engqvist a., k. döös and o. andrejev, 2006

Anders Engqvist, Kristofer Doos and Oleg Andrejev

Modeling Water Exchange and ContaminantTransport through a Baltic Coastal Region

The water exchange of the Baltic coastal zone ischaracterized by its seasonally varying regimes. In thesafety assessment of a potential repository for spentnuclear fuel, it is important to assess the consequencesof a hypothetical leak of radionuclides through the seabedinto a waterborne transport phase. In particular, esti-mates of the associated residence times in the near-shore coastal zone are of interest. There are severalmethods to quantify such measures, of which three arepresented here. Using the coastal location of Forsmark(Sweden) as an example, methods based on passivetracers, particle trajectories, and the average age distri-bution of exogeneous water parcels are compared for arepresentative one-year cycle. Tracer-based methodscan simulate diffusivity more realistically than the othermethods. Trajectory-based methods can handle La-grangian dispersion processes due to advection butneglect diffusion on the sub-grid scale. The methodbased on the concept of average age (AvA) of exoge-neous water can include all such sources simultaneouslynot only boundary water bodies but also various (fresh)-water discharges. Due to the inclusion of sub-griddiffusion this method gives a smoother measure of thewater renewal. It is shown that backward in timetrajectories and AvA-times are basically equipollentmethods, yielding correlated results within the limits setby the diffusivity.

INTRODUCTION

Baltic coastal waters act as an intermediate link of successiveflow-mediated (advective) and mixing-related (diffusive) re-gimes by which waterborne material migrating from thegeosphere may eventually end up in the world oceans, passingthrough the Baltic (Figure 1) on its way. The primaryconnection with the geosphere can be by direct leakage throughthe sea bottom of the coastal zone or via water run-off(discharged diffusely by ground water flow, or discretely bylocalized watersheds such as streams or rivers) entering intosurface layers of the coastal zone. This study is focused entirelyon material assumed to enter through the bottom. In Sweden,two coastal areas (Forsmark and Oskarshamn) are currentlybeing investigated by the Swedish Nuclear Fuel and WasteManagement Co. (SKB) as possible sites for an undergroundrepository for nuclear waste. All aspects of the biosphere (e.g.,surface hydrology, terrestrial, limnic and marine ecosystems)are being investigated in order to assess the potential fate anddistribution of radionuclides in the environment, and exposureof and risk to humans. The Oskarshamn area includes a numberof semi-enclosed embayments that are not well suited for 3D-modelling, but can be modelled as a set of separate 1-dimensional basins interconnected by straits according to (1).From such model results can possibly a two-dimensionaltrajectory analysis be performed in the vertical plane in analogyto the horizontal plane procedure presented in Box 7. TheForsmark area is used here to evaluate the different methods toobtain a useful measures of the water exchange. The focus of thepresent study is to investigate the fate of radionuclides in this

Figure 1. Model grid depicting thebathymetry of the study area. Thebathymetry of the 3D-model of theentire Baltic is inset in the lowerleft corner with an analogousdepth colour legend that spansabout four times the depth indicat-ed by the colourbar. The outerboundary that separates the interi-or of the present study area fromexogeneous coastal water ismarked with solid red lines in thenorth and the south. The innerboundary that delimits the possi-ble discharge area is indicatedwith the solid red line in the center.The scale of the respective axis isindicated as number of grid cells,each with a side of 0.1 nauticalmile. Ten nautical miles corre-spond to about 18.5 km.

Ambio Vol. 35, No. 8, December 2006 435� Royal Swedish Academy of Sciences 2006http://www.ambio.kva.se

latter coastal area in the unlikely but possible event of leak froma repository in the area.

The location of the potential discharge bottom surface areais determined by factors outside oceanographic considerationsand is thus taken as given for the modeling discussed here (pers.comm. Ulrik Kautsky). The local water exchange at this site willdetermine the primary dilution of dissolved substances in thedischarged water, but if the potentially contaminated plumeoriginating from the discharge area is at least temporarilysurrounded by stagnant water, the problem with possibleaccumulation of hazardous material is simply displaced toanother area. The contaminating radionuclides are furthersupposed to be either dissolved or attached to neutrally buoyantparticles, making the plume (defined by water having been incontact with the discharge bottom surface area) non-buoyant.From this perspective it is important that the examined area ofpotential influence is expanded so that material passing throughits boundaries is highly diluted and less likely to re-enter. Thechoice of the study area (Figure 1) has been based on apartitioning of the coast with regard to long-term coastaldevelopment due to anticipated landrise (2). The aim of thepresent study is to assess and compare various measures ofmaterial transfer from this 2-dimensional (2-D) discharge areathrough a related 3D-domain, which subsequently will bereferred to as the study area, here denoted ‘area’, though it is thewater volume contained in that area that is tacitly understood.

The overall objective is to quantify the water exchange of thiscoastal area in such terms that projection into the distant futureis made possible. To this end, various water circulation modelsdriven by reasonably simplified but adequate forcing areemployed, and the resulting massive hydrographic datasetsgenerated over a one-year cycle of a typical year are condensedinto a conceptual form that can serve as a basis forcommunication with other involved disciplines (3). For theForsmark area (Figure 1), the year 1988 was chosen as the mostrepresentative year with regard to the forcing (4), since it cameclose to the long-term average regarding precipitation andtemperature. Physical forcing data pertaining to that year havebeen collected and two nested numerical 3D-models —onelarge-scale comprising the entire Baltic and one with finerspatial resolution of the Forsmark coastal area— have beensimulated (5). The flow-field sampled every hour in this lattermodel has served as the input data for the trajectorycomputations, while the finest temporal resolution (1.2 min-utes), necessitated by numerical stability considerations, is usedfor a plume dilution study and for computation of the alternatemeasure of residence time.

To obtain a direct and communicable appreciation of thespatially complex water exchange of the coastal zone, threedifferent methods are available. The first concerns dilution of atracer substance discharged as a fully specified source regardingits intensity and location. This can be performed both in realityand in a numerical model. In order to facilitate comparison, it ispreferable to use substances that are fully water soluble, lack

state transitions and at the same time are detectable at very low

concentrations. Examples of such studies based on Rhodamine

(e.g. 6) and SF6 (7) are abundant in the literature. Salt also

possesses all of the listed favorable properties and is often used

in this capacity to check the material balance accuracy in a

modeling context (e.g. 8). The disadvantage of salt is that,

except for in the vicinity of freshwater discharge locations, the

salinity contrasts are small, which makes the salinity budgets

less indicative of water exchange events.

