Advances in sequential data assimilation and numerical weather forecasting: an Ensemble Transform Kalman-Bucy Filter, a study on clustering in deterministic Ensemble Square Root Filters, and a test of a new time stepping scheme in an atmospheric model

This dissertation deals with aspects of sequential data assimilation (in particular ensemble Kalman filtering) and numerical weather forecasting. In the first part, the recently formulated Ensemble Kalman-Bucy (EnKBF) filter is revisited. It is shown that the previously used numerical integration scheme fails when the magnitude of the background error covariance grows beyond that of the observational error covariance in the forecast window. Therefore, we present a suitable integration scheme that handles the stiffening of the differential equations involved and doesn’t represent further computational expense. Moreover, a transform-based alternative to the EnKBF is developed: under this scheme, the operations are performed in the ensemble space instead of in the state space. Advantages of this formulation are explained. For the first time, the EnKBF is implemented in an atmospheric model. The second part of this work deals with ensemble clustering, a phenomenon that arises when performing data assimilation using of deterministic ensemble square root filters in highly nonlinear forecast models. Namely, an M-member ensemble detaches into an outlier and a cluster of M-1 members. Previous works may suggest that this issue represents a failure of EnSRFs; this work dispels that notion. It is shown that ensemble clustering can be reverted also due to nonlinear processes, in particular the alternation between nonlinear expansion and compression of the ensemble for different regions of the attractor. Some EnSRFs that use random rotations have been developed to overcome this issue; these formulations are analyzed and their advantages and disadvantages with respect to common EnSRFs are discussed. The third and last part contains the implementation of the Robert-Asselin-Williams (RAW) filter in an atmospheric model. The RAW filter is an improvement to the widely popular Robert-Asselin filter that successfully suppresses spurious computational waves while avoiding any distortion in the mean value of the function. Using statistical significance tests both at the local and field level, it is shown that the climatology of the SPEEDY model is not modified by the changed time stepping scheme; hence, no retuning of the parameterizations is required. It is found the accuracy of the medium-term forecasts is increased by using the RAW filter.

Axonal velocity distributions in neural feld equations

By modelling the average activity of large neuronal populations, continuum mean field models (MFMs) have become an increasingly important theoretical tool for understanding the emergent activity of cortical tissue. In order to be computationally tractable, long-range propagation of activity in MFMs is often approximated with partial differential equations (PDEs). However, PDE approximations in current use correspond to underlying axonal velocity distributions incompatible with experimental measurements. In order to rectify this deficiency, we here introduce novel propagation PDEs that give rise to smooth unimodal distributions of axonal conduction velocities. We also argue that velocities estimated from fibre diameters in slice and from latency measurements, respectively, relate quite differently to such distributions, a significant point for any phenomenological description. Our PDEs are then successfully fit to fibre diameter data from human corpus callosum and rat subcortical white matter. This allows for the first time to simulate long-range conduction in the mammalian brain with realistic, convenient PDEs. Furthermore, the obtained results suggest that the propagation of activity in rat and human differs significantly beyond mere scaling. The dynamical consequences of our new formulation are investigated in the context of a well known neural field model. On the basis of Turing instability analyses, we conclude that pattern formation is more easily initiated using our more realistic propagator. By increasing characteristic conduction velocities, a smooth transition can occur from self-sustaining bulk oscillations to travelling waves of various wavelengths, which may influence axonal growth during development. Our analytic results are also corroborated numerically using simulations on a large spatial grid. Thus we provide here a comprehensive analysis of empirically constrained activity propagation in the context of MFMs, which will allow more realistic studies of mammalian brain activity in the future.

On wave action and phase in the non-canonical Hamiltonian formulation

The long time–evolution of disturbances to slowly–varying solutions of partial differential equations is subject to the adiabatic invariance of the wave action. Generally, this approximate conservation law is obtained under the assumption that the partial differential equations are derived from a variational principle or have a canonical Hamiltonian structure. Here, the wave action conservation is examined for equations that possess a non–canonical (Poisson) Hamiltonian structure. The linear evolution of disturbances in the form of slowly varying wavetrains is studied using a WKB expansion. The properties of the original Hamiltonian system strongly constrain the linear equations that are derived, and this is shown to lead to the adiabatic invariance of a wave action. The connection between this (approximate) invariance and the (exact) conservation laws of pseudo–energy and pseudomomentum that exist when the basic solution is exactly time and space independent is discussed. An evolution equation for the slowly varying phase of the wavetrain is also derived and related to Berry's phase.

Maximum principles for vectorial approximate minimisers on non-convex functionals

We establish Maximum Principles which apply to vectorial approximate minimizers of the general integral functional of Calculus of Variations. Our main result is a version of the Convex Hull Property. The primary advance compared to results already existing in the literature is that we have dropped the quasiconvexity assumption of the integrand in the gradient term. The lack of weak Lower semicontinuity is compensated by introducing a nonlinear convergence technique, based on the approximation of the projection onto a convex set by reflections and on the invariance of the integrand in the gradient term under the Orthogonal Group. Maximum Principles are implied for the relaxed solution in the case of non-existence of minimizers and for minimizing solutions of the Euler–Lagrange system of PDE.

