The boundary layer of the residence time field

The residence time of a tracer in a control domain is usually computed by releasing tracer parcels and registering the time when each of these tracer parcels cross the boundary of the control domain. In this Lagrangian procedure, the particles are discarded or omitted as soon as they leave the control domain. In a Eulerian approach, the same approach can be implemented by integrating forward in time the advection–diffusion equation for a tracer. So far, the conditions to be applied at the boundary of the control domain were uncertain. We show here that it is necessary to prescribe that the tracer concentration vanishes at the boundary of the control domain to ensure the compatibility between the Lagrangian and Eulerian approaches. When we use the Constituent oriented Age and Residence time Theory (CART), this amounts to solving the differential equation for the residence time with boundary conditions forcing the residence time to vanish at the open boundaries of the control domain. Such boundary conditions are likely to induce the development of boundary layers (at outflow boundaries for the tracer concentration and at inflow boundaries for the residence time). The thickness of these boundary layers is of the order of the ratio of the diffusivity to the velocity. They can however be partly smoothed by tidal and other oscillating flows.     

Influence of the turbulence closure scheme on the finite-element simulation of the upwelling in the wake of a shallow-water island

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.     

Normal modes and resonance in Ontario Lacus: a hydrocarbon lake of Titan

Ocean Dynamics - The natural modes of Ontario Lacus surface oscillations, the largest lake in Titan’s southern hemisphere, are simulated and analyzed as they are potentially of broad interest...     

A flux-limiting wetting–drying method for finite-element shallow-water models,with application to the Scheldt Estuary

We present a flux-limiting wetting–drying approach for finite-element discretizations of the shallow-water equations using discontinuous linear elements for the elevation. The key ingredient of the method is the use of limiters for generalized nodal fluxes. This method is implemented into the Second-generation Louvain-la-Neuve Ice-ocean Model (SLIM), and is verified against standard test cases. The method is further applied to the wetting and drying of sand banks in the Scheldt Estuary, which is located in northern Belgium and the southern Netherlands. The results obtained for both the benchmarks and the realistic problem illustrate the accuracy of the method in describing the hydrodynamics in the vicinity of dry areas. In particular, the method strictly conserves mass, and there is no transport through dry areas.     

High-order <Emphasis Type="Italic">h</Emphasis>-adaptive discontinuous Galerkin methods for ocean modelling

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 , the spatial error of discretization of the method can be shown to be of the order of , where 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. An erratum to this article can be found at     

A finite-element,multi-scale model of the Scheldt tributaries,river, estuary and ROFI

We report on the development and validation of a coupled two- and one-dimensional finite-element model for the Scheldt tributaries, river, estuary and region of fresh water influence (ROFI). The hydrodynamic equations are solved on a single, unstructured, multi-scale mesh stretching from the shelf break to the Scheldt tributaries. The tide is forced on the shelf break and propagates upstream in the riverine network. Upstream boundaries lie on sluices or outside of the region of tidal dominance where daily averaged discharges are imposed. Two-dimensional, depth-averaged shallow water equations are solved by means of the discontinuous Galerkin (DG) method over the marine and estuarine parts of the computational domain. In the rivers, however, one-dimensional equations are dealt with using the DG method with the addition of a technique to cope with confluence points. Model parameters are carefully calibrated, leading to the simulation of wind- and tide-forced flows that are in excellent agreement with available data. The diffusivity in the transport equation is calibrated using time series of salinity at various locations in the estuary. Finally, the Lagrangian residual transport in the estuary and the adjacent coastal zone is investigated. This work is a major step towards an integrated model for studying the dynamics of waterborne contaminants and the water renewal timescales in the Scheldt land-sea continuum.     

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.

     

On the computation of the barotropic mode of a free-surface world ocean model

The free-surface formulation of the equations of our world ocean model is briefly described. The barotropic mode equations are solved according to the split-explicit method, using different time steps for the external and internal modes. Because the numerical algorithm is implemented on the B-grid, a spurious, free-surface, two-grid interval mode may develop. This mode must be filtered out. The properties of two filters are theoretically investigated and their actual performance is tested in a series of numerical experiments. It is seen that one of these filters may severely perturb the local mass conservation, rendering it impossible to enforce the impermeability of the surface or the bottom of the ocean. The dynamics of the external mode is also examined, by studying the depth-integrated momentum equations. The depth-integral of the pressure force due to the slope of the ocean surface is approximately balanced by the depth-integral of the force ensuing from the horizontal variations of the density. The depth-integral of the Coriolis force is an order of magnitude smaller, except in the Southern Ocean. Two variational principles are resorted to for computing the fictitious ocean surface elevation corresponding to the approximate equilibrium between the dominant forces of the barotropic momentum equations.     

Age and the time lag method

The time lag method is one of the most straightforward methods used to estimate transit times from experimental data and is therefore widely used. The transit time between two points is estimated from the analysis of time series taken at these two points that suggest the propagation of a signal from one point to the other. To account for the distortion of the signal during its propagation between the two points an optimum time lag can be estimated by the analysis of the cross-correlation of the two time series.