WATER RESOURCES RESEARCH, VOL. 29, NO. 3,PAGES 607-617, MARCH 1993 TidallyAveraged Models for Dispersion in Shallow Water A. W. HEEMINK Tidal Waters Division, R•jkswaterstaat, The Hague, The Netherlands This paper deals with the advection and diffusion of contaminants in tidal waters. The analysis starts with the well-known vertically integrated advection-diffusion equation. In order to derive a tidally averaged model to describe the advection and dispersion processes over long periods and to gain insight into the tidally induced dispersion, a new method to average the tide isintroduced. Here it is assumed that both the dispersion coefficient and the residual flow are sufficiently small. The averaging procedure isderived byusing a random walk model todescribe the dispersion process. This random walk model is shown to beexactly consistent with the advection-diffusion equation. Theaveraging procedure results in equations for theaveraged advective transport aswellasequations for the effective dispersion tensor describing the mixing process that occurs within one tidal cycle. Byusing anumerical model todetermine the tidal movement over one tidal cycle, the tidally averaged advective transport and dispersion tensor in each grid point of the model isapproximated numerically. In this way a tidally induced dispersion map of thearea under study is obtained. 1. INTRODUCTION To predictthe dispersion of pollutants in coastal waters, often the well-known vertically integrated advection- diffusion equation can be employed [Fischer et al., 1979]. Here onemightadopt a Eulerian point of view and solve this equation numerically or one might use a Lagrangian ap- proach and develop a random walk model [Allen, 1982; Heernink, 1990]. In both cases the time step needsto be chosen sufficiently small in orderto be ableto represent the tidal movement accurately. However, since in practice, simulation periods are usually very long compared to the tidal period, both approaches are very time-consuming. Therefore in thispaperwe develop, using a new approach, a tidally averaged model. Using suchan averaged model it is possible to increase the time step an order of magnitude. Most tidally averaged dispersion models are alsobased on theadvection-diffusion equation. Here the advectiveterms are usually determined by averaging the tidal flow over one or more tidal cycles to obtain a Lagrangian or Eulerian residual flow. Furthermore,these models utilize pseudo- dispersion coefficients to accountfor the effect of the tidal motion. However, thesecoefficients are very hard to deter- mine in practicalsituations. Thereforea numberof authors [Bowden, 1965; Younget al., 1982;Kalkwijk, 1987] dis- cussed the dispersiveeffect of simplified two-dimensional harmonic velocity patterns. These studies provide us with important insight into the tidally induced dispersion, but they stilldo not produce accurate results for realistic flow patterns. Toovercome theproblems just described, we introduce a newapproach to average the tide. It can be employedfor tidal flowfields that are determined by using a numerical shallow water flow model. The approach is inspired by the ideas ofKrol[1990], who developed anaveraging procedure for simple quasi-periodic advection-diffusion problems and proved its asymptotic validity. The method wasextended by Heernink [1992]. It is based onthe assumption thatboth the dispersive effect and thenonperiodic residual flow are "suf- Copyright !993by theAmerican Geophysical Union. Paper number 92WR02421. 0043-1397/93/92WR-02421 $05.00 ficiently"small, that is, O(e) with e as a small parameter. Unlike the physically based averaging methods usedup to now, our approach has a mathematicalbasis. If it happens that e is small, the original advection-diffusion equation can be approximated accurately by the averaged advection- diffusionequation. In this case all types of tidal mass transport mechanisms are included [Fisher, 1972]. Further- more, the resultingaveragedmodel is, in some mathematical sense,the best averaged model that can be obtained. On the other hand, if e is notsufficiently small, O(e 2) terms should enter the problem. In this case, however, the averaged model wouldnot be an advection-diffusion equation,andit is fundamentally impossible to model the averaged process by means of an advection-diffusion equation, no matter what averaging method is used. In other words, if e is not sufficiently small, averaging seemsnot a wise thing to do at all. In Krol [1990]and Heemink [1992] the averaging method was introduced and illustrated with some simple examples. In this paper we generalizethe approach and apply it to realistic tidal averaging problems in the Dutch coastal wa- ters. We first describe, in section2, the vertically integrated advection-diffusion equation with a small diffusion coeffi- cient and a small residual flow. Krol derived the averaged model of a simplified advection-diffusion equation. To in- crease our insight into the approach and to generalize the results, we derive in this paper the averaged model by introducing in section 3 a random walk model. This model is shown to be exactly consistent with the advection-diffusion equation described in section 2 in the sense that it produces, except for numerical errors, the sameresultson the concen- tration of pollutants [Heernink, 1990]. In section 4 we first solve the reduced problem, that is, the problem without diffusive effects and without residual flow. With the solution of this problemin mind, the original problem with a small diffusion coefficient and a small residual flow is transformed in section 5 to a model suitable for averaging. The trans- formed problemis averaged in the sections 6 and 7. The tidally averaging procedure results in equations for the averagedadvective transport as well as equationsfor the effective dispersion tensor describing the mixing process that occurs within onetidal cycle. We illustrateour approach 607