Timestep splitting in Lagrangian marine dispersion models

Lagrangian mathematical models, based on random walk methods, are well-established tools for the assessment of discharge dispersion in the aquatic environment. At HR Wallingford, the PLUME-RW suite of Lagrangian dispersion models is in regular use, and is applied with two-dimensional (2D) and three-dimensional (3D) finite element flow models. The range of applications is currently: dissolved pollutant dispersion; for conservative or decaying pollutants, particulate pollutant and suspended cohesive sediment dispersal, deposition and re-suspension, non-cohesive sediment transport, oil slick transport and fate. Whilst Lagrangian models have long been acknowledged as practical and highly flexible tools for assessing pollutant dispersion in the marine environment, there remain limitations in their application, largely associated with timestep sensitivities. In particular, as discussed by Mead [1], sensitivities associated with the representation of vertical mixing processes have been identified. These sensitivities have been considered in recent years by various authors, and both pragmatic and rigorous mathematical solutions have been proposed. This paper presents an approach aimed at improving accuracy over the traditional approach, while avoiding excessively long model run times. Recently, Ross and Sharples [2] have reviewed in detail the criteria for timestep selection in Lagrangian models with spatially-varying diffusivity, and have presented methods for minimising boundary effects and the effects of density discontinuities in such models. The work presented in this paper utilizes the Ross and Sharples [2] analysis, which is referred to hereafter as RS04. In both 2D and 3D models, techniques including the use of shorter timesteps for vertical processes than for horizontal processes are employed. In the context of the present work, the term “2D model” refers to a Lagrangian model which takes the results of a 2D-in-plan, depth-averaged flow model as input data. A “3D model” uses the results of a fully 3D flow model, based on sigma-coordinates. The 2D modules of the PLUME RW suite are actually quasi-3D, as they calculate model particle movements in all three dimensions, using analytic profiles to extrapolate depth-averaged quantities to vertical profiles. Models based on random walk methods employ random number generators to calculate model particle movements associated with turbulent mixing. During the studies described in this paper, an unexpected dependence of model solutions on the sequencing of calls to the random number generator was identified. To the author’s knowledge, this is not a well-known aspect of the use of random number generators in Lagrangian models, so details are given in the paper