In contrast to administering material tracers, there are two

measures of water turnover that are conceptually based on time

rather than mass concentration. The first one is trajectory

analysis in which particles are released and their passing

through defined boundaries is timed. In principle this can be

done not only in models but also in reality if devices capable of

tracking their (3D-) underwater course, so-called ‘floats’ (e.g. 9),

are employed. Such physical objects can be designed so that

they remain on a prescribed density surface and thereby are

subjected to the passive advection corresponding to the length

scales of the physical dimensions of the device.

Box 1. AcronymsADCP Acoustic Doppler Current Profiler instrument

AS3D Name of employed model

ATR Average transit residence (time)

AvA Average age

AvR Average residence (time)

DEM Digital Elevation Model

SKB Swedish Nuclear Fuel and Waste Management Co.

WGS84 World geodesic system (spherical coordinates)

Box 2. From continuous flow totime-discrete 1D cellular automataConsider a cube with volume dV, whose water content is replacedduring the time dt by a steady volume flux of dV/dt. From thedefinition of the ensemble average age (AvA) for this continuousflow it is clear that the AvA and its complementary counterpart‘‘average residence’’ time (AvR), taken over all particles present at agiven instant in the interior, will amount to 0.5dt as long as the flow issteady so that their sum ATR will equal 1 dt. Regarding this scalarcontent as a tracer, the balance is struck by 1 dt gained by aging andthe same amount lost by the net transaction over the boundaries.

A tube can be formed by putting three such cubes together in a row,each one with a volume dV. These cells are enumerated 1 through 3(Fig. a) and are considered to be filled with water in which a tracersubstance is homogeneously dissolved, staying well-mixed at alltimes. The tracer concentration does not in any way affect thepiston-like flow properties through the tube. Outside the tube oneither side there is an ambient compartment assigned to have theorder number zero. Contrary to the continuous flow example abovethe flux is now considered to happen in discrete time steps andencompass the entire cell, the AvA measure is the same for allparticles contained. At every time step, dt, the content of each cellwith a volume dV is shifted from left to right, still representing avolume flow of dV/dt. In Figure b these time-discrete 1D-automatatransitions can be denoted as an advection matrix Aft in which theindices f and t (both running from 0 through 3) are short for from andto, simply instructing from which cell position to take the content andto which cell to put it. This flow is going to the right, and the matrixcan be denoted AR. Flow in the opposite direction, which alsorepresents backward-in-time transition of AR, is consequentlydenoted AL. This can be obtained by simply interchanging theindices: AL¼Atf. This matrix operation, which reverses the directionof time, is called transposing and denoted with a superscript ‘T’, sothat AL¼Atf¼ATft¼ATR. A third possibility is water standing still,AS, represented by an (identity-)matrix with zero values except forthe main diagonal that is all ones.

436 Ambio Vol. 35, No. 8, December 2006� Royal Swedish Academy of Sciences 2006http://www.ambio.kva.se

The trajectory scheme based on Doos (10) and Blanke andRaynaud (11) makes it possible to follow a chosen water massboth forwards and backwards from any region in the ocean. Byassociating each trajectory with a transport measure it isgenerally possible (10) to calculate the water volume transportgiven a sufficient number of trajectories. It is furthermorepossible to integrate the Lagrangian transport vertically orhorizontally in order to construct a stream function associatedwith the studied path. This method was applied to the Baltic byJonsson et al. (12), in which residence times were calculated forthe Bay of Gdansk. These studies made use of the trajectorymethod’s capability of keeping a record of all released waterparticles, which in turn makes it possible to perform statisticalanalysis of, e.g., the particles’ different ages.

The second of the time-oriented methods is based oncomputing the age of water parcels in a reservoir defined by a

closed surface boundary that separates the interior fromexogeneous water outside. The first strict formulation ofreservoir theory was presented by Eriksson (13), in which paperwere formulated the statistical properties that material distrib-uted in a reservoir in steady state must fulfill. This line ofstatistically based analysis has had many sequels in appliedsciences, beginning in the 1970s when ecological concernsinstigated debates about harmful materials and their circulationthrough the environment. The scientific debate was, however,obscured by loose definitions that were often used in a mannerthat led more to confusion than clarity. In order to mitigate theensuing misunderstandings, Bolin and Rodhe (14) sharpenedthe distinction between different types of retention timeconcepts. They pursued their analysis in a general way withthe focus on unevenly distributed materials borne in a flowingmedium, and defined the two concepts of average age (AvA)and average transit residence time (ATR). These and otheracronyms are listed in Box 1. The first concept denotes theaverage time an ensemble of individual particles, present at a

Box 3. AvA and AvR computationfor a 1D tube-flow caseTo compute the AvA-values, each content of each interior cellshould be increased one dt unit as long as it remains in the tube. Asfor a single cube case, the entrance cube in direct contact with theexterior water should be assigned ½ dt units after the first time steptransition. This can be accomplished by prescribing a boundarycondition for the exogeneous water to be held constant at the valueof –½ dt. The dynamics in matrix notation becomes (indices cannow be omitted without causing ambiguity)

an ¼ an�1Aþ dt ; so that an � an�1A ¼ dt :

ðEqs: 1a and 1bÞThe left hand side of Eq. (1b) is commonly denoted Da, thesubstantial difference, giving

Da=dt ¼ 1; ðEq: 2Þwhich is the form given in (8, 19) exempting a diffusive term whichwill be returned to in Box 5 and Box 8. Starting from initial zeroAvA-values, then measured in dt units per volume unit, the initialvector a0 thus equals (�0.5,0,0,0) and this will in successivediscrete time steps develop into the progression a1 ¼(�0.5,0.5,1,1), a2 ¼ (�0.5,0.5,1.5,2), a3 ¼ (�0.5,0.5,1.5,2.5). Thislatter state will be repeated as long as the steady flow to the right ismaintained.

Under steady flow conditions, however, computation of AvAbackwards in time will be equivalent to evaluating AvR, whichstate vector is denoted r. This entity can be computed using thesame boundary condition as for AvA but replacing A with its time-reversed counterpart AT. After three or more iterations thecorresponding state vector will also assume a steady state: r ¼(�0.5,2.5,1.5,0.5). The sum of the a- and r-vectors amounts to ATRand is thus constantly equal to 3 among the interior cells of thetube, which is a reflection of the steady flow. The balance struckbetween aging in the three interior cells and rejuvenating byexchanging aged water with exogeneous water of zero ATR-timemeans that the loss (¼dV/dt�ATR ) should equal the gain in theinterior cells. This is the source strength, presently set to unity,multiplied by the total volume (1�3�dV). Equating these expressionsgives that ATR¼ 3�dt as expected.

