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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.

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.

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.

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.

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.

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 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.

This paper reviews the seminal contributions (and tussles) of Weertman and Lliboutry to the theory of glacier sliding in the 1950s and 1960s, and summarizes later developments up to the present day.

The growth of trees and other plants occurs through the interactive combination of photosynthesis and carbon (and other nutrient) assimilation. Photosynthesis enables the production of carbohydrate that can then be used in growing foliage, whereby photosynthesis is enabled. We construct a mathematical model of carbon uptake and storage, which allows the prediction of the growth dynamics of trees. We find that the simplest model allows uncontrolled foliage production through the positive feedback outlined above, but that leaf shading provides an automatic saturation to carbon assimilation, and hence to foliage production. The model explains the necessity for finite leaf area production at outbreak, and it explains why foliage density reaches a constant value during a growing season, while also non-leaf tissue also continues to grow. It also explains why trees will die when their carbon stores are depleted below a certain threshold, because the cost of foliage growth and maintenance exceeds the dynamic supply of carbon by photosynthesis.

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.

We summarize the present form of the instability theory for drumlin formation, which describes the coupled subglacial flow of ice, water and sediment. This model has evolved over the last 20 years, and is now at the point where it can predict instabilities corresponding to ribbed moraine, drumlins and mega-scale glacial lineations, but efforts to provide numerical solutions of the model have been limited. The present summary adds some slight nuances to previously published versions of the theory, notably concerning the constitutive description of the subglacial water film and its flow. A new numerical method is devised to solve the model, and we show that it can be solved for realistic values of most of the parameters, with the exception of that corresponding to the water film thickness. We show that evolved bedforms can be three-dimensional and of the correct sizes, and we explore to some extent the variation of the solutions with the model’s parameters.

Of several problems associa ted wit h theoretica l expla nation s of the jokul­ hla ups which emerge from the outlet gla cie r Skeioarar jokull of the ice ca p Vatna jokull in sou theast Iceland, the mechani sm of flood init iation is one that has hi therto defied expl anat ion . We provide such an expl ana tion based on a car efu l analysis of the cla ssica l Nye-Ro thlisbe rger model; near the subglacial lake Grimsvotn, the hyd raulic potenti al gradie nt is towards the lake, and there is theref ore a catchme nt boun dary under the ice, whose location dep ends on the subglacial mel twater drai nage charac terist ics. As the con ­ dit ions for a flood approa ch, we show that the wat er divi de migra tes towards the lake, while at the sa me time the lake pre ssure in creases. When the hydra ulic potential gradient to­ wards the lake is low and the refillin g rate is sl ow, the se al will "break" whe n the cat chment boun dary reaches the lake, while the la ke level is st ill below flotati on pressu re, whereas if refillin g is rapid, flo tation can be achie ved bef ore a flood is initia ted. This theory can thus explain why the seal is normally broken when the lake le vel at Grimsvotn is still some 60 m below flotation level. In addit ion , we are abl e to explain why thej okulhla up following the 1996 eruption did not occur until flo tation level was achie ved, and we show how the cyeli­ cit y and magnitude ofjoku lhla ups can be explained wit hin this theory.

A simplified model of a two-dimensional ice sheet is described. It includes basal ice sliding dependent, on the basal water pressure, which itself is described by a simple theory of basal drainage. We show that this simple but sophisticated model predicts surges of the ice mass in realistic circumstances, and we describe these surges by solving the problem numerically. We also are able to describe some parts of the surge analytically. The numerical solution of the model is a delicate matter, and highlights pitfalls to be avoided if more complicated models are to be solved successfully.

Despite their importance in understanding glaciological processes and constraining large-scale flow patterns in palaeo-glaciology, there is little consensus as to how drumlins are formed. Attempts to solve the 'drumlin problem' often fail to address how they are created from an initially flat surface in the absence of obvious cores or obstacles. This is a key strength of the instability theory, which has been described in a suite of physically-based mathematical models and proposes that the coupled flow of ice and till causes spontaneous formation of relief in the till surface. Encouragingly, model predictions of bedform height and length are consistent with observations and, furthermore, the theory has been applied to a range of subglacial bedforms and not just drumlins. However, it has yet to confront the myriad observations relating to the composition and internal structure of drumlins and this could be seen as a major deficiency. This paper is a first attempt to assess whether the instability theory is compatible with the incredible diversity of sediments and structures found within drumlins. We summarise the underlying principles of the theory and then describe and attempt to explain the main types of drumlin composition (e.g. bedrock, till, glaciofluvial sediments, and combinations thereof). Contrary to a view which suggests that the presence of some sedimentary sequences (e.g. horizontally stratified cores) is inconsistent with the theory, we suggest that one would actually expect a diverse range of constituents depending on the inheritance of sediments that pre-date drumlin formation, the duration and variability of ice flow, and the balance between erosion and deposition (till continuity) at the ice-bed interface. We conclude that the instability theory is compatible with (and potentially strengthened by) what is known about drumlin composition and, as such, offers the most complete and promising solution to the drumlin problem to date. © 2012 Elsevier Ltd.

We extend the one-dimensional polymer solution theory of bacterial biofilm growth described by Winstanley et al . (2011 Proc. R. Soc. A 467 , 1449–1467 ( doi:10.1098/rspa.2010.0327 )) to deal with the problem of the growth of a patch of biofilm in more than one lateral dimension. The extension is non-trivial, as it requires consideration of the rheology of the polymer phase. We use a novel asymptotic technique to reduce the model to a free-boundary problem governed by the equations of Stokes flow with non-standard boundary conditions. We then consider the stability of laterally uniform biofilm growth, and show that the model predicts spatial instability; this is confirmed by a direct numerical solution of the governing equations. The instability results in cusp formation at the biofilm surface and provides an explanation for the common observation of patterned biofilm architectures.

