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Mathematics in the Geosciences, September 2001 - June 2002
Michael Ghil (Professor of Atmospheric Sciences and Director, IGPP/UCLA) http://www.atmos.ucla.edu/tcd/ ghil@atmos.ucla.edu or mghil@igpp.ucla.edu
Bifurcations and pattern formation in Earth's fluid envelope Slides
The global climate system is composed of a number of subsystems atmosphere, biosphere, cryosphere, hydrosphere and lithosphere each of which has distinct characteristic times, from days and weeks to centuries and millennia. Each subsystem has its own internal variability, all other things being constant. The nonlinearity of the feedbacks within each subsystem can result in the coexistence of stable equilibria, the presence of self-sustained oscillations, and the possibility of deterministic chaos. Multiple equilibria are discussed in the context of the radiation balance that dominates the thermodynamically open climate system, and the energy-balance and radiative-convective models that describe this balance.
Transitions between multiple equilibria may play a role in climate change on intraseasonal (10100 days) and on multi-million-year time scales. On the interannual and interdecadal time scales of greatest interest to long-term socio-economic planning, it is oscillatory behavior regular or irregular that seems to be most important. Oscillatory behavior is illustrated in models of the oceans¹ thermohaline circulation. It is shown that the bifurcation-theoretical tools of nonlinear dynamics can be applied to fully three-dimensional general circulation models of the ocean with considerable spatial detail and even to coupled ocean-atmosphere models.
Given the complex behavior of the climate system on a full range of time scales, its predictability is discussed and conclusions drawn about the state of our knowledge. Possible approaches for expanding this knowledge and applying it wisely are explored.
References
N.B. References #35 are available at http://www.atmos.ucla.edu/tcd/MG/mg_ref_preprints.html
Eugenia Kalnay (Professor and Chair Department of Meteorology, University of Maryland) ekalnay@atmos.umd.edu http://atmos.umd.edu/~ekalnay
Low dimensionality in the atmospheric dynamics: implications for data assimilation
Joint work with Matteo Corazza, DJ Patil, Rebecca Morss, Brian Hunt, Ed Ott, Ming Cai and Jim Yorke (University of Maryland, College Park, MD 20742-2425).
We introduced a statistic, the BV-dimension, to measure the effective local finite-time dimensionality of the atmosphere. We show that this dimension is often quite low, and suggest that this finding has important implications for data assimilation and the accuracy of weather forecasting (Patil et al, 2001).
The original database for this study was the forecasts of the NCEP global ensemble forecasting system. The initial differences between the control forecast and the perturbed forecasts are called bred vectors. The control and perturbed initial conditions valid at time t=nDt are evolved using the forecast model until time t=(n+1) Dt. The differences between the perturbed and the control forecasts are scaled down to their initial amplitude, and constitute the bred vectors valid at (n+1) Dt. Their growth rate is typically about 1.5/day. The bred vectors are similar by construction to leading Lyapunov vectors except that they have small but finite amplitude, and they are valid at finite times.
The original NCEP ensemble data set has 5 independent bred vectors. We define a local bred vector at each grid point by choosing the 5 by 5 grid points centered at the grid point (a region of about 110km by 1100km), and using the north-south and east-west velocity components at 500mb pressure level to form a 50 dimensional column vector. Since we have k=5 global bred vectors, we also have k local bred vectors at each grid point. We estimate the effective dimensionality of the subspace spanned by the local bred vectors by performing a singular value decomposition (EOF analysis). The k local bred vector columns form a 50xk matrix M. The singular values [IMAGE] of M measure the extent to which the k column unit vectors making up the matrix M point in the direction of [IMAGE]. We define the bred vector dimension as[IMAGE]. For example, if 4 out of the 5 vectors lie along [IMAGE], and one lies along[IMAGE], the BV-dimension would be [IMAGE], less than 2 because one direction is more dominant than the other in representing the original data.
