989) examined the space–time sampling trade-off by providing altimetric data on a gridded map in their model. Their results imply that a finer spatial sampling resolution is more critical for the success of the assimilation than increased time sampling. By assimilating altimeter data into the U.S. Navy’s ocean layer model in the Pacific Ocean, Hogan et al. (1992)showed further that the accuracy of mesoscale variability maps is clearly dominated by the spatial sampling, while the synoptic mapping of mesoscale features is more dependent on the temporal sampling interval. Verron (1990) compared the efficiency of altimetric data assimilation among four sets of orbital parameters in a three-layer QG model. He found that both a 10-day repeat period with 240-km resolution and a 17-day repeat period with 140-km resolution give satisfying results; that is, the model fields converge toward the observations with reasonable accuracy. His results also show that more frequent sampling (10-day case) is of little importance as long as spatial sampling is sufficiently dense and regular.

In fact, the choice of spatial and temporal sampling for any given altimeter depends on the intended application. The aim of this study is to track interannual variability of WBCs and their associated main features—separation, confluence, secondary recirculations, and eastward jet—whose spatial scale is of the order of 100 km. Spatially denser sampling seems preferable in order to map all the WBCs’ features in sufficient detail, while tracking low-frequency variability is weakly affected by the temporal length (order of 10 days) of the measurement. The basic idea is to resolve the analyzed fields based on the observations themselves rather than use the data assimilation scheme as a spatial interpolator, while the known persistence of the vigorous eddies over times exceeding the repeat period (Olson 1980; Sakuma and Ghil 1991) is used to propagate the detailed spatial information forward in time during each repeat period. To test this idea, we adopt the Geosat sampling pattern in conducting most assimilation experiments; the TOPEX/POSEIDON pattern is used only for comparison purposes.

The Geosat and TOPEX/POSEIDON sampling patterns are illustrated in Figs. 10a and 10b, respectively. Here we have made the following simplifications and assumptions: 1) The inclination of ascending tracks is 45 degrees while the nominal inclinations for Geosat and TOPEX/POSEIDON are about 72 and 66 degrees, respectively. Thus, the sampling points coincide with model grid points and no interpolation is needed. This simplification follows Holland and Malanotte-Rizzoli (1989)and Verron (1990). 2) The poleward convergence of the separation distance is only considered between the equator and the southern edge of the domain, being neglected inside the domain itself. This convergence is given, approximately, by the cosine of the latitude; hence, the track separation at 30°N is reduced by about 13% from the equatorial value. Thus, the separation distance at the southern boundary—and within the domain—is about 120 km for Geosat and 280 km for TOPEX/POSEIDON. 3) Along each track, data are sampled at every model grid point; this is rather coarse relative to the actual sampling interval of 7 km for both missions, as the grid points are spaced2Δx = 2Δy apart along the 45° tracks. 4) Data are collected and assimilated into the model simultaneously on a daily basis.

Both abscissa and ordinate in Figs. 10a,b indicate the track sequencing with respect to initial day (day 0). Satellite tracks, either ascending or descending, intersect the domain one after the other as indicated. Typically, Geosat passes the domain 2 or 4 times (descending plus ascending tracks) each day; but there are only two TOPEX/POSEIDON tracks across the domain on any given day. The one-day update interval is a compromise between the broad “windowing” of data covering the entire domain that is still used in operational weather prediction (Daley 1991) and the time-continuous, one-by- one—or satellite patch by satellite patch—processing desirable in truly optimal estimation (Ghil et al. 1979; Ghil and Malanotte-Rizzoli 1991). These simplifications and assumptions are made for modeling convenience and direct comparison between two altimetry missions under similar conditions. They can be easily relaxed in carrying out real-data assimilation.

For the 1½-layer reduced-gravity model used here, the upper-layer thickness h is directly related to the sea surface elevation h′; that is,
i1520-0485-27-1-72-e3-2
This is the simplest case illustrating how surface altimetry information is transferred to the subsurface through the ocean dynamics—the first baroclinic mode here (cf. Hurlburt 1986; Kindle 1986; Thompson 1986). Based on (3.2), we assume that h can be directly derived from the altimetric data. The observation model is, thus, defined as
htkHkhttkϵhtk
where h°(tk) and ht(tk) stand for vector representations of the observational and true h fields, respectively. Generally, the length p of h°(tk) is much less than that of ht(tk), n. The observation matrix Hk maps ht(tk) at model grid points onto h°(tk) at sampling points. The observation error vector ϵh(tk) accounts for uncertainties from geoid, tide, orbit, and atmospheric transmissivity. In this study, Eq. (3.3) states that the observations are sampled from the unperturbed simulations, with observation errors added. We further assume that ϵh(tk) is Gaussian, spatially uncorrelated and serially white, with mean zero and constant variance (ϵh)2 in both time and space:
i1520-0485-27-1-72-e3-4a
where I is the p × p identity matrix. The assumption of spatially uncorrelated ϵh(tk) is not really correct for altimetry data; it is made solely to reduce the computational burden, as it allows us to process observations serially (cf. Bierman 1977; Ghil et al. 1979; Ghil and Malanotte-Rizzoli 1991), 2∼4 tracks every day, and point by point within each track. This assumption can also be removed in more realistic experiments.

