http://www.dspace.cam.ac.uk:80/handle/1810/206446 Fri, 24 May 2013 19:45:47 GMT 2013-05-24T19:45:47Z http://www.dspace.cam.ac.uk:80/handle/1810/244569 Title: Dynamics of quasi-two-dimensional turbulent jets Authors: Landel, Julien Rémy Dominique Gérard Abstract: The study of quasi-two-dimensional turbulent jets is relevant to chemical reactors, the coking process in oil refinement, as well as rivers flowing into lakes or oceans. In the event of a spillage of pollutants into a river, it is critical to understand how these agents disperse with the flow in order to assess damage to the environment. For such flows, characteristic streamwise and cross-stream dimensions can be much larger than the fluid-layer thickness, and so the flow develops in a confined environment. When the distance away from the discharge location is larger than ten times the fluid-layer thickness, the flow is referred to as a quasi-two-dimensional jet. From experimental observations using dyed jets and particle image velocimetry, we find that the structure of a quasi-two-dimensional jet consists of a high-speed meandering core with large counter-rotating eddies developing on alternate sides of the core. The core and eddy structure is self-similar with distance from the discharge location. The Gaussianity of the cross-stream distribution of the time-averaged velocity is due, in part, to the sinuous instability of the core. To understand the transport and dispersion properties of quasi-two-dimensional jets we use a time-dependent advection--diffusion equation, with a mixing length hypothesis accounting for the turbulent eddy diffusivity. The model is supported by experimental releases of dye in jets or numerical releases of virtual passive tracers in experimentally-measured jet velocity fields. We consider the statistical properties of this flow by releasing and then tracking large clusters of virtual particles in the jet velocity field. The probability distributions of two-point properties (such as the distance between two particles) reveal large streamwise dispersion. Owing to this streamwise dispersive effect, a significant amount of tracers can be transported faster than the speed predicted by a simple advection model. Using potential theory, we determine the flow induced by a quasi-two-dimensional jet confined in a rectangular domain. The streamlines of the induced flow predicted by the theory agree with experimental measurements away from the jet boundary. Finally, we investigate the case of a quasi-two-dimensional particle-laden jet. Depending on the bulk concentration of dense particles, we identify different flow regimes. At low concentrations, the jet features the same core and eddy structure observed without the particles, and thus quasi-two-dimensional jet theory can apply to some extent. At larger concentrations, we observe an oscillating instability of the particle-laden jet. Tue, 13 Nov 2012 00:00:00 GMT http://www.dspace.cam.ac.uk:80/handle/1810/244569 2012-11-13T00:00:00Z http://www.dspace.cam.ac.uk:80/handle/1810/244232 Title: Freeze fracturing of elastic porous media Authors: Vlahou, Ioanna Abstract: The physical motivation behind this thesis is the phenomenon of fracturing of rocks and other porous media due to ice growth inside pre-existing faults and large pores. My aim is to explain the basic physical processes taking place inside a freezing elastic porous medium and develop a mathematical model to describe the growth of ice and fracturing of ice-filled cavities. There are two physical processes that can potentially cause high pressures inside a cavity of a porous medium. The expansion of the water by 9% as it freezes causes flow away from the freezing front and through the porous medium, resulting in a water pressure rise inside the cavity. Flow of water towards freezing cavities can occur during the later stages of freezing, when cavities are almost ice-filled, with a thin premelted film separating the ice from the medium. The pressure rise in this case is due to the flux of water into the cavities, which then freezes and increases the overall ice mass. The special geometry of a spherical cavity is initially considered, as a means of comparing how the different processes can contribute to pressure rise inside a cavity. Having established that the expansion of water only contributes to the overall pressure rise in limited situations, I focus attention on the premelting regime and develop a model for the fracturing of a 3D penny-shaped cavity in a porous medium. Integral equations for the pressure and temperature fields are found using Green’s functions, and a boundary element method is used to solve the problem numerically. A similarity solution for a warming environment is discussed, as well as a fully time-dependent problem. I find that the fracture toughness of the medium, the size of pre-existing faults and the undercooling of the environment are the parameters determining the susceptibility of a medium to fracturing. I also explore the dependence of the growth rates on the permeability and elasticity of the medium. Thin and fast-fracturing cracks are found for many types of rocks. I consider how the growth rate can be limited by the existence of pore ice, which decreases the permeability of a medium, and propose an expression for the effective “frozen” permeability. An important further application of the theory developed here is the growth of ice lenses in saturated cohesive soils. I present results for typical soil parameters and find good agreement between our theory and experimental observations of growth rates and minimum undercoolings required for fracturing. Mon, 11 Jun 2012 23:00:00 GMT http://www.dspace.cam.ac.uk:80/handle/1810/244232 2012-06-11T23:00:00Z http://www.dspace.cam.ac.uk:80/handle/1810/244203 Title: On some nonlinear partial differential equations for classical and quantum many body systems Authors: Marahrens, Daniel Abstract: This thesis deals with problems arising in the study of nonlinear partial differential equations arising from many-body problems. It is divided into two parts: The first part concerns the derivation of a nonlinear diffusion equation from a microscopic stochastic process. We give a new method to show that in the hydrodynamic limit, the particle densities of a one-dimensional zero range process on a periodic lattice converge to the solution of a nonlinear diffusion equation. This method allows for the first time an explicit uniform-in-time bound on the rate of convergence in the hydrodynamic limit. We also discuss how to extend this method to the multi-dimensional case. Furthermore we