Local Mixing Determines Spatial Structure of Diahaline Exchange Flow in a Mesotidal Estuary: A Study of Extreme Runoff Conditions

Lloyd Reese aLeibniz Institute for Baltic Sea Research Warnemünde, Rostock, Germany

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Ulf Gräwe aLeibniz Institute for Baltic Sea Research Warnemünde, Rostock, Germany

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Knut Klingbeil aLeibniz Institute for Baltic Sea Research Warnemünde, Rostock, Germany

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Xiangyu Li aLeibniz Institute for Baltic Sea Research Warnemünde, Rostock, Germany
bSouthern Marine Science and Engineering Guangdong Laboratory (Zhuhai), Zhuhai, China

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Marvin Lorenz aLeibniz Institute for Baltic Sea Research Warnemünde, Rostock, Germany

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Hans Burchard aLeibniz Institute for Baltic Sea Research Warnemünde, Rostock, Germany

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Abstract

Salt mixing enables the transport of water between the inflow and outflow layers of estuarine circulation and therefore closes the circulation by driving a diahaline exchange flow. A recently derived universal law links the salt mixing inside an estuarine volume bounded by an isohaline surface to freshwater discharge: it states that on long-term average, the area-integrated mixing across the bounding isohaline is directly proportional to the freshwater discharge entering the estuary. However, even though numerous studies predict that periods of extreme discharge will become more frequent with climate change, the direct impact of such periods on estuarine mixing and circulation has yet to be investigated. Therefore, this numerical modeling study focuses on salinity mixing and diahaline exchange flows during a low-discharge and an extreme high-discharge period. To this end, we apply a realistic numerical setup of the Elbe estuary in northern Germany, using curvilinear coordinates that follow the navigational channel. This is the first time the direct relationship between diahaline exchange flow and salt mixing as well as the spatial distribution of the diahaline exchange flow is shown in a realistic tidal setup. The spatial distribution is highly correlated with the local mixing gradient for salinity, such that inflow occurs near the bottom at the upstream end of the isohaline. Meanwhile, outflow occurs near the surface at its downstream end. Last, increased vertical stratification occurs within the estuary during the high-discharge period, while estuarine-wide mixing strongly converges to the universal law for averaging periods of the discharge event time scale.

Significance Statement

Inside estuaries, such as river mouths, terrestrial freshwater is mixed with salty ocean water. This is accompanied by an estuarine circulation with inflow of saltwater into the estuary and outflow of brackish water toward the ocean. Here, we aim to better understand how salt mixing and estuarine circulation in a tidal estuary react to periods of extreme freshwater discharge. We find that even during extremely high or low discharge, salt mixing follows the freshwater discharge on time scales as short as days, and that estuarine circulation patterns are largely explained by the local distribution of mixing. As extreme runoff events are likely to occur more often with climate change, these findings may help to understand the dynamics inside future estuaries.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Lloyd Reese, nina.reese@io-warnemuende.de

Abstract

Salt mixing enables the transport of water between the inflow and outflow layers of estuarine circulation and therefore closes the circulation by driving a diahaline exchange flow. A recently derived universal law links the salt mixing inside an estuarine volume bounded by an isohaline surface to freshwater discharge: it states that on long-term average, the area-integrated mixing across the bounding isohaline is directly proportional to the freshwater discharge entering the estuary. However, even though numerous studies predict that periods of extreme discharge will become more frequent with climate change, the direct impact of such periods on estuarine mixing and circulation has yet to be investigated. Therefore, this numerical modeling study focuses on salinity mixing and diahaline exchange flows during a low-discharge and an extreme high-discharge period. To this end, we apply a realistic numerical setup of the Elbe estuary in northern Germany, using curvilinear coordinates that follow the navigational channel. This is the first time the direct relationship between diahaline exchange flow and salt mixing as well as the spatial distribution of the diahaline exchange flow is shown in a realistic tidal setup. The spatial distribution is highly correlated with the local mixing gradient for salinity, such that inflow occurs near the bottom at the upstream end of the isohaline. Meanwhile, outflow occurs near the surface at its downstream end. Last, increased vertical stratification occurs within the estuary during the high-discharge period, while estuarine-wide mixing strongly converges to the universal law for averaging periods of the discharge event time scale.

Significance Statement

Inside estuaries, such as river mouths, terrestrial freshwater is mixed with salty ocean water. This is accompanied by an estuarine circulation with inflow of saltwater into the estuary and outflow of brackish water toward the ocean. Here, we aim to better understand how salt mixing and estuarine circulation in a tidal estuary react to periods of extreme freshwater discharge. We find that even during extremely high or low discharge, salt mixing follows the freshwater discharge on time scales as short as days, and that estuarine circulation patterns are largely explained by the local distribution of mixing. As extreme runoff events are likely to occur more often with climate change, these findings may help to understand the dynamics inside future estuaries.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Lloyd Reese, nina.reese@io-warnemuende.de

1. Introduction

Inside a classical estuary, salty ocean water is mixed with terrestrial freshwater. As a result, brackish water is exported from the estuary. Thus, sufficient temporal averaging reveals an inflow of salty water and an outflow of brackish water, known as classical estuarine circulation. This phenomenon was first quantified by Knudsen (1900) for the Baltic Sea. Later, Hansen and Rattray (1965) developed the first theoretical framework for a density-driven estuarine circulation. Further pioneer studies by Chatwin (1976) and Walin (1977) followed.

In recent years, it has become widely established to study estuarine dynamics within an isohaline framework (see, e.g., MacCready and Geyer 2001; Hetland 2005; MacCready 2011; Wang et al. 2017; Burchard et al. 2019). Such a framework had first been introduced to estuarine physics by Walin (1977). Using isohaline instead of vertical coordinates is often beneficial for estuarine studies because the classical estuarine circulation is largely salinity driven. Knowing at which salinities estuarine inflow and outflow occur, and how and where these water masses interact, is therefore of great help for the identification of key mechanisms. MacCready (2011) used such isohaline coordinates along a vertical, cross-channel transect to introduce the total exchange flow (TEF) framework. The TEF framework yields tidally averaged isohaline inflow and outflow per salinity class across this vertical transect. In contrast to an Eulerian (spatial) framework, it properly accounts for tidal salt fluxes. Therefore, when quantifying exchange flows in terms of Knudsen bulk values, the TEF transport-weighted inflow and outflow salinities are more representative of the exchange flow. Building on MacCready (2011), Sutherland et al. (2011) presented the temporal variability of the tidally averaged TEF across a transect in the fjord-like Puget Sound. Later, Wang et al. (2017) studied the along-channel variability of the TEF within the Hudson estuary. Further studies using the TEF framework have been conducted by Chen et al. (2012), Ganju et al. (2012), and many others.

A useful definition for salt mixing in estuaries is the destruction of salinity variance (see Burchard and Rennau 2008; Burchard et al. 2009). According to this definition, mixing is a function of the salinity gradient squared and the eddy diffusivity in each spatial direction. Therefore, it depends not only on turbulence but also on the availability of water masses of different salinities. Consequently, mixing may be strong in a stratified, but somewhat turbulent body of water and will vanish once the water body is homogeneous, i.e., once no further salinity gradients are found, even though turbulence is still present. So far, studies on estuarine mixing, such as Wang et al. (2017), Li et al. (2018), and MacCready et al. (2018), have mostly worked with constant freshwater runoff and focused on the spring–neap cycle as a modulator of salt mixing. However, Wang and Geyer (2018) found a dependence of mixing on the freshwater runoff for scenarios of different constant river discharges. They demonstrated that increased freshwater discharge entering an estuary leads to an increased salinity variance that outweighs the reduced salt intrusion. Additionally, sufficiently long time-averaged salt mixing MT inside an estuarine volume bounded by a TEF transect can be estimated as the product of TEF inflow and outflow salinities Sin, Sout and freshwater runoff Qr (MacCready et al. 2018):
MTSinSoutQr.
Recently, Conroy et al. (2020) showed that the volume-integrated mixing in the relatively small Coos River estuary mostly follows the temporal evolution of the freshwater discharge. In accordance with these findings, Broatch and MacCready (2022) demonstrated that the mixing estimate by MacCready et al. (2018) works well in their realistic simulation of Puget Sound, even for tidal averaging only. Their study showed that mixing in Puget Sound is predominantly driven by freshwater runoff, indicating that the spring–neap variability found in the earlier studies may be of only secondary importance for at least some realistic estuaries.
Instead of a vertical transect, an estuarine volume can also be bounded by an isohaline surface (see MacCready and Geyer 2001; MacCready et al. 2002). Burchard (2020) considered estuarine volumes bounded by such isohaline surfaces to derive a universal law of estuarine mixing. Due to the different choice of the estuarine volume as compared to MacCready et al. (2018), this law yields an even more direct relation between mixing and freshwater runoff: it states that on long-term average and when neglecting surface fluxes, the mixing M(S) inside the estuarine volume is determined from the respective bounding salinity S and average freshwater runoff Qr alone:
M(S)=S2Qr.
The importance of this law has since been demonstrated for an idealized estuary (Burchard et al. 2021) as well as for idealized conditions inside a complex estuary with multiple outlets (Li et al. 2022).

We summarize that freshwater runoff plays a vital role in the estuarine exchange flow, as predicted by the Knudsen (1900) relations, and in estuarine mixing. However, even though studies predict that periods of extremely high and low runoff will become more frequent with climate change (see, e.g., Christensen and Christensen 2004; van Vliet et al. 2013), the direct impact of such events on estuarine circulation and especially on mixing has not yet been studied. Most numerical studies that take extreme discharge values into account have only used idealized forcing, including idealized tides and a constant discharge, such as Wang et al. (2017) or Li et al. (2022). However, events of unusually high discharge are often driven by extreme rainfall events or human intervention such as weirs. They are therefore characterized by a strong variability of discharge in time, which cannot be represented by a choice of constant runoff. The few existing exchange flow and mixing studies using realistic forcing do not focus on extreme discharge events, and either do not include mixing at all (e.g., Sutherland et al. 2011), or do not include spatially resolved mixing or exchange flow analyses (e.g., Conroy et al. 2020; Broatch and MacCready 2022).

