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Acknowledgements. Funding for DIB was provided by NE/R018243/1 REBUS (Resolving Enthalpy Budget to Understand Surging). We present the first general theory of glacier surging that includes both temperate and polythermal glacier surges, based on coupled mass and enthalpy budgets. Enthalpy (in the form of thermal energy and water) is gained at the glacier bed from geothermal heating plus frictional heating (expenditure of potential energy) as a consequence of ice flow. Enthalpy losses occur by conduction and loss of meltwater from the system. Because enthalpy directly impacts flow speeds, mass and enthalpy budgets must simultaneously balance if a glacier is to maintain a steady flow. If not, glaciers undergo out-of-phase mass and enthalpy cycles, manifest as quiescent and surge phases. We illustrate the theory using a lumped element model, which parameterizes key thermodynamic and hydrological processes, including surface-to-bed drainage and distributed and channelized drainage systems. Model output exhibits many of the observed characteristics of polythermal and temperate glacier surges, including the association of surging behaviour with particular combinations of climate (precipitation, temperature), geometry (length, slope) and bed properties (hydraulic conductivity). Enthalpy balance theory explains a broad spectrum of observed surging behaviour in a single framework, and offers an answer to the wider question of why the majority of glaciers do not surge. Publisher PDF

Complex processes occur at the base of glaciers and ice sheets that influence their flow and hydrological behaviour and determine the landforms left behind by glaciation. This chapter considers mathematical models of basal sliding, subglacial water drainage and till deformation, and introduces various subglacial landforms due to the interaction of ice, water and sediment. The treatment extends to models of large-scale phenomena including glacier outburst floods and glacier surges.

Soil treatment units (STUs) receiving effluent from on-site wastewater treatment systems (OWTSs) rely on the gradual development of a microbial biomat/biozone at the infiltrative surface for optimal effluent distribution and pollutant attenuation. Here, we present the first direct measurement of gradual biomat development in the field in STU trenches receiving either primary (PE) or secondary treated effluent (SE) under identical environmental, hydrological and subsoil conditions. Two domestic OWTSs were constructed in Ireland and monitored over a period of >2 years using an automated, three-dimensional network of buried soil water content sensors tracking water flow and retention within the soil underneath the infiltrative surface. While trenches receiving PE expressed signs of biomat formation along the entire length of STU trenches, biomats in trenches receiving SE were significantly muted and did not extend further than 10 m from the inlet at the end of the study. The presence of a mature biomat helped to retain soil moisture above background levels and made the system more resilient towards drought events and desiccation stress but led, in one case, to effluent ponding within the trenches. A growth-limited non-linear model fit revealed that biomats in SE trenches are expected to remain considerably shorter and will not spread along the entire trench design length, even after 10 years of operation, which is contrary to prevalent design assumptions. Muted biomat growth, on the contrary, might lead to localized hydraulic and pollutant overloading and has been shown previously to negatively affect the ability to attenuate pollutants effectively within the soil profile before the effluent reaches the groundwater.

In this paper we revisit the problem of explaining phase transition by a study of a form of the Boltzmann equation, where inter-molecular attraction is included by means of a Vlasov term in the evolution equation for the one particle distribution function. We are able to show that for typical gas densities, a uniform state is unstable if the inter-molecular attraction is large enough. Our analysis relies strongly on the assumption, essential to the derivation of the Boltzmann equation, that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu \ll 1,$$\end{document}ν≪1, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =d/l$$\end{document}ν=d/l is the ratio of the molecular diameter to the mean inter-particle distance; in this case, for fluctuations on the scale of the molecular spacing, the collision term is small, and an explicit approximate solution is possible. We give reasons why we think the resulting approximation is valid, and in conclusion offer some possibilities for extension of the results to finite amplitude.

This paper addresses the problem of extinction in continuous models of population dynamics associated with small numbers of individuals. We begin with an extended discussion of extinction in the particular case of a stochastic logistic model, and how it relates to the corresponding continuous model. Two examples of ‘small number dynamics’ are then considered. The first is what Mollison calls the ‘atto-fox’ problem (in a model of fox rabies), referring to the problematic theoretical occurrence of a predicted rabid fox density of $$10^{-18}$$ 10 - 18 (atto-) per square kilometre. The second is how the production of large numbers of eggs by an individual can reliably lead to the eventual survival of a handful of adults, as it would seem that extinction then becomes a likely possibility. We describe the occurrence of the atto-fox problem in other contexts, such as the microbial ‘yocto-cell’ problem, and we suggest that the modelling resolution is to allow for the existence of a reservoir for the extinctively challenged individuals. This is functionally similar to the concept of a ‘refuge’ in predator–prey systems and represents a state for the individuals in which they are immune from destruction. For what I call the ‘frogspawn’ problem, where only a few individuals survive to adulthood from a large number of eggs, we provide a simple explanation based on a Holling type 3 response and elaborate it by means of a suitable nonlinear age-structured model.

