Editorial Type:
Article
Article Type:
Research Article

The Relation between Exchange Flow and Diahaline Mixing in Estuaries

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

Search for other papers by Hans Burchard in
Current site
Google Scholar
PubMed Close
https://orcid.org/0000-0001-8288-3932
,
Knut Klingbeil Leibniz Institute for Baltic Sea Research Warnemünde, Rostock, Germany

Search for other papers by Knut Klingbeil in
Current site
Google Scholar
PubMed Close
,
Xaver Lange Leibniz Institute for Baltic Sea Research Warnemünde, Rostock, Germany

Search for other papers by Xaver Lange in
Current site
Google Scholar
PubMed Close
,
Xiangyu Li Leibniz Institute for Baltic Sea Research Warnemünde, Rostock, Germany

Search for other papers by Xiangyu Li in
Current site
Google Scholar
PubMed Close
,
Marvin Lorenz Leibniz Institute for Baltic Sea Research Warnemünde, Rostock, Germany

Search for other papers by Marvin Lorenz in
Current site
Google Scholar
PubMed Close
,
Parker MacCready School of Oceanography, University of Washington, Seattle, Washington

Search for other papers by Parker MacCready in
Current site
Google Scholar
PubMed Close
, and
Lloyd Reese Leibniz Institute for Baltic Sea Research Warnemünde, Rostock, Germany

Search for other papers by Lloyd Reese in
Current site
Google Scholar
PubMed Close
Online Publication:
19 Mar 2025
Print Publication:
01 Mar 2025
DOI:
https://doi.org/10.1175/JPO-D-24-0105.1
Page(s):
243–256
Full access

Abstract

Estuaries with excess freshwater discharge are characterized by inflow of saline water from the ocean and outflow of brackish water toward the ocean. The intensity of this so-called estuarine circulation is known to be connected to salinity mixing inside the estuary. Here, the water mass transformation process of mixing is conveniently defined as the destruction of salinity variance within the estuarine volume. In the present study, we derive a new relation that shows how the estuarine exchange flow and the mixing inside the estuary determine each other. The result is a quantitative relation between the inflow volume at a fixed transect across the estuary and the distribution of the diahaline mixing inside the estuary. The concept of dividing salinities between inflowing and outflowing transports is used here. More specifically, the estuarine exchange flow equals the derivative of the mixing (with respect to salinity) on the estuarine part of the dividing salinity isohaline. We show for two realistic example estuaries how the exchange flow and the mixing are spatially distributed on the dividing salinity isohaline. Those estuaries include a channelized tidal estuary and a nontidal industrialized estuary. While the former shows a basic two-layer division between inflow and outflow, the latter has a more complex three-layer structure, due to offshore upwelling and downwelling dynamics. The new theory provides the recipe for analyzing the relation between mixing and exchange flow in any estuary and helps to identify the regions where the mixing occurs that is consistent with the exchange flow.

Significance Statement

Estuaries are coastal water bodies connecting rivers to the ocean by mixing riverine freshwater with inflowing saline ocean water such that outflowing brackish water is generated. This is the key water mass transformation process that characterizes estuaries. To provide the upper parts of estuaries with salt from the ocean, a landward salt transport against the river flow is required, the so-called estuarine circulation. Our study derives a time-averaged relation that quantitatively connects the estuarine mixing and the estuarine circulation in a two-way manner as opposed to the common notion of circulation depending on mixing. This relation is demonstrated for numerical models of two estuaries, a tidal estuary with a simple two-layer circulation and a nontidal estuary with a three-layer circulation driven by high temporal variability of offshore salinity.

© 2025 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: Hans Burchard, hans.burchard@io-warnemuende.de

Abstract

Estuaries with excess freshwater discharge are characterized by inflow of saline water from the ocean and outflow of brackish water toward the ocean. The intensity of this so-called estuarine circulation is known to be connected to salinity mixing inside the estuary. Here, the water mass transformation process of mixing is conveniently defined as the destruction of salinity variance within the estuarine volume. In the present study, we derive a new relation that shows how the estuarine exchange flow and the mixing inside the estuary determine each other. The result is a quantitative relation between the inflow volume at a fixed transect across the estuary and the distribution of the diahaline mixing inside the estuary. The concept of dividing salinities between inflowing and outflowing transports is used here. More specifically, the estuarine exchange flow equals the derivative of the mixing (with respect to salinity) on the estuarine part of the dividing salinity isohaline. We show for two realistic example estuaries how the exchange flow and the mixing are spatially distributed on the dividing salinity isohaline. Those estuaries include a channelized tidal estuary and a nontidal industrialized estuary. While the former shows a basic two-layer division between inflow and outflow, the latter has a more complex three-layer structure, due to offshore upwelling and downwelling dynamics. The new theory provides the recipe for analyzing the relation between mixing and exchange flow in any estuary and helps to identify the regions where the mixing occurs that is consistent with the exchange flow.

Significance Statement

Estuaries are coastal water bodies connecting rivers to the ocean by mixing riverine freshwater with inflowing saline ocean water such that outflowing brackish water is generated. This is the key water mass transformation process that characterizes estuaries. To provide the upper parts of estuaries with salt from the ocean, a landward salt transport against the river flow is required, the so-called estuarine circulation. Our study derives a time-averaged relation that quantitatively connects the estuarine mixing and the estuarine circulation in a two-way manner as opposed to the common notion of circulation depending on mixing. This relation is demonstrated for numerical models of two estuaries, a tidal estuary with a simple two-layer circulation and a nontidal estuary with a three-layer circulation driven by high temporal variability of offshore salinity.

