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Mécanique Rationnelle - Modèle Mathématique de Newton Delhez, Eric Book published by Editions du Perron (2011) Detailed reference viewed: 141 (18 ULg)Absorbing layers for shallow water models Modave, Axel ; ; Delhez, Eric Conference (2010, May 11) Detailed reference viewed: 148 (40 ULg)On the parameters of absorbing layers for shallow water models Modave, Axel ; ; Delhez, Eric in Ocean Dynamics (2010), 60(1), 65-79 Absorbing/sponge layers used as boundary conditions for ocean/marine models are examined in the context of the shallow water equations with the aim to minimize the reflection of outgoing waves at the ... [more ▼] Absorbing/sponge layers used as boundary conditions for ocean/marine models are examined in the context of the shallow water equations with the aim to minimize the reflection of outgoing waves at the boundary of the computationaldomain. The ptimization of the absorption coefficient is not an issue in continuous models, for the reflection coefficient of outgoing waves can then be made as small as we please by increasing the absorption coefficient. The optimization of the parameters of absorbing layers is therefore a purely discrete problem. A balance must be found between the efficient damping of outgoing waves and the limited spatial resolution with which the resulting spatial gradients must be described. Using a one-dimensional model as a test case, the performances of various spatial distributions of the absorption coefficient are compared. Two shifted hyperbolic distributions of the absorption coefficient are derived from theoretical considerations for a pure propagative and a pure advective problems. These distribution show good performances. Their free parameter has a well-defined interpretation and can therefore be determined on a physical basis. The properties of the two shifted hyperbolas are illustrated using the classical two-dimensional problems of the collapse of a Gaussianshaped mound of water and of its advection by a mean current. The good behavior of the resulting boundary scheme remains when a full non-linear dynamics is taken into account. [less ▲] Detailed reference viewed: 213 (92 ULg)Capturing the residence time boundary layer - Application to the Scheldt Estuary. ; ; et al in Ocean Dynamics (2010), 60(3), 535-554 Detailed reference viewed: 17 (2 ULg)Consistent computation of the age of water parcels using CART Mercier, Christophe ; Delhez, Eric in Ocean Modelling (2010), 35 The Constituent-oriented Age and Residence time Theory (CART) provides a flexible and efficient framework to diagnose the dynamics of marine systems. Beside the equation for the concentration of ... [more ▼] The Constituent-oriented Age and Residence time Theory (CART) provides a flexible and efficient framework to diagnose the dynamics of marine systems. Beside the equation for the concentration of appropriate (real or artificial) tracers, the method requires the resolution of differential problems for the so-called age concentration of each of these tracers. Thanks to its Eulerian formulation as an advection/diffusion problem with source terms, the method is easily implemented in existing models. However, some numerical artifacts should be avoided in order to produce physically meaningful results leading to a better understanding of the system under study. In this paper, we address two such issues that are related to the degree of implicitness of the different terms and to the advection scheme. To enforce the consistency between the discrete equations for the concentration of a tracer and for its age concentration, the degree of implicitness must be identical in the source/sink terms of the two equations. However, the ageing term should be computed in a completely explicit (respectively implicit) way if the discretization of the source/sink terms is implicit in time (respectively explicit). A specific attention should also be paid to the advection schemes for the concentration and the age concentration. The raw application of Total Variation Diminishing (TVD) scheme for both equations can lead to the occurrence of artificial local extreme values and spatial oscillations of the age field. While the TVD behavior of the discrete age field cannot be guaranteed, appropriate modifications of the flux/slope limiters used in the TVD schemes can be implemented to enforce a maximum principle that prevents the occurrence of age values outside the physically acceptable range. [less ▲] Detailed reference viewed: 47 (23 ULg)A modified TVD scheme for the advection of two or more variables with consideration for their sum Mercier, Christophe ; Delhez, Eric in Ocean Dynamics (2010), 60 Total variation diminishing (TVD) advection schemes are known to produce results that are free from some of the numerical artifacts (no overshooting, no spurious oscillation, small diffusion) that can ... [more ▼] Total variation diminishing (TVD) advection schemes are known to produce results that are free from some of the numerical artifacts (no overshooting, no spurious oscillation, small diffusion) that can spoil the physical significance of the results. When two or more tracers are advected separately using a TVD scheme, the sum of these variables can however exhibit some inappropriate behaviors. The total variation of the sum will not necessarily be