The prediction of far-field pollutant concentrations using residual currents M. Hartnett, S.Nash, R. Leslie Department of Civil, Structural and Environmental Engineering, Trinity College, Dublin 2, Ireland Email: mhrtnett@tcd. ie Abstract The conventional method used for solute transport modelling involves the application of the two-dimensional, finite difference model, DIVAST (Depth Integrated Velocity and Solute Transport) to a waterbody. The length of time required to run the model is directly proportional to the size of the study area, therefore, for large study areas the process is extremely time-consuming. Since the majority of the run-time is spent computing the hydrodynamics it would obviously be beneficial if the solute transport module could be executed separately. This paper details the application of both the conventional DIVAST model and an independent solute transport model, which is driven by residual currents, to a waterbody, namely Killybegs Harbour, County Donegal, Ireland. The solute concentration results obtained from the residual current method are examined and compared to those predicted by the DIVAST model. The effects of varying some of the controlling parameters within the residual current solute model are also examined. The results from the residual current solute transport model and its comparison with the DIVAST model are presented and discussed. 1 Introduction Solute modelling of bays and estuaries is used to determine the water quality and sediment transport within the waterbodies under a range of varying
282 Environmental Engineering and Management conditions. The calculations used in solute modelling cannot be computed without the required hydrodynamics for the same study area, therefore, a hydrodynamic model must be executed prior to the solute model. These are normally incorporated into the same computer model. Thus the program requires a much longer run-time than if the solute model could be executed independently. Normally this is not a problem for a relatively small study area, where the run-time is only in the order of 3-4 hours, but as the study area becomes larger the run-time increases proportionately and so the process becomes time-consuming and computationally expensive. The numerical model used in this study is a modified form of the model DIVAST (Depth Integrated Velocities And Solute Transport), originally developed by Professor R A Falconer, at the University of Bradford, UK. DIVAST is a comprehensive and versatile model, which uses the finite difference technique, for predicting the water elevation and depth averaged velocity components in the horizontal plane and up to eight user specified water quality constituents and sediment transport fluxes [1]. The water quality constituents which can be simulated include:- salinity, total and faecal coliforms, biochemical oxygen demand, ammonia, nitrate and nitrite nitrogen and dissolved oxygen. The hydrodynamic module is based on the solution of the depth integrated Navier-Stokes equations, while for the water quality and sediment transport module, the advective-diffusion equation is solved for the various water quality constituents. The model is applicable to well-mixed coastal, estuarine, and inland water bodies [1]. For this specific application the original model was split into two basic components:-1) a hydrodynamic model and 2) a solute transport model, both of which could be executed separately. The conventional method for applying the DIVAST model to a study area requires the linked hydrodynamic and solute transport modules to be run over a specified simulation time using a small timestep, usually less than sixty seconds. The small timestep means a longer execution time as the finite difference technique requires all the calculations to be carried out for each grid point, in both the x and y directions, at every timestep. Clearly the number of grid points in the finite difference mesh representing the study area has a direct effect on the run-time, and hence as the study area increases in size so too will the time increase. Therefore, it would obviously be of enormous benefit if the solute model could be run independently whilst using a much larger timestep. The main aim of this study is to compare the results obtained from the
Environmental Engineering and Management 283 combined hydrodynamic and solute transport model with those obtained from a solute transport model run independently for a larger timestep which uses residual currents as the driving force. The study involved the application of both the DIVAST model and the separate solute transport model to a waterbody, namely Killybegs Harbour, which is located in Co Donegal on the north-west coast of Ireland. Both sets of results were then compared and a number of parameters were varied in the solute transport model and their effects, if any, on the results observed. 2 Description of Killybegs Harbour For the purpose of this study an existing model of Killybegs Harbour was used which had previously been developed and calibrated. Killybegs Harbour is a relatively shallow waterbody situated in County Donegal on the north-west coast of Ireland. The harbour is approximately 3.75km long and 0.5km wide with a mean depth of less than 10m and a maximum depth of 23m located at the harbour entrance. It is considered to be a well-mixed waterbody due to its shallowness and the mean spring tidal range of 3.5m induces relatively weak maximum tidal currents of O.lm/s within the harbour. For further information on the study area see [2]. The mathematical formulations employed by this model are the standard Navier-Stokes and advective-diffiision equations and are explained in detail in.[2]. 3 Solute modelling 3.1 Model Calibration As part of the original calibration of the Killybegs Harbour model a dye study was carried out within the study area. The results obtained from this dye study were seen to compare extremely well with the solute plume computed by the DIVAST model, demonstrating the accuracy of the model. 3.2 The conventional method As stated in the introduction the conventional method for solute modelling involves the application of the combined hydrodynamics and solute transport