Furthermore, the mixing estimates from MacCready et al. (2018) and Burchard (2020) both assume “sufficiently long” averaging periods, which may differ between estuaries. For the aims of the present study, it could generally be said that an averaging period is sufficiently long when the computed total mixing converges to the respective mixing estimate. Even though Broatch and MacCready (2022) showed that the MacCready et al. (2018) estimate could generally perform well even for relatively short averaging periods of a single M2 tidal cycle, neither estimate has yet been tested under realistic conditions of extreme runoff. The question remains whether choosing sufficiently short time spans to resolve such extreme events would reproduce these estimates correctly despite potentially rapid changes in discharge intensity. An exemplary event of extremely high runoff occurred in June 2013 in the Elbe estuary in the southeast of the German Bight. It consisted of a strong and fast increase of discharge from about 1000 m3 s−1 to more than 4000 m3 s−1 and occurred over the time scale of roughly 1 month. The Elbe estuary is a rather typical, funnel-shaped mesotidal estuary with a well-defined navigational channel surrounded by tidal flats. Previous studies showed that the high-discharge event led to temporally increased stratification in the German Bight, demonstrating the impact that such events can have on a coastal system (Voynova et al. 2017; Kerimoglu et al. 2020; Chegini et al. 2020). In Kerimoglu et al. (2020), particularly strong mixing occurred in the shallow coastal waters, including the Elbe River mouth. In these regions, vertical mixing played an important role in destratification (Chegini et al. 2020). Even though these findings indicate that mixing is of great importance for coastal dynamics, none of the previous studies focused on the local influence of the high-runoff event on the circulation and mixing inside the Elbe estuary itself, leaving these questions open for future research.

Mixing can be related to the actual transport of water masses by considering a diahaline velocity relative to a moving isohaline that is directed in across-isohaline, i.e., diahaline, direction (see udia in Fig. 1). Such a diahaline transport per unit isohaline area (see Burchard et al. 2021) does not include advective motion because transport across an isohaline can only occur due to mixing. A related, vertically directed property is the local diahaline volume transport per unit horizontal area across an isohaline surface (see udia,zz in Fig. 1). Due to its definition as a velocity relative to the advective isohaline motion, the local diahaline volume transport per unit horizontal area has been described as entrainment velocity in previous studies by Wang et al. (2017) and Li et al. (2022). Wang et al. (2017) linked the entrainment velocity to the diahaline turbulent salt transport, i.e., the turbulent transport of salt across an isohaline. In Li et al. (2022), the diahaline salt transport was related to salt mixing. Recently, Klingbeil and Henell (2023) generalized the framework to support salinity inversions and, in this context, defined the effective vertical diahaline velocity. Moreover, they combined the relations from Wang et al. (2017) and Li et al. (2022), thus directly linking the effective vertical diahaline velocity to the gradient of mixing in isohaline coordinates. As this relation neglects the horizontal diffusion of salt, it has yet to be tested how well it holds for realistic estuaries, where horizontal diffusion might actually be of importance. While Henell et al. (2023) found a good agreement between the mixing gradient and effective vertical diahaline velocity within the microtidal Baltic Sea, it is still unclear how well the estimate holds in a realistic tidal regime: the previously mentioned study by Wang et al. (2017) used only idealized forcing, including idealized tides and a constant, moderate freshwater discharge. Furthermore, it has not yet been shown for a tidal regime how well the estimate holds locally, i.e., for the spatially resolved diahaline exchange flow, and under temporally varying, extreme discharge conditions. So far, the studies by Wang et al. (2017), Li et al. (2022), and Henell et al. (2023) all have revealed a diahaline exchange flow with inflow toward lower salinities and outflow toward higher salinities, similar to the isohaline total exchange flow. The diahaline velocity may therefore provide insight into the spatial distribution of exchange flow under extreme conditions.

Fig. 1.
Fig. 1.

Schematic along-channel section of a tidally averaged estuary, illustrating TEF and diahaline parameters. The bold black line and gray shading indicate the bottom topography, and the angled, bold black line indicates the water surface. The dashed horizontal line shows where z = 0; η is the surface elevation above z = 0, H is the depth of the bottom below z = 0, and Qr is the freshwater runoff. The broad, semitransparent arrows indicate the tidally averaged flow directions. The thin, curved black lines represent isohalines. Blue: the vertical, cross-channel TEF transect at along-channel position x = xT; the estuarine volume VT is bounded by this transect, and the TEF bulk volume transport Qin goes into and Qout goes out of the estuarine volume. Orange: the estuarine volume bounded by the isohaline of salinity s = S. The dashed vertical lines bound a single, finite water column with horizontal area element dAz. The arrows within the water column illustrate the difference between the diahaline velocity udia and the local diasurface volume flux per unit horizontal area in z coordinates, udia,zz. Note that in most estuaries, the isohalines are almost horizontal so that udia and udia,zz are almost identical. In the stably stratified case illustrated here, the effective vertical diahaline velocity udia,zS equals udia,zz. In a more general case, udia,zS has to be calculated from (18).

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0052.1

The present study aims to answer the open questions mentioned above for typical mesotidal estuaries. In detail, these questions are as follows:

  1. Using averaging periods short enough to resolve extreme discharge events, will the mixing estimates by MacCready et al. (2018) and Burchard (2020) still be reproduced properly? Are these estimates therefore applicable to the study of extreme events?

  2. How are extreme discharge events reflected in the total exchange flow (MacCready 2011), diahaline circulation, and salt mixing?

  3. How closely is the two-dimensional distribution of the salinity gradient of mixing linked to the local diahaline exchange flow in a realistic hindcast simulation?

To address these questions, a realistic numerical setup of the Elbe estuary has been created for this study. The mathematical framework required for the TEF, estuarine salt mixing, and diahaline transport is introduced in section 2, followed by the study region itself. Furthermore, the numerical setup is described and validated. In section 3, the setup is applied to demonstrate the impact of the high-discharge event from June 2013 on local circulation and mixing. To this end, the high-discharge month of June 2013 is compared to the low-discharge month of September 2012. Additionally, the temporal evolution of mixing is considered. We draw some final conclusions in section 4.

2. Methods

a. Mathematical framework

1) Total exchange flow

This study applies the TEF framework introduced by MacCready (2011). It is used to analyze residual volume and salt transport across a vertical, cross-channel transect of an estuary (see transect at x = xT in Fig. 1). In addition to the original equations, the more general formulation by Klingbeil and Henell (2023) will be presented here.

With the along-channel velocity u normal to the vertical transect, the isohaline volume transport Q in all salinity classes s > S (where S is the chosen salinity of an isohaline) is then given as
Q(x,S,t)=ATH(szS)udA=ASudA,
with the Heaviside step function,
H(ψ)={0,ψ01,ψ>0,
for an arbitrary function ψ. In (3), t is the temporal argument, sz is the salinity at a given spatial position (x, y, z) on the transect, AT is the total cross-sectional area of the transect, and AS is the cross-sectional area of the transect where s > S. The last term in (3) is the original definition by MacCready (2011). Note that in MacCready (2011), tidal averaging is applied here. However, following Wang et al. (2017) and Klingbeil et al. (2019), averaging can also be applied at a later point without changing the given properties.
To find the respective volume transport per salinity class, q, the derivative of Q with respect to salinity is computed:
q(x,S,t)Q(x,S,t)S.
Here, the negative sign is required such that q is positive for inflow and negative for outflow.
Integrating over the respective inflows and outflows per salinity class then yields the total inflow and outflow across the transect, namely,
Qin(x,t)0Soq(x,S,t)|indS,Qout(x,t)0Soq(x,S,t)|outdS,
where So is the maximum salinity occurring along the transect and 〈⋅〉 indicates temporal averaging, e.g., tidal averaging. This is the original definition for the bulk volume inflow and outflow as given by MacCready (2011). A numerically more robust method is the dividing salinity method (MacCready et al. 2018; Lorenz et al. 2019), where, assuming a classical two-layer exchange flow,
Qin=max(Q),Qout=QinQr.
Here, Qr is the freshwater discharge entering the estuarine volume, averaged over the same period as the other properties. In this study, we use the dividing salinity method. Similarly, the ingoing and outgoing salt transport are defined as
Fin(x,t)0SoSq(x,S,t)|indS,Fout(x,t)0SoSq(x,S,t)|outdS,
which can also be determined from the dividing salinity method, analogous to (7).
From these bulk values, transport-weighted inflow and outflow salinities can be computed:
Sin(x,t)FinQin,Sout(x,t)FoutQout.
The Qin, Qout, Sin, and Sout terms are the total exchange flow properties as given by MacCready (2011).

From these properties, the Knudsen relations (Knudsen 1900) can be obtained for sufficiently long temporal averaging (MacCready 2011; Burchard et al. 2018).

