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Stefano Pierini, Michael Ghil, and Mickael D. Chekroun

Abstract

A low-order quasigeostrophic double-gyre ocean model is subjected to an aperiodic forcing that mimics time dependence dominated by interdecadal variability. This model is used as a prototype of an unstable and nonlinear dynamical system with time-dependent forcing to explore basic features of climate change in the presence of natural variability. The study relies on the theoretical framework of nonautonomous dynamical systems and of their pullback attractors (PBAs), that is, of the time-dependent invariant sets attracting all trajectories initialized in the remote past. The existence of a global PBA is rigorously demonstrated for this weakly dissipative nonlinear model. Ensemble simulations are carried out and the convergence to PBAs is assessed by computing the probability density function (PDF) of localization of the trajectories. A sensitivity analysis with respect to forcing amplitude shows that the PBAs experience large modifications if the underlying autonomous system is dominated by small-amplitude limit cycles, while less dramatic changes occur in a regime characterized by large-amplitude relaxation oscillations. The dependence of the attracting sets on the choice of the ensemble of initial states is then analyzed. Two types of basins of attraction coexist for certain parameter ranges; they contain chaotic and nonchaotic trajectories, respectively. The statistics of the former does not depend on the initial states whereas the trajectories in the latter converge to small portions of the global PBA. This complex scenario requires separate PDFs for chaotic and nonchaotic trajectories. General implications for climate predictability are finally discussed.

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Matthew Huber, James C. McWilliams, and Michael Ghil

Abstract

The authors present properties of turbulent, meridional mixing along isentropic surfaces within the troposphere. Twice-daily wind fields analyses from the European Centre for Medium-Range Weather Forecasts numerical weather prediction model for 1992 are used to calculate Lagrangian trajectories of large ensembles of particles. The ensemble-averaged rms growth of the meridional relative dispersion over the first 10 days after particle release is used to quantify mixing properties. These properties are considered as a function of height in the atmosphere, season, and geographic region. Results are characterized by release latitude and flow regime and compared with simple theories.

All three dispersive regimes—exponential, ballistic, and Richardson–Obukhov—that have been documented in previous studies are found to be important. The extratropics are found to display superdiffusive growth of the relative rms dispersion, consistent with the nonlocal character of midlatitude mixing. The Tropics are characterized by exponential growth of the rms dispersion, consistent with locally constant eddy timescales. Some evidence for zonal inhomogeneity in dispersion growth rates is found.

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Robert Vautard, Kingtse C. Mo, and Michael Ghil

Abstract

Low-frequency variability of large-scale atmospheric dynamics can be represented schematically by a Markov chain of multiple flow regimes. This Markov chain contains useful information for the long-range forecaster, provided that the statistical significance of the associated transition matrix can be reliably tested. Monte Carlo simulation yields a very reliable significance test for the elements of this matrix. The results of this test agree with previously used empirical formulae when each cluster of maps identified as a distinct flow regime is sufficiently large and when they all contain a comparable number of maps. Monte Carlo simulation provides a more reliable way to test the statistical significance of transitions to and from small clusters. It can determine the most likely transitions, as well as the most unlikely ones, with a prescribed level of statistical significance.

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Dmitri Kondrashov, Sergey Kravtsov, and Michael Ghil

Abstract

Signatures of nonlinear dynamics are analyzed by studying the phase-space tendencies of a global baroclinic, quasigeostrophic, three-level (QG3) model with topography. Nonlinear, stochastic, low-order prototypes of the full QG3 model are constructed in the phase space of this model’s empirical orthogonal functions using the empirical model reduction (EMR) approach. The phase-space tendencies of the EMR models closely match the full QG3 model’s tendencies. The component of these tendencies that is not linearly parameterizable is shown to be dominated by the interactions between “resolved” modes rather than by multiplicative “noise” associated with unresolved modes. The method of defining the leading resolved modes and the interactions between them plays a key role in understanding the nature of the QG3 model’s dynamics, whether linear or nonlinear, deterministic or stochastic.

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Blandine L’Hévéder, Francis Codron, and Michael Ghil

Abstract

This paper explores the impact of anomalous northward oceanic heat transport on global climate in a slab ocean setting. To that end, the GCM LMDZ5A of the Laboratoire de Météorologie Dynamique is coupled to a slab ocean, with realistic zonal asymmetries and seasonal cycle. Two simulations with different anomalous surface heating are imposed: 1) uniform heating over the North Atlantic basin and 2) concentrated heating in the Gulf Stream region, with a compensating uniform cooling in the Southern Ocean in both cases. The magnitudes of the heating and of the implied northward interhemispheric heat transport are within the range of current natural variability. Both simulations show global effects that are particularly strong in the tropics, with a northward shift of the intertropical convergence zone (ITCZ) toward the heating anomalies. This shift is accompanied by a northward shift of the storm tracks in both hemispheres. From the comparison between the two simulations with different anomalous surface heating in the North Atlantic, it emerges that the global climate response is nearly insensitive to the spatial distribution of the heating. The cloud response acts as a large positive feedback on the oceanic forcing, mainly because of the low-cloud-induced shortwave anomalies in the extratropics. While previous literature has speculated that the extratropical Q flux may impact the tropics by the way of the transient eddy fluxes, it is explicitly demonstrated here. In the midlatitudes, the authors find a systematic northward shift of the jets, as well as of the associated Ferrel cells, storm tracks, and precipitation bands.