The relationship between the two measures AvA and ATR is givenby Bjorkstrom (12):

AvA ¼ AT R

2þ r2

AT R

2AT R; ðEq: 3Þ

where r2ATR is the variance of the ensemble’s ATR. For an orderly

and steady tube-flow situation, as through a canal, all water parcelspass through in exactly the same time, so the variance equals zero.According to Eq. (3) this means that the AvA-value is exactly half ofthe ATR-value for such a flow regime. This minimum quotientmeans that the exiting water has maximum age and this ratio willbe exceeded when exiting water has any other age distribution,which occurs for example at well-mixed conditions.

Box 4. Fractional cell volumeadvection

It is also possible to loosen the original assumption that the entirecompartment volume is shifted to an adjacent position. The volumeconservation is manifested in the transition matrix A by the demandthat the sum of each row and column should be equal to unity.Maintaining this continuity condition, it is fully permissible to shiftjust a part of each compartment (Fig. a) corresponding to a slowerrate of the volume flux. This procedure involves mixing with knownvolumes of exchanged substances. The exchange can be trackedbackward in time, but this necessitates solving a matrix equation,i.e. finding the inverse matrix A�1 in the mathematical sense,exemplified in Figure b. After three consecutive time steps, theequivalence of a whole original cell has been exchanged. Sincemixing is performed on each of these time steps, the result willdiffer from a mere shifting of the entire cells one step to the right.This difference can in this context be attributed to so-called‘‘numerical diffusion’’. Alternatively, it is also possible to retain thesimplifying idea that the content of entire cells is shifted. This canbe arranged by increasing the total number of cells by partitioningthe original cells according to the transferred fractions, Figure c.The content of the originally defined cell must then be computed asan average of the new ones of which it consists. Note that thetransferred volumes are supposedly known. If the averaging isactually carried out by mixing each of the three original cells afterthe shifting, a diffusive mixing is imposed that matches thenumerical diffusion for the above-mentioned case when thefractional cell volume advection is performed.


given instant inside a predefined domain, has spent inside. Thetime measurement begins for each particle as it enters thedomain. The average should be taken over this simultaneouslypresent ensemble. The time that a particle will remain in thereservoir is defined as its residence time. The average over thesame concomitantly present ensemble is denoted AvR-time. Thesum of corresponding AvR- and the AvA-times gives theaverage transit residence (ATR) time. The relationship betweenthe two latter measures in steady-state conditions is given byBjorkstrom (15). This nomenclature has been adopted byDelhez et al. (16). Note that all these concepts are definedrelative to a specified boundary that separates the particles orwater parcels belonging to the domain from those that areoutside or exogeneous. Any relationship between these time-based measures and water exchange expressed as volume flow isonly valid during steady state conditions so that stableequilibrated distributions can be established. If parts of adomain are exempted from exchange for a period, viz. by waterbeing moved back and forth or by entering a secluded locality,the AvA of these parts will increase monotonously during thistime. The adjustment time to establish quasi-steady equilibria isin proportion to the new established AvA-time, meaning thatduring events with rapid water renewal, e.g., storm periods, anAvA-tracer is rapidly propagated through the studied area. TheAvA-measure is thus a scalar field variable with the dimensiontime per unit volume. It is fully permissible to calculatestatistical properties over partial volumes of the entire AvAdomain.

The time-oriented approach has been shown to have a highpotential for further research and refinement. Deleersnijder etal. (17) presented a theoretical framework for which the AvA ofa set of particles subjected to advection, diffusion anddestruction is determined using mass-weighted averages. Thismakes it possible to distinguish between the transportingmedium (i.e. normally water or air), passive tracers and activetracers such as decaying radioactive material and/or biologicallyreactive constituents. Delhez et al. (13) extended this frameworkto also cover ATR-times for non-stationary flow. Concerningventilation of coastal waters, it is also relevant to focus on therenewal of the flowing medium per se (i.e. sea water) and not onits particle contents. In this context, ATR is often referred to as‘‘hydraulic residence time’’ (18). The AvA concept, originallydefined for a single reservoir, was independently adapted towater circulation models by introducing its volume-specificcounterpart (8, 19), by denoting the length of time a particularwater parcel (or parts thereof) on average has spent within aspecified connected body of water. In numerical models, it canbe straightforwardly computed by assigning just one scalararray denoting the AvA-age. This variable is reset to zero forwater parcels outside the studied domain. The resulting variable(which represents the specific AvA time of the compartments ofthe actual subdivision of the domain) is increased one time-stepunit for each time step the associated water parcel resides in thedomain. In addition to aging, the water parcel is also beingsubjected to passive tracer advection and diffusion. In time, aquasi-steady state between aging (by remaining in the domain)and rejuvenation (by replacing aged water with new water ofzero age) will occur. The combined simplicity and stringency is alikely explanation for the growing interest in using thisapproach.

Notwithstanding a few early contributions based on theATR-time concept, e.g. Pilson (20) using box-models, itsapplication seems to have intensified since the turn of the newmillennium. Engqvist and Andrejev (21) and Andrejev et al. (22,23) applied the original method, while Gustafsson (24) andKhatiwala et al. (25) restricted the active age tracers to thosethat had also been in contact with a boundary of the domain,

starting the counting of age from that event. These latterapproaches thus presuppose that the employed model is capableof distinguishing the different kinds of water, a method whichbears a resemblance to trajectory analysis and can beaccomplished by an additional scalar tracer marker representingthe fraction of each water type in each sub-compartment. TheAvA dynamics scheme should then only be applied to oneparticular fraction of the contents of a compartment at a time,so that the corresponding AvA values represent the residencetime of only this fraction. This method also expands theapplicability to cover the residence time of freshwater (26, 27and 28) that also may be treated as exogenous water with initialage set to zero at its discharge point. Monsen et al. (29) give anencompassing literature review and introduce the concept ofexposure time that applies to particles that enter and leave theresidential domain multiple times without having their time-based scalar measure reset to zero on exiting. This approachdemands a computational domain that is sufficiently greaterthan the studied residence domain that the particles do notreach its boundaries. Meier (30) has analyzed various aspects ofthe Baltic circulation using the AvA method.

