Freshwater flows to the sea: Spatial variability, statistics and scale dependence along coastlines

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1 GEOPHYSICAL RESEARCH LETTERS, VOL. 35, L18401, doi: /2008gl035064, 2008 Freshwater flows to the sea: Spatial variability, statistics and scale dependence along coastlines Georgia Destouni, 1 Yoshihiro Shibuo, 1 and Jerker Jarsjö 1 Received 18 June 2008; revised 4 August 2008; accepted 8 August 2008; published 16 September [1] Beyond the monitoring of main river flows, the discharges of freshwater from land to the sea are typically left unmonitored along long coastline stretches. This study uses uniquely fine-resolved data and determines the spatial variability and statistics of the freshwater fluxes to the sea along two Swedish coastlines. The flux statistics depend greatly on subjective investigation choices of the support (or aggregation) scale of flux measurement, H, and the coastline length resolution, G. For common H and G values and relations, the flux coefficient of variation ranges from 1.5 to 22.5 and there is around 90 95% probability that locally measured or modelled fluxes miss the high-end fluxes that are greater than the arithmetic mean flux and carry most of the total freshwater discharge across the coastline. Quantification of the inland hydrological balance and its distribution over the whole coastal catchment area is needed for objective guidance of coastal discharge interpretations. Citation: Destouni, G., Y. Shibuo, and J. Jarsjö (2008), Freshwater flows to the sea: Spatial variability, statistics and scale dependence along coastlines, Geophys. Res. Lett., 35, L18401, doi: /2008gl Introduction [2] Measured concentrations of distinct inland tracers in coastal waters have revealed large unmonitored waterborne discharges of these tracers to the sea in comparison to the monitored tracer discharges of main rivers [Moore, 1996; Moore et al., 2008]. Near-coastal catchment areas are typically left uncovered by the systematic monitoring of main rivers [Hannerz and Destouni, 2006]. The unmonitored near-coastal areas often have high population pressures that may generate large unmonitored mass loads to the sea, not only of harmless tracers, but also nutrient and pollutant loads of a magnitude similar to or greater than the monitored river loads [Destouni et al., 2008]. Figure 1a illustrates schematically the different hydrological pathways and water flows that may carry these mass loads from the monitored and unmonitored parts of a coastal catchment area to the sea. [3] It is difficult and uncertain to quantify and distinguish among the different unmonitored hydrological discharges to the sea based only on the information from main river monitoring and possible concentration measurements in the coastal waters; see, for example, the different quantifications of submarine groundwater discharge (SGD) to a South Atlantic Bight section obtained by Moore [1996] and by 1 Department of Physical Geography and Quaternary Geology, Stockholm University, Stockholm, Sweden. Copyright 2008 by the American Geophysical Union /08/2008GL Destouni et al. [2008], with the former neglecting and the latter accounting for the unmonitored river and stream discharges to the coast in addition to SGD (Figure 1a). The freshwater flows and their mass loading to the sea across a coastline can to some extent be quantified from direct local flux measurements at the coastline, or from modeling based on various local hydrological, hydrogeological and coastal data in the coastline vicinity [Oberdorfer, 2003; Burnett et al., 2006]. However, the local water fluxes along a coastline may be highly variable [Jarsjö et al., 2008] and the coastline length depends on the resolution of its measurement [Mandelbrot, 1967; Bokuniewicz, 2001]. [4] In view of these flux and coastline length characteristics, this study addresses two main questions with regard to the freshwater discharge quantification: How do locally measurable water fluxes and their spatially aggregated values and statistics vary along a coastline and depend on the spatial scale of measurement? And how well do the local or spatially aggregated flux values and statistics on different scales represent the total freshwater discharge across the coastline to the sea when they are combined with different length measures of the same coastline? We investigate the answers to these questions by use of uniquely fine-resolved spatial data for two Swedish near-coastal catchment areas [Brydsten, 2004; Jarsjö et al., 2008]. 