János Józsa1, Tamás Krámer1 and Hannu Peltoniemi2
1Budapest University of Technology and Economics
Department of Hydraulic and Water Resources
Engineering
Műegyetem rkp. 3. Kmf. 8., H-1111 Budapest,
Hungary
Phone: +36 1 463 1164, Fax: +36 1 463 1879
E-mail: jozsa@vit.bme.hu, kramer@vit.bme.hu,
http://www.vit.bme.hu
2 Environmental Impact Assessment Center of
Finland Ltd
Tekniikantie 21 B, 02150 Espoo, Finland
Phone: +358 9 70018680, Fax: +358 9 70018682
E-mail: hannu.petoniemi@eia.fi, http://www.eia.fi
Abstract: Water exchange mechanisms are assessed via modelling the spatial distribution of the mean residence time in lakes and coastal areas. A grid-based numerical solution of the deterministic advection-diffusion-reaction kinetics equation with the ageing of water as transport variable is applied. The composition of residence time from times spent over separate sub-domains is handled as multi-species transport. Numerical errors arising in the advection approximation at sharp gradients are reduced by robust upwind schemes. The spatial residence time distribution due to through-flow and wind-induced water exchange in a shallow sample lake with littoral reed is analysed by 2-D depth-integrated approach. Apart from the spatial distribution, the normalised volumetric distribution of the residence time is also considered. The impact of planned, large-scale landfill on the residence time in semi-enclosed bays at Helsinki coasts is evaluated by 3-D modelling.
Keywords: residence time, transport modelling, lakes, coastal flows, impact assessment
Water exchange mechanisms in environmental flows are often evaluated on residence time basis. Though advanced transport models are available to calculate the fate of pollutants or sediments, residence time is still a tool for e.g. hydro-biologists to analyse various flow domains (rivers with floodplains, estuaries, coastal areas, semi-enclosed bays, harbours, lakes) with special interest to aquatic habitats. Most of the calculation methods simulate the traditional experimental technique (Spalding, 1958; Danckwerts, 1958), according to which some numerical tracer is released and tracked in a modelled flow field. As an example, a finite-difference model system was used by Hilton et al. (1998) to investigate the residence time of freshwater in Boston’s Inner Harbour. In their investigations the so-called freshwater fraction approach combined with dye release was applied in the simulations in order to reproduce trends observed in field measurements. The spatial distribution of the mean residence time was, however, not considered.
In case of sufficient amount of field data even simple formulae can provide good estimates. Recently Guo and Lordi (2000) have developed a simple method to quantify freshwater input and flushing time in estuaries by using measured flow and salinity data. Their technique, however, is restricted to regions where natural tracers (either fresh or saline water) are available.
Other traditional, inherently Lagrangian calculation methods consist of determining residence time along flow paths (Pollock, 1988; Glasgow et al., 1996). The role of mixing can also be included by using appropriate random walk particle tracking methods (see e.g. Józsa 1989; Uffink, 1991) as a discretised, Monte Carlo solution of the corresponding stochastic differential equation (Arnold, 1974). When the residence time has to be evaluated at few selected sites only, the so-called backward time particle tracking provides an efficient technique by means of calculating the breakthrough-curves at open boundaries for particles injected at the selected site in the reversed velocity field. In order to determine the spatial distribution of the mean residence time, however, a large number of particles are usually needed in order to obtain a solution with low statistical noise.
In this paper, an alternative to the above mentioned solution techniques is used. As recently presented by Józsa and Krámer (2000), the ageing of water while carried by the flow and mixed by turbulence can be jointly expressed in a conventional continuum form by an advection-diffusion-reaction kinetics equation. In the equation the quantity to be transported is the residence time with zero-order “time-source”, that is, ageing term.
A finite difference solution of the governing equations is implemented with robust upwind schemes to reduce numerical errors in the presence of sharp gradients, e.g. at the interface of main stream and recirculation zones. In the paper the method is further developed to handle the residence time as a composition of times spent over sub-domains with different character like the littoral and pelagic zones in shallow lakes. Moreover, a 3-D application is presented as part of the impact assessment study of planned large-scale landfills at the Helsinki coasts.
Applying the conservation law for the “fluxes” of the residence time r(x,y,z,t) in three-dimensional flows and assuming Fickian turbulent diffusion represented by the tensor Dt, the following governing equation holds:
where u(x,y,z,t) is the Reynolds-averaged velocity vector, and the Einstein-type summation rule is used. Once the velocity and diffusion coefficient distributions are known, the solution of this equation can be performed by an appropriate discrete transport solver.
