We describe a compatible finite element discretisation for the shallow water equations on the rotating sphere, concentrating on integrating consistent upwind stabilisation into the framework. Although the prognostic variables are velocity and layer depth, the discretisation has a diagnostic potential vorticity that satisfies a stable upwinded advection equation through a Taylor-Galerkin scheme; this provides a mechanism for dissipating enstrophy at the gridscale whilst retaining optimal order consistency. We also use upwind discontinuous Galerkin schemes for the transport of layer depth. These transport schemes are incorporated into a semi-implicit formulation that is facilitated by a hybridisation method for solving the resulting mixed Helmholtz equation. We demonstrate that our discretisation achieves the expected second order convergence and provide results from some standard rotating sphere test problems. arXiv:1707.00855v2 [math.NA]

Union's Horizon 2020 research and innovation programme (grant agreement no 741112). vii 1 Geophysical fluid dynamics and simulation The hierarchy of models we will build up in this chapter is shown in Figure 1.1. This is just one choice of hierarchy, since for example one can apply the hydrostatic approximation directly to the compressible Euler equations without making Boussinesq /anelastic approximations. Similarly, we do not consider quasi-geostrophic approximations which are valid for rapidly rotating systems; these approximations can be made at any step of our hierarchy. Governing equations: Full, three-dimensional compressible Euler system. Boussinesq/anelastic: Assumption: small variations in density. Impact: no acoustic waves. Hydrostatic: Assumption: fluid is in a thin layer. Impact: No vertical acceleration term, vertical velocity becomes diagnostic from the continuity equation. Shallow water system Assumption: columnar motion (horizontal velocity is independent of height). Impact: equations become two-dimensional, prognostic variables are horizontal velocity and layer height. 1.1.1 The compressible Euler system We start by presenting the compressible Euler equations, which have the fewest approximations amongst our hierarchy. We assume that the air is dry (no moisture), inviscid (no viscous forces), and adiabatic (no sources or diffusion of temperature). The governing equations for a dry, inviscid, adiabatic, compressible fluid in a rotating reference frame with angular velocity Ω may be written in the form

Summary and conclusionsIn a prospective study of 185 players attached to 10 British rugby clubs, 151 injuries were recorded among 98 of them (53%) during a single season. Forwards sustained significantly more injuries than backs. The standard of rugby, players' body weights, degree of fitness, and presence of joint hypermobility did not affect the risk of injury. The leg was the most common site of injury. Head and neck injuries were significantly more common when play was static and on wet pitches.

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This study used a multidisciplinary approach to examine the brains of pediatric road trauma fatalities in the Sydney area over a 3-year period. The brains of 32 children (0-16 years) were examined: 20 pedestrians, nine passengers, and three cyclists. The extent and distribution of brain injury was assessed, peak linear head acceleration determined, and the severity of brain damage was compared to that previously reported for adults using the same scoring method. Skull fractures (20/32) and subarachnoid haemorrhage (22/32) were the commonest head injuries. In general, the neuropathology was similar to that seen in adults, with a high percentage of damage in the corpus callosum and gliding contusions within the subcortical white matter. Intracerebral hemorrhage was relatively rare. For frontal and occipital head impacts, the corpus callosum was the most injured part of the brain, followed by the deep central structures and the temporal lobes, whereas for lateral impacts, the injuries were more evenly distributed. Comparison of the current data for children with the vascular injury sector scores reported for adults suggests that the brains of children are more severely damaged for the same peak linear head acceleration.