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.

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.

Latent variable models for neural data analysis

The brain is perhaps the most complex system to have ever been subjected to rigorous scientific investigation. The scale is staggering: over 10^11 neurons, each making an average of 10^3 synapses, with computation occurring on scales ranging from a single dendritic spine, to an entire cortical area. Slowly, we are beginning to acquire experimental tools that can gather the massive amounts of data needed to characterize this system. However, to understand and interpret these data will also require substantial strides in inferential and statistical techniques. This dissertation attempts to meet this need, extending and applying the modern tools of latent variable modeling to problems in neural data analysis.

It is divided into two parts. The first begins with an exposition of the general techniques of latent variable modeling. A new, extremely general, optimization algorithm is proposed - called Relaxation Expectation Maximization (REM) - that may be used to learn the optimal parameter values of arbitrary latent variable models. This algorithm appears to alleviate the common problem of convergence to local, sub-optimal, likelihood maxima. REM leads to a natural framework for model size selection; in combination with standard model selection techniques the quality of fits may be further improved, while the appropriate model size is automatically and efficiently determined. Next, a new latent variable model, the mixture of sparse hidden Markov models, is introduced, and approximate inference and learning algorithms are derived for it. This model is applied in the second part of the thesis.

The second part brings the technology of part I to bear on two important problems in experimental neuroscience. The first is known as spike sorting; this is the problem of separating the spikes from different neurons embedded within an extracellular recording. The dissertation offers the first thorough statistical analysis of this problem, which then yields the first powerful probabilistic solution. The second problem addressed is that of characterizing the distribution of spike trains recorded from the same neuron under identical experimental conditions. A latent variable model is proposed. Inference and learning in this model leads to new principled algorithms for smoothing and clustering of spike data.

Robot navigation in dense crowds : statistical models and experimental studies of human robot cooperation

This thesis explores the problem of mobile robot navigation in dense human crowds. We begin by considering a fundamental impediment to classical motion planning algorithms called the freezing robot problem: once the environment surpasses a certain level of complexity, the planner decides that all forward paths are unsafe, and the robot freezes in place (or performs unnecessary maneuvers) to avoid collisions. Since a feasible path typically exists, this behavior is suboptimal. Existing approaches have focused on reducing predictive uncertainty by employing higher fidelity individual dynamics models or heuristically limiting the individual predictive covariance to prevent overcautious navigation. We demonstrate that both the individual prediction and the individual predictive uncertainty have little to do with this undesirable navigation behavior. Additionally, we provide evidence that dynamic agents are able to navigate in dense crowds by engaging in joint collision avoidance, cooperatively making room to create feasible trajectories. We accordingly develop interacting Gaussian processes, a prediction density that captures cooperative collision avoidance, and a "multiple goal" extension that models the goal driven nature of human decision making. Navigation naturally emerges as a statistic of this distribution.

Most importantly, we empirically validate our models in the Chandler dining hall at Caltech during peak hours, and in the process, carry out the first extensive quantitative study of robot navigation in dense human crowds (collecting data on 488 runs). The multiple goal interacting Gaussian processes algorithm performs comparably with human teleoperators in crowd densities nearing 1 person/m², while a state of the art noncooperative planner exhibits unsafe behavior more than 3 times as often as the multiple goal extension, and twice as often as the basic interacting Gaussian process approach. Furthermore, a reactive planner based on the widely used dynamic window approach proves insufficient for crowd densities above 0.55 people/m². We also show that our noncooperative planner or our reactive planner capture the salient characteristics of nearly any dynamic navigation algorithm. For inclusive validation purposes, we show that either our non-interacting planner or our reactive planner captures the salient characteristics of nearly any existing dynamic navigation algorithm. Based on these experimental results and theoretical observations, we conclude that a cooperation model is critical for safe and efficient robot navigation in dense human crowds.

