Publications

2006
Kao, Jim, Dawn Flicker, Kayo Ide, and Michael Ghil. 2006. “Estimating model parameters for an impact-produced shock-wave simulation: Optimal use of partial data with the extended Kalman filter.” Journal of Computational Physics 214 (2). Elsevier: 725–737.
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Feliks, Yizhak, Michael Ghil, Israel Ziona, and Météorologie Dynamique. 2006. “Long-range forecasting and the scientific background in Joseph's interpretation to Pharaoh’s dreams.” Proc. 16th Conf. Research Judaea & Samaria. in press. Abstract

Long-range forecasting is today a major area of climate research. Such forecasts affect socioeconomic planning in many fields of activity. There are essentially two approaches to longrange forecasting: one is based on solving the equations that govern atmospheric and ocean dynamics, the other on the statistical properties of past climate records. The present talk is based on the latter, statistical approach. Joseph’s interpretation of Pharaoh’s dreams provides a striking example of long-range planning based on a climate forecast. Joseph interpreted the two dreams as a forecast for seven years of plenty, followed by seven of famine. Based on this forecast, he proposed to Pharaoh a plan for running the agriculture and economy of Egypt. It is not clear from the Biblical story why Pharaoh trusted Joseph’s forecast and appointed him to implement the plan. Our answer to this question is based on ancient and medieval Egypt’s being entirely dependent on the Nile River’s seasonal flooding: when the highest water levels did not cover the arable areas of the river valley, crops were insufficient to feed the population. When successive years of hunger weakened the economy and the state, change of rulers could, and sometimes did ensue. Extreme examples were the fall of the Old Kingdom in 2185 B.C. and the Fatimid conquest of Egypt in 969 A.D. Hence the Egyptians measured the high-water mark of the Nile River for over 5000 years, using different tools. The most advanced of these tools was the nilometer; typical nilometers appear in several mosaics from the Roman and Byzantine period around the Mediterranean, such as the “Nile Festival” mosaic in Zippori (Upper Galilee), Fig. 1. The measurements had a twofold purpose: first to set the annual taxes, which were a function of the high-water mark, for obvious reasons; and second, to provide information for water management, with a view to reduce drought damage. Our analysis of high- and low-water levels for 622–1922 A.D. shows that oscillations with a period of several years occur, with a 7-year oscillation being dominant. We suspect that the origin of this 7-year swing lies in the same periodicity being present in the North Atlantic’s sea-surface temperatures and sea-level pressures. This North Atlantic Oscillation affects the climate of Europe, North America and the Middle East, and might be the ultimate reason for Joseph’s successful climate forecast.

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Kravtsov, S., Andrew W. Robertson, and Michael Ghil. 2006. “Multiple regimes and low-frequency oscillations in the Northern Hemisphere's zonal-mean flow.” Journal of the Atmospheric Sciences 63 (3): 840–860.
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Bellon, G., Michael Ghil, and H. Le Treut. 2006. “Scale separation for moisture-laden regions in the tropical atmosphere.” Geophysical Research Letters 33 (1). Wiley Online Library.
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Kondrashov, Dmitri, and Michael Ghil. 2006. “Spatio-temporal filling of missing points in geophysical data sets.” Nonlinear Processes in Geophysics 13 (2): 151–159. Abstract

The majority of data sets in the geosciences are obtained from observations and measurements of natural systems, rather than in the laboratory. These data sets are often full of gaps, due to to the conditions under which the measurements are made. Missing data give rise to various problems, for example in spectral estimation or in specifying boundary conditions for numerical models. Here we use Singular Spectrum Analysis (SSA) to fill the gaps in several types of data sets. For a univariate record, our procedure uses only temporal correlations in the data to fill in the missing points. For a multivariate record, multi-channel SSA (M-SSA) takes advantage of both spatial and temporal correlations. We iteratively produce estimates of missing data points, which are then used to compute a self-consistent lag-covariance matrix; cross-validation allows us to optimize the window width and number of dominant SSA or M-SSA modes to fill the gaps. The optimal parameters of our procedure depend on the distribution in time (and space) of the missing data, as well as on the variance distribution between oscillatory modes and noise. The algorithm is demonstrated on synthetic examples, as well as on data sets from oceanography, hydrology, atmospheric sciences, and space physics: global sea-surface temperature, flood-water records of the Nile River, the Southern Oscillation Index (SOI), and satellite observations of relativistic electrons.