© 2025 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: Hans Burchard, hans.burchard@io-warnemuende.de
Keywords: Coastal flows; Diapycnal mixing; Mixing; Ocean circulation; Turbulence; Upwelling/downwelling

1. Introduction

This study connects two concepts of estuarine dynamics quantitatively: (i) the exchange flow through a fixed transect across the estuary and (ii) the exchange flow across and mixing on an isohaline surface. Both concepts are based on temporal averaging of the estuarine dynamics. Often, the averaging period is chosen long enough to allow neglecting storage effects in the sense that the average temporal changes of volume and salt content in the estuary are sufficiently small. For tidal flow, averaging over one spring–neap cycle is often sufficient to neglect the storage terms (e.g., Li et al. 2022). For convenience, within this study, the storage terms are neglected by analyzing long-term averaged conditions, but for completeness, all formulations including storage terms are given in the appendix. Nonlinear effects of temporal variability (e.g., spring–neap cycle, variations in runoff, changes in offshore conditions; MacCready et al. 2018; Geyer et al. 2020; Lange et al. 2020; Reese et al. 2024; MacCready and Geyer 2024) are included but not explicitly investigated here. The two estuarine concepts are briefly reviewed here:

  1. (i) Exchange flow across fixed estuarine transects has already been described by Knudsen (1900), see also a review and a translation of Knudsen’s paper (Burchard et al. 2018b). Volume and salt conservation for long-term averaged estuarine conditions can be formulated as
    Qin+Qout+Qr=0,Qinsin+Qoutsout=0,

with the inflow volume transport Qin ≥ 0 across a fixed transect at saltwater inflow salinity sin, the outflow transport volume Qout ≤ 0 at brackish water outflow salinity sout, and the freshwater river discharge Qr ≥ 0 into the estuarine volume (assuming to be of zero salinity), see also the sketch in Fig. 1b. The Knudsen relations can also be formulated including storage terms for water and salt (Burchard et al. 2018b). With this, an estuary can be regarded as a mixing machine (Wang et al. 2017) that mixes fresh riverine and salty ocean water to generate brackish estuarine water. The fundamental function of estuarine mixing has already been mentioned by Knudsen (1900) who stated, As the freshwater spreads out over the seawater, it mixes with it so that the salinity of the surface increases seawards [translation by Burchard et al. (2018b)]. As a measure for estuarine mixing, the decay rate of salinity variance per unit volume χs = 2Kh(∂s/∂x)2 + 2Kh(∂s/∂y)2 + 2Kυ(∂s/∂z)2 had been proposed (Burchard and Rennau 2008, with Kh and Kυ being the horizontal and the vertical eddy diffusivity, respectively). In numerical models, numerical mixing due to truncation errors in the tracer advection schemes occurs in addition to this physical mixing. It can be exactly quantified using the methods by Burchard and Rennau (2008) or Klingbeil et al. (2014). Mixing inside estuaries bounded by a fixed transect could be quantified by MacCready et al. (2018) as
M=sinsoutQr=(sin−sout)sinQin,
where M=⟨∫VχsdV⟩ is the long-term averaged decay rate of salinity variance inside the estuarine volume V (with 〈⋅〉 denoting time averaging). The last equality in (2) is using (1). A mixing formulation including storage terms can be derived (Burchard et al. 2019).
Fig. 1.
Fig. 1.

Sketch explaining the connection between exchange flow and diahaline mixing. (a) Dependence of the diahaline volume transport per unit horizontal area udia,z on the divergence of horizontal transport Ux, for simplicity only shown for the x direction, (b) two-layer flow in x–z coordinates, (c) two-layer flow in x–S coordinates, and (d) three-layer flow in x–z coordinates. The inner part of the isohalines is shown as bold red lines, and the outer part is shown as bold blue lines. Inflowing transports (>0) are shown as red arrows, and outflowing transports (<0) are shown as blue arrows. In (b)–(d), the water volume is marked by light blue shading.

Citation: Journal of Physical Oceanography 55, 3; 10.1175/JPO-D-24-0105.1

For the present study, it is convenient defining Qin as a measure for estuarine exchange flow intensity (Broatch and MacCready 2022; MacCready and Geyer 2024). With this, using (2), zero estuarine mixing M is consistent with zero exchange flow Qin, which can be referred to as the extreme fjord case, where the riverine water entering the fjord is leaving the fjord as freshwater toward the ocean (sout = 0). For given values of sin and Qr, the first part of (2) shows that sout → sin would maximize the mixing M≤sin2Qr, while the second part indicates that in that limiting case, Qin ≫ Qr is possible. An example of this is low river discharge strong-tide basins in the Wadden Sea of the German Bight (Gräwe et al. 2016). A comparable exchange flow estimate had been introduced as the average of the inflow and outflow transports (Geyer et al. 2020; MacCready et al. 2021): Qe=(1/2)(Qin−Qout)=Qin+(1/2)Qr, where the latter equality is using (1). By neglecting the influence of Qr on Qe, again Qin is obtained as a measure for the exchange flow.

To consistently calculate the inflow and outflow volume transports and salinities, MacCready (2011) proposed the total exchange flow (TEF) framework, which resolves inflows and outflows as a function of salinity class. Note that Walin (1977) already laid the foundations for TEF in his Eq. (2). For simple exchange flow situations where inflows occur at higher salinities than outflows, a unique dividing salinity sdiv exists that separates inflows and outflows in salinity space (Lorenz et al. 2019). More complex exchange flows with multiple inflows or outflows consequently have more dividing salinities.

  1. (ii) The concept of analyzing estuarine dynamics on isohaline surfaces has been introduced in the visionary paper by Walin (1977). Similar to the present study, he considered volume and salt conservation below an isohaline surface of a long-term averaged estuary bounded by a fixed transect.