The results (Patil et al, 2001) show that there are large regions where the bred vectors span a subspace of substantially lower dimension than that of the full space.| These low dimensionality regions are dominant in the baroclinic extratropics, typically have a lifetime of 3-7 days, have a well-defined horizontal and vertical structure that spans most of the atmosphere, and tend to move eastward (Fig.1). New results with a large number of ensemble members confirm these results and indicate that the low dimensionality regions are quite robust, and depend only on the verification time (i.e., the underlying flow). Corazza et al (2001) have performed experiments with a data assimilation system based on a quasi-geostrophic model and simulated observations (Morss, 1999, Hamill et al, 2000). A 3D-variational data assimilation scheme for a quasi-geostrophic channel model is used to study the structure of the background error and its relationship to the corresponding bred vectors. The ^Ótrue^Ô evolution of the model atmosphere is defined by an integration of the model and ^Órawinsonde observations^Ô are simulated by randomly perturbing the true state at fixed locations.
It is found that after 3-5 days the bred vectors develop well organized structures which are very similar for the two different norms considered in this paper (potential vorticity norm and streamfunction norm). The results show that the bred vectors do indeed represent well the characteristics of the data assimilation forecast errors, and that the subspace of bred vectors contains most of the forecast error, except in areas where the forecast errors are small. For example, the angle between the 6hr forecast error and the subspace spanned by 10 bred vectors is less than 10o over 90% of the domain, indicating a pattern correlation of more than 98.5% between the forecast error and its projection onto the bred vector subspace.
Case studies using different observational densities are considered to compare the evolution of the Bred Vectors to the spatial structure of the background error. Bred vectors obtained using the ^Ótrue atmosphere^Ô (which would not be possible in an operational center) and analysis are very similar, even when using a low density observing network. This indicates that the bred vectors (and by inference the forecast errors) are more likely dependent on the large scale characteristics of the flow, which are usually captured in an analysis.
In addition, the bred vector dimension (BV-dimension), defined by Patil et al., (2001) is applied to the bred vectors. It is found that the local dimension is usually much smaller (between 2 and 4) than the number of bred vectors, particularly in those areas where the errors are large.
The presence of low-dimensional regions in the perturbations of the basic flow has important implications for data assimilation. At any given time, there is a difference between the true atmospheric state and the model forecast. Assuming that model errors are not the dominant source of errors, in a region of low BV-dimensionality the difference between the true state and the forecast should lie substantially in the low dimensional unstable subspace of the few bred vectors that contribute most strongly to the low BV-dimension. This information should yield a substantial improvement in the forecast: the data assimilation algorithm should correct the model state by moving it closer to the observations along the unstable subspace, since this is where the true state most likely lies.
This can be seen in a simple example based on the 3-dimensional Variational data assimilation (3D-Var) formulation. If we assume that observations [IMAGE] have a diagonal error covariance [IMAGE], and if the local unstable subspace is spanned by [IMAGE], we can assume that locally the background error covariance is of the form [IMAGE] Then the minimum of the cost function for 3D-Var [IMAGE] is attained for the analysis given by [IMAGE] (Kalnay and Toth, 1994).
Note that this involves just scalar products and shows that the correction takes place along the bred vector subspace. The local bred vectors in low dimensionality regions give a representation of the ^Óerrors of the day^Ô in the data assimilation, which depend on the evolving underlying flow.
Preliminary experiments have been conducted with the quasi-geostrophic data assimilation system testing whether it is possible to add ^Óerrors of the day^Ô based on bred vectors to the standard (constant) 3D-Var background error covariance in order to capture these important errors. The results are extremely encouraging, indicating a significant reduction in the analysis errors at a very low computational cost (Figs. 2 and 3).
References:
Corazza, M., E. Kalnay, DJ Patil, R. Morss, M Cai, I. Szunyogh, BR Hunt, E Ott and JA Yorke, 2001: Use of the breeding technique to estimate the structure of the analysis ^Óerrors of the day^Ô. Submitted to Nonlinear Processes in Geophysics.
Hamill, T.M., Snyder, C., and Morss, R.E., 2000: A Comparison of Probabilistic Forecasts from Bred, Singular-Vector and Perturbed Observation Ensembles, Mon. Wea. Rev., 128, 1835--1851. Kalnay, E., and Z. Toth,| 1994:| Removing growing errors in the analysis cycle. Preprints of the Tenth Conference on Numerical Weather Prediction, Amer. Meteor. Soc., 1994, 212-215.
Morss, R. E., 1999: Adaptive observations: Idealized sampling strategies for improving numerical weather prediction. PHD thesis, Massachussetts Institute of technology, 225pp. Patil, D. J. S., B. R. Hunt, E. Kalnay, J. A. Yorke, and E. Ott., 2001: Local Low Dimensionality of Atmospheric Dynamics. Phys. Rev. Lett., 86, 5878.
