Barjenbruch, Ulrich

Hein, Birte

Hein, Hartmut

Mai, Stephan

English

Hydraulic Engineering Repository

Conference Paper, Published VersionHein, Hartmut; Hein, Birte; Mai, Stephan; Barjenbruch, UlrichThe Residence Time in the Elbe River Focussing on theEstuaryZur Verfügung gestellt in Kooperation mit/Provided in Cooperation with:Kuratorium für Forschung im Küsteningenieurwesen (KFKI)Verfügbar unter/Available at: https://hdl.handle.net/20.500.11970/99508Vorgeschlagene Zitierweise/Suggested citation:Hein, Hartmut; Hein, Birte; Mai, Stephan; Barjenbruch, Ulrich (2014): The Residence Time inthe Elbe River Focussing on the Estuary. In: Lehfeldt, Rainer; Kopmann, Rebekka (Hg.):ICHE 2014. Proceedings of the 11th International Conference on Hydroscience &Engineering. Karlsruhe: Bundesanstalt für Wasserbau. S. 837-844.Standardnutzungsbedingungen/Terms of Use:Die Dokumente in HENRY stehen unter der Creative Commons Lizenz CC BY 4.0, sofern keine abweichendenNutzungsbedingungen getroffen wurden. Damit ist sowohl die kommerzielle Nutzung als auch das Teilen, dieWeiterbearbeitung und Speicherung erlaubt. Das Verwenden und das Bearbeiten stehen unter der Bedingung derNamensnennung. Im Einzelfall kann eine restriktivere Lizenz gelten; dann gelten abweichend von den obigenNutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte.Documents in HENRY are made available under the Creative Commons License CC BY 4.0, if no other license isapplicable. Under CC BY 4.0 commercial use and sharing, remixing, transforming, and building upon the materialof the work is permitted. In some cases a different, more restrictive license may apply; if applicable the terms ofthe restrictive license will be binding. 1 INTRODUCTION The ecosystem in an estuary is governed by the physical, chemical and finally biological processes. The rate of water exchange between the river and the coastal sea controls mainly the chemical and biological processes within the area. One helpful parameter combining the physical, chemical and biological pro-cesses is the residence time (Ɵ). Ɵ can be defined as the related time that one body of water remains in the defined section of the river. This sections might be the estuary, one section of the main branch, assort-ed side branches, etc.. Several studies use the residence time to assess the nutrient exports and imports and to estimate primary production (e.g. Lucas et al. 1999, Dettmann, 2001, Hein et al. 2013a). Summa-rizing, the studies demonstrate the importance of this parameter in defining biological timescales. In general three parameters are defined in the context of ecological time scales in estuaries: Age, Ɵ and flushing time. The first one is commonly defined as the time one water parcel travels to one specific loca-tion within one section of the estuary after entering the section through one of its boundaries. This follows that different locations within the section can have different ages. Residence time is how long the water parcel will remain within this region before exiting it from one of its boundaries. If the source of the wa-ter parcel is the lateral boundary Ɵ can also called transit time. Flushing time is defined as the time re-quired reducing the concentration of tracer inside one specific section of the estuary (Zimmerman, 1976). These different parameters are related to the fact that the ways of the water parcel additional to the advec-tion controlled by dispersion processes. The Elbe River is one of the largest rivers in Europe O(1000 km), the tidal influence in the estuary reaches O(150 km) inward from the center of the German Bight to the weir in Geesthacht. The focus of the study is concentrated on the tidal influenced area of the Elbe River. The hydrological regime of the Elbe estuary is dominated by tides, mainly by the M2 tide and their overtides. The residence time in the Elbe River between Schmilka and the mouth near Cuxhaven is estimated by the means of modelling stud-ies and partly compared with measurements of tracer distributions.  The Residence Time in the Elbe River Focussing on the Estuary  H. Hein, B. Hein, S. Mai & U. Barjenbruch Federal Institute of Hydrology, Koblenz, Germany  ABSTRACT: The ecosystem in an estuary is governed by the physical, chemical and finally biological processes. The rate of water exchanges as well between the estuary and the river as between the estuary and the coastal sea controls causally the chemical and biological processes within the area. One relevant parameter combining the physical, chemical and biological processes is the residence time (Ɵ). While the mayor focus is on the tidal influenced area, this study estimates the residence time in the Elbe River from Schmilka toward the mouth near Cuxhaven by the means of numerical modelling. In the estu-ary two different numerical models are used and compared. The influence of the horizontal diffusion is analyzed, based on the permutation of parameters and numerical solutions.     