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Interaction of tidal flow with a complex topography and bathymetry including headlands, islands, coral reefs and shoals create a rich submesoscale field of tidal jets, vortices, unsteady wakes, lee eddies and free shear layers, all of which impact marine ecology. A unique and detailed view of the submesoscale variability in a part of the Great Barrier Reef lagoon, Australia, that includes a number of small islands was obtained by using a 'stereo' pair of 2-m-resolution visible-band images that were acquired just 54 s apart by the WorldView-3 satellite. Near-surface current and vorticity were extracted at a 50-m-resolution from those data using a cross-correlation technique and an optical-flow method, each yielding a similar result. The satellite-derived data are used to test the ability of the second-generation Louvain-la-Neuve ice-ocean model (SLIM), an unstructured-mesh, finite element model for geophysical and environmental flows, to reproduce the details of the currents in the region. The model succeeds in simulating the large-scale (> 1 km) current patterns, such as the main current and the width and magnitude of the jets developing in the gaps between the islands. Moreover, the order of magnitude of the vorticity and the occurrence of some vortices downstream of the islands are correctly reproduced. The smaller scales (< 500 m) are resolved by the model, although various discrepancies with the data are observed. The smallest scales (< 50 m) are unresolved by both the model- and image-derived velocity fields. This study shows that high-resolution models are able to a significant degree to simulate accurately the currents close to a rugged coast. Very-high-resolution satellite oceanography stereo images offer a new way to obtain snapshots of currents near a complex topography that has reefs, islands and shoals, and is a potential resource that could be more widely used to assess the predictive ability of coastal circulation models. [ABSTRACT FROM AUTHOR]

A discontinuous Galerkin nonhydrostatic atmospheric model is used for two-dimensional and three-dimensional simulations. There is a wide range of timescales to be dealt with. To do so, two different implicit/explicit time discretizations are implemented. A stabilization, based upon a reduced-order discretization of the gravity term, is introduced to ensure the balance between pressure and gravity effects. While not affecting significantly the convergence properties of the scheme, this approach allows the simulation of anisotropic flows without generating spurious oscillations, as it happens for a classical discontinuous Galerkin discretization. This approach is shown to be less diffusive than usual spatial filters. A stability analysis demonstrates that the use of this modified scheme discards the instability associated with the usual discretization. Validation against analytical solutions is performed, confirming the good convergence and stability properties of the scheme. Numerical results demonstrate the attractivity of the discontinuous Galerkin method with implicit/explicit time integration for large-scale atmospheric flows. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In the context of the emergence of extra-terrestrial oceanography, we adapted an existing oceanographic model, SLIM (), to the conditions of Titan, a moon of Saturn. The tidal response of the largest southern lake at Titan's surface, namely Ontario Lacus, is simulated. SLIM solves the 2D, depth-averaged shallow water equations on an unstructured mesh using the discontinuous Galerkin finite element method, which allows for high spatial resolution wherever needed. The impact of the wind forcing, the bathymetry, and the bottom friction is also discussed. The predicted maximum tidal range is about 0.56 m in the southern part of the lake, which is more than twice as large as the previous estimates (see Tokano, Ocean Dyn 60:(4) 803-817 (Tokano )). The patterns and magnitude of the current are also markedly different from those of previous studies: the tidal motion is not aligned with the major axis of the lake and the speed is larger nearshore. Indeed, the main tidal component rotates clockwise in an exact period of one Titan day and the tidal currents can reach 0.046 ms close to the shores depending on the geometry and the bathymetry. Except for these specific nearshore regions, the current speed is less than 0.02 ms . Circular patterns can be observed offshore, their rotational direction and size varying along the day. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents analytical test cases for tracer advection–diffusion-decay problems. The test cases are used to validate a finite element, unstructured grid fecal bacteria transport model. The test cases include the following domains: one-dimensional infinitely long river, two-dimensional half plane and two-dimensional infinitely long channel. In this work the bacteria are considered to enter the domain only through point sources. Analytical solutions are derived using either a Dirac delta function or a variable-width Gaussian function as a point source. Both analytical derivations and numerical simulations suggest that the error is maximised at the source. We present formulae for estimating the error caused by replacing a Dirac source with a Gaussian function in the numerical model. Furthermore, numerical simulations suggest that the best approximation for a Dirac source is a Gaussian whose width parameter is one third of the local mesh size. [Copyright &y& Elsevier]

Abstract: We present a new three-dimensional, unstructured mesh finite element shallow-water model. The current configuration is suitable for studying unstratified flows and the evolution of passive tracers. The model has a free surface and is hydrostatic. The mesh is unstructured in the horizontal and extruded towards the seabed in the direction parallel to the local gravity vector to generate a mesh made up of prisms. The mesh moves in the vertical and accommodates the free-surface motions. We describe the numerical treatment of the hydrodynamical equations with the finite element method. A discontinuous representation is used in the vertical for all velocity components. The horizontal velocity components are non-conforming in the horizontal, which is particularly appropriate for advection-dominated flows. The model is validated against a realistic tidal flow around a shallow-water island for which field measurements are available and is shown to operate successfully. The three-dimensional character of the flow is emphasized by use of a passive tracer. We also assess the model’s ability to represent the vertical structure of the horizontal flow field by applying it to a wind-driven flow experiment in an elongated rectangular basin. [Copyright &y& Elsevier]