Fig. 1: Example of the 3-dimensional effective dimension (BVD) of the bred vectors corresponding to 20 March 2000. Blue colors represent a local BVD of about 5 (the number of bred vectors). Red represents a local effective dimensionality close to 1. The vertical slices are computed independently from each other.
[IMAGE]
[IMAGE]
Fig. 2: Example of a year of analysis errors based on the regular 3D-Var data assimilation with an optized constant background error covariance (black, average yellow). It shows strong ^Óerrors of the day^Ô that are not captured by the standard methods. The green analysis errors were obtained by augmenting the constant background error covariance with a sum of 10 bbT, where b are the global bred vectors of the day.
Vladimir Keilis-Borok (Institute of Geophysics and Planetary Physics and Department of Earth and Space Science, University of California, Los Angeles and International Institute for Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences, Moscow) vkb@ess.ucla.edu
Colliding Cascades Models for Earthquake Prediction
1. Colliding cascade (CC) models have been recently introduced to describe development of "critical transitions" (i.e. abrupt overall changes) in hierarchical non-linear ("complex") systems. The models have branching hierarchical structure. The load is applied at the top of the hierarchy and transferred downwards, forming direct cascades. Failures are initiated at the lowest level of hierarchy, and propagate upwards, forming inverse cascades. Direct and inverse cascades collide and interact: loading triggers the failures, failures release and redistribute the load.
2. Three kinds of CC model are developed, different in representation of the loading (differential equations vs. pure cellular automaton) and of interaction between the elements: the interactions are either defined directly , or (according to the concept of Boolean delays equations) replaced by time delays between consecutive switching of the state of an element (loaded vs. unloaded; broken vs. intact).
3. In applications to seismicity loading imitates the impact of tectonic forces, and failures imitated the earthquakes; "a major earthquake" is the failure at the top level of hierarchy. The models reproduce major heuristic constraints, that is basic features of dynamics of seismicity: seismic cycles, magnitude distribution, clustering, and long-range correlations.
4. The CC models reproduce also (for the first time) the wide variety of seismicity patterns premonitory to strong earthquakes. Moreover, premonitory rise of earthquakes' correlation range has been found on such models first, and then - on observations.
The talk summarizes the recent joint studies by A. Gabrielov (Purdue University), M. Ghil (UCLA), V. Keilis-Borok (UCLA&Russian Ac. Sci), W. Newman (UCLA), D. Turcotte (Cornell U), and I. Zaliapin (.(UCLA&Russian Ac. Sci).
Leon Knopoff (Department of Physics and Astronomy and Institute of Geophysics and Planetary Physics, University of California, Los Angeles) lknopoff@ishtar.ess.ucla.edu
Are large earthquakes scaled-up versions of small ones?
Although it is reasonable to assume that space-time patterns of evolution of seismicity depend on the detailed physics of individual fractures, a reasonable attack on the problems of pattern structure have been sidetracked in recent years by the assumption that the problems of earthquake fracture are scale-independent. There is a significant incompatibility between the universality of homogeneity implied in the statistics, and the palpable heterogenous geometry implied in the nature of friction, the topography of natural surfaces, and the nonuniversality of fault geometry. The resolution is to be found in a recent observation that the familiar Gutenberg-Richter frequency law does not hold in its usual form for mainshock earthquakes. The new statistical model in combination with a large number of geophysical observations indicates the coexistence of a physics of self-organization on at least four interactive scales. The new statistical observations can be simulated by a model of self-organization of dynamical fractures on a topographically irregular contact surface under the strong influence of dissipation due to seismic wave radiation. But this restricted model in inadequate to simulate the full range of physics of fracture in large earthquakes.
Vladimir G. Kossobokov (International Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences) volodya@mitp.ru of volodya@ipgp.jussieu.fr
Seismic dynamics prior to and after the great earthquakes worldwide, 1985-2001 Slides: html pdf (5.9MB)
A novel understanding of seismic process, as an essential part of dynamics of a hierarchical system of blocks-and-faults, has already led to reproducible intermediate-term earthquake prediction technique that passed successfully the testing in forward application, 1985-2001. Earthquakes, at least the largest of them, occur after a comparatively large area of lithosphere experiences rise of seismic activity and after smaller earthquakes probe parts of its source. The first happens at intermediate-term scale of years and can be effectively detected. The second arises in a scale of weeks and shorter. It is hard, if possible, to distinguish this stage of precursory seismic rise without an intermediate-term analysis. The decay of aftershock series is evident, although the Omori law fit is poor for the majority of the great earthquakes.
