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Shi Jiang and Michael Ghil

Abstract

Numerical ocean diagnoses and predictions rely on two types of information: model information and data information. Sequential estimation theory shows that the most probable state is a linear combination of the two, weighted according to their error statistics. A Kalman filter technique is applied to a one-layer reduced-gravity linear ocean model in a rectangular midlatitude basin. The model reproduces the main features of the subtropical wind-driven gyre; the filter is used to study the dynamical behavior of the error statistics.

On a midlatitude f plane, the error-correlation patterns among the state variables revealed by the Kalman filter are isotropic and homogeneous and satisfy a geostrophic relation. Introducing the β effect breaks the isotropy and homogeneity of the correlations, inducing behavior that is in agreement with two observational facts: 1) the latitudinal dependence of horizontal correlations and 2) the elliptic correlation shape of the mass field, elongated along the southwest–northeast orientation in the Northern Hemisphere. When a meridional line of observations is assimilated intermittently, the correlation patterns are dynamically adjusted to be wider to the east of the observing line than to the west. This is due to the westward propagation of errors by the model's Rossby wave dynamics.

The influence function of observations, based on the gain matrix of the Kalman filter, is subjected to polar decomposition into an amplitude part and a vector normalized by the amplitude—that is, a solid angle. The amplitude part contains the current observational information and determines the absolute weight given to an observation. The angular part is related to the previous observations only and reflects the structure of relative weights, whose behavior is similar to that of error correlations.

A criterion measuring the relative importance of different types of observations is defined, using Kalman filter techniques and geostrophic-error assumptions. The results from numerical experiments to examine the correctness of this criterion resolve apparent contradictions among the recent results of R. Daley, M. Ghil, and N. A. Phillips.

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Shi Jiang and Michael Ghil

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.

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Shi Jiang, Fei-fei Jin, and Michael Ghil

Abstract

A reduced-gravity shallow-water (SW) model is used to study the nonlinear behavior of western boundary currents (WBCs), with particular emphasis on multiple equilibria and low-frequency variations. When the meridionally symmetric wind stress is sufficiently strong, two steady solutions–nearly antisymmetric about the x axis–are achieved from different initial states. These results imply that 1) the inertial WBCs could overshoot either southward or northward along the western boundary, depending on their initial states; and thus, 2) the WBC separation and eastward jet could occur either north or south of the maximum wind stress line. The two equilibria arise via a perturbed pitchfork bifurcation, as the wind stress increases. A low-order, double-gyre, quasigeostrophic (QG) model is studied analytically to provide further insight into the physical nature of this bifurcation. In this model, the basic state is exactly antisymmetric when the wind stress is symmetric. The perturbations destroying the symmetry of the pitchfork bifurcation can arise, therefore. in the QG model only from the asymmetric components of the wind stress. In the SW model, the antisymmetry of the system's basic response to the symmetric forcing is destroyed already at arbitrarily low wind stress. The pitchfork bifurcation from this basic state to more complex states at high wind stress is accordingly perturbed in the absence of any forcing asymmetry.

Periodic solutions arise by Hopf bifurcation from either steady-state branch of the SW model. A purely periodic solution is studied in detail. The subtropical and subpolar recirculations, separation, and eastward jet exhibit a perfectly periodic oscillation with a period of about 2.8 years. Outside the recirculation zones, the solutions are nearly steady. The alternating anomalies of the upper-layer thickness are periodically generated adjacent to the ridge of the first and strongest downstream meander and are then propagated and advected into the two WBC zones, by Rossby waves and the recirculating currents, respectively. These anomalies periodically change the pressure gradient field near the WBCs and maintain the periodic oscillation. Aperiodic solutions are also studied by either increasing wind forcing or decreasing the viscosity.

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