Beyond eddy diffusivity: an alternative model for turbulent dispersion

Turbulent dispersion proceeds not only much faster but also in a qualitatively different manner than molecular diffusion. Yet, the majority of hydraulic, oceanic and atmospheric models rely on the concept of an eddy diffusivity. It is shown here that an alternative model can be developed to exhibit observed behavior. The new term in the diffusion equation, which is non-local, may be interpreted in terms of the probability density function (pdf) of the turbulent velocity. Different assumptions about this distribution lead to a family of models, one of which is the model proposed here and another, the classical Fickian model of diffusion. A connection is also made with models using fractional calculus.     

Characterization of bedload intermittency near the threshold of motion using a Lagrangian sediment transport model

At the smallest scales of sediment transport in rivers, the coherent structures of the turbulent boundary layer constitute the fundamental mechanisms of bedload transport, locally increasing the instantaneous hydrodynamic forces acting on sediment particles, and mobilizing them downstream. Near the critical threshold for initiating sediment motion, the interactions of the particles with these unsteady coherent structures and with other sediment grains, produce localized transport events with brief episodes of collective motion occurring due to the near-bed velocity fluctuations. Simulations of these flows pose a significant challenge for numerical models aimed at capturing the physical processes and complex non-linear interactions that generate highly intermittent and self-similar bedload transport fluxes. In this investigation we carry out direct numerical simulations of the flow in a rectangular flat-bed channel, at a Reynolds number equal to Re = 3632, coupled with the discrete element method to simulate the dynamics of spherical particles near the bed. We perform two-way coupled Lagrangian simulations of 48,510 sediment particles, with 4851 fixed particles to account for bed roughness. Our simulations consider a total of eight different values of the non-dimensional Shields parameter to study the evolution of transport statistics. From the trajectory and velocity of each sediment particle, we compute the changes in the probability distribution functions of velocities, bed activity, and jump lengths as the Shields number increases. For the lower shear stresses, the intermittency of the global bedload transport flux is described by computing the singularity or multifr actal spectrum of transport, which also characterizes the widespread range of transport event magnitudes. These findings can help to identify the mechanisms of sediment transport at the particle scale. The statistical analysis can also be used as an ingredient to develop larger, upscaled models for predicting mean transport rates, considering the variability of entrainment and deposition that characterizes the transport near the threshold of motion.     

On interacting bee/mite populations: a stochastic model with analysis using cumulant truncation

Parasitism by the Varroa mite has had recent drastic impact on both managed and feral bee colonies. This paper proposes a stochastic population dynamics model for interacting African bee colony and Varroa mite populations. Cumulant truncation procedures are used to obtain approximate transient cumulant functions, unconstrained by the usual assumption of bivariate Normality, for an assumed large-scale model. The apparent size of the variance and skewness functions suggest the importance of the proposed truncation procedure which retains some higher-order cumulants, but determining the accuracy of the approximations is problematical. A smaller-scale bee/Varroa mite model is hence proposed and investigated. The accuracy for the means is exceptional, for the second-order cumulants is moderate, and for some third-order cumulants is poor. Notwithstanding the poor accuracy of a skewness approximation, the saddlepoint approximations for the marginal transient population size distributions are excellent. The cumulant truncation methodology is very general, and research is continuing in its application to this new class of host-parasite models.     

A stochastic model for the use of a country park

The subject of this theoretical study is a country park with a very delicate natural environment, located near large urban areas, as a result of which it receives a heavy inflow of visitors.The problem is that of determining the best organization of the park to achieve a distribution of visitors which is compatible with the protection of the natural environment. Therefore a mathematical model is sought which describes the distribution of visitors in relation to the various zones within the park.In this paper the mathematical model and its subsequent calibration are described.     