Volcanoes of Strombolian type are able to maintain their semi-permanent eruptive states through the constant convective recycling of magma within the conduit leading from the magma chamber. In this paper we study the form of this convection using an analytic model of degassing two-phase flow in a vertical channel. We provide solutions for the flow at small Grashof and large Prandtl numbers, and we suggest that permanent steady-state counter-current convection is only possible if an initial bubbly counter-current flow undergoes a regime transition to a churn-turbulent flow. We also suggest that the magma in the chamber must be under-pressured in order for the flow to be maintained, and that this compromises the assumed form of the flow.

We propose a model for the growth of microbial populations in the presence of a rate-limiting nutrient which accounts for the switching of cells to a dormant phase at low densities in response to decreasing concentration of a putative biochemical signal. We then show that in conditions of nutrient starvation, self-sustained oscillations can occur, thus providing a natural explanation for such phenomena as plankton blooms. However, unlike results of previous studies, the microbial population minima do not become unrealistically small, being buffered during minima by an increased dormant phase population. We also show that this allows microbes to survive in extreme environments for very long periods, consistent with observation. The mechanism provides a natural vehicle for other such sporadic outbreaks, such as viral epidemics.

peer-reviewed Nitrification at the site of a contaminant ammonium plume from a former coal carbonisation plant can be modelled with three competing bacterial populations of Nitrosomonas, Nitrobacter, and Brocadia anammoxidans. Oscillations of chemical species at the site can be explained by a reduced model of ammonium competition between Nitrosomonas and B. anammoxidans which effectively acts as an activator-inhibitor system. Stable oscillations occur in conditions of low nutrient (ammonium) supply and this causes a spatial travelling wave in a borehole profile when diffusion is introduced.

The scientific legacy of George Gabriel Stokes is considered. Certain aspects of Stokes’s research work are reviewed and related to more recent fields of research. These include the Navier–Stokes equations and other approaches to rational continuum mechanics, the issue of existence of solutions, the boundary no-slip condition; Stokes flow and the issue of pendulum drag; the Hele-Shaw cell, viscous fingering, wavelength selection in pattern formation; moving contact lines; the highest water wave, rogue waves, the NLS equation; Stokes lines, exponential asymptotics, dendrite growth, slow manifods, and diffraction.

In a previous paper, we analysed the Keller–Rubinow formulation of Ostwald's supersaturation theory for the formation of Liesegang rings or Liesegang bands, and found that the model is ill-posed, in the sense that after the termination of the first crystal front growth, secondary bands form, as in the experiment, but these are numerically found to be a single grid space wide, and thus an artefact of the numerical method. This ill-posedness is due to the discontinuity in the crystal growth rate, which itself reflects the supersaturation threshold inherent in the theory. Here we show that the ill-posedness can be resolved by the inclusion of a relaxation mechanism describing an impurity coverage fraction, which physically enables the transition in heterogeneous nucleation from precipitate-free impurity to precipitate-covered impurity.

We consider the instability properties of dense granular flow in inclined plane and plane shear geometries as tests for the compressible inertial-dependent rheology. The model, which is a recent generalisation of the incompressible $\unicode[STIX]{x1D707}(I)$ rheology, constitutes a hydrodynamical description of dense granular flow which allows for variability in the solids volume fraction. We perform a full linear stability analysis of the model and compare its predictions to existing experimental data for glass beads on an inclined plane and discrete element simulations of plane shear in the absence of gravity. In the case of the former, we demonstrate that the compressible model can quantitatively predict the instability properties observed experimentally, and, in particular, we find that it performs better than its incompressible counterpart. For the latter, the qualitative behaviour of the plane shear instability is also well captured by the compressible model.

