Finite-differences advection term

An effective approximation to Equation 1 is obtained by segmenting the water body of interest into n volume elements of volume Vj and representing the derivatives in Equation 1 by differences. Let V be the n X n diagonal matrix of volumes V, A, the n X n matrix of dispersive and advective transport terms SPy the n vector of source terms SPjy averaged over the volume Vjy- and P, the n vector of concentrations P which are the concentrations in the volumes. Then the finite difference equations can be expressed as a vector differential equation... 

The question of what controls the asymptotic decay rate and how is it related to characteristic properties of the velocity field has been an area of active research recently, and uncovered the existence of two possible mechanisms leading to different estimates of the decay rate. Each of these can be dominant depending on the particular system. One theoretical approach focuses on the small scale structure of the concentration field, and relates it to the Lagrangian stretching histories encountered along the trajectories of the fluid parcels. This leads to an estimate of the decay rate based on the distribution of finite-time Lyapunov exponents of the chaotic advection. Details of this type of description can be found in Antonsen et al. (1996) Balkovsky and Fouxon (1999) Thiffeault (2008). Here we give a simplified version of this approach in term of the filament model based... 

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