The observation noise ϵh has to be specified carefully. Notice that our main purpose is to track the variability of the WBCs. Their amplitude h′ in most oceans is about 35 cm, while the accuracy of actual altimeters is about 3∼10 cm (Arnault and Perigaud 1992; Wunsch and Stammer 1994). The ratio of noise to the WBCs’ signal is of the order of 1:10; this small ratio provides us with an opportunity to address the low-frequency variability of the WBCs. For the model’s periodic case here, however, the maximum surface variability is about ±10 cm (±30 m in h). The ratio of the actual altimetric-data noise to this weaker periodic signal is as high as 4:10, so the altimetry data are too noisy to be useful in tracking this model signal. To make the model situation more comparable to the real one, we introduce an equivalent observation accuracy, which is determined from the model’s variability multiplied by a fairly realistic (at least for TOPEX/POSEIDON) noise-to-signal ratio of 1:10 (see also discussion at the beginning of section 5c). Thus, the compatible observation error corresponding to the periodic case is around 1.1 cm; the equivalent error in h is 3.3 m. By analogy, we choose a larger ϵh= 2.6 cm (∼ 7.7 m in h) for the aperiodic case because of the relatively stronger signal (about ±23 cm in h′ or ±70 m in h).

4. Data assimilation scheme

a. IOI method

The suboptimal IOI scheme (Daley 1992; Jiang 1994;Todling and Cohn 1994) is used to update the error- corrupted h field. The velocity fields can also be improved using the updated h field based on geostrophic relations, since oceanic motions at these scales and latitudes are nearly geostrophic. The IOI scheme has two parts:

  1. State update
    i1520-0485-27-1-72-e4-1a
  2. Covariance update
    i1520-0485-27-1-72-e4-1c
    where superscripts f and a denote forecast and analysis fields and the forecast and analysis error covariance matrices are defined by
    Bf,atkEhf,atkhttkhf,akhttkT
    The matrix-valued operator M is the discrete dynamics operator describing the system (2.1); the weight matrix KIOIk, acting on the height measurements only, is multiplied by the geostrophy matrix G that projects the hfield onto the V field. Here Ba(t0) = 0 is specified since the model’s initial state is identical to the true solution. The notation here follows Ide et al.’s (1996) recent proposal for unified notation in data assimilation, for both dynamic meteorology and physical oceanography.

The major difference between the IOI and the EKF resides in the forecast-error covariance equation (4.1c). In the IOI, Bf(tk) increases linearly at a rate given by Q(tk) between two updating times, while in the EKF Bf(tk) is propagated by the model dynamics and is also inflated with Q(tk). In other words, Bf(tk) in IOI lacks the knowledge of model dynamics.

On the other hand, the IOI method is distinct from conventional OI as it accounts better for the error growth in Bf(tk). In OI, Bf(tk) is given by
BftkDftk½COIDftk½
where COI is a time-constant correlation matrix, while the diagonal variance matrix Df(tk) = diagBf(tk) is time dependent. In the National Meteorological Center’s previously operational OI scheme (McPherson et al. 1979), Df(tk) was allowed to evolve according to
Dftk+1DakΩ
where Da(tk) = diagBa(tk) and Ω measures the growth rate of the forecast-error variance. This operational Ωwas specified on a purely empirical basis. If we assume that the model error η is the only source of Ω—that is, Ω = Dt(tk)—Eqs. (4.2) and (4.3) yield
i1520-0485-27-1-72-e4-4
this is, in general, different from (4.1c) in IOI. Sequential estimation theory suggests that Eq. (4.1c) describes better the growth mechanism of forecast errors than Eq. (4.4) (Todling and Cohn 1994). Daley’s (1992)version of IOI, applied to the barotropic vorticity equation, differs slightly from that implemented here in its way of propagating the forecast-error covariances; still, we chose not to use a different designation for the present method [see also Jiang (1994)] to avoid overloading the literature with terminology.

