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Michael Ghil

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Michael Ghil

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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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Michael Ghil and Ricardo Todling

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Sequential data assimilation schemes approaching true optimality for sizable atmospheric models are becoming a reality. The behavior of the Kalman filter (KF) under difficult conditions needs therefore to be understood. In this two-part paper the authors implemented a KF for a two-dimensional shallow-water model with one or two layers. The model is linearized about a basic flow that depends on latitude; this permits the one-layer (1-L) case to be barotropically unstable. Constant vertical shear in the two-layer (2-L) case induces baroclinic instability.

The stable and unstable 1-L cases were studied in Part I. In the unstable case, even a very small number of observations can keep the forecast and analysis errors from the exponential growth induced by the flow's instability. In Part II, the authors now consider the 2-L, baroclinically stable and unstable cases. Simple experiments show that both cases are, quite similar to their barotropic counterparts. Once again, the KF is shown to keep the estimated flow's error bars bounded, even when a small number of observations—taken with realistic frequency—is utilized.

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Ricardo Todling and Michael Ghil

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Sequential data assimilation schemes approaching true optimality for sizable atmospheric models are becoming a reality. The behavior of the Kalman filter (KF) under difficult conditions needs therefore to be understood. In this two-part paper we implement a KF for a two-dimensional shallow-water model, with one or two layers. The model is linearized about a basic flow that depends on latitude; this permits the one-layer (1-L) case to be barotropically unstable. Constant vertical shear in the two-layer (2-L) case induces baroclinic instability.

A model-error covariance matrix for the KF simulations is constructed based on the hypothesis that an ensemble of slow modes dominates the errors. In the 1-L case, the system is stable for a meridionally constant basic flow. Assuming equipartition of energy in the construction of the model-error covariance matrix has a deleterious effect on the process of data assimilation in both the stable and unstable cases. Estimation errors are found to be smaller for a model-error spectrum that decays exponentially with wavenumber than an equipartition spectrum. Then the model-error covariance matrix for the 2-L model is also obtained using a decaying-energy spectrum.

The barotropically unstable 1-L case is studied for a basic velocity profile that has a cosine-square shape. Given this linear instability, forecast errors grow exponentially when no observations are present. The KF keeps the errors bounded, even when very few observations are available. The best placement of a single observation is determined in this simple situation and shown to be where the instability is strongest. The 2-L case and a comparison with the performance of a currently operational data assimilation scheme will appear in Part II.

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Nathan Paldor and Michael Ghil

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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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Hisanori Itoh and Michael Ghil

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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

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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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Masahide Kimoto and Michael Ghil

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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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Michael Ghil and Kingtse Mo

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In Part II of this two-part article, we complete the systematic examination of oscillatory modes in the global atmosphere by studying 12 years of 500 mb geopotential heights in the Southern Hemisphere. As in Part I, for the tropics and Northern Hemisphere extratropics, the data were band-pass filtered to focus on intraseasonal (IS) phenomena, and spatial EOFs were obtained. The leading principal components were subjected to singular spectrum analysis (SSA), in order to identify nonlinear IS oscillations with high statistical confidence.

In the Southern Hemisphere, the dominant mode has a period of 23 days, with spatial patterns carried by the second and third winter EOF of the IS band. It has a zonal wavenumber-four structure. The 40-day mode is second, and dominated by wavenumbers three and four, while a 16-day mode is too weak to separate its spatial behavior from the previous two. The IS dynamics in the Southern Hemisphere is more complex and dominated by shorter wavenumbers than the Northern Hemisphere. No statistically significant correlations between the Southern Hemisphere and the tropics or the Northern Hemisphere are apparent in the IS band.

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Fei Chen and Michael Ghil

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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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