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.

Ring-opening metathesis of bulky norbornene monomers and the radical-mediated hydrophosphonation of olefins

This thesis discusses two major topics: the ring-opening metathesis polymerization (ROMP) of bulky monomers and the radical-mediated hydrophosphonation of olefins. The research into the ROMP of bulky monomers is further divided into three chapters: wedge-shaped monomers, the alternating copolymerization of 1-methyloxanorbornene derivatives with cyclooctene, and the kinetic resolution polymerization of 1-methyloxanorbornene derivatives. The wedge-shaped monomers can be polymerized into diblock copolymers that possess photonic crystal properties. The alternating copolymerization of 1-methyloxanorbornene derivatives with cyclooctene is performed with > 90% alternation via two different routes: typical alternating copolymerization and a sequence editing approach. The kinetic resolution polymerization of these same 1-methyloxanorbornene monomers achieves only modest selectivity (S=4), but there is evidence that the growing polymer chain forms a helix that influences the selectivity of the resolution. The last topic is the radical-mediated hydrophosphonation of olefins. This synthetic method provides access to Wittig reagents that are capable of highly cis-selective olefinations of aldehydes.

Palladium-catalyzed asymmetric allylic alkylation: insights, application toward cyclopentanoid and cycloheptanoid molecules, and the total synthesis of several daucane sesquiterpenes

The asymmetric construction of quaternary stereocenters is a topic of great interest in the organic chemistry community given their prevalence in natural products and biologically active molecules. Over the last decade, the Stoltz group has pursued the synthesis of this challenging motif via a palladium-catalyzed allylic alkylation using chiral phosphinooxazoline (PHOX) ligands. Recent results indicate that the alkylation of lactams and imides consistently proceeds with enantioselectivities substantially higher than any other substrate class previously examined in this system. This observation prompted exploration of the characteristics that distinguish these molecules as superior alkylation substrates, resulting in newfound insights and marked improvements in the allylic alkylation of carbocyclic compounds.

General routes to cyclopentanoid and cycloheptanoid core structures have been developed that incorporate the palladium-catalyzed allylic alkylation as a key transformation. The unique reactivity of α-quaternary vinylogous esters upon addition of hydride or organometallic reagents enables divergent access to γ-quaternary acylcyclopentenes or cycloheptenones through respective ring contraction or carbonyl transposition pathways. Derivatization of the resulting molecules provides a series of mono-, bi-, and tricyclic systems that can serve as valuable intermediates for the total synthesis of complex natural products.

The allylic alkylation and ring contraction methodology has been employed to prepare variably functionalized bicyclo[5.3.0]decane molecules and enables the enantioselective total syntheses of daucene, daucenal, epoxydaucenal B, and 14-p-anisoyloxydauc-4,8-diene. This route overcomes the challenge of accessing β-substituted acylcyclopentenes by employing a siloxyenone to effect the Grignard addition and ring opening in a single step. Subsequent ring-closing metathesis and aldol reactions form the hydroazulene core of these targets. Derivatization of a key enone intermediate allows access to either the daucane sesquiterpene or sphenobolane diterpene carbon skeletons, as well as other oxygenated scaffolds.

Analyses of planetary atmospheres across the spectrum : from Titan to exoplanets

Planetary atmospheres exist in a seemingly endless variety of physical and chemical environments. There are an equally diverse number of methods by which we can study and characterize atmospheric composition. In order to better understand the fundamental chemistry and physical processes underlying all planetary atmospheres, my research of the past four years has focused on two distinct topics. First, I focused on the data analysis and spectral retrieval of observations obtained by the Ultraviolet Imaging Spectrograph (UVIS) instrument onboard the Cassini spacecraft while in orbit around Saturn. These observations consisted of stellar occultation measurements of Titan's upper atmosphere, probing the chemical composition in the region 300 to 1500 km above Titan's surface. I examined the relative abundances of Titan's two most prevalent chemical species, nitrogen and methane. I also focused on the aerosols that are formed through chemistry involving these two major species, and determined the vertical profiles of aerosol particles as a function of time and latitude. Moving beyond our own solar system, my second topic of investigation involved analysis of infra-red light curves from the Spitzer space telescope, obtained as it measured the light from stars hosting planets of their own. I focused on both transit and eclipse modeling during Spitzer data reduction and analysis. In my initial work, I utilized the data to search for transits of planets a few Earth masses in size. In more recent research, I analyzed secondary eclipses of three exoplanets and constrained the range of possible temperatures and compositions of their atmospheres.

Structural and clumped-isotope constraints on the mechanisms of displacement along low-angle detachments

Despite years of research on low-angle detachments, much about them remains enigmatic. This thesis addresses some of the uncertainty regarding two particular detachments, the Mormon Peak detachment in Nevada and the Heart Mountain detachment in Wyoming and Montana.

