Probabilistic foundation of nonlocal diffusion and formulation and analysis for elliptic problems on uncertain domains

2011 Summer.Includes bibliographical references.In the first part of this dissertation, we study the nonlocal diffusion equation with so-called Lévy measure ν as the master equation for a pure-jump Lévy process. In the case ν ∈ L1(R), a relationship to fractional diffusion is established in a limit of vanishing nonlocality, which implies the convergence of a compound Poisson process to a stable process. In the case ν ∉ L1(R), the smoothing of the nonlocal operator is shown to correspond precisely to the activity of the underlying Lévy process and the variation of its sample paths. We introduce volume-constrained nonlocal diffusion equations and demonstrate that they are the master equations for Lévy processes restricted to a bounded domain. The ensuing variational formulation and conforming finite element method provide a powerful tool for studying both Lévy processes and fractional diffusion on bounded, non-simple geometries with volume constraints. In the second part of this dissertation, we consider the problem of estimating the distribution of a quantity of interest computed from the solution of an elliptic partial differential equation posed on a domain Ω(θ) ⊂ R2 with a randomly perturbed boundary, where (θ) is a random vector with given probability structure. We construct a piecewise smooth transformation from a partition of Ω(θ) to a reference domain Ω in order to avoid the complications associated with solving the problems on Ω(θ). The domain decomposition formulation is exploited by localizing the effect of the randomness to boundary elements in order to achieve a computationally efficient Monte Carlo sampling procedure. An a posteriori error analysis for the approximate distribution, which includes a deterministic error for each sample and a stochastic error from the effect of sampling, is also presented. We thus provide an efficient means to estimate the distribution of a quantity of interest via a Monte Carlo sampling procedure while also providing a posteriori error estimates for each sample

Magnetic helicity fluxes in interface and flux transport dynamos

Dynamos in the Sun and other bodies tend to produce magnetic fields that possess magnetic helicity of opposite sign at large and small scales, respectively. The build-up of magnetic helicity at small scales provides an important saturation mechanism. In order to understand the nature of the solar dynamo we need to understand the details of the saturation mechanism in spherical geometry. In particular, we want to understand the effects of magnetic helicity fluxes from turbulence and meridional circulation. We consider a model with just radial shear confined to a thin layer (tachocline) at the bottom of the convection zone. The kinetic alpha owing to helical turbulence is assumed to be localized in a region above the convection zone. The dynamical quenching formalism is used to describe the build-up of mean magnetic helicity in the model, which results in a magnetic alpha effect that feeds back on the kinetic alpha effect. In some cases we compare with results obtained using a simple algebraic alpha quenching formula. In agreement with earlier findings, the magnetic alpha effect in the dynamical alpha quenching formalism has the opposite sign compared with the kinetic alpha effect and leads to a catastrophic decrease of the saturation field strength with increasing magnetic Reynolds numbers. However, at high latitudes this quenching effect can lead to secondary dynamo waves that propagate poleward due to the opposite sign of alpha. Magnetic helicity fluxes both from turbulent mixing and from meridional circulation alleviate catastrophic quenching.Comment: 9 pages, 14 figures, submitted to A &

Zonal Flow Magnetic Field Interaction in the Semi-Conducting Region of Giant Planets

All four giant planets in the Solar System feature zonal flows on the order of 100 m/s in the cloud deck, and large-scale intrinsic magnetic fields on the order of 1 Gauss near the surface. The vertical structure of the zonal flows remains obscure. The end-member scenarios are shallow flows confined in the radiative atmosphere and deep flows throughout the entire planet. The electrical conductivity increases rapidly yet smoothly as a function of depth inside Jupiter and Saturn. Deep zonal flows will inevitably interact with the magnetic field, at depth with even modest electrical conductivity. Here we investigate the interaction between zonal flows and magnetic fields in the semi-conducting region of giant planets. Employing mean-field electrodynamics, we show that the interaction will generate detectable poloidal magnetic field perturbations spatially correlated with the deep zonal flows. Assuming the peak amplitude of the dynamo alpha-effect to be 0.1 mm/s, deep zonal flows on the order of 0.1 - 1 m/s in the semi-conducting region of Jupiter and Saturn would generate poloidal magnetic perturbations on the order of 0.01% - 1% of the background dipole field. These poloidal perturbations should be detectable with the in-situ magnetic field measurements from the Juno mission and the Cassini Grand Finale. This implies that magnetic field measurements can be employed to constrain the properties of deep zonal flows in the semi-conducting region of giant planets.Comment: 38 pages, 12 figures, revised submission to Icaru