The major relation derived by Walin (1977) is (using our notation)
Q(S,x)=Qdia(S,x)=−∂Jest(S,x)∂S,
where Q(S, x) is the volume transport into the estuary across this fixed transect at position x for all salinities larger than S, Qdia(S, x) is the diahaline transport through the isohaline surface of salinity S that is situated on the estuarine side from the transect, and Jest(S, x) is the upward diffusive salt transport across this part of the isohaline. Walin (1977) stated about this relation [his Eq. (7)]: Eq. (3) represents in the most simple way how the deep water supply is related to the overall vertical (i.e., cross isohaline) mixing properties in the basin. The key relation of our study will be similar to (3) for the special case of S = sdiv, giving
Q[S=sdiv(x),x]=Qin(x).
Based on the pioneering work of Walin (1977), a water mass transformation (WMT) framework for mostly larger oceanic scales has been constructed by several authors (see, e.g., Zika et al. 2012; Hieronymus et al. 2014; Pemberton et al. 2015; Groeskamp et al. 2019).
The nucleus of the estuarine mixing theory presented here is the local relation between the diahaline volume transport per unit horizontal area udia,z(S, x, y), see Fig. 1a, and the local mixing per salinity class m(S, x, y):
udia,z=12∂m∂S,
which establishes the dependence of residual estuarine circulation on estuarine mixing on the local scale (Klingbeil and Henell 2023). The diahaline volume transport per unit horizontal area udia,z(S, x, y) is defined as the time-averaged local contribution to the diahaline transport Qdia(S,x)=−∫Aest(S,x)udia,z(S,x,y)dA, where Aest(S, x) is the horizontal area covered by the isohaline inside the estuary (Fig. 1c). Similarly, the local mixing per salinity class m(S, x, y) is defined as the local distribution of the mixing per salinity class: 𝕞est(S,x)=−∂⟨∫Vest(S,x)χs(x,y,z,t)dV⟩/∂S=∫Aest(S,x)m(S,x,y)dA, with the estuarine volume Vest(S, x) for all salinities s ≥ S (see Figs. 1b,c and Klingbeil and Henell 2023, for details).
For a stable and monotone salinity stratification, the relation (5) can be directly derived from the one-dimensional vertical salinity equation:
∂s∂t+w∂s∂z−∂∂z(Kυ∂s∂z)=0,
with the physical vertical velocity w by transformation into salinity coordinates. Under those conditions, after time averaging,
udia,z(S)=−⟨∂z∂S∂S∂t+w⟩=−∂∂S⟨Kυ∂S∂z⟩=12∂m∂S
results (Wang et al. 2017; Klingbeil and Henell 2023). In this monotone case and for long-term averages, udia,z can be simply calculated from the divergence of the time-averaged vertically integrated transport under the isohaline, see Fig. 1a for an explanatory sketch and (A1). For a rigorous derivation for the general three-dimensional case with nonmonotone salinity distribution in three dimensions, see Klingbeil and Henell (2023). Note that a relation similar to (5) has been derived (McDougall 1984; Marshall et al. 1999) and has been used in several ocean modeling studies (see, e.g., Ferrari et al. 2016; McDougall and Ferrari 2017). It can be shown that the mixing per salinity class equals twice the upward diffusive salt transport across the isohaline (Burchard et al. 2021):
𝕞est(S)=2Jest(S).
The isohaline framework has been used by several authors to investigate mixing and exchange flow processes in tidal estuaries, such as the role of turbulent diahaline salt fluxes for maintaining estuarine salt intrusions (MacCready and Geyer 2001) and the role of advective and diffusive diahaline transports for explaining the long-term isohaline structure of tidal estuaries (MacCready et al. 2002) and turbulent mixing in river plumes (Hetland 2005). Based on the work of Walin (1977), Burchard (2020) derived a universal law of estuarine mixing, postulating that the long-term averaged mixing per salinity class is twice the salinity of the respective salinity class times the river runoff. This relation has been reproduced by a number of numerical model experiments in several tidal estuaries, showing that typically monthly averaging is sufficient to reproduce the law (Burchard et al. 2021; Li et al. 2022; Reese et al. 2024).

The local relation (5) between diahaline exchange flow and the derivative of mixing with respect to salinity has been investigated for estuarine segments by Wang et al. (2017). Two-dimensional fields of diahaline exchange flow and local diahaline mixing have been computed (Li et al. 2022; Henell et al. 2023; Reese et al. 2024). Across a given isohaline surface, water generally leaves an estuarine volume further downstream the estuary near the surface and enters further upstream close to the bottom. Large differences occur between different estuaries.

The research gap that we intend to fill with the present theory is to derive relations which explicitly equate exchange flow and mixing in estuarine systems and show their applicability to different estuarine systems. Such a framework can then be used to identify mixing hotspots that are consistent with the estuarine circulation and to further investigate those critical regions and processes that explain the water mass transformations and overturning circulation in estuarine systems. The mixing relation (2) by MacCready et al. (2018) has been an important step toward such a quantification, but it still contains the bulk inflowing and outflowing TEF salinities as parameters. We are here combining the mixing and exchange flow concepts for fixed transects and specified isohalines laid out above in (i) and (ii) to derive such a relation.

The paper is organized as follows: After this introduction, the numerical model applied for this study is described (section 2a) and the estuarine model setups are explained (section 2b). Afterward, the key relations between mixing and exchange flow, (9) for estuaries with two-layer exchange flow and (10) for estuaries with three-layer exchange flow, are presented in section 3. The accuracy of these relations is then demonstrated for the two-layered tidal Elbe estuary (section 3a) and the three-layered nontidal Warnow estuary (section 3b). Implications of the results are discussed in section 4. A detailed derivation of (9) and (10) including storage terms can be found in the appendix at the end of the paper. The appendix also contains a table (Table A1) with all variables used throughout the paper.

2. Methods

a. Numerical model

For all simulations, the General Estuarine Transport Model (GETM, www.getm.eu; Burchard and Bolding 2002) is applied, which is a finite-volume, structured-grid model with explicit mode splitting and vertically adaptive coordinates (Hofmeister et al. 2010; Gräwe et al. 2015a; Klingbeil et al. 2018). Vertical turbulent mixing which plays an essential role in this study is parameterized by means of a k–ε turbulence closure model with an algebraic second-moment closure (Umlauf and Burchard 2005), using the turbulence module of the General Ocean Turbulence Model (GOTM, www.gotm.net; Burchard et al. 1999; Umlauf et al. 2005). The properties on isohaline surfaces are diagnosed by a binning in salinity classes using a prescribed constant ΔS (see Burchard et al. 2021, for details).

b. Estuarine model setups

The model simulations for the two estuaries are using realistic bathymetry, lateral boundary conditions, atmospheric forcing, and river runoff. For both model simulations, surface freshwater fluxes due to precipitation and evaporation were applied but did not play a significant role in the freshwater budget of the estuaries. Both temperature and salinity fields were calculated in these simulations.

For the Elbe estuary simulations, a bathymetry has been used that is representative for the year 2010, and since no major dredgings have been carried out in the three subsequent years, it is also representative for the analyzed months September 2012 and June 2013. These bathymetry data have then been interpolated onto a curvilinear grid that is following the navigational channel, resulting in a horizontal resolution of 200–400 m in along-channel direction and 50–100 m in cross-channel direction. In the vertical, 30 equidistant σ levels were used. Lateral boundary conditions have been extracted from a 600-m resolution simulation of the southern North Sea (Gräwe et al. 2015a). Atmospheric forcing has been provided by the German Weather Service with a horizontal resolution of about 7 km and a temporal resolution of 3 h. For further details of the Elbe estuary model setup, see Reese et al. (2024).

For the Warnow estuary simulations, a Cartesian grid with a constant resolution of 20 m × 20 m has been used. Lateral boundary conditions are downscaled from a 600-m resolution model of the western Baltic Sea (Gräwe et al. 2015a) via a 200-m resolution intermediate model. In the vertical, 25 equidistant σ levels were used. For the atmospheric forcing, the same data source as for the Elbe model was used. The model simulations were initialized at the beginning of 2013, such that sufficient spin-off time was provided for this analysis of the year 2014. Daily averaged river runoff from the Warnow River in the south was the only lateral freshwater source to the model. For further details of the Warnow estuary model setup, see Lange et al. (2020).