b. Eigenstructure analysis

To help understand the scheme’s numerical performance in the next section, we carry out an analytic study for some highly idealized situations. To start, we examine the eigenvectors and eigenvalues of the error correlation matrix C. For simplicity, we consider the correlation matrix described by (2.4) for a one-dimensional model with LD = 100 km and n = 50 grid points (equal to the total number of grid points in the x-direction of our model). The eigenvector decomposition of the symmetric and positive definite C is
i1520-0485-27-1-72-e4-5a
where
i1520-0485-27-1-72-e4-5b
is a diagonal matrix with the eigenvalues λi > 0 of Cas its entries and
Ee1eien
is the matrix with the orthonormal eigenvectors as its columns.
Table 2 lists the ten largest eigenvalues λi ordered by descending size, λ1 > λ2 > ··· > λ10 > 0. Note that λi decays exponentially, in agreement with the spatially continuous spectral transform of (2.4):
i1520-0485-27-1-72-e4-6
where m is the wavenumber. Because each diagonal entry of C is unity, it follows that Σ50iλi = 50. Figure 11 shows the ten gravest eigenmodes (Arabic numbering) corresponding to the ten largest eigenvalues. These eigenmodes exhibit the “oscillation property” familiar from Sturm–Liouville problems: the first one (heavy solid curve) has no zero, the second mode has one, and so on. They form statistically independent error patterns;the estimate h is uncertain to the extent of adding to each eigenmode ei a random coefficient of variance λi. Since the ten largest eigenvalues, Σ10iλi = 49.95, represent 99.9% of the total variance, C is well represented by these ten gravest modes.

In order to reduce the forecast-error variance corresponding to the n′ < n largest eigenmodes, n′ equally spaced observations are required, provided that these observations are uncorrelated and ocean behavior does not change rapidly during the repeat period. Table 3shows how much of the relative variance associated with C is captured by different orbital patterns. Notice that TOPEX/POSEIDON and Geosat-D describe the same percentage of total variance, 99.2%.

The spectral response of the IOI algorithm to the altimetric data along a satellite track is also interesting. The variance in the oceanic signal falls off with wavenumber (Wunsch 1981). On the other hand, the variance of observation errors is equally distributed among all scales if these errors are not correlated. The IOI algorithm acts as a filter that extracts the large-scale signal while suppressing the small-scale observational noise. To demonstrate this, we again use the one-dimensional model above for the correlation matrix but lay it out along the track with n = 50 observation points (i.e., assume Hk = I).

First, we project the h variable and error covariances of IOI from the real space onto the orthonormal basis of E:
i1520-0485-27-1-72-e4-7a
i1520-0485-27-1-72-e4-7b
i1520-0485-27-1-72-e4-7c
here the hatted variables are the expansion coefficients for the field in mode space, the time index k has been dropped, and ETE = I has been used, while (σf,a,oh)2 is the error variance for the forecast, analysis, and observation fields, respectively. After projection onto the Espace, the errors in each component i are uncorrelated so that they can be estimated independently, after which they can be recombined by the inverse projection ET. Thus, substituting (4.7) into (4.1), the optimal estimates can be written for component i in the transparent form:
i1520-0485-27-1-72-e4-8a
i1520-0485-27-1-72-e4-8b
where
i1520-0485-27-1-72-e4-8c

Since λi and ν are positive, we have 0 < Kfi < 1. The analysis field can always be improved; that is, σa< σf (Ghil et al. 1981). The degree of this improvement depends on the scale of the error structure: the larger the eigenvalue λi and the graver the ith eigenmode, the greater the error reduction in σa. For those modes with scales so small that λiν and Kfi ∼ 1, there is almost no improvement in σa over σf.

The scale dependence of the two competing weighting factors Kfi and Koi is different: For longer scales with λi> ν, Koi is larger thanKfi and the analysis field ĥai extracts more information from the observation field ĥoi; otherwise,ĥai relies more on ĥfi. For λiν, there is almost no improvement in ĥai—by and large, only the model’s large-scale motions are improved by the observations. Fortunately, these are the motions that dominate the low-frequency variability of interest here.

5. Results of tracking experiments

In this section, we begin to track periodic and aperiodic solutions from altimetry data at 18 months, when the model solutions have been seriously disturbed by noise (section 2b). The altimetry data are generated from the unperturbed simulations with observation errors added (section 3). In theory (Ghil and Malanotte-Rizzoli 1991; Miller 1990), the perturbed run should be regarded as the “truth” since the real ocean is not known and can thus be described as the sum of the purely deterministic model and a stochastic forcing term. In this study, we chose the unperturbed run as the truth because it is more striking to track the exact perodicity of one of its solutions. For consistency, the same procedure is followed in the aperiodic experiment. The assumptions involved in the usual theoretical setup do matter for the optimality of the EKF that is derived based on them; they matter less when dealing with a semiempirical, suboptimal filter like the IOI.

Our main goals are to 1) recover the system’s lost periodicity with the correct phase, 2) capture the correct time of occurrence of the persistent patterns (jet or eddy) in the aperiodic case, 3) improve the solutions’ predictability past the observing time interval for both cases, and 4) compare results between the Geosat and TOPEX/POSEIDON observing stencils. The former stencil is used in sections 5a and 5b below, with the latter added in section 5c.

a. Periodic case

In the upper and middle panels of Fig. 2, we have seen that the pure periodicity of the true solution can be lost due to the stochastic pumping of errors. After observations start to be assimilated at 18 months, the lower panel of Fig. 2 shows that the periodic oscillation of h anomalies is recaptured, although small, rapid fluctuations are still visible. The time evolution of the domain-averaged rms errors is shown in Fig. 7a (curves marked “IOI”).