2. Materials and Methods [5] We use reported data on coastline location [Brydsten, 2004] and modeled coastal fluxes [Jarsjö et al., 2008] for the two Swedish near-coastal catchment areas Forsmark and Simpevarp (Figures 1b 1c). These areas were previously hydrologically unmonitored. However, detailed spatial input data for hydrological modelling and runoff data from a few inland observation stations have recently become available from intensive investigations of these areas as possible sites of a Swedish geological repository of high-level radioactive nuclear waste [Werner et al., 2006]. The accuracy of the present analysis benefits in particular from the high spatial resolution (10 m 10 m) of the digital elevation model and coastline representation at zero elevation that underlie the data. The latter was obtained by a combination of fieldmeasured differential global positioning system (DGPS) data on the shoreline location at zero elevation and manual digitalisation of the shoreline location with infrared orthophoto [Brydsten, 2004]. [6] We quantify first the coastline length L between the two coastal endpoints of each catchment area for different resolution G [dimension L] of the coastline length. The finest resolution G = 10 m of this study extends previously reported L(G) results, which had finest resolutions G of tenths of kilometres [Mandelbrot, 1967] down to hundreds L of5

2 Figure 1. Schematic illustration of (a) the different monitored and unmonitored hydrological pathways from coastal catchment areas to the sea and the Swedish coastal catchment areas (b) Forsmark and (c) Simpevarp. In Figure 1a the solid and dotted black lines are the water divides of the monitored and the unmonitored (shaded grey) parts of the coastal catchment areas, respectively, the red filled circles show the most-near coastal monitoring stations that define these parts, the straight flow arrows at and across the coastline show the monitored (blue) and the unmonitored (orange) freshwater discharges from land to sea, the curved blue arrows across the coastline show the re-circulated seawater component of submarine groundwater discharge, the blue lines within the catchments show the rivers and streams, and the arrows within the catchments and at the coastline show the diffuse fields of groundwater flow into streams and rivers. Figures 1b and 1c show field-controlled streams (blue lines), water divides between streams (dashed black lines), lakes (blue), wetlands (blue-green) and the coastline (solid black line). of meters [Bokuniewicz, 2001]. The coastline length L is important because it determines the average flux per unit coastline length q(g)=q tot /L(G)[L 2 T 1 ] for any given total annual average freshwater flow Q tot [L 3 T 1 ]. The total flow Q tot is determined by the annual average precipitation surplus (precipitation minus evapotranspiration) that drains from the whole coastal catchment area, across the coastline L, tothe sea. For the Forsmark and Simpevarp areas, Q tot has been quantified to be m 3 /day and m 3 /day, respectively, by hydrological modelling with different approaches, which yielded consistent Q tot results and local runoff values that also were consistent with the recent inland runoff observations [Jarsjö etal., 2008]. However, since the coastline length L varies depending on the resolution G [Mandelbrot, 1967; Bokuniewicz, 2001], different estimates of the average coastal flux q(g) =Q tot / L(G) are obtained depending on how L is resolved. [7] Furthermore, we quantify locally measurable fluxes i i as q m = Q m / K [L 2 T 1 ], where Q m i [L 3 T 1 ] is measured or modelled volumetric water flow through any coastline fraction i of length H < L and H then denotes the spatial support scale of q i i m. A local flux q m along an unmonitored coastline may be either a stream flux or SGD (Figure 1a) [Jarsjö etal., 2008; Destouni et al., 2008]. Stream fluxes are open to direct and relatively straightforward measurement, with the measurement support scale representing a whole stream cross-section or some part of it. The