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Ghil, Michael, and Ilya Zaliapin. 2006. “Une Nouvelle source de Fractales: Les Equations Booléennes avec Retard, et leurs Applications aux Sciences de la Planete.” L'irruption Des Géométries Fractales Dans Les Sciences,une Apologie De L'oeuvre De Benoît Mandelbrot, 161–187. Paris: Editions de l'Académie Européenne Interdisciplinaire des Sciences.
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2005
Moore, JC, Aslak Grinsted, and S Jevrejeva. 2005. “New tools for analyzing time series relationships and trends.” Eos, Transactions American Geophysical Union 86 (24). Wiley Online Library: 226–232. Abstract
Geophysical studies are plagued by short and noisy time series. These time series are typically nonstationary contain various long-period quasi-periodic components, and have rather low signal-to-noise ratios and/or poor spatial sampling. Classic examples of these time series are tide gauge records, which are influenced by ocean and atmospheric circulation patterns, twentieth-century warming, and other long-term variability. Remarkable progress recently has been made in the statistical analysis of time series. Ghil et al. [2002] presented a general review of several advanced statistical methods with a solid theoretical foundation. This present article highlights several new approaches that are easy to use and that may be of general interest.
Kravtsov, S., Andrew W. Robertson, and Michael Ghil. 2005. “Bimodal behavior in the zonal mean flow of a baroclinic beta-channel model.” Journal of the Atmospheric Sciences 62 (6): 1746–1769.
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Zhang, Yunyan, Bjorn Stevens, and Michael Ghil. 2005. “On the diurnal cycle and susceptibility to aerosol concentration in a stratocumulus-topped mixed layer.” Quarterly Journal of the Royal Meteorological Society 131 (608). Wiley Online Library: 1567–1583.
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Kondrashov, Dmitri, S Kravtsov, Andrew W. Robertson, and Michael Ghil. 2005. “A hierarchy of data-based ENSO models.” Journal of climate 18 (21): 4425–4444. Abstract

Global sea surface temperature (SST) evolution is analyzed by constructing predictive models that best describe the dataset’s statistics. These inverse models assume that the system’s variability is driven by spatially coherent, additive noise that is white in time and are constructed in the phase space of the dataset’s leading empirical orthogonal functions. Multiple linear regression has been widely used to obtain inverse stochastic models; it is generalized here in two ways. First, the dynamics is allowed to be nonlinear by using polynomial regression. Second, a multilevel extension of classic regression allows the additive noise to be correlated in time; to do so, the residual stochastic forcing at a given level is modeled as a function of variables at this level and the preceding ones. The number of variables, as well as the order of nonlinearity, is determined by optimizing model performance. The two-level linear and quadratic models have a better El Niño–Southern Oscillation (ENSO) hindcast skill than their one-level counterparts. Estimates of skewness and kurtosis of the models’ simulated Niño-3 index reveal that the quadratic model reproduces better the observed asymmetry between the positive El Niño and negative La Niña events. The benefits of the quadratic model are less clear in terms of its overall, cross-validated hindcast skill; this model outperforms, however, the linear one in predicting the magnitude of extreme SST anomalies. Seasonal ENSO dependence is captured by incorporating additive, as well as multiplicative forcing with a 12-month period into the first level of each model. The quasi-quadrennial ENSO oscillatory mode is robustly simulated by all models. The “spring barrier” of ENSO forecast skill is explained by Floquet and singular vector analysis, which show that the leading ENSO mode becomes strongly damped in summer, while nonnormal optimum growth has a strong peak in December.

