Nonlinear oscillations in discrete and continuous systems

The first thesis topic is a perturbation method for resonantly coupled nonlinear oscillators. By successive near-identity transformations of the original equations, one obtains new equations with simple structure that describe the long time evolution of the motion. This technique is related to two-timing in that secular terms are suppressed in the transformation equations. The method has some important advantages. Appropriate time scalings are generated naturally by the method, and don't need to be guessed as in two-timing. Furthermore, by continuing the procedure to higher order, one extends (formally) the time scale of valid approximation. Examples illustrate these claims. Using this method, we investigate resonance in conservative, non-conservative and time dependent problems. Each example is chosen to highlight a certain aspect of the method.

The second thesis topic concerns the coupling of nonlinear chemical oscillators. The first problem is the propagation of chemical waves of an oscillating reaction in a diffusive medium. Using two-timing, we derive a nonlinear equation that determines how spatial variations in the phase of the oscillations evolves in time. This result is the key to understanding the propagation of chemical waves. In particular, we use it to account for certain experimental observations on the Belusov-Zhabotinskii reaction.

Next, we analyse the interaction between a pair of coupled chemical oscillators. This time, we derive an equation for the phase shift, which measures how much the oscillators are out of phase. This result is the key to understanding M. Marek's and I. Stuchl's results on coupled reactor systems. In particular, our model accounts for synchronization and its bifurcation into rhythm splitting.

Finally, we analyse large systems of coupled chemical oscillators. Using a continuum approximation, we demonstrate mechanisms that cause auto-synchronization in such systems.

Characterization of diverse megathrust fault behavior related to seismic supercycles, Mentawai Islands, Sumatra

Long paleoseismic histories are necessary for understanding the full range of behavior of faults, as the most destructive events often have recurrence intervals longer than local recorded history. The Sunda megathrust, the interface along which the Australian plate subducts beneath Southeast Asia, provides an ideal natural laboratory for determining a detailed paleoseismic history over many seismic cycles. The outer-arc islands above the seismogenic portion of the megathrust cyclically rise and subside in response to processes on the underlying megathrust, providing uncommonly good illumination of megathrust behavior. Furthermore, the growth histories of coral microatolls, which record tectonic uplift and subsidence via relative sea level, can be used to investigate the detailed coseismic and interseismic deformation patterns. One particularly interesting area is the Mentawai segment of the megathrust, which has been shown to characteristically fail in a series of ruptures over decades, rather than a single end-to-end rupture. This behavior has been termed a seismic “supercycle.” Prior to the current rupture sequence, which began in 2007, the segment previously ruptured during the 14th century, the late 16th to late 17th century, and most recently during historical earthquakes in 1797 and 1833. In this study, we examine each of these previous supercycles in turn.

First, we expand upon previous analysis of the 1797–1833 rupture sequence with a comprehensive review of previously published coral microatoll data and the addition of a significant amount of new data. We present detailed maps of coseismic uplift during the two great earthquakes and of interseismic deformation during the periods 1755–1833 and 1950–1997 and models of the corresponding slip and coupling on the underlying megathrust. We derive magnitudes of Mw 8.7–9.0 for the two historical earthquakes, and determine that the 1797 earthquake fundamentally changed the state of coupling on the fault for decades afterward. We conclude that while major earthquakes generally do not involve rupture of the entire Mentawai segment, they undoubtedly influence the progression of subsequent ruptures, even beyond their own rupture area. This concept is of vital importance for monitoring and forecasting the progression of the modern rupture sequence.

Turning our attention to the 14th century, we present evidence of a shallow slip event in approximately A.D. 1314, which preceded the “conventional” megathrust rupture sequence. We calculate a suite of slip models, slightly deeper and/or larger than the 2010 Pagai Islands earthquake, that are consistent with the large amount of subsidence recorded at our study site. Sea-level records from older coral microatolls suggest that these events occur at least once every millennium, but likely far less frequently than their great downdip neighbors. The revelation that shallow slip events are important contributors to the seismic cycle of the Mentawai segment further complicates our understanding of this subduction megathrust and our assessment of the region’s exposure to seismic and tsunami hazards.

Finally, we present an outline of the complex intervening rupture sequence that took place in the 16th and 17th centuries, which involved at least five distinct uplift events. We conclude that each of the supercycles had unique features, and all of the types of fault behavior we observe are consistent with highly heterogeneous frictional properties of the megathrust beneath the south-central Mentawai Islands. We conclude that the heterogeneous distribution of asperities produces terminations and overlap zones between fault ruptures, resulting in the seismic “supercycle” phenomenon.

Wind-driven desertification: process modeling, remote monitoring, and forecasting

Arid and semiarid landscapes comprise nearly a third of the Earth's total land surface. These areas are coming under increasing land use pressures. Despite their low productivity these lands are not barren. Rather, they consist of fragile ecosystems vulnerable to anthropogenic disturbance.

The purpose of this thesis is threefold: (I) to develop and test a process model of wind-driven desertification, (II) to evaluate next-generation process-relevant remote monitoring strategies for use in arid and semiarid regions, and (III) to identify elements for effective management of the world's drylands.