Constraints on the geometry and kinematics of emplacement of the Mormon Peak detachment are provided by detailed geologic mapping of the Meadow Valley Mountains, along with an analysis of structural data within the allochthon in the Mormon Mountains. Identifiable structures well suited to constrain the kinematics of the detachment include a newly mapped, Sevier-age monoclinal flexure in the hanging wall of the detachment. This flexure, including the syncline at its base and the anticline at its top, can be readily matched to the base and top of the frontal Sevier thrust ramp, which is exposed in the footwall of the detachment to the east in the Mormon Mountains and Tule Springs Hills. The ~12 km of offset of these structural markers precludes the radial sliding hypothesis for emplacement of the allochthon.

The role of fluids in the slip along faults is a widely investigated topic, but the use of carbonate clumped-isotope thermometry to investigate these fluids is new. Faults rocks from within ~1 m of the Mormon Peak detachment, including veins, breccias, gouges, and host rocks, were analyzed for carbon, oxygen, and clumped-isotope measurements. The data indicate that much of the carbonate breccia and gouge material along the detachment is comminuted host rock, as expected. Measurements in vein material indicate that the fluid system is dominated by meteoric water, whose temperature indicates circulation to substantial depths (c. 4 km) in the upper crust near the fault zone.

Slip along the subhorizontal Heart Mountain detachment is particularly enigmatic, and many different mechanisms for failure have been proposed, predominantly involving catastrophic failure. Textural evidence of multiple slip events is abundant, and include multiple brecciation events and cross-cutting clastic dikes. Footwall deformation is observed in numerous exposures of the detachment. Stylolitic surfaces and alteration textures within and around “banded grains” previously interpreted to be an indicator of high-temperature fluidization along the fault suggest their formation instead via low-temperature dissolution and alteration processes. There is abundant textural evidence of the significant role of fluids along the detachment via pressure solution. The process of pressure solution creep may be responsible for enabling multiple slip events on the low-angle detachment, via a local rotation of the stress field.

Clumped-isotope thermometry of fault rocks associated with the Heart Mountain detachment indicates that despite its location on the flanks of a volcano that was active during slip, the majority of carbonate along the Heart Mountain detachment does not record significant heating above ambient temperatures (c. 40-70°C). Instead, cold meteoric fluids infiltrated the detachment breccia, and carbonate precipitated under ambient temperatures controlled by structural depth. Locally, fault gouge does preserve hot temperatures (>200°C), as is observed in both the Mormon Peak detachment and Heart Mountain detachment areas. Samples with very hot temperatures attributable to frictional shear heating are present but rare. They appear to be best preserved in hanging wall structures related to the detachment, rather than along the main detachment.

Evidence is presented for the prevalence of relatively cold, meteoric fluids along both shallow crustal detachments studied, and for protracted histories of slip along both detachments. Frictional heating is evident from both areas, but is a minor component of the preserved fault rock record. Pressure solution is evident, and might play a role in initiating slip on the Heart Mountain fault, and possibly other low-angle detachments.

Observations and Modeling of Tropical Planetary Atmospheres

This thesis is a comprised of three different projects within the topic of tropical atmospheric dynamics. First, I analyze observations of thermal radiation from Saturn’s atmosphere and from them, determine the latitudinal distribution of ammonia vapor near the 1.5-bar pressure level. The most prominent feature of the observations is the high brightness temperature of Saturn’s subtropical latitudes on either side of the equator. After comparing the observations to a microwave radiative transfer model, I find that these subtropical bands require very low ammonia relative humidity below the ammonia cloud layer in order to achieve the high brightness temperatures observed. We suggest that these bright subtropical bands represent dry zones created by a meridionally overturning circulation.

Second, I use a dry atmospheric general circulation model to study equatorial superrotation in terrestrial atmospheres. A wide range of atmospheres are simulated by varying three parameters: the pole-equator radiative equilibrium temperature contrast, the convective lapse rate, and the planetary rotation rate. A scaling theory is developed that establishes conditions under which superrotation occurs in terrestrial atmospheres. The scaling arguments show that superrotation is favored when the off-equatorial baroclinicity and planetary rotation rates are low. Similarly, superrotation is favored when the convective heating strengthens, which may account for the superrotation seen in extreme global-warming simulations.

Third, I use a moist slab-ocean general circulation model to study the impact of a zonally-symmetric continent on the distribution of monsoonal precipitation. I show that adding a hemispheric asymmetry in surface heat capacity is sufficient to cause symmetry breaking in both the spatial and temporal distribution of precipitation. This spatial symmetry breaking can be understood from a large-scale energetic perspective, while the temporal symmetry breaking requires consideration of the dynamical response to the heat capacity asymmetry and the seasonal cycle of insolation. Interestingly, the idealized monsoonal precipitation bears resemblance to precipitation in the Indian monsoon sector, suggesting that this work may provide insight into the causes of the temporally asymmetric distribution of precipitation over southeast Asia.