Quantifying Water Retention Time in Non-tidal Coastal Waters Using Statistical and Mass Balance Models

Quantifying Water Retention Time in Non-tidal CoastalWaters Using Statistical and Mass Balance Models

Peter H. Dimberg & Andreas C. Bryhn

Received: 7 April 2014 /Accepted: 28 May 2014# Springer International Publishing Switzerland 2014

Abstract The water retention time (sometimes calledresidence time) in coastal areas is an indicator of coastalhydrodynamics which can be used to quantify the localtransportation of dissolved and suspended pollutants.This study has used dynamic and statistical models toexplore what governs the water retention time in non-tidal coastal waters of the Baltic Sea. If freshwater inputdivided by the cross-section area between the coastalwater and the sea was below a certain threshold, fresh-water had no notable impact on the retention time.Moreover, statistical models were developed forpredicting surface water retention time and total waterretention time from coastal water volume, cross-sectionarea and freshwater discharge. This study can be usefulfor managers who need to determine where abatementmeasures should be focused in order to be as effective aspossibly against coastal water pollution.

Keywords Retention time . Coastal area . Baltic Sea .

Statistical model .Mass balance

1 Introduction

The influence of hydrodynamics and water retentiontime is important in many aspects of coastal environ-mental management. For instance, protected coastalbays with long water retention time may serve as im-portant spawning areas for coastal fish (Snickars et al.2010). The extent of water exchange with the sea canaffect the species composition of macrophytes (Thorne-Miller et al. 1983). Perhaps even more important from amanagement perspective is that hydrodynamics entailstransportation of dissolved or suspended pollutants andother substances. If the water retention time in a coastalwater body is short due to a large water discharge fromthe catchment, the concentration of pollutants in coastalwater is likely to resemble the concentration in thedischarge. In this case, remedial action is in many casesbest undertaken in the catchment. If the water retentiontime is short due to a large water influx from the sea, itmay be better to address the pollution situation in thewhole sea, in case there are high pollutant concentra-tions there. If water retention time is long, it is particu-larly important to study all possible pollutant transpor-tation paths: from the catchment, from the atmosphere,from the sediment and from the sea. Thus, water reten-tion time may be decisive for which action alternative isthe most effective one against water pollution(Håkanson and Bryhn 2008; Liu et al. 2011; Grifollet al. 2013).

The Baltic Sea (N. Europe) is a large, stratified, non-tidal estuarine sea with small temporal changes in salin-ity and an increasing salinity gradient from the north to

Water Air Soil Pollut (2014) 225:2020DOI 10.1007/s11270-014-2020-z

P. H. Dimberg (*)Department of Earth Sciences, Uppsala University,Villav. 16, 752 36 Uppsala, Swedene-mail: [email protected]

A. C. BryhnDepartment of Aquatic Resources, Swedish University ofAgricultural Sciences,Skolgatan 6, 742 42 Öregrund, Sweden

the south. The mean salinity is 7.4 psu. The salt entersthe Baltic Sea through two straits in Denmark and onestrait between Denmark and Sweden. The deep wateronly receives substantial additions of salt during the socalled major Baltic inflows, which most often occurduring the winter and only during certain years (Meier2007). The large-scale water circulation of the BalticSea has been studied quite intensively during the lastdecades, although the extent, causes and effects of var-iations in water exchange in the coastal zone are lesswell known. In most coastal waters of the Baltic, thewater quality is determined by conditions in the outsidesea. Exceptions are enclosed coastal waters with a verylimited exchange with the sea, and coastal waters with alarge freshwater input which contains pollutants fromthe catchment. Statistical models which calculate thewater retention time in Baltic coastal waters have, how-ever, hitherto not taken discharge from the catchmentinto account (Håkanson and Bryhn 2008). In this paper,we will determine a limit concerning freshwater dis-charge when such models can or cannot be used.

This study has three main aims: (i) to determine therelative impact from water discharge and precipitationon water retention time in Baltic coastal waters com-pared to the impact fromwater inflow from the open sea,(ii) to derive statistical models to predict retention timesof the whole water body and of surface water usingmorphological parameters along with water dischargesfrom drainage area and precipitation on the coast and(iii) to calculate the distribution of relative retention timeand retention time in Baltic coastal waters located inSweden.

2 Material and Methods

The study is based on a dynamic salt model and statis-tical modelling, and the details concerning these modelswill be described in this section. Moreover, input dataand study areas will be described. In this study, retentiontimes are meant to describe the average time it takes fora water molecule to leave the water body for the firsttime (Delhez et al. 2014). The water retention time iscalculated by dividing the total water outflow by thetotal volume; also, surface water time is calculated bydividing total surface water outflow by the volume ofthe surface water (Persson et al. 1994; Håkanson andEklund 2007; Håkanson and Lindgren 2010).

2.1 Model Structure and Equations

Knudsen’s theorem, which is a basic mass balance ofsalt and water, can be used to calculate water retentiontime in coastal waters (Knudsen 1900). Salinity insideand outside the coastal water boundary are used toobtain a mass balance of salt along with the watercontribution from area of drainage area (ADA).However, to be able to calculate reliable retention times,it is also important to include dynamic responses withinthe coastal water, e.g. diffusion, and mixing betweenwater layers. Håkanson and Lindgren (2010) usedKnudsen’s approach and included important processesof salt (e.g. diffusion from deep water (DW) to surfacewater (SW) and mixing between DW and SW) to cal-culate water retention time for five sub-basins in theBaltic Sea. The same model structure will be used inthis study (Fig. 1); however, to facilitate its use and tomake the model applicable for other coastal waters,other mathematical equations are included. For instance,the wave base (Wb) separating surface water from deepwater and water temperature derived from latitude werecalculated using two different equations. Theseequations and others may be found in the studypresented by Håkanson and Eklund (2007). Some equa-tions are provided below, and Appendix A presents theremaining equations and model variables for the massbalance model of salt used in this study.

