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.

A perturbation procedure for nonlinear oscillations (The dynamics of two oscillators with weak nonlinear coupling)

This thesis considers in detail the dynamics of two oscillators with weak nonlinear coupling. There are three classes of such problems: non-resonant, where the Poincaré procedure is valid to the order considered; weakly resonant, where the Poincaré procedure breaks down because small divisors appear (but do not affect the O(1) term) and strongly resonant, where small divisors appear and lead to O(1) corrections. A perturbation method based on Cole's two-timing procedure is introduced. It avoids the small divisor problem in a straightforward manner, gives accurate answers which are valid for long times, and appears capable of handling all three types of problems with no change in the basic approach.

One example of each type is studied with the aid of this procedure: for the nonresonant case the answer is equivalent to the Poincaré result; for the weakly resonant case the analytic form of the answer is found to depend (smoothly) on the difference between the initial energies of the two oscillators; for the strongly resonant case we find that the amplitudes of the two oscillators vary slowly with time as elliptic functions of ϵ t, where ϵ is the (small) coupling parameter.

Our results suggest that, as one might expect, the dynamical behavior of such systems varies smoothly with changes in the ratio of the fundamental frequencies of the two oscillators. Thus the pathological behavior of Whittaker's adelphic integrals as the frequency ratio is varied appears to be due to the fact that Whittaker ignored the small divisor problem. The energy sharing properties of these systems appear to depend strongly on the initial conditions, so that the systems not ergodic.

The perturbation procedure appears to be applicable to a wide variety of other problems in addition to those considered here.

Constraints on the carbon cycle and climate during the early evolution of animals

One of the greatest challenges in science lies in disentangling causality in complex, coupled systems. This is illustrated no better than in the dynamic interplay between the Earth and life. The early evolution and diversification of animals occurred within a backdrop of global change, yet reconstructing the potential role of the environment in this evolutionary transition is challenging. In the 200 million years from the end-Cryogenian to the Ordovician, enigmatic Ediacaran fauna explored body plans, animals diversified and began to biomineralize, forever changing the ocean's chemical cycles, and the biological community in shallow marine ecosystems transitioned from a microbial one to an animal one.

In the following dissertation, a multi-faceted approach combining macro- and micro-scale analyses is presented that draws on the sedimentology, geochemistry and paleontology of the rocks that span this transition to better constrain the potential environmental changes during this interval.

In Chapter 1, the potential of clumped isotope thermometry in deep time is explored by assessing the importance of burial and diagenesis on the thermometer. Eocene- to Precambrian-aged carbonates from the Sultanate of Oman were analyzed from current burial depths of 350-5850 meters. Two end-member styles of diagenesis independent of burial depth were observed.

Chapters 2, 3 and 4 explore the fallibility of the Ediacaran carbon isotope record and aspects of the sedimentology and geochemistry of the rocks preserving the largest negative carbon isotope excursion on record---the Shuram Excursion. Chapter 2 documents the importance of temperature, fluid composition and mineralogy on the delta 18-O min record and interrogates the bulk trace metal signal. Chapter 3 explores the spatial variability in delta 13-C recorded in the transgressive Johnnie Oolite and finds a north-to-south trend recording the onset of the excursion. Chapter 4 investigates the nature of seafloor precipitation during this excursion and more broadly. We document the potential importance of microbial respiratory reactions on the carbonate chemistry of the sediment-water interface through time.

Chapter 5 investigates the latest Precambrian sedimentary record in carbonates from the Sultanate of Oman, including how delta 13-C and delta 34-S CAS vary across depositional and depth gradients. A new model for the correlation of the Buah and Ara formations across Oman is presented. Isotopic results indicate delta 13-C varies with relative eustatic change and delta 34-S CAS may vary in absolute magnitude across Oman.

Chapter 6 investigates the secular rise in delta 18-Omin in the early Paleozoic by using clumped isotope geochemistry on calcitic and phosphatic fossils from the Cambrian and Ordovician. Results do not indicate extreme delta 18-O seawater depletion and instead suggest warmer equatorial temperatures across the early Paleozoic.

Storm track response to perturbations in climate

This thesis advances our understanding of midlatitude storm tracks and how they respond to perturbations in the climate system. The midlatitude storm tracks are regions of maximal turbulent kinetic energy in the atmosphere. Through them, the bulk of the atmospheric transport of energy, water vapor, and angular momentum occurs in midlatitudes. Therefore, they are important regulators of climate, controlling basic features such as the distribution of surface temperatures, precipitation, and winds in midlatitudes. Storm tracks are robustly projected to shift poleward in global-warming simulations with current climate models. Yet the reasons for this shift have remained unclear. Here we show that this shift occurs even in extremely idealized (but still three-dimensional) simulations of dry atmospheres. We use these simulations to develop an understanding of the processes responsible for the shift and develop a conceptual model that accounts for it.

