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- Author or Editor: Michael Ghil x
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Abstract
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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.
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.
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.
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.
Abstract
We have examined systematically oscillatory modes in the Northern Hemisphere and in the tropics. The 700 mb heights were used to analyze extratropical oscillations, and the outgoing longwave radiation to study tropical oscillations in convection. All datasets were band-pass filtered to focus on the intraseasonal (IS) band of 10–120 days. Leading spatial patterns of variability were obtained by applying EOF analysis to these IS data. The leading principal components (PCs) were subjected to singular spectrum analysis (SSA). SSA is a statistical technique related to EOF analysis, but in the time domain, rather than the spatial domain. It helps identify nonlinear oscillations in short and noisy time series.
In the Northern Hemisphere, there are two important modes of oscillation with periods near 48 and 23 days, respectively. The 48-day mode is the most important of the two. It has both traveling and standing components, and is dominated by a zonal wavenumber two. The 23-day mode has the spatial structure and propagation properties described by Branstator and by Kushnir.
In the tropics, the 40–50 day oscillation documented by Madden and Julian, Weickmann, Lau, their colleagues, and many other authors dominates the Indian and Pacific oceans from 60°E to the date line. From 170°W to 90°W, however, a 24–28 day oscillation is equally strong. The extratropical modes are often independent of, and sometimes lead, the tropical modes.
Abstract
We have examined systematically oscillatory modes in the Northern Hemisphere and in the tropics. The 700 mb heights were used to analyze extratropical oscillations, and the outgoing longwave radiation to study tropical oscillations in convection. All datasets were band-pass filtered to focus on the intraseasonal (IS) band of 10–120 days. Leading spatial patterns of variability were obtained by applying EOF analysis to these IS data. The leading principal components (PCs) were subjected to singular spectrum analysis (SSA). SSA is a statistical technique related to EOF analysis, but in the time domain, rather than the spatial domain. It helps identify nonlinear oscillations in short and noisy time series.
In the Northern Hemisphere, there are two important modes of oscillation with periods near 48 and 23 days, respectively. The 48-day mode is the most important of the two. It has both traveling and standing components, and is dominated by a zonal wavenumber two. The 23-day mode has the spatial structure and propagation properties described by Branstator and by Kushnir.
In the tropics, the 40–50 day oscillation documented by Madden and Julian, Weickmann, Lau, their colleagues, and many other authors dominates the Indian and Pacific oceans from 60°E to the date line. From 170°W to 90°W, however, a 24–28 day oscillation is equally strong. The extratropical modes are often independent of, and sometimes lead, the tropical modes.
Abstract
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.
Abstract
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.
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.
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.
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.
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.
Abstract
A mixture model is a flexible probability density estimation technique, consisting of a linear combination of k component densities. Such a model is applied to estimate clustering in Northern Hemisphere (NH) 700-mb geopotential height anomalies. A key feature of this approach is its ability to estimate a posterior probability distribution for k, the number of clusters, given the data and the model. The number of clusters that is most likely to fit the data is thus determined objectively.
A dataset of 44 winters of NH 700-mb fields is projected onto its two leading empirical orthogonal functions (EOFs) and analyzed using mixtures of Gaussian components. Cross-validated likelihood is used to determine the best value of k, the number of clusters. The posterior probability so determined peaks at k = 3 and thus yields clear evidence for three clusters in the NH 700-mb data. The three-cluster result is found to be robust with respect to variations in data preprocessing and data analysis parameters. The spatial patterns of the three clusters’ centroids bear a high degree of qualitative similarity to the three clusters obtained independently by Cheng and Wallace, using hierarchical clustering on 500-mb NH winter data: the Gulf of Alaska ridge, the high over southern Greenland, and the enhanced climatological ridge over the Rockies.
Separating the 700-mb data into Pacific (PAC) and Atlantic (ATL) sector maps reveals that the optimal k value is 2 for both the PAC and ATL sectors. The respective clusters consist of Kimoto and Ghil’s Pacific–North American (PNA) and reverse PNA regimes, as well as the zonal and blocked phases of the North Atlantic oscillation. The connections between our sectorial and hemispheric results are discussed from the perspective of large-scale atmospheric dynamics.
Abstract
A mixture model is a flexible probability density estimation technique, consisting of a linear combination of k component densities. Such a model is applied to estimate clustering in Northern Hemisphere (NH) 700-mb geopotential height anomalies. A key feature of this approach is its ability to estimate a posterior probability distribution for k, the number of clusters, given the data and the model. The number of clusters that is most likely to fit the data is thus determined objectively.
A dataset of 44 winters of NH 700-mb fields is projected onto its two leading empirical orthogonal functions (EOFs) and analyzed using mixtures of Gaussian components. Cross-validated likelihood is used to determine the best value of k, the number of clusters. The posterior probability so determined peaks at k = 3 and thus yields clear evidence for three clusters in the NH 700-mb data. The three-cluster result is found to be robust with respect to variations in data preprocessing and data analysis parameters. The spatial patterns of the three clusters’ centroids bear a high degree of qualitative similarity to the three clusters obtained independently by Cheng and Wallace, using hierarchical clustering on 500-mb NH winter data: the Gulf of Alaska ridge, the high over southern Greenland, and the enhanced climatological ridge over the Rockies.