present an argument, which seems to be new in the context of hydrodynamic limits, how to deduce the convergence of the microscopic entropy and Fisher information towards the corresponding macroscopic quantities from the validity of the hydrodynamic limit and the initial convergence of the entropy. The second part deals with problems arising in the analysis of nonlinear Schrödinger equations of Gross-Pitaevskii type. First, we consider the Cauchy problem for (energy-subcritical) nonlinear Schrödinger equations with sub-quadratic external potentials and an additional angular momentum rotation term. This equation is a well-known model for superfluid quantum gases in rotating traps. We prove global existence (in the energy space) for defocusing nonlinearities without any restriction on the rotation frequency, generalizing earlier results given in the literature. Moreover, we find that the rotation term has a considerable influence in proving finite time blow-up in the focusing case. Finally, a mathematical framework for optimal bilinear control of nonlinear Schrödinger equations arising in the description of Bose-Einstein condensates is presented. The obtained results generalize earlier efforts found in the literature in several aspects. In particular, the cost induced by the physical work load over the control process is taken into account rather then often used L^2- or H^1-norms for the cost of the control action. We prove well-posedness of the problem and existence of an optimal control. In addition, the first order optimality system is rigorously derived. Also a numerical solution method is proposed, which is based on a Newton type iteration, and used to solve several coherent quantum control problems. Tue, 13 Nov 2012 00:00:00 GMT http://www.dspace.cam.ac.uk:80/handle/1810/244203 2012-11-13T00:00:00Z http://www.dspace.cam.ac.uk:80/handle/1810/243913 Title: Stability of fluid-loaded structures Authors: Arzoumanian, Sevag Hrair Abstract: It is known theoretically that infinitely long fluid loaded plates in mean flow exhibit a range of unusual phenomena in the 'long time' limit. These include convective instability, absolute instability and negative energy waves which are destabilized by dissipation. However, structures are necessarily of finite length and may have discontinuities. We have undertaken an analytical and computational study to investigate the response of finite plates, and of plates with local inhomogeneities, to ascertain if these unusual effects might be realized in practice. Analytically, we adopt Crighton & Oswell’s (1991) structural acoustics approach and take a "wave scattering" --as opposed to a "modal superposition"-- view of the fluttering plate problem. First, we derive the energy balance relations for the extended plate (i.e., plate with two sided flow, spring foundation and plate pretension) and define a generalized wave impedance valid for both positive energy waves (PEW) and negative energy waves (NEW). Next, we solve for the scattering coefficients of localized plate discontinuities using a multipole source approach. Our solutions are exact and include the nearfields due to fluid-loading effects. We introduce the concept of power normalized scattering coefficients, and show that overall power is conserved during the scattering process if the sign of the wave energy is preserved. We argue that energy conservation, combined with the presence of NEWs on the plate, are responsible for the phenomenon of over-scattering, or of amplified reflection/transmission. These are scattering processes that draw energy from the mean flow into the plate. Next, we use the Wiener-Hopf technique to solve for the scattering coefficients of a variety of plate leading and trailing edge conditions -- including the flag like configuration of a free trailing edge with wake. We find that the edges are over-reflective in the frequency range where NEWs are present. The exception is a free trailing edge with wake where, remarkably, the wake is found to absorb almost all of the incident wave energy. We use combinations of these upstream and downstream edge reflection matrices to solve for the complex resonance frequencies of long, finite plates immersed in mean flow. Finally, we construct the response of a finite plate by a superposition of infinite plate propagating waves continuously scattering off the plate ends. We solve for the unstable resonance frequencies and temporal growth rates for long plates. We derive upper and lower bounds on the unstable growth rates of finite plates with given edge conditions. We find that a flag-like configuration of a clamped leading edge and a free trailing edge with wake is destabilized for sub-critical flow speeds only for very long plate lengths and only in the presence of convectively unstable waves. We present a comparison between direct computational results and the infinite plate theory. In particular, the resonance response of a moderately sized plate is shown to be in excellent agreement with the long plate analytical predictions. Mon, 06 Jun 2011 23:00:00 GMT http://www.dspace.cam.ac.uk:80/handle/1810/243913 2011-06-06T23:00:00Z http://www.dspace.cam.ac.uk:80/handle/1810/243623 Title: Differential systems, moving frames, structure-preserving submersions and geometrical problems in physics Authors: Hu, Ziyang Abstract: The present work applies the theories of exterior differential systems, method of equivalence and moving frames to the study of geometrical problems arising in physics, especially the class of problems that can be described as “structure-preserving submer- sions”. A novel feature of our approach is the formulation of an algorithm which we have named “the method of involutive seeds”. By using this method, we can rapidly determine the number of free functions that we must specify in order to completely specify the problem, which we will call the “degree of arbitrariness” of the problem, and which for many physical systems is linked to the physical degree of freedom. This algorithm is especially helpful in dealing with systems with many constraints such as structure-preserving submersions. We also give other examples of calculations using this algorithm: in particular, we used it to investigate the degree of arbitrariness of the theory of general very special relativity, based on Riemannian geometry with a holonomy constraint, and thus argue that such a theory is not a suitable candidate for a physical theory. As for structure-preserving submersions, which we propose as a generalisation for Riemannian submersions to other geometrical structures, after investigating the prop- erties