We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

Approximate Riemann Solvers

The Mar Menor lagoon has been subjected to high anthropogenic pressures. Among them, in the early 1970s, dredging and enlargement of one of the inlets to make a navigable channel has had strong consequences on the hydrology, ecology, and fisheries of the lagoon. In recent years, changes in agricultural practices have induced an eutrophication process, leading to loss of water quality. As a solution, some management proposals have included dredging of the inlets in order to increase the water renewal. However, these proposals did not take into account the negative effects of previous experiences nor the consequences on environmental conditions of the lagoon and therefore on biological processes. The purpose of this work is to assess the effect that proposed mitigation measures, could have on the hydrodynamic conditions and discuss its possible ecological consequences. A three-dimensional (3D) numerical model has been used to simulate the lagoon under different dredging scenarios, covering different dredging depths and extensions. The simulated current fields and fluxes through the inlets, as well as water renewal times have been compared for the different scenarios. It is found that some of the considered scenarios take the system beyond the threshold of sustainability, where the modified current dynamics could affect sediment transport, beach dynamics and fishing capacities in a significant way. Water exchange with the Mediterranean is also strongly affected, with consequences for species connectivity, and a homogenization of the water renewal times that could lead to loss of ecosystem heterogeneity and structural complexity. The study demonstrates the utility of numerical models as effective tools for the management of coastal areas.

Assessing the Hydrodynamic Response of the Mar Menor Lagoon to Dredging Inlets Interventions through Numerical Modelling

Modelling the impact of dredging inlets on the salinity and temperature regimes in coastal lagoons

Meshless numerical model based on radial basis function (RBF) method to simulate the Rayleigh–Taylor instability (RTI)

Numerical Investigation of New Alternative Nador Lagoon Inlet Relocation

This paper presents details of finite volume and finite element numerical models based on unstructured triangular meshes that are used to solve the two-dimensional nonlinear shallow water equations (SWEs). The finite volume scheme uses Roe's approximate Riemann solver to evaluate the convection terms. Second order accuracy is achieved by means of the MUSCL approach with MinMod and VanAlbada limiters. The finite element model utilizes the Lax–Wendroff two-step scheme, which is second-order in space and time. The models are validated and their relative performance compared for several benchmark problems, including a hydraulic jump, and flows in converging and converging–diverging channels.

A comparative study of finite volume and finite element on some transcritical free surface flow problems

Numerical simulations are presented of the flow hydrodynamics and hypothetical contaminant dispersion patterns in Nador Lagoon, a shallow lagoon with a barrier island situated on the coast of Morocco. It is found that the natural circulation forced by the tidal flow in the lagoon is greatly affected by the development of an artificial inlet in the barrier island. The case study demonstrates the potential use of modern computational hydraulics as a tool integrated in the decision support system designed to manage a lagoon ecosystem.

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Numerical survey of contaminant transport and self-cleansing of water in nador lagoon, morocco

A balanced adaptive scheme is proposed for the numerical solution of the coupled non-linear shallow water equations and depth-averaged advection-diffusion pollutant transport equation. The scheme uses the Roe approximate Riemann solver with centred discretization for advection terms and the Vazquez scheme for source terms. It is designed to handle non-uniform bed topography on triangular unstructured meshes, while satisfying the conservation property. Dynamic mesh adaptation criteria are based on the local pollutant concentration gradients. The model is validated for steady flow over irregular bed topography, recirculation due to a sidewall expansion in a frictionless channel, and pollution advection in a flat-bottomed channel. An idealised application to the simulation of pollution dispersion in the Bay of Tangier, Morocco is presented, which demonstrates the capability of the dynamically adaptive grid model to represent water quality scenarios in a bay of non-uniform bed topography and complicated shoreline.

Balanced adaptive simulation of pollutant transport in Bay of Tangier

A well balanced adaptive scheme is proposed for the numerical solution of the coupled non-linear shallow water equations and depth-averaged advection-diffusion pollutant transport equation. The scheme uses the Roe approximate Riemann solver with upwind discretization for advection terms and the Vazquez scheme for source terms. It is designed to handle non-uniform bed topography on triangular unstructured meshes, while satisfying the conservation property. Dynamic mesh adaptation criteria are based on the local pollutant concentration gradients. The model is validated for steady flow over irregular bed topography, recirculation due to a sidewall expansion in a frictionless channel, and pollution advection in a flat-bottomed channel. An idealized application to the simulation of pollutant dispersion in the Bay of Tangier, Morocco is presented, which demonstrates the capability of the dynamically adaptive grid model to represent water quality scenarios in a bay of non-uniform bed topography and complicated shoreline.

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Elmiloud Chaabelasri

Papers

Numerical Investigation of New Alternative Nador Lagoon Inlet Relocation

A comparative study of finite volume and finite element on some transcritical free surface flow problems

Numerical survey of contaminant transport and self-cleansing of water in nador lagoon, morocco

Balanced adaptive simulation of pollutant transport in Bay of Tangier

Well Balanced Adaptive Simulation of Pollutant Transport by Shallow Water Flows: Application to the Bay of Tangier