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

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Abstract

The water retention time (sometimes called residence time) in coastal areas is an indicator of coastal hydrodynamics which can be used to quantify the local transportation of dissolved and suspended pollutants. This study has used dynamic and statistical models to explore what governs the water retention time in non-tidal coastal waters of the Baltic Sea. If freshwater input divided by the cross-section area between the coastal water and the sea was below a certain threshold, freshwater had no notable impact on the retention time. Moreover, statistical models were developed for predicting surface water retention time and total water retention time from coastal water volume, cross-section area and freshwater discharge. This study can be useful for managers who need to determine where abatement measures should be focused in order to be as effective as possibly against coastal water pollution.

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Acknowledgments

The authors would like to thank one anonymous peer-reviewer who greatly helped improving this article. The Swedish Meteorological and Hydrological Institute (SMHI) is also acknowledged for making the data available on their web page.

Author information

Authors and Affiliations

  1. Department of Earth Sciences, Uppsala University, Villav. 16, 752 36, Uppsala, Sweden

    Peter H. Dimberg

  2. Department of Aquatic Resources, Swedish University of Agricultural Sciences, Skolgatan 6, 742 42, Öregrund, Sweden

    Andreas C. Bryhn

Authors
  1. Peter H. Dimberg
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  2. Andreas C. Bryhn
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Corresponding author

Correspondence to Peter H. Dimberg.

Appendix

Appendix

Equations and model variables used in the mass balance model for salt. F for flow (kg/month), SW for surface water, DW for deep water, R for rate (1/month), C for concentration (psu = kg/m3) and M for mass (kg). Model structure and variables are well explained in the studies Håkanson and Eklund (2007), Håkanson and Bryhn (2008), and Håkanson and Lindgren (2010).

1.1 Surface Water

M SW(t) = M SW(t − dt) + (F DWSWx + F Trib + F SWIn + F DiffDWSW + F Direct fallout − F SWDWx − F SWOut)·dt

Inflows

F DWSWx = M DW·R x·V SW/V DW; mixing flow from DW to SW (kg/month)

F Trib = Q Trib·C Trib; tributary inflow (kg/month)

F SWIn = Q SWIn·C SWSea; SW inflow from sea (kg/month)

F DiffDWSW = R DiffDWSW·M DW; diffusive flow from DW to SW (kg/month)

F Direct fallout = Q Direct fallout·C Prec; precipitation on coastal area (kg/month)

Outflows

F SWDWx = M SW·R x; mixing flow from SW to DW (kg/month)

F SWOut = Q SWOut·C SW; SW outflow to sea (kg/month)

1.2 Deep Water

M DW(t) = M DW(t − dt) + (F SWDWx + F DWIn − F DWSWx − F DiffDWSW − F DWOut)·dt

Inflows

F SWDWx = M SW·R x; mixing flow from SW to DW (kg/month)

F DWIn = C DWSea·Q DWIn; DW inflow from sea (kg/month)

Outflows

F DWSWx = M DW·R x·V SW/V DW; mixing flow from DW to SW (kg/month)

F DiffDWSW = R DiffDWSW·M DW; diffusive flow from DW to SW (kg/month)

F DWOut = C DW·Q DWin; DW outflow to sea (kg/month)

1.3 Model Variables

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

\( {\mathrm{Area}}_A=\mathrm{Coastal}\;\mathrm{area}\cdot {\left(\frac{D_{\max }-\mathrm{Wb}}{D_{\max }+\mathrm{Wb}\cdot {e}^{3-{V_{\mathrm{d}}}^{1.5}}}\right)}^{\frac{0.5}{\mathrm{Form}\;\mathrm{factor}}};\mathrm{area}\mathrm{ofaccumulation}\ \mathrm{sediment}\ \left({\mathrm{m}}^2\right) \)

C DW = M DW/V DW; DW salinity (psu)

C DWemp = Empirical salinity in coast DW (psu)

C DWSea = Empirical salinity in sea DW (psu)

Coastal area = Observed coastal area (m2)

ConstDiff = 0.05; Diffusion constant (1/month)

C Prec = Empirical salinity; salinity in precipitation (psu)

C SW = M SW/V SW; SW salinity (psu)

C SWemp = Empirical salinity in coast SW (psu)

C SWSea = Empirical salinity in sea SW (psu)

C Trib = 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 sediments (−)

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

Mean annual temp = (35 − ((750/(90 − Latitude0.85)1.29)); Annual temperature in SW (°C)

Mean depth = Observed mean depth (m)

Precipitation = Empirical precipitation (mm/month)

Q DirectFallout = 0.001·Coastal area·Precipitation; Water flow from precipitation on SW (m3/month)

Q DWin = Q in·(1 − DC); DW flow from sea (m3/month)

Q in = Inflow from sea (tuned against empirical salinity data in respective coast) (m3/month)

Q SWin = Q in·DC; SW flow from sea (m3/month)

Q SWOut = (Q SWin + Q Trib + Q DirectFallout); SW outflow to sea (m3/month)

Q Trib = Empirical inflow from ADA (m3/month)

R DiffDWSW = if C SW > C DW, then 0 else ConstDiff·(C DW − C SW)/1; diffusion rate for DW to SW (1/month)

R mixdef = ET; default mixing rate [1/month]

R x = if (C DW > C SW), then R mixdef·(1/(1 + C DW − C SW))2 else R mixdef; mixing rate (1/month)

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

Section area = Observed section area (m2)

Stratification limit = if Mean annual temp > 17 or dynamic ratio > 3.8 or Mean annual temp < 4, then 0 else TempSW1; moderator for temperature in DW

T coast = Volume/(Q DWin + Q SWOut); coastal retention time (month)

T coast day = T coast·30; coastal retention time (day)

T DW coast = V DW/Q DWin; DW retention time (month)

T DW coast day = T DW 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 temperature in DW (°C)

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

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

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

T SW coast = V SW/Q SWOut; SW retention time (month)

T SW coast day = T SW 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 then Wb limit·Max depth else (YEx1)·(45.7·(Coastal area·10−6)0.5)/(21.4 + (Coastal area·10−6)0.5); Moderator for wave base (m)

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

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

V DW = Volume − V SW; DW volume (m3)

Volume = Coastal area·Mean depth; coastal volume (m3)

V SW = if Wave base = 0 or TempSW = 100, then volume else (Volume − Area A ·(Max depth − Wave base)·Form factor/3); SW volume (m3)

YEx1 = if YEx2 > 10, then 10 else YEx2; moderator for 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))); moderator 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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Dimberg, P.H., Bryhn, A.C. Quantifying Water Retention Time in Non-tidal Coastal Waters Using Statistical and Mass Balance Models. Water Air Soil Pollut 225, 2020 (2014). https://doi.org/10.1007/s11270-014-2020-z

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