284 Environmental Engineering and Management DIVAST model to the study area followed by calibration and validation of the model. In order to achieve accurate results the program must be run using a small timestep. The simulation time must also be of sufficient length to have allowed the velocity time-traces to overcome any cold-start effects and the solute concentration time-traces to have reached steady state with the timetraces following a regular sinusoidal pattern. The simulation time required for the model to have reached steady state may be in the order of tens or even hundreds of hours. This large simulation time combined with the small timestep significantly affects the length of the program run-time. A main factor affecting run-time length is the size of the study area since the time is directly related to the number of grid points. Therefore as the size increases, so the time of execution lengthens accordingly. For example, a study area of approximately 500,000 grid points could take up to two days to run on a PC before steady state conditions are obtained. 3.3 Residual current method The motion of the estuarine waters in a harbour is affected by many factors and, therefore, in general, a water particle will not return to its starting point at the end of a tidal cycle. This resultant displacement after a tidal cycle is usually expressed as a residual current. The residual current for a particular point is evaluated by summing algebraiclly the components in the coordinate directions of the computed velocities at each timestep [3]. Since these residual currents essentially tell us by how much and in which direction a particle will be displaced, in relation to its starting point, at the end of a tidal cycle, they should therefore enable us to model solute motion in a waterbody. To enable solute modelling using the residual current method the DIVAST model was separated into its two basic components, namely the hydrodynamic and the solute transport modules, which could then be run as two separate programs. Since the solute model cannot run without the appropriate hydrodynamics, the hydrodynamic model was executedfirst,using a small timestep, with the required data output at a specified time. As the residual current represents the mean velocity of a particle over a tidal cycle, the value for residual current at a specific grid point should always remain the same provided the model has reached steady state. This implies that we can use the same set of residual currents at each timestep. In this case, the hydrodynamic model was run for thirty tidal cycles ensuring that steady state had been
Environmental Engineering and Management 285 reached, and the necessary data outputted at the end of the thirthieth cycle. The solute transport model was then run at a much larger timestep for the same simulation time while reading in the hydrodynamics when required, using residual velocities in place of instantaneous velocities. Figure 1 shows the study area. Figure 1. Plan view of Killybegs Harbour showing outfall and output points
286 Environmental Engineering and Management 4 Results For this study the conventional DIVAST model was run using a timestep of 15 seconds and a simulation time of thirty tidal cycles (375 hours), while the residual solute transport model was executed using a timestep of one hour and the same simulation time. In both cases the solute being modelled was BOD. The solute concentrations were output at each timestep at the four output points shown in Figure 1, and time-concentration curves were plotted for each of these points. 4.1 Comparison of results Figure 2 shows the time-concentration plot for output point A in the harbour It can be seen that the concentrations rise fairly rapidly at first, up until the fourth or fifth tidal cycle, and then continue to rise more gradually to a maximum value of 0.145 mg/1. To compare the methods, the graphs obtained from the residual current method were superimposed onto their counterparts obtained using the conventional DIVAST model. Figure 3 shows both plots for point A. 150 200 250 Time (hrs) 300 350 400 Figure 2. Time-concentration plot for point A using residual current method
0.25 Environmental Engineering and Management 287 100 200 300 400 Time (hrs) Figure 3. Chart comparing concentration results from both methods From this graph we see that initially the residual mean concentration, BOD 2, increases at a much quicker rate than the conventional mean concentration, BOD 1, for approximately the first six tidal cycles. Within this region the BOD levels computed by the residual current method are roughly twice the corresponding mean values obtained from the conventional method. This error may be due to the fact that during this early period of the simulation, the hydrodynamics used to calculate the solute concentrations in the combined DIVAST model have not reached steady state whereas the hydrodynamics used to run the residual current method are not outputted from the hydrodynamics model until it has reached steady state From the end of the sixth tidal cycle onwards the concentrations from the residual solute model and the mean BOD levels from the conventional model can be seen to converge towards a similar value, so that at the end of the thirtieth cycle the values are almost identical, with BOD 2 having a value of 0.145 mg/1 compared to an average value, over the thirtieth cycle, of 0.1498 mg/1 for BOD 1, a difference of 0.0048 mg/1. The following table, Table 1, gives the corresponding values, in mg/1, computed by both methods for the thirtieth tidal cycle at each of the four output points as well as the associated percentage errors.
