Anders Engqvist

Modeling Water Exchange and Contaminant Transport through a Baltic Coastal Region

Abstract The water exchange of the Baltic coastal zone is characterized by its seasonally varying regimes. In the safety assessment of a potential repository for spent nuclear fuel, it is important to assess the consequences of a hypothetical leak of radionuclides through the seabed into a waterborne transport phase. In particular, estimates of the associated residence times in the near-shore coastal zone are of interest. There are several methods to quantify such measures, of which three are presented here. Using the coastal location of Forsmark (Sweden) as an example, methods based on passive tracers, particle trajectories, and the average age distribution of exogeneous water parcels are compared for a representative one-year cycle. Tracer-based methods can simulate diffusivity more realistically than the other methods. Trajectory-based methods can handle Lagrangian dispersion processes due to advection but neglect diffusion on the sub-grid scale. The method based on the concept of average age (AvA) of exogeneous water can include all such sources simultaneously not only boundary water bodies but also various (fresh)water discharges. Due to the inclusion of sub-grid diffusion this method gives a smoother measure of the water renewal. It is shown that backward in time trajectories and AvA-times are basically equipollent methods, yielding correlated results within the limits set by the diffusivity.

Water exchange and density structure in a multi-basin estuary

Abstract A mathematical model of the Himmerfja¨rd estuary, divided into four basins, has been formulated and validated against measured data for 1986. The structure of each sub-basin is assumed to be horizontally homogeneous with vertical mean velocities based upon in- and outflows from adjacent basins and freshwater supply to each basin. The horizontal water exchange is formulated as a quasi-stationary Bernoulli flow, driven by horizontal pressure gradients over the sounds and instantaneously interleaved at a neutral buoyancy level. The salinity and temperature profiles measured outside the mouth of the estuary serve as forcing, as do the water level changes, the freshwater run-off and the local wind. Inherent in the model assumptions of the horizontal exchange over the sounds is that only a fraction, α, of the pressure gradient is used to accelerate each stratum. Variation of the α-value shows that the best statistical fit is found for α = 0.15 when compared with water exchange estimates based on measurements in one of the internal sounds for almost an entire month. Using this α-value, in combination with standard mixing parameters and hypsographical data, the model satisfactorily captures the major features of the salinity and temperature profiles development for the year 1986. This is substantiated by statistical analysis of the salinity profiles in the sub-basins for which different measures of similarity between simulated and measured data give the best fit for the same α-value as above.

Water exchange of the Stockholm archipelago—a cascade framework modelling approach

Abstract The Stockholm archipelago spans roughly a semicircular area with a radius of approximately 60 km, traditionally partitioned into three parts: the inner, the middle and the outer archipelago. This subdivision coincides with differing water exchange regimes. The inner and middle archipelagos are characterised by comparatively larger basins which are interconnected by a limited number of straits. This configuration is well suited for a discrete basin (DB-) model approach by partitioning the area into a set of sub-basins that are only resolved vertically. The advantage of this approach over 3D-models is the possibility for enhanced vertical resolution and improved strait exchange formulation, outweighing the disadvantage of neglected horizontal gradients within the basins. In the inner archipelago the dominating exchange process is estuarine circulation, induced by the marked freshwater discharge and the vertical mixing. In the outer and middle archipelagos the density fluctuations due to Ekman pumping along the Baltic boundary interface produce another type of baroclinic process that clearly dominates. Measurements to adequately resolve these density variations do not exist. Missing forcing data are provided by linking the middle archipelagos boundary straits to a 3D-model of the Baltic with a grid resolution of 0.5 nautical miles (n.m.). This fine resolution model (FR-domain) is in turn driven by the atmospheric forcing and the density variation at the rectangular boundary of the FR-domain which acceptably resolves both the interfacial straits and the outer archipelagos complex hypsography. Massive computing resources would be demanded if the FR-domain were extended to comprise the entire Baltic. The FR-domain is thus interfaced with an existing coarse resolution model of the entire Baltic (CR-domain) with a grid size of 5 n.m., the open boundary of which is located in the Kattegat. This 3-fold model set-up has been run for one whole year (1992) with a one-year spin-up time to make up for the lack of initial data. The model concept is at this stage to be regarded as a framework for further development in anticipation of improved formulations, particularly for the strait exchange formulation. Therefore only primary validation experiments and a few sensitivity analyses have been performed.

