Superfast non-linear diffusion: capillary transport in particulate porous media

The migration of liquids in porous media, such as sand, has been commonly considered at high saturation levels with liquid pathways at pore dimensions. In this letter we reveal a low saturation regime observed in our experiments with droplets of extremely low volatility liquids deposited on sand. In this regime the liquid is mostly found within the grain surface roughness and in the capillary bridges formed at the contacts between the grains. The bridges act as variable-volume reservoirs and the flow is driven by the capillary pressure arising at the wetting front according to the roughness length scales. We propose that this migration (spreading) is the result of interplay between the bridge volume adjustment to this pressure distribution and viscous losses of a creeping flow within the roughness. The net macroscopic result is a special case of non-linear diffusion described by a superfast diffusion equation (SFDE) for saturation with distinctive mathematical character. We obtain solutions to a moving boundary problem defined by SFDE that robustly convey a time power law of spreading as seen in our experiments.

The unsteady flow of a weakly compressible fluid in a thin porous layer II: three-dimensional theory

We consider the problem of determining the pressure and velocity fields for a weakly compressible fluid flowing in a three-dimensional layer, composed of an inhomogeneous, anisotropic porous medium, with vertical side walls and variable upper and lower boundaries, in the presence of vertical wells injecting and/or extracting fluid. Numerical solution of this three-dimensional evolution problem may be expensive, particularly in the case that the depth scale of the layer h is small compared to the horizontal length scale l, a situation which occurs frequently in the application to oil and gas reservoir recovery and which leads to significant stiffness in the numerical problem. Under the assumption that $\epsilon\propto h/l\ll 1$, we show that, to leading order in $\epsilon$, the pressure field varies only in the horizontal directions away from the wells (the outer region). We construct asymptotic expansions in $\epsilon$ in both the inner (near the wells) and outer regions and use the asymptotic matching principle to derive expressions for all significant process quantities. The only computations required are for the solution of non-stiff linear, elliptic, two-dimensional boundary-value, and eigenvalue problems. This approach, via the method of matched asymptotic expansions, takes advantage of the small aspect ratio of the layer, $\epsilon$, at precisely the stage where full numerical computations become stiff, and also reveals the detailed structure of the dynamics of the flow, both in the neighbourhood of wells and away from wells.

Effect of earthworm Eisenia fetida and wetland plants on nitrification and denitrification potentials in vertical flow constructed wetland.

The response of nitrification potentials, denitrification potentials, and N removal efficiency to the introduction of earthworms and wetland plants in a vertical flow constructed wetland system was investigated. Addition of earthworms increased nitrification and denitrification potentials of substrate in non-vegetated constructed wetland by 236% and 8%, respectively; it increased nitrification and denitrification potentials in rhizosphere in vegetated constructed wetland (Phragmites austrail, Typha augustifolia and Canna indica), 105% and 5%, 187% and 12%, and 268% and 15% respectively. Denitrification potentials in rhizosphere of three wetland plants were not significantly different, but nitrification potentials in rhizosphere followed the order of C. indica > T. augustifolia > P. australis when addition of earthworms into constructed wetland. Addition of earthworms to the vegetated constructed significantly increased the total number of bacteria and fungi of substrates (P < 0.05). The total number of bacteria was significantly correlated with nitrification potentials (r = 913, P < 0.01) and denitrification potentials (r = 840, P < 0.01), respectively. The N concentration of stems and leaves of C. indica were significantly higher in the constructed wetland with earthworms (P < 0.05). Earthworms had greater impact on nitrification potentials than denitrification potentials. The removal efficiency of N was improved via stimulated nitrification potentials by earthworms and higher N uptake by wetland plants.

Nonlinear stability of multilayer quasi-geostrophic flow

New nonlinear stability theorems are derived for disturbances to steady basic flows in the context of the multilayer quasi-geostrophic equations. These theorems are analogues of Arnol’d's second stability theorem, the latter applying to the two-dimensional Euler equations. Explicit upper bounds are obtained on both the disturbance energy and disturbance potential enstrophy in terms of the initial disturbance fields. An important feature of the present analysis is that the disturbances are allowed to have non-zero circulation. While Arnol’d's stability method relies on the energy–Casimir invariant being sign-definite, the new criteria can be applied to cases where it is sign-indefinite because of the disturbance circulations. A version of Andrews’ theorem is established for this problem, and uniform potential vorticity flow is shown to be nonlinearly stable. The special case of two-layer flow is treated in detail, with particular attention paid to the Phillips model of baroclinic instability. It is found that the short-wave portion of the marginal stability curve found in linear theory is precisely captured by the new nonlinear stability criteria.