A simple Lagrangian model to simulate temporal variability of algae in the Elbe River

We present a five-year (1997–2001) numerical simulation of daily mean chlorophyll a concentrations at station Geesthacht Weir on the lower Elbe River (Germany) using an extremely simple Lagrangian model driven by (a) water discharge, global radiation, water temperature, and (b) silica observations at station Schmilka in the upper reach of the Elbe River. Notwithstanding the lack of many mechanistic details, the model is able to reproduce observed chlorophyll a variability surprisingly well, including a number of sharp valleys and ascents/descents in the observed time series. The model's success is based on the assumption of three key effects: prevailing light conditions, sporadic limitation of algal growth due to lack of silica and algae loss rates that increase above an empirically specified temperature threshold of 20 °C. Trimmed-down model versions are studied to analyse the model's success in terms of these mechanisms.In each of the five years the model consistently fails, however, to properly simulate characteristic steep increases of chlorophyll a concentrations after pronounced spring minima. Curing this model deficiency by global model re-calibration was found to be impossible. However, suspension of silica consumption by algae for up to 10 days in spring is shown to serve as a successful placeholder for processes that are disregarded in the model but apparently play an important role in the distinctly marked period of model failure. For the remainder of the year the very simple model was found to be adequate.     

A stochastic Lagrangian micromixing model for the dispersion of reactive scalars in turbulent flows: role of concentration fluctuations and improvements to the conserved scalar theory under non-homogeneous conditions

Several reaction schemes, based on the conserved scalar theory, are implemented within a stochastic Lagrangian micromixing model to simulate the dispersion of reactive scalars in turbulent flows. In particular, the formulation of the reaction-dominated limit (RDL) reaction scheme is here extended to improve the model performance under non-homogeneous conditions (NHRDL scheme). The validation of the stochastic model is obtained by comparison with the available measurements of reactive pollutant concentrations in a grid-generated turbulent flow. This test case describes the dispersion of two atmospheric reactant species (NO and O₃) and their reaction product (NO₂) in an unbounded turbulent flow. Model inter-comparisons are also assessed, by considering the results of state-of-the-art models for pollutant dispersion. The present validation shows that RDL reaction scheme provides a systematic overestimation (relative error of ca. 85% around the centreline) in computing the local reactant consumption/production rate, whereas the NHRDL scheme drastically reduces this gap (relative error lower than 5% around the centreline). In terms of NO₂ production (or reactant consumption), neglecting concentration fluctuations determines overestimations of the product mean of around 100% and a NO₂ local production of one order of magnitude higher than the reference simulation. In terms of standard deviations, the concentration fluctuations of both the passive and reactive scalars are generally of the same order of magnitude or up to 1 or 2 orders of magnitudes higher than the corresponding ensemble mean values, except for the background reactant close to the plume edges. The study highlights the importance of modelling pollutant reactions depending on the instantaneous instead of the mean concentrations of the reactants, thus quantifying the role of the turbulent fluctuations of concentration, in terms of scalar statistics (mean, standard deviation, intensity of fluctuations, skewness and kurtosis of concentration, segregation coefficient, simulated reaction rate). This stochastic particle method represents an efficient numerical technique to solve the convection–diffusion equation for reactive scalars and involves several application fields: micro-scale air quality (urban and street-canyon scales), accidental releases, impact of odours, water quality and fluid flow industrial processes (e.g. combustion).     

Local population dynamics of an invasive tree species with a complex life-history cycle: A stochastic matrix model

Biological invasions are widely accepted as having a major impact on ecosystem functioning worldwide, giving urgency to a better understanding of the factors that control their spread. Modelling tools have been developed for this purpose but are often discrete-space, discrete-time spatial-mechanistic models that adopt a computer simulation approach and resist mathematical analysis. We constructed a simple demographic matrix model to explore the local population dynamics of an invasive species with a complex life history and whose invasive success depends on resource availability, which occurs stochastically. As a case study we focused on the American black cherry (Prunus serotina Ehrh.), a gap-dependent tree able both to constitute a long-living seedling bank under unfavourable light conditions and to resprout vigorously once cut-down, which is invading European temperate forests. The model used was a stage-classified matrix population model (i.e., Lefkovitch matrix), integrating environmental stochasticity. Stochastic matrix projection analysis was combined with elasticity analysis and stochastic simulations to search for the species’ ‘Achille heel’. As expected, the population growth rate (i.e., Lyapunov exponent), which measures the risk of P. serotina invasion at the stand scale, increased with light frequency. There was a critical value above which the population of P. serotina explodes and below which it locally goes extinct. The resprouting capacity usually speed up the invasion but appeared to play a minor role. The mean duration of stand invasion was measured and important life stage transitions that mostly contribute to the local stochastic growth rate were identified. Some relevant management implications are discussed and the interest of such models for the understanding of demographic characteristics of invasive species is stressed.     