non- increasing and local artificial oscillations and extrema can appear. We show that these can be avoided with only minor perturbations of the original solution by adjusting the slope limiters used for the different variables. If the sum of these variables has some physical significance, for instance as refinement of a larger model compartment, the correction procedure introduced in this paper should be used to ensure a physically meaningful solution. [less ▲] Detailed reference viewed: 45 (25 ULg)Residence time and exposure time of sinking phytoplankton in the euphotic layer Delhez, Eric ; in Journal of Theoretical Biology (2010), 262 Detailed reference viewed: 62 (20 ULg)Really TVD advection schemes for the depth-integrated transport equation Mercier, Christophe ; Delhez, Eric in Ocean Modelling (2010), 33 Detailed reference viewed: 94 (54 ULg)Really TVD advection schemes for shelf seas Mercier, Christophe ; Delhez, Eric Conference (2010) During the last decade large efforts have been devoted to the development of high-resolution schemes to solve advection problems. High-resolution conservative numerical schemes satisfying conservative ... [more ▼] During the last decade large efforts have been devoted to the development of high-resolution schemes to solve advection problems. High-resolution conservative numerical schemes satisfying conservative, monotonicity preserving and shock-capturing properties are nowadays widely used in ocean modeling. Among these, TVD schemes, based on the concept of Total Variation Diminishing (TVD), were progressively adopted because of their good behavior that guarantees a solution free from numerical artifacts (no overshooting, no spurious oscillation, small diffusion) that can spoil the physical significance of the results. Most of the TVD schemes and associated limiters have been originally developed in idealized one-dimensional flows described by a linear advection. In finite volume marine models, one has however often to deal with the depth integrated advection equation. This formulation is usually preferred because of its conservative form that is particularly suited to numerical treatment using a finite volume approach. Conservative numerical schemes can be easily formulated to ensure that the total mass of the advected quantity is conserved. This property is very valuable in the context of environmental studies for which a strict equilibrium of the mass budget of pollutants is often more relevant that the raw accuracy of the integration scheme. In the same context, the numerical scheme should also produce neither new local extremum nor negative concentrations, i.e. it should be monotonicity preserving which is implied by the TVD property. The development of TVD schemes for the resolution of advection equations written in the conservative form is however not trivial. Numerical experiments show that the blind application to the depth-integrated equation of the usual TVD schemes and associated flux limiters introduced in the context of linear advection can lead to non-TVD solutions in presence of complex geometries. Spatial and/or temporal variations of the local bathymetry can indeed break the TVD property of the usual schemes. Really TVD schemes can be recovered by taking into account the local depth and its variations in the formulation of the flux limiters. Using this approach, a generalized superbee limiter is introduced and validated. [less ▲] Detailed reference viewed: 80 (9 ULg)Age and the time lag method Delhez, Eric ; in Continental Shelf Research (2008), 28(8), 1057-1067 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 ... [more ▼] 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. This study clarifies the relation between the transit time estimated by the time lag method and the well-defined concept of the age of a water mass. It is shown, through simplified process models, that the time lag method systematically underestimates the true mean age. The error can be quantified by means of a dimensionless parameter which is the inverse of a Peclet number based on a characteristic length given by the ratio of the velocity of the flow to the frequency of the signal. (c) 2008 Elsevier Ltd. All rights reserved. [less ▲] Detailed reference viewed: 133 (30 ULg)Timescale- and Tracer-Based Methods for Understanding the Results of Complex Marine Models ; Delhez, Eric in Estuarine Coastal & Shelf Science (2007), 74 Detailed reference viewed: 24 (1 ULg)Timescale- and tracer-based methods for understanding the results of complex marine models ; Delhez, Eric in Estuarine Coastal & Shelf Science (2007), 74(4), - Detailed reference viewed: 47 (8 ULg)Overshootings and spurious oscillations caused by biharmonic mixing Delhez, Eric ; in Ocean Modelling (2007), 17(3), 183-198 Biharmonic mixing is often used in large scale numerical models of the ocean because of its scale selectivity; it effectively damps small scale noise and leaves the large scale dynamics nearly unaffected ... [more ▼] Biharmonic mixing is often used in large scale numerical models of the ocean because of its scale selectivity; it effectively damps small scale noise and leaves the large scale dynamics nearly unaffected. The biharmonic operator lacks however positiveness and monotonicity and can therefore produce unphysical results exhibiting spurious overshootings and oscillations. This