3. Results

The key relation that is derived for this study is a direct quantitative relation between the inflow transport Qin across a fixed cross section and the S derivative of the diahaline mixing integrated over the horizontal area covered by the estuarine part of the dividing salinity isohaline 𝕞est(sdiv):
Qin(x)=−12∂𝕞est[sdiv(x),x]∂S,
see Fig. 1b (in vertical Cartesian coordinates) and Fig. 1c (in salinity coordinates). Equation (9) is directly obtained by inserting S = sdiv into (3), derived by Walin (1977) and combining it with (8). Equation (9) can also be derived by lateral integration of the local mixing relation (5) over the area of the dividing salinity isohaline that is located upstream of the transect, see Figs. 1b and 1c and in the appendix, where (A14) shows a version of the key relation (9) including the storage term.
With (9), we have derived for the first time a direct relation between estuarine mixing and exchange flow intensity for two-layer estuaries. This convenient relation can easily be extended to estuaries with a multilayer flow through multiple channels. In some situations, substantial deviations from the two-layer Knudsen type of flow shown in Figs. 1b and 1c can occur (Lorenz et al. 2019). In Fig. 1d, a situation is shown for a three-layer exchange flow with inflow at high and low salinities Qin,high and Qin,low, respectively, and one outflow Qout at intermediate salinities. Now, two dividing salinities occur sdiv,high and sdiv,low. The transports across the isohalines of these dividing salinities on the estuarine side are given by Qdia,high = Qin,high and Qdia,low = −(Qin,low + Qr), respectively. The mixing integrated over the estuarine parts of each of these isohalines are denoted as 𝕞est,high=𝕞est(sdiv,high) and 𝕞est,low=𝕞est(sdiv,low). Given the previous relation (9) and assuming the geometry as shown in Fig. 1d as well as long-term averaging (negligible storage terms), we obtain the following relations between exchange flow and mixing for the three-layer flow:
Qin,high=−12∂𝕞est,high∂S,−(Qin,low+Qr)=−12∂𝕞est,low∂S,
where we dropped the explicit dependence on x for convenience. Based on this geometrical principle, relations between exchange flow and mixing can be constructed also for situations with more than three layers. A version of (10) including storage terms is given in (A15).

In the following sections, we explore high-resolution model results for a tidal estuary with two-layer exchange flow (Elbe estuary; see section 3a) and a nontidal estuary with three-layer exchange flow (Warnow estuary; see section 3b). The locations and bathymetries of the estuaries are shown in Fig. 2. For all simulations, the GETM (getm.eu; Burchard and Bolding 2002) is applied. The simulations are realistic and validated, using both salinity and temperature as active tracers (see details in section 2).

Fig. 2.
Fig. 2.

Locations and bathymetries of the two estuaries investigated in the present study. (a) Map of Europe with an outline of the map area of (b) indicated; (b) Map of northern Germany with the complete model domains for the two estuaries; white lines indicate the open boundaries; (c) Bathymetry of the lower Elbe estuary, with a white line marking the open boundary and a red line marking the Cuxhaven transect; and (d) Bathymetry of the Warnow estuary, with a red line marking the Warnow transect.

Citation: Journal of Physical Oceanography 55, 3; 10.1175/JPO-D-24-0105.1

a. Tidal two-layer estuary

The Elbe estuary is investigated here as a typical two-layer estuary with inflow at high salinities and outflow at low salinities (Reese et al. 2024). It is a mesotidal estuary with a pronounced navigational channel (Fig. 2c) and is characterized by a dominant semidiurnal tide and a pronounced spring–neap cycle. The tidal influence has an upstream limit that is given by a weir in Geesthacht located about 150 km upstream of the mouth. The salt intrusion is highly dependent on the river discharge which can vary by more than one order of magnitude between <100 and >2000 m3 s−1 (Kappenberg and Grabemann 2001). At the upstream limit of the salt intrusion, a pronounced estuarine turbidity maximum typically occurs (Postma and Kalle 1955; Kappenberg et al. 1995; Burchard et al. 2004, 2018a). The mixing in the Elbe estuary is dominated by the strongly variable river runoff rather than by the spring–neap cycle (Reese et al. 2024).

With its elongated structure, the Elbe estuary is highly suitable for an analysis of mixing and exchange flow at all cross sections along its longitudinal axis. The results for the Elbe estuary have been extracted from the model study by Reese et al. (2024) who carried out numerical simulations with a curvilinear grid that follows the navigational channel. Results for the two-layer mixing relation (9) along the estuary and mixing and exchange flow analysis with respect to a transect at the mouth of the estuary (Cuxhaven transect) are shown in Fig. 3 with averaging intervals ΔT being two spring–neap cycles for comparably low (Qr ≈ 350 m3 s−1 in September 2012) and exceptionally high (Qr ≈ 2500 m3 s−1 in June 2013) monthly mean river runoff. For both months, the numerically calculated budget of (9) is largely closed along the range of more than 70 km along the estuary (Figs. 3a,b). For the low-runoff month (Fig. 3a), the storage term makes a small contribution to the budget at some locations, but for the high run-off (Fig. 3b), the storage term is negligible all over the estuary. For both months, the estuarine circulation across the Cuxhaven transect is a classical two-layer circulation, as shown by the TEF profiles in Figs. 3c and 3d. The dividing salinity is much higher for the low-runoff month (sdiv = 21.0 g kg−1) than for the high-runoff month (sdiv = 12.6 g kg−1) because the salt intrusion has propagated farther into the estuary for low runoff. Interestingly, the exchange flow intensity across the Cuxhaven transect as quantified by Qin is slightly higher for the low runoff (Qin = 1354 m3 s−1) than for high runoff (Qin = 1179 m3 s−1), probably due to the much lower runoff velocity in September 2012. During high discharge, the entire circulation cell is shifted downstream, such that in June 2013, the position of the transect relative to the along-channel development of the circulation is further upstream as in September 2012. There, the salt intrusion reaches farther upstream such that the circulation is already quite strong in the region of the transect. In June 2013, however, the transect is close to the end of the salt intrusion such that the estuarine circulation is not fully developed. For both months, the distribution of the local mixing per salinity class m is spread over the entire area covered by the isohaline of the dividing salinity, but highest values are above steep bathymetry at the edges of the navigational channel (Figs. 3e,f). Generally, the level of mixing is higher for the high-runoff month June 2013, which is in accordance with the universal law of estuarine mixing (Burchard 2020), as shown for this Elbe simulation (Reese et al. 2024). Seemingly, the Cuxhaven transect divides the udia,z distribution into inflow toward lower salinities near the bottom (−udia,z > 0) landwards of the transect and outflow toward higher salinities near the surface (−udia,z < 0) seawards of the transect (Figs. 3g,h). This can be explained by the fact that cross-transect inflows occur for salinities higher than the dividing salinity, and therefore, inflows should dominate the estuarine part of the dividing salinity isohaline as sketched in Figs. 1b and 1c.