Fig. 1 Schematic illustration of salt fluxes in coastal waters

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There has been no tuning in the model to fit themodelled salinity values with the empirical salinitiesexcept for two instances: (1) the inflow from the outsidesea and (2) a distribution coefficient (DC) to separate themagnitude of inflow from SW and DW. This was nec-essary in order to estimate the contribution of waterfrom the outside sea. After the tuning, when themodelled salinity corresponded with the empirical sa-linity, retention times were calculated for each month,for the respective coastal water, where data were avail-able. A comparison was made of the retention timeswhen water discharges and precipitation to the coastwere neglected; i.e. assuming that water discharge andprecipitation equal to zero in the model.

One method to separate SW from DW is to calculatethe Wb (Eq. 1; Håkanson and Bryhn 2008). The wavebase is the limit between transportation and accumula-tion sediment, where transportation sediments are occa-sionally affected by wind and wave activity and accu-mulation sediments underlie calm waters and have a netaccumulation of particles.

Wb ¼ 45:7⋅ffiffiffiffiffiffiffiffiffiffiArea

p

21:4þ ffiffiffiffiffiffiffiffiffiffiArea

p ð1Þ

where Wb (m) is the wave base, and Area (km2) is thecoastal surface water area

The topographical bottleneck principle (Eq. 2;Håkanson and Bryhn 2008) was used to determine therelative openness to the sea, i.e. exposure (Ex; unitless).A high Ex value indicates a large inflow from the sea,whereas small Ex values indicate a relatively enclosedarea. A small Ex and a large drainage area (ADA; inkm2) imply that a significant part of the water retentiontime is affected by water discharge from the ADA.

Ex ¼ 100⋅At

Areað2Þ

where Ex is the exposure to the sea and At is the verticalsection area. Both At and Area are expressed in the sameunit.

The dynamic ratio (DR; unitless; Eq. 3) was used toestablish the influence of wind and wave activities in thewater. DR plays an important role in quantifying thestratification limit in aquatic systems (Håkanson andBryhn 2008).

DR ¼ffiffiffiffiffiffiffiffiffiffiArea

p

Dmð3Þ

where DR is the dynamic ratio, area is expressed in km2

and Dm (m) is the mean depth.The form of the coast (volume development (Vd)

unitless) can be described by a form factor (Eq. 4;Håkanson and Bryhn 2008).

V d ¼ 3⋅Dm

Dmaxð4Þ

where Dmax (m) is the maximum depth. If a system is“u-shaped” the Vd is equal to 3, i.e. Dm=Dmax. Vd isimportant for estimating the area of accumulation sedi-ments (Eq. 5) which in turn can be used to calculate thevolume of surface and deep waters (Eqs. 6 and 7).Equations 5 and 6 have been derived from hypsographicrelationships between Vd, depth, areas and volumes ofaquatic systems (Håkanson and Eklund 2007).

AreaA ¼ Area⋅Dmax−Wb

Dmax þWb⋅e3−V d1:5

� �0:5Vd ð5Þ

V SW ¼ V−AreaA⋅V d Dmax−Wbð Þ

3ð6Þ

VDW ¼ V−V SW ð7ÞwhereDmax (m) is the maximum depth, AreaA (m

2) is areaof accumulation sediments, V (m3) is volume, VSW (m3) issurface water volume and VDW (m3) is deep water volume.

2.2 Data on Swedish Coastal Waters

Two different datasets were used in this study. The firstdataset contained seven different coastal waters locatedin Sweden (latitude 56 to 63°N; Fig. 2; Table 1) tocalculate retention times using the mass balance of salt.Two coastal waters consisted of sub-areas, i.e. Draget(Draget, Svartviksfjärden, Sundsvallsfjärden andAlnösundet) and Ullångersfjärden (Ullångersfjärden,Dockstafjärden and Norrfjärden). The coastal water sur-face area ranged from 10.3 to 98.1 km2 with a meandepth ranging from 4 to 48.9 m. ADA ranged from 76 to13,438 km2 and indicated that water discharges fromADA had a wide variability between the seven coastalwaters (Abrahamsson and Håkanson 1998).

Observed salinity (psu) data for the seven coastalwaters were taken from 4 to 8 years where 3 to11 months were included in 1-year cycle (Table 2).

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Ronnebyfjärden had measurements from 2004 to 2010,Slätbaken from 2005 to 2011, Gävlefjärden from 2005to 2012, Hudiksvallsfjärden from 2005 to 2011, Dragetfrom 2008 to 2011, Gaviksfjärden from 2006 to 2012and Ullångersfjärden from 2008 to 2011. Salinities werecalculated as median values for each available month inthe surface and deep waters from several depths (ifpossible), i.e. where data were available and whereverthe Wb did not exceed Dmax. Corresponding measure-ments of salinity were collected in the sea outside eachcoastal area (Table 3). Deep water concentrations in thesea were used if Dmax of the coastal inlet exceeded theWb. Otherwise, only surface water salinities were used.Overall, the salinity was higher and had a smaller

variability (coefficient of variation; CV) in the DWcompared to SW (Tables 2 and 3).

Water discharges were calculated from ADA to eachcoast for corresponding years to when salinity data wererecorded (Table 4; SMHI 2014a). The water dischargeswere modelled values from the Swedish Meteorologicaland Hydrological Institute’s (SMHI) model S-HYPEand consisted of data from entire Sweden, specified foreach drainage area. The water discharge ranged from0.8·106 m3/month (Gaviksf järden, July) to402.8·106 m3/month (Draget, June). Presented valueswere incorporated into the mass balance model of saltand henceforth used to tune modelled salinity data withthe empirical data. Data on precipitation to the coastalwaters (Table 5) were processed in the same manner.Precipitation ranged from 22 mm (Slätbaken, April) to116 mm (Gävlefjärden and Ullångersfjärden, August).

The second dataset used in the study concerned 313Swedish non-tidal coastal waters, and this dataset wasused for calculating percentual distributions of relativeretention time and retention time among coastal waters.These 313 coastal waters had a wide variability ofmorphology and inflow of water, i.e. area values rangingfrom 0.63 to 410.2 km2, V from 0.00007 to 10.22 km3,Dm from approximately 0 to 51 m, Dmax from 1 to131 m, ADA from 0.03 to 26,800 km2, At from 29 to1,232,522 m2, mean monthly water discharge from518.4 to 12.3·108 m3/month (SMHI 2003). Since therewas a lack of precipitation data on coastal waters, theprecipitation was estimated using water discharges fromADA divided by 0.38 as it can be assumed that about62 % of the precipitation on ADAwill be evaporated atthese latitudes (Chow 1988; Abrahamsson andHåkanson 1998). Evaporation from the Baltic Sea issmaller in magnitude than precipitation (Omstedt et al.1997) and will be assumed to be negligible in the watermass balances. The influence on results of theseassumptions about precipitation and evaporation willbe discussed. The mean precipitation on the coastalareas ranged from 32 to 128 mm for typical months.

