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A numerical approach to estimating the mean exposure of a tracer to a scalar property is applied to predict the light exposure of sinking phytoplankton. In our application, effective light exposure is defined as the time integral of a light limitation factor commonly used in phytoplankton models. Solutions from an adjoint approach have previously been published for a set of one-dimensional sinking phytoplankton scenarios. We illustrate that a simple extension to a standard advection–diffusion model produces similar results. Specifically, we present numerical solutions using property tracking in a one-dimensional advection–diffusion model for multiple tracers. Solutions are calculated for a range of eddy diffusivity distributions and compared with the published solutions. The consistency of the numerical solutions with the published solutions provides validation of the property-tracking approach. While the adjoint method solution is much less computationally intensive for the test cases, the property-tracking approach can be applied in multidimensional time-varying applications with an arbitrary distribution of sinking speed, diffusivity, and turbidity for which an adjoint solution has not been developed. Our intention is for this example application and corroboration of the “property-tracking” approach to inspire readers to envision additional applications for this approach.

Offshore wind farms (OWFs) are an important source of renewable energy accounting for 2.3% of the European Union's electricity demand. Yet their impact on the environment needs to be assessed. Here, we couple a hydrodynamic (including tides and waves) and sediment transport model with a description of the organic carbon and mineral particle dynamics in the water column and sediments. The model is applied to the Belgian Coastal Zone (BCZ) where OWFs currently occupy 7% of its surface area which is estimated to double in the next 5 years. The impact of OWFs on the environment is represented through the filtration of the water column and fecal pellets production by the blue mussel, the dominant fouling organism. Our model simulations show that the impact of biodeposition on the mud particle sedimentation and on sediment composition is small compared to the fluxes associated with tidal deposition and resuspension and the lateral inputs. In contrast, the total organic carbon (TOC) flux to the sediment is significantly altered inside the OWF perimeters and TOC deposition is increased up to 50% in an area 5 km around the monopiles. Further away, the TOC flux to the bottom decreases with a notable effect up to 30 km away. The major changes are found along the direction of the main residual current and tidal ellipse's major axis. In addition, sub-mesoscale gyres act as retention areas with increased carbon deposition. A future OWF in the BCZ will be located close to gravel beds in a Natura 2000 area, considered as vulnerable habitats and biodiversity hotspots. The different scenarios for this OWF, varying in turbine number and positioning, are compared in terms of impact on the carbon and mineral particle deposition flux in the BCZ and, particularly, to these gravel beds. The scenarios show that the number of turbines has only a slight impact on the TOC deposition flux, unlike their positioning that significantly alters the TOC flux to the gravel beds. The TOC deposition flux exceeds 50%, when the turbines are placed next to the gravel beds; while a limited increase is simulated, when the turbines are located the farthest possible from them.

The concept of age is widely used to quantify the transport rate of tracers - or pollutants - in the environment. The age focuses only on the time taken to reach a given location and disregards other aspects of the path followed by the tracer parcel. To keep track of the subregions visited by the tracer parcel along this path, partial ages are defined as the time spent in the different subregions. Partial ages can be computed in an Eulerian framework in much the same way as the usual age by extending the Constituent oriented Age and Residence Time theory (CART, www.climate.be/CART ). In addition to the derivation of theoretical results and properties of partial ages, applications to a 1D model with lateral/transient storage, to the 1D advection-diffusion equation and to the diagnosis of the ventilation of the deep ocean are provided. They demonstrate the versatility of the concept of partial age and the potential new insights that can be gained with it.

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.

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 computational domain. The optimization 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 Gaussian-shaped 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.

The age of a water particle in a shallow man-made reservoir is defined as the time elapsed since it entered it. Analyzing this diagnostic timescale provides valuable information for optimally sizing and operating such structures. Here, the constituent-oriented age and residence time theory (CART) is used to obtain not only the mean age, but also the water age distribution function at each location. The method is applied to 10 different shallow reservoirs of simple geometry (rectangular), in a steady-state framework. The results show that complex, multimodal water age distributions are found, implying that focusing solely on simple statistics (e.g., mean or median age) fails to reflect the complexity of the actual distribution of water age. The latter relates to the fast or slow pathways that water particles may take for traveling from the inlet to the outlet of the reservoirs.

At high Peclet number, the residence time exhibits a boundary layer adjacent to incoming open boundaries. In a Eulerian model, not resolving this boundary layer can generate spurious oscillations that can propagate into the area of interest. However, resolving this boundary layer would require an unacceptably high spatial resolution. Therefore, alternative methods are needed in which no grid refinement is required to capture the key aspects of the physics of the residence time boundary layer. An extended finite element method representation and a boundary layer parameterisation are presented and tested herein. It is also explained how to preserve local consistency in reversed time simulations so as to avoid the generation of spurious residence time extrema. Finally, the boundary layer parameterisation is applied to the computation of the residence time in the Scheldt Estuary (Belgium/The Netherlands). This timescale is simulated by means of a depth-integrated, finite element, unstructured mesh model, with a high space-time resolution. It is seen that the residence time temporal variations are mainly affected by the semi-diurnal tides. However, the spring-neap variability also impacts the residence time, particularly in the sandbank and shallow areas. Seasonal variability is also observed, which is induced by the fluctuations over the year of the upstream flows. In general, the residence time is an increasing function of the distance to the mouth of the estuary. However, smaller-scale fluctuations are also present: they are caused by local bathymetric features and their impact on the hydrodynamics.

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.[PUBLICATION ABSTRACT]

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 profile 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.[PUBLICATION ABSTRACT]

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.