On the mathematical stability of stratified flow models with local turbulence closure schemes

Occasionally, numerical simulations using local turbulence closure schemes to estimate vertical turbulent fluxes exhibit small-scale oscillations in space, causing the eddy coefficients to vary over several orders of magnitude on short distances. Theoretical developments suggest that these spurious oscillations are essentially due to the way the eddy coefficients depend on the vertical gradient of the model’s variables. An instability criterion is derived based on the assumptions that the artefacts under study are due to the development of small-amplitude, small time- and space-scale perturbations of a smooth solution. The relevance of this criterion is demonstrated by applying it to a series a closure schemes, ranging from the Pacanowski–Philander formulas to the Mellor–Yamada level 2.5 model.     

Spatial variability in growth and reproduction of the Pacific oyster Crassostrea gigas (Thunberg, 1793) along the west European coast

The Pacific oyster Crassostrea gigas was introduced in Europe for commercial purposes in the mid 1960s. It was initially thought that low winter temperatures would restrain this species' reproduction and settlement; however, its present distribution in areas where no introduction has taken place suggests that natural invasion and expansion has occurred. Along the European coast, wild populations of Pacific oysters are already found from northern Germany to southern Portugal. Whether C. gigas will continue to further expand through northern waters will depend on its physiological performance. In this study, the performance of wild oyster populations has been studied in terms of growth and reproduction at three stations: La Rochelle (France; 46°N), Yerseke (Oosterschelde estuary, The Netherlands, 51°N), and Texel (Wadden Sea estuary, The Netherlands, 53°N). The French population had the lowest somatic-shell mass ratio and an increase in maximum shell length, somatic and gonadal mass was observed from France to the Netherlands. In addition, mean oocyte diameter decreased significantly from south to north. The combination of increasing gonadal mass and decreasing oocyte volume suggests an increasing reproductive output in terms of egg numbers from France to The Netherlands. Differences in temperature between locations will at least be partly responsible for the observed patterns; however, other environmental factors (such as food availability, predation pressure, sediment type and/or seston concentration) cannot be excluded. Since smaller eggs (oocytes) are thought to have a longer development time, the environmental conditions along the Dutch coast may result in increased larval dispersal and possibly in further population expansion.     

Growth and reproduction of the bivalve Spisula subtruncata (da Costa) in Dutch coastal waters

The bivalve Spisula subtruncata is usually abundant in shallow coastal waters along the Dutch coast. However, its biomass has been decreasing since 1995. In order to assess whether reproductive failure may be the cause of the observed decline over the last decades, the energy investment in reproduction of a population of S. subtruncata from central Dutch coastal waters was studied. The population studied consisted of individuals of up to four years old. Shell length reached maximum values of around 32 mm and individual total body, somatic and gonadal ash-free dry mass reached maximum values of about 278 mg AFDM, 252 mg AFDM and 76 mg AFDM, respectively. A clear seasonal cycle in somatic and gonadal mass was observed. Somatic and gonadal mass indices increased in early spring and reached maximum values during summer, followed by a decrease to minimum values at the beginning of the following year. Spawning was in June–July and settlement of spat seems to have occurred in July–August. Mean oocyte diameter was 57.43 ± 0.03 μm, corresponding to a volume of 98972 μm³. These results suggested that reproductive failure was not the cause of the current population decline. Most likely, unsuccessful settlement of spat and/or severe predation during the first months of life were responsible for the observed patterns.     

Seasonal variability in somatic and reproductive investment of the bivalve Scrobicularia plana (da Costa, 1778) along a latitudinal gradient

Monthly investment in soma and gonads in the bivalve Scrobicularia plana is described for three populations along its distributional range: Minho estuary, Portugal; Westerschelde estuary, The Netherlands and Buvika estuary, Norway. Seasonal cycles in body mass (BMI), somatic mass (SMI) and gonadal mass (GMI) indices were observed for all populations. In Portugal, BMI and SMI peaked in mid-autumn, while in The Netherlands both indices were at their highest in mid-spring. Norway showed a different pattern with two distinct peaks: one in mid-autumn and a second peak in spring. GMI reached maximum values in July in Portugal and Netherlands and in June in Norway. Overall, mean BMI and SMI were lower in Portugal while mean GMI was lower in Norway. The spawning period lasted the whole summer in Portugal, but was shorter (only two months) in The Netherlands and Norway. The reproductive investment in The Netherlands was significantly higher than in Portugal and Norway, with the lowest values being observed in Norway. Differences in annual cycles between populations were attributed to environmental factors, namely temperature and food availability. Temperature seems important in shaping the reproductive pattern with more northern populations showing shorter reproductive periods starting later in the year, and a lower reproductive output. In addition, winter water temperatures can explain the lower mean body and somatic mass values observed in Portugal. Food availability influenced the physiological performance of the species with peaks in somatic mass coinciding with phytoplankton blooms. This relation between physiological performance and environmental factors influences S. plana distribution, densities and even survival, with natural consequences on its commercial importance.     