2) Salt mixing

Estuarine exchange flow is related to the mixing of water masses (MacCready et al. 2018). Following Burchard and Rennau (2008), mixing is defined here as the destruction of salinity variance:
χ(x,y,z,t)=2Kh(xsz)2+2Kh(ysz)2+2Kυ(zsz)2.
This is the sink term of the salinity variance budget equation; see, e.g., Burchard and Rennau (2008). It therefore directly quantifies the decay rate of salinity variance. Here, Kh is the horizontal and Kυ is the vertical eddy diffusivity of salt s, which may vary in space and time. The term ∂i represents the partial derivative with respect to a coordinate xi. Under this definition, mixing can only occur when the water body is not fully mixed, and will increase quadratically with stratification for a given turbulent diffusivity.
The time-averaged total mixing MT inside an estuarine volume bounded by a vertical TEF transect, VT (see Fig. 1), can be found as the volume integral of χ:
MT=VTχdV.
Assuming long-term averaged conditions and neglecting surface volume fluxes, MacCready et al. (2018) approximated MT as
MTSinSoutQr.
The exact averaging period required for a convergence of the actual mixing with the estimate (12), and therefore the exact definition of what “long-term averaging” is, may vary between estuaries. Instead of considering a volume bounded by a vertical transect, we can also use an estuarine volume V(S) that is bounded by an isohaline surface of salinity S (see Fig. 1). The total mixing inside V(S) is found as a function of the bounding salinity S:
M(S)=V(S)χdV.
Using (13), Burchard (2020) showed that for long-term averaging (defined as above) and when neglecting surface volume fluxes, this property could also be computed from the bounding salinity and freshwater runoff:
M(S)=S2Qr.
If formulated for the mixing per salinity class, m(S) = ∂SM(S), this is the universal law of estuarine mixing:
m(S)=2SQr.
Again, this equation is valid only for long-term averaging.
The local mixing per salinity class inside a single water column, mxy, is defined such that its area integral yields the mixing per salinity class for the full isohaline surface:
m(S)=Az(S)mxy(x,y,S)dAz,
where Az(S) is the horizontal projection of the isohaline surface A(S) and dAz is an infinitesimal horizontal area element. For the exact definition of mxy, see Klingbeil and Henell (2023).

3) Diahaline transport

While the TEF framework considers only isohaline properties, i.e., fluxes and transports along a given isohaline, it is also possible to consider the diahaline direction, i.e., across an isohaline. The diahaline velocity udia is the relative velocity of an isohaline to the three-dimensional flow velocity vector u:
udia(x,y,z,t)=(uus)sz|sz|.
Here, us is the velocity of a given point on the isohaline surface and includes advective motion. For an illustration of udia, see Fig. 1.
For a single water column, the effective vertical diahaline velocity udia,zS,
udia,zS(x,y,S,t)=Hηδ(Ssz)|sz|udiadz,
describes the diahaline volume transport per unit horizontal area across a given isohaline S (Klingbeil and Henell 2023). The udia,zS is a vertically directed property and positive in the direction of higher salinity. For stable salt stratification, udia,zS is identical to the negative entrainment velocity we used in Wang et al. (2017) and Li et al. (2022). The Dirac function δ(x) is introduced here as the derivative of the Heaviside step function. It is used to find an average value for udia,zS in cases where the salinity S appears at multiple positions in the water column. We will refer to this as the term effective. From the effective vertical diahaline velocity, we can compute the integrated diahaline volume transport
Qdia(S,t)=udia,zSdAz,
which is equivalent to the entrainment flux Qe from Wang et al. (2017).
Klingbeil and Henell (2023) linked the effective vertical diahaline velocity to the S derivative of the local mixing, as long as horizontal diffusive fluxes are neglected:
udia,zS=12mxyS.
Last, the local mixing mxy computed from a numerical, hydrodynamic simulation can be split into a physical component mxy,phy, originating from the turbulence parameterization, and a numerical component mxy,num, caused by the discretization error of advection schemes (Burchard and Rennau 2008; Klingbeil et al. 2014):
mxy=mxy,phy+mxy,num.

b. Study area

We have performed a hydrodynamic simulation of the Elbe estuary for this study. This partially to well-mixed, mesotidal estuary (see, e.g., Kappenberg and Fanger 2007; Stanev et al. 2019) is located in the southeast of the German Bight and consists of a well-defined, deep navigational channel widely surrounded by tidal flats (see Figs. 2a,b). The dominant tidal constituent is the M2 tide. Inside the estuary, the tidal intrusion is limited by a weir at Geesthacht, marking the upstream end of the so-called tidal Elbe (see Fig. 2c). The end of the salt intrusion inside the tidal Elbe is usually located between Glückstadt and Stadersand (Boehlich and Strotmann 2008). However, it can extend almost as far upstream as Schulau (see Fig. 2a) for particularly low freshwater discharge (Bergemann 1995). Overall, the salt intrusion is strongly dependent on the freshwater discharge (see, e.g., Grabemann et al. 1995; Kappenberg and Grabemann 2001). Aside from its ecological function, the Elbe estuary is also of significant importance for international maritime trade and shipping, with ports in Cuxhaven, Brunsbüttel, and the city of Hamburg (see, e.g., Boehlich and Strotmann 2008).

Fig. 2.
Fig. 2.

Model domain of the numerical tidal Elbe setup. (a) Setup topography showing the deep navigational channel and locations of the observational stations used for calibration and validation. Gauges provide information for the tidal analysis. The full names of the salt and temperature stations are LZ4a Steinriff, Cuxhaven Alte Liebe, LZ2a Neufeldreede, and LZ1b Krummendeich. Station D4 (Rhinplate Nord) is the only station where stratification data for salinity and temperature were available. Water outside the model domain is colored in blue. The label “NOK” marks the Kiel Canal. The transects T1 and T2 bound the along-channel section that is studied in section 3b. (b) Location of the estuary on the North Sea coast. (c) Curvilinear grid used for the setup, with (d),(e) details shown as indicated by the black rectangles. The thick gray line marks the open boundary of the setup. The main discharge into the estuary enters the model domain at Geesthacht, with the average discharge from August 2012 to December 2013 in parentheses (m3 s−1). Gray arrows show the location of each tributary that is included in the model, and their average discharge (m3 s−1) for the same period as Geesthacht. In (c) and (d), only every third grid line is shown. Yellow dots with numbers show the distance from the upstream end of the setup at cell centers between the 158th and 159th along-channel grid lines, in the following referred to as “model-km”.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0052.1

The freshwater runoff reaching the tidal Elbe via Geesthacht may vary strongly depending on conditions in its catchment area but averages around 700 m3 s−1 (Strotmann 2015). In June 2013, massive rainfalls in the Elbe catchment area (see, e.g., Grams et al. 2014) led to an extreme event with up to 4000 m3 s−1 of freshwater runoff passing a gauge in Neu Darchau, slightly upstream of Geesthacht. In comparison, periods of particularly low runoff may occur predominantly in summer and early autumn (see Strotmann 2015). An exemplary runoff for a dry summer is the average of 300 m3 s−1 that occurred in the month of September 2012, which was more than 100 m3 s−1 below the average September runoff for the period from 1925 to 2012 (Strotmann 2014). In addition to the main runoff, the tidal Elbe has several small tributaries entering between Geesthacht and the German Bight. However, their runoff is small compared to the freshwater input from Geesthacht. The strong variability of the total freshwater runoff is illustrated in Fig. 3b for the period from August 2012 to December 2013.

Fig. 3.
Fig. 3.

Time series from August 2012 to December 2013 for different forcing and validation parameters. (a) Magnitude (black line) and oceanographic direction (arrows; point in the direction in which the wind is blowing) of the wind forcing used for the setup, spatially averaged over the Elbe River mouth region. The wind magnitude has been filtered with a 1-day running mean, while wind direction was filtered with a 7-day running mean and is shown with about one arrow per day. The wind forcing is given as the velocity vector at 10-m height. (b) Daily averaged runoff from all sources of freshwater discharge included in the setup, compared to the runoff of the tributaries only. (c) Modeled and observed water temperature 2.5 m above ground at Cuxhaven Alte Liebe. (d) Modeled and observed surface (bold; Ssurf) and bottom (dashed; Sbott) salinity at D4 Rhinplate Nord. (e) Modeled and observed stratification SbottSsurf at D4 Rhinplate Nord. In (c)–(e), the transparent colors (light orange, gray) represent tidally varying data and opaque colors (dark orange, black) represent 30-h low-pass-filtered, subtidal data. The gray-shaded areas in all panels mark the months of September 2012 and June 2013, which are the focus of this study.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0052.1

Due to the relatively simple, funnel-shaped topography of the estuary and its intermediate properties such as discharge intensity and tidal range, the tidal Elbe provides a good representation of realistic, medium to large tidal estuaries. We will therefore use it to compare the effects of low-runoff (September 2012) and extremely high-runoff events (June 2013) on estuarine circulation, salt mixing, and diahaline properties in realistic mesotidal estuaries.

c. Numerical simulations

Our numerical setup uses the General Estuarine Transport Model (GETM; Burchard and Bolding 2002). The hydrostatic model kernel of GETM is used for the present study, which applies the hydrostatic and Boussinesq assumptions (Klingbeil and Burchard 2013). GETM is designed explicitly for coastal ocean and estuarine applications (Klingbeil et al. 2018). It allows for an efficient splitting of the baroclinic and barotropic modes and includes a state-of-the-art drying-and-flooding algorithm. The latter is particularly suited for the Elbe estuary with its large tidal flats, as previously demonstrated by Burchard et al. (2004). Turbulence closure is provided through coupling with the General Ocean Turbulence Model (GOTM; Burchard et al. 1999; Li et al. 2021), using a k–ε two-equation model (Umlauf and Burchard 2005) with a second-order closure (Canuto et al. 2001; Burchard and Bolding 2001). The present study applies a second-order Superbee advection scheme with minimal numerical mixing (Mohammadi-Aragh et al. 2015) on a C-grid. The setup was run from February 2012 to December 2013 at a barotropic time step of 2 s and a baroclinic time step of 20 s. The first 2 months of 2012 were reserved for spinup, the period from April 2012 to December 2012 was used for calibration, and the analysis takes place for the period from August 2012 to December 2013.

The model domain used in this study reaches from the weir at Geesthacht to the southeastern German Bight, as shown in Fig. 2. The model topography shown in Fig. 2a consists of data for the year 2010 from the DGM-W 2010 tidal Elbe dataset (Wasserstraßen- und Schifffahrtsamt Hamburg 2011) as well as a German Bight dataset (Bundesamt für Seeschifffahrt und Hydrographie 2019; Heyer and Schrottke 2015). Between 2010 and 2013, no major dredgings were conducted along the navigational channel.