SGD fluxes are more difficult to measure directly and their possible support scales vary depending on the measurement or model estimation method. Overviews of different methods for direct measurement or indirect model estimation of SGD [Oberdorfer, 2003; Burnett et al., 2006] show that support scales may range from the order of 100 m for local estimates (by direct seepage meter measurement, or indirect quantification from piezometer measurements of hydraulic heads in sediments at different depths or local geochemical or geophysical tracer measurements in the coastal water) to larger length scales for combinations of local estimates in transects. Here, the spatial variability and statistics of i i q m = Q m / H are quantified for different support scales H, by departing from the reported cell by cell (10 m 10 m) model results of local water fluxes along the Forsmark and Simpevarp coastlines (Jarsjö etal. s [2008] Figure 6 illustrates explicitly the basic local flux, q i m, data on the 10 m coastline resolution and flux support scale) and systematically aggregating them over different H. 3. Results [8] Figure 2a shows the coastline length L between the two coastal endpoints of the Forsmark and the Simpevarp catchment area for different coastline length resolution G, 2of5

3 along with a fitted power-law function L(G) G 1-D. The fitted function yields fractal dimension D = 1.33 for the Forsmark and D = 1.31 for the Simpevarp coastline, which are consistent with the findings in previous studies that show similar fractal dimensions extending over much coarser coastline length resolution G [Mandelbrot, 1967; Bokuniewicz, 2001]. However, the present fine-resolution results show a deviation from the self-similar power-law behaviour of L(G) as G approaches 10 m, toward a maximum coastline length of around 50 km for Forsmark and 90 km for Simpevarp for G =10m. On the coarse resolution end of the L(G) results, the minimum coastline lengths 8 km for Forsmark and 13.5 km for Simpevarp are obtained from the straight lines between the two endpoints of each coastal catchment. [9] Due to its dependence on L(G), also the average flux q(g) =Q tot / L(G) G D-1 exhibits self-similar power law behaviour (Figure 2b), with exponent D-1 = 0.33 for Forsmark and D-1 = 0.31 for Simpevarp. In analogy with L(G), the flux q(g) deviates from this self-similarity as G approaches the finest resolution of 10 m. Depending on how L(G) is quantified, the range of different possible Figure 3. Cumulative distributions for (a) Forsmark and (b) Simpevarp and (c) the coefficient of variation CV for both areas of the local fluxes q m (H) along the coastlines of each catchment area, for different spatial support scale H of the local flux measurement. The large q m variability and distribution asymmetry yield large differences between the arithmetic mean (filled circles in Figures 3a 3b) and the geometric mean (filled squares in Figures 3a 3b) values of q m for all investigated H values. Figure 2. The (a) coastline length L(G) and (b) average flux q(g) (flow per unit coastline length), plotted versus the resolution G of coastline length measurement. For G around and greater than 100 m, L(G) G 1-D (solid lines in Figure 2a) with fractal dimension D = 1.33 for the Forsmark and D = 1.31 for the Simpevarp coastline, and q(g) G D-1 (solid lines in Figure 2b) with exponent D 1 = 0.33 for Forsmark and D 1 = 0.31 for Simpevarp. Q tot -based estimates of average flux is q = m 2 /d for Forsmark and q = m 2 /d for Simpevarp. [10] Figures 3a 3b illustrate further the cumulative distributions of locally measurable fluxes q m (H) = Q m / H along the Forsmark and Simpevarp coastlines. In general, the results show large q m variability, highly asymmetric q m distributions, and large dependence of the q m statistics on the measurement support (or aggregation) scale H. The large q m variability and distribution asymmetry yield large differences between the arithmetic mean value (q m = ðq m =HÞ; filled circles in Figures 3a 3b) and the geometric mean value (q g m = exp [lnðq m =HÞ]; filled squares in Figures 3a 3b) of the local fluxes q m. [11] Figure 3c quantifies the spatial q m variability in terms of the coefficient of variation CV[q m (H)] (the ratio between the standard deviation and the arithmetic mean q m ) and shows that CV[q m ] decreases as H increases and exhibits self-similar power law dependence on H, with exponent 0.68 for Forsmark and 0.67 for Simpevarp. The decrease of CV[q m ] with increasing H means that the difference between the arithmetic and the geometric mean 3of5