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Simonnet, Eric, Michael Ghil, and Henk Dijkstra. 2005. “Homoclinic bifurcations in the quasi-geostrophic double-gyre circulation.” Journal of Marine Research 63 (5). Sears Foundation for Marine Research: 931–956. Abstract

The wind-driven double-gyre circulation in a rectangular basin goes through several dynamical regimes as the amount of lateral friction is decreased. This paper studies the transition to irregular flow in the double-gyre circulation by applying dynamical systems methodology to a quasi-geostrophic, equivalent-barotropic model with a 10-km resolution. The origin of the irregularities, in space and time, is the occurrence of homoclinic bifurcations that involve phase-space behavior far from stationary solutions. The connection between these homoclinic bifurcations and earlier transitions, which occur at larger lateral friction, is explained. The earlier transitions, such as pitchfork and asymmetric Hopf bifurcation, only involve the nonlinear saturation of linear instabilities, while the homoclinic bifurcations are associated with genuinely nonlinear behavior. The sequence of bifurcations—pitchfork, Hopf, and homoclinic—is independent of the lateral friction and may be described as the unfolding of a singularity that occurs in the frictionless, Hamiltonian limit of the governing equations. Two distinct chaotic regimes are identified: Lorenz chaos at relatively large lateral friction versus Shilnikov chaos at relatively small lateral friction. Both types of homoclinic bifurcations induce chaotic behavior of the recirculation gyres that is dominated by relaxation oscillations with a well-defined period. The relevance of these results to the mid-latitude oceans' observed low-frequency variations is discussed. A previously documented 7-year peak in observed North-Atlantic variability is shown to exist across a hierarchy of models that share the gyre modes and homoclinic bifurcations discussed herein.

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Dijkstra, Henk A., and Michael Ghil. 2005. “Low-frequency variability of the large-scale ocean circulation: a dynamical systems approach.” Reviews of Geophysics 43. Abstract

Oceanic variability on interannual, interdecadal, and longer timescales plays a key role in climate variability and climate change. Paleoclimatic records suggest major changes in the location and rate of deepwater formation in the Atlantic and Southern oceans on timescales from millennia to millions of years. Instrumental records of increasing duration and spatial coverage document substantial variability in the path and intensity of ocean surface currents on timescales of months to decades. We review recent theoretical and numerical results that help explain the physical processes governing the large-scale ocean circulation and its intrinsic variability. To do so, we apply systematically the methods of dynamical systems theory. The dynamical systems approach is proving successful for more and more detailed and realistic models, up to and including oceanic and coupled ocean-atmosphere general circulation models. In this approach one follows the road from simple, highly symmetric model solutions, through a “bifurcation tree,” toward the observed, complex behavior of the system under investigation. The observed variability can be shown to have its roots in simple transitions from a circulation with high symmetry in space and regularity in time to circulations with successively lower symmetry in space and less regularity in time. This road of successive bifurcations leads through multiple equilibria to oscillatory and eventually chaotic solutions. Key features of this approach are illustrated in detail for simplified models of two basic problems of the ocean circulation. First, a barotropic model is used to capture major features of the wind-driven ocean circulation and of the changes in its behavior as wind stress increases. Second, a zonally averaged model is used to show how the thermohaline ocean circulation changes as buoyancy fluxes at the surface increase. For the wind-driven circulation, multiple separation patterns of a “Gulf-Stream like” eastward jet are obtained. These multiple equilibria are followed by subannual and interannual oscillations of the jet and of the entire basin's circulation. The multiple equilibria of the thermohaline circulation include deepwater formation near the equator, near either pole or both, as well as intermediate possibilities that bear some degree of resemblance to the currently observed Atlantic overturning pattern. Some of these multiple equilibria are subject, in turn, to oscillatory instabilities with timescales of decades, centuries, and millennia. Interdecadal and centennial oscillations are the ones of greatest interest in the current debate on global warming and on the relative roles of natural and anthropogenic variability in it. They involve the physics of the truly three-dimensional coupling between the wind-driven and thermohaline circulation. To arrive at this three-dimensional picture, the bifurcation tree is sketched out for increasingly complex models for both the wind-driven and the thermohaline circulation.