problematic behaviour cannot be avoided by the addition of an ordinary Laplacian diffusion term. It appears in both continuous and discrete approaches/solutions in both unbounded and bounded domains. The overshootings and oscillations are induced by the strong damping of the smaller scale modes and are therefore comparable to the Gibbs' phenomenon. With appropriate boundary conditions, the variance of the field decreases monotonically and the oscillations are expected to remain small. The lack of positiveness is however a severe drawback for (dynamic) tracer studies. (C) 2007 Elsevier Ltd. All rights reserved. [less ▲] Detailed reference viewed: 73 (7 ULg)Unstructured, anisotropic mesh generation for the Northwestern European continental shelf, the continental slope and the neighbouring ocean ; ; Delhez, Eric et al in Continental Shelf Research (2007), 27(9), 1344-1356 A new mesh refinement strategy for generating high quality unstructured meshes of the Northwestern European continental shelf, the continental slope and the neighbouring ocean is presented. Our objective ... [more ▼] A new mesh refinement strategy for generating high quality unstructured meshes of the Northwestern European continental shelf, the continental slope and the neighbouring ocean is presented. Our objective is to demonstrate the ability of anisotropic unstructured meshes to adequately address the challenge of simulating the hydrodynamics occurring in these three regions within a unique mesh. The refinement criteria blend several hydrodynamic considerations as the tidal wave propagation on the continental shelf and the hydrostatic consistency condition in steep areas. Several meshes illustrate both the validity and the efficiency of the refinement strategy. The selection of the refinement parameters is discussed. Finally, an attempt is made to take into account tidal ellipses, providing another cause for anisotropy in the mesh. (c) 2007 Elsevier Ltd. All rights reserved. [less ▲] Detailed reference viewed: 56 (9 ULg)Diagnosis of the sediment transport in the Belgian Coastal Zone Mercier, Christophe ; Delhez, Eric in Estuarine Coastal & Shelf Science (2007), 74(4), 670-683 Estimating the age of particles in marine environment constitutes an invaluable tool to understand the interactions between complex flows and sediment dynamics, particularly in highly energetic coastal ... [more ▼] Estimating the age of particles in marine environment constitutes an invaluable tool to understand the interactions between complex flows and sediment dynamics, particularly in highly energetic coastal areas such as the Belgian Coastal Zone (Southern Bight in the North Sea). To this end, the Constituent Age and Residence time Theory-CART-introduced by Delhez, E.J.M., Campin, J.-M., Hirst, A.C., Deleersnijder, E. [1999a. Toward a general theory of the age in ocean modelling. Ocean Modelling 1, 17-27] for passive water constituents is extended to describe the sediment dynamics. It is then used in combination with a three-dimensional coupled hydrodynamic-sediment transport model to investigate sediment processes in the Belgian Coastal Zone focusing on two complementary aspects of the sediment dynamics: the internal sediment motion and redistribution within the Belgian coast; and the horizontal transport. (C) 2007 Elsevier Ltd. All rights reserved. [less ▲] Detailed reference viewed: 130 (42 ULg)Three dimensional sediment transport model of the Belgian coastal zone: application of the CART theory Mercier, Christophe ; Delhez, Eric Conference (2006) Suspended sediment processes in the Belgian coastal zone are discussed by presenting the first results given by a 3D coupled hydrodynamic-sediment transport model covering the Southern Bight in North Sea ... [more ▼] Suspended sediment processes in the Belgian coastal zone are discussed by presenting the first results given by a 3D coupled hydrodynamic-sediment transport model covering the Southern Bight in North Sea. The hydrodynamic sub-model is three-dimensional, baroclinic and includes a refined turbulence closure. In the sediment transport sub-model, the dynamic of sediments is described by an evolution equation of the sediment concentration in the water column and an equation for the sediment load on the seabed. Sedimentation is modeled through the addition of a sedimentation velocity in the vertical advection term. For the evaluation of deposition and erosion terms, formu- lae based on the calculation of the bottom stress under the combined effect of prevailing currents and waves are used. For more realism, sediment loads has also been split into different sediment classes with different sedimentation, deposition and erosion characteristics. The equations are solved by a finite volume method based on an Arakawa C grid, using sigma-transform and mode-splitting numerical methods. The horizontal resolution is 500x500 m and 10 vertical sigma-layers are used. Advection is handled using a TVD scheme with superbee flux limiter. Boundary conditions are provided by coupling this model to a 3D coupled hydrodynamic-sediment transport model of the North-Western European Continental Shelf. The results are validated with measurements carried out by Haecon NV and are compared with those obtained with the MU-STM model. A sensitivity analysis of the model to critical parameters of erosion, deposition and sedimentation