Exploring fisheries strategies for the western English Channel using an ecosystem model

An ecosystem model of the western English Channel ecosystem in 1994 was used to explore the effects of the use of a fishing policy optimization routine on profits, number of jobs and ecosystem structure. The optimization for single objective led to the specialization of the fishing fleet, with some fleet types being almost excluded. The profits and mainly the job optimizations led to big changes in the ecosystem structure, with loss of diversity, but the overall biomass of all vertebrate groups represented in the model increased considerably. For the objective focusing on ecosystem structure, there was an increase in biodiversity, with many long-lived groups predicted to increase, although the overall vertebrate biomass suffered just a small increase. An “ideal” mixed policy configuration was found when slightly greater weight was given to ecosystem structure than was given to profits and jobs. This scenario led to an overall reduction in effort but also to increased profits and biodiversity, while keeping the number of jobs at the same level as the baseline estimates. The results of the optimizations showed that the average trophic level of the catches is quite resistant to changes in the underlying system structure. On the other hand, despite the high level of aggregation of the model structure, a biodiversity index estimated by the model presented large changes as a function of the weights placed on the single policy functions, reflecting the changes in the system structure. The output of the application of the fishing optimization presented here should be considered in qualitative rather than in quantitative terms as an aid and part contribution to the complicated discussions on future long term management actions. Nonetheless it points to an overall reduction in fishing capacity, an objective widely accepted within the scientific community, while keeping the fishery in a profitable state.     

The influence of hydrodynamic processes on zooplankton transport and distributions in the North Western Mediterranean: Estimates from a Lagrangian model

A Lagrangian module has been developed and coupled with the 3D circulation model Symphonie to study the influence of hydrodynamic processes on zooplankton transport and distributions in the North Western Mediterranean (NWM). Individuals are released every 3 days from March to August 2001 in two initial areas: around the DYFAMED sampling station in the central Ligurian Sea and in the Rhône river plume. Then the individuals are tracked for 40 days either as passive particles or with a simple diel vertical migration (DVM) pattern. The simulations suggest strong seasonal patterns in the distributions of the individuals released around the DYFAMED sampling station. Individuals spread all over the NWM basin after 40 days but different patterns occur depending on the season, the initial depths of release and the capacity of DVM. Offshore-shelf transport only occurs in April and May with particles ending up in the Gulf of Lions (GoL) in low concentrations. In other months, the Northern Current (NC) can be considered as a barrier for particles entering the GoL from the offshore sea. A quarter to a half of passive individuals released in the Rhône river plume remain in the GoL. The rest is transported by the NC towards the Catalan Sea. Applying a simple DVM scheme does not increase the retention of particles on the shelf.     

Litter decomposition and nitrogen mineralization from an annual to a monthly model

The forest litter decomposition model (FLDM) described in this paper provides an important basis for assessing the impacts of forest management on seasonal stream water quality and export of dissolved organic carbon (DOC). By definition, models with annual time steps are unable to capture seasonal, within-year variation. In order to simulate seasonal variation in litter decomposition and DOC production and export, we have modified an existing annual FLDM to account for monthly dynamics of decomposition and residual mass in experimental litterbags placed in 21 different forests across Canada.The original annual FLDM was formulated with three main litter pools (fast, slow, and very slow decomposing litter) to address the fact that forest litter is naturally composed of a mixture of organic compounds that decompose at different rates. The annual FLDM was shown to provide better simulations than more complex models like CENTURY and SOMM.The revised monthly model retains the original structure of the annual FLDM, but separates litter decomposition from nitrogen (N) mineralization. In the model, monthly soil temperature, soil moisture, and mean January soil temperature are shown to be the most important controlling variables of within-year variation in decomposition. Use of the three variables in a process-based definition of litter decomposition is a significant departure from the empirical definition in the annual model. The revised model is shown to give similar calculations of residual mass and N concentration as the annual model (r² = 0.91, 0.78), despite producing very different timeseries of decomposition over six years. It is shown from a modelling perspective that (i) forest litter decomposition is independent of N mineralization, whereas N mineralization is dependent on litter decomposition, and (ii) mean January soil temperature defines litter decomposition in the summer because of winter-temperatures’ role in modifying forest-floor microorganism community composition and functioning in the following summer.     