A mathematical model of crystallization in an emulsion

A mathematical model incorporating many of the important processes at work in the crystallization of emulsions is presented. The model describes nucleation within the discontinuous domain of an emulsion, precipitation in the continuous domain, transport of monomers between the two domains, and formation and subsequent growth of crystals in both domains. The model is formulated as an autonomous system of nonlinear, coupled ordinary differential equations. The description of nucleation and precipitation is based upon the Becker–Döring equations of classical nucleation theory. A particular feature of the model is that the number of particles of all species present is explicitly conserved; this differs from work that employs Arrhenius descriptions of nucleation rate. Since the model includes many physical effects, it is analyzed in stages so that the role of each process may be understood. When precipitation occurs in the continuous domain, the concentration of monomers falls below the equilibrium concentration at the surface of the drops of the discontinuous domain. This leads to a transport of monomers from the drops into the continuous domain that are then incorporated into crystals and nuclei. Since the formation of crystals is irreversible and their subsequent growth inevitable, crystals forming in the continuous domain effectively act as a sink for monomers “sucking” monomers from the drops. In this case, numerical calculations are presented which are consistent with experimental observations. In the case in which critical crystal formation does not occur, the stationary solution is found and a linear stability analysis is performed. Bifurcation diagrams describing the loci of stationary solutions, which may be multiple, are numerically calculated.

A moving mesh approach for modelling avascular tumour growth

A key step in many numerical schemes for time-dependent partial differential equations with moving boundaries is to rescale the problem to a fixed numerical mesh. An alternative approach is to use a moving mesh that can be adapted to focus on specific features of the model. In this paper we present and discuss two different velocity-based moving mesh methods applied to a two-phase model of avascular tumour growth formulated by Breward et al. (2002) J. Math. Biol. 45(2), 125-152. Each method has one moving node which tracks the moving boundary. The first moving mesh method uses a mesh velocity proportional to the boundary velocity. The second moving mesh method uses local conservation of volume fraction of cells (masses). Our results demonstrate that these moving mesh methods produce accurate results, offering higher resolution where desired whilst preserving the balance of fluxes and sources in the governing equations.

Mathematical model for NF-kappaB-driven proliferation of adult neural stem cells

Neural stem cells (NSCs) are early precursors of neuronal and glial cells. NSCs are capable of generating identical progeny through virtually unlimited numbers of cell divisions (cell proliferation), producing daughter cells committed to differentiation. Nuclear factor kappa B (NF-kappaB) is an inducible, ubiquitous transcription factor also expressed in neurones, glia and neural stem cells. Recently, several pieces of evidence have been provided for a central role of NF-kappaB in NSC proliferation control. Here, we propose a novel mathematical model for NF-kappaB-driven proliferation of NSCs. We have been able to reconstruct the molecular pathway of activation and inactivation of NF-kappaB and its influence on cell proliferation by a system of nonlinear ordinary differential equations. Then we use a combination of analytical and numerical techniques to study the model dynamics. The results obtained are illustrated by computer simulations and are, in general, in accordance with biological findings reported by several independent laboratories. The model is able to both explain and predict experimental data. Understanding of proliferation mechanisms in NSCs may provide a novel outlook in both potential use in therapeutic approaches, and basic research as well.

Modelling the rheology of anisotropic particles adsorbed on a two-dimensional fluid interface

We present a general approach based on nonequilibrium thermodynamics for bridging the gap between a well-defined microscopic model and the macroscopic rheology of particle-stabilised interfaces. Our approach is illustrated by starting with a microscopic model of hard ellipsoids confined to a planar surface, which is intended to simply represent a particle-stabilised fluid–fluid interface. More complex microscopic models can be readily handled using the methods outlined in this paper. From the aforementioned microscopic starting point, we obtain the macroscopic, constitutive equations using a combination of systematic coarse-graining, computer experiments and Hamiltonian dynamics. Exemplary numerical solutions of the constitutive equations are given for a variety of experimentally relevant flow situations to explore the rheological behaviour of our model. In particular, we calculate the shear and dilatational moduli of the interface over a wide range of surface coverages, ranging from the dilute isotropic regime, to the concentrated nematic regime.

On the structure of infinity-harmonic maps

Let H ∈ C 2(ℝ N×n ), H ≥ 0. The PDE system arises as the Euler-Lagrange PDE of vectorial variational problems for the functional E ∞(u, Ω) = ‖H(Du)‖ L ∞(Ω) defined on maps u: Ω ⊆ ℝ n → ℝ N . (1) first appeared in the author's recent work. The scalar case though has a long history initiated by Aronsson. Herein we study the solutions of (1) with emphasis on the case of n = 2 ≤ N with H the Euclidean norm on ℝ N×n , which we call the “∞-Laplacian”. By establishing a rigidity theorem for rank-one maps of independent interest, we analyse a phenomenon of separation of the solutions to phases with qualitatively different behaviour. As a corollary, we extend to N ≥ 2 the Aronsson-Evans-Yu theorem regarding non existence of zeros of |Du| and prove a maximum principle. We further characterise all H for which (1) is elliptic and also study the initial value problem for the ODE system arising for n = 1 but with H(·, u, u′) depending on all the arguments.