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Yizhak Feliks, Michael Ghil, and Andrew W. Robertson

Abstract

Spectral analyses of the sea surface temperature (SST) in the Simple Ocean Data Analysis (SODA) reanalysis for the past half-century identify prominent and statistically significant interannual oscillations in two regions along the Gulf Stream front over the North Atlantic. A model of the atmospheric marine boundary layer coupled to a baroclinic quasigeostrophic model of the free atmosphere is then forced with the SST history from the SODA reanalysis. Two extreme states are found in the atmospheric simulations: 1) an eastward extension of the westerly jet associated with the front, which occurs mainly during boreal winter, and 2) a quiescent state of very weak flow found predominantly in the summer. This vacillation of the oceanic-front-induced jet in the model is found to exhibit periodicities similar to those identified in the observed Gulf Stream SST front itself. In addition, a close correspondence is found between interannual spectral peaks in the observed North Atlantic Oscillation (NAO) index and the SODA-induced oscillations in the atmospheric model. In particular, significant oscillatory modes with periods of 8.5, 4.2, and 2.8 yr are found in both observed and simulated indices and are shown to be highly synchronized and of similar energy in both time series. These oscillatory modes in the simulations are shown to be suppressed when either (i) the Gulf Stream front or (ii) its interannual oscillations are omitted from the SST field. Moreover, these modes also disappear when (iii) the SST front is spatially smoothed, thus confirming that they are indeed induced by the oceanic front.

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Bruno Deremble, Guillaume Lapeyre, and Michael Ghil

Abstract

To understand the atmospheric response to a midlatitude oceanic front, this paper uses a quasigeostrophic (QG) model with moist processes. A well-known, three-level QG model on the sphere has been modified to include such processes in an aquaplanet setting. Its response is analyzed in terms of the upper-level atmospheric jet for sea surface temperature (SST) fronts of different profiles and located at different latitudes.

When the SST front is sufficiently strong, it tends to anchor the mean atmospheric jet, suggesting that the jet’s spatial location and pattern are mainly affected by the latitude of the SST front. Changes in the jet’s pattern are studied, focusing on surface sensible heat flux and on moisture effects through latent heat release. It is found that latent heat release due to moist processes is modified when the SST front is changed, and this is responsible for the meridional displacement of the jet. Moreover, both latent heat release and surface sensible heat flux contribute to the jet’s strengthening. These results highlight the role of SST fronts and moist processes in affecting the characteristics of the midlatitude jet stream and of its associated storm track, particularly their positions.

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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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Robert N. Miller, Michael Ghil, and François Gauthiez

Abstract

Advanced data assimilation methods are applied to simple but highly nonlinear problems. The dynamical systems studied here are the stochastically forced double well and the Lorenz model. In both systems, linear approximation of the dynamics about the critical points near which regime transitions occur is not always sufficient to track their occurrence or nonoccurrence.

Straightforward application of the extended Kalman filter yields mixed results. The ability of the extended Kalman filter to track transitions of the double-well system from one stable critical point to the other depends on the frequency and accuracy of the observations relative to the mean-square amplitude of the stochastic forcing. The ability of the filter to track the chaotic trajectories of the Lorenz model is limited to short times, as is the ability of strong-constraint variational methods. Examples are given to illustrate the difficulties involved, and qualitative explanations for these difficulties are provided.

Three generalizations of the extended Kalman filter are described. The first is based on inspection of the innovation sequence, that is, the successive differences between observations and forecasts; it works very well for the double-well problem. The second, an extension to fourth-order moments, yields excellent results for the Lorenz model but will be unwieldy when applied to models with high-dimensional state spaces. A third, more practical method—based on an empirical statistical model derived from a Monte Carlo simulation-is formulated, and shown to work very well.

Weak-constraint methods can be made to perform satisfactorily in the context of these simple models, but such methods do not seem to generalize easily to practical models of the atmosphere and ocean. In particular, it is shown that the equations derived in the weak variational formulation are difficult to solve conveniently for large systems.

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Dmitri Kondrashov, Chaojiao Sun, and Michael Ghil

Abstract

The parameter estimation problem for the coupled ocean–atmosphere system in the tropical Pacific Ocean is investigated using an advanced sequential estimator [i.e., the extended Kalman filter (EKF)]. The intermediate coupled model (ICM) used in this paper consists of a prognostic upper-ocean model and a diagnostic atmospheric model. Model errors arise from the uncertainty in atmospheric wind stress. First, the state and parameters are estimated in an identical-twin framework, based on incomplete and inaccurate observations of the model state. Two parameters are estimated by including them into an augmented state vector. Model-generated oceanic datasets are assimilated to produce a time-continuous, dynamically consistent description of the model’s El Niño–Southern Oscillation (ENSO). State estimation without correcting erroneous parameter values still permits recovering the true state to a certain extent, depending on the quality and accuracy of the observations and the size of the discrepancy in the parameters. Estimating both state and parameter values simultaneously, though, produces much better results. Next, real sea surface temperatures observations from the tropical Pacific are assimilated for a 30-yr period (1975–2004). Estimating both the state and parameters by the EKF method helps to track the observations better, even when the ICM is not capable of simulating all the details of the observed state. Furthermore, unobserved ocean variables, such as zonal currents, are improved when model parameters are estimated. A key advantage of using this augmented-state approach is that the incremental cost of applying the EKF to joint state and parameter estimation is small relative to the cost of state estimation alone. A similar approach generalizes various reduced-state approximations of the EKF and could improve simulations and forecasts using large, realistic models.

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