Spatial interpolation of air temperature using environmental context: Application to a crop model

The air temperature is one of the main input data in models for water balance monitoring or crop models for yield prediction. The different phenological stages of plant growth are generally defined according to cumulated air temperature from the sowing date. When these crop models are used at the regional scale, the meteorological stations providing input climatic data are not spatially dense enough or in a similar environment to reflect the crop local climate. Hence spatial interpolation methods must be used. Climatic data, particularly air temperature, are influenced by local environment. Measurements show that the air above dry surfaces is warmer than above wet areas. We propose a method taking into account the environment of the meteorological stations in order to improve spatial interpolation of air temperature. The aim of this study is to assess the impact of these corrected climatic data in crop models. The proposed method is an external drift kriging where the Kriging system is modified to correct local environment effects. The environment of the meteorological stations was characterized using a land use map summarized in a small number of classes considered as a factor influencing local temperature. This method was applied to a region in south-east France (150×250 km) where daily temperatures were measured on 150 weather stations for two years. Environment classes were extracted from the CORINE Landcover map obtained from remote sensing data. Categorical external drift kriging was compared to ordinary kriging by a cross validation study. The gain in precision was assessed for different environment classes and for summer days. We then performed a sensitivity study of air temperature with the crop model STICS. The influence of interpolation corrections on the main outputs as yield or harvest date is discussed. We showed that the method works well for air temperature in summer and can lead to significant correction for yield prediction. For example, we observed by cross validation a bias reduction of 0.5 to 1.0°C (exceptionally 2.5°C for some class), which corresponds to differences in yield prediction from 0.6 to 1.5 t/ha.     

Application of a multi-step computation scheme to an intratidal estuarine water quality model

A computation scheme, which involves multi-step computation of the decoupled mass-balance equations in water quality models, was proposed by Park and Kuo (1996. Water Res. 30, 2255–2264). Using a simple hypothetical model, they explained the concept and demonstrated the accuracy and computational efficiency of the multi-step scheme. This paper describes the concept and the implementation of the multi-step computation scheme in a more generalized format for both intratidal and intertidal estuarine water quality models. The scheme is applied to a prototype estuary using a laterally-integrated two-dimensional intratidal model. The computational efficiency of the scheme is demonstrated by comparing the model results using the multi-step scheme with those using the traditional single-step explicit numerical solution of the kinetic processes.     

Application of an evolutionary algorithm to the inverse parameter estimation of an individual-based model

Inverse parameter estimation of individual-based models (IBMs) is a research area which is still in its infancy, in a context where conventional statistical methods are not well suited to confront this type of models with data. In this paper, we propose an original evolutionary algorithm which is designed for the calibration of complex IBMs, i.e. characterized by high stochasticity, parameter uncertainty and numerous non-linear interactions between parameters and model output. Our algorithm corresponds to a variant of the population-based incremental learning (PBIL) genetic algorithm, with a specific “optimal individual” operator. The method is presented in detail and applied to the individual-based model OSMOSE. The performance of the algorithm is evaluated and estimated parameters are compared with an independent manual calibration. The results show that automated and convergent methods for inverse parameter estimation are a significant improvement to existing ad hoc methods for the calibration of IBMs.     