The weak curvature of the Elbe navigational channel allowed for the creation of a horizontal curvilinear grid that closely follows the thalweg (see Figs. 2c–e). This grants a convenient analysis of cross-channel transects along each respective, transversal grid line. Additionally, a thalweg-following grid will likely lead to reduced numerical mixing compared to a Cartesian grid since the main flow will, on average, cross fewer grid lines. The grid was created with the Delft3D curvilinear grid generation tool “rgfgrid” (Deltares 2020). The grid consists of 616 transversal (cross-channel) and 305 longitudinal (along-channel) grid lines of varying resolution with a focus on the navigational channel. On average, this leads to a horizontal grid resolution of 200–400 m in the along-channel direction. Across the navigational channel, where the grid is focused, the resulting cross-channel resolution ranges mostly between 50 and 100 m, while it becomes as coarse as 400 m away from the focus in the German Bight. Cell center line 158 closely follows the thalweg from the open boundary to transect T2 in Fig. 2 and is therefore suited as a transect along the thalweg. In the vertical, 30 equidistant sigma layers have been prescribed, leading to a resolution finer than one meter, even in the deepest parts of the navigational channel.

Along the open boundary in the German Bight, we prescribe downscaled parameters from a 600-m German Bight setup (Gräwe et al. 2015) using the Flather boundary condition (Flather and Davies 1975). This boundary forcing includes a half-hourly resolved, realistic tidal forcing in the form of surface elevations and hourly resolved vertical salinity and temperature profiles. A sponge layer with a width of three grid cells provides a smooth transition from the open boundary to the interior of the model domain.

Meteorological forcing consists of downscaled, 3-h-resolved data from the operational forecast model ICON by the German Weather Service (Zängl et al. 2015). It has a resolution of about 7 km and includes precipitation, wind direction, and magnitude 10 m above ground, dry air and dewpoint temperature 2 m above ground, and total cloud cover. The area-averaged magnitude and direction of the included wind forcing are displayed in Fig. 3a.

The main freshwater runoff is prescribed at the upstream end of the setup. It consists of daily averaged observations from Neu Darchau (Wasserstraßen- und Schifffahrtsamt Magdeburg 2021) that have been shifted by 1 day, as the observed runoff from Neu Darchau passes the weir in Geesthacht with a delay of about that duration (see Boehlich and Strotmann 2008). Additionally, all Elbe tributaries downstream of Hamburg are included, as they may interfere with the area of salt intrusion despite their overall low discharge. The river Ilmenau upstream of Hamburg is also included because it is one of the largest tributaries of the tidal Elbe. For the daily averages used, see Strotmann (2014, 2015), Landesamt für Landwirtschaft, Umwelt und ländliche Räume Schleswig-Holstein (2022), and Niedersächsicher Landesbetrieb für Wasserwirtschaft, Küsten-und Naturschutz (2022). Additional data and weighting factors have been received via personal communication from the Bundesanstalt für Wasserbau (BAW). The weighting factors are required to estimate the runoff of each tributary at the point where they enter the tidal Elbe from the datasets mentioned above. For the Kiel Canal and Rhin, only constant discharge values of 19.1 and 0.9 m3 s−1, respectively, were available, while the combined Medem and Hadelner Kanal discharge was given as monthly averages. All other tributaries were provided with daily averaged discharge. The locations and average discharge of all tributaries that are included in the setup are depicted in Fig. 2c. The summed-up discharge from all included sources is shown in Fig. 3b for the analysis period from August 2012 to December 2013. It can be seen that the contribution of the tributaries to the total discharge is small within our model. The discharge salinities are set to a constant 0 g kg−1 for a more convenient interpretation of the salt dynamics. In reality, the freshwater runoff at Geesthacht has a finite salinity of about 0.35 g kg−1 that may vary slightly with the runoff (see Strotmann 2015).

d. Observational data

Setup calibration and validation were performed by comparing model output and observational data. For the location, names, and types of the observational stations used, we refer to Fig. 2a. The gauge data used for a tidal analysis are provided at 1-min resolution, while salt and temperature measurements are given at 5-min resolution. Measurements of salt and temperature were located 2.5 m above ground. Only for station D4 Rhinplate, surface and bottom salt and temperature data were available from 2012 to 2013, allowing for additional analysis of vertical stratification. All data are freely available via Wasserstraßen- und Schifffahrtsamt Elbe-Nordsee and Hamburg Port Authority (2023) and Wasserstraßen- und Schifffahrtsamt Elbe-Nordsee (2023).

e. Validation of model performance

As previously stated, the full model run covers the period from February 2012 to December 2013. Within this period, the setup reaches a semibalanced state after about 1 month of spinup. The setup was calibrated with model output for April 2012 to December 2012. Calibration was performed with respect to the M2 and M4 tidal range and M2 tidal phase as well as salinity and temperature 2.5 m above ground. The spatial distribution of the bottom roughness length z0 was used as a calibration parameter for the tides. To this end, z0 was set to be proportional to the water depth. This is based on the real bottom roughness distribution within the Elbe estuary. However, this alone did not sufficiently dissipate the tidal energy upstream of Brunsbüttel, where the river contracts. To counter this, an artificial, steep linear increase of z0 upstream of Brunsbüttel was superimposed to the depth-dependent distribution. The resulting bottom roughness length ranges from 1 × 10−4 to 6 × 10−3 m. The salt intrusion was not found to be sensitive to the choice of z0 and was instead calibrated by setting the constant parameter for horizontal physical diffusivity Kh to a value of 20 m2 s−1. Temperature calibration was limited by the low resolution of forcing data; see below.

Model validation is performed for the period from August 2012 to December 2013, which is also used for our study in section 3. To quantify the performance of our model as compared to observational data, we apply a number of statistical parameters: for the tidal data, we use the relative deviation of the modeled tidal amplitude Amod from the observed amplitude Aobs in percent,
ΔArel=AmodAobsAobs×100,
the difference between the modeled tidal phase ϕmod and the observed phase ϕobs in minutes,
Δϕ=(ϕmodϕobs)×Ttide360°,
with the period Ttide of each respective tidal constituent in minutes, as well as the root-mean-square error (RMSE) of the 30-h low-pass-filtered (subtidal) surface elevation data:
RMSE=[1Ni=1N(ηmod,iηobs,i)2]1/2.
Here, N is the total number of samples and ηmod,i and ηobs,i are the modeled and observed data at each point in time, respectively. To quantify the model reproduction of observed temperature, salinity, and stratification, we use the RMSE for the nonfiltered data as well as the Pearson R2, computed from the correlation coefficient R:
R=i=1N(ηmod,iη¯mod)(ηobs,iη¯obs)i=1N(ηmod,iη¯mod)2i=1N(ηobs,iη¯obs)2.
Here, η¯mod and η¯obs are the mean of the modeled and observed data, respectively.

The statistics of the calibrated model regarding tidal analysis, salinity, temperature, and stratification are summarized in Tables 1 and 2. Slightly larger deviations of the computed tidal M2 and M4 amplitudes from the observations, as well as an increased RMSE of the subtidal sea level variability, mostly occur in the less well-resolved setup region upstream of Glückstadt (Fig. 4, Table 1). For St. Pauli and Zollenspieker, this is accompanied by an underestimation of the M2, M4, and S2 tidal phase. The S2 tidal amplitude tends to be overestimated by the setup, with the strongest overestimation between Brunsbüttel and Schulau.

Table 1.

Statistical summary of the model performance from August 2012 to December 2013 regarding M2, M4, and S2 tidal amplitude and phase compared to observations at different stations along the Elbe estuary. For the location of each station, see Fig. 2a. The full name of Cuxhaven Sth. is Cuxhaven Steubenhöft. The subtidal root-mean-square error (RMSE) between the simulated and observed subtidal range has been computed from the 30-h low-pass-filtered data.

Table 1.
Table 2.

Statistical summary of the model performance from August 2012 to December 2013 regarding salinity S, salt stratification SbottSsurf, temperature T, and temperature stratification TbottTsurf compared to observations at different stations along the Elbe estuary. The full names of Cuxhaven AL, LZ2a, LZ1b, and D4 are Cuxhaven Alte Liebe, LZ2a Neufeldreede, LZ1b Krummendeich, and D4 Rhinplate Nord, respectively. All values shown are for the nonfiltered (i.e., tidal) data. Temperature and salt stratification have been computed as the difference between surface and bottom values. Note that stratification data were available only for station D4 Rhinplate Nord, where bottom, surface, and stratification values are shown in separate rows.

Table 2.
Fig. 4.
Fig. 4.

Comparison of M2, M4, and S2 (a) tidal amplitude and (b) tidal phase from the tidal Elbe setup with observational data from different gauges along the channel, which are located at the vertical lines and named along the x axis. For the model data, the tidal amplitude and phase have been computed along the navigational channel. The full name of the leftmost gauge is Cuxhaven Steubenhöft. The tidal analysis has been performed for the analysis period from August 2012 to December 2013.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0052.1

Salinity also tends to be overestimated by the setup, as can be seen for all stations shown in Fig. 5 as well as in Table 2. This overestimation increases in upstream direction. However, with a Pearson correlation coefficient of R2 ≈ 0.8 for each station, the overall agreement between simulated and observed salinity is still high. There is no notable difference between the correlation of salinity throughout the tidal cycle and low-pass-filtered salinity, meaning that tidal variability and subtidal changes in salinity are reproduced at a similar quality.

Fig. 5.
Fig. 5.

Correlation of observed and simulated salinity 2.5 m above ground at the stations (a) LZ4a Steinriff, (b) Cuhaven Alte Liebe, (c) LZ2a Neufeldreede, and (d) LZ1b Krummendeich for the analysis period from August 2012 to December 2013. Gray dots represent tidally varying salinity data, while the blue dots represent data that have been low-pass-filtered to remove M2 tidal variability. The stations are given in an order of increasingly upstream positions.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0052.1

The simulated water temperature follows the observed annual cycle as well as short-term variability, such as the decrease in temperature in late June 2013 (see Fig. 3c). However, the setup underestimates temperatures, especially in spring and summer. This discrepancy increases in upstream direction, as can be seen from the increasing RMSE (see Table 2).