and degrees of arbitrariness of the general construction we use it to study the problem of flows, especially rigid flows in relativity. We generalise the classical Her- glotz–Noether theorem, which states that rotational rigid flow in Minkowski spacetime must be isometric, to all dimensions and to all conformally flat spacetimes in all di- mensions, and also to shear-free flows in conformally flat spacetimes; we generalize a partial result of the Ellis conjecture that a self-gravitating shear-free perfect fluid in geodesic motion must be either expansion-free or vorticity-free to all dimensions, and we will see clearly the origin of this result from the group structure of spacetime; we discuss an approach for the general Ellis conjecture, and show the relation between the Herglotz–Noether theorem and the Ellis conjecture; we show that for a free point particle lagrangian to have a Galilean boost symmetry, it is necessary and sufficient that we have a totally flat direction decoupled from the rest; finally, we give a rough, heuristic reasoning for why some of the Pauli reductions in which we attempt to get a larger gauge group than usually allowed from dimensional reduction are consistent. Mon, 02 Jul 2012 23:00:00 GMT http://www.dspace.cam.ac.uk:80/handle/1810/243623 2012-07-02T23:00:00Z http://www.dspace.cam.ac.uk:80/handle/1810/243244 Title: Turbulence ingestion noise of open rotors Authors: Robison, Rosalyn Aruna Venner Abstract: Renewed interest in open rotor aeroengines, due to their fuel efficiency, has driven renewed interest in all aspects of the noise they generate. Noise due to the ingestion of distorted atmospheric turbulence, known as Unsteady Distortion Noise (UDN), is likely to be higher for open rotors than for conventional turbofan engines since the rotors are fully exposed to oncoming turbulence and lack ducting to attenuate the radiated sound. However, UDN has received less attention to date, particularly in wind-tunnel and flight testing programmes. In this thesis a new prediction scheme for UDN is described, which allows inclusion of many key features of real open rotors which have not previously been investigated theoretically. Detailed features of the mean flow induced by the rotor, the form of atmospheric turbulence, asymmetries due to installation features, and the effect of rotor incidence are all considered. Parameter studies are conducted in each of these cases to investigate their effect upon UDN in typical static testing and flight conditions. A thorough review of the technological issues of most relevance and previous theoretical work on all types of turbulence-blade interaction noise is first undertaken. The prediction scheme is then developed for the case in which the mean flow into the rotor is axisymmetric. This shows excellent qualitative agreement with previous findings, with increased streamtube contraction resulting in a more tonal noise spectrum. The theoretical framework involves using Rapid Distortion Theory to calculate the distortion of an isotropic turbulence field (such as given by the von Karman spectrum) by the mean flow induced by the rotor (such as given by actuator disk theory), leading to an expression for the velocity incident upon the leading edge of the rotor blades. Strip theory is then used to calculate the pressure jumps across the blades, input as the forcing term in the far-field wave equation. Models are derived for open rotor-induced flow which account for the variation of blade circulation with radius, and the presence of the rotor hub and rear blade row. An investigation of appropriate turbulence models and realistic turbulence parameters is also undertaken. A key finding is that the heights of the tonal peaks are determined by the overall magnitude of the induced streamtube contraction (dependent on the total thrust generated) whereas the precise form of distortion (affected by the detailed components of the mean flow and the form of atmospheric turbulence present) alters the resulting broadband level. The prediction scheme is formulated in such a way as to facilitate extension to the asymmetric case, which is also fully derived. The model is applied in the first instance to the case of two adjacent rotors and then to the case of a single rotor at incidence. Under flight conditions, when distortion is reduced but UDN can still contribute a significant broadband component to overall noise levels, asymmetry is found to increase broadband levels around 1 Blade Passing Frequency but reduce levels elsewhere. Mon, 09 Apr 2012 23:00:00 GMT http://www.dspace.cam.ac.uk:80/handle/1810/243244 2012-04-09T23:00:00Z http://www.dspace.cam.ac.uk:80/handle/1810/241497 Title: Thermodynamic and hydrodynamic behaviour of interacting Fermi gases Authors: Goulko, Olga Abstract: Fermionic matter is ubiquitous in nature, from the electrons in metals and semiconductors or the neutrons in the inner crust of neutron stars, to gases of fermionic atoms, like 40K or 6Li that can be created and studied under laboratory conditions. It is especially interesting to study these systems at very low temperatures, where we enter the world of quantum mechanical phenomena. Due to the Fermi-Dirac statistics, a dilute system of spin-polarised fermions exhibits no interactions and can be viewed as an ideal Fermi gas. However, interactions play a crucial role for fermions of several spin species. This thesis addresses several questions concerning interacting Fermi gases, in particular the transition between the normal and the superfluid phase and dynamical properties at higher temperatures. First we will look at the unitary Fermi gas: a two-component system of fermions interacting with divergent scattering length. This system is particularly interesting as it exhibits universal behaviour. Due to the strong interactions perturbation theory is inapplicable and no exact theoretical description is available. I will describe the Determinant Diagrammatic Monte Carlo algorithm with which the unitary Fermi gas can be studied from first principles. This algorithm fails in the presence of a spin imbalance (unequal number of particles in the two components) due to a sign problem. I will show how to apply reweighting techniques to generalise the algorithm to the imbalanced case, and present results for the critical temperature and other thermodynamic observables at the critical point, namely the chemical potential, the energy per particle and the contact density. These are the first numerical results for the imbalanced unitary Fermi gas at finite temperature. I will also show how temperatures beyond the critical