In typical shallow
water environment the depth-integrated description of the flow and the related
transport processes can often provide acceptable approximation. After
integral-averaging over the depth h, the depth-averaged version of the
residence time rh(x,y,t)
with U(x,y,t) depth-averaged
velocity vector and D(x,y,t)
dispersion tensor can be written as follows:
In case it is needed to distinguish the contribution of sub-domains with significantly different character to the residence time, a kind of multi-species version of the above mentioned equation can be applied as follows:
in which f
indicates the species responsible for counting the contribution of sub-domain
f to the residence time. The distribution
of the mean residence time for the whole domain consisting of m sub-domains can be obtained by simple
summation:
.
Note that this approach does not give the original entry location and the paths of the water masses inside the domain, but gives an overall picture on the age-distribution.
In this paper the numerical solution using finite differences on an equidistant grid has been adopted. In the numerical solution some unusual features of the residence time distributions have to be handled with care. For instance in case of through-flow with recirculation the residence times in the recirculating areas are expected to be much longer than elsewhere, resulting in sharp gradients in the interface regions. As an extreme case, in the absence of dispersion in the model even discontinuity with ever-growing residence time inside the recirculation zone would occur. All this requires advection schemes with substantially reduced numerical errors at such locations. Of the existing schemes, the second order method described by Koren (1993) has been implemented here. Time marching is used both for unsteady and steady-state calculations.
Two applications are presented to demonstrate the features and the usefulness of the approach. In the first case a simple dish-shaped shallow lake, partly covered by reed, is considered in 2-D depth-averaged model environment with pure through-flow then with superimposed wind effect. In the second example the impact of planned, large-scale landfill on the water exchange in semi-enclosed bays at Helsinki coasts is assessed by 3-D flow and transport modelling.
A dish-shaped shallow lake seen in Figure 1 has been defined in order to investigate the applicability of the multi-species version of the residence time modelling. Steady-state flow patterns for constant 80 m3/s through-flow as well as for constant 12 m/s NW wind superimposed onto it have been analysed. Special attention has been paid to the partition of water ageing in the reed-covered littoral and in the pelagic zones.
The calculated flow patterns are shown in Figure 2. The spatial distribution of the mean residence time over the littoral and pelagic zones as well as the overall distribution are presented in Figures 3–5. As can be seen the wind can substantially modify the flow pattern thus the residence time. Note that the flow field has been calculated by taking into account reasonable spatial wind shear stress irregularities as an influence of the reed stands, and also the enhanced hydraulic resistance of the reed. Figure 6 presents the so-called normalised volumetric residence time distribution for the two cases analysed above as well as for the situation of NE wind, not presented here in details. Note that if it was a straight channel with uniform through-flow then the distribution would be also uniform. Further evaluation of the volumetric distribution by using its first four moments is underway.
Fig.1 Dish shaped test lake. Contour lines represent the depth with 0.25 m interval. Figures are in metres.
Fig.2 Horizontal velocity field. Left: Through-flow only. Right: NW wind superimposed to through-flow.
Fig.3 Distribution of the littoral zones to the residence time. Left: Through-flow only. Right: NW wind superimposed to through-flow.
Fig.4 Distribution of the pelagic zones to the residence time. Left: Through-flow only. Right: NW wind superimposed to through-flow.
Fig.5 Overall residence time. Left: Through-flow only. Right: NW wind superimposed to through-flow.
Fig.6 Normalised volumetric
residence time distribution in the littoral and the pelagic zones. (a): No wind;
(b): NW wind; (c): NE wind.
The irregular coastal area of Finland includes a number of semi-enclosed bays the water exchange of which is usually induced by storm surges and baroclinic effects. As was pointed out by Stipa (1999) water exchange in such cases is a combination of various factors and can considerably change both in space and time. The 3-D version of the residence time calculation outlined in this paper was used to estimate the effect of some planned landfills on the water exchange in Helsinki coastal area. The fast growing area has a shortage of non-built land, and land filling is one possibility to cure this problem. The purpose of the study was to map the most vulnerable areas and to give guidelines for the future city planning. The Helsinki sea area is characterised by fractal coastline, multiple small islands and bays. It is very shallow and practically non-tidal. The water is only slightly saline and the coastal areas freeze annually. Despite the closeness of the capital area the bays and islands are rich in wild life, especially birds.
The applied flow model is a 3D hydrostatic primitive equation finite difference model developed at the Environmental Impact Assessment Centre of Finland Ltd. (Koponen et al., 1992; Kokkila, 1995). The model grid is four times nested from the whole Gulf of Finland grid to the local fine resolution one. Horizontal grid cell sizes range from 5000 to 50 m. The calculated time period was one and half months from July to August in 2000. The model was driven by measured wind data from one wind station, whereas measured sea level changes were set at the open boundary in the western part of the Gulf of Finland. As a first approach barotropic conditions were adopted. The flow model was validated against measured currents and water levels in the study area. The transport problem was approximated on the same grid.