Application of a spatial meta-population model with stochastic parameters to the management of the invasive grass Nassella trichotoma in North Canterbury, New Zealand

Optimising the management of invasive plants requires the identification of the population size outcomes for alternative management strategies. Mathematical models can be useful tools for making such management strategy comparisons. In this paper we develop a generic landscape meta-population model and apply it to the weedy grass, Nassella trichotoma, an invasive species occupying approximately 800 land parcels, predominantly pastoral farms, in the Hurunui district, North Canterbury, New Zealand. Empirical evidence reveals that this meta-population is currently stable (at a median density of 6 plants ha⁻¹) under a community strategy requiring manual removal (termed ‘grubbing’) of plants annually from all land parcels. Reduction in population size requires an alternative management strategy. Field data, collected over a 12 year period, were used to provide stochastic parameter values for land parcel size, carrying capacity, rates of local population growth and grubbing.The model reveals that at steady state, the most significant contribution to population growth on a land parcel comes from within the land parcel itself; the expected annual per capita growth on an individual land parcel is 4 orders of magnitude greater than the expected annual contribution from plants arising from other land parcels. This result implies that many of the farms currently supporting N. trichotoma may pose little or no threat to, nor are threatened themselves by, other farms infested by the weed. However, the steady state distribution (of the weed's population density) was sensitive to the spread rate, revealing a need for data on this process. It was also sensitive to how any increase in the grubbing rate is distributed; increasing it via a uniform distribution U(0, 1) where all rates between 0 and 100% year⁻¹ are equally probable did not affect the steady state, whereas increasing the rates via the uniform distribution U(0.25, 0.75) resulted in fewer farms with high population densities. In general the model provides a basis for exploring the effects of a wide range of alternative grubbing strategies on population growth in N. trichotoma.     

Lip-reading in remote subjects: an attempt to quantify and separate ingestion, breathing and vocalisation in free-living animals using penguins as a model

A new mandibular sensor is presented here based on the use of a Hall sensor, attached to one mandible, opposite a magnet, attached to the other mandible. Changes in sensor voltage, proportional to magnetic field strength, and thus inter-mandibular angle, are recorded in a logger. This system was tested on seven captive Adélie penguins (Pygoscelis adeliae) and three gentoo penguins (Pygoscelis papua) during: (1) feeding trials on land, where birds were given known quantities and types of food; and (2) trials in water where birds were allowed to swim and dive freely. In addition, six free-living Magellanic penguins (Spheniscus magellanicus) were equipped with the system for single foraging trips. Angular signatures were looked for in instances when both captive and free-living birds might open their beaks, and it was discovered that five major behaviours could be identified: ingestion, breathing, calling, head shaking and preening. Captive feeding trials showed that prey mass could be determined with reasonable accuracy (r²=0.92), and there was some indication that prey type could be resolved if recording frequency were high enough. Vocalisations in Adélie penguins (arc calls) took <0.7 s for mean maximum beak angles of 4.2° (SD 1.3), and were distinguished by their relatively gradual change in beak angle and by their high degree of symmetry. Beak shakings were distinguishable by their short duration (multiple peaks of <0.5 s) and minimal maximum angle (<0.5°). Preening behaviour was apparent due to multiple decreasing peaks (angles <8°). Breathing could be subdivided into that during porpoising, where a characteristic double peak in beak angle was recorded, and that during normal surface rests between dives. During porpoising, only the primary peak (mean maximum beak angle 25.1°, SD 4.7) occurred when the bird was out of the water (mean maximum for second peak 5.9°, SD 4.1). During normal surface rests in free-living birds, breaths could be distinguished as a series of beak openings and closures, showing variation in amplitude and frequency according to an apparent recovery from the previous dive and preparation for the subsequent dive to come. The mandibular measuring system presented shows considerable promise for elucidating many hitherto intractable aspects of the behaviour of free-living animals.     

Estimating the spatio-temporal risk of disease epidemics using a bioclimatic niche model

There are presently few tools available for estimating epidemic risks from forest pathogens, and hence informing pro-active disease management. In this study we demonstrated that a bioclimatic niche model can be used to examine questions of epidemic risk in temperate eucalypt plantations. The bioclimatic niche model, CLIMEX, was used to identify regional variation in climate suitability for Mycosphaerella leaf disease (MLD), a major cause of foliage damage in temperate eucalypt plantations around the world. Using historical observations of MLD damage, we were able to convert the relative score of climatic suitability generated by CLIMEX into a severity ranking ranging from low to high, providing for the first time a direct link between risk and impact, and allowing us to explore disease severity in a way meaningful to forest managers. We determined that the ‘Compare Years’ function in CLIMEX could be used for site-specific risk assessment to identify severity, frequency and seasonality of MLD epidemics. We explored appropriate scales of risk assessment for forest managers. Applying the CLIMEX model of MLD using a 0.25° or coarser grid size to areas of sharp topographic relief frequently misrepresented the risk posed by MLD, because considerable variation occurred between individual forest sites encompassed within a single grid cell. This highlighted the need for site-specific risk assessment to address many questions pertinent to managing risk in plantations.     