George Molchan (International Institute of Earthquake Prediction Theory & Math Geoscience) molchan@mitp.ru
Fractality in Physical Models: Probability Problems Slides
We consider two objects: a simple sedimentation model in geology and the inviscid Burgers equation. In both of these cases we consider the problem of calculating fractal dimensions or multifractal characteristics of these physical objects. The problems are reduced to calculation of fine asymptotics for self-similar random processes, in particular, to the calculation of the probability nonexceedance of a fixed level for fractional Brownian motion (FBM or integrals of FBM) on a very long interval. Solved and unsolved problems will be discussed.
William I. Newman (Department of Earth and Space Sciences, Physics and Astronomy, and Mathematics, University of California-Los Angeles) win@ucla.edu
Earthquakes as a Nonlinear Dynamical Process Slides
Nonlinear processes dominate seismicity and, to complicate matters, we do not have first-principle equations that describe the behavior. While atmospheric scientists have the Navier Stokes equation to work with, solid earth geophysicists do not have---nor will ever have---an equivalent set of equations that describe, for example, the Sierra Nevadas. The "laws" of fracture mechanics, for example, are phenomenological. Nevertheless, we see many forms of universal behavior---nature seems to be unconcerned with the geologic details, but adheres to scaling laws independent of rheology, geology, geometry, the weather, Congress, .... I will propose how the application of geophysical intuition into these problems can facilitate the developmeant of robust phenomenological models that can provide some important insights into these complex problems. In this lecture, I will review a number of nonlinear dynamical themes and resulting models that have helped improve our understanding of the complexity present in seismic processes.
Norbert Schorghofer (Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology) norbert@segovia.mit.edu
Periodically Spaced Channels in Geomorphology Slides
Joint work with Arshad Kudrolli and Daniel H. Rothman.
Periodically spaced fluvial channels are observed in various geomorphic settings, such as mountain belts, fault blocks, submarine canyons, and Martian gullies. We study this phenomenon in porous sand using field observations, sapping experiments, numerical simulations, and analytical theory. In field observations we observe channels initiated by surface phenomena, but growing due to water provided from underground. The periodicity is established in the initial stage of channel formation. Periodic channels are also reproduced in a tabletop experiment with a dozen and more parallel channels observed. As a result of these observations, the problem is formulated in terms of flow in a porous medium with moving interfaces. Small deformations of the underground watertable provide an efficient instability mechanism. The length-scale of the spacing between channels is discussed.
Joseph Tribbia (National Center for Atmospheric Research (NCAR)) tribbia@ucar.edu
Competing Theoretical Frameworks for Atmospheric Variability: Quasi-geostrophic Turbulence vs Linear Stochastic Dynamics
Up until recently, the standard explanation of atmospheric patterns of variability relied upon the dynamical systems paradigm of successive bifurcations from laminar stationary states, to periodic motions, to low order chaos and on to high dimensional, (quasi-geostrophic) turbulence. This view can be applied both to medium scale weather fluctuations and planetary scale low frequency anomalies. The latter can also be viewed, however, from the perspective of planetary scales induced by the nonlinear organization of medium scale fluctuations.
Recently this view has been challenged and an alternative explanation has been seriously proposed in which all transient atmospheric patterns-from weather to planetary scale-are the result of random forcing. The background state upon which this transient motion is excited is the laminar stationary background obtained in the bifurcation paradigm, but this flow is never linearly unstable. Thus no bifurcation actually occurs. My talk will pursue a determination from models and observations as to which paradigm best fits atmospheric variability.
Anastasios Tsonis (University of Wisconsin-Milwaukee) aatsonis@csd.uwm.edu
Spatio-temporal properties of the extratropical atmospheric circulation Slides
The atmospheric general circulation often enters into regimes that cause weather anomalies to persist over areas of the globe. By considering 500-hPa measurements we demonstrate the existence of scale invariance in the variability of extratropical atmospheric circulation anomalies over the whole range of timescales resolved by the available data, from a week to a decade. We find that this scale invariance is consistent with atmospheric dynamics and indicates that the memory of the climate system is not confined only to large scales but extends to small scales as well. By investigating the hemispheric structure of the 500-hPa fields in the last 34 years we were able to link this scale invariance to anomaly patterns that exhibit strong spatial coherence and a seemingly decadal variability. We relate these findings to climate processes considered in the recent literature and we discuss the implications of such a property of the general circulation for modeling and prediction of the climate system response.