The Anelastic Approximation: Magnetic Buoyancy and Magnetoconvection

In this thesis I discuss a series of anelastic approximations and detail the assumptions used in the derivation. I derive an entropy and temperature formulation of the anelastic approximation along with a simplification to the entropy formulation introduced by Lantz (1992) and independently by Braginsky & Roberts (1995). I assess range of applicability of the anelastic approximation, which is often used in describing the dynamics of geophysical and astrophysical flows. I consider two linear problems: magnetoconvection and magnetic buoyancy and compare the fully compressible solutions with those determined by solving the anelastic problem. I further compare the Lantz-Braginsky simplification with the full anelastic formulation which I find to work well if and only if the atmosphere is nearly adiabatic. I find that for the magnetoconvection problem the anelastic approximation works well if the departure from adiabaticity is small (as expected) and determine where the approximation breaks down. When the magnetic field is large then the anelastic approximation produces results which are markedly different from the fully compressible results. I also investigate the effects of altering the boundary conditions from isothermal to isentropic and the effect of stratification on how some of the parameters scale with the Chandrasekhar number. The results for magnetic buoyancy are less straight-forward, with the accuracy of the approximation being determined by the growth rate of the instability. I argue that these results make it difficult to assess a priori whether the anelastic approximation will provide an accurate approximation to the fully compressible system for stably stratified problems. Thus, unlike the magnetoconvection problem, for magnetic buoyancy it is difficult to provide general rules as to when the anelastic approximation can be used. When the instability grows quickly or the magnetic field is large the results do not compare well with the fully compressible equations. I outline a method for a two-dimensional non-linear time-stepping computer program and explain some problems with current non-linear programs

The Zoo of Non-Fourier Heat Conduction Models

The Fourier heat conduction model is valid for most macroscopic problems. However, it fails when the wave nature of the heat propagation or time lags become dominant and the memory or/and spatial non-local effects significant -- in ultrafast heating (pulsed laser heating and melting), rapid solidification of liquid metals, processes in glassy polymers near the glass transition temperature, in heat transfer at nanoscale, in heat transfer in a solid state laser medium at the high pump density or under the ultra-short pulse duration, in granular and porous materials including polysilicon, at extremely high values of the heat flux, in heat transfer in biological tissues. In common materials the relaxation time ranges from $10^{-8}$ to $10^{-14}$ sec, however, it could be as high as 1 sec in the degenerate cores of aged stars and its reported values in granular and biological objects varies up to 30 sec. The paper considers numerous non-Fourier heat conduction models that incorporate time non-locality for materials with memory (hereditary materials, including fractional hereditary materials) or/and spatial non-locality, i.e. materials with non-homogeneous inner structure

Astrophysical turbulence modeling

The role of turbulence in various astrophysical settings is reviewed. Among the differences to laboratory and atmospheric turbulence we highlight the ubiquitous presence of magnetic fields that are generally produced and maintained by dynamo action. The extreme temperature and density contrasts and stratifications are emphasized in connection with turbulence in the interstellar medium and in stars with outer convection zones, respectively. In many cases turbulence plays an essential role in facilitating enhanced transport of mass, momentum, energy, and magnetic fields in terms of the corresponding coarse-grained mean fields. Those transport properties are usually strongly modified by anisotropies and often completely new effects emerge in such a description that have no correspondence in terms of the original (non coarse-grained) fields.Comment: 88 pages, 26 figures, published in Reports on Progress in Physic

Heat equations beyond Fourier: from heat waves to thermal metamaterials

In the past decades, numerous heat conduction models beyond Fourier have been developed to account for the large gradients, fast phenomena, wave propagation, or heterogeneous material structure, such as being typical for biological systems, superlattices, or thermal metamaterials. It became a challenge to orient among the models, mainly due to their various thermodynamic backgrounds and possible compatibility issues. Additionally, in light of the recent findings on the field of non-Fourier heat conduction, it is not even straightforward how to interpret and utilize a non-Fourier heat equation, primarily when one aims to thermally design the material structure to construct the new generation of thermal metamaterials. Adding that numerous modeling strategies can be found in the literature accompanying different interpretations even for the same heat equation makes it even more difficult to orient ourselves and find a comprehensive picture of this field of research. Therefore, this review aims to ease the orientation among advanced heat equations beyond Fourier by discussing properties concerning their possible practical applications in light of experiments. We start from the simplest model with basic principles and notions, then proceed toward the more complex models related to coupled phenomena such as ballistic heat conduction. We do not enter the often complicated technical details of each thermodynamic framework but do not aim to compare each approach. However, we still briefly present their background to highlight their origin and the limitations acting on the models. Additionally, the field of non-Fourier heat conduction has become quite segmented, and that paper also aims to provide a common ground, a comprehensive mutual understanding of the basics of each model, together with what phenomenon they can be applied to