Simon D. Watt

Deterministic KPZ model for stromatolite laminae

The deterministic variant of the Kardar–Parisi–Zhang equation for the evolution of a growing interface is used to model patterning produced by successive laminations in certain stromatolites. Algebraic solutions of the model together with numerical simulations are employed to fit model parameters consistent with digital recordings of individual stromatolite laminae. Numerical values for model parameters, related to lateral growth, vertical growth and surface diffusion, provide a set of indices which may prove useful for classifying different stromatolites.

The accurate dynamic modelling of contaminant dispersion in channels

Throw some dye into a flowing river; how is the dye going to disperse along the river? The dynamics may be modelled by Taylor’s one-dimensional model of shear dispersion. We discuss the creation and utility of models of the dispersion of the dye, using invariant manifold theory. The models are effective exponentially quickly. We describe a one-mode centre manifold and a two-mode and a three-mode invariant manifold model. From the analysis, we decide under what conditions these three models are valid. Using the geometric picture provided by the theory of invariant manifolds, we find a complete set of initial conditions for the models so that the evolution of the full and approximate systems are identical after a short time. Throughout, a comparison of the three models is made to identify the practical situations when each model will be preferred.

Linear and nonlinear dynamics of cylindrically and spherically expanding detonation waves

The nonlinear stability of cylindrically and spherically expanding detonation waves is investigated using numerical simulations for both directly (blast) initiated detonations and cases where the simulations are initialized by placing quasi-steady solutions corresponding to different initial shock radii onto the grid. First, high-resolution one-dimensional (axially or radially symmetric) simulations of pulsating detonations are performed. Emphasis is on comparing with the predictions of a recent one-dimensional linear stability analysis of weakly curved detonation waves. The simulations show that, in agreement with the linear analysis, increasing curvature has a rapid destabilizing effect on detonation waves. The initial size and growth rate of the pulsation amplitude decreases as the radius where the detonation first forms increases. The pulsations may reach a saturated nonlinear behaviour as the amplitude grows, such that the subsequent evolution is independent of the initial conditions. As the wave expands outwards towards higher (and hence more stable) radii, the nature of the saturated nonlinear dynamics evolves to that of more stable behaviour (e.g. the amplitude of the saturated nonlinear oscillation decreases, or for sufficiently unstable cases, the oscillations evolve from multi-mode to period-doubled to limit-cycle-type behaviour). For parameter regimes where the planar detonation is stable, the linear stability prediction of the neutrally stable curvature gives a good prediction of the location of the maximum amplitude (provided the stability boundary is reached before the oscillations saturate) and of the critical radius of formation above which no oscillations are seen. The linear analysis also predicts very well the dependence of the period on the radius, even in the saturated nonlinear regimes. Secondly, preliminary two-dimensional numerical simulations of expanding cellular detonations are performed, but it is shown that resolved and accurate calculations of the cellular dynamics are currently computationally prohibitive, even with a dynamically adaptive numerical scheme.

A theoretical explanation of the influence of char formation on the ignition of polymers

A mathematical model is described for the behaviour of polymer combustion with char formation in thermally-thick conditions. A non-competitive single-step reaction is used, converting the polymer to char and volatiles. By examining this model numerically, the ignition behaviour of this system can be expressed as a function of the char yield. This simplified analytical approach yields good agreement with the numerical solution.

Combustion pseudo-waves in a system with reactant consumption and heat loss

A model for the combustion of a reactant with heat loss in an infinite two-dimensional layer or an infinitely long circular cylinder is derived. Using center manifold techniques the model is shown to reduce to a one-dimensional model on an infinite region for both geometries. For an exothermic first-order reaction with Arrhenius temperature dependence, a numerical method is used to calculate solutions and comparisons are made with the no-reactant consumption case. Traveling pseudo-waves are shown to exist and their speeds determined. A first-order estimate of the reaction zone thickness, obtained by perturbation methods, is shown to be in excellent agreement with the full numerical solution.

One-dimensional linear stability of curved detonations

In this paper, a one–dimensional stability analysis of weakly curved, quasi–steady detonation waves is performed using a numerical shooting method, for an idealized detonation with a single irreversible reaction. Neutral stability boundaries are determined and shown in an activation temperature–curvature diagram, and the dependence of the complex growth rates on curvature is investigated for several cases. It is shown that increasing curvature destabilizes detonation waves, and hence curved detonations can be unstable even when the planar front is stable. Even a small increase in curvature can significantly destabilize the wave. It is also shown that curved detonations are always unstable sufficiently near the critical curvature above which there are no underlying quasi–steady solutions.

Dimensional reduction of a bushfire model

Throw a match into a vegetation layer. What happens to the fire as time goes on? The spread from the ground to canopy, and along the terrain may be modelled by a simple two-dimensional reaction-diffusion system. We discuss the use of centre manifold theory to simplify a two-dimensional model of fire spread down to a one-dimensional model of fire propagation along the layer. This not only shows how to justify such simple one-dimensional models, it is also possible, for example, to determine the reaction wave speed analytically.

Continuum model for radial interface growth

A stochastic partial differential equation along the lines of the Kardar–Parisi–Zhang equation is introduced for the evolution of a growing interface in a radial geometry. Regular polygon solutions as well as radially symmetric solutions are identified in the deterministic limit. The polygon solutions, of relevance to on-lattice Eden growth from a seed in the zero-noise limit, are unstable in the continuum in favour of the symmetric solutions. The asymptotic surface width scaling for stochastic radial interface growth is investigated through numerical simulations and found to be characterized by the same scaling exponent as that for stochastic growth on a substrate.

Modelling pasture mass through time in a managed grazing system subject to perturbations resulting from complexity in natural biological processes

Abstract Variation is a characteristic of biological systems which may be due to both inaccuracies in measurement and the complexity of the interactions involved. The complexity of the interactions means that any abstraction of a biological system into a model often includes considerable uncertainty. Traditionally this uncertainty has been modelled by probabilistic methods. When dynamical systems are used to describe the evolution of biological processes it is necessary to consider the variation resulting from the impact of variables that are ignored in the formulation. Typically these variables are ignored because they act on a time scale faster than that of the abstraction. However, these fast variables can still affect the evolution of the system. This fast variable affect can be modelled using stochastic differential equations which incorporate the uncertainty due to complexity in the natural system. Solutions of stochastic differential equations may differ from the corresponding deterministic equations. Stochastic differential equations introduce a number of modelling issues not present in the deterministic case. These issues are reviewed using a model of a pasture grazing system. It is shown that the mode of the probability density which is the solution of the stochastic differential equation is an important statistic in the dynamical case. The importance of the first passage time in modelling complex ecological systems subject to uncertainty is also discussed.

The construction of zonal models of dispersion in channels via matched centre manifolds

Taylors model of dispersion simply describes the long-term spread of material along a pipe, channel or river. However, often we need multi-mode models to resolve finer details in space and time. Here we construct zonal models of dispersion via the new principle of matching their long-term evolution with that of the original problem. Using centre manifold techniques this is done straightforwardly and systematically. Furthermore, this approach provides correct initial and boundary conditions for the zonal models. We expect the proposed principle of matched centre manifold evolution to be useful in a wide range of modelling problems.