Within the first repeat cycle of the assimilation starting at 18 months, the rms error of the h component already drops below the observational error level, labeled by the “Obs” arrow on the ordinate. It quickly settles thereafter into a nearly flat equilibrium with small-amplitude fluctuations. These fluctuations correspond to a continuous increase of the rms error, due to the presence of model errors, and to an instantaneous reduction at each observation time (viz., at the end of each day). During every repeat cycle, the instantaneous reduction varies day by day since daily track position and measurement density are different. Overall, the rms error of the h field in this experiment is well controlled by the quantity of data available. Furthermore, since the velocity fields are also updated via the geostrophic relations, the rms errors of these fields are reduced accordingly (not shown).

Snapshots for the updated h field appear in Fig. 12, for the same months as in Figs. 1 and 6. By comparing these three figures, we see that the large-scale circulation patterns, which were disturbed by system noise, track the unperturbed periodic solution rather well. Differences—particularly in the Sverdrup region where the mean features are weaker—are still noticeable at smaller scales; this is due to the scale dependence of the error reduction, as discussed in section 4b. Strongly nonlinear features, like the WBCs, eastward jet, recirculations, and downstream meander, exhibit significant convergence toward the true ones. This improvement is further illustrated by contrasting Fig. 9c with 9a. The simulation error in the updated h field is decreased over the whole domain, with the more drastic reduction in the strongly nonlinear region.

This result has an important implication for the predictability of numerical ocean models that, by necessity, have limited resolution and hence misrepresent subgrid scales, idealized here as stochastic perturbations. As we know, the model’s predictability is quite sensitive to initial errors, particularly so in the highly nonlinear area (section 2). Tracking the model solutions with altimetry data can effectively suppress the initial errors building up and trapped in that area in the absence of observations. Accordingly, predictability is improved for model forecasts starting after the data stream stops, on a daily, weekly or monthly basis.

b. Aperiodic case

In section 2b, we showed that there exist two persistent patterns in the aperiodic solution. The extended-jet pattern is always associated with high total energy and lasts about 1–3 years, while the eddy-meandering pattern corresponds to low total energy and persists about 3–6 months. The system’s variability is characterized by the alternating occurrence of these two persistent patterns and the transition between them (intermediate energy states). Figure 5a indicates that model error displaces the timing of these persistent patterns; in this particular case, they are almost out of phase with respect to the true solution. As shown in Fig. 5b, however, the total energy of the updated model follows the true energy curve very well, as soon as observations are starting to be processed at 18 months; the superimposed noiselike fluctuations do not affect this overall tracking of the slower, deterministically aperiodic evolution: the true phase of the persistent patterns is recaptured from altimetry data.

When comparing the h anomalies among the upper, middle, and lower panels in Fig. 3, we find that the longer and shorter episodes have been shifted back to their true positions. The noisy features in the Sverdrup region are essentially filtered out, leaving the correct features of the true oscillation behind. The horizontal structure of the updated h field is illustrated in Fig. 13at 6 selected months, as in Figs. 4 and 8. The two persistent patterns, jet extension (at 81 months) and eddy meandering (at 39 and 106 months), are well reconstructed from altimetry data. A similar improvement in the updated h field is also seen during other interesting stages, such as the maximum rms error stages (at 30, 39, and 73 months), minimum rms error stages (at 90 months), and transition stages between long and short episodes (at 39 and 106 months, see Fig. 3).

As in the periodic case, the rms error of the h component is sharply reduced to a nearly constant level, well below the observational error mark (see the “Obs”arrow in Fig. 7b). Although the rms error increases each day, due to the system noise, it decreases again when altimetry data become available by the end of that day. Figure 9d shows the estimation error in the assimilated h field at 120 months. As in the periodic case, the simulation error is diminished over the entire domain, but most strikingly so in the highly nonlinear region (contour intervals are 10 m in panel b and 1 m in panel d). However, it is still greater than for the periodic case (see Fig. 9c) due to less accurate observations, the stronger nonlinearity, and the ensuing greater energy and irregularity of the flow. Once more, the remarkable reduction of the simulation error in the strongly nonlinear region can effectively improve forecast accuracy, past the time at which the data stream stops.

c. Comparison between Geosat and TOPEX/POSEIDON

The TOPEX/POSEIDON altimeter is more accurate than the one on Geosat: the accuracy of the former is of 3∼5 cm, while that of the latter is of 5∼10 cm, depending on the scale of the measured features. On the other hand, the sampling patterns of these two altimetry missions are quite different, as discussed in section 3(see Fig. 10). Our intention here is to examine the efficiency of the two sampling patterns in tracking the low-frequency variability of WBCs, as well as smaller- scale eddy and meander features. For simplicity, we assume that Geosat and TOPEX/POSEIDON have the same equivalent accuracy, of 2.6 cm for the aperiodic case. An additional reason to do so is that the more accurate GFO mission will be launched shortly [Barry et al. (1995), as updated on the GFO web site at this time, http://GFO.bmpcoe.org].