MATERIAL AND METHODS

The Forsmark area is located in Oregrundsgrepen, a funnel-likeopen-ended embayment with the wider end toward the north(Figure 1). The narrow southern end is also shallower with athreshold of approximately 25 m. There are notable densityfluctuations over a yearly cycle mainly due to the collectivedischarge of all the rivers into the Bothnian Bay. Via the straitof Oregrund (Oregrundsund), a connection is made to thesouthern basins, forming a buffer zone to the study area. Thebasins of the buffer zone are connected to the Baltic by onemain strait. On its offshore side, the wide interface to the Balticcontains a few areas that are charted in less detail. From amodeling point of view, it is advantageous that the lack ofdetailed bathymetric information is mitigated by a wide andopen boundary zone. Two streams discharge into the inner partof Kallrigafjarden with a combined freshwater flux of 9.8 m3 s�1

as a yearly mean for 1988. The flow varies considerably with amarked peak in the springtime. A notable estuarine circulationmode is thus present in Kallrigafjarden embayment most of thetime. From studies in other Nordic coastal embayments, thebaroclinic exchanges on upwelling events are the most efficientwater exchange processes (31, 32).

The 3D-domain grid has been computed from a digitalelevation model (DEM) based on national digitized charts,complemented with shoreline information from economicalmaps. The grid has been specified in spherical coordinatesWGS84 with the constraint that to be considered as a wet gridcell, at least 50% of the covered area must consist of water.

The Forsmark coastal area was resolved horizontally into2413 241 grid cells, each with a side length of 0.1 nautical mile(or approximately 185 m). The final choice of the actual modelarea (Figure 1) includes a large section east of Graso island thatseems unlikely to influence the strait connection to the southernsection of model area that connects to Oregrundsgrepenthrough Oregrundsund. The Forsmark area has been parti-tioned into nine sub-basins according to anticipated catchmentareas (2) that will occur in the long-term future due to landrise.A few of these are also identified as possible sites for near-futuredischarge of radionuclides from the projected repository. Theselected discharge area, Figure 1, has been chosen from twosuch representative sub-basins in which the contaminant may bedischarged. These sub-basins were deemed equivalent from amodelling point of view and the choice fell on the somewhat lessseclude one. In the distant future many more sub-basins are


eligible for potential discharge of radionuclides. The union of allthese sub-basins forms the study area and is also indicated in thesame figure. This has two boundaries to the external coastalwater, a wide interface in the north and a narrow one in thesouth. An appreciation of the variability of water movementsacross this wide boundary can be seen in Figure 2.

For contemporary coastal oceanography of an open (incontrast to landlocked) coastal section, three-dimensional (3D)models represent the state of the art. When the aspect ratio(vertical scale to horizontal scale quotient) is sufficiently smallerthan unity, the hydrostatic approximation applies, and thenumerically more efficient shallow water equations can beemployed. When more detailed horizontal resolution is re-quired, this simplification may ultimately need to be aban-doned, resulting in considerably increased computational effort.Forcing of the coastal zone model also requires informationabout the sea level and density fluctuations at the boundarywith the Baltic Sea. Since only a few such sufficiently spatiallyand temporally dense measurements are available, the requiredboundary forcing data is achieved by coupling two 3D-modelsin a cascade arrangement along geometrically simple interfaces,so that information on the large-scale Baltic oceanography ispropagated into the better resolved fine-scale coastal areas(Figure 1). This coupling of two cascaded AS3D-models wasperformed by Engqvist and Andrejev (21), in which previousoceanographic investigations and modeling efforts in theForsmark area were also reviewed.

The Baltic (AS3D) model employed in this study (34, 35) hasbeen developed for the main purpose of providing insight intothe circulation of the central Baltic. Its present horizontalresolution is 20 3 20 (nautical miles) based on the Warnemundebathymetric data. The horizontal eddy diffusivity is nominally

set to 30 m2 s�1, consistent with assuming the grid cells to bewell mixed. This model is presently involved in several ongoingBaltic hydrographic studies (22, 23). A thorough testing of thismodel in comparison with measured data (21) revealed thatalong an interface to a model area comprising the Stockholmarchipelago, the measured salinity and temperature profileswere acceptably well reproduced, thus strongly increasing theconfidence of the realism of the AS3D-model as provider ofboundary data. The complete set of equations of the AS3D-model including boundary formulation and numerical scheme isgiven in Andrejev and Sokolov (36). In the present version, theBaltic area including the section of its Kattegat boundary isrepresented by a 3153 363 cell grid. The integration time step isnormally 1 hour except on storm occasions, when it may belowered to some appropriate fraction of an hour.

A particular constraint put on this modeling approach is thedemand that it should be possible to project the computationinto a distant future for which climate scenarios could onlylikely produce a prognosis of shifting air temperatures, whileother factors determining the heat exchange (insolation, relativehumidity and nebulosity) would probably be only poorlydetermined, if at all. Attempts at such projection are made inEngqvist and Andrejev (37). The heat exchange and radiationbalance with the atmosphere is mainly determined by the airtemperature and has been calculated following Lane andPrandle (38). Likewise, the ice formation and melting processesare formulated in a simple but straightforward manner. Thiswould be a liability if the main concern were to correctly predictthe ice situation (e.g. 39), but this is not the present objective.

The basic ideas behind the computation of AvA, AvR andtrajectory times are simple and have been formulated above. Adescription of the involved 3D-model numerical schemes for

Figure 2. ADCP current measurement across an E/W-transect in the vicinity of the northern boundary recorded on 2004-07-2. The transectstarts from a point near Orskar north of the boundary of the study area and continues approximately 2 nautical miles to the west, thusspanning about 20 grid cells. This distance was covered in 1.1 hour. The implied spatial variability of the currents across this section of thewide interface is considerable.


Figure 3. a) A snap shot corresponding to 1988-03-14 of the modeled plume when tracer is constantly supplied from the bottom of the gridcells of the release area, rendering an instantiation of the 3D-character of the dilution process. The inset picture shows the tracerconcentration along the indicated transect with the imposed boundary condition of zero concentration visible as the straight line verticalinterface between the two darkest shades of blue. b) The left panel shows the resulting depth and time averaged plume computed using theAS3D model. The plume is released evenly distributed over the discharge area and subjected to both vertical and horizontal diffusion inaddition to advection. The time average is taken over 366 daily snapshots. The right panel shows the depth integrated density of the


computation of the time development of the correspondingscalar fields becomes quite technical. In order to give readersunfamiliar to oceanography an introduction to the correspond-ing computational schemes, these procedures have beensummarized in Boxes 2 through 8 for the 1D, 2D and 3Dcases. This is done in the form of a simplified flow model(cellular automata) that from a first stage of exchanging equalvolumes between its grid cells in discrete time steps is graduallymodified towards the complexity of a numerical 3D-circulationmodel, while keeping the mathematical formalism as simple aspossible. Frisch et al. (40) and Wolfram (41) demonstrate theintricate and realistic transient velocity fields that can resultfrom only a few simple rules describing how interacting ‘atomic’particles behave in a 2D-space.