For the simulation period from 2012 to 2013, only the station “D4 Rhinplate Nord” was available to provide observational data for salt and temperature stratification (see Fig. 3e). This station is located close to the upstream end of the salt intrusion, where the salinity is low. Consequently, the M2-filtered salt stratification is weak and rarely exceeds 0.5 g kg−1. The location of this station complicates the validation of the model performance with respect to stratification, as the model stratification at this point will strongly depend on the simulated end of the salt intrusion: slight differences in the location of the salt intrusion between the model and observations may lead to large deviations in stratification. Consequently, the correlation between modeled and observed salt stratification is low, with a Pearson R2 of only about 0.3 (see Table 2). Overall, the setup underestimates the observed salt stratification at Rhinplate but performs slightly better at times where the low-pass-filtered salinity is closer to the observed salinity (Figs. 3d,e). During September 2012 and June 2013, which will be the focus of this study, the modeled stratification is relatively close to the observations. Temperature stratification (not shown) lacks some skill in the setup and is underestimated at station D4 (Rhinplate), with a low Pearson correlation of 0.2 (see Table 2). Despite the low correlation, the RMSE for temperature stratification is only at 0.1°C because temperature stratification is generally weak at Rhinplate: the observed top to bottom temperature difference rarely exceeds 1°C on tidal and 0.2°C on subtidal time scales.

Regarding the tidal analysis, the stations where the dominant M2 tide is less well reproduced are located far outside the area of salt intrusion and are, therefore, of little interest to the present study. The model tendency to overestimate the S2 tide is apparently countered in upstream direction by increasing tidal dissipation. The remaining differences between the model and observations can be explained by local topography and bottom roughness differences due to the limited model resolution.

The discrepancies between the modeled and observed salinity seem to be of a more complex origin. While the increasing overestimation of salinity in upstream direction indicates that the simulated salt intrusion is not sufficiently mixed with freshwater, the underestimation of stratification at Rhinplate implies that mixing is too strong. We conclude that vertical mixing might be overestimated in our simulation, leading to potentially reduced stratification. However, without observational data further downstream of the end of the salt intrusion, it is impossible to properly quantify the true extent of stratification underestimation. To explain the too-strong saltintrusion, it is consequently possible that horizontal fronts experience too little mixing in the simulation, despite our calibration with a rather high horizontal diffusivity of Kh = 20 m2 s−1. A comparative model run with a strongly reduced diffusivity Kh = 0.001 m2 s−1 yielded an unrealistically strong salt intrusion within the estuary and increased numerical mixing due to increased stratification, while the qualitative nature of our results regarding mixing and estuarine and diahaline circulation presented in section 3 was not changed. The respective versions of Figs. 5 and 3 as well as all figures from section 3 for the low-diffusivity case are included as Figs. S1–S8 in the online supplemental material. These results imply that our choice of Kh does not directly impact the dynamics discussed in section 3, which are dominantly driven by vertical stratification and mixing.

The station D4 Rhinplate Nord differs from the other stations in that it shows a tendency of salinity underestimation instead of overestimation: due to the finite salinity of the real Elbe freshwater, the observed salinities at this station do not fall below about 0.25 g kg−1, while our simulation uses a freshwater runoff of 0 g kg−1. This explains the underestimation of salinity from February 2013 to August 2013 (see Fig. 3d) despite the model tendency to overestimate the salt intrusion: during this phase, the runoff was relatively high (Fig. 3b), and stratification was low, so the observed salinity > 0 g kg−1 can likely be attributed to the real freshwater runoff. In phases where the simulated salt intrusion is slightly too strong, but the freshwater influence is still present, this may result in a seemingly correct reproduction of observed salinities, such as in September–December 2012.

To explain the temperature discrepancy between the model and observations, the 7 km × 7 km resolved DWD weather forcing data were compared to a different dataset from a 1 km × 1 km resolved atmospheric model (Norwegian Meteorological Institute 2022). The comparison indicated that the 7-km data resolution cannot correctly reproduce air temperatures directly above the Elbe River. While both datasets agreed above land, significant differences could be observed above the river, where the 7-km data tended to prescribe temperatures closer to those found on land. Unfortunately, the high-resolution data were not available for the period from 2012 to 2013. The underestimation of temperature stratification may have the same cause. However, due to the generally low subtidal stratification, temperature is considered here to be only of minor importance for the estuarine dynamics inside the tidal Elbe.

Despite the overlap of the calibration and analysis periods, no clear bias in the model performance was found between the end of 2012 (analysis and calibration) and the year 2013 (only analysis).

From the above-presented performance of our numerical setup, we conclude that our simulation is suited for the aims of this study. Nonetheless, it should be kept in mind that the strength of vertical stratification might be slightly underestimated, while horizontal salinity gradients may be overestimated.

3. Results and discussion

In the following, TEF, mixing, and diahaline properties will be compared for the high-runoff month of June 2013 and the low-runoff month of September 2012. Unless mentioned otherwise, these properties have been averaged over two spring–neap cycles, i.e., over the first 29.5 days of each month. This will be referred to as spring–neap averaged. In section 3c, this choice of averaging period is motivated and tested in more detail. Mixing will additionally be analyzed as an M2-averaged, temporally varying property from August 2012 to December 2013.

a. Salt distribution

To illustrate the general situation regarding salinity and stratification inside the Elbe estuary for each month, we first introduce the salinity distribution in September 2012 and June 2013, shown in Fig. 6. Since the spring–neap variability of salt intrusion within each month was found to be much smaller than the differences between both months, we only consider the spring–neap averaged distributions here. We find that the high-runoff event in June 2013 has resulted in a significant freshening of the estuary compared to the low-runoff period in September 2012. This is true for both the surface salinity and the vertical salt distribution along the thalweg. In June, the salt intrusion has moved about 30 km downstream compared to September (see Figs. 6c,d). Furthermore, increased isohaline areas and increased vertical stratification are found in June. A time-resolved analysis of stratification at Cuxhaven Steubenhöft (see Fig. 2a), defined as the difference between bottom and surface salinity as before in Fig. 3e, revealed that in our simulation, this vertical stratification is persistent throughout the entire tidal cycle from 4 to 24 June 2013 (not shown). During low tide, stratification reaches up to 10 g kg−1, while a reduction of up to 8 g kg−1 occurs during high tide. In September 2012, persistent stratification only occurred for two 3-day periods after neap tide and is generally much weaker than in June 2013.

Fig. 6.
Fig. 6.

Simulated, spring–neap averaged salinity distribution inside the Elbe estuary. (a),(b) Surface salinity in September 2012 and June 2013, respectively. (c),(d) Vertical salinity distribution along the navigational channel in September 2012 and June 2013, respectively. Bold white lines are isohalines at even salinities, and dashed white lines are isohalines at odd salinities. In (a) and (b), the gray outline corresponds to the 12-m isobath. The black outline in (c) and (d) indicates the bottom topography. Transects T1 and T2 enclose the estuarine volume discussed in section 3b.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0052.1

The freshening of the estuary and coastal waters during the extreme runoff event in June 2013 is consistent with earlier studies of this event, including observational data presented by Voynova et al. (2017) as well as numerical simulations of the German Bight by Kerimoglu et al. (2020) and Chegini et al. (2020). However, the increased stratification found in our simulation for June 2013 contrasts the findings by Kerimoglu et al. (2020) and Chegini et al. (2020). These studies reported an increased vertical stratification in the deeper parts of the German Bight, but no persistent stratification was found within the estuary itself. It is likely that the comparatively large-scale setups of the German Bight used for these studies could not resolve important processes within the Elbe estuary, as mentioned by Chegini et al. (2020) themselves.

Furthermore, it should be noted that both Kerimoglu et al. (2020) and Chegini et al. (2020) found that the stratification dynamics in the German Bight were not only runoff-driven but also strongly dependent on wind direction and magnitude. Within the area corresponding to the model domain of the present study, however, the results by Kerimoglu et al. (2020) showed that the distribution of surface salinity depended almost exclusively on freshwater runoff instead of wind, meaning that our model domain only includes the discharge dominated section of the river mouth. Additionally, the wind magnitude and direction above the Elbe River mouth were similar in September 2012 and June 2013 and the magnitude did not reach the strength of the autumn and winter storms from 2013 (see Fig. 3a). In the following, we will therefore compare September 2012 and June 2013 without discussing wind conditions. As shown in the following paragraphs, this is reasonable with respect to spring–neap averaged mixing dynamics, which can primarily be attributed to freshwater discharge.

b. Influence of freshwater discharge on the estuarine circulation

To analyze differences in the estuarine circulation between September 2012 and June 2013, we consider the along-channel (x-) variability of TEF properties, mixing, and effective vertical diahaline velocity between the transects T1 and T2 from Fig. 2a. We find that all properties become nonzero further downstream in June than in September, which can be attributed to the reduced salt intrusion in June. In both months, the sum of inflow and outflow is close to the spring–neap averaged freshwater runoff; see Figs. 7a and 7b. The sum slightly underestimates the magnitude of the runoff in September, while it is overestimated in June. These deviations are due to slow mean water level variations in the German Bight which are included in the boundary conditions. At the same time, the bulk inflow and outflow salinities are considerably fresher in June, accompanied by a strongly increased difference Δs = sinsout toward the downstream end of the channel section (Figs. 7c,d). This is consistent with the findings from the previous section as well as the broader salinity range covered by the TEF inflow and outflow (Figs. 8c,d). The longitudinal distribution of Q(x, S) shown in Figs. 8a and 8b can be interpreted as a streamfunction for water transport in salinity space. It is of similar shape in both months.

Fig. 7.
Fig. 7.