The experiments to compare between the Geosat and TOPEX/POSEIDON flight patterns consists of four cases: 1) full Geosat tracks (C1), 2) full TOPEX/POSEIDON tracks (C2), 3) Geosat descending tracks only (C3), and 4) TOPEX/POSEIDON ascending tracks only (C4). Note that the Geosat descending tracks are parallel to the TOPEX/POSEIDON ascending tracks. We perform these experiments for the aperiodic case from month 33 to 39, during which the meandering pattern is persistent in the true solution.

Figure 14 shows the h field at the end of the 6-month assimilation cycle. Comparing to the true solution in Fig. 4b, the basic meandering features are well captured in all four cases. The successful tracking in C4 is, in fact, somewhat surprising since only one TOPEX/POSEIDON ascending track crosses the domain each day;this success must be due to the slowly evolving dynamics and to the IOI scheme’s being sufficient for the tracking. However, as shown in Fig. 15, the simulation error in C4 is still largest because the C4 sampling captures a smaller fraction of the total error variance. The differences in estimation error between C1, C2, and C3 are not appreciable.

In Fig. 16, we display the evolution for 6 months of rms errors in h, normalized by the observation error (7.7 m). The rms errors for all cases drop rapidly below the observation error level after the first repeat period and reach statistical equilibrium thereafter. The increase visible in all four curves between months 3 and 4 of the comparison (i.e., months 36–37 of the full experiment) coincides with the eddies being most active and the total energy lowest.

The following remarks can be made by examining Fig. 16. 1) The full Geosat sampling pattern (C1: heavy solid) is most efficient in tracking the meandering aperiodic solution, as measured by average rms errors: the estimation error drops fastest and reaches a statistical equilibrium before the others do. The partial TOPEX/POSEIDON coverage (C4: light dashed) has the poorest performance, indicating that finer spatial coverage is more important than a shorter repeat period, in agreement with Holland and Malanotte-Rizzoli (1989). 2) Although the experiment with partial Geosat coverage (C3: heavy dashed) samples data more frequently than the one with full TOPEX/POSEIDON coverage (C2: light solid), it is difficult to discriminate between them after the initial repeat period. Note that these two sampling patterns describe theoretically the same fraction of total error structure (see Table 3). Thus, when the sampling can represent a large fraction of total error variance, say, greater than 90%, the denser temporal sampling does not seem to help much in reducing the rms error; at least, our numerical experiments do not appear to distinguish between 10- and 17-day repeating orbits, in agreement with Verron (1990). Note that the aperiodic case in this study is still weakly nonlinear and the eddies produced by the 20-km resolution of our SW model have fairly large scales, whereas the eddy-resolving models used by Holland and Malanotte-Rizzoli (1989) and Verron (1990) are strongly nonlinear and capture smaller scales, so that the eddies studied by these authors evolve much faster and more vigorously than in the present case. 3) The difference between C2 and C4 is greater than that between C1 and C3. This means that Geosat performance is less affected by the loss of either descending or ascending tracks than is the TOPEX/POSEIDON. 4) The rms error reduction by C1 is not much different from C2 after 4 months of data assimilation. Clearly, once the flight pattern has enough coverage (say, more than eight tracks for each repeat period), increasing spatial sampling gives only a slight improvement. Below such a threshold, denser sampling improves performance considerably, as shown by the difference between C3 and C4. This result is consistent with those of Hogan et al. (1992).

We also conducted such a comparison among sampling patterns from month 78 to 84, when the jet is persistent. The results (not shown) are quite similar to those for the the meandering episode above.

6. Summary and discussion

An ocean model is a concise mathematical statement of our dynamical understanding of the oceans’ circulation. Of course this understanding is far from complete. But to the extent that a numerical model is a faithful representation of important aspects of the dynamics, it will be a sensible and powerful tool to spread the observed information in space and time and to further monitor the real ocean circulation. Recent numerical studies of western boundary currents (WBCs) indicate that the system’s intrinsic nonlinearity can cause low-frequency variability in the wind-driven, double- gyre circulation (Cessi and Ierley 1995; JJG; McCalpin and Haidvogel 1996; Speich et al. 1995). The potential impact of this midlatitude oceanic variability on interannual climate changes was discussed by Speich and Ghil (1994). Our incomplete knowledge of the system’s dynamics should be accounted for by admitting model errors, which can distort the inferred variability.

In this paper, we have examined how periodic and aperiodic solutions of our deterministic, nonlinear, reduced-gravity SW model are perturbed by model errors, assumed to be random. For the periodic case, we have shown that the periodicity of the undisturbed solution is lost after about 10 months of stochastic perturbation (Fig. 2b). Instantaneous pictures of the height field (Fig. 6) are all distorted with respect to the true field (Fig. 1). The globally averaged rms error of the h field (Fig. 7a) grows parabolically at the beginning and then saturates and fluctuates in time due to the nonlinear effects.