The Anderson–May model of human parasite infections and specifically that for the intestinal worm Ascaris lumbricoides is reconsidered, with a view to deriving the observed characteristic negative binomial distribution which is frequently found in human communities. The means to obtaining this result lies in reformulating the continuous Anderson–May model as a stochastic process involving two essential populations, the density of mature worms in the gut, and the density of mature eggs in the environment. The resulting partial differential equation for the generating function of the joint probability distribution of eggs and worms can be partially solved in the appropriate limit where the worm lifetime is much greater than that of the mature eggs in the environment. Allowing for a mean field nonlinearity, and for egg immigration from neighbouring communities, a negative binomial worm distribution can be predicted, whose parameters are determined by those in the continuous Anderson–May model; this result assumes no variability in predisposition to the infection.\ud

We mathematically model the uptake of phosphorus by a soil community consisting of a plant and two bacterial groups: copiotrophs and oligotrophs. Four equilibrium states emerge, one for each of the species monopolising the resource and dominating the community and one with coexistence of all species. We show that the dynamics are controlled by the ratio of chemical adsorption to bacterial death permitting either oscillatory states or quasi-steady uptake. We show how a steady state can emerge which has soil and plant nutrient content unresponsive to increased fertilization. However, the additional fertilization supports the copiotrophs leading to community reassembly. Our results demonstrate the importance of time-series measurements in nutrient uptake experiments.

The theory of chaos in one-dimensional maps is described. We start with the basic bifurcations, following which there is an extended discussion of period-doubling sequences and the Feigenbaum conjectures. There follows an extended discussion of kneading theory, culminating with a statement of Sarkovskii’s ordering of the periodic orbits of unimodal maps.

We provide an analytic estimate for the size of the bulbs adjoining the main cardioid of the Mandelbrot set. The bulbs are approximate circles, and are associated with the stability regions in the complex parameter μ-space of period-q orbits of the underlying map z → z 2 − μ . For the (p, q) orbit with winding number p/q, the associated stability bulb is an approximate circle with radius 1 q 2 sin π p q .

The increased use of sulfate fertilizers to compensate for soil sulphur (S) limitation in agricultural soils may affect soil microbes and micro-fauna involved in S mobilization. Here, columns with podzolic soil material and ryegrass (Lolium perenne) were fertilized with 0, 5, 10 and 20 kg ha−1 (S0/S5/S10/S20) inorganic sulfate-S alongside a full complement of other nutrients. In the S10 and S20 columns, significantly higher amounts of sulfate were present in soil solution. After two grass cuts (14 weeks in total), there was a significant decrease in arylsulfatase activity, bacterial-feeding nematode abundances and mycorrhizal colonization in the S10 and S20 columns compared to the S0. Bacterial, fungal and AM community structures shifted significantly across the treatments. After final harvest, the S10 and S20 columns had significantly higher grass dry matter yield and uptake of S, N, K, Ca and Mg compared to the S0. While the overall bacterial diversity was reduced in the S20 treatment, abundance (asfA) and diversity (ssuD and atsA) of bacterial genes involved in S cycling were not significantly affected by one-time sulfate fertilization. These results indicate that short-term sulfate fertilization benefits to plant growth outweighed the negative feedback from parts of the soil biota. To improve nutrient use efficiencies in a sustainable manner, future studies should consider alternative S fertilizers which may be beneficial to both, the soil biota and plants in the long-term.

In the original article, the second author's name was incorrect in the metadata. The given name is T. Deirdre, and the family name is Hollingsworth.

This study evaluated the effect of one-time phosphate fertilization on the soil microbiota, its cycling of phosphorus (P) and grass growth. Soil columns were established in a greenhouse using a P-limited Irish soil (index 1), planted with Lolium perenne and fertilized with 0 (control), 5 (quarter), 10 (half) and 20 (full) kg P ha−1 as inorganic phosphate. Only traces of phosphate in soil solution were detected over the 14 week experiment, even after phosphate fertilization. Grass dry matter yield between treatments was not significantly different. Full phosphate fertilization significantly reduced the arbuscular mycorrhization (AM) rate, bacterial- and fungal-feeding nematode population, bacterial phoD gene abundance, but increased alkaline and acid phosphatase activities at the time of harvest. Full and half P treatments significantly shifted the bacterial, fungal and AM community structures compared to the control. Furthermore, the control had a significantly higher relative abundance of bacterial genera including Bacillus, Bradyrhizobium, Paenibacillus, Nocardioides and Balneimonas, that have been associated with P mobilization in the past, when compared to the full phosphate treatment. These results suggest that a positive effect of a single phosphate application on plant growth in a soil can be cancelled out by its negative effect on the soil microbiota and their ecosystem services.