Abstract: A three-dimensional finite-element model is used to investigate the tidal flow around Rattray Island, Great Barrier Reef, Australia. Field measurements and visual observations show both stable eddies developing at rising and falling tide in the wake of the island. The water turbidity suggests intense upwelling able to carry bed sediments upwards. Based on previous numerical studies, it remains unclear at this point whether the most intense upwelling occurs near the centre of the eddies or off the island''s tips, closer to the island. All these studies resorted to a very simple turbulence closure, with a zero-equation model whereby the coefficient of vertical viscosity is computed via an algebraic expression. In this work, we aim at studying the influence of the turbulence closure on model results, with emphasis on the prediction of vertical motions. The Mellor and Yamada level 2.5 closure scheme is used and an increase in the intensity of vertical transport is observed. This increase is partly explained by the fact that the Mellor and Yamada model takes into account the hysteresis effect in the time variation of turbulence variables. The influence of the advection of turbulence variables is estimated to be negligible. By a better representation of transient coastal phenomena, the Mellor and Yamada level 2.5 turbulence closure improves the model to a significant degree. [Copyright &y& Elsevier]

In this paper, we present an h-adaptive discontinuous Galerkin formulation of the shallow water equations. For a discontinuous Galerkin scheme using polynomials up to order $$ p $$ , the spatial error of discretization of the method can be shown to be of the order of $$ h^{{p + 1}} $$ , where $$h$$ is the mesh spacing. It can be shown by rigorous error analysis that the discontinuous Galerkin method discretization error can be related to the amplitude of the inter-element jumps. Therefore, we use the information contained in jumps to build error metrics and size field. Results are presented for ocean modelling problems. A first experiment shows that the theoretical convergence rate is reached with the discontinuous Galerkin high-order h-adaptive method applied to the Stommel wind-driven gyre. A second experiment shows the propagation of an anticyclonic eddy in the Gulf of Mexico. [ABSTRACT FROM AUTHOR]

The main goal of this work is to appraise the finite element method in the way it represents barotropic instabilities. To that end, three different formulations are employed. The free-surface formulation solves the primitive shallow-water equations and is of predominant use for ocean modeling. The vorticity–stream function and velocity–pressure formulations resort to the rigid-lid approximation and are presented because theoretical results are based on the same approximation. The growth rates for all three formulations are compared for hyperbolic tangent and piecewise linear shear flows. Structured and unstructured meshes are utilized. The investigation is also extended to time scales that allow for instability meanders to unfold, permitting the formation of eddies. We find that all three finite element formulations accurately represent barotropic instablities. In particular, convergence of growth rates toward theoretical ones is observed in all cases. It is also shown that the use of unstructured meshes allows for decreasing the computational cost while achieving greater accuracy. Overall, we find that the finite element method for free-surface models is effective at representing barotropic instabilities when it is combined with an appropriate advection scheme and, most importantly, adapted meshes. [ABSTRACT FROM AUTHOR]

Hall and Haine [J. Mar. Syst., in press] briefly addressed the problem of estimating the age of irreducible fluid elements or water parcels in a one-dimensional flow with constant velocity and diffusivity. Herein further developments are achieved on this subject. The age of every water parcel is set to zero at the moment it passes through the point x=0, where x is an appropriate space coordinate. As time progresses, the age of the water is seen to increase unboundedly upstream of the point x=0, and tend to the steady-state value &z.sfnc;x/u&z.sfnc; downstream of the point x=0, where u is the water velocity. By contrast, the age of the water parcels that have touched at least once the point x=0 is symmetric with respect to the point x=0 is smaller than the water age at any time and position, and tends to &z.sfnc;x/u&z.sfnc; as time progresses. Asymptotic expansions are derived for large times. [Copyright &y& Elsevier]

Abstract: A number of recent studies suggest that human and animal mobility patterns exhibit scale-free, Lévy-flight dynamics. However, current reaction-diffusion epidemics models do not account for the superdiffusive spread of modern epidemics due to Lévy flights. We have developed a SIR model to simulate the spatial spread of a hypothetical epidemic driven by long-range displacements in the infective and susceptible populations. The model has been obtained by replacing the second-order diffusion operator by a fractional-order operator. Theoretical developments and numerical simulations show that fractional-order diffusion leads to an exponential acceleration of the epidemic''s front and a power-law decay of the front''s leading tail. Our results indicate the potential of fractional-order reaction-diffusion models to represent modern epidemics. [Copyright &y& Elsevier]

Abstract: The accuracy and efficiency of an Eulerian method is assessed by solving the non-linear shallow water equations and compared with the performances of an existing semi-Lagrangian method. Both methods use a linear non-conforming finite element discretization for velocity and a linear conforming finite element discretization for surface elevation. This finite element pair is known to be computationally efficient and free of pressure modes. The model equations are carefully derived and a comparison is performed by simulating the propagation of slow Rossby waves in the Gulf of Mexico. Simulations show that the Eulerian model performs well and gives results comparable to high order semi-Lagrangian schemes using kriging interpolators. [Copyright &y& Elsevier]