based on the concept of residence time was carried out. [less ▲] Detailed reference viewed: 81 (6 ULg)The residence time of settling particles in the surface mixed layer ; Beckers, Jean-Marie ; Delhez, Eric in Environmental Fluid Mechanics (2006), 6(1), 25-42 The transport from the upper mixed layer into the pycnocline of particles with negative buoyancy is considered. Assuming the hydrodynamic parameters to be time-independent, an adjoint model is resorted to ... [more ▼] The transport from the upper mixed layer into the pycnocline of particles with negative buoyancy is considered. Assuming the hydrodynamic parameters to be time-independent, an adjoint model is resorted to that provides a general expression of the residence time in the mixed layer of the constituent under study. It is seen that the residence time decreases as the settling velocity increases or the diffusivity decreases. Furthermore, it is demonstrated that the residence time must be larger than z/w and smaller than h/w, where z, h and w denote the distance to the pycnocline, the thickness of the mixed layer and the sinking velocity. In the vicinity of the pycnocline, the residence time is not necessarily zero; its behaviour critically depends on the eddy diffusivity pro. le in this region. Closed-form solutions are obtained for constant and quadratic diffusivity profiles, which allows for an analysis of the sensitivity of the residence time to the Peclet number. Finally, an approximate value is suggested of the depth-averaged value of the residence time. [less ▲] Detailed reference viewed: 60 (12 ULg)On the behaviour of the residence time at the bottom of the mixed layer ; Beckers, Jean-Marie ; Delhez, Eric in Environmental Fluid Mechanics (2006), 6(6), 541-547 To understand why the findings of Deleersnijder et al. [(2006), Environ Fluid Mech 6: 25-42]-the residence time in the mixed layer in not necessarily zero at the pycnocline-are consistent with those of ... [more ▼] To understand why the findings of Deleersnijder et al. [(2006), Environ Fluid Mech 6: 25-42]-the residence time in the mixed layer in not necessarily zero at the pycnocline-are consistent with those of Delhez and Deleersnijder [(2006), Ocean Dyn 56:139-150]-the residence time in a control domain vanishes at the open boundaries of this control domain-, it is necessary to consider a control domain that includes part of the pycnocline, in which the eddy diffusivity is assumed to be zero. Then, depending on the behaviour of the eddy diffusivity near the bottom of the mixed layer, the residence time may be seen to exhibit a discontinuity at the interface between the mixed layer and the pycnocline. If such a discontinuity exists, the residence time is non-zero in the former and zero in the latter. This is illustrated by analytical solutions obtained under the assumption that the eddy diffusivity is constant in the mixed layer. [less ▲] Detailed reference viewed: 23 (8 ULg)The boundary layer of the residence time field Delhez, Eric ; in Ocean Dynamics (2006), 56(2), 139-150 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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 57 (10 ULg)Study of the seasonal cycle of the biogeochemical processes in the Ligurian Sea using a ID interdisciplinary model Raick, Caroline ; Delhez, Eric ; et al in Journal of Marine Systems (2005), 55(3-4), 177-203 A one-dimensional coupled physical-biogeochemical model has been built to study the pelagic food web of the Ligurian Sea (NW Mediterranean Sea). The physical model is the turbulent closure model (version ... [more ▼] A one-dimensional coupled physical-biogeochemical model has been built to study the pelagic food web of the Ligurian Sea (NW Mediterranean Sea). The physical model is the turbulent closure model (version I D) developed at the GeoHydrodynamics and Environmental Laboratory (GHER) of the University of Liege. The ecosystem model contains 19 state variables describing the carbon and nitrogen cycles of the pelagic food web. Phytoplankton and zooplankton are both divided in three size-based compartments and the model includes an explicit representation of the microbial loop including bacteria, dissolved organic matter, nano-, and microzooplankton. The internal carbon/nitrogen ratio is assumed variable for phytoplankton and detritus, and constant for zooplankton and bacteria. Silicate is considered as a potential limiting nutrient of phytoplankton's growth. The aggregation model described by Kriest and Evans in (Proc. Ind. Acad. Sci., Earth Planet. Sci. 109 (4) (2000) 453) is used to evaluate the sinking rate of particulate detritus. The model is forced at the air-sea interface by meteorological data coming from the "Cote d'Azur" Meteorological Buoy. The dynamics of atmospheric fluxes in the Mediterranean Sea (DYFAMED) time-series data obtained during the year 2000 are used to calibrate and validate the biological model. The comparison of model results within in situ DYFAMED data shows that although some processes are not represented by the model, such as horizontal and vertical advections, model results are overall in agreement with observations and differences observed can be explained with environmental conditions. (c) 2004 Elsevier B.V. All rights reserved. [less ▲] Detailed reference viewed: 73 (15 ULg) |
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