In rivers the discharge itself is the main transport process, which influence Ɵ. However, in estuaries the processes that transport one parcel of water depends on several lower-frequency residual flows, e.g. river flows, baroclinic currents or wind driven currents. Ultimately, the transport regime is dominated by tidal flows . Keywords: Residence Time, Tracer Concentrations, Numerical Modelling, Elbe River, Elbe Estuary ICHE 2014, Hamburg - Lehfeldt & Kopmann (eds) - © 2014 Bundesanstalt für Wasserbau ISBN 978-3-939230-32-8837use of a virtual tracer, which can be calculated by solving the advection-diffusion equation. Because, they are mass-conservative, Eulerian solutions are state of the art in this context.  CADPtC 2∇⋅−∇⋅−−=∂∂Cv   (1) Equation (1) shows the advection-diffusion equation which was solved in this study. Here c, t, P, D, v, A are the concentration, the time, the source, the sinks, the velocities and the Diffusion coefficient. The time variable concentration of the tracer Ct is zero at the start of the numerical model. The tracer concentration is set to at the upper boundary. Hence, P becomes positive. The first interesting point in time is when Ct1 > Cmin. Cmin is defined as the mean standard deviation of tracer concentration during one tidal cycle or inside one day in the riverine sections. The third important time is when Ct2 falls again be-low Cmin. The residence time Ɵ can simply be calculated by Ɵ = t2-t1. For the calculation of the two scales (tidal and non-tidal) in the estuary the tracer concentration are low-pass filtered on the time scale. A second method is the estimation of Ɵ without tracer, but with integrating the current-field (Hein, 2013b).  ∫−=trttta vLt /)(θ   (2) Here, La is the length-scale, defined by the expansion of the section of interest. In our study La ≈ O(50) km, defined roughly by the extension of the port of Hamburg. The velocity vt is the mean velocity. Addi-tionally Ɵ is calculated with vt reduced with one standard deviation, representing the 0.68 confidence in-terval of the velocity field.  4 BENEATH THE NUMERICAL RESOLUTION Dispersion of any passive or active suspended substances in coastal waters is one of the most challenging in hydrodynamic modelling. Coastal or estuarine topographic structures build a complex system on a wide spectrum of length scales. It is impractical to resolve all these scales in the numerical resolved grid, because there is always a minimum limit of the grid size. This or the use of inadequate numerical solu-tions leads to artificial diffusion. The state of the art method to avoid such nastiness is the use of a least second order eulerian solutions for the transport part of equation (1). For this study we used both, the first order upwind scheme and the Lax-Wendroff scheme (Lax and Wendroff, 1960); last is combined with a total variation diminishing (TVG) limiter function. As there is no general optimum TVG function and various limiters have different characteristics. The theoretical de-ductions for the limiter functions can be found in Waterson and Deconinck (1995), van Leer (1979) and Roe (1986). The combined advection schemes using the TVD function found by van Leer (1979) are compared with the upwind advection scheme. In HYDRAX/QSim the transport-equation is solved by the QUICKEST scheme (Leonard, 1979).   However, in complex topographies one important decisive step for successful modelling is to determi-nate the limit between the numerically resolved and unresolved scales. In modern hydrodynamic models horizontal sub-grid scale transports follows an eddy viscosity concept, expressed by a stochastic model, commonly that by Smagorinsky (1963). This allows estimating the sub-grid exchange rate which bases on the characteristic length scale of the numerical unresolved scales. The Smagorinsky Subgrid Model (SSM) is based mainly on the relevance of momentum dissipation as motivated by the elasticity theory. An introduction into the elasticity theory applied to dislocation is given by Hull and Bacon (1984). Ro-bust numerical schemes for the computation of the SSM use two continuity terms and one bi-directional shear-term (Hein, 2008).  In general, the SSM assume that the topographic length scale is the same as the grid-size, but in com-plex geometries this may cause ineffective use of the computable resources. Therefore, in this study addi-tional an implementation of a meso-scale topographic model (MSTM) is done. This allows to resolve the hydrodynamic numerically on a coarse fixed and therefore scale-conform and effective grid and the con-sideration of typical mesoscale topographic features like sand dunes, dams or goyes. The meso-scale in-formation is used to improve the length-scale concept of the SSM in form of a topographic scaling factor 839 Figure 3. Spreading of the numerical tracer in one section of the Elbe estuary at km 605 and km 645. Black: HAMSOM, grey, dashed: HYDRAX/QSIM. 5.3 Comparison of different numerical methods The influence of the numerical scheme is tested by perturbation, which is shown in figure 4. Imaged are the concentration at km 645, the tidal signal is filtered out in the same manner than in Figure 2. Three model runs from HAMSOM are compared: The first order upwind scheme, the TVD Scheme and the TVD Scheme including physical diffusion. The physical diffusion is estimated by equation (3) with cs (cs = 0.2). The difference of the three runs is that they are different in the magnitude of diffusion. The up-wind scheme is most diffusive, because of the inherent numerical diffusion. From the TVD Scheme it is known that there is almost no numerical diffusion, which is added in the third run by the use of the esti-mations of A with the SSM (equation (3)).  