2.3 Interpretation of the Results

Below, the procedure to derive statistical models forrelative retention times and retention times are described.In aquatic science, it is common that statistical models aresignificant if the p value is below 0.05, and Prairie (1996)showed that when the r2 is above 0.65, the model hassufficient predictive power. The same thresholds (r2>

Fig. 2 Seven coastal areas used in the mass balance of salt. RRonnebyfjärden, S Slätbaken, G1 Gävlef järden, HHudiksvallsfjärden, D Draget, G2 Gaviksfjärden, UUllångersfjärden


0.65 and p<0.05) have been used in this study to deter-mine whether a model had sufficient predictive power.

2.3.1 Relative Retention Times

Relative retention times (T*/T and TSW*/TSW; *=nowater contribution from ADA and precipitation) were

used to analyse how much the retention time was affect-ed bywater contribution from other sources than the sea.The relative retention times were regressed against thetotal water inflow from ADA and precipitation(QADA+Prec) divided by At (QADA+Prec/At). Equationsderived from correlations between T*/T and QADA+Prec/At and between TSW*/TSW and QADA+Prec/At from the

Table 1 Morphometry of seven coastal waters

Name Latitude (°N) Coastal area (km2) Dm (m) Dmax (m) At (m2) ADAa (km2)

Ronnebyfjärden 56.2c 10.3b 4b 12b 11,020b 1,199b

Slätbaken 58.5c 15.4b 11b 44b 249b 1,001b

Gävlefjärden 60.4c 19.6c 7.2c 15c 6,414b 3,420d

Hudiksvallsfjärden 61.4c 25.3b 8b 23b 7,076b 198b

Draget 62.3c 33.2b 19b 56b 56,882b 13,438b

Gaviksfjärden 62.9c 22.9b 34b 94b 31,300b 76b

Ullångersfjärden 63.0c 98.1b 48.9b 121b 406,880b 410b

Dm mean depth, Dmax max depth, At section area, ADA area of drainage areaa Data not needed to run the modelb SMHI (2003)c SMHI (2014a)d Calculated using geographic information systems

Table 2 Salinity (psu) inside seven coastal waters. Median values

Name January February March April May June July August September October November December Mean CV

Ronnebyfjärden

SW 6.81 6.99 6.94 7.12 7.11 7.02 7.00 0.02

Slätbaken

SW 3.70 3.55 3.80 4.30 4.20 4.40 4.60 4.08 0.10

DW 5.08 4.80 4.75 5.00 5.00 4.90 4.90 4.92 0.02

Gävlefjärden

SW 4.50 4.95 5.00 4.75 4.68 4.70 4.80 4.77 0.04

Hudiksvallsfjärden

SW 5.15 4.83 4.85 4.90 4.90 5.05 4.95 0.03

DW 5.30 5.05 5.00 5.15 5.10 5.10 5.12 0.02

Draget

SW 4.95 4.00 4.60 4.52 0.11

DW 5.20 5.00 5.10 5.10 0.02

Gaviksfjärden

SW 5.22 5.14 5.14 4.98 4.80 4.68 4.69 4.71 5.02 5.08 5.20 4.97 0.04

DW 5.58 5.49 5.52 5.51 5.64 5.74 5.72 5.88 5.79 5.71 5.39 5.63 0.03

Ullångersfjärden

SW 4.84 4.64 4.62 4.82 5.17 4.82 0.05

DW 5.78 5.64 5.48 5.76 5.22 5.58 0.04

SW surface water, DW deep water, CV coefficient of variation (standard deviation/mean value)Data from SMHI (2014b)


first dataset (Tables 1, 2, 3, 4 and 5) were then testedfor the second dataset (313 other Swedish coastalwaters) and also used to detect a ratio limit ofQADA+Prec/At to determine at which level the contribu-tion from QADA+Prec can be considered insignificant.Subsequently, the equations were used to calculate therelative impact of water inflow from the sea andQADA+Prec on data from the second dataset (with 313coastal waters). Relative retention times were also cal-culated for a subset of 120 (out of the 313) coastalwaters which were within the derived models range,i.e. QADA+Prec from 2.4·106 to 4.0·108 m3/month andAt from 249 to 406,880 m2.

2.3.2 Retention Times

Multiple regressions were created for T and TSW using avolume-related exposure (ExV; Eq. 8) andQADA+Prec asindependent variables.

ExV ¼ 100⋅At

Volumeð8Þ

Where ExV (1/m) is the volume-related exposure andAt (m2) is the section area; volume has the unit m3.

ExV is based on the same principle as Ex (Eq. 2) butconsiders the whole water body to explain the totalexposure of the coast.

Table 3 Salinity (psu) in the sea outside seven coastal waters. Median values


Ronnebyfjärden

SW 7.58 7.43 7.60 7.42 7.49 7.50 7.50 0.01

Slätbaken

SW 7.00 6.77 6.75 6.53 6.43 6.54 6.81 6.69 0.03

Gävlefjärden

SW 5.38 5.41 5.42 5.33 5.09 5.04 5.17 5.26 0.03

Hudiksvallsfjärden

SW 5.43 5.36 5.13 5.09 4.98 5.18 5.20 0.03

Draget

SW 5.37 5.10 4.95 5.14 0.04

Gaviksfjärden

SW 5.33 5.32 5.17 5.25 5.03 4.76 4.80 5.03 5.12 5.08 5.36 5.11 0.04

DW 5.55 5.53 5.68 5.57 5.59 5.70 5.79 5.85 5.80 5.65 5.53 5.66 0.02

Ullångersfjärden

SW 5.34 5.04 4.81 4.84 5.32 5.07 0.05

DW 5.61 5.65 5.74 5.77 5.58 5.67 0.01

SW surface water, DW deep water, CV coefficient of variation (standard deviation/mean value)Data from SMHI (2014b)