point can be accessed and present results for the equation of state and the temperature dependence of the contact density. At sufficiently high temperatures a semiclassical description captures all relevant physical features of the system. The dynamics of an interacting Fermi gas can then be studied via a numerical simulation of the Boltzmann equation. I will describe such a numerical setup and apply it to study the collision of two spin-polarised fermionic clouds. When the two components are separated in an elongated harmonic trap and then released, they collide and for sufficiently strong interactions can bounce off each other several times. I will discuss the different types of the qualitative behaviour, show how they can be interpreted in terms of the equilibrium properties of the system, and explain how they relate to the coupling between different excitation modes. I will also demonstrate how transport coefficients, for instance the spin drag, can be extracted from the numerical data. Tue, 10 Jan 2012 00:00:00 GMT http://www.dspace.cam.ac.uk:80/handle/1810/241497 2012-01-10T00:00:00Z http://www.dspace.cam.ac.uk:80/handle/1810/241495 Title: On some partial differential equation models in socio-economic contexts - analysis and numerical simulations Authors: Pietschmann, Jan-Frederik Abstract: This thesis deals with the analysis and numerical simulation of different partial differential equation models arising in socioeconomic sciences. It is divided into two parts: The first part deals with a mean-field price formation model introduced by Lasry and Lions in 2007. This model describes the dynamic behaviour of the price of a good being traded between a group of buyers and a group of vendors. Existence (locally in time) of smooth solutions is established, and obstructions to proving a global existence result are examined. Also, properties of a regularised version of the model are explored and numerical examples are shown. Furthermore, the possibility of reconstructing the initial datum given a number of observations, regarding the price and the transaction rate, is considered. Using a variational approach, the problem can be expressed as a non-linear constrained minimization problem. We show that the initial datum is uniquely determined by the price (identifiability). Furthermore, a numerical scheme is implemented and a variety of examples are presented. The second part of this thesis treats two different models describing the motion of (large) human crowds. For the first model, introduced by R.L. Hughes in 2002, several regularised versions are considered. Existence and uniqueness of entropy solutions are proven using the technique of vanishing viscosity. In one space dimension, the dynamic behaviour of solutions of the original model is explored for some special cases. These results are compared to numerical simulations. Moreover, we consider a discrete cellular automaton model introduced by A. Kirchner and A. Schadschneider in 2002. By (formally) passing to the continuum limit, we obtain a system of partial differential equations. Some analytical properties, such as linear stability of stationary states, are examined and extensive numerical simulations show capabilities and limitations of the model in both the discrete and continuous setting. Tue, 10 Jan 2012 00:00:00 GMT http://www.dspace.cam.ac.uk:80/handle/1810/241495 2012-01-10T00:00:00Z http://www.dspace.cam.ac.uk:80/handle/1810/241092 Title: Exploring nonlinear regression methods, with application to association studies Authors: Speed, Douglas Christopher Abstract: The field of nonlinear regression is a long way from reaching a consensus. Once a method decides to explore nonlinear combinations of predictors, a number of questions are raised, such as what nonlinear combinations to permit and how best to search the resulting model space. Genetic Association Studies comprise an area that stands to gain greatly from the development of more sophisticated regression methods. While these studies’ ability to interrogate the genome has advanced rapidly over recent years, it is thought that a lack of suitable regression tools prevents them from achieving their full potential. I have tried to investigate the area of regression in a methodical manner. In Chapter 1, I explain the regression problem and outline existing methods. I observe that both linear and nonlinear methods can be categorised according to the restrictions enforced by their underlying model assumptions and speculate that a method with as few restrictions as possible might prove more powerful. In order to design such a method, I begin by assuming each predictor is tertiary (takes no more than three distinct values). In Chapters 2 and 3, I propose the method Sparse Partitioning. Its name derives from the way it searches for high scoring partitions of the predictor set, where each partition defines groups of predictors that jointly contribute towards the response. A sparsity assumption supposes most predictors belong in the “null group” indicating they have no effect on the outcome. In Chapter 4, I compare the performance of Sparse Partitioning to existing methods using simulated and real data. The results highlight how greatly a method’s power depends on the validity of its model assumptions. For this reason, Sparse Partitioning appears to offer a robust alternative to current methods, as its lack of restrictions allows it to maintain power in scenarios where other methods will fail. Sparse Partitioning relies on Markov chain Monte Carlo estimation, which limits the size of problem on which it can be used. Therefore, in Chapter 5, I propose a deterministic version of the method which, although less powerful, is not affected by convergence issues. In Chapter 6, I describe Bayesian Projection Pursuit, which adds spline fitting into the method to cope with non-tertiary predictors. Mon, 11 Jul 2011 23:00:00 GMT http://www.dspace.cam.ac.uk:80/handle/1810/241092 2011-07-11T23:00:00Z http://www.dspace.cam.ac.uk:80/handle/1810/241041 Title: The application of automated perturbation theory to lattice QCD Authors: Monahan, Christopher John Abstract: Predictions of heavy quark parameters are an integral component of precision tests of the Standard Model of particle physics. Experimental measurements of electroweak processes involving heavy hadrons provide stringent tests of Cabibbo-Kobayashi-Maskawa (CKM) matrix unitarity and serve as a probe of new physics. Hadronic matrix elements parameterise the strong dynamics of these interactions and these matrix elements must be calculated nonperturbatively. Lattice quantum