We develop theoretically a description of a possible subglacial drainage mechanism for glaciers and ice sheets moving over saturated, deformable till. The model is based on the plausible assumption that flow of water in a thin film at the ice-till interface is unstable to the formation of a channelized drainage system, and is restricted to the case in which meltwater cannot escape through the till to an underlying aquifer. In describing the physics of such channelized drainage, we have generalized and extended Röthlisberger’s model of channels cut into basal ice to include “canals” cut into the till, paying particular attention to the role of sediment properties and the mechanics of sediment transport. We show that sediment-floored Röthlisberger (R) channels can exist for high effective pressures, and wide, shallow, ice-roofed canals cut into the till for low effective pressures. Canals should form a distributed, non-arborescent system, unlike R channels. For steep slopes typical of alpine glaciers, both drainage systems can exist, but with the water pressure lower in the R channels than in the canals; the canal drainage should therefore be unstable in the presence of channels. For small slopes typical of ice sheets, only canals can exist and we therefore predict that, if channelized meltwater flow occurs under ice sheets moving over deformable till, it takes the form of shallow, distributed canals at low effective pressure, similar to that measured at Ice Stream B in West Antarctica. Geologic evidence derived from land forms and deposits left by the Pleistocene ice sheets in North America and Europe is also consistent with predictions of the model.

Purpose The polishing of cut lead glass crystal is effected through the dowsing of the glass in a mixture of two separate acids, which between them etch the surface and as a result cause it to be become smooth. In order to characterise the resultant polishing the rate of surface etching must be known, but when this involves multicomponent surface reactions it becomes unclear what this rate actually is. Methods We develop a differential equation based discrete model to determine the effective etching rate by means of an atomic scale model of the etching process. Results We calculate the etching rate numerically and provide an approximate asymptotic estimate. Conclusions The natural extension of this work would be to develop a continuum advection-diffusion model.

The deformation of subglacial till is instrumental in causing certain glacier surges, the motion of rapid ice streams in ice sheets, and ice-sheet surges which are associated with Heinrich events, and consequent rapid climatic shifts, during the last ice age. It may also be the means whereby drumlins are formed, and these in turn may act as a brake on large-scale ice flow. It is therefore important in building models to understand the rate at which till deforms, and how this controls the basal ice velocity. In recent years, two paradigms have emerged. On the one hand, theoreticians have tended to use a viscous rheology, though this lacks quantitative support. On the other hand, field and laboratory studies suggest that till behaves plastically. In this paper I will examine some of the dynamic consequences of this latter assumption, and show how the dichotomy between viscous and plastic may be less clear-cut than previously thought.

Polysilicon fuses are one time programmable memory elements which allow the calibration of integrated circuits at wafer and package level. We present a zero dimensional lumped parameter model of the programming of fuses made from a combination of tungsten silicide and polycrystalline silicon. The components of the model are an electrical model, a thermal model and a flow model. The electrical model describes the temperature and geometry dependent resistance of the fuse. The thermal model describes the heating and melting of the fuse and its surroundings. The flow model describes the disconnection of the fuse by electromigration driven flow of silica. The model generates quantitatively accurate results and reproduces trends with applied voltage and fuse size.

We here describe the process by which a complex model set of equations and boundary conditions may be rationally reduced to a simpler and more manageable set by the. processes of non-dimensionalization and asymptotic approximation. Such a reduced model (derived elsewhere) is then presented for an incompressible, two-dimensional ice flow. It consists of two coupled equations for the stream function and enthalpy variable, together with a complex set of boundary conditions.The important dimensionless parameters which arise are given, and various limiting values of these are commented on. Nye’s (1960) equation for kinematic waves may be reproduced, and a non-linear analysis of this reveals that disturbances reach the glacier snout in finite time, and are uniformly bounded there: in the particular case considered here, one can also show that the temperature field is stable.It is shown that the effect of introducing a (realistic) sliding law which is continuously dependent on the temperature has a major effect on the bedrock temperature profile.Lastly we consider seasonal waves using a kinematic wave equation based on a plausible form of the sliding law when cavitation is present. The main observed features are qualitatively reproduced.

We present a model for the determination of a sliding law in the presence of subglacial cavitation. This law determines the basal stress at a clean ice‒bedrock interface in terms of the velocity and effective pressure. The method is based on an exact solution of the Nye—Kamb (linearly viscous) sliding problem with cavities, and uses ideas of Lliboutry (1979) to construct, via renormalization methods, an approximate law for general bedrock form. We show that, for a bedrock whose spectrum has a power‒law behaviour, one obtains a sliding law which gives the basal shear stress proportional to a power of the velocity, and to a power of the effective pressure.The effect of subglacial cavitation on the drainage system is examined, using recent ideas of Kamb. For sufficiently high velocities, drainage through a Röthlisberger tunnel system is unstable, and drainage takes place through the linked system of cavities. This leads to a reduction of the effective pressure, and by taking account of this, one can rewrite the sliding law in terms of stress and velocity only.This sliding law can be multi‒valued, and it is suggested that this underlies the dynamic phenomenon of surges.

Recent observations of glacier sliding at temperatures below the melting point are discussed. It is pointed out that these observations can be simply explained by including solid friction in the sliding law. Furthermore, we re-emphasize the point that such sub-temperate sliding has an important effect on the basal boundary conditions which should be applied in model studies of ice sheets and glaciers.