Table 4 Water discharge from ADA to the coastal waters. The water discharges have the unit ·106 m3 and are calculated as mean values(SMHI 2014a)

Name January February March April May June July August September October November December

Ronnebyfjärden 32.8 33.8 19.8 14.9 12.5 27.6

Slätbaken 19.0 35.1 10.0 8.9 11.7 6.4 16.8

Gävlefjärden 94.8 85.0 81.4 106.3 48.2 50.2 54.1

Hudiksvallsfjärden 5.6 3.5 3.1 2.0 1.5 4.3

Draget 314.5 402.8 354.1

Gaviksfjärden 2.1 1.6 6.0 6.1 1.4 0.8 1.1 2.0 3.0 4.0 3.9

Ullångersfjärden 7.3 7.2 8.6 13.4 13.5


Data from the mass balance of salt were used, i.e. 45data points (representing each month available for theseven areas in Tables 1, 2, 3, 4 and 5) for each multipleregression. Variables were transformed using differenttransformations (e.g. log, ln or sqrt) to obtain normaldistribution. Other variables than ExV (e.g., At, Ex,volume) were tested as independent variables forexplaining the cross-systems variation in retentiontimes.

The multiple regressions were then used to calculateT and TSW in 313 coastal waters. The same dataset as theone used to calculate relative retention times was used inthis case. The distribution and number of availableretention times were calculated. Retention times werealso calculated for a subset of 75 (out of the 313) coastalwaters which were within the derived model ranges, i.e.QADA+Prec from 2.4·106 to 4.0·108 m3/month and ExVfrom 0.0001 to 0.027 m−1.

3 Results

3.1 Modelled Results Using Mass Balance of Salt

The relative modelled salinity had an r2=0.96 whencompared with relative empirical salinity values(Fig. 3). The inflow and outflow of water had similarvariability for the seven coastal waters (Table 6).However, the differences between inflow and outflowof water ranged from 0 to 48 % (Gaviksfjärden andSlätbaken, respectively), while the magnitude of waterflow between the investigated areas differed by approx-imately 1,000 times (Slätbaken and Ullångersfjärden).The retention times increased by 0–51.0% for the wholewater body and by 0–96.9 % for the surface water(Ullångersfjärden and Slätbaken, respectively) whenthe inflow QADA+Prec was eliminated in simulations.

The variability of the retention times between monthswas approximately zero for Ronnebyfjärden,Gaviksfjärden and Ullångersfjärden’s whole water bodyand surface water, and the highest difference was foundfor Slätbaken, Gävlefjärden and Draget when the inflowQADA+Prec was eliminated in simulations (Table 7).

3.2 Relative Retention Times

While both At and Ex were tested as independent var-iables, QADA+Prec normalised with At was found to bethe best predictor of relative retention times (T*/T andTSW*/TSW) with sufficient predictive power (r2=0.88and r2=0.86, respectively; p<0.001; Fig. 4). Retentiontime was found to be significantly affected by QADA+

Prec if QADA+Prec was relatively high compared to At.Conversely, a high At compared toQADA+Prec indicatedthat the retention time was much more affected byinflow from the sea than by QADA+Prec. The equationfor T*/T (Fig. 4) showed that when QADA+Prec/At wasbelow 64.5 m/month (ratio limit), the section area waslarge enough to imply a completely dominating inflowfrom the open sea (i.e. T*/T=1). For TSW*/TSW theratio-limit of QADA+Prec/At was 66.5 m/month. Usingthe equations from Fig. 4, it was found that out of the313 Swedish coastal waters, 46 % had no impact fromQADA+Prec on the retention times (T and TSW; Table 8).QADA+Prec had an effect by up to 5 % on T among 40 %of the coastal waters and among 37 % of the waters onthe TSWof the coastal waters. Similar results were foundfor coastal waters within derived model ranges(Table 9).

3.3 Retention Times

Both ExV and QADA+Prec could be used to significantestimate T (r2=0.94; n=45, p<0.0001; Eq. 9) and TSW

Table 5 Mean precipitation (mm) on the coastal waters (SMHI 2014a)

Name January February March April May June July August September October November December

Ronnebyfjärden 55 41 58 115 51 72

Slätbaken 45 24 69 97 111 62 60

Gävlefjärden 42 40 34 48 72 116 74

Hudiksvallsfjärden 37 49 65 92 48 55

Draget 38 64 108

Gaviksfjärden 75 42 41 42 44 70 107 81 87 75 93

Ullångersfjärden 36 48 116 68 77


(r2=0.93; n=45; p<0.0001; Eq. 10). Each determinantvariable in Eqs. 9 and 10 had a p value below 0.0001.Using ExV instead of Ex was important for coastal areas

which had a volume-area relationship that contradictedtraditional relationships where a small volume gives asmall area and a large volume gives a large area.

Fig. 3 Relative modelled salinitycorrelated with relative empiricalsalinity for both SW and DW inseven non-tidal coastal waters. N(number of monthly data)=77,p<0.001

Table 6 Coastal in- and outflow of water from and to the sea. Unit in ·108 m3


Ronnebyfjärden

In 2.7 4.1 2.1 2.7 2.1 2.7 2.7 0.27

Out 3.0 4.4 2.3 2.9 2.2 3.0 3.0 0.27

Slätbaken

In 0.20 0.35 0.35 0.38 0.32 0.32 0.38 0.33 0.19

Out 0.40 0.70 0.46 0.48 0.45 0.39 0.56 0.49 0.22

Gävlefjärden

In 4.5 8.3 8.5 7.6 5.3 6.4 5.8 6.6 0.23

Out 5.5 9.2 9.3 8.7 5.8 6.9 6.4 7.4 0.22

Hudiksvallsfjärden

In 3.5 3.5 1.5 1.5 3.5 3.5 2.8 0.36

Out 3.6 3.5 1.5 1.5 3.5 3.6 2.9 0.37

Draget

In 51 25 51 42 0.35

Out 54 29 55 46 0.32

Gaviksfjärden

In 13.0 13.0 13.0 13.0 13.0 15.0 10.0 13.0 13.0 13.0 13.0 12.9 0.09

Out 13.0 13.0 13.1 13.1 13.0 15.0 10.0 13.0 13.0 13.1 13.0 12.9 0.09

Ullångersfjärden

In 130.0 130.0 130.0 130.0 130.0 130.0 0.00

Out 130.1 130.1 130.2 130.2 130.2 130.2 0.00

CV coefficient of variation (standard deviation/mean value)


log Tð Þ ¼ 1:7748−0:5884⋅log ExVð Þ−0:2851⋅log Qð Þð9Þ

log TSWð Þ ¼ 1:7547−0:5656⋅log ExVð Þ−0:2733⋅log Qð Þð10Þ

Where Q is the water discharge from ADA plusprecipitation on coastal waters (m3/month).