chromodynamics (QCD) provides the framework for nonperturbative calculations of QCD processes. Current lattices are too coarse to directly simulate b quarks. Therefore an effective theory, nonrelativistic QCD (NRQCD), is used to discretise the heavy quarks. High precision simulations are required so systematic uncertainties are removed by improving the NRQCD action. Precise simulations also require improved sea quark actions, such as the highly-improved staggered quark (HISQ) action. The renormalisation parameters of these actions cannot be feasibly determined by hand and thus automated procedures have been developed. In this dissertation I apply automated lattice pertubartion theory to a number of heavy quark calculations. I first review the fundamentals of lattice QCD and the construction of lattice NRQCD. I then motivate and discuss lattice perturbation theory in detail, focussing on the tools and techniques that I use in this dissertation. I calculate the two-loop tadpole improvement factors for improved gluons with improved light quarks. I then compute the renormalisation parameters of NRQCD. I use a mix of analytic and numerical methods to extract the one-loop radiative corrections to the higher order kinetic operators in the NRQCD action. I then employ a fully automated procedure to calculate the heavy quark energy shift at two-loops. I use this result to extract a new prediction of the mass of the b quark from lattice NRQCD simulations by the HPQCD collaboration. I also review the calculation of the radiative corrections to the chromo-magnetic operator in the NRQCD action. This computation is the first outcome of our implementation of background field gauge for automated lattice perturbation theory. Finally, I calculate the heavy-light currents for highly-improved NRQCD heavy quarks with massless HISQ light quarks and discuss the application of these results to nonperturbative studies by the HPQCD collaboration. Tue, 08 Nov 2011 00:00:00 GMT http://www.dspace.cam.ac.uk:80/handle/1810/241041 2011-11-08T00:00:00Z http://www.dspace.cam.ac.uk:80/handle/1810/240629 Title: Spiky strings and the AdS/CFT correspondence Authors: Losi, Manuel Abstract: In this dissertation, we explore some aspects of semiclassical type IIB string theory on AdS3 × S1 and on pure AdS3 in the limit of large angular momentum S. We first focus on the integrability technique known as finite-gap formalism for strings in AdS3 × S1, leading to the definition of a hyperelliptic Riemann surface, the spectral curve, which encodes, albeit in a rather implicit fashion, the semiclassical spectrum of a very large family of string solutions. Then, we show that, in the large angular momentum limit, the spectral curve separates into two distinct surfaces, allowing the derivation of an explicit expression for the spectrum, which is correspondingly characterised by two separate branches. The latter may be interpreted in terms of two kinds of spikes appearing on the strings: “large” spikes, yielding an infinite contribution to the energy and angular momentum of the string, and “small” spikes, representing finite excitations over the background of the “large” spikes. According to the AdS/CFT correspondence, strings moving in AdS3 × S1 should be dual to single trace operators in the sl(2) sector of N = 4 super Yang- Mills theory. The corresponding one-loop spectrum in perturbation theory may also be computed through integrability methods and, in the large conformal spin limit S → ∞ (equivalent to the AdS3 angular momentum in string theory) is also expressed in terms of a spectral curve and characterised in terms of the so-called holes. We show that, with the appropriate identifications and with the usual extrapolation from weak to strong ’t Hooft coupling described by the cusp anomalous dimension, the large-S spectra of gauge theory and of string theory coincide. Furthermore, we explain how “small” and “large” holes may be identified with “small” and “large” spikes. Finally, we discuss several explicit spiky string solutions in AdS3 which, at the leading semiclassical order, display the previously studied finite-gap spectrum. We compute the spectral curves of these strings in the large S limit, finding that they correspond to specific regions of the moduli space of the finite-gap curves. We also explain how “large” spikes may be used in order to extract a discrete system of degrees of freedom from string theory, which can then be matched with the degrees of freedom of the dual gauge theory operators, and how “small” spikes are in fact very similar to the Giant Magnons living in R × S2. Mon, 10 Oct 2011 23:00:00 GMT http://www.dspace.cam.ac.uk:80/handle/1810/240629 2011-10-10T23:00:00Z http://www.dspace.cam.ac.uk:80/handle/1810/240581 Title: Tidal interactions between planets and stars Authors: Barker, Adrian John Abstract: Since the first discovery of an extrasolar planet around a solar-type star, observers have detected over 500 planets outside the solar system. Many of these planets have Jovian masses and orbit their host stars in orbits of only a few days, the so-called “Hot Jupiters”. At such close proximity to their parent stars, strong tidal interactions between the two bodies are expected to cause significant secular spin-orbit evolution. This thesis tackles two problems regarding the tidal evolution of short-period extrasolar planets. In the first part, we adopt a simple model of the orbit-averaged effects of tidal friction, to study the tidal evolution of planets on inclined orbits. We also analyse the effects of stellar magnetic braking. We then discuss the implications of our results for the importance of Rossiter-Mclaughlin effect observations. In the second part, we study the mechanisms of tidal dissipation in solar-type stars. In particular, internal gravity waves are launched at the interface of the convection and radiation zones of such a star, by the tidal forcing of a short-period planet. The fate of these waves as they approach the centre of the star is studied, primarily using numerical simulations, in both two and three dimensions. We find that the waves undergo instability and break above a critical amplitude. A model for the tidal dissipation that results from this process is presented, and its validity is verified by numerical integrations of the linear tidal response, in an extensive set of stellar models. The dissipation is efficient, and varies by less than an order of magnitude between all solar- type stars, throughout their main-sequence lifetimes, for a given planetary orbit. The implications of this mechanism for the survival of short-period