Fig. 3.
Fig. 3.

Mixing and exchange flow results for the Elbe estuary for (a),(c),(e),(g) low (September 2012) and (b),(d),(f),(h) high (July 2013) runoff situations. The average discharge Qr for each month is given in the axis titles. (a),(b) Left-hand side (black line) and right-hand side (solid gray line) of the mixing–exchange flow relation, see (9), taken along the Elbe estuary; dashed gray lines represent the right-hand side of relation (9) corrected by the storage term ΔVest/ΔT (ΔT denoting the averaging period); the Cuxhaven transect [see (e)–(h) for its location] at 139 km is indicated as dashed red line; (c),(d) TEF q(S) = −∂Q/∂S across the Cuxhaven transect; the amounts of Qin (red) and Qout (blue) are indicated as numbers and graphically as shaded areas. The dividing salinity is marked as a thin dashed line; (e),(f) diahaline mixing per salinity class, m on the sdiv isohaline of the Cuxhaven transect (bold red line); the mean relative isohaline heights above ground of σ = −0.8 (solid), σ = −0.5 (dash–dotted), and σ = −0.2 (dotted); (g),(h) as in (e) and (f), but for the diahaline volume transport per unit horizontal area udia,z across the sdiv isohaline of the Cuxhaven transect; incoming (positive) transports are marked in red, and outflowing (negative) transports are marked in blue.

Citation: Journal of Physical Oceanography 55, 3; 10.1175/JPO-D-24-0105.1

b. Nontidal three-layer estuary

The Warnow estuary is a small industrialized estuary of the essentially nontidal and brackish western Baltic Sea (Fig. 2d). The Baltic Sea itself is strongly salinity stratified all year around, such that its coasts are characterized by regular wind-driven upwelling events during all seasons (Lehmann and Myrberg 2008). Since the western Baltic Sea is strongly stratified in salinity, wind-driven upwelling (during easterly winds) and downwelling (during westerly winds) dynamics drive a high variability of surface salinity at its southern coast (Fennel and Sturm 1992). At the mouth of the Warnow estuary, salinity ranges between 10 g kg−1 during downwelling and 20 g kg−1 during upwelling. In addition, upwelling waters can be 5°–10°C colder than the downwelling waters. Therefore, during upwelling, the salinity outside the estuary is higher than inside, driving an estuarine circulation with inflows at high salinities and outflows at low salinities. During downwelling, the salinity outside the estuary can drop by more than 10 g kg−1 within less than 1 day to values below inside the estuary, supported by a temperature increase, such that a density-driven antiestuarine circulation is occurring (Lange et al. 2020). In addition, during the so-called major Baltic inflows (MBIs), huge amounts of high-salinity waters pass through the western Baltic Sea (Gräwe et al. 2015b), such that during the MBI occurring at the end of 2014, salinity in the Warnow estuary increased considerably (Lange et al. 2020).

Based on a TEF analysis of the entire year 2014 using the simulations by Lange et al. (2020), the three-layer circulation of the Warnow can be clearly seen in Fig. 4a. The upper circulation cell for low salinities is formed during downwelling, and the lower cell with high salinities is formed during upwelling (Fig. 4b). The two dividing salinities are sdiv,high = 16.8 g kg−1 and sdiv,low = 11.2 g kg−1. Other than in the Elbe scenario (section 3a), the isohalines of the dividing salinities extend far beyond the limits of the model domain (Figs. 4c,d). On both isohalines, there is considerable mixing also visible in the western Baltic Sea adjacent to the Warnow estuary. In fact, the isohaline of sdiv,low covers large parts of the central Baltic Sea (Wüst 1957; Henell et al. 2023). The spatial distribution of the local mixing per salinity class m inside the estuary is clearly focused above steep topography along the edge of the dredged navigational channel. For the sdiv,low isohaline, mixing levels are generally higher. The spatial distribution of udia,z is clearly distinct across the two isohalines (Figs. 4e,f). For the sdiv,high isohaline, inflow dominates, such that −udia,z > 0 in most locations, meaning that volume flows into the estuary from high to low salinities (Fig. 4f). For the sdiv,low isohaline, a diahaline exchange flow is visible (Fig. 4e) inside the estuary. At the southern part of the estuary, inflow toward low salinities dominates, and at the northward part, outflow toward high salinities dominates.

Fig. 4.
Fig. 4.

Mixing and exchange flow results for the Warnow estuary during the year 2014. (a) TEF across the Warnow transect shown in (c)–(f) averaged over the entire year; (b) temporally resolved TEF; local mixing per salinity class m on the dividing salinity isohalines for (c) sdiv,low and (d) sdiv,high; and diahaline volume transport per unit horizontal area udia,z on the dividing salinity isohalines for (e) sdiv,low and (f) sdiv,high. In (a) and (b), the horizontal lines indicate the two dividing salinities.

Citation: Journal of Physical Oceanography 55, 3; 10.1175/JPO-D-24-0105.1

Despite the complex patterns of mixing, inflow, and outflow inside the estuary, the numerical calculation of the transport budgets according to (10) averaged over the year 2014 shows that it is nearly closed, see Table 1. The difference between the discrete evaluations of the left-hand side and the right-hand side is about 3% for the high-salinity isohaline of sdiv,high and about 2% for the isohaline of sdiv,low. Calculations show that storage terms are largely negligible due to the long averaging interval of 1 year.

Table 1.

Volume transports across the Warnow transect shown in Figs. 4c–f (m3 s−1) through interfaces for the three-layer estuarine circulation in the Warnow estuary averaged over the year 2014. The numbers in columns 2 and 3 reflect the left-hand side and those in column 4 reflect the right-hand side of (10). The numbers in column 5 give the difference between the left-hand and the right-hand sides of this equation as the numerical calculation error. The second and third rows show the values for high and low dividing salinity isohalines, respectively.
Table 1.