Observations have long associated ice streams with the presence of meltwater at the bed. More recently, theoretical models have been able to reproduce ice-stream behaviour as a consequence of the coupled dynamics of ice and subglacial meltwater. In this paper we analyse the properties of ice streams that form in a coupled model of ice flow and subglacial hydrology. We see that there is a natural length scale defining ice stream separation and width. This arises as a result of the balance between effective pressure gradients driving meltwater away from ice streams and the enhanced water production in the streams due to the fast ice flow. We further discuss how the model interacts with topography and we show that small perturbations to a uniform bed have a strong effect on where ice streams emerge in the model. However, in many cases ice streams then evolve to be closer to the dimensions defined by the natural length scale of the unperturbed system. The non-dimensional parameter that defines this length scale is therefore of fundamental importance in the model.

In this paper we undertake a quantitative analysis of the dynamic process by which ice underneath a dry porous debris layer melts. We show that the incorporation of debris-layer airflow into a theoretical model of glacial melting can capture the empirically observed features of the so-called Østrem curve (a plot of the melt rate as a function of debris depth). Specifically, we show that the turning point in the Østrem curve can be caused by two distinct mechanisms: the increase in the proportion of ice that is debris-covered and/or a reduction in the evaporative heat flux as the debris layer thickens. This second effect causes an increased melt rate because the reduction in (latent) energy used for evaporation increases the amount of energy available for melting. Our model provides an explicit prediction for the melt rate and the temperature distribution within the debris layer, and provides insight into the relative importance of the two effects responsible for the maximum in the Østrem curve. We use the data of Nicholson and Benn (2006) to show that our model is consistent with existing empirical measurements.

Debris-covered glaciers are prone to the formation of a number of supraglacial geomorphological features, and generally speaking, their upper surfaces are far from level surfaces. Some of these features are due to radiation screening or enhancing properties of the debris cover, but theoretical explanations of the consequent surface forms are in their infancy. In this paper we consider a theoretical model for the formation of ‘ice sails’, which are regularly spaced bare ice features which are found on debris-covered glaciers in the Karakoram.

We study the model of Keller & Rubinow (Keller & Rubinow 1981J. Chem. Phys74, 5000–5007. (doi:10.1063/1.441752)) describing the formation of Liesegang rings due to Ostwald's supersaturation mechanism. Keller and Rubinow provided an approximate solution both for the growth and equilibration of the first band, and also for the formation of secondary bands, based on a presumed asymptotic limit. However, they did not provide a parametric basis for the assumptions in their solution, nor did they provide any numerical corroboration, particularly of the secondary band formation. Here, we provide a different asymptotic solution, based on a specific parametric limit, and we show that the growth and subsequent cessation of the first band can be explained. We also show that the model is unable to explain the formation of finite width secondary bands, and we confirm this result by numerical computation. We conclude that the model is not fully posed, lacking a transition variable which can describe the hysteretic switch across the nucleation threshold.

We propose a mechanism previously developed as a hypothetical cause of the initiation of subduction in the Earth's mantle, to describe a situation where such subduction may occur transiently, at irregular intervals of time. It has been suggested that tectonics on Venus may be described by such a scenario. In our model, a subduction event is followed by resumption of high Rayleigh number mantle convection below a stagnant lithosphere which thickens due to conductive cooling. As it thickens, differential buoyancy causes large lithospheric stresses which eventually lead to (plastic) failure in the upper portions of the lithosphere. This plastic zone thickens faster than the lithosphere, so that at some critical time, it reaches the base of the lithosphere. At this point, the effective lithosphere viscosity decreases to that of the underlying mantle, and subduction can occur. We suggest that this is mechanistically consistent with the postulated Venusian tectonic style.

Decompression melting of hot upwelling rock in the mantle creates a region of partial melt comprising a porous solid matrix through which magma rises buoyantly. Magma transport and the compensating matrix deformation are commonly described by two-phase compaction models, but melt production is less often incorporated. Melting is driven by the necessity to maintain thermodynamic equilibrium between mineral grains in the partial melt; the position and amount of partial melting that occur are thus thermodynamically determined. We present a consistent model for the ascent of a one-dimensional column of rock and provide solutions that reveal where and how much partial melting occurs, the positions of the boundaries of the partial melt being determined by conserving energy across them. Thermodynamic equilibrium of the boundary between partial melt and the solid lithosphere requires a boundary condition on the effective pressure (solid pressure minus melt pressure), which suggests that large effective stresses, and hence fracture, are likely to occur near the base of the lithosphere. Matrix compaction, melt separation and temperature in the partially molten region are all dependent on the effective pressure, a fact that can lead to interesting oscillatory boundary-layer structures. © 2008 The Royal Society.