extrasolar planets is discussed, and we propose a possible explanation for the survival of all of the extrasolar planets currently observed in short-period orbits around F, G and K stars. We then perform a stability analysis of a standing internal gravity wave near the centre of a solar-type star, to understand the early stages of the wave breaking process in more detail, and to determine whether the waves are subject to weaker parametric instabilities, below the critical amplitude required for wave breaking. We discuss the relevance of our results to our explanation for the survival of short-period planets presented in the second part of this thesis. Finally, we propose an alternative mechanism of tidal dissipation, involving the gradual radiative damping of the waves. Based on a simple estimate, it appears that this occurs even for low mass planets. However, it is in conflict with current observations since it would threaten the survival of all planets in orbits shorter than 2 days. We discuss some hydrodynamic instabilities and magnetic stresses which may prevent this process. Mon, 06 Jun 2011 23:00:00 GMT http://www.dspace.cam.ac.uk:80/handle/1810/240581 2011-06-06T23:00:00Z http://www.dspace.cam.ac.uk:80/handle/1810/240580 Title: New approaches to higher-dimensional general relativity Authors: Durkee, Mark N. Abstract: This thesis considers various aspects of general relativity in more than four spacetime dimensions. Firstly, I review the generalization to higher dimensions of the algebraic classification of the Weyl tensor and the Newman-Penrose formalism. In four dimensions, these techniques have proved useful for studying many aspects of general relativity, and it is hoped that their higher dimensional generalizations will prove equally useful in the future. Unfortunately, many calculations using the Newman-Penrose formalism can be unnecessarily complicated. To address this, I describe new work introducing a higher-dimensional generalization of the so-called Geroch-Held-Penrose formalism, which allows for a partially covariant reformulation of general relativity. This approach provides great simplifications for many calculations involving spacetimes which admit one or two preferred null directions. The next chapter describes the proof of an important result regarding algebraic classification in higher dimensions. The classification is based upon the existence of a particular null direction that is aligned with the Weyl tensor of the geometry in some appropriate sense. In four dimensions, it is known that a null vector field is such a multiple Weyl aligned null direction (WAND) if and only if it is tangent to a shearfree null geodesic congruence. This is not the case in higher dimensions. However, I have formulated and proved a partial generalization of the result to arbitrary dimension, namely that a spacetime admits a multiple WAND if and only if it admits a geodesic multiple WAND. Moving onto more physical applications, I describe how the formalism that we have developed can be applied to study certain aspects of the stability of extremal black holes in arbitrary dimension. The final chapter of the thesis has a rather different flavour. I give a detailed analysis of the properties of a particular solution to the Einstein equations in five dimensions: the Pomeransky-Sen'kov doubly spinning black ring. I study geodesic motion around this black ring and demonstrate the separability of the Hamilton-Jacobi equation for null, zero energy geodesics. I show that this unexpected separability can be understood in terms of a symmetry described by a conformal Killing tensor on a four dimensional spacetime obtained by a Kaluza-Klein reduction of the original black ring spacetime. Mon, 06 Jun 2011 23:00:00 GMT http://www.dspace.cam.ac.uk:80/handle/1810/240580 2011-06-06T23:00:00Z http://www.dspace.cam.ac.uk:80/handle/1810/240578 Title: Geometric aspects of gauge and spacetime symmetries Authors: Gielen, Steffen C. M. Abstract: We investigate several problems in relativity and particle physics where symmetries play a central role; in all cases geometric properties of Lie groups and their quotients are related to physical effects. The first part is concerned with symmetries in gravity. We apply the theory of Lie group deformations to isometry groups of exact solutions in general relativity, relating the algebraic properties of these groups to physical properties of the spacetimes. We then make group deformation local, generalising deformed special relativity (DSR) by describing gravity as a gauge theory of the de Sitter group. We find that in our construction Minkowski space has a connection with torsion; physical effects of torsion seem to rule out the proposed framework as a viable theory. A third chapter discusses a formulation of gravity as a topological BF theory with added linear constraints that reduce the symmetries of the topological theory to those of general relativity. We discretise our constructions and compare to a similar construction by Plebanski which uses quadratic constraints. In the second part we study CP violation in the electroweak sector of the standard model and certain extensions of it. We quantify fine-tuning in the observed magnitude of CP violation by determining a natural measure on the space of CKM matrices, a double quotient of SU(3), introducing different possible choices and comparing their predictions for CP violation. While one generically faces a fine-tuning problem, in the standard model the problem is removed by a measure that incorporates the observed quark masses, which suggests a close relation between a mass hierarchy and suppression of CP violation. Going beyond the standard model by adding a left-right symmetry spoils the result, leaving us to conclude that such additional symmetries appear less natural. Mon, 06 Jun 2011 23:00:00 GMT http://www.dspace.cam.ac.uk:80/handle/1810/240578 2011-06-06T23:00:00Z http://www.dspace.cam.ac.uk:80/handle/1810/238679 Title: Scattering of internal gravity waves Authors: Leaman Nye, Abigail Abstract: Internal gravity waves play a fundamental role in the dynamics of stably stratified regions of the atmosphere and ocean. In addition to the radiation of momentum and energy remote from generation sites, internal waves drive vertical transport of heat and mass through the ocean by wave breaking and the mixing subsequently produced. Identifying regions where internal gravity waves contribute to ocean mixing and quantifying this mixing are therefore important for accurate climate and weather predictions. Field studies report significantly enhanced measurements