Iterations of different values within the derivedmodels ranges of ExV and Q showed that the retentiontimes were most sensitive for variations in ExV (Fig. 5).The median for the 75 coastal waters within derived

Table 7 Water retention time (days) in coastal waters


Ronnebyfjärden

TSW 4.1 2.8 5.4 4.4 5.6 4.2 4.4 0.23

T 4.1 2.8 5.4 4.3 5.5 4.1 4.4 0.23

TSW* 4.6 3.1 6.0 4.6 6.0 4.6 4.8 0.23

T * 4.6 3.0 5.9 4.6 5.9 4.6 4.8 0.23

Slätbaken

TSW 105.0 59.0 109.2 106.2 106.1 133.9 85.0 100.6 0.23

T 128.0 72.2 110.3 105.1 111.9 129.3 91.2 106.9 0.19

TSW* 312.0 178.3 178.3 164.2 195.0 195.0 164.2 198.1 0.26

T * 254.1 145.2 145.2 133.7 158.8 158.8 133.7 161.4 0.26

Gävlefjärden

TSW 7.8 4.7 4.6 4.9 7.4 6.2 6.7 6.0 0.22

T 7.7 4.6 4.5 4.9 7.3 6.1 6.7 6.0 0.22

TSW* 9.5 5.1 5.0 5.6 8.1 6.7 7.4 6.8 0.25

T * 9.4 5.1 5.0 5.6 8.0 6.6 7.3 6.7 0.24

Hudiksvallsfjärden

TSW 15.7 15.8 36.1 36.2 15.8 15.7 22.6 0.47

T 17.0 17.1 39.3 39.3 17.2 17.1 24.5 0.47

TSW* 16.0 16.0 37.3 37.3 16.0 16.0 23.1 0.48

T * 17.4 17.4 40.5 40.5 17.4 17.4 25.1 0.48

Draget

TSW 4.6 8.1 4.5 5.7 0.36

T 3.5 6.5 3.5 4.5 0.38

TSW* 5.1 10.3 5.1 6.8 0.44

T * 3.7 7.6 3.7 5.0 0.45

Gaviksfjärden

TSW 20.0 20.0 19.9 19.9 19.9 19.8 19.9 20.1 20.1 20.0 20.0 20.0 0.00

T 18.1 18.1 18.1 18.1 18.1 17.9 18.0 18.2 18.2 18.2 18.2 18.1 0.01

TSW* 20.1 20.1 20.1 20.1 20.0 19.9 20.0 20.2 20.2 20.2 20.1 20.1 0.00

T * 18.2 18.2 18.2 18.1 18.1 18.0 18.1 18.3 18.3 18.2 18.2 18.2 0.00

Ullångersfjärden

TSW 11.1 11.1 11.1 11.1 11.1 11.1 0.00

T 11.1 11.1 11.1 11.1 11.1 11.1 0.00

TSW* 11.1 11.1 11.1 11.1 11.1 11.1 0.00

T * 11.1 11.1 11.1 11.1 11.1 11.1 0.00

TSW surface water, Twhole water body, CV coefficient of variation*Retention time when influence of water discharge and precipitation has been eliminated. Range ofmean T 4.4 to 106.9. Range ofmean TSW4.4 to 100.6


model ranges was for ExV 0.011 m−1 and forQADA+Prec

4.8·106 m3/month and gives a typical retention time ofapproximately 10 days according to Fig. 5.

Retention times for the 313 coastal waters were cal-culated using Eqs. 9 and 10. There were overall nearly

50 % of the coastal waters which had a retention timebetween 4 and 10 days for both the whole water bodyand surface water. There were approximately 19 % ofcoastal waters which had retention times above 15 daysand 2.5 % above 50 days (Table 10). Approximately50 % of coastal waters within the derived model range

Fig. 4 Relative change inretention times when neglectingmonthly water discharges andprecipitation on seven non-tidalcoastal waters. At is the sectionarea (m2), Q is the monthly waterdischarge (m3/month) from ADAand precipitation (m3/month), T(days) is the retention time for thewhole water body, and TSW (days)is the retention time for surfacewater. The retention time whenneglecting water discharges andprecipitation. aRelationship for T.b Relationship for TSW. N(number of monthly data)=45,p<0.001

Table 8 Percent of coastal waters (out of 313) and their relativeimpact from QADA+Prec on the retention times T and TSW. Medianfor T*/T=1.00028, for TSW*/TSW=1.00029

Relative impact (T*/T and TSW*/TSW) T*/T (%) TSW*/TSW (%)

1 45.0 45.7

1–1.05 40.3 37.4

1.05–1.10 6.4 4.8

1.10–1.15 2.2 4.5

1.15–1.20 1.9 2.2

1.20–1.25 1.0 0.6

1.25–1.30 1.0 1.3

1.30–1.35 0.3 1.0

>1.35 1.3 2.6

The retention time when neglecting water discharge and precipi-tation. Equations from Fig. 4

Table 9 Percent of coastal waters (out of 120; coastal areas withinthe derived model range) and their relative impact fromQADA+Prec

on the retention times T and TSW. Median for T*/T=1.00543, forTSW*/TSW=1.00714

Relative impact (T*/T and TSW*/TSW) T*/T (%) TSW*/TSW (%)

1 32.5 32.5

1–1.05 45.8 42.5

1.05–1.10 11.7 6.7

1.10–1.15 5.0 8.3

1.15–1.20 4.2 5.0

>1.20 0.8 5.0

The retention time when neglecting water discharge and precipi-tation. Equations from Fig. 4


had a retention time between 4 and 10 days for both thewhole water body and surface water (Table 11).