We demonstrate that changes in the convective static stability in the deep tropics alone can drive remote shifts in the midlatitude storm tracks. Through simulations with a dry idealized general circulation model (GCM), midlatitude storm tracks are shown to be located where the mean available potential energy (MAPE, a measure of the potential energy available to be converted into kinetic energy) is maximal. As the climate varies, even if only driven by tropical static stability changes, the MAPE maximum shifts primarily because of shifts of the maximum of near-surface meridional temperature gradients. The temperature gradients shift in response to changes in the width of the tropical Hadley circulation, whose width is affected by the tropical static stability. Storm tracks generally shift in tandem with shifts of the subtropical terminus of the Hadley circulation.

We develop a one-dimensional diffusive energy-balance model that links changes in the Hadley circulation to midlatitude temperature gradients and so to the storm tracks. It is the first conceptual model to incorporate a dynamical coupling between the tropical Hadley circulation and midlatitude turbulent energy transport. Numerical and analytical solutions of the model elucidate the circumstances of when and how the storm tracks shift in tandem with the terminus of the Hadley circulation. They illustrate how an increase of only the convective static stability in the deep tropics can lead to an expansion of the Hadley circulation and a poleward shift of storm tracks.

The simulations with the idealized GCM and the conceptual energy-balance model demonstrate a clear link between Hadley circulation dynamics and midlatitude storm track position. With the help of the hierarchy of models presented in this thesis, we obtain a closed theory of storm track shifts in dry climates. The relevance of this theory for more realistic moist climates is discussed.

Simulations and mechanisms of subtropical low-cloud response to climate change

This thesis focuses on improving the simulation skills and the theoretical understanding of the subtropical low cloud response to climate change.

First, an energetically consistent forcing framework is designed and implemented for the large eddy simulation (LES) of the low-cloud response to climate change. The three representative current-day subtropical low cloud regimes of cumulus (Cu), cumulus-over-stratocumulus, and stratocumulus (Sc) are all well simulated with this framework, and results are comparable to the conventional fixed-SST approach. However, the cumulus response to climate warming subject to energetic constraints differs significantly from the conventional approach with fixed SST. Under the energetic constraint, the subtropics warm less than the tropics, since longwave (LW) cooling is more efficient with the drier subtropical free troposphere. The surface latent heat flux (LHF) also increases only weakly subject to the surface energetic constraint. Both factors contribute to an increased estimated inversion strength (EIS), and decreased inversion height. The decreased Cu-depth contributes to a decrease of liquid water path (LWP) and weak positive cloud feedback. The conventional fixed-SST approach instead simulates a strong increase in LHF and deepening of the Cu layer, leading to a weakly negative cloud feedback. This illustrates the importance of energetic constraints to the simulation and understanding of the sign and magnitude of low-cloud feedback.

Second, an extended eddy-diffusivity mass-flux (EDMF) closure for the unified representation of sub-grid scale (SGS) turbulence and convection processes in general circulation models (GCM) is presented. The inclusion of prognostic terms and the elimination of the infinitesimal updraft fraction assumption makes it more flexible for implementation in models across different scales. This framework can be consistently extended to formulate multiple updrafts and downdrafts, as well as variances and covariances. It has been verified with LES in different boundary layer regimes in the current climate, and further development and implementation of this closure may help to improve our simulation skills and understanding of low-cloud feedback through GCMs.

Wave induced oscillations in harbors of arbitrary shape

Theoretical and experimental studies were conducted to investigate the wave induced oscillations in an arbitrary shaped harbor with constant depth which is connected to the open-sea.

A theory termed the “arbitrary shaped harbor” theory is developed. The solution of the Helmholtz equation, ∇²f + k²f = 0, is formulated as an integral equation; an approximate method is employed to solve the integral equation by converting it to a matrix equation. The final solution is obtained by equating, at the harbor entrance, the wave amplitude and its normal derivative obtained from the solutions for the regions outside and inside the harbor.

Two special theories called the circular harbor theory and the rectangular harbor theory are also developed. The coordinates inside a circular and a rectangular harbor are separable; therefore, the solution for the region inside these harbors is obtained by the method of separation of variables. For the solution in the open-sea region, the same method is used as that employed for the arbitrary shaped harbor theory. The final solution is also obtained by a matching procedure similar to that used for the arbitrary shaped harbor theory. These two special theories provide a useful analytical check on the arbitrary shaped harbor theory.

