Impact of mass lumping on gravity and Rossby waves in 2D finite-element shallow-water models

The goal of this study is to evaluate the effect of mass lumping on the dispersion properties of four finite-element velocity/surface-elevation pairs that are used to approximate the linear shallow-water equations. For each pair, the dispersion relation, obtained using the mass lumping technique, is computed and analysed for both gravity and Rossby waves. The dispersion relations are compared with those obtained for the consistent schemes (without lumping) and the continuous case. The P0-P1, RT0 and P-P1 pairs are shown to preserve good dispersive properties when the mass matrix is lumped. Test problems to simulate fast gravity and slow Rossby waves are in good agreement with the analytical results.

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.

Consistent approximate models of the global atmosphere: shallow, deep, hydrostatic, quasi-hydrostatic and non-hydrostatic

We study global atmosphere models that are at least as accurate as the hydrostatic primitive equations (HPEs), reviewing known results and reporting some new ones. The HPEs make spherical geopotential and shallow atmosphere approximations in addition to the hydrostatic approximation. As is well known, a consistent application of the shallow atmosphere approximation requires omission of those Coriolis terms that vary as the cosine of latitude and of certain other terms in the components of the momentum equation. An approximate model is here regarded as consistent if it formally preserves conservation principles for axial angular momentum, energy and potential vorticity, and (following R. Müller) if its momentum component equations have Lagrange's form. Within these criteria, four consistent approximate global models, including the HPEs themselves, are identified in a height-coordinate framework. The four models, each of which includes the spherical geopotential approximation, correspond to whether the shallow atmosphere and hydrostatic (or quasi-hydrostatic) approximations are individually made or not made. Restrictions on representing the spatial variation of apparent gravity occur. Solution methods and the situation in a pressure-coordinate framework are discussed. © Crown copyright 2005.

A comparison of two methods for developing the linearization of a shallow-water model

We develop the linearization of a semi-implicit semi-Lagrangian model of the one-dimensional shallow-water equations using two different methods. The usual tangent linear model, formed by linearizing the discrete nonlinear model, is compared with a model formed by first linearizing the continuous nonlinear equations and then discretizing. Both models are shown to perform equally well for finite perturbations. However, the asymptotic behaviour of the two models differs as the perturbation size is reduced. This leads to difficulties in showing that the models are correctly coded using the standard tests. To overcome this difficulty we propose a new method for testing linear models, which we demonstrate both theoretically and numerically. © Crown copyright, 2003. Royal Meteorological Society

A self-repairing, supramolecular polymer system: healability as a consequence of donor-acceptor pi-pi stacking interactions

A novel supramolecular polymer system, in which the terminal pyrenyl groups of a polyamide intercalate into the chain-folds of a polyimide via electronically-complementary pi-pi stacking, shows both enhanced mechanical properties relative to those of its individual components and facile healing characteristics as a result of the thermoreversibility of non-covalent interactions.

A general synthesis of macrocyclic pi-electron-acceptor systems

Cyclocondensations of aromatic diamines with 1,1'-bis(2,4-dinitrophenyl)-4,4'-bipyridinium salts afford doubly or quadruply charged, macrocyclic, N,N'-diarylbipyridinium cations. These are tolerant of a wide range of acids, bases, and nucleophiles, although they appear to undergo reversible, one-electron reduction by tertiary amines. Single-crystal X-ray analysis demonstrates the presence of a macrocycle conformation in which the 4,4'-bipyridinium and 4,4'-biphenylenedisulfonyl residues are suitably spaced and aligned for complexation with pi-donor arenes, and NMR studies in solution indeed confirm binding to 1,5-bis[hydroxy(ethoxy)ethoxy]naphthalene.

Prebiotic properties of alternansucrase maltose-acceptor oligosaccharides

alpha-(1-6) and alpha-(1-3)-linked oligosaccharides were obtained from the reaction between sucrose and maltose, catalyzed by an alternansucrase isolated from Leuconostoc mesenteroides NRRL B-21297 and separated using a Bio-Gel P2 column in six fractions. Fractions 1, 2, and 3 were mainly composed of DP3, DP4, and DP5, respectively. However, fractions 4, 5, and 6 consisted of mixtures from DP5 to IDP9, and they are identified here as DP5.7, DP6.7, and DP7.4, respectively. Potential prebiotic properties of these oligosaccharides were tested using pure and mixed cultures. Generally, in pure studies, most of the tested bacteria failed to grow or grew poorly using the DP6.7 and DP7.4 fractions and showed the greatest growth on DP3. Growth of fecal bacteria on the maltose-acceptor products was tested following an in vitro fermentation method. DP3 showed the highest prebiotic effect, followed by DP4 and DP6.7, whereas DP7.4 did not present any prebiotic activity.