4 Discussion

4.1 Mass Balance of Salt

The relative modelled values of salinity using the massbalance of salt showed a high predictive power (r2=0.96;p<0.001; Fig. 3) when regressed against the relativeempirical values. This also means that the mass balance

of salt used in this study can be used on other coastalwaters as well, as long as there are sufficient measure-ments of salinity data. During the modelling proce-dure, we noticed that the retention time (and implic-itly the inflow of water) was sensitive and dependenton having reliable data of the salinity. It is thereforeimportant to have salinity data on several depths (i.e.in the SW and DW) which corresponds to the sametime period as the water discharge; otherwise, there isa risk that the model output will have a too large errorto describe retention times in a meaningful way. Themodelled values (Fig. 3) can be considered reliable

Fig. 5 3D-plot for retention times in the whole water body andsurface water (days) using Eqs. 9 and 10. The volume-relatedexposure (ExV) ranged from 0.0001 to 0.028 m−1 and Q

(QADA+Prec) from 2.4·106 to 3.8·108 m3/month. a Retention time(T; days), b Surface water retention time (TSW; days)


since the modelled values fit the observations quitewell (r2=0.96; p<0.001; Fig. 3).

4.2 Inflow and Outflow of Water and Their Impacton Retention Times

The reason why the differences between monthly inflowand outflow of water were close to zero (Table 6) forseveral coastal waters is that the water discharge andprecipitation contribution is relatively small comparedto the inflow from the open sea. This means that thevariability of retention times (Table 7) between monthsis mainly an effect of temporal variations of waterinflow from the sea in such areas. The difference invariability for retention times between months was largefor coastal waters having small area and At compared tothe ADA, which was shown by the difference in CVvalues before and after the impact from precipitation andfreshwater discharge on the retention times (consideringand not considering the contribution of water dischargeand precipitation; Tables 1 and 7; see Slätbaken,Gävlefjärden and Draget). In other words, if At/ADAand Area/ADA are small, the variability of retentiontimes from different months is significantly affected byQADA+Prec. This also regards the total impact on reten-tion time; for example that the TSW increased by 97 %for Slätbaken when the inflow from QADA+Prec waseliminated in simulations (Table 7).

4.3 Relative Retention Times

The model for calculating relative retention times for Tand TSW (Fig. 4) can be applied on coastal waters as

long as the water discharge and precipitation are known.It is possible to estimate the water discharge usingsimple relationships considering ADA (Abrahamssonand Håkanson 1998), which could, however, potentiallydecrease the predictive power. In this study, the waterdischarge was known and used. The precipitation oncoastal waters was estimated using a relationship be-tween water discharge and an evaporation constant(Chow 1988; Abrahamsson and Håkanson 1998).However, since the water contribution from precipita-tion on coastal waters was for the most cases marginalbecause of a large ADA and At (i.e. water inflow fromprecipitation<<water inflow from ADA and At) com-pared to the coastal area, the precipitation estimate wasnegligible in most cases. The same should apply toevaporation which is even smaller in magnitude thanprecipitation (Omstedt et al. 1997). For those caseswhere the coastal area was relatively large comparedto the ADA and At, the estimates of the relative reten-tion times might be improved by using real measuredprecipitation and evaporation. It is, however, worth tonote that the modelled precipitation was often consistentwith empirical precipitation data in areas where suchdata were available. Using relative retention timeswould be preferable to model managers compared tousing relative salinities (salinity inside coast and outsidesea), since it is possible to predict relative responses andrelate it to QADA+Prec and At which is not possible byusing salinity data exclusively. A sensitivity analysis ofthe target coast is therefore possible by using the pre-sented equations (Fig. 4).

Even though the 313 non-tidal coastal waters are farfrom all in Sweden, they can be used to roughly estimatethe percentage of coastal waters in Sweden that have aretention time which are relatively intensively affectedby water discharge and precipitation (Table 8).Important uncertainties to mention are that the 313coastal waters may not represent all the others non-tidal coastal waters located in Sweden. Moreover, theresults for coastal waters which were outside the derivedmodel ranges should be interpreted with caution butwere nevertheless used to capture the overall distribu-tion for non-tidal coastal waters in Sweden. There wasonly a small deviance in percentual distribution betweenthe subset of 120 coastal waters within model ranges(Table 9) and the total dataset of 313 coastal waters(Table 8). The same regards the results of the retentiontimes (Tables 10 and 11; see Section “RetentionTimes”). According to the results, approximately 46 %

Table 10 Percentual distribution of water retention time in 313non-tidal coastal waters. Median for T=8.6 (range 0.7 to 123) daysand for TSW=9.0 (range 0.8 to 115) days

Retention time (days) T (%) TSW (%)

≤2 2.2 2.2

2–4 8.9 7.7

4–6 9.9 9.6

6–10 39.6 39.0

10–15 20.4 22.7

15–20 4.8 4.5

20–30 7.0 7.3

30–50 4.5 4.8

>50 2.6 2.2


of the coastal waters had retention times which were notsignificantly affected by water discharges from land andprecipitation; and approximately 85 % of the coastalwaters had retention times which were affected by lessthan 5 % by freshwater discharge and precipitation(Table 8). This implies that most non-tidal coastal watersin Sweden may have retention times that are affectedentirely or almost entirely by the inflow of water fromthe open sea (corroborating results in Håkanson andBryhn 2008). Also, that it may be more effective toaddress pollution in the open sea compared to in theADA for 85 % of the coastal waters since QADA+Prec donot contribute withmore than 5% to the T in those areas.Since water transports pollutants, T influences the reten-tion time of pollutants in coastal waters (Håkanson andBryhn 2008). It is worth to emphasise that the relativeretention times reflect how much the T is affected byQADA+Prec but that this percentage can be used to esti-mate relative response in coastal waters depending onwhere the abatement action will be implemented.Moreover, approximately 8 % of coastal waters T and12 % of coastal waters TSW were affected by more than10 % of the water inflow from the ADA and precipita-tion (Table 8); which mainly concerned coastal waterswith relatively small area and At compared to ADA.This means that in order for local land-based actionagainst water pollution to be effective in coastalwaters, it is essential to single out those waters withlarge ADA compared to At, in addition to freshwaterdischarges with very high concentrations of pollutants.In the remaining cases, it may be wiser to addresspollution problems by considering the total pollution

input to the whole sea and to determine which majorfluxes in the sea determine concentrations of pollutants.This largely corroborates findings in Håkanson andBryhn (2008), Liu et al. (2011) and Grifoll et al.(2013). The same conclusion concerns the coastal wa-ters which were within the derived model ranges wherethe retention times of approximately 76 % of the coastalwaters were affected by less than 5 % of water contri-bution from ADA and precipitation (Table 9).

