Abstract
In this study, optimal diffusion coefficients for Lake Zirahuén, Mexico, were found under particular conditions based on images taken with a drone of a dye release experiment. First, the dye patch concentration was discretized using image processing tools, and it was then approximated by an ellipse, finding the optimal major and minor axes. The inverse problem was implemented by comparing these observational data with the concentration obtained numerically from the 2D advection–diffusion equation, varying the diffusion tensor. When the tensor was isotropic, values of K11=K22≈0.003 m2/s were found; when nonequal coefficients were considered, it was found that K11≈0.005 m2/s and K22≈0.002 m2/s, and the cross-term K12 influenced the results of the orientation of the ellipse. It is important to mention that, with this simple technique, the parameter estimation had consequences of great importance as the value for the diffusion coefficient was bounded significantly under particular conditions for this site of study.
Highlights
Lakes are vitally important components that provide essential ecosystem services, such as water for drinking, and food supply and sites for recreation and tourism [1]
The methodology applied in this study tools to obtain theestimation area of theofdye based on an optimization technique, and the consists of (i) the the patch dye distribution using image processing tools to (ii) obtain solution problem for the advection–diffusion in order to the area of of the the nonlinear dye patchinverse based on an optimization technique, and (ii)equation the solution of the nonlinear inversediffusion problem for the advection–diffusion in order to find against the optimal find the optimal coefficients by comparing equation this analytical solution the diffusion coefficients by comparing this against the observational dye observational dye patch distribution; seeanalytical
A perfect match with the ellipse approximation was not expected, as there would be more physical factors to consider in reality, for example, if there was no wind, the diffusion of a point source in a uniform medium should be perfectly circular; there are other physical mechanisms that make the diffusion parameter variable according to the conditions such as turbulence, enhanced by waves and its secondary circulations (e.g., Stokes drift and Langmuir circulations)
Summary
Lakes are vitally important components that provide essential ecosystem services, such as water for drinking, and food supply and sites for recreation and tourism [1]. Observations and monitoring are important aspects of ecosystem services that can be used to map the distribution of sediments and can be used as a target for numerical models to predict sediment movement and spreading, in particular, knowing that the diffusion coefficient is essential for the numerical modeling of such transport processes [6,7,8,9]. The transport of pollutants can be described by many factors, including advection, diffusion, dispersion, reaction, dilution or mixing, retardation, and decay. These factors are usually incorporated into transport equations, which describe phenomena such as mass transfer or heat, fluid, waves, and momentum transfer [10,11]. Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations
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