Boundary integral equations on unbounded rough surfaces: Fredholmness and the finite section method

We consider a class of boundary integral equations that arise in the study of strongly elliptic BVPs in unbounded domains of the form $D = \{(x, z)\in \mathbb{R}^{n+1} : x\in \mathbb{R}^n, z > f(x)\}$ where $f : \mathbb{R}^n \to\mathbb{R}$ is a sufficiently smooth bounded and continuous function. A number of specific problems of this type, for example acoustic scattering problems, problems involving elastic waves, and problems in potential theory, have been reformulated as second kind integral equations $u+Ku = v$ in the space $BC$ of bounded, continuous functions. Having recourse to the so-called limit operator method, we address two questions for the operator $A = I + K$ under consideration, with an emphasis on the function space setting $BC$. Firstly, under which conditions is $A$ a Fredholm operator, and, secondly, when is the finite section method applicable to $A$?

On the energetics of stratified turbulent mixing, irreversible thermodynamics, Boussinesq models and the ocean heat engine controversy

In this paper, the available potential energy (APE) framework of Winters et al. (J. Fluid Mech., vol. 289, 1995, p. 115) is extended to the fully compressible Navier– Stokes equations, with the aims of clarifying (i) the nature of the energy conversions taking place in turbulent thermally stratified fluids; and (ii) the role of surface buoyancy fluxes in the Munk & Wunsch (Deep-Sea Res., vol. 45, 1998, p. 1977) constraint on the mechanical energy sources of stirring required to maintain diapycnal mixing in the oceans. The new framework reveals that the observed turbulent rate of increase in the background gravitational potential energy GPEr , commonly thought to occur at the expense of the diffusively dissipated APE, actually occurs at the expense of internal energy, as in the laminar case. The APE dissipated by molecular diffusion, on the other hand, is found to be converted into internal energy (IE), similar to the viscously dissipated kinetic energy KE. Turbulent stirring, therefore, does not introduce a new APE/GPEr mechanical-to-mechanical energy conversion, but simply enhances the existing IE/GPEr conversion rate, in addition to enhancing the viscous dissipation and the entropy production rates. This, in turn, implies that molecular diffusion contributes to the dissipation of the available mechanical energy ME =APE +KE, along with viscous dissipation. This result has important implications for the interpretation of the concepts of mixing efficiency γmixing and flux Richardson number Rf , for which new physically based definitions are proposed and contrasted with previous definitions. The new framework allows for a more rigorous and general re-derivation from the first principles of Munk & Wunsch (1998, hereafter MW98)’s constraint, also valid for a non-Boussinesq ocean: G(KE) ≈ 1 − ξ Rf ξ Rf Wr, forcing = 1 + (1 − ξ )γmixing ξ γmixing Wr, forcing , where G(KE) is the work rate done by the mechanical forcing, Wr, forcing is the rate of loss of GPEr due to high-latitude cooling and ξ is a nonlinearity parameter such that ξ =1 for a linear equation of state (as considered by MW98), but ξ <1 otherwise. The most important result is that G(APE), the work rate done by the surface buoyancy fluxes, must be numerically as large as Wr, forcing and, therefore, as important as the mechanical forcing in stirring and driving the oceans. As a consequence, the overall mixing efficiency of the oceans is likely to be larger than the value γmixing =0.2 presently used, thereby possibly eliminating the apparent shortfall in mechanical stirring energy that results from using γmixing =0.2 in the above formula.

Voronoi, Delaunay, and Block-Structured Mesh Refinement for Solution of the Shallow-Water Equations on the Sphere

Alternative meshes of the sphere and adaptive mesh refinement could be immensely beneficial for weather and climate forecasts, but it is not clear how mesh refinement should be achieved. A finite-volume model that solves the shallow-water equations on any mesh of the surface of the sphere is presented. The accuracy and cost effectiveness of four quasi-uniform meshes of the sphere are compared: a cubed sphere, reduced latitude–longitude, hexagonal–icosahedral, and triangular–icosahedral. On some standard shallow-water tests, the hexagonal–icosahedral mesh performs best and the reduced latitude–longitude mesh performs well only when the flow is aligned with the mesh. The inclusion of a refined mesh over a disc-shaped region is achieved using either gradual Delaunay, gradual Voronoi, or abrupt 2:1 block-structured refinement. These refined regions can actually degrade global accuracy, presumably because of changes in wave dispersion where the mesh is highly nonuniform. However, using gradual refinement to resolve a mountain in an otherwise coarse mesh can improve accuracy for the same cost. The model prognostic variables are height and momentum collocated at cell centers, and (to remove grid-scale oscillations of the A grid) the mass flux between cells is advanced from the old momentum using the momentum equation. Quadratic and upwind biased cubic differencing methods are used as explicit corrections to a fast implicit solution that uses linear differencing.

Is the Indian Ocean SST variability a homogeneous diffusion process.

Whereas the predominance of El Niño Southern Oscillation (ENSO) mode in the tropical Pacific sea surface temperature (SST) variability is well established, no such consensus seems to have been reached by climate scientists regarding the Indian Ocean. While a number of researchers think that the Indian Ocean SST variability is dominated by an active dipolar-type mode of variability, similar to ENSO, others suggest that the variability is mostly passive and behaves like an autocorrelated noise. For example, it is suggested recently that the Indian Ocean SST variability is consistent with the null hypothesis of a homogeneous diffusion process. However, the existence of the basin-wide warming trend represents a deviation from a homogeneous diffusion process, which needs to be considered. An efficient way of detrending, based on differencing, is introduced and applied to the Hadley Centre ice and SST. The filtered SST anomalies over the basin (23.5N-29.5S, 30.5E-119.5E) are then analysed and found to be inconsistent with the null hypothesis on intraseasonal and interannual timescales. The same differencing method is then applied to the smaller tropical Indian Ocean domain. This smaller domain is also inconsistent with the null hypothesis on intraseasonal and interannual timescales. In particular, it is found that the leading mode of variability yields the Indian Ocean dipole, and departs significantly from the null hypothesis only in the autumn season.