of turbulence near ‘rough’ ocean topography compared with those recorded in the ocean interior or near more gradually varying topography (e.g. Toole et al. 1997, J. Geophys. Res. 102). Such observations suggest that interaction of waves with rough topography may act to skew wave energy spectra to high wavenumbers and hence promote wave breaking and fluid mixing. This thesis examines the high wavenumber scatter and spatial partitioning of wave energy at ‘rough’ topography containing features that are of similar scales to those characterising incident waves. The research presented here includes laboratory experiments using synthetic schlieren and PIV to visualise two-dimensional wavefields produced by small amplitude oscillations of cylinders within linear salt-water stratifications. Interactions of wavefields with planar slopes and smoothly varying sinusoidal topography are compared with those with square-wave, sawtooth and pseudo knife-edge profiles, which have discontinuous slopes. Far-field structures of scattered wavefields are compared with linear analytical models. Scatter to high wavenumbers is found to be controlled predominantly by the relative slopes and characterising length scales of the incident wavefield and topography, as well as the shape and aspect ratio of the topographic profile. Wave energy becomes highly focused and the spectra skewed to higher wavenumbers by ‘critical’ regions, where the topographic slope is comparable with the slope of the incident wave energy vector, and at sharp corners, where topographic slope is not defined. Contrary to linear geometric ray tracing predictions (Longuet-Higgins 1969, J. Fluid Mech. 37), a significant back-scattered field can be achieved in near-critical conditions as well as a forward scattered wavefield in supercritical conditions, where the slope of the boundary is steeper than that of the incident wave. Results suggest that interaction with rough benthic topography could efficiently convert wave energy to higher wavenumbers and promote fluid mixing in such ocean regions. Mon, 18 Apr 2011 23:00:00 GMT http://www.dspace.cam.ac.uk:80/handle/1810/238679 2011-04-18T23:00:00Z http://www.dspace.cam.ac.uk:80/handle/1810/237379 Title: The effects of stochastic forces on the evolution of planetary systems and Saturn's rings Authors: Rein, Hanno Abstract: The increasing number of discovered extra-solar planets opens a new opportunity for studies of the formation of planetary systems. Their diversity keeps challenging the long-standing theories which were based on data primarily from our own solar system. Resonant planetary systems are of particular interest because their dynamical configuration provides constraints on the otherwise unobservable formation and migration phase. In this thesis, formation scenarios for the planetary systems HD128311 and HD45364 are presented. N-body simulations of two planets and two dimensional hydrodynamical simulations of proto-planetary discs are used to realistically model the convergent migration phase and the capture into resonance. The results indicate that the proto-planetary disc initially has a larger surface density than previously thought. Proto-planets are exposed to stochastic forces, generated by density fluctuations in a turbulent disc. A generic model of both a single planet, and two planets in mean motion resonance, being stochastically forced is presented and applied to the system GJ876. It turns out that GJ876 is stable for reasonable strengths of the stochastic forces, but systems with lighter planets can get disrupted. Even if a resonance is not completely disrupted, stochastic forces create characteristic, observable libration patterns. As a further application, the stochastic migration of small bodies in Saturn’s rings is studied. Analytic predictions of collisional and gravitational interactions of a moonlet with ring particles are compared to realistic three dimensional collisional N-body simulations with up to a million particles. Estimates of both the migration rate and the eccentricity evolution of embedded moonlets are confirmed. The random walk of the moonlet is fast enough to be directly observable by the Cassini spacecraft. Turbulence in the proto-stellar disc also plays an important role during the early phases of the planet formation process. In the core accretion model, small, metre-sized particles are getting concentrated in pressure maxima and will eventually undergo a rapid gravitational collapse to form a gravitationally bound planetesimal. Due to the large separation of scales, this process is very hard to model numerically. A scaled method is presented, that allows for the correct treatment of self-gravity for a marginally collisional system by taking into account the relevant small scale processes. Interestingly, this system is dynamically very similar to Saturn’s rings. Mon, 11 Oct 2010 23:00:00 GMT http://www.dspace.cam.ac.uk:80/handle/1810/237379 2010-10-11T23:00:00Z http://www.dspace.cam.ac.uk:80/handle/1810/237241 Title: Instabilities and transport in magnetized plasmas Authors: Rosin, Mark Abstract: In a magnetized plasma, naturally occurring pressure anisotropies facilitate in- stabilities that are expected to modify the transport properties of the system. In this thesis we examine two such instabilities and, where appropriate, their effects on transport. First we consider the collisional (fluid) magnetized magnetorotational instability (MRI) in the presence of the Braginskii viscosity. We conduct a global linear analysis of the instability in a galactic rotation profile for three magnetic field configurations: purely azimuthal, purely vertical and slightly pitched. Our analysis, numerical and asymptotic, shows that the first two represent singular configurations where the Braginskii viscosity’s primary role is dissipative and the maximum growth rate is proportional to the Reynolds number when this is small. For a weak pitched field, the Braginskii viscosity is destabilising and when its effects dominate over the Lorentz force, the growth rate of the MRI can be up to 2√2 times faster than the inviscid limit. If the field is strong, an over-stability develops and both the real and imaginary parts of the frequency increase with the coefficient of the viscosity. Second, in the context of the ICM of galaxy clusters, we consider the pressure-anisotropy-driven firehose instability. The linear instability is fast (∼ ion cyclotron period) and small-scale (ion Larmor radius ρi) and so fluid theory is inapplicable. We determine its nonlinear evolution in an ab initio kinetic calculation (for parallel gradients