Finally, we produce a large database of ground truth pedestrian crowd data. We make this ground truth database publicly available for further scientific study of crowd prediction models, learning from demonstration algorithms, and human robot interaction models in general.

Transition metal clusters supported by multinucleating ligand frameworks as models of biological active sites

This dissertation describes efforts to model biological active sites with small molecule clusters. The approach used took advantage of a multinucleating ligand to control the structure and nuclearity of the product complexes, allowing the study of many different homo- and heterometallic clusters. Chapter 2 describes the synthesis of the multinucleating hexapyridyl trialkoxy ligand used throughout this thesis and the synthesis of trinuclear first row transition metal complexes supported by this framework, with an emphasis on tricopper systems as models of biological multicopper oxidases. The magnetic susceptibility of these complexes were studied, and a linear relation was found between the Cu-O(alkoxide)-Cu angles and the antiferromagnetic coupling between copper centers. The triiron(II) and trizinc(II) complexes of the ligand were also isolated and structurally characterized.

Chapter 3 describes the synthesis of a series of heterometallic tetranuclear manganese dioxido complexes with various incorporated apical redox-inactive metal cations (M = Na⁺, Ca²⁺, Sr²⁺, Zn²⁺, Y³⁺). Chapter 4 presents the synthesis of heterometallic trimanganese(IV) tetraoxido complexes structurally related to the CaMn₃ subsite of the oxygen-evolving complex (OEC) of Photosystem II. The reduction potentials of these complexes were studied, and it was found that each isostructural series displays a linear correlation between the reduction potentials and the Lewis acidities of the incorporated redox-inactive metals. The slopes of the plotted lines for both the dioxido and tetraoxido clusters are the same, suggesting a more general relationship between the electrochemical potentials of heterometallic manganese oxido clusters and their “spectator” cations. Additionally, these studies suggest that Ca²⁺ plays a role in modulating the redox potential of the OEC for water oxidation.

Chapter 5 presents studies of the effects of the redox-inactive metals on the reactivities of the heterometallic manganese complexes discussed in Chapters 3 and 4. Oxygen atom transfer from the clusters to phosphines is studied; although the reactivity is kinetically controlled in the tetraoxido clusters, the dioxido clusters with more Lewis acidic metal ions (Y³⁺ vs. Ca²⁺) appear to be more reactive. Investigations of hydrogen atom transfer and electron transfer rates are also discussed.

Appendix A describes the synthesis, and metallation reactions of a new dinucleating bis(N-heterocyclic carbene)ligand framework. Dicopper(I) and dicobalt(II) complexes of this ligand were prepared and structurally characterized. A dinickel(I) dichloride complex was synthesized, reduced, and found to activate carbon dioxide. Appendix B describes preliminary efforts to desymmetrize the manganese oxido clusters via functionalization of the basal multinucleating ligand used in the preceding sections of this dissertation. Finally, Appendix C presents some partially characterized side products and unexpected structures that were isolated throughout the course of these studies.

I. Quantum mechanics of one-dimensional two-particle models. II. A quantum mechanical treatment of inelastic collisions

Part I

Solutions of Schrödinger’s equation for system of two particles bound in various stationary one-dimensional potential wells and repelling each other with a Coulomb force are obtained by the method of finite differences. The general properties of such systems are worked out in detail for the case of two electrons in an infinite square well. For small well widths (1-10 a.u.) the energy levels lie above those of the noninteresting particle model by as much as a factor of 4, although excitation energies are only half again as great. The analytical form of the solutions is obtained and it is shown that every eigenstate is doubly degenerate due to the “pathological” nature of the one-dimensional Coulomb potential. This degeneracy is verified numerically by the finite-difference method. The properties of the square-well system are compared with those of the free-electron and hard-sphere models; perturbation and variational treatments are also carried out using the hard-sphere Hamiltonian as a zeroth-order approximation. The lowest several finite-difference eigenvalues converge from below with decreasing mesh size to energies below those of the “best” linear variational function consisting of hard-sphere eigenfunctions. The finite-difference solutions in general yield expectation values and matrix elements as accurate as those obtained using the “best” variational function.

