Stochastic modelling of turbulent diffusion

The modelling of scalar diffusion in turbulent flows represents a fundamental question in eco-hydrology, eco-hydraulics, biosphere-atmosphere exchange, etc. Therefore, the modelling of scalar turbulence in environmental problems has received a constantattention in the past few decades, particularly in the atmospheric boundary-layer studies. Despite this great amount of work, several questions remain unsolved. Most of these works develop a lagrangian dispersion model (ldm) based on the langevin equation. However, to apply the langevin equation with success, it is necessary to observe the time series to be modelled on a time scale on which it is first order markovian. Several techniques have been developed to determine the Markov order of a time series, and applications can be found in various scientific fields. Most of the works one can find have been devoted to analyze the case of Markov chains, a particular class of Markov processes in which the domain of the observed stochastic variable is a discrete set of states. As a consequence, if one deals with continuous processes, it is necessary to discretize the state space. This operation does not always give good results, particularly when the signal amplitude is wide. In fact, these methods require a low number of possible discrete states, demanding then a coarse description of the time series. Moreover, the use of these techniques has some problematic aspects. First, they were proposed to test low order processes, whereas in many cases one is interested in dealing with processes characterized by having high Markov orders on the time scale at which they are measured. Second, these techniques often require to make assumptions on the Markov order of the time series, which is unknown, and to check which hypothesis gives the best results. This requires to assume a priori the set of possible values of the Markov order of the process, which is often a difficult task when dealing with real-valued time series. To overcome these problems, in the recent years a new method has been proposed in turbulence research. In particular, to measure the Markov order of a turbulent time series, renner et al. (2001b) used a statistical test, named the “wilcoxon test” after its developer. This test can be applied to continuous time series, without being limited to low Markov orders and without making assumptions on its value. However, it is rather complex to implement and its performances are affected by the presence of a large number of parameters. In this work, we propose a new test to assess the Markov order of a process, that allows one to overcome some of the above difficulties. This test is different from the previous techniques in that it does not consider directly the conditional pdf that appears in the definition of the Markov properties, but only its moments. In particular, we focus on the expected value of the conditional pdf. This test evaluates then a new measure of the memory of the process that is called the expected value Markov (evm) order. Once established the time scales on which the time seris is first order markovian, it is necessary to determine the drift and diffusion coefficients of the langevin equation of lagrangian dispersion models. The functional forms of the drift and diffusion coefficients are usually chosen following physically-based models, and only the coefficients of this models are determined from the experimental data. However, the functional forms of the drift and diffusion terms can be easily determined directly from the time series, employing the finite-difference form of their definition together with suitable interpolations of the resulting trends. This technique has been applied with success also to turbulent time series. In this work we present further improvements of the various methods in use and track a footpath for determining the drift and diffusion values from turbulent time series. The final step is building a lagrangian dispersion model of turbulent diffusion based on the langevin equation that has been determined in the previous steps. However, some points still have to be made clearer, particularly when one deals with a non-homogeneous case, such as the flow over a vegetation canopy, both in the atmosphere and in water. In order to try to shed some light on the the critical aspects of applying ldms to canopy turbulence, we build a one-dimensional ldm at different levels in a canopy with the previously determined drift and diffusion coefficients and we evaluate the different behavior that they have on these different levels by comparing them with those obtained, for the same vertical profile, from the different levels of approximation of thomson’s one-dimensional model. This comparison allows then one to understand the influence of turbulence intensity and vertical inhomogeneity without physical reasoning. In fact, different levels of approximation in Thomson’s model correspond to different hypotheses on their influence.