4. Discussion and conclusions

The key result of this study is the first derivation of an exact quantitative relation between exchange flow and mixing in estuaries, highlighting how essential water mass transformations are for understanding estuaries. Estuarine exchange flow is defined here relative to fixed transects across the estuary. Since the generally continuous transition between estuary and river plume prohibits the specification of a distinct seaward limit of an estuary, the theory is formulated for an arbitrary estuarine transect. In this formulation, the estuarine circulation in simple estuaries with a two-layer circulation equals the S derivative of the mixing integrated over the dividing salinity isohaline landwards of the transect, as shown in (9). This emphasizes the importance of the dividing salinity sdiv for each transect. So far, sdiv has only been used for the interpretation of total exchange flow profiles and the accurate calculation of Qin and Qout as well as sin and sout (MacCready et al. 2018; Lorenz et al. 2019; Broatch and MacCready 2022). For the mixing–exchange flow relation (9), estuarine circulation needs to be quantified as the inflowing volume transport calculated in salinity coordinates, following the total exchange flow (TEF) analysis framework established by MacCready (2011). Mixing needs to be quantified for the same quantity (salinity) that defines the coordinate of the circulation analysis to obtain this simple key relation (Klingbeil and Henell 2023). Using other coordinate systems as reference for the inflowing volume transport such as Eulerian (e.g., z or σ coordinates) would give different results (MacCready 2011; Burchard et al. 2018b) and violate the relation (9). Such an Eulerian exchange flow analysis is common (Hansen and Rattray 1965; Geyer and MacCready 2014; Lange et al. 2020) but does not establish any relation to estuarine water mass transformation processes. While for the two-layer estuarine exchange flow, mixing gradients are analyzed on the isohaline with the unique dividing salinity, more complicated multilayer exchange flow requires analysis of mixing gradients on multiple dividing isohalines as shown in relation (10).

Going back to the example of the two-layer circulation, it is remarkable that the time-averaged exchange flow across a fixed transect is quantified by the S derivative of mixing on only one single isohaline surface, ignoring the mixing occurring at other salinities in the estuary. However, it could be argued that mixing and other hydrodynamic processes at all salinities inside the estuary (i.e., landward of the transect) contribute to adjusting the time-averaged mixing and position of the dividing salinity isohaline in a way that relation (9) is fulfilled. Similar arguments hold for the more complex multilayer estuaries following relation (10) for three layers or similarly derived multilayer relations.

It is evident in both estuaries that the mixing is substantially elevated over sloping bathymetry along navigational channels. This has been described before for tidal estuaries as the effect of lateral differential advection (Lacy et al. 2003; Geyer et al. 2020): cross-flow lateral shear in combination with an along-flow salinity gradient leads to a cross-flow salinity gradient being much stronger than the along-flow gradient. The cross-flow salinity gradient induces a cross-flow density gradient leading to lateral exchange flow generating a vertical salinity gradient. Along-flow tidally induced turbulence (high Kυ) in conjunction with the vertical salinity gradient over the slopes (high |∂s/∂z|) leads to substantially enhanced mixing χs ≈ 2 Kυ (∂s/∂z)2. In nontidal estuaries such as the Warnow estuary, this process has not yet been investigated, but similar processes might be at work there as well.

Our findings give the picture of a subtle balance between mixing and exchange flow in estuaries. The common notion that estuarine exchange flow is related to mixing (MacCready et al. 2018) which implies a one-way relation should be reformulated to mixing and estuarine exchange flow are in balance for sufficiently long time averaging. The bulk mixing relation (2) derived by MacCready et al. (2018) supports this latter argument but still contains the inflowing and outflowing salinities sin and sout as variables in the mixing M–exchange flow Qin relation. Furthermore, relation (2) is only approximate, whereas relation (9) is exact (for long-term averaging and negligible surface freshwater fluxes).

The examples presented in this paper are limited to two-layer and three-layer estuaries with only one connection to the ocean. In the real world, many other types exist. Estuaries might, for example, have a circulation with more than three layers. By cutting a transect through the central Baltic Sea, Lorenz et al. (2019) obtained an exchange flow with four layers and three dividing salinities that were accurately identifiable using their numerical methods. The S derivatives of mixing can now be quantified on those isohalines and mixing hotspot locations can be identified. A straightforward extension of the three-layer key relation (10) could then be applied to check for consistency with the intensities of the volume transports (i.e., the complex exchange flow) in the four layers.

Many estuarine systems have more than one connection to the ocean such as the Baltic Sea (with the Sound, the Great Belt, and the Little Belt as connections, see, e.g., Burchard et al. 2018b) or the Gulf of St. Lawrence with the major connection through the Cabot Strait and a minor connection through the Belle Isle Strait (Dickie and Trites 1983). Also, those multientrance estuarine systems can be investigated by means of the relations derived here. In a model setup for such a system, transects would be chosen across each of the entrances, individual TEF profiles for each entrance would be calculated, and then these TEF profiles would be added to each other. In a simple case, a two-layer exchange flow would result, such that (9) could be applied for an analysis. However, it would be more likely that the combined TEF profile has more layers, such that the three-layer relation (10) or a derived more complex relation would have to be applied. In cases where one of the entrances dominates (such as in the case of the Gulf of St. Lawrence), however, only a small disturbance of the dominating TEF profile would be expected. In the case of the Baltic Sea, a diahaline exchange flow and a mixing analysis have recently been conducted by Henell et al. (2023) for selected isohalines. It would be straightforward to repeat that analysis for the dividing isohalines identified by a combined TEF analysis for all three entrance channels. The caveat of the Henell et al. (2023) study was, however, the relatively short averaging period of 2 years which is much shorter than the overturning time scale of the Baltic Sea of a decade or more (Reissmann et al. 2009), such that storage terms might dominate the balance.

The required length of the averaging periods to fulfill relations (9) and (10) depends on the time scales of mixing and exchange flow processes in the estuaries under investigation. The monthly averages (essentially two spring–neap cycles) for the Elbe estuary show that for the low-runoff scenario (Fig. 3a), storage terms still contribute slightly to the volume transport budgets at some locations, showing that the adjustment time scales for this case are longer than 1 month. In contrast, in the high-runoff scenario (Fig. 3b), storage terms are sufficiently small.

