A discontinuous finite element baroclinic marine model on unstructured prismatic meshes

We describe the space discretization of a three-dimensional baroclinic finite element model, based upon a discontinuous Galerkin method, while the companion paper (Comblen et al. 2010a) describes the discretization in time. We solve the hydrostatic Boussinesq equations governing marine flows on a mesh made up of triangles extruded from the surface toward the seabed to obtain prismatic three-dimensional elements. Diffusion is implemented using the symmetric interior penalty method. The tracer equation is consistent with the continuity equation. A Lax–Friedrichs flux is used to take into account internal wave propagation. By way of illustration, a flow exhibiting internal waves in the lee of an isolated seamount on the sphere is simulated. This enables us to show the advantages of using an unstructured mesh, where the resolution is higher in areas where the flow varies rapidly in space, the mesh being coarser far from the region of interest. The solution exhibits the expected wave structure. Linear and quadratic shape functions are used, and the extension to higher-order discretization is straightforward.     

Review of solutions for 3D hydrodynamic modeling applied to aquaculture in South Pacific atoll lagoons

A workshop organized in French Polynesia in November 2004 allowed reviewing the current methods to model the three-dimensional hydrodynamic circulation in semi-enclosed atoll lagoons for aquaculture applications. Mollusk (e.g. pearl oyster, clam) aquaculture is a major source of income for South Pacific countries such as French Polynesia or Cook Islands. This aquaculture now requires a better understanding of circulation patterns to improve the spatial use of the lagoons, especially to define the best area to set larvae collectors. The pelagic larval duration of the relevant species (<20 days) and the size of the semi-closed lagoons (few hundreds of km2) drive the specifications of the model in terms of the spatial and temporal scale. It is considered that, in contrast with fish, mollusk larvae movements are limited and that their cycle occurs completely in the lagoon, without an oceanic stage. Atolls where aquaculture is productive are generally well-bounded, or semi-closed, without significant large and deep openings to the ocean. Nevertheless part of the lagoon circulation is driven by oceanic water inputs through the rim, ocean swells, tides and winds. Therefore, boundary conditions of the lagoon system are defined by the spatial structure of a very shallow rim (exposition and number of hoas), the deep ocean swell climate, tides and wind regimes. To obtain a realistic 3D numerical model of lagoon circulation with adequate forcing, it is thus necessary to connect in an interdisciplinary way a variety of methods (models, remote sensing and in situ data collection) to accurately represent the different components of the lagoon system and its specific boundary conditions. We review here the current methods and tools used to address these different components for a hypothetical atoll of the Tuamotu Archipelago (French Polynesia), representative of the semi-closed lagoons of the South Pacific Ocean. We hope this paper will serve as a guide for similar studies elsewhere and we provide guidelines in terms of costs for all the different stages involved.     

Submesoscale tidal eddies in the wake of coral islands and reefs: satellite data and numerical modelling

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.     

Large amplitude, leaky, island-generated, internal waves around Palau, Micronesia

Three years of temperature data along two transects extending to 90 m depth, at Palau, Micronesia, show twice-a-day thermocline vertical displacements of commonly 50–100 m, and on one occasion 270 m. The internal wave occurred at a number of frequencies. There were a number of spectral peaks at diurnal and semi-diurnal frequencies, as well as intermediate and sub-inertial frequencies, less so at the inertial frequency. At Palau the waves generally did not travel around the island because there was no coherence between internal waves on either side of the island. The internal waves at a site 30 km offshore were out-of-phase with those on the island slopes, suggesting that the waves were generated on the island slope and then radiated away. Palau Island was thus a source of internal wave energy for the surrounding ocean. A numerical model suggests that the tidal and low-frequency currents flowing around the island form internal waves with maximum wave amplitude on the island slope and that these waves radiate away from the island. The model also suggests that the headland at the southern tip of Palau prevents the internal waves to rotate around the island. The large temperature fluctuations (commonly daily fluctuations ≈10 °C, peaking at 20 °C) appear responsible for generating a thermal stress responsible for a biologically depauperate biological community on the island slopes at depths between 60 and 120 m depth.     

Comparison of free-surface and rigid-lid finite element models of barotropic instabilities

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.     

Numerical study of tides in Ontario Lacus,a hydrocarbon lake on the surface of the Saturnian moon Titan

In the context of the emergence of extra-terrestrial oceanography, we adapted an existing oceanographic model, SLIM (www.climate.be/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 10.1007/s10236-010-0285-3 (Tokano 2010)). 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 ^?1 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 ^?1. Circular patterns can be observed offshore, their rotational direction and size varying along the day.     

Residence and exposure times : when diffusion does not matter

Under constant hydrodynamic conditions and assuming horizontal homogeneity, negatively buoyant particles released at the surface of the water column have a mean residence time in the surface mixed layer of h/w, where h is the thickness of the latter and w (?>?0) is the sinking velocity Deleersnijder (Environ Fluid Mech 6(6):541–547, 2006a). The residence time does not depend on the diffusivity and equals the settling timescale. We show that this behavior is a result of the particular boundary conditions of the problem and that it is related to a similar property of the exposure time in a one-dimensional infinite domain. In 1-D advection–diffusion problem with a constant and uniform velocity, the exposure time—which is a generalization of the residence time measuring the total time spent by a particle in a control domain allowing the particle to leave and reenter the control domain—is also equal to the advection timescale at the upstream boundary of the control domain. To explain this result, the concept of point exposure is introduced; the point exposure is the time integral of the concentration at a given location. It measures the integrated influence of a point release at a given location and is related to the concept of number of visits of the theory of random walks. We show that the point exposure takes a constant value downstream the point of release, even when the diffusivity varies in space. The analysis of this result reveals also that the integrated downstream transport of a passive tracer is only effected by advection. While the diffusion flux differs from zero at all times, its integrated value is strictly zero.     

Application of modified Patankar schemes to stiff biogeochemical models for the water column

In this paper, we apply recently developed positivity preserving and conservative Modified Patankar-type solvers for ordinary differential equations to a simple stiff biogeochemical model for the water column. The performance of this scheme is compared to schemes which are not unconditionally positivity preserving (the first-order Euler and the second- and fourth-order Runge–Kutta schemes) and to schemes which are not conservative (the first- and second-order Patankar schemes). The biogeochemical model chosen as a test ground is a standard nutrient–phytoplankton–zooplankton–detritus (NPZD) model, which has been made stiff by substantially decreasing the half saturation concentration for nutrients. For evaluating the stiffness of the biogeochemical model, so-called numerical time scales are defined which are obtained empirically by applying high-resolution numerical schemes. For all ODE solvers under investigation, the temporal error is analysed for a simple exponential decay law. The performance of all schemes is compared to a high-resolution high-order reference solution. As a result, the second-order modified Patankar–Runge–Kutta scheme gives a good agreement with the reference solution even for time steps 10 times longer than the shortest numerical time scale of the problem. Other schemes do either compute negative values for non-negative state variables (fully explicit schemes), violate conservation (the Patankar schemes) or show low accuracy (all first-order schemes).