Along-channel variability of spring–neap averaged TEF properties in (left) September 2012 and (right) June 2013. (a),(b) TEF inflow and outflow and their sum Qin + Qout, and spring–neap averaged freshwater runoff Qr. (c),(d) Respective TEF inflow and outflow salinities and their difference ΔS = SinSout. (e),(f) Total mixing MT (11) inside the estuarine volume bounded by a vertical transect at each respective model-km, its numerical and physical contributions MT,num and MT,phy, and the mixing estimate (12) from MacCready et al. (2018). The runoff Qr is a step-like function of the along-channel distance x, as tributaries are added in downstream direction starting at the position x where they enter the estuary.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0052.1

Fig. 8.
Fig. 8.

Spring–neap averaged TEF, mixing, and diahaline properties vs along-channel position x and salinity class S in (left) September 2012 and (right) June 2013. (a),(b) Volume transport Q(S) for salinities s > S. (c),(d) Volume transport per salinity class q(S). The red and white lines indicate the TEF inflow and outflow salinities, respectively. The black line represents the average of TEF inflow and outflow salinity, i.e., S¯=(1/2)(Sin+Sout). (e),(f) Local mixing per salinity class, integrated in cross-channel direction. (g),(h) Negative effective vertical diahaline velocity as estimated from (20) (Klingbeil and Henell 2023). (i),(j) Negative effective vertical diahaline velocity as computed online with GETM.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0052.1

Next, we compare the volume-integrated mixing inside the estuarine volume bounded by a TEF transect at position x to the mixing estimate (12) from MacCready et al. (2018) (see Figs. 7e,f). In September, the simulated mixing is underestimated by the estimate (12) by up to 10%, which is within the range of accuracy presented by Burchard et al. (2019) for idealized estuaries. In June, the estimate and total mixing agree even better, with only a small overestimation. Numerical mixing is small in both months so that the total mixing MT closely represents physical mixing. The numerical mixing is weakly negative due to the choice of advection scheme; see Klingbeil et al. (2014) for an in-depth discussion of the negative mixing tendencies of the Superbee advection scheme. Most notably, however, mixing increases much faster with volume in June than in September. A similar behavior is found for the cross-channel integrated local mixing per salinity class,
mx(x,S)=yminymaxmxy(x,y,S)dy,
shown in Figs. 8e and 8f. In (26), ymin(x) and ymax(x) are the bounding cross-channel coordinates of the river section. Here, the overall mixing intensity is strongly increased in June. In both months, local maxima of mixing can be found at model-km 122, 127, and 140. In September, an additional maximum occurs at model-km 110, which in June is in the freshwater range and therefore not visible. Mixing is strongest for salinity classes between the inflow and outflow layers. It tends to span a broader salinity range in places where local maxima occur, particularly at the 110-km maximum.

The cross-channel integrated estimate (20) for the effective vertical diahaline velocity from Klingbeil and Henell (2023), by definition, yields local extrema in the same x-locations as the cross-channel integrated local mixing; see Figs. 8g and 8h. Results are very similar to the directly computed distribution of udia,zSdy shown in Figs. 8i and 8j. In particular, we can see a layer of outward-directed diahaline transport at low salinities and a layer of inward-directed diahaline transport at high salinities. The separation of the two layers closely follows the average of the flux-weighted TEF (isohaline) bulk salinities, (Sin + Sout)/2. Overall, the effective vertical diahaline velocity is of greater magnitude in June than in September.

In a modeling study of the Hudson River estuary with idealized external forcing, Wang et al. (2017) focused on the along-channel variability of the isohaline transport q(x, S), entrainment velocity (here, udia,zS), and diffusive salt flux (i.e., salt mixing) within a single spring–neap cycle. The spring–neap averaged along-channel profiles yielded results similar to the ones found in this study, with a two-layer total exchange flow. Wang et al. (2017) also located the strongest mixing in between the inflow and outflow layers, with local extrema due to topography effects. The mechanisms causing such local extrema were studied in detail by Warner et al. (2020). Most notably, they showed that the locations and mechanisms of strong mixing vary between flood and ebb. As in our study, the entrainment velocity in Wang et al. (2017) displayed local extrema in the same along-channel locations as mixing. Here, we find a similar causality with topography: At model-km 110, the Kiel Canal connects with the estuary (see Fig. 2a), providing a topographical irregularity as well as an additional inflow of freshwater. The mixing of this new freshwater source explains the wide salinity range of this extremum, occurring at lower salinity classes than the remaining extrema. The extrema at 122 and 127 km coincide with the location of two particularly deep sections of the navigational channel (see Figs. 6c,d) as well as the upstream end of a side branch of the navigational channel, the so-called Medem branch (see Fig. 2c). At 140 km, this branch reconnects with the navigational channel, explaining the last of the occurring extrema.

Whereas the study by Wang et al. (2017) focused on the along-channel variability of estuarine circulation within a single spring–neap cycle and at constant runoff, we focus here on the impact of extreme runoff. From our results, it is evident that the freshening of the estuary in June 2013 applies to all TEF, mixing, and diahaline parameters. Additionally, the broader salinity range of Q(x, S) and q(x, S) in June directly results from the increased vertical salt stratification. The estuarine circulation, therefore, reacts strongly to the increased runoff but without changing the basic shape of its streamfunction Q(x, S) and its two-layer structure. Regarding the TEF bulk values, such a runoff dependency has, e.g., also been found for the Hudson and Merrimack estuaries (Chen et al. 2012).

Our results show that the strengthened isohaline outflow q(x, S) during high runoff is accompanied by increased mixing intensity. The increase in effective vertical diahaline velocity links this increased mixing with the diahaline circulation: The excellent agreement between relation (20) and the directly computed diahaline velocity in both months (see Figs. 8g–j) implies that horizontal diffusion is of minor importance in the Elbe estuary. The S gradient of the local mixing per salinity class (here, mx) therefore directly forces the diahaline circulation.

Last, from Figs. 7e and 7f, it follows that the mixing estimate (12) holds well for the high-runoff event in June 2013, while slightly larger deviations occur in September 2012. In a numerical study of mixing in the fjord-like Puget Sound, Broatch and MacCready (2022) found an excellent agreement between the estimate and actual mixing even for only tidally averaged properties. They argued that in their study, mixing was mostly limited to the river mouths far inside the estuary, while salinity variance from the seaward boundary only played a minor role. In our study, mixing appears to be strongly runoff-dependent for the high-runoff period in June 2013. Short estuarine turnover times would be expected for such periods, with only short-lived volume and salt storage along the seaward estuarine boundary. Under such conditions, the estimate should indeed perform similarly well as in Broatch and MacCready (2022). In September, however, when runoff is low, storage effects may gain some importance due to increased turnover times.

c. Universal law of estuarine mixing

In contrast to the mixing estimate (12) from MacCready et al. (2018), the universal law (15), which describes the mixing across a single isohaline surface, depends only on the freshwater runoff and is independent of the TEF inflow and outflow salinities, which were found to vary strongly between September 2012 and June 2013. Because the universal law refers to an estuarine volume bounded by isohalines instead of a vertical transect, it is also independent of the spatial location of the salt intrusion. A good agreement between the universal law and the simulated mixing after sufficiently long averaging therefore directly illustrates the dependence of mixing within such a volume on freshwater runoff. As mentioned by Burchard (2020), it may depend on different factors, such as the type of estuary, how long exactly a “sufficiently long” averaging period would be. In the case of the extreme runoff events considered in this study, we try to find averaging periods that properly cover the extent of each event while still allowing for a convergence of the simulated mixing with the universal law. To this end, only a limited number of averaging periods are suited in the first place: To properly resolve the extreme event, averaging should at maximum be performed for the time span of the event. In the case of the Elbe high-runoff event, this is about 1 month. Shorter averaging periods should not consist of incomplete cycles of the dominant M2 or spring–neap tides. Otherwise, it would not be possible to distinguish between effects of the respective tidal cycle and effects of the extreme event. Consequently, we here compare the conformance of simulated mixing to the universal law for averaging periods of two M2 cycles (about 1 day), one spring–neap cycle (about 2 weeks), and two spring–neap cycles (about 4 weeks). For better compatibility of the 1-day and 2-week averaging periods to the 1-month period, we choose time frames at different stages of each event. For the low-runoff period in September, the runoff is relatively constant, so that the 1-day averaging periods are about equally spaced throughout the month. Placement has been chosen to cover phases of increasing (S-M2-3) and right after decreasing (S-M2-1) runoff as well as the day of greatest change (S-M2-2); see Fig. 9a. For the June event, the spread of different runoff situations is much greater (Fig. 9b), so an additional fourth 1-day window was chosen. The four windows cover the comparatively low, but increasing runoff in the beginning of June (J-M2-1), the day of strongest runoff increase (J-M2-2), the day of maximum runoff (J-M2-3), and a day of decreasing runoff toward the end of the month (J-M2-4). For the 2-week averaging periods, we chose time frames covering the first two weeks (SN-1), center two weeks (SN-2), and last two weeks of each month (SN-3), respectively.

Fig. 9.
Fig. 9.

Performance of the universal law (15) in the Elbe estuary in (left) September 2012 and (right) June 2013. (a),(b) Daily-averaged freshwater discharge. Vertical lines show the placement of the 1-day averaging periods displayed in (c) and (d). Colored bars show the placement of the single spring–neap cycle averaging periods displayed in (e) and (f). From top to bottom, these averaging periods cover the first two weeks (SN-1), center two weeks (SN-2), and last two weeks (SN-3) of the first 29.5 days of each month. Note that the y axis in (b) has been scaled with a factor 10 compared to (a). (c)–(h) Simulated mixing per salinity class m(S) as computed for the entire model domain of the tidal Elbe setup (bold lines) compared to the universal law (dashed lines) for averaging periods of two M2 cycles in (c) and (d), one spring–neap cycle in (e) and (f), and two spring–neap cycles in (g) and (h). In (g) and (h), the physical and numerical fractions mphy and mnum of the total simulated mixing are also shown. The gray-shaded areas indicate salinities where the isohalines are not fully inside the model domain, computed for the 1-month averaging period.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0052.1

The universal law yields mixing across isohalines completely inside the model domain. In our numerical simulation, isohalines at higher salinity classes will leave parts of the model domain, reducing the integration surface and thus leading to a deviation of the simulated mixing from the universal law. To quantify which salinity classes leave the model domain to a significant amount, we consider the averaged, integrated diahaline volume transport (19). For isohalines fully inside the model domain, this quantity should be close to Qr. Consequently, all salinity classes where the integrated diahaline volume transport diverges from Qr leave the domain to such a degree that the simulated mixing can no longer be compared to the universal law. In the case of the funnel-shaped Elbe estuary, this is identical to the salinity classes where the average isohaline surface area does not increase monotonously toward higher salinity classes anymore. In the following, we will only compare the universal law to the mixing in salinity classes that are fully inside the model domain.