Donald L. Turcotte (Department of Geological Sciences Cornell University) Turcotte@Geology.Cornell.edu
Self-organized criticality: What is it and what is it good for Slides
The concept of self-organized criticality (SOC) was introduced to explain the behavior of the "sand-pile" model. Other models that exhibit this behavior are the "forest-fire" model and the "slider-block" model. Each of these models can be associated with a serious natural hazard: landslides, forest and wild fires, and earthquakes. The forest-fire model is also closely related to the site-percolation model which exhibits classical critical point behavior. Self-organized criticality can be understood in terms of a self-similar cascade of cluster growth. The growth process is identical in terms of self-similarity to the branching in diffusion-limited aggregation (DLA).
Eli Tziperman (Department of Environmental Sciences Weizmann Institute of Science) eli@beach.weizmann.ac.il http://www.weizmann.ac.il/~eli
A sea ice switch mechanism for the glacial cycles Slides: pdf gzipped postscript
A novel mechanism and model for the glacial-interglacial oscillations will be described. The dominant 100 kyr oscillation in the model land ice volume has the familiar saw-tooth shape of climate proxy records. The glacial oscillation in the proposed mechanism is a nonlinear relaxation oscillation. The transition from the slow to fast phases of the oscillation results from the behavior of sea ice which controls, via its albedo and insulating effects, the atmospheric poleward moisture fluxes and therefore the precipitation that enable the land ice-sheet growth. This control, and the rapid growth and melting of the sea ice, allow the sea ice to rapidly switch the climate system from a slow growing ice-sheet phase to a fast retreating ice-sheet phase, and to shape the oscillation's saw-tooth structure. Milankovitch insolation changes due to variations in the earth orbit around the sun act as a pacemaker, setting the phase of the oscillation via nonlinear phase locking by directly controlling summer melting of ice sheets.
The proposed sea ice switch mechanism for the glacial cycles also results in a natural explanation for the transition from 41 kyr glacial cycles to 100 kyr cycles about one million years ago. The transition according to this explanation is due to a bifurcation of the climate system that resulted in the activation of the sea ice switch at that time.
John A. Whitehead (Woods Hole Oceanographic Institution) jwhitehead@whoi.edu
Morphological instabilities, oscillations, fingers, channels, and multiple equilibria in mantle upwelling, volcanoes, glaciers, and oceans
We review convections rolls, cells, patches of cells, and contrast them with global tectonic plates. This introduces the instability of temperature-dependent viscous flow as it enters a cold region. For one-dimensional flow, multiple equilibrium is found. With elasticity, oscillations are present. Laboratory experiments show oscillatory instability. Glacial surges and some earthquakes have similarities. Volatile presence also produces oscillations. Fingers form in channels and perhaps networks form in porous igneous flows. Multiple equilibrium flows and oscillations are found in models and in laboratory experiments with temperature and salinity. The chimney problem, hydraulic jumps and inertia. Mixing rate has an important and perhaps fundamental effect on global thermohaline transitions.
Ilya Zaliapin (Institute of Geophysics and Planetary Physics, University of California, Los Angeles and International Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences) zal@ess.ucla.edu
Long-range correlations of seismicity prior to strong earthquake. Simple model vs.complex observations Slides pdf (6.8MB)
It is demonstrated that long-range correlations of seismicity precede occurrence of strong earthquakes in Southern California. This is in agreement with the finding made originally for a simple hierarchical model of seismicity. This model uses a Boolean delay equation concept to model the interaction of colliding cascades (direct cascade of loading and inverse cascade of fracturing) that govern the dynamics of seismicity. Distant branches of the system become correlated prior to a fracture at the highest level of the hierarchy. Precursor "Accord" that depicts this phenomenon is rigorously defined for the model. With the same definition precursor "Accord" is applied to major branches of S. California's fault network. Its good and stable performance in retrospective prediction of major earthquakes is demonstrated.
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