Bakaninbreen in Svalbard and Trapridge Glacier in Yukon Territory, Canada, are two prominent examples of surging glaciers which are thought to be controlled by their thermal regime. Both glaciers have developed large bulges which have propagated forward as travelling wave fronts, and which are thought to divide relatively stagnant downstream cold-based ice from faster-moving warm-based upstream ice. Additionally, both glaciers are underlain by a wet, metres thick layer of deforming till. We develop a simple model for the cyclic surging behaviour of these glaciers, which interrelates the motion of the ice and till through a description of the subglacial hydrology. We find that oscillations (surges) can occur if the subglacial hydrological transmissivity is sufficiently low and the till layer is sufficiently thin, and we suggest that these oscillations are associated with the development and propagation of a travelling wave front down the glacier. We therefore interpret the travelling wave fronts on both Trapridge Glacier and Bakaninbreen as manifestations of surges. In addition, we find that the violence of the surge in the model is associated with the resistance to ice flow offered by undulations in the bed, and the efficiency with which occasional hydrological events can release water accumulated at the glacier sole.

One challenge in improving our understanding of ice-stream dynamics is to develop models of the spatial and temporal transition from ice-sheet to ice-stream flow. We address this with a new, vertically integrated, higher-order formulation for ice-sheet dynamics that captures the leading-order physics of low aspect ratio, viscous fluid flow, regardless of the amount of slip at the bed. The theory introduces a parameter, λ, which approximates the ratio of the basal stress to the shear stress scale, providing a measure of the relative importance of sliding and internal deformation. Our model is able to simultaneously describe the dynamics of both a slow-moving sheet and rapidly flowing ice streams. To test the formulation, we apply a triple-valued sliding law as the basal boundary condition and obtain numerical solutions that can be compared with previous work. We investigate the sensitivity of flow regimes and shear margin width to parameter variation, deriving a scaling for the latter. We also consider a double-valued sliding law, which enforces a constant, low basal stress beneath the ice stream. Comparisons of the resultant stress fields illustrate the different stress balances that can maintain ice-stream flow.

We have undertaken a study of the complex Lorenz equations x = −σx + σy . y = (r − z)x − ay . z = −bz + 12(x∗y + xy∗) . where x and y are complex and z is real. The complex parameters r and a are defined by r = r1 + ir2; a = 1 − ie and σ and b are real. Behaviour remarkably different from the real Lorenz model occurs. Only the origin is a fixed point except for the special case e + r2 = 0. We have been able to determine analytically two critical values of r1, namely r1c and r1c . The origin is a stable fixed point for 0 r1c, a Hopf bifurcation to a limit cycle occurs. We have an exact analytic solution for this limit cycle which is always stable if σ + 1 then this limit is only stable in the region r1c rlc, a transition to a finite amplitude oscillation about the limit cycle occurs. The nature of this bifurcation is studied in detail by using a multiple time scale analysis to derive the Stuart-Landau amplitude equation from the original equations in a frame rotating with the limit cycle frequency. This latter bifurcation is either a sub- or super-critical Hopf-like bifurcation to a doubly periodic motion, the direction of bifurcation depending on the parameter values. The nature of the bifurcation is complicated by the existence of a zero eigenvalue.

In this paper, we present a model that explains the prepatterning of lymphatic vessel morphology in collagen gels. This model is derived using the theory of two phase rubber material due to Flory and coworkers and it consists of two coupled fourth order partial differential equations describing the evolution of the collagen volume fraction, and the evolution of the proton concentration in a collagen implant; as described in exper- iments of Boardman and Swartz (Circ. Res. 92, 801-808, 2003). Using linear stability analysis, we find that above a critical level of proton concentration, spatial patterns form due to small perturbations in the initially uniform steady state. Using a long wavelength reduction, we can reduce the two coupled partial differential equations to one fourth order equation that is very similar to the Cahn-Hilliard equation; however, it has more com- plex nonlinearities and degeneracies. We present the results of numerical simulations and discuss the biological implications of our model.