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Kravtsov, S, Dmitri Kondrashov, and M Ghil. 2005. “Multilevel regression modeling of nonlinear processes: Derivation and applications to climatic variability.” Journal of Climate 18 (21): 4404–4424. Abstract

Predictive models are constructed to best describe an observed field’s statistics within a given class of nonlinear dynamics driven by a spatially coherent noise that is white in time. For linear dynamics, such inverse stochastic models are obtained by multiple linear regression (MLR). Nonlinear dynamics, when more appropriate, is accommodated by applying multiple polynomial regression (MPR) instead; the resulting model uses polynomial predictors, but the dependence on the regression parameters is linear in both MPR and MLR. The basic concepts are illustrated using the Lorenz convection model, the classical double-well problem, and a three-well problem in two space dimensions. Given a data sample that is long enough, MPR successfully reconstructs the model coefficients in the former two cases, while the resulting inverse model captures the three-regime structure of the system’s probability density function (PDF) in the latter case. A novel multilevel generalization of the classic regression procedure is introduced next. In this generalization, the residual stochastic forcing at a given level is subsequently modeled as a function of variables at this level and all the preceding ones. The number of levels is determined so that the lag-0 covariance of the residual forcing converges to a constant matrix, while its lag-1 covariance vanishes. This method has been applied to the output of a three-layer, quasigeostrophic model and to the analysis of Northern Hemisphere wintertime geopotential height anomalies. In both cases, the inverse model simulations reproduce well the multiregime structure of the PDF constructed in the subspace spanned by the dataset’s leading empirical orthogonal functions, as well as the detailed spectrum of the dataset’s temporal evolution. These encouraging results are interpreted in terms of the modeled low-frequency flow’s feedback on the statistics of the subgrid-scale processes.

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Koo, Seongjoon, Andrew W. Robertson, and Michael Ghil. 2005. “A Multiple-Regime Approach to Atmospheric Zonal-Flow Vacillation”.
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Kondrashov, Dmitri, Yizhak Feliks, and Michael Ghil. 2005. “Oscillatory modes of extended Nile River records (A.D. 622–1922).” Geophysical Research Letters 32 (10). AGU: L10702. Abstract

The historical records of the low- and high-water levels of the Nile River are among the longest climatic records that have near-annual resolution. There are few gaps in the first part of the records (A.D. 622-1470) and larger gaps later (A.D. 1471-1922). We apply advanced spectral methods, Singular-Spectrum Analysis (SSA) and the Multi-Taper Method (MTM), to fill the gaps and to locate interannual and interdecadal periodicities. The gap filling uses a novel, iterative version of SSA. Our analysis reveals several statistically significant features of the records: a nonlinear, data-adaptive trend that includes a 256-year cycle, a quasi-quadriennial (4.2-year) and a quasi-biennial (2.2-year) mode, as well as additional periodicities of 64, 19, 12, and, most strikingly, 7 years. The quasi-quadriennial and quasi-biennial modes support the long-established connection between the Nile River discharge and the El-Niño/Southern Oscillation (ENSO) phenomenon in the Indo-Pacific Ocean. The longest periods might be of astronomical origin. The 7-year periodicity, possibly related to the biblical cycle of lean and fat years, seems to be due to North Atlantic influences.

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Stevens, Bjorn, Yunyan Zhang, and Michael Ghil. 2005. “Stochastic effects in the representation of stratocumulus-topped mixed layers.” Proc. ECMWF Workshop on Representation of Sub-grid Processes Using Stochastic-Dynamic Models, 79–90. Shinfield Park, Reading, UK.
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Ghil, Michael, Tian Ma, and Shouhong Wang. 2005. “Structural Bifurcation of 2-D Nondivergent Flows with Dirichlet Boundary Conditions: Applications to Boundary-Layer Separation.” SIAM J. Appl. Math. 65 (5). Society for Industrial & Applied Mathematics (SIAM): 1576–1596.
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2004
Feliks, Yizhak, Michael Ghil, and Eric Simonnet. 2004. “Low-frequency variability in the midlatitude atmosphere induced by an oceanic thermal front.” Journal of the atmospheric sciences 61 (9): 961–981.
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Feliks, Yizhak, Michael Ghil, and Eric Simonnet. 2004. “Low-frequency variability in the midlatitude atmosphere induced by an oceanic thermal front.” Journal of the Atmospheric Sciences 61 (9): 961–981. 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.
Ghil, Michael, Jian-Guo Liu, Cheng Wang, and Shouhong Wang. 2004. “Boundary-layer separation and adverse pressure gradient for 2-D viscous incompressible flow.” Physica D: Nonlinear Phenomena 197 (1-2). Elsevier BV: 149–173.
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