The time-averaged theory developed here characterizes the average dynamic state of an estuary, as contrasted to a typical state. For example, for the tidal estuary (section 3a), the two-layer flow situation is not present during ebb or flood and is only weakly detectable during short times of flow reversal during slack tides (Schulz et al. 2020). It requires at least tidal averaging to obtain the two-layer exchange flow. Substantial differences in exchange flow occur between neap and spring tides (Chen et al. 2012), as well as between high and low runoff (Figs. 3c,d). The three-layer flow (section 3b) in the nontidal estuary is only occurring for short times in between the upwelling and downwelling dominated periods. Otherwise, there are mainly characteristic estuarine (upwelling) and antiestuarine (downwelling) exchange flow situations present. The analysis method presented here is suitable to analyze these different states of the estuaries (spring, neap, upwelling, downwelling) separately. The more unsteady these states are, the more relevant storage effects would be. Storage terms can easily be added to the analysis method (as indicated for the low runoff case in the tidal estuary, see Fig. 3a), such that it is suitable for in-depth investigations of transient estuarine states.

The theory derived here provides the fundamental recipe for analyzing the relation between mixing and exchange flow in any estuary. It helps to identify the regions where the mixing occurs that is consistent with the exchange flow. Although only the mixing on the dividing salinity isohaline is included in the mixing–exchange flow relation (9), mixing elsewhere in the estuary indirectly influences the distribution of all isohalines and as such feeds back to the mixing of the dividing salinity isohaline.

Generally, observations are intrinsically prone to undersampling, such that mixing or overturning circulation is hard to observe directly. The sampling requirements for TEF profiles directly derived from mooring observations across transects have been presented by Lemagie et al. (2022) for three different tidal estuaries. However, in most estuarine systems, such observations would be too demanding due to the complexity of the estuaries or limitations in equipment and personnel. Mixing observations covering the estuaries would be even more demanding. Instead of purely observational approaches, a numerical model for the estuarine system under investigation needs to be constrained by critical observations, e.g., of water levels, temperature, salinity, and velocity profiles. If available, turbulence and mixing observations should be added. Once the numerical model is sufficiently calibrated against the observations, e.g., by adjusting bed roughness, boundary conditions, and by reducing numerical errors, the model results can be considered as a sufficient representation of reality (in the sense of a digital twin). In that way, it becomes clear that observations are essential for setting up a representative numerical model. Specifically, observations close to the key processes are important such as those representing water mass transformation processes or overturning circulation. Tracer release experiments as they have, for example, been carried out for the Baltic Sea (Holtermann et al. 2012, 2014) or for the Gulf of St. Lawrence (Stevens et al. 2024) are such critical experimental observations that have been successfully reproduced by numerical models, although the reproduction of tracer spreading depends on available numerical resolution and accuracy. Once a sufficient numerical model simulation is available, model results will be analyzed using the new mixing relation to motivate identifying the temporally and spatially varying mixing. In a next step, suitable transects need to be chosen, dividing salinities to be identified and exchange flow and mixing on those critical isohalines to be quantified. Once mixing hotspots are identified, the model results can in turn motivate research cruises to such potentially critical areas to verify the processes causing high mixing rates.

Acknowledgments.

H. Burchard and L. Reese are supported by the Research Training Group GRK 2000 Baltic TRANSCOAST (240942083), funded by the German Research Foundation. The work of X. Lange and X. Li is supported by the projects CoastalFutures (03F0911B) and ElbeXtreme (03F0954F), respectively, both funded by the German Federal Ministry of Research and Education. The work of H. Burchard, K. Klingbeil, and M. Lorenz is a contribution to the Collaborative Research Centre TRR 181 Energy Transfers in Atmosphere and Ocean (274762653), funded by the German Research Foundation. The scientific work is linked to the small strategic institute expansion at the IOW [shore to basin (S2B)]. We are grateful for the constructive in-depth comments of the two reviewers Samuel W. Stevens (The University of British Columbia, Vancouver) and Nick Nidzieko (the University of California, Santa Barbara) on our original manuscript.

Data availability statement.

The model simulations that are analyzed here have been described in detail by Lange et al. (2020) and Reese et al. (2024); see the data availability statements in those publications.

APPENDIX

Calculating Exchange Flow Intensity from Estuarine Mixing

For the present study, we consider the long-term averaged state of an estuarine system which is stably stratified in salinity, i.e., the time-averaged salinity decreases with distance from the bottom. For an arbitrary water column in this system located at a fixed horizontal position (x, y), integration of the continuity equation from the bottom to an isohaline of salinity S reads as (assuming incompressibility)
ΔH(S,x,y)ΔT+∂Ux(S,x,y)∂x+∂Uy(S,x,y)∂y=udia,z,(S,x,y),
with the time-averaged horizontal transports in x and y direction Ux(S, x, y) and Uy(S, x, y), respectively, vertically integrated from the bottom to the isohaline height, the change in isohaline height ΔH(S, x, y) divided by the averaging duration ΔT, and the time-averaged diahaline volume transport per unit horizontal area udia,z(S, x, y) (defined positive downward, toward increasing salinity), see Fig. 1a. We call an estuarine system long-term averaged when ΔH is negligibly small everywhere, i.e., when |ΔH|/ΔT ≪ |∂Ux/∂x + ∂Uy/∂y| for all positions (x, y). Note that the x axis is pointing into the estuary, see Fig. 1.

To develop the present theory, (A1) and (5) will be integrated along the part of isohaline surfaces that is situated on the estuarine side of an arbitrary transect across the estuary.