On Arnol'd's second nonlinear stability theorem for two-dimensional quasi-geostrophic flow

Arnol'd's second hydrodynamical stability theorem, proven originally for the two-dimensional Euler equations, can establish nonlinear stability of steady flows that are maxima of a suitably chosen energy-Casimir invariant. The usual derivations of this theorem require an assumption of zero disturbance circulation. In the present work an analogue of Arnol'd's second theorem is developed in the more general case of two-dimensional quasi-geostrophic flow, with the important feature that the disturbances are allowed to have non-zero circulation. New nonlinear stability criteria are derived, and explicit bounds are obtained on both the disturbance energy and potential enstrophy which are expressed in terms of the initial disturbance fields. While Arnol'd's stability method relies on the second variation of the energy-Casimir invariant being sign-definite, the new criteria can be applied to cases where the second variation is sign-indefinite because of the disturbance circulations. A version of Andrews' theorem is also established for this problem.

An exact local conservation theorem for finite-amplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and on Arnol'd's stability theorems

Disturbances of arbitrary amplitude are superposed on a basic flow which is assumed to be steady and either (a) two-dimensional, homogeneous, and incompressible (rotating or non-rotating) or (b) stably stratified and quasi-geostrophic. Flow over shallow topography is allowed in either case. The basic flow, as well as the disturbance, is assumed to be subject neither to external forcing nor to dissipative processes like viscosity. An exact, local ‘wave-activity conservation theorem’ is derived in which the density A and flux F are second-order ‘wave properties’ or ‘disturbance properties’, meaning that they are O(a2) in magnitude as disturbance amplitude a [rightward arrow] 0, and that they are evaluable correct to O(a2) from linear theory, to O(a3) from second-order theory, and so on to higher orders in a. For a disturbance in the form of a single, slowly varying, non-stationary Rossby wavetrain, $\overline{F}/\overline{A}$ reduces approximately to the Rossby-wave group velocity, where (${}^{-}$) is an appropriate averaging operator. F and A have the formal appearance of Eulerian quantities, but generally involve a multivalued function the correct branch of which requires a certain amount of Lagrangian information for its determination. It is shown that, in a certain sense, the construction of conservable, quasi-Eulerian wave properties like A is unique and that the multivaluedness is inescapable in general. The connection with the concepts of pseudoenergy (quasi-energy), pseudomomentum (quasi-momentum), and ‘Eliassen-Palm wave activity’ is noted. The relationship of this and similar conservation theorems to dynamical fundamentals and to Arnol'd's nonlinear stability theorems is discussed in the light of recent advances in Hamiltonian dynamics. These show where such conservation theorems come from and how to construct them in other cases. An elementary proof of the Hamiltonian structure of two-dimensional Eulerian vortex dynamics is put on record, with explicit attention to the boundary conditions. The connection between Arnol'd's second stability theorem and the suppression of shear and self-tuning resonant instabilities by boundary constraints is discussed, and a finite-amplitude counterpart to Rayleigh's inflection-point theorem noted

On the generalized eigenvalue problem for the Rossby wave vertical velocity in the presence of mean flow and topography

In a series of papers, Killworth and Blundell have proposed to study the effects of a background mean flow and topography on Rossby wave propagation by means of a generalized eigenvalue problem formulated in terms of the vertical velocity, obtained from a linearization of the primitive equations of motion. However, it has been known for a number of years that this eigenvalue problem contains an error, which Killworth was prevented from correcting himself by his unfortunate passing and whose correction is therefore taken up in this note. Here, the author shows in the context of quasigeostrophic (QG) theory that the error can ulti- mately be traced to the fact that the eigenvalue problem for the vertical velocity is fundamentally a non- linear one (the eigenvalue appears both in the numerator and denominator), unlike that for the pressure. The reason that this nonlinear term is lacking in the Killworth and Blundell theory comes from neglecting the depth dependence of a depth-dependent term. This nonlinear term is shown on idealized examples to alter significantly the Rossby wave dispersion relation in the high-wavenumber regime but is otherwise irrelevant in the long-wave limit, in which case the eigenvalue problems for the vertical velocity and pressure are both linear. In the general dispersive case, however, one should first solve the generalized eigenvalue problem for the pressure vertical structure and, if needed, diagnose the vertical velocity vertical structure from the latter.

