Convective modes reveal the incoherence of the Southern Polar Vortex.

The Southern Polar Vortex (SPV) is prominent over Antarctica in the Austral winter, and typically associated with a region of low temperature, low ozone concentration, negative potential vorticity, and polar stratospheric clouds. Seasonal and unexpected changes in the SPV have a profound influence on global weather. A methodology which identifies the SPV's coherence and breakup using only wind and pressure data is developed and validated against temperature, ozone and potential vorticity data. The process identifies "convective modes", each with an assigned "coherence" value, which form building blocks for the observed spatial variation of the SPV. Analysis and interpretation are presented for 4 years with quite different known behavior of the SPV: 1999 (a relatively standard year), 2002 (when the SPV split into two), 2019 (an atmospheric warming year which led to an early dissipation in the SPV), and the most recent year 2022 (which was influenced by submarine volcano eruptions and a prolonged La Niña event). In decomposing convective effects into modes with quantifiable coherence, this study solidifies connections between wind velocities and atmospheric variables while providing new tools to study the evolution of coherent structures and signal the occurrence of atypical geophysical events.

Thin-film lubrication model for biofilm expansion under strong adhesion.

Understanding microbial biofilm growth is important to public health because biofilms are a leading cause of persistent clinical infections. In this paper, we develop a thin-film model for microbial biofilm growth on a solid substratum to which it adheres strongly. We model biofilms as two-phase viscous fluid mixtures of living cells and extracellular fluid. The model explicitly tracks the movement, depletion, and uptake of nutrients and incorporates cell proliferation via a nutrient-dependent source term. Notably, our thin-film reduction is two dimensional and includes the vertical dependence of cell volume fraction. Numerical solutions show that this vertical dependence is weak for biologically feasible parameters, reinforcing results from previous models in which this dependence was neglected. We exploit this weak dependence by writing and solving a simplified one-dimensional model that is computationally more efficient than the full model. We use both the one- and two-dimensional models to predict how model parameters affect expansion speed and biofilm thickness. This analysis reveals that expansion speed depends on cell proliferation, nutrient availability, cell-cell adhesion on the upper surface, and slip on the biofilm-substratum interface. Our numerical solutions provide a means to qualitatively distinguish between the extensional flow and lubrication regimes, and quantitative predictions that can be tested in future experiments.

A thin-film extensional flow model for biofilm expansion by sliding motility.

In the presence of glycoproteins, bacterial and yeast biofilms are hypothesized to expand by sliding motility. This involves a sheet of cells spreading as a unit, facilitated by cell proliferation and weak adhesion to the substratum. In this paper, we derive an extensional flow model for biofilm expansion by sliding motility to test this hypothesis. We model the biofilm as a two-phase (living cells and an extracellular matrix) viscous fluid mixture, and model nutrient depletion and uptake from the substratum. Applying the thin-film approximation simplifies the model, and reduces it to one-dimensional axisymmetric form. Comparison with Saccharomyces cerevisiae mat formation experiments reveals good agreement between experimental expansion speed and numerical solutions to the model with O ( 1 ) parameters estimated from experiments. This confirms that sliding motility is a possible mechanism for yeast biofilm expansion. Having established the biological relevance of the model, we then demonstrate how the model parameters affect expansion speed, enabling us to predict biofilm expansion for different experimental conditions. Finally, we show that our model can explain the ridge formation observed in some biofilms. This is especially true if surface tension is low, as hypothesized for sliding motility.

Estimating stable and unstable sets and their role as transport barriers in stochastic flows.

We consider the situation of a large-scale stationary flow subjected to small-scale fluctuations. Assuming that the stable and unstable manifolds of the large-scale flow are known, we quantify the mean behavior and stochastic fluctuations of particles close to the unperturbed stable and unstable manifolds and their evolution in time. The mean defines a smooth curve in physical space, while the variance provides a time- and space-dependent quantitative estimate where particles are likely to be found. This allows us to quantify transport properties such as the expected volume of mixing as the result of the stochastic fluctuations of the transport barriers. We corroborate our analytical findings with numerical simulations in both compressible and incompressible flow situations. We moreover demonstrate the intimate connection of our results with finite-time Lyapunov exponent fields, and with spatial mixing regions.

Diffusion-Limited Growth of Microbial Colonies.

The emergence of diffusion-limited growth (DLG) within a microbial colony on a solid substrate is studied using a combination of mathematical modelling and experiments. Using an agent-based model of the interaction between microbial cells and a diffusing nutrient, it is shown that growth directed towards a nutrient source may be used as an indicator that DLG is influencing the colony morphology. A continuous reaction-diffusion model for microbial growth is employed to identify the parameter regime in which DLG is expected to arise. Comparisons between the model and experimental data are used to argue that the bacterium Bacillus subtilis can undergo DLG, while the yeast Saccharomyces cerevisiae cannot, and thus the non-uniform growth exhibited by this yeast must be caused by the pseudohyphal growth mode rather than limited nutrient availability. Experiments testing directly for DLG features in yeast colonies are used to confirm this hypothesis.

Nutrient-limited growth with non-linear cell diffusion as a mechanism for floral pattern formation in yeast biofilms.