From Boxes 2–8, it should be clear that the followingrelationships must hold for this kind of simplified cellularautomata:

� Backward-in-time trajectories should be analogous to AvAcomputations;

� Diffusion can be handled by the forward schemes for AvAcomputations, but not for trajectories;

� Time-reversed computations for any method can only beproperly performed if sub-grid diffusive exchange is com-pletely disregarded.

The validity of these relationships when the correspondingcomputations are applied to the full complexity of the AS3D-model is tested. The first issue concerns accounting for acontinuously supplied tracer. Then follows a comparison ofcalculating estimates of the water exchange from the source inthe release area relative to the entire study area. This isperformed with both the AvA and the trajectory methods.Finally the concordance of AvA-estimates and back-in-timetracked trajectories is evaluated.

RESULTS

The first test concerns the modeling of a tracer source withconstant discharge intensity in time which is evenly distributedover the bottom layer of the release area. Figure 3a gives asnapshot of the emanating plume on a randomly chosen date,computed with the full resolution of the model. A comparisonof the one-year average depth-integrated particle concentrationis given in Figure 3b, in which the left panel is computed usingthe full resolution and diffusion of the AS3D-model, while theright panel is based on the same flow-field sampled every hourwith no diffusion. The total mass contained in the study areaaccording to these two methods is compared in Figure 3c,showing a clear covariation between the curves but alsosignificant differences regarding the total amounts.

With a ‘spin-up’ time of one month corresponding to theforcing of December 1987, the AvA times were computed for allgrid cells in the study area following the simple proceduredescribed above. This computation includes discharge of thetwo rivers that is also treated as exogeneous water. Twice amonth the resulting AvA-values were saved, and from thesevalues the average AvA over one year was computed for thewhole area, presented for separate layers in Figure 4a. Thesurface layer clearly reveals the rejuvenation of water in theKallrigafjarden due to the freshwater discharge. The standarddeviation of these 25 instances is given in Figure 4b. The AvA-value distribution of the 109 grid cells that comprise thedischarge area are presented for each of these 25 occasions inFigure 4c.

Trajectories were started every hour from the 109 grid cellsof the discharge area. They were followed both forward andbackward in time in order to track the paths by which the waterenters and exits the study area. A selection of trajectories ispresented in Figure 5a showing the large variety of possiblewater travel routes. It is graphically not possible to present allthe possible individual trajectories starting from the dischargearea until they leave the study area. It is nevertheless possible to

Figure 3. (continued) trajectories of particles regardless of depth,continuously released from all cells of the discharge area adjacentto the bottom at the same rate throughout the year. Both colorlegends are normed to the maximum occurring density in order tofacilitate comparison. 3(c) Total amounts of tracer and trajectoryparticles inside the study area. The tracer is computed with the 3D-model’s full temporal resolution including diffusion and presentedwith daily values. The additional diffusive loss results in thenoticeable difference to the number of particles discharged at acomparable rate.

Box 5. Diffusion

Fundamentally different to advective exchange is the mixing thatoccurs without leaving associated information of the exchangedvolumes involved. This so-called Fickian diffusion has a directiondefined by the concentration difference so that it is always directedfrom a higher concentration to a lower. The rate is in proportion tothis difference. In almost all applied oceanographic contexts theeddy-diffusion driven by turbulent irregular flows dominates overmolecular diffusion, driven by thermodynamic unevenly distributedmolecular velocities. The fundamental difference compared toadvective exchange is that diffusion necessarily presupposes bothspatial and temporal gradients that cannot be made fullycompatible with the assumptions of well-mixed cells and exchangeevents taking place in discrete time steps. The figure above showsan example of the corresponding matrix expression, D, when cell 1and 2 are subjected to a mutually diffusive exchange. The indicatedexchanged volumes are fictive in the sense that they are notexplicitly computed in model. In principle such approximateequivalent volumes could be computed and saved separatelytogether with the advection field, but it can be shown by inspectinga property of the involved matrices (i.e. the modulus of so-calledeigenvectors) that even small disturbances in backtracking will leadto rapidly growing errors.


sum over the millions of trajectories and to calculate the averageroutes by constructing a Lagrangian stream function, which ispresented in left panel of Figure 5b. The flow here is along thestreamlines with a constant volume transport between the linesshowing that most of the flow exits through the north and inparticular the northeast. Only about 10% exits through thenarrow channel in the south. The equivalent path for the waterflowing into the discharge area is presented by the Lagrangianstream function in middle panel of Figure 5b, which iscalculated by following the trajectories from the release areaand backward in time. This shows that most of the water entersfrom the northwest. The total transport through the dischargearea is the sum of these two stream functions and is presented inthe right–hand panel.

The travel time of these trajectories is as individual as theirpaths. It is thus possible to attribute a spectrum of ages even toa single grid cell. Ideally one should follow each water moleculeand calculate its path and time evolution but this is inherentlyimpossible since no sub-grid scale diffusion process is accountedfor. The time evolution of the decay of number of trajectories inthe study area, which have been released in the discharge areaand traced until they leave the study area, is represented by theblack line in Figure 6a. It shows that the number of trajectoriesin the area decays exponentially in time with an e-folding time(i.e. when about 63% of the trajectories have left the basin) of 22days, which is basically the average residence time (AvR).

The same calculation has been performed on the backwardtrajectories and is presented by the purple line in Figure 6a. It

has a time evolution very similar to the forwardtrajectories, but with an e-folding time of 19days, which is the equivalent of the average age(AvA) of the water, again averaged over oneyear.

By adding the forward and backward timefor each trajectory, it is possible to calculate thetime evolution of the total time spent in thestudy area for trajectories that have passed atleast once through the discharge area and this isindicated by the yellow line in Figure 6a. The e-folding time for this is 45 days and is equivalentto the so-called average transient time (ATR)averaged over one year. The sum of AvA andAvR equals ATR for each individual pair ofconnected forward and backward trajectories. InFigure 6a these matching pairs become scram-bled because of the sorting process according toincreased times along the x-axis.

The time evolution of the water massresidence in Figure 6a has been performed fortrajectories that have been released every hourduring the whole year and then traced forwardand backward until they respectively enter andexit the domain. The time evolution is hence anaverage over a matching number of releasesduring the year.