Flow structure and near-field dispersion in arrays of building-like obstacles

Dispersion in the near-field region of localised releases in urban areas is difficult to predict because of the strong influence of individual buildings. Effects include upstream dispersion, trapping of material into building wakes and enhanced concentration fluctuations. As a result, concentration patterns are highly variable in time and mean profiles in the near field are strongly non-Gaussian. These aspects of near-field dispersion are documented by analysing data from direct numerical simulations in arrays of building-like obstacles and are related to the underlying flow structure. The mean flow structure around the buildings is found to exert a strong influence over the dispersion of material in the near field. Diverging streamlines around buildings enhance lateral dispersion. Entrainment of material into building wakes in the very near field gives rise to secondary sources, which then affect the subsequent dispersion pattern. High levels of concentration fluctuations are also found in this very near field; the fluctuation intensity is of order 2 to 5.

A study of anticipatory non-autonomous systems

Rhythms are manifested ubiquitously in dynamical biological processes. These fundamental processes which are necessary for the survival of living organisms include metabolism, breathing, heart beat, and, above all, the circadian rhythm coupled to the diurnal cycle. Thus, in mathematical biology, biological processes are often represented as linear or nonlinear oscillators. In the framework of nonlinear and dissipative systems (ie. the flow of energy, substances, or sensory information), they generate stable internal oscillations as a response to environmental input and, in turn, utilise such output as a means of coupling with the environment.

Incoherent scatter radar observations of non-Maxwellian ion velocity distributions in the auroral F-region

Observations by the EISCAT experiments “POLAR” and Common Programme CP-3 reveal non-Maxwellian ion velocity distributions in the auroral F-region ionosphere. Analysis of data from three periods is presented. During the first period, convection velocities are large (≈2 km s-1) and constant over part of a CP-3 latitude scan; the second period is one of POLAR data containing a short-lived (<1 min.) burst of rapid (>1.5 km s-1) flow. We concentrate on these two periods as they allow the study of a great many features of the ion-neutral interactions which drive the plasma non-thermal and provide the best available experimental test for models of the 3-dimensional ion velocity distribution function. The third period is included to illustrate the fact that non-thermal plasma frequently exists in the auroral ionosphere: the data, also from the POLAR experiment, cover a three-hour period of typical auroral zone flow and analysis reveals that the ion distribution varies from Maxwellian to the threshold of a toroidal form.

Scattered power from non-thermal, F-region plasma observed by EISCAT—evidence for coherent echoes?

Three rapid, poleward bursts of plasma flow, observed by the U.K.-POLAR EISCAT experiment, are studied in detail. In all three cases the large ion velocities (> 1 kms−1) are shown to drive the ion velocity distribution into a non-Maxwellian form, identified by the characteristic shape of the observed spectra and the fact that analysis of the spectra with the assumption of a Maxwellian distribution leads to excessive rises in apparent ion temperature, and an anticorrelation of apparent electron and ion temperatures. For all three periods the total scattered power is shown to rise with apparent ion temperature by up to 6 dB more than is expected for an isotropic Maxwellian plasma of constant density and by an even larger factor than that expected for non-thermal plasma. The anomalous increases in power are only observed at the lower altitudes (< 300 km). At greater altitudes the rise in power is roughly consistent with that simulated numerically for homogeneous, anisotropic, non-Maxwellian plasma of constant density, viewed using the U.K.-POLAR aspect angle. The spectra at times of anomalously high power are found to be asymmetric, showing an enhancement near the downward Doppler-shifted ion-acoustic frequency. Although it is not possible to eliminate completely rapid plasma density fluctuations as a cause of these power increases, such effects cannot explain the observed spectra and the correlation of power and apparent ion temperature without an unlikely set of coincidences. The observations are made along a beam direction which is as much as 16.5° from orthogonality with the geomagnetic field. Nevertheless, some form of coherent-like echo contamination of the incoherent scatter spectrum is the most satisfactory explanation of these data.

Model predictions of the occurrence of non-Maxwellian plasmas, and analysis of their effects on EISCAT data

The recent identification of non-thermal plasmas using EISCAT data has been made possible by their occurrence during large, short-lived flow bursts. For steady, yet rapid, ion convection the only available signature is the shape of the spectrum, which is unreliable because it is open to distortion by noise and sampling uncertainty and can be mimicked by other phenomena. Nevertheless, spectral shape does give an indication of the presence of non-thermal plasma, and the characteristic shape has been observed for long periods (of the order of an hour or more) in some experiments. To evaluate this type of event properly one needs to compare it to what would be expected theoretically. Predictions have been made using the coupled thermosphere-ionosphere model developed at University College London and the University of Sheffield to show where and when non-Maxwellian plasmas would be expected in the auroral zone. Geometrical and other factors then govern whether these are detectable by radar. The results are applicable to any incoherent scatter radar in this area, but the work presented here concentrates on predictions with regard to experiments on the EISCAT facility.