There exist two relatively persistent patterns of the true h field in the aperiodic case (Fig. 10b in JJG). One shows strong jet penetration (Fig. 4d here) and is associated with a high-energy state (Fig. 5), the other is characterized by eddies and large meanders (e.g., Fig. 4b) and a low-energy state (Fig. 5). The former can last 1–3 years, whereas the latter has shorter durations of 3–6 months. These phenomena are similar to high and low index cycles in the atmospheric circulation (e.g., Namias 1950; Ghil and Childress 1987, §§ 5.5 and 6.5). The system’s low-frequency variability is characterized in this case by the alternating occurrence of these two persistent patterns and the transitions between them. When the system is perturbed by random noise, the timing of eddy and jet patterns is reversed with respect to the unperturbed solution.

In both the periodic and aperiodic case, the simulation errors are trapped in the highly nonlinear region around the separation of the WBCs and the birth of the eastward jet from their confluence (Figs. 9a and 9b). Since the system’s low-frequency variability is sensitive to the errors in this region, it is important for the system’s subsequent predictability to suppress the growth of these errors. To do so, observations are needed, and have been used in the remainder of this study.

Satellite altimeters offer a promising capability for routinely providing global observations suitable for tracking the oceanic circulation and its variability. WBC systems, in particular, such as the eastward jet and recirculating gyres, are associated with pronounced sea surface variations, which have been shown to be detectable in altimetric records (e.g., Cheney and Marsh 1981). We have performed tracking experiments using simulated Geosat data and the improved optimal interpolation method (Jiang 1994; Todling and Cohn 1994). The IOI method is better than the conventional “optimal interpolation” method of operational weather prediction (Daley 1991) since it uses a more realistic model of error growth but is still suboptimal because the forecast- error covariance is not propagated by the full model dynamics as in the extended Kalman filter (EKF).

We summarize the tracking results for both periodic and aperiodic cases as follows: 1) For the periodic case, the periodicity that had been lost due to model errors is recaptured (Figs. 2c and 12); 2) for the aperiodic case, the timing of persistent episodes with high and low energy is also recaptured (Figs. 3c, 5b, and 13); 3) the domain-averaged rms errors of the h field drop below the observational error level by a factor of about 4:10 (Fig. 7b); 3) the simulation errors in the h field at large scales are effectively suppressed, particularly in the strongly nonlinear region; and 4) the analyzed fields (Figs. 12 and 13) are well estimated by the assimilation cycle. These results could change for more complex ocean models or for model errors in wind stress, as well as heat flux (Miller and Cane 1989; Hao and Ghil 1994) but they seem, at least, to provide some guidance on the benefits of data assimilation in ocean problems with a high degree of nonlinearity.

We have also compared data assimilation performance, measured by an average rms level (Fig. 16), between Geosat and TOPEX/POSEIDON flight patterns (Fig. 10), while assuming, for simplicity, that the accuracies of the two altimeters are the same. Both missions are able to reduce the rms errors below the observation error level, when using the IOI method. However, differences between them are apparent too (see also Figs. 14 and 15). When both ascending and descending tracks are available, Geosat does a noticeably better job than TOPEX/POSEIDON during the first 4 months. After that, the difference in performance between these two sampling patterns is rather small; Geosat is only slightly better. When either ascending or descending tracks are used, Geosat is clearly better than TOPEX/POSEIDON. The number of Geosat descending tracks per unit time is the same as that of TOPEX/POSEIDON full tracks (Table 3); more frequent sampling by TOPEX/POSEIDON does not make much of a difference with respect to the less frequent sampling for Geosat with descending tracks only (see light solid and heavy dashed lines in Fig. 16).

It follows from our numerical experiments that spatially dense sampling is preferable to temporally frequent sampling, at least within the repeat-period window of 10–17 days considered and for the slowly evolving dynamical model used here. These results agree with Holland and Malanotte-Rizzoli (1989) and Verron (1990). However, for the spatial sampling there is a threshold, beyond which the improvement is very limited as sampling resolution is increased. The threshold is defined by the number of tracks that capture the number of largest-scale eigenmodes describing about 90% of the forecast-error correlation structure.

The present study mainly focuses on tracking rather than predicting the WBCs’s behavior using both model and observed information. Effectively tracking the WBCs’s variability will eventually help to forecast it. Future work will investigate 1) real altimetry data and more realistic model configurations, 2) experiments at higher resolution, and 3) application of the extended Kalman filter to the nonlinear stochastic model. Work on these problems can contribute to improve the assimilation of oceanic data into numerical models.