Previous experiments have shown that mature yeast mat biofilms develop a floral morphology, characterised by the formation of petal-like structures. In this work, we investigate the hypothesis that nutrient-limited growth is the mechanism by which these floral patterns form. To do this, we use a combination of experiments and mathematical analysis. In mat formation experiments of the yeast species Saccharomyces cerevisiae, we observe that mats expand radially at a roughly constant speed, and eventually undergo a transition from circular to floral morphology. To determine the extent to which nutrient-limited growth can explain these features, we adopt a previously proposed mathematical model for yeast growth. The model consists of a coupled system of reaction-diffusion equations for the yeast cell density and nutrient concentration, with a non-linear, degenerate diffusion term for cell spread. Using geometric singular perturbation theory and numerics, we show that the model admits travelling wave solutions in one dimension, which enables us to infer the diffusion ratio from experimental data. We then use a linear stability analysis to show that two-dimensional planar travelling wave solutions for feasible experimental parameters are linearly unstable to non-planar perturbations. This provides a potential mechanism by which petals can form, and allows us to predict the characteristic petal width. There is good agreement between these predictions, numerical solutions to the model, and experimental data. We therefore conclude that the non-linear cell diffusion mechanism provides a possible explanation for pattern formation in yeast mat biofilms, without the need to invoke other mechanisms such as flow of extracellular fluid, cell adhesion, or changes to cellular shape or behaviour.

Stochastic uncertainty of advected curves in finite-time unsteady flows.

Identifying coherent structures in unsteady flows acting over a finite-time is a well-established research area, in part due to the applicability to realistic velocities obtained from experimental, observational, or numerically generated data. More recently, there is an emerging need to understand the impact of small-scale uncertainties on larger scale structures; for example, the "stochastic parametrization" problem in climate models. This article establishes a rigorous tool in this direction, specifically quantifying the uncertainty of advected curves in the presence of small stochasticity. Explicit expressions are derived for the expectation and the variance of the curves' location. The velocity field may be unsteady and compressible, and the Wiener process driving the stochasticity can have general spatiotemporal dependence. Monte Carlo simulations are used to verify the uncertainty expressions.

Influences of Allee effects in the spreading of malignant tumours.

A recent study by Korolev et al. [Nat. Rev. Cancer, 14:371-379, 2014] evidences that the Allee effect-in its strong form, the requirement of a minimum density for cell growth-is important in the spreading of cancerous tumours. We present one of the first mathematical models of tumour invasion that incorporates the Allee effect. Based on analysis of the existence of travelling wave solutions to this model, we argue that it is an improvement on previous models of its kind. We show that, with the strong Allee effect, the model admits biologically relevant travelling wave solutions, with well-defined edges. Furthermore, we uncover an experimentally observed biphasic relationship between the invasion speed of the tumour and the background extracellular matrix density.

Accurate control of hyperbolic trajectories in any dimension.

The unsteady (nonautonomous) analog of a hyperbolic fixed point is a hyperbolic trajectory, whose importance is underscored by its attached stable and unstable manifolds, which have relevance in fluid flow barriers, chaotic basin boundaries, and the long-term behavior of the system. We develop a method for obtaining the unsteady control velocity which forces a hyperbolic trajectory to follow a user-prescribed variation with time. Our method is applicable in any dimension, and accuracy to any order is achievable. We demonstrate and validate our method by (1) controlling the fixed point at the origin of the Lorenz system, for example, obtaining a user-defined nonautonomous attractor, and (2) the saddle points in a droplet flow, using localized control which generates global transport.

Energy constrained transport maximization across a fluid interface.

With enhancing mixing in micro- or nanofluidic applications in mind, the problem of maximizing fluid transport across a fluid interface subject to an available energy budget is examined. The optimum cross-interface perturbing velocity is obtained explicitly in the time-periodic instance using an Euler-Lagrange constrained optimization approach. Numerical investigations which calculate transferred lobe areas and cross-interface flux are used to verify that the predicted strategy achieves optimum transport. Explicit active protocols for achieving this optimal transport are suggested.

Optimal frequency for microfluidic mixing across a fluid interface.

A new analytical tool for determining the optimum frequency for a micromixing strategy to mix two fluids across their interface is presented. The frequency dependence of the flux is characterized in terms of a Fourier transform related to the apparatus geometry. Illustrative microfluidic mixing examples based on electromagnetic forcing and fluid pumping strategies are presented.

Invasions with density-dependent ecological parameters.

The speed and the minimum carrying capacity needed for a successful population expansion into new territory are addressed using a reaction-diffusion model. The model is able to encapsulate a rich collection of ecological behaviours, including the Allee effect, resource depletion due to consumption, dispersal adaptation due to population pressure, biological control agents, and a range of breeding suppression mechanisms such as embryonic diapause, delayed development and sperm storage. It is shown how many of these phenomena can be characterised as density-dependence in a few fundamental ecological parameters. With the help of a powerful mathematical technique recently developed by Balasuriya and Gottwald (J. Math. Biol. 61, pp. 377-399, 2010), explicit formulae for the effect on the speed and minimum carrying capacity are obtained.

Wavespeed in reaction-diffusion systems, with applications to chemotaxis and population pressure.

We present a method based on the Melnikov function used in dynamical systems theory to determine the wavespeed of travelling waves in perturbed reaction-diffusion systems. We study reaction-diffusion systems which are subject to weak nontrivial perturbations in the reaction kinetics, in the diffusion coefficient, or with weak active advection. We find explicit formulae for the wavespeed and illustrate our theory with two examples; one in which chemotaxis gives rise to nonlinear advection and a second example in which a positive population pressure results in both a density-dependent diffusion coefficient and a nonlinear advection. Based on our theoretical results we suggest an experiment to distinguish between chemotactic and population pressure in bacterial colonies.