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Nathan Paldor and Michael Ghil

Abstract

Finite-wavelength instabilities of a coupled density front with zero potential vorticity are found for the single-layer and the two-layer problems. These instabilities result from the resonance between two distinct waves whose real phase speeds coalesce. In the single-layer problem, the range of wavenumbers over which the coalescence takes place decreases with increasing wavenumber; consequently, the instability exponents and the growth rates also decrease. For shallow lower layers, the coalescence range increases with increasing wavenumber; at large wavenumbers, the coalescence range becomes continuous, while the instability exponent is approaching a constant value. The growth rate in the two-layer problem increases, therefore, linearly with wavenumber and the short waves fastest. These short-wave instabilities are qualitatively reminiscent of small-scale features along coastal fronts and in laboratory experiments.

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Fei Chen and Michael Ghil

Abstract

An idealized North Atlantic Ocean model is forced by climatological wind stress, restoring temperature, and a diagnosed salinity flux. Both centennial and interdecadal oscillations are sustained in the model if the diagnosed salinity flux is characterized by net evaporation in high latitudes. To investigate further the role of salinity fluxes two different linear profiles are imposed: one has net evaporation in high latitudes and the other net precipitation. The first salinity flux induces a purely interdecadal oscillation in the model, while the second one causes a millennial and a decadal-to-interdecadal oscillation. Next, the authors consider a boundary condition for temperature expressed as the sum of a fixed heat flux and a restoring term. Constant heat flux characterized by net cooling in high latitudes leads to an interdecadal oscillation similar to the one caused by net evaporation.

Both the decadal-to-interdecadal and the purely interdecadal oscillation are upper-ocean phenomena. Inter-decadal anomalies are mainly confined to high latitudes, with their center moving anticlockwise near the north-west corner of the model domain; they are amplified and sink in that region. Decadal-to-interdecadal anomalies are mainly confined to midlatitudes, advected eastward by the mean flow, and disappear near the cast coast.

The physical mechanisms for the two oscillations are different. The interdecadal oscillation is caused by surface-density variations in northern high latitudes; variations are due to either net evaporation from the applied salinity flux or constant cooling from the applied heat flux. The decadal-to-interdecadal oscillation is a by-product of deep-water warming, due to the strong braking effect of salinity forcing on thermal forcing: surface saline water from the subtropics overlies continuously warming intermediate water to provide a favorable environment for the decadal-to-interdecadal oscillation. Further analysis implies that in a fully coupled ocean-atmosphere situation the decadal-to-interdecadal oscillation is less likely to exist.

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Yizhak Feliks and Michael Ghil

Abstract

The instability of the downwelling front along the southern coast of Asia Minor is studied with a multimode quasigeostrophic model. Linear analysis shows that the most unstable wave has a length of about 100 km, The wavelength depends only very weakly on the transversal scale of the front. The wave period is larger by an order of magnitude than the e-folding time; that is, rapid local growth occurs with little propagation. The growth rate is proportional to the maximum of the speed of the downwelling westward jet.

The evolution of the frontal waves can be divided into three stages. At first, the evolution is mainly due to linear instability; the second stage is characterized by closed eddy formation; and finally, isolated eddies separate from the front and penetrate into the open sea. The largest amount of available potential energy is transferred to kinetic energy and into the barotropic mode during the second, eddy-forming stage, when several dipoles develop in this mode. The formation of anticyclonic eddies is due to advection of the ridges of the unstable wave's first baroclinic mode by the barotropic dipole. The baroclinic eddies ride on the barotropic dipoles. The propagation of such dipole-rider systems is determined mainly by the evolution of the corresponding barotropic dipole.

These results suggest that the warm- and salty-core eddies observed in the Eastern Mediterranean are due, at least in part, to the instability of the downwelling front along the basin's northeastern coastline. There is both qualitative and quantitative similarity between the observed and calculated eddies in their radius (35–50 km), thermal structure, and distribution along the coast.

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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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Zheng Hao and Michael Ghil

Abstract

A major error source in the numerical simulation of tropical oceans is the uncertainty in wind stress forcing. A reduced-gravity shallow-water model has been used to test how assimilated ocean data correct simulation errors caused by erroneous wind stress in the tropics. The geometry of the basin is rectangular and symmetric about the equator, and the long-wave approximation is applied. All experiments are of the identical-twin type: the “observations” are generated by sampling the desired reference solution, and the data are assimilated by optimal interpolation into the same model, with wind stress forcing different from that in the reference case.