Acknowledgments

It is a pleasure to thank Z. Hao, K. Ide, P. Malanotte-Rizzoli, R. N. Miller, A. Or, D. Stammer, R. Todling, and C. Wunsch for helpful discussions and suggestions. SJ is greatly indebted to P. Malanotte-Rizzoli for letting him complete this project while at MIT. The two reviewers’ comments helped improve the presentation. This research was supported at UCLA by ONR Grant N00014-89-J-1845 (MG and SJ) and at MIT by a NOAA Postdoctoral Fellowship in Climate and Global Change (SJ), administered by UCAR. MG also benefitted from the support of the Condorcet Visiting Chair he held while on sabbatical at the Ecole Normale Supérieure, Paris.

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APPENDIX

Generation of Model Noise η(tk)

Given the model-error covariance matrix Q(tk), we generate η(tk) as follows:

1) A normally distributed random vector σh(tk) is produced by a random-number generator; σh(tk) has zero mean and unit covariance at zero lag:
i1520-0485-27-1-72-ea1a
Technically, the mean of pseudorandom numbers is not equal to zero; that is,
i1520-0485-27-1-72-eqa1
where Nt is the total sample length. How much the mean differs from zero depends on the quality of each random- number generator. Hence, Eq. (A.1a) is not satisfied and, furthermore, the random vector sequence is not white; that is, the true correlation between the vectors in the sequence is not zero. The former problem can be overcome by shifting each random sample by its actual mean, −σh, thus guaranteeing (A.1a). This simple trick also improves the situation with respect to (A.1b), but it is impossible to remove all long periods from pseudorandom sequence generation.
2) Since Q(tk) is symmetric and positive definite, its Cholesky decomposition (cf. Maindonald 1984) yields
QtkLLT
where L is a real lower triangular matrix.
3) The vector ηh(tk) of h errors is given by
ηhtkLσhtk
which has the statistical properties (2.2a,b), since
EηhtkLEσhtk
i1520-0485-27-1-72-ea-4b

4) The velocity component ηv(tk) is derived from ηh(tk) through the geostrophic relations. Figure A1shows one sample of the model noise η(tk).

Fig. 1.
Fig. 1.

Snapshots of the true (reference) h field for the periodic case at (a) 30, (b) 36, (c) 42, (d) 48, (e) 54, and (f) 60 mo. Solid curves stand for h ≥ 500 m and dashed curves for h < 500 m; the contour interval (CI) is 5 m.

Citation: Journal of Physical Oceanography 27, 1; 10.1175/1520-0485(1997)027<0072:TNSWSA>2.0.CO;2

Fig. 2.
Fig. 2.

Time evolution of h anomalies along x = 200 km for 72 mo: upper panel for true h anomalies (periodic); middle panel for noise-corrupted anomalies; and lower panel for assimilation of altimetric data, starting at 18 mo (CI = 7 m). The zero-anomaly contour has been omitted for clarity.

Citation: Journal of Physical Oceanography 27, 1; 10.1175/1520-0485(1997)027<0072:TNSWSA>2.0.CO;2

Fig. 3.
Fig. 3.

As in Fig. 2 but for an aperiodic case extended to 120 mo (CI = 10 m.)

Citation: Journal of Physical Oceanography 27, 1; 10.1175/1520-0485(1997)027<0072:TNSWSA>2.0.CO;2

Fig. 4.
Fig. 4.

Snapshots of the true (reference) h field for the aperiodic case at (a) 30, (b) 39, (c) 73, (d) 81, (e) 90, and (f) 106 mo. All curves as in Fig. 1 but with CI = 10 m.

Citation: Journal of Physical Oceanography 27, 1; 10.1175/1520-0485(1997)027<0072:TNSWSA>2.0.CO;2

Fig. 5.
Fig. 5.

Time evolution of spatially averaged total energy (unit: 1012Jm2) for the aperiodic case: (a) Simulation results with no data—thick line for true case, thin line for noise-perturbed case; (b) assimilation results for altimetry data being processed starting at 18 mo.

Citation: Journal of Physical Oceanography 27, 1; 10.1175/1520-0485(1997)027<0072:TNSWSA>2.0.CO;2

Fig. 6.
Fig. 6.

As in Fig. 1 but for the error-contaminated simulation.

Citation: Journal of Physical Oceanography 27, 1; 10.1175/1520-0485(1997)027<0072:TNSWSA>2.0.CO;2

Fig. 7.
Fig. 7.

The evolution of rms errors for h component averaged over the whole domain: (a) periodic and (b) aperiodic cases. The rms errors are normalized by rms errors at the 72nd (periodic case) and 120th (aperiodic case) month (cf. ho value in the inset), respectively. “Mod”indicates the model’s rms error growth, with no observations being assimilated, while “IOI” shows the rms error reduction when the model is updated by altimeter data on a daily basis, starting at 18 mo. The arrow labeled “Obs” marks the observational error level in h.

Citation: Journal of Physical Oceanography 27, 1; 10.1175/1520-0485(1997)027<0072:TNSWSA>2.0.CO;2

Fig. 8.
Fig. 8.