In Figure 6b, the corresponding curves arepresented separately so that one curve representsthe time evolution for a particular day when thetrajectories have been released. There is there-fore one curve for each day of the year. Thismakes it possible to see that, when currents arestrong, the water will sometimes reside in thestudy area for less than 10 days, whereas whenthe currents are weak, the decay time may bemore than 100 days. Since each curve consists oftrajectory times sorted according to their mag-nitude, it is not feasible to infer the point in time

at which important water exchange events took place. Suchinformation is obscured by sorting the data according to timeduration.

DISCUSSION

The philosophical difference between the compared methodscould perhaps best be described as dependent on the purpose ofthe modelling. From a material turnover perspective, as forecologically oriented models, one would naturally be inclined toregard each grid cell as well-mixed, i.e. containing the samehomogeneously distributed scalar concentration throughout.The trajectory-oriented point of view is that each scalar value,in particular the velocity components, represents a discreteinstantiation of a continuously varying scalar field. The majorcomputational difference, however, concerns the eddy diffusion,as this is disregarded in the trajectory calculations. In a calmlyflowing major river, trajectories of real particles will divergeonly slowly over time, primarily due to dispersion induced byfrictional effects and secondarily manifested as induced vorticityphenomena. A patch of tracer will remain on the same side ofthe river unless the river bends and (secondary) vorticity inducesa flow component across the major flow direction. This is incontrast to a surging turbulent stream for which the eddydiffusivity alone may spread the particles across the main floweven in perfectly straight canals. This diffusive mixing varies ona time scale that is linearly inversely proportional to the (eddy-)diffusivity coefficient. It should, in principle, be possible to

Box 6. 2-D flow regimes

The grid cells can be arranged in a plane, which allows more degrees of freedomfor the exchange between the individual cells. Ordering the cells anew, for exampleadopting the numbering convention in the figure above, the previously introducedmatrix notation can still be used for describing the different A-variants that now canbe formed. Capital indices are now used in order to distinguish these and the actualflow regime at time t is denoted Ax(t ) where the index X could stand for flowsflowing in the South, East, North, West directions. For the reversed direction flowcases it is clear that the matrix representation can be obtained by transposingindices so that AN ¼ AT

S, but AN is not equal to the mathematical inverse AS�1.

The third example of an advection matrix, AC, in the figure above is different in thesense that all circulation is internal and no exchange is performed with exogenouswater. This means that for the entire 2D-area both AvA and AvR for each cellincrease with 1 dt unit each time step and ATR with twice this rate, an indicationthat a transient phase has been started. Concerning AvA and AvR, Eq. (2) stillholds, but with the actual succession matrix at time t (Ax(t )) substituted. Thedynamics can be summarized with x denoting the a- or r-vectors and X indicatingany advection pattern like the ones described.

xn ¼ x0AXð1ÞAXð2ÞAXð3Þ . . . AXðnÞ þ 1 ¼ x0 Pn

t¼1AXðtÞ þ 1: ðEq: 4Þ


account for diffusivity when performing forward in timetrajectory analysis, by adding a random component to the flowfields that are linearly interpolated from the advective fluxes atthe sides of each grid cell in the model. This would, however,radically increase the computational effort and also reduce theelegance of the method.

An ecological model often resolves a coastal zone intoseveral sub-basins for which the water exchange and itsdissolved or suspended content can be estimated betweenindividual pairs of adjacent sub-basins. This type of problemis probably most appropriately approached by utilizing thevolume fluxes directly or by merging the interlinked ecologicaland oceanographic processes into a single model context.

The present scope has focused on a particular problem inwhich the source of contamination is located in the bottomlayer of a near-shore area. It is obvious that the concern is notthe ventilation of the near-bottom volume that the actualrelease site defines, but instead the residence time in theneighboring coastal region where discharged radionuclides arelikely to be incorporated into the foodweb. The longer thatradioactive particles are resident in the coastal zone, the higherthe likelihood of their being taken up into the biota and thusentering the foodweb. The average rate of water renewal can beestimated from the AvA. The yearly averages for the innermostand less ventilated parts of the study area are less than aboutthree weeks, which is short in comparison to the time-scales ofone year used in ecological models (42) in which biomass ismodelled as the carbon content in various ecological aggregates.Since the import of carbon by advection turns out to besignificantly greater than the internal carbon turn-over, thismeans that the biomass accumulation of carbon is predomi-nantly derived from exogenous material. Increased waterexchange lowers the risk of internally released contaminantsto enter the local food-web of sessile organisms. Once thecontaminants leave the study area they become subjected tolarger-scale forcing and the rate dilution is accelerated.

The rendering of yearly mean AvA-fields in Figure 4adisplays what was already found in (33), namely that the meanhorizontal circulation mode in Oregrundsgrepen Bay is acounterclockwise gyre which is even more clearly expressed inFigure 5b. The relative decrease of mean AvA in layers towardbottom in the centre of the Bay reveals that dense bottomcurrents is a normally occurring mode of exchange. This alsocorroborates the results in (33). The rendering of thecorresponding standard deviation of the AvA-fields indicatesthat this entity peaks close to the location of the discharge area.This means that in the vicinity of this site, the maximumcontrast of water exchange intensity is encountered. Transitionsbetween more or less stagnant periods and such with moreintense water exchange may have significant consequences forthe ecological dynamics. Because the winter of 1988 wascomparatively mild (4), the stagnation during winter ice coverregime is not as conspicuous in Figure 4c as would have beenexpected an ordinary year.

The horizontal eddy-diffusion is set to 20 m2s�1 in the fine-resolution grid of the Forsmark area model, a value that hasbeen chosen as a compromise between several demands thatdimensional considerations place on turbulent flows. Thedifference of the slope of the three curves in Figure 3c wouldgive an indication of the rate of the diffusive exchange

Figure 4 a) One-year time average of AvA for each grid cell of theentire study area, computed from 25 instances during the year.Each panel presents one of nine model strata (z). Note therejuvenation of the water of the inner Kallrigafjarden and by thebottom water entering from the north. b) Standard deviation (S.D.)corresponding to a). Note that variability of the surface layers ishighest at the discharge area, indicating that the water exchange atthis location is event-oriented with periods of intensified exchange

interspersed over the year. c) Average age (AvA)-distribution overthe 109 grid cells of the discharge area on 25 semi-monthlyinstances in 1988. This distribution is compared in Figure 7 to thecorresponding set of trajectories emanating from the same area butcomputed in reverse time and tracked until they leave the study area.The horizontal axis spans 0–50 days for all diagrams.


component, provided periods of relatively feeble advectiveexchange could be identified. As a safeguard recommendation,however, the maximum concentration at any level would servebetter than the average depth-integrated concentration present-ed in Figure 3b.