4.4 Retention Times

Most coastal waters in this study had a retention timebelow 10 days (Tables 10 and 11), which is consistentwith previous studies of Swedish coastal waters in theBaltic Sea (e.g. Persson et al. 1994; Håkanson andEklund 2007). This was also demonstrated for coastalwaters which were within the derived ranges of themodels (Fig. 5; Table 11). Therefore, the majority ofcoastal waters were less sensitive to potential local pol-lution due to an extensive water exchange with the opensea, i.e. dissolved pollutants would for most watersleave the water body within 10 days. However, therewere some waters where local pollutants could stay inthe coastal water for several months; therefore, an extracaution and monitoring would be recommended forthose waters. It should be noted that pollutants can alsobe bound to particles and sink to the bottom and therebystay in the ecosystem for longer time than the waterretention time (Håkanson and Eklund 2007).

Precipitation was one of the driving variables in theproduced models, along with water discharges from theADA (Eqs. 9 and 10). However, for most coastal waters,the precipitation can be assumed to be zero (<<waterdischarge) and apparently had a very small impact onthe results. This means that it may often be the ExValong with water discharges from ADA that has thehighest impact on retention times. However, for coastswhere area is large and At small compared to the ADA,precipitation could have an impact on the retention time,even though the impact would still be within inherentuncertainties from the derived statistical models. It is,however, not possible to use Eqs. 9 and 10 to explore therelative impact of water inflow on retention times fromthe open sea and ADA, mainly because the models arenot derived for those types of data. In other words, themodels can be used to explore typical retention timesbased on ExV and QADA+Prec, but are not derived toexplore the relative retention times. To explore the

Table 11 Percentual distribution of water retention time in 75non-tidal coastal waters (which are within the derived modelranges). Median for T=9.5 days (range 3 to 56) and for TSW=9.9(range 3.2 to 54) days

Retention time (days) T (%) TSW (%)

≤2 0 0

2–4 2.7 2.7

4–6 9.3 6.7

6–10 45.3 42.7

10–15 22.7 28.0

15–20 6.7 5.3

20–30 9.3 10.7

30–50 2.7 2.7

>50 1.3 1.3


relative impact on retention times, equations from Fig. 4are recommended since they are derived from thosetypes of data.

5 Conclusions

This study has investigated the relative impact that waterdischarges from the drainage area and precipitation haveon the retention time of coastal waters. The distributionof retention time for 313 coastal waters, located inSweden, has also been calculated. Most of the studiedcoastal waters were only marginally affected by waterinflows from drainage area and precipitation on thecoast. The retention time was for most cases (approxi-mately 85 % of the coastal waters) almost entirelyaffected by the water inflow from the open sea. Mostcoastal waters had a retention time of up to 9 days whichmade them rather insensitive to potential local pollutionand very sensitive to pollution from the sea. However,coastal waters with a relatively small volume and sec-tion area compared to the drainage area could in somecases receive a significantly impacting pollution loadfrom water discharge and precipitation during the reten-tion time and are therefore more sensitive for potentialpollution. Model equations presented in this study canbe used to fairly easily calculate the relative impact thatwater inflows from drainage area and precipitation haveon the retention time compared to the inflow from theopen sea. Model equations from this study can also beused to calculate retention times using the total volume,section area, inflow of water from drainage area andprecipitation on the coast. These developed statisticalmodels can be useful for managers who need to deter-mine which type of abatement action (local, regional orglobal) is the most effective against pollution of a par-ticular coastal water body in non-tidal areas. Whilemodels developed in this study can provide a first hintabout which non-tidal coastal waters are very sensitiveto pollution from the catchment and which ones are not,a more definite answer can be provided by a massbalance model which simulates fluxes and masses ofthe pollutant in question.

Acknowledgments The authors would like to thank one anon-ymous peer-reviewer who greatly helped improving this arti-cle. The Swedish Meteorological and Hydrological Institute(SMHI) is also acknowledged for making the data availableon their web page.

Appendix

Equations and model variables used in the mass balancemodel for salt. F for flow (kg/month), SW for surfacewater, DW for deep water, R for rate (1/month), C forconcentration (psu=kg/m3) and M for mass (kg). Modelstructure and variables are well explained in the studiesHåkanson and Eklund (2007), Håkanson and Bryhn(2008), and Håkanson and Lindgren (2010).

Surface Water

MSW( t) =MSW( t− dt) + (FDWSWx +FTrib +FSWIn +FDiffDWSW+FDirect fallout−FSWDWx−FSWOut)·dt

InflowsFDWSWx=MDW·Rx·VSW/VDW; mixing flow from DW

to SW (kg/month)FTrib=QTrib·CTrib; tributary inflow (kg/month)FSWIn =QSWIn·CSWSea; SW inflow from sea

(kg/month)FDiffDWSW=RDiffDWSW·MDW; diffusive flow from

DW to SW (kg/month)FDirect fallout=QDirect fallout·CPrec; precipitation on

coastal area (kg/month)OutflowsFSWDWx=MSW·Rx; mixing flow from SW to DW

(kg/month)FSWOut=QSWOut·CSW; SWoutflow to sea (kg/month)