Experiments were conducted to verify the theories in a wave basin 15 ft wide by 31 ft long with an effective system of wave energy dissipators mounted along the boundary to simulate the open-sea condition.

Four harbors were investigated theoretically and experimentally: circular harbors with a 10° opening and a 60° opening, a rectangular harbor, and a model of the East and West Basins of Long Beach Harbor located in Long Beach, California.

Theoretical solutions for these four harbors using the arbitrary shaped harbor theory were obtained. In addition, the theoretical solutions for the circular harbors and the rectangular harbor using the two special theories were also obtained. In each case, the theories have proven to agree well with the experimental data.

It is found that: (1) the resonant frequencies for a specific harbor are predicted correctly by the theory, although the amplification factors at resonance are somewhat larger than those found experimentally,(2) for the circular harbors, as the width of the harbor entrance increases, the amplification at resonance decreases, but the wave number bandwidth at resonance increases, (3) each peak in the curve of entrance velocity vs incident wave period corresponds to a distinct mode of resonant oscillation inside the harbor, thus the velocity at the harbor entrance appears to be a good indicator for resonance in harbors of complicated shape, (4) the results show that the present theory can be applied with confidence to prototype harbors with relatively uniform depth and reflective interior boundaries.

Stationary Eddies and Zonal Variations of the Global Hydrological Cycle in a Changing Climate

This thesis advances our physical understanding of the sensitivity of the hydrological cycle to global warming. Specifically, it focuses on changes in the longitudinal (zonal) variation of precipitation minus evaporation (P - E), which is predominantly controlled by planetary-scale stationary eddies. By studying idealized general circulation model (GCM) experiments with zonally varying boundary conditions, this thesis examines the mechanisms controlling the strength of stationary-eddy circulations and their role in the hydrological cycle. The overarching goal of this research is to understand the cause of changes in regional P - E with global warming. An understanding of such changes can be useful for impact studies focusing on water availability, ecosystem management, and flood risk.

Based on a moisture-budget analysis of ERA-Interim data, we establish an approximation for zonally anomalous P - E in terms of surface moisture content and stationary-eddy vertical motion in the lower troposphere. Part of the success of this approximation comes from our finding that transient-eddy moisture fluxes partially cancel the effect of stationary-eddy moisture advection, allowing divergent circulations to dominate the moisture budget. The lower-tropospheric vertical motion is related to horizontal motion in stationary eddies by Sverdrup and Ekman balance. These moisture- and vorticity-budget balances also hold in idealized and comprehensive GCM simulations across a range of climates.

By examining climate changes in the idealized and comprehensive GCM simulations, we are able to show the utility of the vertical motion P - E approximation for splitting changes in zonally anomalous P - E into thermodynamic and dynamic components. Shifts in divergent stationary-eddy circulations dominate changes in zonally anomalous P - E. This limits the local utility of the "wet gets wetter, dry gets drier” idea, where existing P - E patterns are amplified with warming by the increase in atmospheric moisture content, with atmospheric circulations held fixed. The increase in atmospheric moisture content manifests instead in an increase in the amplitude of the zonally anomalous hydrological cycle as measured by the zonal variance of P - E. However, dynamic changes, particularly the slowdown of divergent stationary-eddy circulations, limit the strengthening of the zonally anomalous hydrological cycle. In certain idealized cases, dynamic changes are even strong enough to reverse the tendency towards "wet gets wetter, dry gets drier” with warming.

Motivated by the importance of stationary-eddy vertical velocities in the moisture budget analysis, we examine controls on the amplitude of stationary eddies across a wide range of climates in an idealized GCM with simple topographic and ocean-heating zonal asymmetries. An analysis of the thermodynamic equation in the vicinity of topographic forcing reveals the importance of on-slope surface winds, the midlatitude isentropic slope, and latent heating in setting the amplitude of stationary waves. The response of stationary eddies to climate change is determined primarily by the strength of zonal surface winds hitting the mountain. The sensitivity of stationary-eddies to this surface forcing increases with climate change as the slope of midlatitude isentropes decreases. However, latent heating also plays an important role in damping the stationary-eddy response, and this damping becomes stronger with warming as the atmospheric moisture content increases. We find that the response of tropical overturning circulations forced by ocean heat-flux convergence is described by changes in the vertical structure of moist static energy and deep convection. This is used to derive simple scalings for the Walker circulation strength that capture the monotonic decrease with warming found in our idealized simulations.

Through the work of this thesis, the advances made in understanding the amplitude of stationary-waves in a changing climate can be directly applied to better understand and predict changes in the zonally anomalous hydrological cycle.