As in Fig. 4 but for the error-contaminated simulation.

Citation: Journal of Physical Oceanography 27, 1; 10.1175/1520-0485(1997)027<0072:TNSWSA>2.0.CO;2

Fig. 9.
Fig. 9.

The h-simulation error (hmodelhtrue) at the end of each experiment: (a) and (c) for periodic case, (b) and (d) for aperiodic case; (a) and (b) without assimilating observations, (c) and (d) with data assimilation. CIs are 10 m for (a,b) and 1 m for (c,d); the zero contour is omitted for clarity.

Citation: Journal of Physical Oceanography 27, 1; 10.1175/1520-0485(1997)027<0072:TNSWSA>2.0.CO;2

Fig. 10.
Fig. 10.

Idealized altimetry networks used in the data assimilation experiments; the repeat periods and track separations correspond to (a) Geosat and (b) TOPEX/POSEIDON. Model-simulated sea surface height variability is sampled along ascending (solid lines) and descending (dotted lines) groundtracks. The daily sequence in which satellite tracks cross the domain is numbered outside the frame (only ascending tracks are numbered). The background contours represent two typical persistent patterns for the aperiodic case: (a) jet and (b) eddy-meander patterns.

Citation: Journal of Physical Oceanography 27, 1; 10.1175/1520-0485(1997)027<0072:TNSWSA>2.0.CO;2

Fig. 11.
Fig. 11.

Ten normalized eigenmodes of the correlation matrix C [Eq. (2.4)], corresponding to the ten largest eigenvalues, in a one-dimensional case with 50 grid points.

Citation: Journal of Physical Oceanography 27, 1; 10.1175/1520-0485(1997)027<0072:TNSWSA>2.0.CO;2

Fig. 12.
Fig. 12.

As in Fig. 1 (periodic case) but for data-updated h fields.

Citation: Journal of Physical Oceanography 27, 1; 10.1175/1520-0485(1997)027<0072:TNSWSA>2.0.CO;2

Fig. 13.
Fig. 13.

As in Fig. 4 (aperiodic case) but for data-updated h fields.

Citation: Journal of Physical Oceanography 27, 1; 10.1175/1520-0485(1997)027<0072:TNSWSA>2.0.CO;2

Fig. 14.
Fig. 14.

Snapshots of h field updated for 6 months by altimetry data with (a) combined ascending and descending Geosat tracks (GEO), (b) combined ascending and descending TOPEX/POSEIDON tracks (TPX), (c) descending Geosat tracks only (GEO-D), and (d) ascending TOPEX/POSEIDON tracks only (TPX-A). All curves as in Fig. 4 (aperiodic case).

Citation: Journal of Physical Oceanography 27, 1; 10.1175/1520-0485(1997)027<0072:TNSWSA>2.0.CO;2

Fig. 15.
Fig. 15.

Simulation error fields corresponding to Fig. 14. All curves as in Fig. 9 but with CI = 2 m.

Citation: Journal of Physical Oceanography 27, 1; 10.1175/1520-0485(1997)027<0072:TNSWSA>2.0.CO;2

Fig. 16.
Fig. 16.

The 6-mo evolution of the global rms errors between the updated and the true h fields for assimilation experiments that differ in temporal and spatial coverage: heavy solid for combined ascending and descending Geosat tracks, light solid for combined ascending and descending TOPEX/POSEIDON tracks, heavy dashed for descending Geosat tracks only, and light dashed for ascending TOPEX/POSEIDON tracks only. All cases are normalized by the observation error, which equals 7.7 m in this case.

Citation: Journal of Physical Oceanography 27, 1; 10.1175/1520-0485(1997)027<0072:TNSWSA>2.0.CO;2

i1520-0485-27-1-72-f17

Fig. A1. Fields of normally distributed random numbers: (a) σh(tk), no spatial correlation; (b) ηh(tk)(= Lσh(tk)), with the spatial correlation given by Eq. (2.4) and the correlation scale LD = 100 km; (c) u; and (d) v components of ηv(tk), geostrophically projected from ηh(tk). CIs are 1.0 for (a), 0.5 for (b), 0.3 for (c), and (d).

Citation: Journal of Physical Oceanography 27, 1; 10.1175/1520-0485(1997)027<0072:TNSWSA>2.0.CO;2

Table 1.

Model parameters.

Table 1.
Table 2.

Ten largest eigenvalues λi of correlation matrix C. Leading variance: Σ10iλi = 49.95; total variance: Σ50iλi = 50.

Table 2.
Table 3.

Fraction of Gaussian correlation structure C captured by various altimetric observing patterns. Geosat-D denotes Geosat descending tracks only; TOPEX-A denotes TOPEX ascending tracks only. Modes captured and variance associated with them shown for correlation matrix C (see text for details).

Table 3.

* Preliminary results of this investigation were presented at the Second WMO International Symposium on Assimilation of Observations in Meteorology and Oceanography, Tokyo, Japan, March 1995.

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