We summarize some recently obtained results on real and complex Lorenz equations and discuss their possible significance in relation to real fluid dynamical processes.

The derivation of the quasi-geostrophic potential vorticity equation of mathematical meteorology is usually done using fairly sophisticated techniques of perturbation theory, but stops short of deriving self-consistently the stratification parameter of the mean atmospheric state. In this note we suggest how this should be done within the confines of the theory, and as a consequence we raise the possibility that the atmosphere could become globally unstable, with dramatic consequences. © 2011 Taylor and Francis.

This paper describes the oxidation of iron-rich uranium-bearing rocks by infiltration of ground-water. A reaction-diffusion model is set up to describe the sequence of reactions involving iron oxidation, uranium oxidation and reduction, sulphuric acid production, and dissolution of the host rock that occur. On a geological timescale of millions of years, the reactions occur very fast in very thin reaction fronts. It is shown that the redox front that separates oxidized (orange) rock from reduced (black) rock must actually consist of two separate fronts that move together, at which the two separate processes of uranium oxidation and iron reduction occur, respectively. Between these fronts, a high concentration of uranium is predicted. The mechanics of this process are not specific to uranium-mediated redox reactions, but apply generally and may be used to explain the formation of concentrated ore deposits in extended veins. On the long timescales of relevance, a quasi-static response results, and the probl...

The recent article by Schoof (J. Fluid Mech., vol. 712, 2012, pp. 552–578) provides a technically demanding solution to the problem of determining ice-stream margin evolution. It is important in opening the way to the future theoretical description of how the ice sheets will melt and sea level will rise as the climate warms. But the sophistication of the mathematics should not operate as a mask to an examination of the credibility of the model.

Asymptotic methods for singularly perturbed delay differential equations are in many ways more challenging to implement than for ordinary differential equations. In this paper, four examples of delayed systems which occur in practical models are considered: the delayed recruitment equation, relaxation oscillations in stem cell control, the delayed logistic equation, and density wave oscillations in boilers, the last of these being a problem of concern in engineering two-phase flows. The ways in which asymptotic methods can be used vary from the straightforward to the perverse, and illustrate the general technical difficulties that delay equations provide for the central technique of the applied mathematician. © Springer 2006.

We study a model for channels of magma flow within mantle undergoing decompression melting. Cylindrical conduits in a viscous, porous, compacting matrix are considered, and it is found that the dynamics of the conduit walls are governed by the competition between melting (caused by decompression) and viscous closure (caused by the reduced pressure in the conduit). There are many similarities with the Röthlisberger channels which transport melt water beneath glaciers. Pressure in these mantle conduits is very nearly magmastatic, and ascent velocities on the order of 100 m a-1 are predicted. Flow from the surrounding porous partially molten matrix into the low-pressure channel is considered and can supply a continual source of melt. The accumulation region is on the order of the compaction length, and the residual matrix is reduced to very low melt fractions, typically

Subglacial floods (jökulhlaups) are well documented as occurring beneath present day glaciers and ice caps. In addition, it is known that massive floods have occurred from ice-dammed lakes proximal to the Laurentide ice sheet during the last ice age, and it has been suggested that at least one such flood below the waning ice sheet was responsible for a dramatic cooling event some 8000 years ago. We propose that drainage of lakes from beneath ice sheets will generally occur in a time-periodic fashion, and that such floods can be of severe magnitude. Such hydraulic eruptions are likely to have caused severe climatic disturbances in the past, and may well do so in the future.

In the last few years, the efforts to elucidate the mechanisms underlying the segmentation clock in various vertebrate species have multiplied. Early evidence suggested that oscillations are caused by one of the genes under the Notch signalling pathway (like those of the her or Hes families). Recently, Aulehla et al. [Wnt3a plays a major role in the segmentation clock controlling somitogenesis. Dev. Cell 4, 395–406] discovered that Axin2 (a gene under the Wnt3a signalling pathway) also oscillates in the presomitic mesoderm (PSM) of mice embryos and proposed some mechanisms through which the Notch and Wnt3a pathways may interact. They further suggested that a decreasing concentration of Wnt3a along the PSM may be the gradient the segmentation clock interacts with to form somites. These results were reviewed by Rida et al. [A notch feeling of somite segmentation and beyond. Dev. Biol. 265, 2–22], who introduced a complex clockwork comprising genes Hes1, Lfng (under the Notch pathway), and Axin2, as well as their multiple interactions. In the present work we develop a mathematical model based on the Rida et al. review and use it to tackle some of the questions raided by the Aulehla et al. paper: can the Axin2 feedback loop constitute a clock? Could a decreasing Wnt3a signaling constitute the wavefront, where phase is recorded and the spatial pattern laid down? What is the master oscillator? r 2007 Elsevier Ltd. All rights reserved.