The effect of geometry and tidal forcing on hydrodynamics and net sediment transport in semi-enclosed tidal basins

A new depth-averaged exploratory model has been developed to investigate the hydrodynamics and the tidally averaged sediment transport in a semi-enclosed tidal basin. This model comprises the two-dimensional (2DH) dynamics in a tidal basin that consists of a channel of arbitrary length, flanked by tidal flats, in which the water motion is being driven by an asymmetric tidal forcing at the seaward side. The equations are discretized in space by means of the finite element method and solved in the frequency domain. In this study, the lateral variations of the tidal asymmetry and the tidally averaged sediment transport are analyzed, as well as their sensitivity to changes in basin geometry and external overtides. The Coriolis force is taken into account. It is found that the length of the tidal basin and, to a lesser extent, the tidal flat area and the convergence length determine the behaviour of the tidally averaged velocity and the overtides and consequently control the strength and the direction of the tidally averaged sediment transport. Furthermore, the externally prescribed overtides can have a major influence on tidal asymmetry in the basin, depending on their amplitude and phase. Finally, for sufficiently wide tidal basins, the Coriolis force generates significant lateral dynamics.     

Differential reproductive strategies of two bivalves in the Dutch Wadden Sea

Cerastoderma edule and Mya arenaria are two common bivalve species in European waters. Longevity and maximum size are much greater in the latter species. Because comparison of species life-history strategies states that a long life span (i.e. high annual survival) generally goes with lower fecundity, we hypothesise that reproductive output would be lower in M. arenaria than in C. edule. In the present paper, we studied the reproductive strategies of these two species in an intertidal and a subtidal area of the western Dutch Wadden Sea, by following seasonal changes in absolute and relative weights of somatic and gonadal tissues in these bivalves. Starting of spawning was similar in the two species, around May, except for intertidal M. arenaria, which initiated spawning in August. Individual energy investment in reproduction was similar for the two species but, unlike M. arenaria, C. edule spawned completely, releasing all energy of gonadal mass in the form of gametes. Mya arenaria used the gonad not only for reproduction but also for storage. In the intertidal area, we found a trade-off between longevity and reproduction, i.e. maximum reproductive output (expressed as a proportion of body mass) was higher in C. edule than in M. arenaria. However, since body size is larger and life span longer in M. arenaria than in C. edule, mean lifetime reproductive output per individual must be higher in the first than in the latter. Based on the differences in reproductive strategies of these two species, we hypothesise that the negative effects of warming climate on bivalve population dynamics in the Wadden Sea will be stronger in C. edule than in M. arenaria.     

The AquaDEB project (phase I): Analysing the physiological flexibility of aquatic species and connecting physiological diversity to ecological and evolutionary processes by using Dynamic Energy Budgets

The European Research Project AquaDEB (2007–2011, http://www.ifremer.fr/aquadeb/) is joining skills and expertise of some French and Dutch research institutes and universities to analyse the physiological flexibility of aquatic organisms and to link it to ecological and evolutionary processes within a common theoretical framework for quantitative bioenergetics [Kooijman, S.A.L.M., 2000. Dynamic energy and mass budgets in biological systems. Cambridge University Press, Cambridge]. The main scientific objectives in AquaDEB are i) to study and compare the sensitivity of aquatic species (mainly molluscs and fish) to environmental variability of natural or human origin, and ii) to evaluate the related consequences at different biological levels (individual, population, ecosystem) and temporal scales (life cycle, population dynamics, evolution). At mid-term life, the AquaDEB collaboration has already yielded interesting results by quantifying bio-energetic processes of various aquatic species (e.g. molluscs, fish, crustaceans, algae) with a single mathematical framework. It has also allowed to federate scientists with different backgrounds, e.g. mathematics, microbiology, ecology, chemistry, and working in different fields, e.g. aquaculture, fisheries, ecology, agronomy, ecotoxicology, climate change. For the two coming years, the focus of the AquaDEB collaboration will be in priority: (i) to compare energetic and physiological strategies among species through the DEB parameter values and to identify the factors responsible for any differences in bioenergetics and physiology; and to compare dynamic (DEB) versus static (SEB) energy models to study the physiological performance of aquatic species; (ii) to consider different scenarios of environmental disruption (excess of nutrients, diffuse or massive pollution, exploitation by man, climate change) to forecast effects on growth, reproduction and survival of key species; (iii) to scale up the models for a few species from the individual level up to the level of evolutionary processes.