4 for Forsmark and H = 37 m for Simpevarp) and different coastline resolutions G. For G < H, the results are similar to those for G = H (Figure 4a), while for G > H, which is relevant for many practical applications, not only the g geometric mean q m but also the arithmetic mean q m underestimates q(g), and thereby also Q tot when multiplied with a coastline length L(G > H). For coastline resolution G 1 km, the underestimation of Q tot by the product q g m (H = m) L(G = 1 km) is nearly two orders of magnitude. Figure 4. The relations of the arithmetic and of the geometric mean value of local flux q m (H) totheq tot -based average flux q(g) =Q tot /L(G) for (a) H = G and (b) H 6¼ G, where Q tot is the total freshwater discharge, L(G) isthe coastline length, G is the coastline resolution and H is the q m measurement support scale. of q m decreases as H increases. However, both CV[q m ] and the difference between q m and q g m remain generally high, with the CV ranging from about 1700% and 2250% of the smallest q m (for the smallest H of the order of 10 m) to about 150% and 250% of the largest q m (for the largest H of about 1 km) for Forsmark and Simpevarp, respectively. [12] Figure 4a illustrates how the arithmetic mean q m (H) and the geometric mean q g m (H) of local q m values compare with q(g) =Q tot /L(G) if the measurement support scale H and the coastline resolution G are equal. Under this condition, the arithmetic mean q m (H) reproduces the Q tot -based average flux q(g) reasonably well. This also means that the product between q m (H) and the coastline length L(G = H) reproduces well the total flow Q tot across the coastline to the sea. In contrast, the product between the geometric mean q g m (H) and the coastline length L(G = H) generally underestimates Q tot. This underestimation is as large as an order of magnitude or more for G = H around 100 m or smaller. [13] Local flux measurements may also have different support scale H than the coastline length resolution G. Figure 4b shows the relations of the arithmetic mean q m and the geometric mean q g m to the Q tot -based average q(g) = Q tot /L(G) for fixed measurement support scale H (H =28m 4. Discussion and Conclusion [14] This study has quantified the diffuse nature and large spatial variability of local water fluxes along coastlines and shown that the spatial flux statistics depend greatly on the subjective choices of measurement support scale H and coastline length resolution G. Average values and variability measures (CV, as well as standard deviation) of the local water fluxes exhibit self-similar power-law dependence on G and H. However, the average flux for fine coastline resolution (small G) deviates from this global self-similarity. Furthermore, the spatial flux distribution around the mean flux for any given H (Figures 2a 2b) is not any scaleinvariant power-law distribution, but reflects the types of asymmetric stochastic distributions that often describe spatial variability of hydraulic conductivity and water fluxes in subsurface water systems [Dagan, 1989; Rubin, 2003]. [15] The distribution asymmetry implies that that there is much higher probability that locally measured q m values are smaller (around 90 95% probability; Figures 2a 2b) than that they are greater (remaining 5 10% probability) than the arithmetic mean q m. Local measurement of fluxes, like SGD, along an unmonitored coastline is thus likely to mostly sample the relatively low and miss the high-end q m values, yielding sample means that are closer to the geometric (q g m ) than to the arithmetic (q m ) mean value of q m. This may lead to large underestimation of freshwater discharges to the sea. [16] The total freshwater discharge, Q tot, is partitioned between the monitored river discharges and the unmonitored surface water discharges and freshwater component of SGD (Figure 1a) [Jarsjö etal., 2008]. The coastal mass loading of inland tracers is carried by Q tot and an additional, seawater component of SGD (Figure 1a) [Destouni and