With higher diffusions the magnitude of the concentration increases (Figure 4a). This result seems in-tuitively difficult to explain. However, on explanation might be that higher diffusivity the tidal variation decreases stronger with time (Figure 4b). Vice versa considered this follows that the dispersion effects of the tidal movements are more important for the extension of Ɵ.     Figure 4. Spreading of the numerical tracer in the Elbe estuary at km 645. a) Tides filtered out. b) tidal signal only.  Black,  lines: Lax-Wenddroff; grey, dashed: Lax-Wenddroff + SSM; black, dotted: Upwind. 841 Figure 5. Time series of Ɵ, calculated by equation (2). Dispersion can be differentiated from diffusion that it is caused by irregular flow patterns and it is as-sumed to be a macroscopic phenomenon. In contrast diffusion is more a microscopic phenomenon caused by random motions beneath the resolved scale. The complicated geometry in the Port of Hamburg causes the strong irregular Eulerian velocity field. Side channels and a vast number of harbor basins "randomiz-ing" (Zimmermann, 1986) the velocity field. Zimmermann (1986) calls this a "Lagrangian chaos" and ad-dressed that its parameterization in a dispersion coefficient is far from clear.  5.4 Alternative method to calculate Ɵ The second method to estimate Ɵ, is to calculate the value of Ɵ without a tracer, but with the integration of the current-field (equation 2). This method has the advantage that it is computable efficient and the changes in time of Ɵ can be calculated. Figure 5 shows the estimated Ɵ during the years 2006 and 2007. The residence time is estimated in the range between O(20 days) and O(40 days). Ɵ undergoes a strong seasonal cycle, with increased residence times during the summer season. In the Elbe River during sum-mer the discharges are less than during winter. The results indicate one reasonable cause the variations of the oxygen budget in the port of Hamburg. 6 CONCLUSION & OUTLOOK In brief, it has been found that the residence time changes significantly in magnitude and structure in the tidal influenced area. It can be concluded, that the main transport processes in the estuary are far more re-lated to dispersion processes than to the mean advection processes. In contrast, mean advection is the main process in the riverine part of main rivers (Deng et al., 2010). Ɵ depends on both temporal and spa-tial scales. In terms of the spatial scale, the distribution of Ɵ is channel-size dependent (Deng et al., 2010), together with the tidal dispersion effects this explains the changes in the estuary.  In the Elbe Estuary Ɵ is a mutable of Ɵ in the riverine part and additionally diffusion processes be-come important, both ensure certain consequences for the tracer simulations: Numerical schemes with ra-ther low artificial diffusion are appropriate, as well as the eddy diffusion tensor must be estimated by sub-grid schemes. However, Ɵ depends in the same way in both, the riverine and the estuarine part of the Elbe River, on river discharges.  Future work is mainly the validation of the results with observations; particularly the observations of conductivity in the Elbe estuary may help to find reasonable parameterizations of the Smagorinski con-stant (cs). Second, the calculation of Ɵ without tracer should be done by the use of distributions which are more representative for the current field than the normal distribution can represent the distribution of the currents. 842NOTATION Ɵ    Residence time C     Treacer concentration SSM   Smagorinsky subgrid model MSTM   Meso-scale topographic model TVG   Total variation diminishing limiter function A     Horizontal diffusion coefficient cs     Smagorinsky constant REFERENCES Anderson, E. J., & Phanikumar, M. S. (2011) Surface storage dynamics in large rivers: Comparing three‐dimensional particle transport, one‐dimensional fractional derivative, and multirate transient storage models. Water Resources Research, 47(9). Backhaus J. O. (1985) A three-dimensional model for the simulation of shelf sea dynamics. Dt. Hyd. Z.38, 165–187. Deng, Z. Q., Jung, H. S., & Ghimire, B. (2010) Effect of channel size on solute residence time distributions in rivers. Advances in Water Resources, 33(9), 1118-1127. 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The Residence Time in the Elbe River Focussing on the Estuary

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The Residence Time in the Elbe River Focussing on the Estuary

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