Non-Maxwellian ion velocity distributions observed using EISCAT

Recent observations from the EISCAT incoherent scatter radar have revealed bursts of poleward ion flow in the dayside auroral ionosphere which are consistent with the ionospheric signature of flux transfer events at the magnetopause. These bursts frequently contain ion drifts which exceed the neutral thermal speed and, because the neutral thermospheric wind is incapable of responding sufficiently rapidly, toroidal, non-Maxwellian ion velocity distributions are expected. The EISCAT observations are made with high time resolution (15 seconds) and at a large angle to the geomagnetic field (73.5°), allowing the non-Maxwellian nature of the distribution to be observed remotely for the first time. The observed features are also strongly suggestive of a toroidal distribution: characteristic spectral shape, increased scattered power (both consistent with reduced Landau damping and enhanced electric field fluctuations) and excessively high line-of-sight ion temperatures deduced if a Maxwellian distribution is assumed. These remote sensing observations allow the evolution of the distributions to be observed. They are found to be non-Maxwellian whenever the ion drift exceeds the neutral thermal speed, indicating that such distributions can exist over the time scale of the flow burst events (several minutes).

Characterising phosphorus and nitrate inputs to a rural river using high-frequency concentration-flow relationships

The total reactive phosphorus (TRP) and nitrate concentrations of the River Enborne, southern England, were monitored at hourly interval between January 2010 and December 2011. The relationships between these high-frequency nutrient concentration signals and flow were used to infer changes in nutrient source and dynamics through the annual cycle and each individual storm event, by studying hysteresis patterns. TRP concentrations exhibited strong dilution patterns with increasing flow, and predominantly clockwise hysteresis through storm events. Despite the Enborne catchment being relatively rural for southern England, TRP inputs were dominated by constant, non-rain-related inputs from sewage treatment works (STW) for the majority of the year, producing the highest phosphorus concentrations through the spring–summer growing season. At higher river flows, the majority of the TRP load was derived from within-channel remobilisation of phosphorus from the bed sediment, much of which was also derived from STW inputs. Therefore, future phosphorus mitigation measures should focus on STW improvements. Agricultural diffuse TRP inputs were only evident during storms in the May of each year, probably relating to manure application to land. The nitrate concentration–flow relationship produced a series of dilution curves, indicating major inputs from groundwater and to a lesser extent STW. Significant diffuse agricultural inputs with anticlockwise hysteresis trajectories were observed during the first major storms of the winter period. The simultaneous investigation of high-frequency time series data, concentration–flow relationships and hysteresis behaviour through multiple storms for both phosphorus and nitrate offers a simple and innovative approach for providing new insights into nutrient sources and dynamics.

Endothelial function and insulin sensitivity during acute non-esterified fatty acid elevation: effects of fat composition and gender

Background and Aims: We have reported that adverse effects on flow-mediated dilation of an acute elevation of non-esterified fatty acids rich in saturated fat (SFA) are reversed following addition of long-chain (LC) n-3 polyunsaturated fatty acids (PUFA), and hypothesised that these effects may be mediated through alterations in insulin signalling pathways. In a subgroup, we explored the effects of raised NEFA enriched with SFA, with or without LC n-3 PUFA, on whole body insulin sensitivity (SI) and responsiveness of the endothelium to insulin infusion. Methods and Results: Thirty adults (mean age 27.8 y, BMI 23.2 kg/m2) consumed oral fat loads on separate occasions with continuous heparin infusion to elevate NEFA between 60-390 min. For the final 150 min, a hyperinsulinaemic-euglycaemic clamp was performed, whilst FMD and circulating markers of endothelial function were measured at baseline, pre-clamp (240 min) and post-clamp (390 min). NEFA elevation during the SFA-rich drinks was associated with impaired FMD (P=0.027) whilst SFA+LC n-3 PUFA improved FMD at 240 min (P=0.003). In males, insulin infusion attenuated the increase in FMD with SFA+LC n-3 PUFA (P=0.049), with SI 10% greater with SFA+LC n-3 PUFA than SFA (P=0.041). Conclusion: This study provides evidence that NEFA composition during acute elevation influences both FMD and SI, with some indication of a difference by gender. However our findings are not consistent with the hypothesis that the effects of fatty acids on endothelial function and SI operate through a common pathway. Trial registered at clinicaltrials.gov, NCT01351324.