A two-fluid model for annular two-phase flow is presented, which incorporates realistic phase interaction terms corresponding to turbulence in the gas phase, interphase pressure differences, and profile effects (nonuniform velocity profiles); this model avoids the conundrum of illposedness associated with the simplest averaged model (D. A. Drew, Continuum modelling of two-phase flows, in Theory of Dispersed Flow, R. E. Meyer, ed., Academic Press, New York, 1983). In this paper it is shown that an appropriately scaled form of this model is capable of significant (asymptotic) simplification and in its reduced form is able to predict the phenomena of flooding and flow reversal in annular flow, both qualitatively and quantitatively.

We develop a predictive model which has the ability to explain a postulated style of episodic plate tectonics on Venus, through the periodic occurrence of lithospheric subduction events. Present-day incipient subduction zones are associated with the existence of arcuate trenches on the Venusian lithosphere. These trenches resemble terrestrial subduction zones, and occur at the rim of coronae, uplift features thought to be due to deep-mantle convective plumes. The model we adopt represents the lithosphere as the thermal boundary layer which lies above a convective plume. We assume a temperature-dependent nonlinear viscoelastic rheology, and we assume a stress-based criterion for plastic yield. In developing this latter criterion, we are led to a re-interpretation of the strength envelope which is commonly used in analysing lithospheric stress, and we propose that the plastic yield strength has meaning (and is finite) below the lithosphere, using behaviour in the Earth as our 'laboratory' justification for this view. An inferred yield stress on the Earth is ca. 300 bar (30 MPa). Our model then shows that a thickening lithosphere becomes progressively more fluid as the stresses induced by the buoyant convective plume become large. Failure occurs when the effective lithosphere viscosity becomes equal to that of the underlying mantle. We show that reasonable expected values of yield stress in the range 100-200 bar (10-20 MPa) for Venusian mantle rocks are consistent within the framework of the model with radii of coronal trenches in the range 100-1200 km, and with the approximate time (200-800 Myr) which they may take to develop.

Several hematological diseases are characterised by oscillations of various blood cell populations. Two of these are a variant of chronic myelogenous leukemia (CML) and cyclical neutropenia (CN). These oscillations typically have long periods ranging from 20 to 60 days, despite the fact that the stem cell cycling time is thought to be of the order of 2-3 days. Clinical data from humans and laboratory data from the grey collie animal model of CN is suggestive of the idea that these long period oscillations may also contain higher frequency spiky oscillations. We show how such oscillations can be understood in the context of slow periodic stem cell oscillations, by analysing a two component differential-delay equation model of stem cell and neutrophil populations.

The continuous model of Anderson et al. (1981), Nature 289 , 765–771, is successful in describing certain characteristics of rabies epizootics, in particular, the secondary recurrences which follow the initial outbreak; however, it also predicts the occurrence of exponentially small minima in the infected population, which would realistically imply extinction of the virus. Here we show that inclusion of a more realistic distribution of incubation times in the model can explain why extinction will not occur, and we give explicit parametric estimates for the minimum infected fox density which will occur in the model, in terms of the incubation time distribution.

We propose a two-dimensional model of a valley glacier in order to reconsider the question of whether thermal runaway could be a viable mechanism for the onset of creep instability in surging glaciers. We do this by providing an approximate solution for the temperature field based on the idea that shear is concentrated at the glacier bed. With this assumption, we show that a closed-form evolution equation for the glacier profile exists. While this is well known for isoviscous flows, it has not been previously derived for variable viscosity flows. During the process of deriving this equation, we show that thermal runaway does not occur.We provide numerical solutions of the model, and are led to infer that enhanced basal heating owing to refreezing of surface meltwater is an essential constituent in raising the bed temperature to the melting point. © 2009 The Royal Society.

We provide and analyse a model for the growth of bacterial biofilms based on the concept of an extracellular polymeric substance as a polymer solution, whose viscoelastic rheology is described by the classical Flory-Huggins theory. We show that one-dimensional solutions exist, which take the form at large times of travelling waves, and we characterize their form and speed in terms of the describing parameters of the problem. Numerical solutions of the time-dependent problem converge to the travelling wave solutions. © 2010 The Royal Society.