Figures 7 shows the mean residence time distribution after 45 days at the northernmost bay Laajalahti at half meter below the surface calculated in the present situation, and the relative changes with one landfilling scenario. The calculation period was not long enough for the mean residence time to reach equilibrium at least at the most stagnant area at the north-west part of Laajalahti Bay. Nevertheless, it is clearly seen in the figure of the relative change of the residence time that the landfills reduce the water exchange of the bay. The reduction is between 10 to 50%. Outside the Laajalahti Bay the relative increase has rather little practical importance as the biggest changes are at areas with very short residence time, but it is interesting to notice the capturing effect of new bays formed by land filling.
Fig.7 Left: Residence time before landfill. Right: Relative change in residence time due to landfill. Landfills are marked with black areas.
The outlined residence time calculation method have been successfully applied both in its depth-integrated 2-D and 3-D version. Partitioned to pelagic and littoral zones both spatial and normalised volumetric distributions could be obtained for a shallow sample lake. The impact of planned landfills on the water exchange of a semi-enclosed bay in Helsinki area could also be reasonably evaluated by using a 3-D modelling approach. The utilisation of the various moments of the normalised volumetric residence time distribution in flow field evaluation is one of the subjects for further research.
Acknowledgements
This work was part of the project “Measurement and parameterisation of free surface flows” No. T030792 supported by the National Scientific Research Fund of Hungary, and the “Development and comparative analysis of advanced hydroinformatics methods in surface water environment research” SF-18/99 Hungarian-Finnish Intergovernmental S&T Cooperation.
Arnold, L.: Stochastic Differential Equations: Theory and Applications. John Wiley and Sons, 1974.
Danckwerts, P. V.: Local Residence-times in continuous-flow systems. Chemical Engineering Science, Vol. 9, pp. 78-79, 1958.
Glasgow, C., A. K. Parrott, Handscomb, D. C.: Particle Tracking Methods for Residence Time Calculations in Incompressible flow. Oxford University Computing Laboratory, Numerical Analysis Group, Report No. 96/05, 1996.
Guo, Q., Lordi, G. P.: Method for quantifying freshwater input and flushing time in estuaries. Journal of Environmental Engineering, ASCE, Vol. 126(7), pp. 675-683, 2000.
Hilton, A. B. C., McGillivary, D. L., Adams, E. E.: Residence Time of Freshwater in Boston’s Inner Harbor. Journal of Waterway, Port, Coastal and Ocean Engineering, ASCE, 124(2), pp. 82-89, 1998.
Józsa, J.: 2-D particle model for predicting depth-integrated pollutant and surface oil slick transport in rivers. Proc. International Conference on Hydraulic and Environmental Modelling of Coastal, Estuarine and River Waters, Bradford, U.K., Gower, pp. 332-340, 1989.
Józsa, J., Krámer, T.: Residence time modelling as advection-diffusion with zero-order reaction kinetics. Proc. Hydroinformatics 2000, 4th International Conference, Cedar Rapids, Iowa, USA, IIHR, University of Iowa, 2000.
Kokkila, T.: Suomenlahden virtausten mallintaminen. M.Sc. Thesis (in Finnish), Helsinki University of Technology, Department of Technical Physics, Helsinki, Finland, 1995.
Koponen, J., Alasaarela, E., Lehtinen, K., Sarkkula, J., Simbierowicz, P., Vepsä, H., Virtanen, M.: Modelling dynamics of large sea area. Publications of the Water and Environment Research Institute, Helsinki, Finland, Vol. 7, pp. 1-91, 1992.
Koren, B.: A robust upwind discretisation method for advection, diffusion and source terms. In: Vreugdenhil, C.B., and Koren, B. (eds), Numerical methods for advection diffusion problems, Vieweg, pp. 117-137, 1993.
Pollock, D.W.: Semi-analytical computation of path lines for finite-difference models. Ground Water, Vol. 26(6), pp. 743-750, 1988.
Spalding, D. B.: A note on mean residence-times in steady flows of arbitrary complexity. Chemical Engineering Science, Vol. 9, pp. 74-77, 1958.
Stipa, T.: Water exchange and mixing in a semi-enclosed coastal basin (Pohja Bay). Boreal Environment Research, Vol. 4(4), pp. 307-317, Helsinki, Finland, 1999.
Uffink, G.J.M.: Application of the Kolmogorov’s backward equation in random walk simulations of groundwater contaminant transport. In: Kobus, W., and W. Kinzelbach (eds), Contaminant Transport in Groundwater, Balkema, 1989.