The Wheal Jane wetlands model for bioremediation of acid mine drainage

Acid mine drainage (AMD) is a widespread environmental problem associated with both working and abandoned mining operations. As part of an overall strategy to determine a long-term treatment option for AMD, a pilot passive treatment plant was constructed in 1994 at Wheal Jane Mine in Cornwall, UK. The plant consists of three separate systems, each containing aerobic reed beds, anaerobic cell and rock filters, and represents the largest European experimental facility of its kind. The systems only differ by the type of pretreatment utilised to increase the pH of the influent minewater (pH <4): lime dosed (LD), anoxic limestone drain (ALD) and lime free (LF), which receives no form of pretreatment. Historical data (1994-1997) indicate median Fe reduction between 55% and 92%, sulphate removal in the range of 3-38% and removal of target metals (cadmium, copper and zinc) below detection limits, depending on pretreatment and flow rates through the system. A new model to simulate the processes and dynamics of the wetlands systems is described, as well as the application of the model to experimental data collected at the pilot plant. The model is process based, and utilises reaction kinetic approaches based on experimental microbial techniques rather than an equilibrium approach to metal precipitation. The model is dynamic and utilises numerical integration routines to solve a set of differential equations that describe the behaviour of 20 variables over the 17 pilot plant cells on a daily basis. The model outputs at each cell boundary are evaluated and compared with the measured data, and the model is demonstrated to provide a good representation of the complex behaviour of the wetland system for a wide range of variables. (C) 2004 Elsevier B.V/ All rights reserved.

Linear and nonlinear generalized Fourier transforms

This article presents an overview of a transform method for solving linear and integrable nonlinear partial differential equations. This new transform method, proposed by Fokas, yields a generalization and unification of various fundamental mathematical techniques and, in particular, it yields an extension of the Fourier transform method.

Unification of the probe and singular sources methods for the inverse boundary value problem by the no-response test

In this article, we use the no-response test idea, introduced in Luke and Potthast (2003) and Potthast (Preprint) and the inverse obstacle problem, to identify the interface of the discontinuity of the coefficient gamma of the equation del (.) gamma(x)del + c(x) with piecewise regular gamma and bounded function c(x). We use infinitely many Cauchy data as measurement and give a reconstructive method to localize the interface. We will base this multiwave version of the no-response test on two different proofs. The first one contains a pointwise estimate as used by the singular sources method. The second one is built on an energy (or an integral) estimate which is the basis of the probe method. As a conclusion of this, the probe and the singular sources methods are equivalent regarding their convergence and the no-response test can be seen as a unified framework for these methods. As a further contribution, we provide a formula to reconstruct the values of the jump of gamma(x), x is an element of partial derivative D at the boundary. A second consequence of this formula is that the blow-up rate of the indicator functions of the probe and singular sources methods at the interface is given by the order of the singularity of the fundamental solution.

Multi-mode approximations to wave scattering by an uneven bed

Approximations to the scattering of linear surface gravity waves on water of varying quiescent depth are Investigated by means of a variational approach. Previous authors have used wave modes associated with the constant depth case to approximate the velocity potential, leading to a system of coupled differential equations. Here it is shown that a transformation of the dependent variables results in a much simplified differential equation system which in turn leads to a new multi-mode 'mild-slope' approximation. Further, the effect of adding a bed mode is examined and clarified. A systematic analytic method is presented for evaluating inner products that arise and numerical experiments for two-dimensional scattering are used to examine the performance of the new approximations.

Boundary value problems for the N-wave interaction equations

We consider boundary value problems for the N-wave interaction equations in one and two space dimensions, posed for x [greater-or-equal, slanted] 0 and x,y [greater-or-equal, slanted] 0, respectively. Following the recent work of Fokas, we develop an inverse scattering formalism to solve these problems by considering the simultaneous spectral analysis of the two ordinary differential equations in the associated Lax pair. The solution of the boundary value problems is obtained through the solution of a local Riemann–Hilbert problem in the one-dimensional case, and a nonlocal Riemann–Hilbert problem in the two-dimensional case.

Initial boundary value problems for the nonlinear Schrödinger equation

A new spectral method for solving initial boundary value problems for linear and integrable nonlinear partial differential equations in two independent variables is applied to the nonlinear Schrödinger equation and to its linearized version in the domain {x≥l(t), t≥0}. We show that there exist two cases: (a) if l″(t)<0, then the solution of the linear or nonlinear equations can be obtained by solving the respective scalar or matrix Riemann-Hilbert problem, which is defined on a time-dependent contour; (b) if l″(t)>0, then the Riemann-Hilbert problem is replaced by a respective scalar or matrix problem on a time-independent domain. In both cases, the solution is expressed in a spectrally decomposed form.