In this paper, three types of wind stress errors are considered: errors of timing only, as well as persistent errors, systematic or stochastic. The relative usefulness of thermocline depth and current observations, and the effect of data distribution on state estimation are examined. The role of equatorial ocean waves in the process of data assimilation is also studied.

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Shi Jiang and Michael Ghil

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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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Fei Chen and Michael Ghil

Abstract

A hybrid coupled ocean–atmosphere model is used to investigate low-frequency variability in the climate system. The model's atmospheric component is a Budyko-Sellers-North, two-dimensional energy-balance model; the oceanic component is a simplified general circulation model. The coupled model is confined to an idealized, rectangular North Atlantic basin. In the present model version, the ocean density depends exclusively on temperature.

An interdecadal oscillation with a period of 40–50 years is found in the hybrid coupled model when model parameters are within the climatological range, even though density does not depend on salinity. This interdecadal oscillation is characterized by a pair of vortices of opposite signs, that grow and decay in quadrature with each other in the ocean's upper layer; their centers follow each other anticlockwise through the northwestern quadrant of the model domain.

The interdecadal oscillation's physical mechanism resembles that of the interdecadal oscillation analyzed in an earlier, uncoupled model by the same authors. Central to the mechanism is the prescribed component in the surface heat fluxes. In this coupled model, the prescribed forcing component comes from solar radiation. Surface-density variations in high latitudes drive the oscillation and are due to the cooling effect of atmospheric forcing there.

Sensitivity studies are performed by adjusting two free parameters in the model: the atmospheric thermal diffusion coefficient and air-sea coupling coefficient. The 40–50 year oscillation arises, by Hopf bifurcation as the model parameters cross the neutral stability curve. The resulting limit cycle is fairly robust, exists in a wide parameter range, and responds more to the diffusion parameter than the coupling parameter. Larger values of both parameters reduce the amplitude of the interdecadal oscillation, but neither affects crucially its period.

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Nathan Paldor and Michael Ghil

Abstract

The linear instability of a zonal geostrophic jet with a cosh−2 meridional profile on an f plane is investigated in a reduced-gravity, shallow-water model. The stability theory developed here extends classic quasigeostrophic theory to cases where the change of active-layer depth across the jet is not necessarily small. A shooting method is used to integrate the equations describing the cross-stream structure of the alongstream wave perturbations. The phase speeds of these waves are determined by the boundary conditions of regularity at infinity. Regions exist in parameter space where the waves that propagate along the jet will grow exponentially with time. The wavelength of the most unstable waves is 2π R, where R is the internal deformation radius on the deep side, and their e-folding time is about 25 days.

The upper-layer thickness of the basic state in the system has a spatial structure resembling that of the isopycnals across the Gulf Stream. The unstable waves obtained in the present analysis have a wavelength that is in agreement with some recent observations—based on infrared imaging of the sea surface temperature field—of the fastest- growing meanders’ wavelength. Calculated growth rates fall toward the low end of the range of values obtained from these infrared observations on the temporal evolution of Gulf Stream meanders.

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Shi Jiang, Fei-fei Jin, and Michael Ghil

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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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Eric Simonnet, Michael Ghil, Kayo Ide, Roger Temam, and Shouhong Wang

Abstract

Successive bifurcations—from steady states through periodic to aperiodic solutions—are studied in a shallow-water, reduced-gravity, 2½-layer model of the midlatitude ocean circulation subject to time-independent wind stress. The bifurcation sequence is studied in detail for a rectangular basin with an idealized spatial pattern of wind stress. The aperiodic behavior is studied also in a North Atlantic–shaped basin with realistic continental contours. The bifurcation sequence in the rectangular basin is studied in Part I, the present article. It follows essentially the one reported for single-layer quasigeostrophic and 1½-layer shallow-water models. As the intensity of the north–south-symmetric, zonal wind stress is increased, the nearly symmetric double-gyre circulation is destabilized through a perturbed pitchfork bifurcation. The low-stress steady solution, with its nearly equal subtropical and subpolar gyres, is replaced by an approximately mirror-symmetric pair of stable equilibria. The two solution branches so obtained are named after the inertial recirculation cell that is stronger, subtropical or subpolar, respectively. This perturbed pitchfork bifurcation and the associated Hopf bifurcations are robust to changes in the interface friction between the two active layers and the thickness H 2 of the lower active layer. They persist in the presence of asymmetries in the wind stress and of changes in the model's spatial resolution and finite-difference scheme. Time-dependent model behavior in the rectangular basin, as well as in the more realistic, North Atlantic–shaped one, is studied in Part II.

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