Self-similar multi-layer exchange flow through a contraction

The multi-layer exchange equations for gravitationally driven flows between two basins with stable Boussinesq type of stratification in discrete layers, specified far upstream on either side of a connecting strait, result in a hydraulic control condition that must be satisfied at the narrowest part of the contraction, the control point. If one stagnant layer is present at the control point, the control condition that applies to all layers collectively may be separated into two such conditions that apply independently to two groups of layers going in opposite directions separated by the stagnant layer. Such bidirectional flow regimes exist if the structure of the prespecified density profiles permits each of the opposing groups to vertically reduce their thickness by the ratio 2/3 relative to their upstream thicknesses, leaving space for the stagnant layer to protrude through the contraction. Under these restrictions, the bidirectional flow is controlled by the fastest propagating wave mode and the stationary solution then relies on the superposition of two previously known unidirectional self-similar flow regimes that are completely decoupled. Techniques for their numerical computation are presented. The transition into loosely coupled and fully coupled flow is discussed. The decoupling principle also applies when several non-adjacent stagnant layers are simultaneously present at control in which case multiple groups of decoupled layers flow in alternating directions.

Unidirectional stratified flow through a non-rectangular channel

A self-similar solution describing stratified flow through a non-rectangular channel is derived. The solution shown here is an extension of Woods (1968) solution for stratified withdrawal through a rectangular channel. We consider a restricted set of geometries (where the bottom of the channel is constrained to be flat) and calculate the flow, assuming first multi-layer stratification, and second continuous stratification. In the case of two-layer flow we prove that the self-similar solution is the only possible solution. The analytical solutions are corroborated by three-dimensional numerical model simulations.

An analytical integration method of computing diurnal primary production from steele's light response curve☆

Abstract In recent literature on the photosynthesis-light response in aquatic habitats the mathematical formulation proposed by Steele (1962) is one of the most frequently utilized. Its explicit depth and time integrals are stated but only approximate and/or graphic solutions are attempted for its integration in time. In this paper an exact analytical solution is presented as well as an efficient algorithm for its numerical evaluation, given an a priori relative error. The method requires assumptions of diurnal constancy for a few parameters. Its applicability, particularly in ecosystems simulations, is discussed in comparison to alternative methods. It is concluded that this method of analytical integration is advantageous in combining high numerical accuracy with comparatively little computational effort.

Modelling the effects of a pumping program for increasing water circulation in a semi-enclosed bay in the Stockholm archipelago

Brunnsviken in the inner Stockholm archipelago, close to the City of Stockholm and popular for recreational activities, is a semi-enclosed bay with a very narrow passage to the adjacent archipelago and consequently has a limited water exchange. Various attempts have been made over the past three decades to improve the water quality in the bay. Since 1986, the drinking water authorities of Stockholm withdraw bottom water by pumping it from one of the deep basins of the bay through a pipe to be eventually discharged into the nearest embayment of the archipelago. There are, however, some questions regarding the cost-benefit aspect of this strategy. In particular, it is unclear if the location and the rate of pumping are well chosen in order to increase the ventilation of the bay at large. In addition, it should be possible to eventually optimize the pumping schedule, so that these energy-demanding and thus costly efforts are concentrated to times when they act in concert with natural forcing to increase water exchange. To help improve the pumping program, a modelling project was started in spring 2004. First, a one-dimensional (1-D) layer model that resolves the bay into one basin with multiple-layer stratification was attempted. Second, a three-dimensional (3-D) model with the capacity to be run under non-hydrostatic assumptions was set up at high resolution to study a number of scenarios. Both models were forced by wind, river discharge, surface temperature and the exchange driven by density fluctuations across the boundary to the adjacent archipelago. In the 3-D model, the pumping is included as a virtual divergence of the flow at the location of the pipe. The main results are that the 1-D model performs considerably better than the 3-D model in simulating the measured salinity profiles, even though it does not resolve the basin in the horizontal direction. The poor performance of the 3-D model with regard to salinity is however mainly due to boundary problems: too little inflow of saline water through the narrow and shallow entrance channel, and underestimation of the freshwater supply. The dynamics of the thermocline is better captured by the 3-D model but can certainly also be improved.