Deep Water

MDW(t)=MDW(t−dt)+(FSWDWx+FDWIn−FDWSWx−FDiffDWSW−FDWOut)·dt

InflowsFSWDWx=MSW·Rx; mixing flow from SW to DW

(kg/month)FDWIn=CDWSea·QDWIn; DW inflow from sea

(kg/month)OutflowsFDWSWx=MDW·Rx·VSW/VDW; mixing flow from DW

to SW (kg/month)FDiffDWSW=RDiffDWSW·MDW; diffusive flow from

DW to SW (kg/month)FDWOut=CDW·QDWin; DWoutflow to sea (kg/month)

Model Variables

A=(AreaA/Coastal area); ratio of accumulation sediment(−)


AreaA ¼Coastal area⋅ Dmax−Wb

DmaxþWb⋅e3−Vd1:5

� � 0:5Form factor

; area of

accumulation sediment m2ð ÞCDW=MDW/VDW; DW salinity (psu)CDWemp=Empirical salinity in coast DW (psu)CDWSea=Empirical salinity in sea DW (psu)Coastal area=Observed coastal area (m2)ConstDiff=0.05; Diffusion constant (1/month)CPrec=Empirical salinity; salinity in precipitation

(psu)CSW=MSW/VSW; SW salinity (psu)CSWemp=Empirical salinity in coast SW (psu)CSWSea=Empirical salinity in sea SW (psu)CTrib=0; Salinity in tributary (psu)DC=0.6; distribution coefficient from sea (tuned

against empirical salinity data in respective coastal) (−)Dynamic ratio=(Coastal area·10−6)0.5/Mean depth;

influence of wind and wave activity on sediments (−)ET=(1−A); Ratio of erosion and transportation sed-

iments (−)Exposure=(100·Section area·10−6)/(Coastal ar-

ea·10−6); Coastal exposure to the sea (−)Form factor=3·Mean depth/Max depth; the form of

the coast (−)Latitude=Observed latitude (°N)Max depth=Observed max depth (m)M e a n a n n u a l t e m p = ( 3 5 − ( ( 7 5 0 / ( 9 0 −

Latitude0.85)1.29)); Annual temperature in SW (°C)Mean depth=Observed mean depth (m)Precipitation=Empirical precipitation (mm/month)QDirectFallout=0.001·Coastal area·Precipitation; Water

flow from precipitation on SW (m3/month)QDWin=Qin·(1−DC); DW flow from sea (m3/month)Qin=Inflow from sea (tuned against empirical salin-

ity data in respective coast) (m3/month)QSWin=Qin·DC; SW flow from sea (m3/month)QSWOut=(QSWin+QTrib+QDirectFallout); SW outflow

to sea (m3/month)QTrib=Empirical inflow from ADA (m3/month)RD i f f DW SW = i f C SW > CDW, t h e n 0 e l s e

ConstDiff·(CDW−CSW)/1; diffusion rate for DW toSW (1/month)

Rmixdef=ET; default mixing rate [1/month]Rx= if (CDW>CSW), then Rmixdef·(1/(1+CDW−

CSW))2 else Rmixdef; mixing rate (1/month)

Seasonal variability in SWtemp=Mean annualtemp+SMTH1 (Seasonal variability norm for temp,(Mean annual temp/6), 0); Predicted mean monthlytemperature (°C)

Section area=Observed section area (m2)Stratification limit=if Mean annual temp>17 or dy-

namic ratio>3.8 or Mean annual temp<4, then 0 elseTempSW1; moderator for temperature in DW

Tcoast=Volume/(QDWin+QSWOut); coastal retentiontime (month)

Tcoast day=Tcoast·30; coastal retention time (day)TDW coast=VDW/QDWin; DW retention time (month)TDW coast day=TDW coast·30; DW retention time (day)TempDW=if TempDW2<4, then 4 else TempDW2;

moderator for temperature in DW (°C)TempDW1=if Stratification limit=0 or Mean depth

<2 then 0 else (4+0.2·TempSW1); moderator for tem-perature in DW (°C)

TempDW2=if ABS (TempSW1−TempDW1)<4,then (TempSW1+TempDW1)/2 else TempDW1; mod-erator for temperature in DW (°C)

TempSW=if ABS (TempSW1−TempDW1)<4, then(TempSW1+TempDW1)/2 else TempSW1; moderatorfor temperature in SW (°C)

TempSW1=if Seasonal variability in SWtemp<0,then 0 else Seasonal variability in SWtemp; moderatorfor temperature in SW (°C)

TSW coast=VSW/QSWOut; SW retention time (month)TSW coast day=TSW coast·30; SW retention time (day)Wave base=(If (YEx1)·(45.7·(Coastal area·10−6)0.5/

(21.4+(Coastal area·10−6)0.5>Wb limit·Max depth thenWb limit·Max depth else (YEx1)·(45.7·(Coastal ar-ea·10−6)0.5)/(21.4+(Coastal area·10−6)0.5); Moderatorfor wave base (m)

Wb=(45.7·(Coastal area·10−6)0.5)/(21.4+(Coastal ar-ea·10−6)0.5); wave base (m)

Wblimit=if Wb>0.95·Max depth, then YWb else 1;moderator for wave base (−)

VDW=Volume−VSW; DW volume (m3)Volume=Coastal area·Mean depth; coastal volume (m3)VSW=if Wave base=0 or TempSW=100, then vol-

ume else (Volume−AreaA ·(Max depth−Wavebase)·Form factor/3); SW volume (m3)

YEx1=if YEx2>10, then 10 else YEx2; moderatorfor exposure

YEx2=if Exposure<0.003, then 1 else (Exposure/0.003)0.25; moderator for exposure

YWb=(0.95/(1+0.05·((Max depth/Wb)−1))); mod-erator for wave base

Seasonal variability norm for temp=GRAPH (MOD(time, 12))

(1.00, −8.00), (2.00, −2.00), (3.00, 0.00), (4.00,2.00), (5.00, 8.00), (6.00, 20.0), (7.00, 8.00), (8.00,


2.00), (9.00, 0.00), (10.0, −2.00), (11.0, −8.00), (12.0,−20.0); normal monthly surface water temperatures (°C)

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http://vattenwebb.smhi.se/modelarea/

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Quantifying Water Retention Time in Non-tidal Coastal Waters Using Statistical and Mass Balance Models