Stochastic Stability of Open-Ocean Deep Convection

Open-ocean deep convection is a highly variable and strongly nonlinear process that plays an essential role in the global ocean circulation. A new view of its stability is presented here, in which variability, as parameterised by stochastic forcing, is central. The use of an idealised deep convection box model allows analytical solutions and straightforward conceptual understand-ing, while retaining the main features of deep convection dynamics. In contrast to the generally abrupt stability changes in deterministic systems, measures of stochastic stability change smoothly in response to varying forcing parameters. These stochastic stability measures depend chie y on the residence times of the system in dierent regions of phase space, which need not contain a stable steady state in the deterministic sense. Deep convection can occur frequently even for parameter ranges in which it is deterministically unstable; this eect is denoted wandering unimodality. The stochastic stability concepts are readily applied to other components of the climate system. The results highlight the need to take climate variability into account when analysing the stability of a climate state.

The genetic architecture of the human cerebral cortex

The cerebral cortex underlies our complex cognitive capabilities, yet little is known about the specific genetic loci that influence human cortical structure. To identify genetic variants that affect cortical structure, we conducted a genome-wide association meta-analysis of brain magnetic resonance imaging data from 51,665 individuals. We analyzed the surface area and average thickness of the whole cortex and 34 regions with known functional specializations. We identified 199 significant loci and found significant enrichment for loci influencing total surface area within regulatory elements that are active during prenatal cortical development, supporting the radial unit hypothesis. Loci that affect regional surface area cluster near genes in Wnt signaling pathways, which influence progenitor expansion and areal identity. Variation in cortical structure is genetically correlated with cognitive function, Parkinson's disease, insomnia, depression, neuroticism, and attention deficit hyperactivity disorder

Nonlinea principal component analysis of climate data

A nonlinear generalisation of Principal Component Analysis (PCA), denoted Nonlinear Principal Component Analysis (NLPCA), is introduced and applied to the analysis of climate data. This method is implemented using a 5-layer feed-forward neural network introduced originally in the chemical engineering literature. The method is described and details of its implementation are addressed. It is found empirically that NLPCA partitions variance in the same fashion as does PCA, that is, that the sum of the total variance of the NLPCA approximation with the total variance of the residual from the original data is equal to the total variance of the original data. An important distinction is drawn between a modal P-dimensional NLPCA analysis, in which P successive 1D approximations are determined iteratively so that the approximation is the sum of P nonlinear functions of one variable, and a nonmodal analysis, in which the P-dimensional NLPCA approximation is determined as a nonlinear non-additive function of P variables. Nonlinear Principal Component Analysis is first applied to a data set sampled from the Lorenz attractor. It is found that the NLPCA approximations are much more representative of the data than are the corresponding PCA approximations. In particular, the 1D and 2D NLPCA approximations explain 76% and 99.5% of the total variance, respectively, in contrast to 60% and 95% explained by the 1D and 2D PCA approximations. When applied to a data set consisting of monthly-averaged tropical Pacific Ocean sea surface temperatures (SST), the modal 1D NLPCA approximation describes average variability associated with the El Nino/Southern Oscillation (ENSO) phenomenon, as does the 1D PCA approximation. The NLPCA approximation, however, characterises the asymmetry in spatial pattern of SST anomalies between average warm and cold events (manifested in the skewness of the distribution) in a manner that the PCA approximation cannot. The second NLPCA mode of SST is found to characterise differences in ENSO variability between individual events, and in particular is consistent with the celebrated 1977 "regime shift". A 2D nonmodal NLPCA approximation is determined, the interpretation of which is complicated by the fact that a secondary feature extraction problem has to be carried out to interpret the results. It is found that this approximation contains much the same information as that provided by the modal analysis. A modal NLPC analysis of tropical Indo-Pacific sea level pressure (SLP) finds that the first mode describes average ENSO variability in this field, and also characterises an asymmetry in SLP fields between average warm and cold events. No robust nonlinear mode beyond the first could be found. Nonlinear Principal Component Analysis is used to find the optimal nonlinear approximation to SLP data produced by a 1001 year integration of the Canadian Centre for Climate Modelling and Analysis (CCCma) coupled general circulation model (CGCM1). This approximation's associated time series is strongly bimodal and partitions the data into two distinct regimes. The first and more persistent regime describes a standing oscillation whose signature in the mid-troposphere is alternating amplification and attenuation of the climatological ridge over Northern Europe. The second and more episodic regime describes mid-tropospheric split-flow south of Greenland. Essentially the same structure is found in the 1D NLPCA approximation of the 500mb height field itself. In a 500 year integration with atmospheric CO2 at four times pre-industrial concentrations, the occupation statistics of these preferred modes of variability change, such that the episodic split-flow regime occurs less frequently while the standing oscillation regime occurs more frequently. Finally, a generalisation of Kramer’s NLPCA using a 7-layer autoassociative neural network is introduced to address the inability of Kramer’s original network to find P-dimensional structure topologically different from the unit cube in RP. The example of an ellipse is considered, and it is shown that the approximation produced by the 7-layer network is a substantial improvement over that produced by the 5-layer network. [Scientific formulae used in this abstract could not be reproduced.]Science, Faculty ofEarth, Ocean and Atmospheric Sciences, Department ofGraduat