The basic difference between the AvA measure compared totrajectory-based estimates of the residence time is that the AvAprocedure rapidly dilutes aged water, while trajectories contin-uously keep track of old parcels of water. A grid cell containingAvA tracer that has associated with it an age of 10 days couldthus either consist of a plume that had crossed from theboundary on an orderly broad front and thus be of pure 10-dage; or it could just as well consist of 90% 1-d old water mixedwith 10% 91-d old. Within the time frame set by the actual AvA-time values, transient changes in circulation are propagated intothe domain of the grid cells. In order to achieve sufficientstatistical data for making an analogous inference usingtrajectories, one must thus begin earlier than the point in timefor which the assessment of the water exchange status is to beassessed. How much earlier is impossible to state, becauseparticles that remain for both short and long times will exist. Itseems irrational to base statements about the intensity of the

water exchange during a certain period of time, say the first halfof November 1998, on statistics of the fate of particles trackedoutside the same time frame. Only the restricted subset oftrajectories corresponding to particles present in the study areashould be selected to be included in a diagram pertaining to thisparticular 15-day period. If the water exchange during thisperiod was considerably intense as is indicated by Figure 4c,then the selected curves corresponding to Figure 6a will displayconsiderably lowered AvA and AvR times. Such a selection canbe performed but demands additional computation. Thisargument together with the accumulated uncertainty due toneglected turbulent sub-grid processes add to the disadvantageof using the trajectory method as an indicator of transient flowregimes.

Time-reversed tracer tracking is not to be recommended fortheoretical reasons, because time-reversed diffusion represents,from a mathematical point of view, an ill-posed problem thatpossesses not one unique solution, but a multitude. Consider anend state with equal concentrations—within the resolution ofthe measurement method—in two cells that have been indiffusive exchange with each other. What were the concentra-tions at an earlier time? It is not even possible to deduce which


one had the higher concentra-tion in a preceding state, andmuch less to calculate what thedifferent concentrations were.The fundamental disparity be-tween advective and diffusivefluxes is that the former isinvariably associated with atransfer of a known volumewhile diffusive fluxes are not,but presuppose gradients bothin space and time. The diffusionprocess acts over such gradi-ents, the existence of which arein principle not compatible withthe assumption of well-mixedgrid cells. The resolution of thispredicament is that diffusivematerial transfer is treated, inthe context of most numericalmodels, as if it consisted of aseries of mutual exchanges ofappropriately determined vol-umes between adjacent gridcells, reducing and eventuallyeliminating the concentrationdifferences. This exchange,however, is not possible toinclude in the advective ex-change flow since it operateson a sub-grid scale. The corre-sponding fictive exchanged vol-umes can thus not be includedin the advective fields, which inturn means that diffusive effectscannot be directly accounted forwhen computing trajectories ei-ther forward or backward intime.

Trajectories can be launchedcontinuously in time and theirstarting point can be arbitrarily

Box 7. Trajectory calculationA trajectory can be tracked starting from any grid cell. If the cell, say number j (restricted in the presentcase to j¼ 1,2,3), is chosen, we want to follow this particular water parcel from its starting point in timeprogression until it leaves compartment 3. This can be achieved numerically by assigning 0.5dt as theinitial scalar value of compartment j. All corresponding values of the other compartments are set to zero.The initial state vector will be T ¼ (0,0,0.5,0) if j ¼ 2 and the scalar time is measured in dt-units. Forobvious reasons the trajectory dynamics will be the same as for AvA computation going forward in timeusing A, but also the same as for AvR going backwards using AT:

DT=dt ¼ 1; if ti ¼ 0;0; i f ti � 0;

�ðEq: 5Þ

With this numerical arrangement only the original j-element that is greater than zero will keep track of thetime and when this parcel exits the last cell in the row (cell 3), the accumulated time is noted, which forsteady flows will give exactly the same value as the AvR-time of the starting cell position. It is stillobvious – as long as the transition involves whole grid cells – that AvA and backtracked trajectory timeswill yield identical times since they follow identical paths with the same time difference.

2-D trajectory computationThe lack of information about diffusively exchanged volumes is the main feature that distinguishes theAvA method from the trajectory method that solely utilizes advective flow information of the exchange.Advection may involve whole or fractional exchanged cell volumes and can be represented by velocitiesat the center point of the interfacial boundaries. Velocities at all other locations within the 2D-area cannow be interpolated (some areas near the corners need extrapolation) and in this new interpretation a2D-velocity field is established for every point in the domain (see figure above). The velocities can alsobe interpolated between the discrete intermediate times. Inserting the 3 3 3 array in a Cartesiancoordinate system, particles can be released at any random point and their trajectory, r, as a function oftime (t ) corresponding to a momentary position in a coordinate system (x,y), can be integrated accordingto

rðtÞ ¼ ðxðtÞ; yðtÞÞ ¼Zt

0

uðsÞ�ds ¼Zt

0

ðuðsÞ; vðsÞÞds; ðEq: 6Þ

where u and v are the formerly discrete velocities interpolated so that they are defined in the continuousplane representing the interior of boxes. The consequence is that it is possible to realize an infinitenumber of different trajectories.

Figure 6. a) Black curve: the time evolution of decay of the number of trajectories in the study area, those trajectories having been releasedevery hour during the whole year in the discharge area, and traced forward in time until they leave the study area, giving the averageresidence time (AvR). The purple curve is the backward in time case corresponding to the average age (AvA). The yellow curve is the totaltime spent in the study area for trajectories that have passed at least once through the discharge area. This is equivalent to the averagetransient time (ATR). b) The average transient time (ATR) evolution with one curve for each day of release of trajectories. The average ofthese 365 lines is the yearly average of the yellow line in a).


chosen. If the source is evenly distributed over the boundarysurfaces, those that enter the study area can be traced until afraction of them reach the appointed release area. Theirdifference in arrival time will then form a statistical basis fromwhich the renewal AvA time can be inferred. It would then bemore rational to track the particle flow backward in timestarting from the release area. In the assumed absence ofdiffusion this should exactly replicate the forward trajectories.In Figure 7 the close relationship is convincingly demonstrated(correlation coefficient ¼ 0.90, N ¼ 25) between AvA-measurescomputed with back-tracked trajectories vs. the standard AvAmethod. Due to diffusion, there is a notably greater varianceamong the trajectory times than for the standard AvA method.The chosen diffusivity 20 m2s�1 is arguably somewhat high butnumerical experiments indicate that the qualitative featurespresently discussed remain mainly unchanged if this parametervalue is halved. All the different water parcels simultaneouslypresent in a grid cell may, on the other hand, by backtrackingtrajectories, in time be attributed an individual age, giving adistribution curve comprising the age of all released particles,while the AvA-method can only give their average. In thiscontext, the risk of entering into the food-web is measured inexposure time, not concentration. This means that thetrajectory-based ATR times that neglect sub-grid turbulentdiffusive exchange, can serve as an upper bound of the averagelikelihood that released radionuclides enter the foodweb. Forstationary flow regimes the time-based measures are possible torelate to water exchange. The longer the ATR time is, the lessintense is the ventilation. Adding the diffusive component ofventilation, this would shorten the AvR time and increase therate of dilution. This means that without diffusion one wouldobtain a higher concentration of contaminants. Finally, as aparticle-oriented method, individual trajectories can also becombined with positive or negative buoyancy forces. Thisoption enhances the versatility of the method and makes itpossible to account for e.g. sedimentation processes.