Implementing a new model to measure and assess eco-effectiveness as an indicator of sustainability

A model able to measure eco-effectiveness, which was theoretically proposed and illustrated in a previous article, is here developed and implemented for three case studies (USA, EU-15 and Japan) in order to verify its potential usefulness in the natural resources field. The results obtained by the application of the 'loss function' – ability to measure the distance between a real and an ideal situation – show that the model can be seen as a fundamental basis to assess the exploitation intensity of resources among different countries/areas and/or over time, especially if new measures and best environmental practices are adopted to increase resource productivity and to reduce environmental burdens. Since a main issue in the model implementation is related to the subjective choice of weights to be given to the various resources in the loss function (materials, fossil fuels and biomass), a graphical approach based on a 'weighting triangle' is also proposed to overcome the problem of previous weight selection. This tool permits us to make a distinction between a 'non-sustainability' area and a 'sustainability' area, based on whether an increase or a decrease of the loss function is more plausible.     

Simulation of NO and N2O emissions from a spruce forest during a freeze/thaw event using an N-flux submodel from the PnET-N-DNDC model integrated to CoupModel

The amount of nitrogen gases (N₂O, NO and N₂) emitted from forest soils depends on interactions between soil properties, climatic factors and soil management. To increase the understanding of nitrogen processes in soil ecosystems, two dynamic models, CoupModel (coupled heat and mass transfer model for soil–plant–atmosphere systems) and the denitrification–decomposition (DNDC) model were selected. Both are dynamic models with different submodels for soil, vegetation, hydrology and climate system. CoupModel has a higher degree of detail on soil physical and abiotic components, whereas the DNDC model contains details of microbiological processes involved in production of nitrogen gases. To improve the previous simple submodel of nitrogen emission in CoupModel, we included a submodel corresponding to the forest version of DNDC containing photosynthesis/evapotranspiration-nitrogen (PnET-N-DNDC model).     

Sensitivity of normal theory methods to model misspecification in the calculation of upper confidence limits on the risk function for continuous responses

Normal theory procedures for calculating upper confidence limits (UCL) on the risk function for continuous responses work well when the data come from a normal distribution. However, if the data come from an alternative distribution, the application of the normal theory procedures may lead serious over- or under-coverage depending upon the alternative distribution. In this paper we conduct simulation studies to investigate the sensitivity of three normal theory UCL procedures to departures from normality. Data from several gamma, reciprocal gamma, and lognormal distributions are considered. The normal theory procedures are applied to both the raw data and the log-transformed data.     

Study of the nutrient and plankton dynamics in Lake Tanganyika using a reduced-gravity model

An eco-hydrodynamic (ECOH) model is proposed for Lake Tanganyika to study the plankton productivity. The hydrodynamic sub-model solves the non-linear, reduced-gravity equations in which wind is the dominant forcing. The ecological sub-model for the epilimnion comprises nutrients, primary production, phytoplankton biomass and zooplankton biomass. In the absence of significant terrestrial input of nutrients, the nutrient loss is compensated for by seasonal, wind-driven, turbulent entrainment of nutrient-rich hypolimnion water into the epilimnion, which gives rise to high plankton productivity twice in the year, during the transition between two seasons. Model simulations predict well the seasonal contrasts of the measured physical and ecological parameters. Numerical tests indicate that the half saturation constant for grazing by zooplankton and the fish predation rate on zooplankton affect the zooplankton biomass measurably more than that of phytoplankton biomass. This work has implications for the application of this model to predict the climatological biological productivity of Lake Tanganyika.