In developing the process model of wind-driven desertification in arid and semiarid lands, field, remote sensing, and modeling observations from a degraded Mojave Desert shrubland are used. This model focuses on aeolian removal and transport of dust, sand, and litter as the primary mechanisms of degradation: killing plants by burial and abrasion, interrupting natural processes of nutrient accumulation, and allowing the loss of soil resources by abiotic transport. This model is tested in field sampling experiments at two sites and is extended by Fourier Transform and geostatistical analysis of high-resolution imagery from one site.

Next, the use of hyperspectral remote sensing data is evaluated as a substantive input to dryland remote monitoring strategies. In particular, the efficacy of spectral mixture analysis (SMA) in discriminating vegetation and soil types and detennining vegetation cover is investigated. The results indicate that hyperspectral data may be less useful than often thought in determining vegetation parameters. Its usefulness in determining soil parameters, however, may be leveraged by developing simple multispectral classification tools that can be used to monitor desertification.

Finally, the elements required for effective monitoring and management of arid and semiarid lands are discussed. Several large-scale multi-site field experiments are proposed to clarify the role of wind as a landscape and degradation process in dry lands. The role of remote sensing in monitoring the world's drylands is discussed in terms of optimal remote sensing platform characteristics and surface phenomena which may be monitored in order to identify areas at risk of desertification. A desertification indicator is proposed that unifies consideration of environmental and human variables.

The Madden-Julian Oscillation : observation, modeling, and theory

The Madden-Julian Oscillation (MJO) is a pattern of intense rainfall and associated planetary-scale circulations in the tropical atmosphere, with a recurrence interval of 30-90 days. Although the MJO was first discovered 40 years ago, it is still a challenge to simulate the MJO in general circulation models (GCMs), and even with simple models it is difficult to agree on the basic mechanisms. This deficiency is mainly due to our poor understanding of moist convection—deep cumulus clouds and thunderstorms, which occur at scales that are smaller than the resolution elements of the GCMs. Moist convection is the most important mechanism for transporting energy from the ocean to the atmosphere. Success in simulating the MJO will improve our understanding of moist convection and thereby improve weather and climate forecasting.

We address this fundamental subject by analyzing observational datasets, constructing a hierarchy of numerical models, and developing theories. Parameters of the models are taken from observation, and the simulated MJO fits the data without further adjustments. The major findings include: 1) the MJO may be an ensemble of convection events linked together by small-scale high-frequency inertia-gravity waves; 2) the eastward propagation of the MJO is determined by the difference between the eastward and westward phase speeds of the waves; 3) the planetary scale of the MJO is the length over which temperature anomalies can be effectively smoothed by gravity waves; 4) the strength of the MJO increases with the typical strength of convection, which increases in a warming climate; 5) the horizontal scale of the MJO increases with the spatial frequency of convection; and 6) triggered convection, where potential energy accumulates until a threshold is reached, is important in simulating the MJO. Our findings challenge previous paradigms, which consider the MJO as a large-scale mode, and point to ways for improving the climate models.

Towards in situ Single Cell Systems Biology

Systems-level studies of biological systems rely on observations taken at a resolution lower than the essential unit of biology, the cell. Recent technical advances in DNA sequencing have enabled measurements of the transcriptomes in single cells excised from their environment, but it remains a daunting technical problem to reconstruct in situ gene expression patterns from sequencing data. In this thesis I develop methods for the routine, quantitative in situ measurement of gene expression using fluorescence microscopy.

The number of molecular species that can be measured simultaneously by fluorescence microscopy is limited by the pallet of spectrally distinct fluorophores. Thus, fluorescence microscopy is traditionally limited to the simultaneous measurement of only five labeled biomolecules at a time. The two methods described in this thesis, super-resolution barcoding and temporal barcoding, represent strategies for overcoming this limitation to monitor expression of many genes in a single cell. Super-resolution barcoding employs optical super-resolution microscopy (SRM) and combinatorial labeling via-smFISH (single molecule fluorescence in situ hybridization) to uniquely label individual mRNA species with distinct barcodes resolvable at nanometer resolution. This method dramatically increases the optical space in a cell, allowing a large numbers of barcodes to be visualized simultaneously. As a proof of principle this technology was used to study the S. cerevisiae calcium stress response. The second method, sequential barcoding, reads out a temporal barcode through multiple rounds of oligonucleotide hybridization to the same mRNA. The multiplexing capacity of sequential barcoding increases exponentially with the number of rounds of hybridization, allowing over a hundred genes to be profiled in only a few rounds of hybridization.

The utility of sequential barcoding was further demonstrated by adapting this method to study gene expression in mammalian tissues. Mammalian tissues suffer both from a large amount of auto-fluorescence and light scattering, making detection of smFISH probes on mRNA difficult. An amplified single molecule detection technology, smHCR (single molecule hairpin chain reaction), was developed to allow for the quantification of mRNA in tissue. This technology is demonstrated in combination with light sheet microscopy and background reducing tissue clearing technology, enabling whole-organ sequential barcoding to monitor in situ gene expression directly in intact mammalian tissue.

The methods presented in this thesis, specifically sequential barcoding and smHCR, enable multiplexed transcriptional observations in any tissue of interest. These technologies will serve as a general platform for future transcriptomic studies of complex tissues.