It is proposed that the formation of the subglacial bedforms known as drumlins occurs through an instability associated with the flow of ice over a wet deformable till. We pose a mathematical model that describes this instability, and we solve a simplified version of the model numerically in order to establish the form of finite-amplitude two-dimensional waveforms. A feature of the solutions is that cavities frequently form downstream of the bedforms; we allow the model to cater for this possibility and we provide an efficient numerical method to solve the resulting free boundary problem.

We have measured grain size distributions of the results of laboratory decompression explosions of volcanic rock. The resulting distributions can be approximately represented by gamma distributions of weight per cent as a function of ϕ = − log 2 ⁡ d , where d is the grain size in millimetres measured by sieving, with a superimposed long tail associated with the production of fines. We provide a description of the observations based on sequential fragmentation theory, which we develop for the particular case of ‘self-similar’ fragmentation kernels, and we show that the corresponding evolution equation for the distribution can be explicitly solved, yielding the long-time lognormal distribution associated with Kolmogorov's fragmentation theory. Particular features of the experimental data, notably time evolution, advection, truncation and fines production, are described and predicted within the constraints of a generalized, ‘reductive’ fragmentation model, and it is shown that the gamma distribution of coarse particles is a natural consequence of an assumed uniform fragmentation kernel. We further show that an explicit model for fines production during fracturing can lead to a second gamma distribution, and that the sum of the two provides a good fit to the observed data.

The aim of this paper is to gain an understanding of the factors that influence the degradation of bags of milk powder when they are shipped around the world. We discuss two approaches, mathematical modelling and experimental design. We model the spatial and temporal variation of humidity, oxygen and temperature and its effect on the shelf life of the bags of milk powder. We show that the diffusion of these quantities is so rapid that they can be assumed to be uniform throughout the bags. This justifies the simplification of the model to the form of a system of ordinary differential equations involving time. The experimental design discussed in this paper involves a statistical approach to measuring the different factors that influence the degradation of milk powder over time.

Analytical approximations have generated many insights into the dynamics of epidemics, but there is only one well-known approximation which describes the dynamics of the whole epidemic. In addition, most of the well-known approximations for different aspects of the dynamics are for the classic susceptible–infected–recovered model, in which the infectious period is exponentially distributed. Whilst this assumption is useful, it is somewhat unrealistic. Equally reasonable assumptions are that the infectious period is finite and fixed or that there is a distribution of infectious periods centred round a nonzero mean. We investigate the effect of these different assumptions on the dynamics of the epidemic by deriving approximations to the whole epidemic curve. We show how the well-known sech-squared approximation for the infective population in ‘weak’ epidemics (where the basic reproduction rate $$R_0\approx 1$$ ) can be extended to the case of an arbitrary distribution of infectious periods having finite second moment, including as examples fixed and gamma-distributed infectious periods. Further, we show how to approximate the time course of a ‘strong’ epidemic, where $$R_0\gg 1$$ , demonstrating the importance of estimating the infectious period distribution early in an epidemic.

Plate tectonics on the Earth is a surface manifestation of convection within the Earth’s mantle, a subject which is as yet improperly understood, and it has motivated the study of various forms of buoyancy-driven thermal convection. The early success of the high Rayleigh number constant viscosity theory was later tempered by the absence of plate motion when the viscosity is more realistically strongly temperature dependent, and the process of subduction represents a continuing principal conundrum in the application of convection theory to the Earth. A similar problem appears to arise if the equally strong pressure dependence of viscosity is considered, since the classical isothermal core convection theory would then imply a strongly variable viscosity in the convective core, which is inconsistent with results from post-glacial rebound studies. In this paper we address the problem of determining the asymptotic structure of high Rayleigh number convection when the viscosity is strongly temperature and pressure dependent, i. e., thermobaroviscous. By a method akin to lid-stripping, we are able to extend numerical computations to extremely high viscosity contrasts, and we show that the convective cells take the form of narrow, vertically-oriented fingers. We are then able to determine the asymptotic structure of the solution, and it agrees well with the numerical results. Beneath a stagnant lid, there is a vigorous convection in the upper part of the cell, and a more sluggish, higher viscosity flow in the lower part of the cell. We then offer some comments on the possible meaning and interpretation of these results for planetary mantle convection.