Stabilization of climate regimes by noise in a simple model of the thermohaline circulation

ABSTRACT Salinity dynamics in a simple two-box model of the thermohaline circulation (THC) is considered. The model parameterizes fluctuating eddy transport and buoyancy forcing by two independent stochastic processes. The associated stationary probability density function is calculated analytically, and its structure is analyzed in the space of the three parameters of the model. It is found that over a broad range of model parameters in which the stationary density is technically bimodal, the population of one regime is very much larger than that of the other, so the system behaves effectively unimodally. This preferential population of one regime is denoted stabilization. This phenomenon is only relevant if the timescale of THC variability is less than the mean residence times of the destabilized regime, so that the system may be described by its stationary probability density. These average residence times are calculated, and it is found that stabilization occurs over a broad range of parameter values. The stabilization phenomenon has important consequences for the stability of the THC. It is shown that the inclusion of stochastic processes in the model results in random hysteresis responses to steady changes in freshwater forcing, such that the transitions between regimes generally occur some distance away from the bifurcation points at which transitions occur in the deterministic model

Relativistic few-body quantum mechanics

This thesis develops relativistic quantum mechanical models with a finite number of degrees of freedom and the scattering theories associated with these models. Starting from a consideration of the Poincare Group and its irreducible unitary representations, we develop such representations on Hilbert Spaces of physical states of one, two, and three particles. In the two- and three- particle cases, we consider systems in which the particles are non-interacting and in which the particles experience mutual interactions. We are also careful to ensure that for the three-body system, the formalism predicts that subsystems separated by infinite spatial distances behave independently. We next develop the Faddeev equations, which simplify the solution of multi-channel scattering equations. These are specialised to the three-body system introduced earlier and a series solution of the Faddeev Equations is obtained. A simple mechanical model is introduced to provide a heuristic understanding of this solution. The series solution is also expressed in a diagrammatic form complementary to this mechanical model. A system in which particle production and annihilation are allowed is then introduced by working on an Hilbert Space which is the direct sum of the two- and three-body Hilbert Spaces considered earlier. It is found that in this 2-3 system, as the mass operator and the number operators do not commute, it is not possible for a system to simultaneously have a sharply defined mass and number of particles. The Faddeev Equations for this system are then considered, and a series solution of these equations is developed and discussed. It is also shown that the particle production and annihilation potential has a non-trivial effect on pure two-body and three-body scattering. In the last chapter we consider an attempt to derive from a more elementary field theory, using the dressing transformation, a form for the potential coupling the two- and three-body sectors of the Hilbert Space in the 2-3 system. It is found that this method is inherently ambiguous and is not, therefore, able to provide such information.Science, Faculty ofPhysics and Astronomy, Department ofGraduat