Diffusion of the synthetic pyrethroid permethrin into bed-sediments

Bed-sediments are a sink for many micro-organic contaminants in aquatic environments. The impact of toxic contaminants on benthic fauna often depends on their spatial distribution, and the fate of the parent compounds and their metabolites. The distribution of a synthetic pyrethroid, permethrin, a compound known to be toxic to aquatic invertebrates, was studied using river bed-sediments in lotic flume channels. trans/cis-Permethrin diagnostic ratios were used to quantify the photoisomerization of the trans isomer in water. Rates were affected by the presence of sediment particles and colloids when compared to distilled water alone. Two experiments in dark/light conditions with replicate channels were undertaken using natural sediment, previously contaminated with permethrin, to examine the effect of the growth of an algal biofilm at the sediment-water interface on diffusive fluxes of permethrin into the sediment. After 42 days, the bulk water was removed, allowing a fine sectioning of the sediment bed (i.e., every mm down to 5 mm and then 5-10 mm, then every 10 mm down to 50 mm). Permethrin was detected in all cases down to a depth of 5-10 mm, in agreement with estimates by the Millington and Quirk model, and measurements of concentrations in pore water produced a distribution coefficient (K-d) for each section, High K-d's were observed for the top layers, mainly as a result of high organic matter and specific surface area. Concentrations in the algal biofilm measured at the end of the experiment under light conditions, and increases in concentration in the top 1 mm of the sediment, demonstrated that algal/bacterial biofilm material was responsible for high K-d's at the sediment surface, and for the retardation of permethrin diffusion. This specific partition of permethrin to fine sediment particles and algae may enhance its threat to benthic invertebrates. In addition,the analysis of trans/cis-permethrin isomer ratios in sediment showed greater losses of trans-permethrin in the experiment under light conditions, which may have also resulted from enhanced biological activity at the sediment surface.

Illustration of micro-scale advection using grid-pattern mini-lysimeters

Twenty-five small soil-filled perspex boxes arranged in a square, with dwarf sunflowers growing in them, were used to study micro-scale advection. Hydrological heterogeneity was introduced by applying two different amounts of irrigation water (low-irrigation, L, versus high-irrigation, H). The nine central boxes (4 H, 4 L and I bare box) were precision weighing lysimeters, yielding diurnal measurements of evaporation. After the onset of soil water stress, a large difference in latent heat flux (up to 4-fold) was observed between the lysimeters of the H and L treatments, mainly caused by large differences between H and L stomatal conductance values. This resulted in micro-advection, causing H soil-sunflower systems to evaporate well above equilibrium latent heat flux. The occurrence of micro-advective enhancement was reflected in large values of the Priestley-Taylor constant (often larger than 2.0) and generally negative values of sensible heat flux for the H treatment. (c) 2005 Elsevier B.V. All rights reserved.

The effect of aqueous diffusion on the fractionation of chlorine and bromine stable isotopes

Diffusive isotopic fractionation factors are important in order to understand natural processes and have practical application in radioactive waste storage and carbon dioxide sequestration. We determined the isotope fractionation factors and the effective diffusion coefficients of chloride and bromide ions during aqueous diffusion in polyacrylamide gel. Diffusion was determined as functions of temperature, time and concentration. The effect of temperature is relatively large on the diffusion coefficient (D) but only small on isotope fractionation. For chlorine, the ratio, D-35cl/D-37cl varied from 1.00128 +/- 0.00017 (1 sigma) at 2 degrees C to 1.00192 +/- 0.00015 at 80 degrees C. For bromine, D-79Br/D-81Br varied from 1.00098 +/- 0.00009 at 2 degrees C to 1.0064 +/- 0.00013 at 21 degrees C and 1.00078 +/- 0.00018 (1 sigma) at 80 degrees C. There were no significant effects on the isotope fractionation due to concentration. The lack of sensitivity of the diffusive isotope fractionation to anything at the most common temperatures (0 to 30 C) makes it particularly valuable for application to understanding processes in geological environments and an important natural tracer in order to understand fluid transport processes. (C) 2009 Elsevier Ltd. All rights reserved.

A comparison of conservative upwind difference schemes for the Euler equations

Numerical results are presented and compared for three conservative upwind difference schemes for the Euler equations when applied to two standard test problems. This includes consideration of the effect of treating part of the flux balance as a source, and a comparison of different averaging of the flow variables. Two of the schemes are also shown to be equivalent in their implementation, while being different in construction and having different approximate Jacobians. (C) 2006 Elsevier Ltd. All rights reserved.

Upwind solution of singular differential equations arising from steady channel flows

We study ordinary nonlinear singular differential equations which arise from steady conservation laws with source terms. An example of steady conservation laws which leads to those scalar equations is the Saint–Venant equations. The numerical solution of these scalar equations is sought by using the ideas of upwinding and discretisation of source terms. Both the Engquist–Osher scheme and the Roe scheme are used with different strategies for discretising the source terms.