Sensitivity of the solution of the Elder problem to density, velocity and numerical perturbations

In this paper the Elder problem is studied with the purpose of evaluating the inherent instabilities associated with the numerical solution of this problem. Our focus is first on the question of the existence of a unique numerical solution for this problem, and second on the grid density and fluid density requirements necessary for a unique numerical solution. In particular we have investigated the instability issues associated with the numerical solution of the Elder problem from the following perspectives: (i) physical instability issues associated with density differences; (ii) sensitivity of the numerical solution to idealization irregularities; and, (iii) the importance of a precise velocity field calculation and the association of this process with the grid density levels that is necessary to solve the Elder problem accurately. In the study discussed here we have used a finite element Galerkin model we have developed for solving density-dependent flow and transport problems, which will be identified as TechFlow. In our study, the numerical results of Frolkovic and de Schepper [Frolkovic, P. and H. de Schepper, 2001. Numerical modeling of convection dominated transport coupled with density-driven flow in porous media, Adv. Water Resour., 24, 63-72.] were replicated using the grid density employed in their work. We were also successful in duplicating the same result with a less dense grid but with more computational effort based on a global velocity estimation process we have adopted. Our results indicate that the global velocity estimation approach recommended by Yeh [Yeh, G.-T., 1981. On the computation of Darcian velocity and mass balance in finite element modelling of groundwater flow, Water Resour. Res., 17(5), 1529-1534.] allows the use of less dense grids while obtaining the same accuracy that can be achieved with denser grids. We have also observed that the regularity of the elements in the discretization of the solution domain does make a difference in obtaining a unique stationary solution for this problem. The results of our study also indicate that the density differences are critical in the solution of the Elder problem and that high density differences lead to the physical instability that is inherent with this problem. Other than the physical instability associated with the level of density differences used in the Elder problem, the following two points should be considered in solving the Elder problem in a consistent manner: (i) strict attention should be paid to the vertical grid Peclet number in developing the criteria for convergent grid selection; and, (ii) with a globally continuous velocity calculation stable solutions can be obtained at lower grid densities.     

Relationships between diversity of grassland vegetation, field characteristics and land use management practices assessed at the farm level

In order to assess the impact of policies to encourage extensification in less favoured areas and improve our knowledge of extensive livestock systems, we analyzed relationships between the diversity of grassland vegetation and land use management practices and field characteristics. This study, conducted on a mountainous area in the centre of France, was based on 149 fields, mainly of natural grasslands belonging to 7 farmers. Regression analyses were performed to analyze the relations between the grassland vegetation types (five types established from the list of dominant species), management practices (cutting versus grazing and fertilization) and the topographic (altitude and aspect) and topologic (slope, distance and surface area) characteristics of the fields. The land use management rules used by the farmers were studied by specifying the grazing management rules of the herd (dairy cows), as well as those for conserved forage (mainly hay or silage) and were identified from observations mentioned on the “grazing schedules”, as well as from interviews at the beginning, in the middle and at the end of the study period. The statistical analysis showed that neither the topographic characteristics of the fields nor the distance from the cowshed or surface area were correlated with the grassland vegetation types. It was the management practices used, largely determined by the field slope, which determined the grassland vegetation type. On the other hand, farmers’ statements showed that the grazing and cutting management rules were mostly determined by the slope of the fields and the distance from the cowshed and, to a lesser extent, by the altitude and aspect. These results showed that the farmers take into consideration environmental differences when choosing fields to allocate for grazing and cutting at different seasons, particularly when they are constrained by these features. Nevertheless, when the constraints were minimal, a wide diversity of grassland vegetation types was also observed. This diversity was a result of attributing different functions to the fields which led to different management practices (defoliation methods and fertilization) and, thus, to different grassland vegetation types. Consequently, for farms where animal feed requirements vary according to the time of the year and the type of animal, we suggest that diversity in the grassland vegetation types is a sound component of these livestock systems.     

Modelling above-ground herbage mass for a wide range of grassland community types

Whereas it is recognized that management of plant diversity can be the key to reconciling production and environmental aims, most grassland models are tailored for high-value grass species. We proposed to adapt a mono-specific grass model to take into account specific features of species-rich permanent grasslands, especially over the reproductive phase. To this end, we used the concept of plant functional type (PFT), i.e. the grouping of plant species according to plant traits determined by the response of plant species to different management practices (land use and fertilization) and characterizing of agronomic properties of the corresponding species. In the model, weather and nutrient availability act upon rates of biophysical processes (radiation capture and use, plant senescence). These rates are modified over times due to PFT-specific parameters determined experimentally which represent the different strategies of plant species regarding growth. The integration of these parameters into the model made it possible to predict herbage biomass accumulation rate under different management practices for a wide range of plant communities differing in their PFT composition. The model was evaluated in two steps, first by analyzing separately the effects of PFT and an indicator of nutrient availability on herbage accumulation and then by conducting a sensitivity analysis. It was validated using two independent datasets; a cutting experiment running over the whole growing season to examine the consistency of the model outputs under different cutting regimes, and a monitoring of meadows and pastures in spring over a whole growth cycle to assess the model’s ability to reproduce growth curves. Although a good fit was observed between the simulated and observed data, the few discrepancies noticed between field data and predicted values were attributed mainly to the potential presence of non-grass species. More specifically, we noticed that nutrient (mainly nitrogen) availability is the main driver of plant growth rate, and that PFT determines the times at which this rate changes in relation to the phenological characteristics of species present. We concluded that integration of the PFT concept into the initial mono-specific growth model is especially suited to evaluating the consequences of management practices on species-rich permanent grasslands to meet feed production targets.