288 Environmental Engineering and Management Table 1. Comparison of results from both methods Output Points 1)1 VAST Res. Current % Error Point A Point B Point C Point D 0.1498 0.1921 0.2200 0.2276 0.1450 0.1789 0.2235 0.2316 3.20 6.87 1.59 1.78 As can be seen from the table, the residually calculated solute concentrations for the thirtieth tidal cycle compare very well with their DIVAST counterparts with the largest error, 6.87 %, occurring at output point B For the other three output points the concentrations obtained from the independent solute transport model are all within ± 3.5 % of those computed by the combined hydrodynamic and solute transport model which is well within acceptable error levels. As stated earlier the solute transport model was run using a timestep of one hour requiring a program execution time of approximately ten minutes. This compares with a run-time of approximately six hours using the conventional DIVAST model. Therefore, the residual current method has an obvious advantage over the conventional method with regards to modelling time, speeding up the process enormously especially where large study areas are being modelled. 4.2 Effects of varying parameters Once the residual current-driven solute transport model was seen to compare favourably with the combined hydrodynamic and solute transport model the effects of varying some of the controlling parameters in the datafilewere then observed. The parameters varied included the timestep, the dispersion and diffusion coefficients. The model was run using three different timesteps : 60 minutes, 30 minutes and 15 minutes. When executed using the 60 minute timestep the solute concentration time-trace plots for points C and D showed areas of instability over thefirsttidal cycle, as can be seen from Figure 4 and 5. When the model was run using the smaller timesteps these areas of instability no longer occurred as would be expected see Figure. 6 and 7.
Environmental Engineering and Management 289 BOD (mg/l) Q P 2 P 01 -* Ul K) b P -* P rg o Cn -» en K> 01 t I- 8 S O O 8 8 ON O %' JBOD(mg/l, Q P ^ P k) o1 -* cn N) cn BOD (mg/l) b P '-* P ro oi -* tn to cn o. s 1- =" o? 1 S "8 II ON O
290 Environmental Engineering and Management It was also noted that over the early period of the simulation time where the time-concentration curve rises steeply the concentration levels decrease as the timestep is reduced. For the change in timestep from 60 min. to 30 min the values decreased by as much as 30 % over thefirsttwo tidal cycles with the reduction getting smaller, as time increased, with less than a 5 % reduction by the end of the sixth cycle (75 hours), and continuing until both sets of results converged. A similar trend was also observed for the reduction in timestep from 30 min. to 15 min. albeit on a smaller scale, beginning with reductions of approximately 10% over thefirsttidal cycle which decreased to less than 5% by the end of the second cycle. The dispersion coefficient, y, was varied from 0 to 50 with the model run for nine different values within this range. No significant effect was observed on the solute concentrations at any of the output points with the values only changing very slightly for each run. The largest differences recorded were obviously between the runs using 0 and 50 for y with the greatest of these being 0.0031 mg/1, only 2.13 % of the total concentration. The diffusion coefficient, 6, was varied between 0.01 and 20.0, with the program executed for 5 different values within this range. As was the case with the variations of the dispersion coefficient there were little or no discernible effects on the concentration levels with the largest difference only in the region of 3 % of the total solute concentration. 5 Conclusions The application of both the conventional DIVAST model and a residual current-driven solute transport model to Killybegs Harbour, Co Donegal, Ireland, has been explained. The results obtained from both model simulations were compared and discussed and the effects of varying some of the controlling parameters in the solute transport model were observed. The main findings of this study can be summarised as follows : The solute concentrations computed by the residual current-driven solute transport model, although not very accurate over the early period of the simulation time, compare very well with the means of the concentrations obtained from the conventional DIVAST model as time increases.
Environmental Engineering and Management 291 The model execution time is significantly less for the solute transport model than for the combined DIVAST model providing a much faster modelling process. The accuracy of the results from the independent solute model increases as the timestep is reduced The variation of the dispersion and diffusion coefficients has little or no effect on the solute concentrations predicted by the residual current solute model. 6 References 1. Falconer, R.A., & Chen, Y, "An improved representation of flooding and drying and wind stress effects in a 2-D tidal numerical model", Proc. of the Inst. of Civil Engineers, Part 2, Research and Theory, Vol. 91,1991. 2. Hartnett, ML, Morris, C, Leslie, R, "An assessment of some parameters affecting an environmental model of a natural harbour", Proceedings of Environmental Engineering & Management 98, Barcelona, In Press. 3. Gilroy, J.P., "A highly efficient computational scheme for offshore circulation studies", MSc Thesis, University College Galway, Ireland, 1981.