The system of two electrons in a parabolic well is also treated by finite differences. In this system it is possible to separate the center-of-mass motion and hence to effect a considerable numerical simplification. It is shown that the pathological one-dimensional Coulomb potential gives rise to doubly degenerate eigenstates for the parabolic well in exactly the same manner as for the infinite square well.

Part II

A general method of treating inelastic collisions quantum mechanically is developed and applied to several one-dimensional models. The formalism is first developed for nonreactive “vibrational” excitations of a bound system by an incident free particle. It is then extended to treat simple exchange reactions of the form A + BC →AB + C. The method consists essentially of finding a set of linearly independent solutions of the Schrödinger equation such that each solution of the set satisfies a distinct, yet arbitrary boundary condition specified in the asymptotic region. These linearly independent solutions are then combined to form a total scattering wavefunction having the correct asymptotic form. The method of finite differences is used to determine the linearly independent functions.

The theory is applied to the impulsive collision of a free particle with a particle bound in (1) an infinite square well and (2) a parabolic well. Calculated transition probabilities agree well with previously obtained values.

Several models for the exchange reaction involving three identical particles are also treated: (1) infinite-square-well potential surface, in which all three particles interact as hard spheres and each two-particle subsystem (i.e. BC and AB) is bound by an attractive infinite-square-well potential; (2) truncated parabolic potential surface, in which the two-particle subsystems are bound by a harmonic oscillator potential which becomes infinite for interparticle separations greater than a certain value; (3) parabolic (untruncated) surface. Although there are no published values with which to compare our reaction probabilities, several independent checks on internal consistency indicate that the results are reliable.

Relativistic quark models and the current algebra

In this thesis we are concerned with finding representations of the algebra of SU(3) vector and axial-vector charge densities at infinite momentum (the "current algebra") to describe the mesons, idealizing the real continua of multiparticle states as a series of discrete resonances of zero width. Such representations would describe the masses and quantum numbers of the mesons, the shapes of their Regge trajectories, their electromagnetic and weak form factors, and (approximately, through the PCAC hypothesis) pion emission or absorption amplitudes.

We assume that the mesons have internal degrees of freedom equivalent to being made of two quarks (one an antiquark) and look for models in which the mass is SU(3)-independent and the current is a sum of contributions from the individual quarks. Requiring that the current algebra, as well as conditions of relativistic invariance, be satisfied turns out to be very restrictive, and, in fact, no model has been found which satisfies all requirements and gives a reasonable mass spectrum. We show that using more general mass and current operators but keeping the same internal degrees of freedom will not make the problem any more solvable. In particular, in order for any two-quark solution to exist it must be possible to solve the "factorized SU(2) problem," in which the currents are isospin currents and are carried by only one of the component quarks (as in the K meson and its excited states).

In the free-quark model the currents at infinite momentum are found using a manifestly covariant formalism and are shown to satisfy the current algebra, but the mass spectrum is unrealistic. We then consider a pair of quarks bound by a potential, finding the current as a power series in 1/m where m is the quark mass. Here it is found impossible to satisfy the algebra and relativistic invariance with the type of potential tried, because the current contributions from the two quarks do not commute with each other to order 1/m³. However, it may be possible to solve the factorized SU(2) problem with this model.

The factorized problem can be solved exactly in the case where all mesons have the same mass, using a covariant formulation in terms of an internal Lorentz group. For a more realistic, nondegenerate mass there is difficulty in covariantly solving even the factorized problem; one model is described which almost works but appears to require particles of spacelike 4-momentum, which seem unphysical.

Although the search for a completely satisfactory model has been unsuccessful, the techniques used here might eventually reveal a working model. There is also a possibility of satisfying a weaker form of the current algebra with existing models.