The results displayed in Figs. 9c–h show that in September, where the runoff is mostly uniform, the computed mixing is similar to the universal law almost independently of the averaging period. Only for the 1-day averaging periods S-M2-2 and S-M2-3, some deviation from the universal law occurs at higher salinity classes, which implies storage effects. For the remaining 1-day period S-M2-1, as well as all longer averaging periods, the simulated mixing is very close to the universal law. In contrast to September, the computed mixing in June only converges toward the universal law with an increasing length of the averaging period. During periods of greatest change, the deviation from the universal law is greater than during periods of smaller change, as can be seen for the 1-day averaging J-M2-2 as well as the 2-week averaging during the first half of the month J-SN-1. For the 1-month averaging period, the computed total mixing per salinity class is close to the universal law for both months for isohalines fully contained inside the setup volume (see Figs. 9g,h). The absolute deviation from the universal law is found to be about the same in both months. In September, the actual mixing is slightly underestimated, while in June, it is slightly overestimated.

The comparison of the different averaging periods demonstrates that even in a single estuary, the period required for conformance to the universal law may vary strongly depending on the state of the estuary. For periods of uniform freshwater discharge, even an averaging period as short as a day may lead to an excellent estimation of the actual mixing. However, we note that our setup was forced with daily-averaged freshwater discharge, so shorter-scale changes of discharge are not included. Inside a real estuary, the required averaging period might therefore be slightly longer than a day. For sudden discharge events such as in June, daily averages yield a rough estimation of the actual mixing, but longer averaging of at least one spring–neap cycle (2 weeks) is necessary for a more accurate prediction. However, for 2-week averaging, the exact quality of the estimate still depends on the placement of the period within the event. A 1-month averaging period removes this dependency, as it covers the full event and reproduces the universal law well. We therefore consider the averaging period of 1 month to be a good choice for the analysis of the high-discharge event and for the comparison to the low-discharge period in September 2012. The results for 1-month averaging demonstrate that the universal law (15), just as the mixing estimate (12) before, holds well under the extreme runoff conditions in June. It also holds well in September, demonstrating that mixing in the Elbe estuary is indeed strongly runoff-dependent on spring–neap averaged time scales. As before for the estimate (12), deviations from the universal law can most likely be explained with storage effects.

d. Temporal variability of mixing

So far, we have demonstrated that the spring–neap averaged mixing is stronger in June than in September and strongly runoff-dependent in both months. To put these findings into further context, we now consider the temporal evolution of total mixing per salinity class and of the universal law. We find that even the tidally averaged mixing adjusts to changes of the discharge, but with an overlying variation on shorter time scales of several days that was found to coincide with variations of wind magnitude (see Figs. 3a and 10a). Periods of increased mixing can be found from January to April 2013 and in June 2013 and coincide with increased runoff. The extreme runoff event in June is accompanied by the strongest mixing within the entire considered time period, meaning it is also an extreme mixing event. The periods of generally increased mixing occur throughout all salinity classes (Fig. 10b). During weaker runoff in September and October 2013, an additional signal of increased mixing becomes visible around the transition from neap to spring tides. This apparent spring–neap signal is more pronounced when mixing per salinity class is normalized with the universal law (15); see Fig. 10c. There, the signal occurs during low runoff as brief periods of strong underprediction by the universal law, i.e., m(S, t)/(2SQr) > 1. A comparative model run with a reduced constant horizontal diffusivity Kh = 0.001 m2 s−1 yielded a more pronounced spring–neap signal due to the presence of stronger vertical stratification, but the freshwater discharge still remained the source of strongest modulation of salt mixing (see Fig. S7). Additional periods of strong underprediction in Fig. 10c were found to coincide with storms, especially from January to April 2013; see also Fig. 3a. During periods of increased mixing associated with high runoff, on the other hand, m(S, t)/(2SQr) mainly varies between values of 0.8 and 1.4, meaning that mixing roughly follows the universal law. The river flood is immediately succeeded by a prolonged period of relatively strong and consistent underestimation in late July and early August 2013 (see Figs. 10a,c).

Fig. 10.
Fig. 10.

Temporal evolution of M2-averaged mixing inside the Elbe estuary from August 2012 to December 2013. (a) Mixing at a salinity S = 13 g kg−1, showing numerical mixing and total mixing m = mphy + mnum, and freshwater runoff Qr. The freshwater runoff has been linearly interpolated from daily averages to the M2-cycle data points of mixing. The axis for the freshwater runoff has been scaled with the universal law (15) to fit the proportions of the mixing axis. The black rectangle marks the period of prolonged mixing underestimation by the universal law seen also in (c). (b) Total mixing per salinity class m(S, t) for all salinity classes. (c) Total mixing per salinity class m(S, t) normalized with the universal law (15). The dashed magenta lines indicate spring tides. The white line in (b) and (c) is the maximum salinity Slim still fully inside the model domain for the area surrounding the navigational channel. The northeastern and southwestern parts of the open boundary have not been considered for Slim because there, negligible amounts of freshwater enter the model domain without notably influencing mixing and circulation inside the estuary.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0052.1

In consistency with Fig. 9, the results from Figs. 10a and 10c show that the universal law is a reasonably good, first approximation even for the temporal evolution of mixing with a varying freshwater runoff. This underlines the key role of freshwater runoff with respect to mixing in the Elbe estuary. For Puget Sound, Broatch and MacCready (2022) demonstrated that the spring–neap cycle and wind forcing are clearly of secondary importance. Here, we find a similar situation for the Elbe, where wind mostly has a modulating effect. However, we also see that the universal law tends to fail for strong wind events, i.e., periods of increased wind amplitude that last only up to a few days. An example is the strong underestimation of mixing by the universal law in late October 2013. During such periods, previously built-up stratification can be degraded, volume storage changed, and salt storage can be increased due to large-scale effects in the German Bight. Such processes are not represented in the freshwater runoff and thus the universal law, leading to the reported mismatch.

Earlier studies that used idealized, constant runoff scenarios found a strong spring–neap dependence of mixing, such as Wang et al. (2017), MacCready et al. (2018), Wang and Geyer (2018), and Li et al. (2022). For the Changjiang estuary, which has a powerful tidal spring–neap signal, a spring–neap variability of mixing was found even for a realistic runoff scenario (Li et al. 2018). However, the study focused on a relatively short period of three spring–neap cycles and did not put mixing into the context of the freshwater runoff. To study a high- and a low-runoff scenario in the Pearl River estuary, Li et al. (2022) used idealized spring–neap modulated tidal forcing. Similar to the present study, they found an increased spring–neap variability of the conformity of mixing to the universal law during the low-runoff scenario. The magnitude of the spring–neap variability of mixing is expected to be similar throughout the year. When normalized with the universal law, it becomes more visible during low runoff, when the relative deviation with respect to the magnitude of the runoff is largest.

Last, the period of increased underestimation by the universal law in July and August 2013, after the extreme event in June, possibly results from the destruction left over of stratification that was built up during the high-runoff period. It supports our assumption that the slight overestimation of mixing by the universal law in June, as found in Fig. 9h, coincided with a decrease in salt storage, which could also be interpreted as a relative increase in freshwater storage. Such a freshwater storage would lead to additional mixing as the system returns to normal. As the mixing overestimation occurs throughout all salinity classes, it is unlikely that it is solely caused by open boundary forcing. Nonetheless, the increased stratification in the German Bight reported by Voynova et al. (2017) until the end of August 2013 may still have played some part in the overestimation via open boundary forcing.

In addition to the findings from Broatch and MacCready (2022) and earlier studies, we conclude that the accuracy of mixing estimates in the Elbe estuary is not independent of freshwater runoff: for phases of low runoff, the choice of the averaging period can strongly impact the accuracy of an estimate, e.g., due to increased visibility of the spring–neap cycle. Nonetheless, on time scales of weeks to months, freshwater runoff is the dominant driver of the temporal evolution of mixing in the Elbe estuary.

e. Horizontal dynamics

In the previous sections, we have shown that during the extreme runoff event in June 2013, water is mixed as predicted by the universal law. To better understand the nature and location of underlying mechanisms, the following section will focus on the horizontal distribution of mixing and diahaline velocity in September 2012 and June 2013. The isohaline S = 13 g kg−1 is chosen as a representative salinity class because it covers the area of the river mouth without leaving the model domain in either month. Therefore, dynamics are covered within the deep navigational channel as well as on the surrounding tidal flats.

Due to the increased stratification in June, the isohaline surface and local mixing mxy are distributed over a significantly increased area compared to September (Figs. 11a,b). The overall magnitude of local mixing is greater in June than in September. For both months, but especially in June, mixing is mostly confined to the river channel since little stratification, and therefore little mixing and entrainment, occurs on the tidal flats. In detail, the strongest mixing mxy occurs along the banks of the navigational channel and in other locations where the topography gradient is enhanced.

Fig. 11.
Fig. 11.