Prieto, 2003; Smith, 2004; Prieto and Destouni, 2005; Shibuo et al., 2006]; detection of this total loading in the coastal waters [Moore, 1996; Moore et al., 2008] does not distinguish between the different unmonitored, fresh- and seawater components of the partly overlapping Q tot and SGD [Destouni et al., 2008]. This study has shown that distinction of the coastal discharge components by a few direct, local measurements may be misleading. Modeling [Shibuo et al., 2007; Jarsjö et al., 2008] and possible additional, spaceborne measurements [Alsdorf et al., 2007; Syed et al., 2007] of the inland hydrological balance and its distribution over the whole coastal catchment area are then also needed for objective interpretation and quantification of freshwater discharges to the coast. [17] Acknowledgments. Financial support for this work has been provided by the Swedish Research Council (VR) and the Swedish Nuclear Fuel and Waste Management Company (SKB). The work has been carried 4of5

5 out within the Bert Bolin Centre for Climate Research, which is supported by a Linnaeus grant from VR and the Swedish Research Council Formas. References Alsdorf, D. E., E. Rodríguez, and D. P. Lettenmaier (2007), Measuring surface water from space, Rev. Geophys., 45, RG2002, doi: / 2006RG Bokuniewicz, H. (2001), Toward a coastal ground-water typology, J. Sea Res., 46, Brydsten, L. (2004), A method for construction of digital elevation models for site investigation program in Forsmark and Simpevarp, Rep. P-04 03, Swed. Nucl. Fuel Waste Manage. Co., Stockholm. Burnett, W. C., et al. (2006), Quantifying submarine groundwater discharge in the coastal zone via multiple methods, Sci. Total Environ., 367, Dagan, G. (1989), Flow and Transport in Porous Formations, Springer, Berlin. Destouni, G., and C. Prieto (2003), On the possibility for generic modelling of submarine groundwater discharge, Biogeochemistry, 66, Destouni, G., F. Hannerz, C. Prieto, J. Jarsjo, and Y. Shibuo (2008), Small unmonitored near-coastal catchment areas yielding large mass loading to the sea, Global Biogeochem. Cycles, doi: /2008gb003287, in press. Hannerz, F., and G. Destouni (2006), Spatial characterization of the Baltic Sea Drainage Basin and its unmonitored catchments, Ambio, 35, Jarsjö, J., Y. Shibuo, and G. Destouni (2008), Spatial distribution of unmonitored inland water discharges to the sea, J. Hydrol., 348, Mandelbrot, B. (1967), How long is coast of Britain: Statistical self-similarity and fractional dimension, Science, 156, Moore, W. S. (1996), Large groundwater inputs to coastal waters revealed by 226Ra enrichments, Nature, 380, Moore, W. S., J. L. Sarmiento, and R. M. Key (2008), Submarine groundwater discharge revealed by 228 Ra distribution in the upper Atlantic Ocean, Nature Geosci., 1, Oberdorfer, J. (2003), Hydrogeologic modeling of submarine groundwater discharge: Comparison to other quantitative methods, Biogeochemistry, 66, Prieto, C., and G. Destouni (2005), Quantifying hydrological and tidal influences on groundwater discharges into coastal waters, Water Resour. Res., 41, W12427, doi: /2004wr Rubin, Y. (2003), Applied Stochastic Hydrogeology, Oxford Univ. Press, Oxford, U. K. Shibuo, Y., J. Jarsjö, and G. Destouni (2006), Bathymetry-topography effects on saltwater-fresh groundwater interactions around the shrinking Aral Sea, Water Resour. Res., 42, W11410, doi: /2005wr Shibuo, Y., J. Jarsjö, and G. Destouni (2007), Hydrological responses to climate change and irrigation in the Aral Sea drainage basin, Geophys. Res. Lett., 34, L21406, doi: /2007gl Smith, A. J. (2004), Mixed convection and density-dependent seawater circulation in coastal aquifers, Water Resour. Res., 40, W08309, doi: /2003wr Syed, T. H., J. S Famiglietti, V. Zlotnicki, and M. Rodell (2007), Contemporary estimates of Pan-Arctic freshwater discharge from GRACE and reanalysis, Geophys. Res. Lett., 34, L19404, doi: / 2007GL Werner, K., E. Bosson, and S. Berglund (2006), Analysis of water flow paths: Methodology and example calculations for a potential geological repository in Sweden, Ambio, 35, G. Destouni, J. Jarsjö, and Y. Shibuo, Department of Physical Geography and Quaternary Geology, Stockholm University, SE Stockholm, Sweden. (georgia.destouni@natgeo.su.se) 5of5

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