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Michael Ghil

Abstract

We study a diffusive energy-balance climate model, governed by a nonlinear parabolic partial differential equation. Three positive steady-state solutions of this equation are found; they correspond to three possible climates of our planet: an interglacial (nearly identical to the present climate), a glacial, and a completely ice-covered earth. We consider also models similar to the main one studied, and determine the number of their steady states. All the models have albedo continuously varying with latitude and temperature, and entirely diffusive horizontal heat transfer. The diffusion is taken to be nonlinear as well as linear.

We investigate the stability under small perturbations of the main model's climates. A stability criterion is derived, and its application shows that the “present climate” and the “deep freeze” are stable, whereas the model's glacial is unstable. A variational principle is introduced to confirm the results of this stability analysis.

We examine the dependence of the number of steady states and of their stability on the average solar radiation. The main result is that for a sufficient decrease in solar radiation (∼2%) the glacial and interglacial solutions disappear, leaving the ice-covered earth as the only possible climate.

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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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Masahide Kimoto and Michael Ghil

Abstract

This paper presents an observational analysis of recurrent flow patterns in the Northern Hemisphere (NH) winter, based on a 37-year series of daily 700-mb height anomalies. Large-scale anomaly patterns that appear repeatedly and persist beyond synoptic time scales are identified by searching for local maxima of probability density in a phase subspace, which is spanned by the leading empirical orthogenal functions (EOFs).

By using an angular probability density function (PDF), we focus on the shape, not magnitude, of the anomaly patterns. The PDF estimate is nonparametric; that is, our algorithm makes no a priori assumption on symmetry with respect to the climatological mean as in one-point correlation and rotated EOF analyses. The local density maxima are searched by iterative bump hunting.

Based on observed partial decoupling between the Pacific (PAC) and the Atlantic-Eurasian (ATL) sectors, the classification algorithm is applied separately to each of the two. Seven PAC and six ATL patterns are obtained. Anomaly maps that belong to the neighborhood of each PDF peak are associated with distinct flow regimes. These include regional blocked and zonal flows, and wave train-like anomaly patterns, some of them well known from previous studies, others revealed by our analysis for the first time.

Successive appearances of flow regimes are generally separated by unclassifiable, transient periods. A Markov chain describes transitions between different flow regimes; highly likely, as well as unlikely routes of transition exist. Chains of preferred transitions may be related to the existence of oscillatory modes in the NH extratropics.

A synoptic characterization of onsets and breaks for the flow regimes obtained is given by compositing. In situ evolutions of anomaly patterns, slow westward shifts of high-latitude anomaly centers, and successive down-stream increase of anomaly magnitudes are the typical signatures of such events.

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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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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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Andreas Groth and Michael Ghil

Abstract

Singular spectrum analysis (SSA) along with its multivariate extension (M-SSA) provides an efficient way to identify weak oscillatory behavior in high-dimensional data. To prevent the misinterpretation of stochastic fluctuations in short time series as oscillations, Monte Carlo (MC)–type hypothesis tests provide objective criteria for the statistical significance of the oscillatory behavior. Procrustes target rotation is introduced here as a key method for refining previously available MC tests. The proposed modification helps reduce the risk of type-I errors, and it is shown to improve the test’s discriminating power. The reliability of the proposed methodology is examined in an idealized setting for a cluster of harmonic oscillators immersed in red noise. Furthermore, the common method of data compression into a few leading principal components, prior to M-SSA, is reexamined, and its possibly negative effects are discussed. Finally, the generalized Procrustes test is applied to the analysis of interannual variability in the North Atlantic’s sea surface temperature and sea level pressure fields. The results of this analysis provide further evidence for shared mechanisms of variability between the Gulf Stream and the North Atlantic Oscillation in the interannual frequency band.

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Masahide Kimoto and Michael Ghil

Abstract

Recurrent and persistent flow patterns are identified by examining multivariate probability density functions (PDFs) in the phase space of large-scale atmospheric motions. This idea is pursued systematically here in the hope of clarifying the extent to which intraseasonal variability can be described and understood in terms of multiple flow regimes.

Bivariate PDFs of the Northern Hemisphere (NH) wintertime anomaly heights at 700 mb are examined in the present paper, using a 37-year dataset. The two-dimensional phase plane is defined by the two leading empirical orthogonal functions (EOFs) of the anomaly fields. PDFs on this plane exhibit synoptically intriguing and statistically significant inhomogeneities on the periphery of the distribution. It is shown that these inhomogeneities are due to the existence of persistent and recurrent anomaly patterns, well-known as dominant teleconnection patterns; that is, the Pacific/North American (PNA) pattern, its reverse, and zonal and blocked phases of the North Atlantic Oscillation (NAO). It is argued that the inhomogeneities are obscured when PDFs are examined in a smaller-dimensional subspace than dynamically desired.

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Hisanori Itoh and Michael Ghil

Abstract

Numerical experiments are performed to clarify the excitation mechanism of mixed Rossby-gravity waves (Yanai waves) in the tropical troposphere, as well as the selection of zonal wavenumbers 4–5 and of the five-day period. The model used is governed by the primitive equations on an equatorial β-plane. Moisture budgets are calculated explicitly.

A nonlinear wave-CISK mechanism produces Yanai waves with the same spectral peaks in wavenumber and frequency as observed. In the absence of antisymmetric lateral forcing, these peaks do not appear distinctly, because the symmetric equatorially trapped modes, i.e., Kelvin-like waves having different spectral peaks, are dominant. It is the lateral antisymmetric forcing which puts the peaks characterizing the antisymmetric Yanai waves in evidence.

It appears that Yanai waves of very small wavenumbers (1–3) cannot have large amplitudes because their frequencies are too large for moisture to be effectively supplied for the convection associated with these waves. Symmetric Kelvin modes are dominant in the absence of forcing asymmetries due at least in part to the difference in the nature of heating between symmetric and antisymmetric modes: precipitation, and hence heating, is not normally distributed. Given a strongly skewed distribution of heating, it can be shown that symmetric modes are excited more effectively. Finally, our results indicate that the vertical wavenumber, and hence the period of Yanai waves are selected by the height of cumulus convection, while the lateral forcing selects the horizontal wavenumber within a certain band provided by the nonlinear wave-CISK mechanism.

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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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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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