(a),(b) Horizontal distribution of local mixing mxy(S, x, y). (c),(d) Negative effective vertical diahaline velocity as estimated from the local mixing gradient from (20). (e),(f) Negative effective vertical diahaline velocity as computed online with GETM. All properties are shown for S = 13 g kg−1 and have been spring–neap averaged for (left) September 2012 and (right) June 2013, respectively. The bold gray line marks the open boundary of the setup, thin gray lines represent the 12-m isobath, and the black lines represent the relative height of the S = 13 g kg−1 isohaline above ground at intervals of 0.2 (bold), 0.5 (dash–dot), and 0.8 (dotted). The inset figures in (a), (b), (e), and (f) show details as indicated by the black rectangles, with the 12-m isobath as a bold black line. In the insets in (a) and (e), an additional 6-m isobath was added as a dashed line.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0052.1

Regarding the effective vertical diahaline velocity, we also find that the estimate (20) by Klingbeil and Henell (2023) holds well, aside from a slightly noisy quality due to the computation of the salinity gradient (Figs. 11c–f). In fact, the estimated and directly computed udia,zS correlate with a Pearson correlation coefficient R2 of 0.76 and 0.69 in September 2012 and June 2013, respectively. Like with mixing, the magnitude of the diahaline velocity is increased along strong topography gradients and, aside from the steep sidewalls, low inside the navigational channel itself. This is particularly obvious in the detail panels in Figs. 11e and 11f. It can generally be seen that udia,zS is negative in the downstream region of the isohaline surface (outflow toward higher salinities) and positive in the upstream region (inflow toward lower salinities).

The relatively clear separation of positive and negative diahaline velocities suggests a well-defined two-layer exchange flow with a downstream, near-surface export of freshwater toward the North Sea and an upstream, near-bottom import of ocean water into the estuary. This diahaline exchange flow is strongly topography-dependent and mostly confined to the navigational channel. This is consistent with findings by Li et al. (2022) for the Pearl River estuary during a low-runoff scenario where the river plume did not reach any offshore islands. However, in their study, the diahaline velocity did not appear to be quite as strongly limited to channels, and there was no clear pattern of increased mixing along channel sidewalls. The topography dependence of mixing can be related to lateral differential advection, which is associated with a lateral shear between the inner channel and shoals. This is possibly caused by a process similar to the one described in Warner et al. (2020). They explained increased mixing near the channel shoals of the Hudson estuary during ebb tide with the build-up of salinity fronts in such locations. Since the diahaline velocity indeed correlates well with the local mixing gradient, we can conclude from relation (20) that high diahaline velocities occur where the local mixing varies strongly throughout the salinity classes. Consequently, the diahaline exchange flow is mainly confined to the channel shoals and other strong topography gradients as described above. Therefore, only a small part of the total estuarine area is responsible for the majority of the water exchange across salinity classes.

Last, various potential mechanisms exist that adjust volume- or isohaline surface-integrated mixing to increased runoff inside an estuary. Namely, isohaline surfaces can be increased by moving them to areas with larger cross sections or entirely out of the river, or local mixing can be increased due to higher eddy diffusivities (Burchard 2020). In our study, the strongly increased isohaline area during high runoff in combination with increased local mixing intensity indicates that both mechanisms may be of importance.

4. Summary and conclusions

In this study, we have analyzed the impact of extreme discharge conditions on estuarine circulation, diahaline exchange flow, and salt mixing inside the Elbe estuary. Relatively short averaging periods from two M2 tidal cycles up to two spring–neap cycles allowed the comparison of dynamics during a period of low and extremely high runoff, respectively. It was shown that the simulated mixing converges toward the estimate by Burchard (2020) with the length of the averaging period. An averaging period of two spring–neap cycles was found to properly reproduce the total exchange flow (TEF) as well as the mixing estimates from MacCready et al. (2018) and Burchard (2020). At the same time, each extreme event was still clearly represented in the resulting parameters. With our results, we have demonstrated that the estuarine system of the Elbe River adjusts relatively fast to changes of the freshwater discharge. Consequently, the TEF framework and the mixing estimates from MacCready et al. (2018) and Burchard (2020) can be suitable tools for the study of estuarine dynamics even under extreme discharge conditions. In addition to earlier findings by Voynova et al. (2017), Kerimoglu et al. (2020), and Chegini et al. (2020) for the German Bight, we have further found that the high-runoff event in June also led to increased stratification within the Elbe estuary itself. We have shown that the estuarine circulation within the Elbe estuary consists of a classic two-layer flow with high-salinity inflow and low-salinity outflow. All parameters of the isohaline exchange flow, i.e., the TEF, were found to be freshened during the high-runoff event in June 2013 compared to the low-runoff period of September 2012. As a result, salt intrusion varied strongly between low- and high-runoff periods. As demonstrated before by Broatch and MacCready (2022) for the Salish Sea, we have found here that freshwater runoff strongly drives mixing in the Elbe estuary. For periods of lower runoff, we could additionally identify a potential weak modulation of mixing by the spring–neap cycle which gained relevance in a comparative scenario where horizontal diffusivity was reduced. The diahaline exchange flow, quantified by the effective vertical diahaline velocity udia,zS, was shown here for the first time in a spatially resolved manner and under realistic conditions for a tidal estuary. It is well defined within the Elbe estuary and consists of a near-surface outflow near the downstream end of the isohaline surface and a near-bottom inflow near the upstream end. The spatial distribution of the diahaline exchange flow follows the salinity gradient of local mixing, as given by the estimate from Klingbeil and Henell (2023). It could therefore be concluded that horizontal diffusion is of minor importance for mixing in the Elbe estuary, even under extreme discharge conditions. This means that the diahaline exchange flow is driven by vertical rather than horizontal mixing. Our results further show that due to its relation to the local mixing, the majority of the diahaline exchange flow is confined to regions of strong topography gradients. This provides a first insight into potential mechanisms behind the spatial structure of the diahaline exchange flow and estuarine salt mixing inside mesotidal estuaries, which are possibly related to the mixing due to the build-up of salinity fronts described by Warner et al. (2020). As a next step, finding a link between the diahaline exchange flow and the (isohaline) total exchange flow could directly quantify the connection between salt mixing and estuarine circulation.

Last, with the expected increased frequency of both high- and low-runoff events (see, e.g., Christensen and Christensen 2004; van Vliet et al. 2013), we conclude that typical mesotidal estuaries such as the tidal Elbe will likely be subject to more frequent changes between periods of freshening and temporary stratification and well-mixed periods of high salt intrusion. It might be of interest for future studies to investigate the impact that such changes may have on the estuarine sediment dynamics and ecosystem.

Acknowledgments.

We thank the employees at the Bundesanstalt für Wasserbau (BAW) for their aid in the creation of our setup, in particular Julia Benndorf, Jessica Kelln, Frank Kösters, and Marissa Törber. Hans Burchard and Lloyd Reese were supported by the Research Training Group Baltic TRANSCOAST GRK 2000, funded by the German Research Foundation. The work of Knut Klingbeil is a contribution to the project M5 of the Collaborative Research Centre TRR 181 Energy Transfers in Atmosphere and Ocean (project 274762653), funded by the German Research Foundation (DFG). Marvin Lorenz was supported by ECAS-BALTIC funded by the German Federal Ministry of Education and Research (BMBF, funding code 03F0860H). Xiangyu Li was supported by the project Processes Impacting on Estuarine Turbidity Zones in tidal estuaries (PIETZ) funded by the German Research Foundation under the Grant BU 1199/24-1. We thank two anonymous reviewers for their valuable comments that have greatly improved the contents and quality of this manuscript.

Data availability statement.

The GETM source code used for this work is archived on Zenodo (https://doi.org/10.5281/zenodo.7741730). Boundary forcing as well as initial salt and temperature distributions was retrieved from the numerical setup presented in Gräwe et al. (2015). Meteorological forcing was provided from the ICON model by the German Weather Service (Zängl et al. 2015), which cannot be made public due to its proprietary nature. Therefore, the numerical setup cannot be made public in a fully functional state. Instead, the full tidal Elbe setup used in this study, including forcing and initial distributions, is available on request via the corresponding author of this study. The numerical data from the tidal Elbe setup that have been used for this study are archived at https://doi.org/10.5281/zenodo.10230005 and https://doi.org/10.5281/zenodo.10230235. The setup topography as well as freshwater runoff of the main river and tributaries as used in this study is available through https://doi.org/10.5281/zenodo.10222958. For rivers where daily-averaged data were available, the data were retrieved from the following public domain resources: http://www.portal-tideelbe.de (Wasserstraßen- und Schifffahrtsamt Magdeburg 2021), https://hsi-sh.de/pegelsuche.html (Landesamt für Landwirtschaft, Umwelt und ländliche Räume Schleswig-Holstein 2022), and http://www.wasserdaten.niedersachsen.de/cadenza/pages/home/welcome.xhtml (Niedersächsicher Landesbetrieb für Wasserwirtschaft, Küsten-und Naturschutz 2022). Constant and monthly averaged freshwater runoff data from the remaining tributaries as well as weighting factors for gauge data were provided by the Bundesanstalt für Wasserbau via personal communication. Observational data used for the calibration and validation of the numerical setup are freely available through http://www.portal-tideelbe.de (Wasserstraßen- und Schifffahrtsamt Elbe-Nordsee and Hamburg Port Authority 2023; Wasserstraßen- und Schifffahrtsamt Elbe-Nordsee 2023). For the creation of the numerical grid of our setup, the Delft3D rgfgrid GUI (Deltares 2020) was used. The software is available through https://oss.deltares.nl/web/delft3d/home and requires a license that can be freely requested on the same website. All figures in this study have been created with Matplotlib version 3.3.4 (Hunter 2007; Caswell et al. 2021), freely available through https://matplotlib.org. The respective Python scripts are available through https://doi.org/10.5281/zenodo.10222958. The map in Fig. 2b was created with cartopy v0.17.0 (Met Office 2015; Elson et al. 2018), available through http://scitools.org.uk/cartopy, and using the NaturalEarthFeature function (made with Natural Earth. Free vector and raster map data at naturalearthdata.com). A tidal analysis was performed using the pytides package (Cox 2018), available through https://github.com/sahitono/pytides. TEF bulk values were computed with pyTEF (Lorenz and Boergel 2021), available through https://florianboergel.github.io/pyTEF.

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Supplementary Materials

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