Separating the 700-mb data into Pacific (PAC) and Atlantic (ATL) sector maps reveals that the optimal k value is 2 for both the PAC and ATL sectors. The respective clusters consist of Kimoto and Ghil’s Pacific–North American (PNA) and reverse PNA regimes, as well as the zonal and blocked phases of the North Atlantic oscillation. The connections between our sectorial and hemispheric results are discussed from the perspective of large-scale atmospheric dynamics.
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.
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.
Abstract
This study examines the flow induced in a highly idealized atmospheric model by an east–west-oriented oceanic thermal front. The model has a linear marine boundary layer coupled to a quasigeostrophic, equivalent- barotropic free atmosphere. The vertical velocity at the top of the boundary layer drives the flow in the free atmosphere and produces an eastward jet, parallel to the oceanic front's isotherms. A large gyre develops on either side of this jet, cyclonic to the north and anticyclonic to the south of it. As the jet intensifies during spinup from rest, it becomes unstable. The most unstable wave has a length of about 500 km, it evolves into a meander, and eddies detach from the eastern edge of each gyre.
The dependence of the atmospheric dynamics on the strength T∗ of the oceanic front is studied. The Gulf Stream and Kuroshio fronts correspond roughly, in the scaling used here, to T∗ ≅ 7°C. For weak fronts, T∗ ≤ 4°C, the circulation is steady and exhibits two large, antisymmetric gyres separated by a westerly zonal jet. As the front strengthens, 4 < T∗ < 5, the solution undergoes Hopf bifurcation to become periodic in time, with a period of 30 days, and spatially asymmetric. The bifurcation is due to the westerly jet's barotropic instability, which has a symmetric spatial pattern. The addition of this pattern to the antisymmetric mean results in the overall asymmetry of the full solution. The spatial scale and amplitude of the symmetric, internally generated, and antisymmetric, forced mode increase with the strength T∗ of the oceanic front. For T∗ ≥ 5°C, the solution becomes chaotic, but a dominant period still stands out above the broadband noise. This dominant period increases with T∗ overall, but the increase is not monotonic.
The oceanic front's intensity dictates the mean speed of the atmospheric jet. Two energy regimes are obtained. 1) In the low-energy regime, the SST front, and hence the atmospheric jet, are weak; in this regime, small meanders develop along the jet axis, and the dominant period is about 25 days. 2) In the high-energy regime, the SST front and the jet are strong; in it, large meanders and eddies develop along the jet, and the dominant oscillation has a period of about 70 days. The physical nature of the two types of oscillations is discussed, as are possible transitions between them when T∗ changes on very long time scales. The results are placed in the context of previous theories of ocean front effects on atmospheric flows, in which baroclinic phenomena are dominant.
Abstract
This study examines the flow induced in a highly idealized atmospheric model by an east–west-oriented oceanic thermal front. The model has a linear marine boundary layer coupled to a quasigeostrophic, equivalent- barotropic free atmosphere. The vertical velocity at the top of the boundary layer drives the flow in the free atmosphere and produces an eastward jet, parallel to the oceanic front's isotherms. A large gyre develops on either side of this jet, cyclonic to the north and anticyclonic to the south of it. As the jet intensifies during spinup from rest, it becomes unstable. The most unstable wave has a length of about 500 km, it evolves into a meander, and eddies detach from the eastern edge of each gyre.
The dependence of the atmospheric dynamics on the strength T∗ of the oceanic front is studied. The Gulf Stream and Kuroshio fronts correspond roughly, in the scaling used here, to T∗ ≅ 7°C. For weak fronts, T∗ ≤ 4°C, the circulation is steady and exhibits two large, antisymmetric gyres separated by a westerly zonal jet. As the front strengthens, 4 < T∗ < 5, the solution undergoes Hopf bifurcation to become periodic in time, with a period of 30 days, and spatially asymmetric. The bifurcation is due to the westerly jet's barotropic instability, which has a symmetric spatial pattern. The addition of this pattern to the antisymmetric mean results in the overall asymmetry of the full solution. The spatial scale and amplitude of the symmetric, internally generated, and antisymmetric, forced mode increase with the strength T∗ of the oceanic front. For T∗ ≥ 5°C, the solution becomes chaotic, but a dominant period still stands out above the broadband noise. This dominant period increases with T∗ overall, but the increase is not monotonic.
The oceanic front's intensity dictates the mean speed of the atmospheric jet. Two energy regimes are obtained. 1) In the low-energy regime, the SST front, and hence the atmospheric jet, are weak; in this regime, small meanders develop along the jet axis, and the dominant period is about 25 days. 2) In the high-energy regime, the SST front and the jet are strong; in it, large meanders and eddies develop along the jet, and the dominant oscillation has a period of about 70 days. The physical nature of the two types of oscillations is discussed, as are possible transitions between them when T∗ changes on very long time scales. The results are placed in the context of previous theories of ocean front effects on atmospheric flows, in which baroclinic phenomena are dominant.