only). We use a particular physical asymptotic ordering to derive a closed nonlinear equation for the firehose turbulence, which we solve. We find secular (∝ t) growth of magnetic fluctuations and a k−∥3 spectrum, starting at scales >~ ρi. When a parallel ion heat flux is present, the parallel firehose instability mutates into the new gyrothermal instability. Its nonlinear evolution also involves secular magnetic energy growth, but its spectrum is eventually dominated by modes with a maximal scale ∼ρilT/λmfp,(lT is the parallel temperature gradient scale). Throughout we discuss implications for modelling, transport and other areas of magnetized plasma physics. Tue, 08 Feb 2011 00:00:00 GMT http://www.dspace.cam.ac.uk:80/handle/1810/237241 2011-02-08T00:00:00Z http://www.dspace.cam.ac.uk:80/handle/1810/230114 Title: The solar tachocline: a self-consistent model of magnetic confinement Authors: Wood, Toby Abstract: In this dissertation we consider the dynamics of the solar interior, with particular focus on angular momentum balance and magnetic field confinement within the tachocline. In Part I we review current knowledge of the Sun's rotation. We summarise the main mechanisms by which angular momentum is transported within the Sun, and discuss the difficulties in reconciling the observed uniform rotation of the radiative interior with purely hydrodynamical theories. Following Gough & McIntyre (1998) we conclude that a global-scale interior magnetic field provides the most plausible explanation for the observed uniform rotation, provided that it is confined within the tachocline. We discuss potential mechanisms for magnetic field confinement, assuming that the field has a roughly axial-dipolar structure. In particular, we argue that the field is confined, in high latitudes, by a laminar downwelling flow driven by turbulence in the tachocline and convection zone above. In Part II we describe how the magnetic confinement picture is affected by the presence of compositional stratification in the "helium settling layer" below the convection zone. We use scaling arguments to estimate the rate at which the settling layer forms, and verify our predictions with a simple numerical model. We discuss the implications for lithium depletion in the convection zone. In Part III we present numerical results showing how the Sun's interior magnetic field can be confined, in the polar regions, while maintaining uniform rotation within the radiative envelope. These results come from solving the full, nonlinear equations numerically. We also show how these results can be understood in terms of a reduced, analytical model that is asymptotically valid in the parameter regime of relevance to the solar tachocline. In Part IV we discuss how our high-latitude model can be extended to a global model of magnetic confinement within the tachocline. Tue, 11 Jan 2011 00:00:00 GMT http://www.dspace.cam.ac.uk:80/handle/1810/230114 2011-01-11T00:00:00Z http://www.dspace.cam.ac.uk:80/handle/1810/228707 Title: Spreading of viscous fluids and granular materials on slopes Authors: Takagi, Daisuke Abstract: Materials can flow down a slope in a wide range of geophysical and industrial contexts, including lava flows on volcanoes and thin films on coated surfaces. The aim of my research is to provide quantitative insight into these forms of motion and their dependence on effects of the topography, the volume and the rheology of the flowing structure. Numerous different problems are investigated through mathematical models, which are developed analytically and confirmed by laboratory experiments. The initial advance of long lava flows is studied by considering the flow of viscous fluid released on sloping channels. A scaling analysis, in agreement with analog experiments and field data, offers a practical tool for predicting the advance of lava flows and conducting hazard analysis. A simple and powerful theory predicts the structure of flows resulting from any time-dependent release of fluid down a slope. Results obtained by the method of characteristics reveal how the speed of the advancing front depends importantly on the rate of fluid supplied at an earlier time. Viscous flows on surfaces with different shapes are described by similarity solutions to address problems motivated by engineering as well as geophysical applications. Pouring viscous fluid out of a container can be a frustratingly slow process depending on the shape and the degree of tipping of the container. The discharge rate of the fluid is analysed in simple cases, shedding light on how containers can be emptied most quickly in cosmetic and food industries. In a separate study motivated by coating industries, thin films are shown to evolve with uniform thickness as they drain near the top of a horizontal cylinder or sphere. The leading edge eventually splits into rivulets as predicted theoretically and confirmed by experiments. Debris flows can develop levees and trigger avalanches which are studied by considering dense granular flows down a rough inclined plane. Granular materials released down a slope can produce a flowing structure confined by levees or trigger avalanches at regular intervals, depending on the steady rate of supply. The experimental results are discussed using theoretical ideas of shallow granular flows. Finally, materials flowing in long and slender ducts are investigated theoretically to better understand the digestive and urinary systems in biology. The materials are pumped in an elastic tube by translating waves of muscular contraction and relaxation. The deformation of the tube is predicted by solving a free-boundary problem, a similar mathematical exercise to predicting the moving boundaries of materials spreading on slopes. Tue, 16 Nov 2010 00:00:00 GMT http://www.dspace.cam.ac.uk:80/handle/1810/228707 2010-11-16T00:00:00Z http://www.dspace.cam.ac.uk:80/handle/1810/227571 Title: Classical and thermodynamic stability of black holes Authors: Monteiro, Ricardo Abstract: We consider the stability of black holes within both classical general relativity and the semiclassical thermodynamic description. In particular, we study linearised perturbations and their contribution to the gravitational partition function, addressing technical issues for charged (Reissner-Nordstrom) and rotating (Kerr-AdS) black holes. Exploring the connection between classical and thermodynamic stability, we find classical instabilities of Myers-Perry black holes and bifurcations to new black hole families. Mon, 05 Jul 2010 23:00:00 GMT http://www.dspace.cam.ac.uk:80/handle/1810/227571 2010-07-05T23:00:00Z