CONCLUSIONS AND SUMMARY

As an instrument for indicating transient changes of the generalcirculation, the AvA method seems more efficient and reacts ontime scale that is determined by the length scale of the domaindivided by the velocity scale of the advective events. Thisincludes exogeneous water that does not pass over the verticalmodel boundaries but may come from other sources such asfreshwater discharge or entering through pipes. Such sources ofexogeneous water can certainly also be analyzed with trajectorytracking, but this requires a special study for each source (16,43). When performing such an analysis, a distinct advantage isthat comprehensive stream functions ensue as a spin-off.

Trajectory analysis is an appropriate and well-suited methodfor analyzing the present particular problem with a defined 2-Drelease area and a 3D-study site. It may also, in certaincircumstances, be advantageous that the forward-in-timecalculated residence time does not involve diffusive transports,since this gives a conservative over-estimate of the resultingplume concentration.

From a computational point of view, trajectory analysis canbe done off-line provided that an appropriately frequently time-resolved flow field is saved. AvA-fields can on the other hand be

Figure 7. Average age (AvA) shown with red lines of which the solidline represents the spatial average of the discharge area’s 109bottom layer grid cells and the dashed lines 6 1 S.D. These arecomputed online with the full temporal model resolution indischarge area for 25 instances that are nearly equidistantlyseparated in time during the year cycle. These curves are comparedto the average time trajectories of exogenous water needed to travelto arrive at the bottom layer of the discharge area on the indicateddays. Computations are made entirely offline. The curve represent-ing the average is in solid black and 6 1 S.D. is dashed. Thecorrelation coefficient between simultaneous semi-monthly samplesof the two averages is 0.90 (N¼25).

Box 8. 3D-models

Stacking 2D-arrays on top of each other produces yet additionaldegrees of freedom for the water movements. By extending thesame kind of cell numbering convention as for the 2D-case, there isno hindrance to utilize the same formalism as above to describe thewater movements (see figure above). In fact, the AS3D model of thepresent Forsmark study area has 241 grid cells in the horizontal x-and y- directions corresponding to a distance of 185 m or one tenthof a nautical mile. In the vertical (z) direction the thicknesses of thegrid layers vary, but are approximately 10 m, thinner towards thesurface. If all these cells represented water, it would mean that thecorresponding A-matrix should have 2323240 3 2323240 rows andcolumns. Less than half this number are ‘wet’ cells so the rank of thematrix is in fact only (900055 3 900055), due to the bathymetriccurve with diminishing horizontal areas towards the bottom. Withthe exception of the slight complications that unequal grid cellvolumes and a varying sea surface involves, accurate accounting ofthe transacted volumes can be fully described according to thesame formalism so that Eqs. (4) and (5) still apply. Time-wise theflow fields are integrated with a 1.2 min time step. This time stepthus also applies to the AvA-values that are computed within themodel frame. The trajectories are computed out of a subset of thevelocity fields that were saved every hour. This motivates theapproximative setting of AvA of exogeneous water to zero and notthe more exact -½ dt . Fickian type of (eddy-)diffusion does occurboth in reality and in 3D-model, however. This adds a diffusive termso that the complete AvA-equation becomes (8, 19):

Da=dt ¼ 1þ aDðtÞ; ðEq: 7Þ

where D(t) is the time-varying diffusion matrix, varying indepen-dently of the contained scalar property a for AvA, since withdiffusion processes, active, forward-in-time AvA computation canalways be performed. Not knowing the volumes associated with thediffusive exchange makes backtracking in time of the flows inprinciple impossible. Making the reasonable assumption that thespreading is randomly distributed, then the average over manyrealizations should converge to the advective flow, even though thedivergence from this average would grow immensely with time.


computed with full model temporal resolution. With itsdiffusion-related conservative nature it suffices that the AvA-fields are only saved with a frequency that is in concord with theimposed large-scale changes in the forcing. The backwardtrajectory and the AvA methods give mainly qualitativelysimilar results. They are demonstrated to possess a high degreeof correlation although their ensemble averages deviate, thedifference being mainly due to the diffusion.

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44. We thank Jonas Nycander for discussion on time-reversal of diffusive processes andSwedish Nuclear Fuel and Waste Management Co. for funding.

45. All SKB reports mentioned in this article can be downloaded from http://www.skb.se

Anders Engqvist, Ph.D, works with coastal oceanography atthe Department of Systems Ecology at Stockholm University,and is presently involved with modeling the water exchange ofthe Stockholm, Aland and Aboland archipelagos and theirnutrient dynamics. His address: Department of SystemsEcology, Stockholm University, SE-106 91 Stockholm, Swe-den.E-mail: [email protected]

Kristofer Doos is an associate professor in physical oceanog-raphy at the Department of Meteorology at StockholmUniversity. He has previously worked at Southampton Ocean-ography Centre and at the Institute of OceanographicSciences in the U.K. He has a PhD in oceanography fromUniversite de Pierre et Marie Curie in Paris. His main researchhas been on the overturning circulation and the Lagrangiantracking of water masses in the World Ocean, Southern Oceanand the Baltic Sea. His address: Department of Meteorology,Stockholm University, SE-10691 Stockholm.E-mail: [email protected]

Oleg Andrejev, Ph.D. is affiliated to the Finnish Institute ofMarine Research and works with physical oceanography,specializing in 3D-modelling and associated numerical meth-ods in particular data assimilation. Together with AlexanderSokolov, he developed the AS3D baroclinic model. He is theauthor of about 70 scientific papers and books. His address:Finnish Institute of Marine Research, P.O. Box 33, FIN-00931Helsinki, Finland.E-mail: [email protected]


engqvist a., k. döös and o. andrejev, 2006

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