Lateral integration of (A1) along constant S over constant width W results in
ΔA(S,x)ΔT+∂Q(S,x)∂x−Udia(S,x)=0,
with the accumulated volume transport across the transect at location x for all salinities larger than S,
Q(S,x)=∫WUx(S,x,y)dy,
the laterally integrated diahaline transport across a line with constant x and S,
Udia(S,x)=∫Wudia,z(S,x,y)dy,
and the cross-sectional area for all s ≥ S,
A(S,x)=∫WH(S,x,y)dy.
Note that the assumption of a constant width W is no restriction, since practically W could be taken as the maximum width such that for locations with smaller width, velocities and mixing rates are assumed to be zero outside the estuary (Burchard et al. 2018a). With the volume transport per salinity class across a fixed transect located at position x,
q(S,x)=−∂Q(S,x)∂S,
being the total exchange flow (TEF) as introduced by MacCready (2011), and in (A2), it becomes clear that for long-term averaging (ΔA/ΔT → 0), Q is the streamfunction of the estuarine overturning circulation in (S, x) space (MacCready 2011) established by (−∂Q/∂S, ∂Q/∂x) = (q, Udia).
Integration of (A2) from an arbitrary transect location x < 0 to x = 0 with Q(S, 0) = 0 for all S (freshwater range of the estuary) leads to
Q(S,x)=−∫x0Udia(S,ξ)dξ+ΔVest(S,x)ΔT,
where
Vest(S,x)=∫x0A(S,ξ)dξ
is the volume on the estuarine side of the transect with all s ≥ S.
Note that (A7) is a special case of the derivation by Wang et al. (2017) who carried out the integration between two transects of nonzero salinity. Combining (5), (A4), and (A7) gives
Q(S,x)=−12∂∂S∫x0∫Wm(S,ξ,y)dydξ+ΔVest(S,x)ΔT.
When defining
𝕞est(S,x)=∫x0∫Wm(S,ξ,y)dydξ
as the mixing per salinity class on part of an isohaline with salinity S occurring inside an estuarine volume bounded by a transect fixed at position x, then we obtain
Q(S,x)=−12∂𝕞est(S,x)∂S+ΔVest(S,x)ΔT.
Relation (A11) shows the direct interdependence of the total exchange flow profile and the derivative of the mixing per salinity class in the estuarine volume bounded by the transect, corrected by the storage term. This has indirectly been shown by Wang et al. (2017) in their Figs. 3 and 4.
In an estuarine system where the time-averaged salinity is stably stratified, a single dividing salinity sdiv(x) exists with q > 0 for S > sdiv and q ≤ 0 for S ≤ sdiv, such that we can define
Qin(x)=∫sdiv(x)Smax⁡q(S,x)dS>0,Qout(x)=∫0sdiv(x)q(S,x)dS≤0,
see Lorenz et al. (2019) and Fig. 1. It should be noted that (A12) gives
Qin(x)=Q[sdiv(x),x].
In (A12), Qin(x) and Qout(x) are the inflow and outflow volume transports defined in (1) and displayed in Fig. 1b. Evaluating (A11) at S = sdiv(x),
Qin(x)=−12∂𝕞est[sdiv(x),x]∂S+ΔVest[sdiv(x),x]ΔT,
including the storage term. When additionally assuming sufficient long-term averaging, i.e., vanishing of the storage term ΔVest/ΔT, we obtain the simplified version (9).
In a similar way, storage terms can also be added to the mixing–exchange flow relation (10) for the three-layer flow (see also Fig. 1d):
Qin,high=−12∂𝕞est,high∂S+ΔVest,highΔT,Qin,low+Qr=12∂𝕞est,low∂S+ΔVest,lowΔT.
The most important variables of this study, including their meanings and units, are shown in Table A1.
Table A1.

List of variables including their meanings and units.
Table A1.

REFERENCES

  • Broatch, E. M., and P. MacCready, 2022: Mixing in a salinity variance budget of the Salish Sea is controlled by river flow. J. Phys. Oceanogr., 52, 2305–2323, https://doi.org/10.1175/JPO-D-21-0227.1.

    • Search Google Scholar
    • Export Citation

  • Burchard, H., 2020: A universal law of estuarine mixing. J. Phys. Oceanogr., 50, 81–93, https://doi.org/10.1175/JPO-D-19-0014.1.

  • Burchard, H., and K. Bolding, 2002: GETM – A general estuarine transport model. Scientific documentation, European Commission Tech. Rep. EUR 20253 EN, 157 pp., https://op.europa.eu/en/publication-detail/-/publication/5506bf19-e076-4d4b-8648-dedd06efbb38.

  • Burchard, H., and H. Rennau, 2008: Comparative quantification of physically and numerically induced mixing in ocean models. Ocean Modell., 20, 293–311, https://doi.org/10.1016/j.ocemod.2007.10.003.

    • Search Google Scholar
    • Export Citation

  • Burchard, H., K. Bolding, and M. R. Villarreal, 1999: GOTM – A general ocean turbulence model. Theory, applications and test cases, European Commission Tech. Rep. EUR 18745 EN, 103 pp., https://op.europa.eu/en/publication-detail/-/publication/5b512e12-367d-11ea-ba6e-01aa75ed71a1/language-en.

  • Burchard, H., K. Bolding, and M. R. Villarreal, 2004: Three-dimensional modelling of estuarine turbidity maxima in a tidal estuary. Ocean Dyn., 54, 250–265, https://doi.org/10.1007/s10236-003-0073-4.

    • Search Google Scholar
    • Export Citation

  • Burchard, H., H. M. Schuttelaars, and D. K. Ralston, 2018a: Sediment trapping in estuaries. Annu. Rev. Mar. Sci., 10, 371–395, https://doi.org/10.1146/annurev-marine-010816-060535.

    • Search Google Scholar
    • Export Citation

  • Burchard, H., and Coauthors, 2018b: The Knudsen theorem and the Total Exchange Flow analysis framework applied to the Baltic Sea. Prog. Oceanogr., 165, 268–286, https://doi.org/10.1016/j.pocean.2018.04.004.

    • Search Google Scholar
    • Export Citation

  • Burchard, H., X. Lange, K. Klingbeil, and P. MacCready, 2019: Mixing estimates for estuaries. J. Phys. Oceanogr., 49, 631–648, https://doi.org/10.1175/JPO-D-18-0147.1.

    • Search Google Scholar
    • Export Citation

  • Burchard, H., U. Gräwe, K. Klingbeil, N. Koganti, X. Lange, and M. Lorenz, 2021: Effective diahaline diffusivities in estuaries. J. Adv. Model. Earth Syst., 13, e2020MS002307, https://doi.org/10.1029/2020MS002307.

    • Search Google Scholar
    • Export Citation

  • Chen, S.-N., W. R. Geyer, D. K. Ralston, and J. A. Lerczak, 2012: Estuarine exchange flow quantified with isohaline coordinates: Contrasting long and short estuaries. J. Phys. Oceanogr., 42, 748–763, https://doi.org/10.1175/JPO-D-11-086.1.

    • Search Google Scholar
    • Export Citation

  • Dickie, L. M., and R. W. Trites, 1983: The Gulf of St. Lawrence. Estuaries and Enclosed Seas, B. H. Ketchum, Ed., Ecosystems of the World, Vol. 26, Elsevier Science, 403–425.

  • Fennel, W., and M. Sturm, 1992: Dynamics of the western Baltic. J. Mar. Syst., 3, 183–205, https://doi.org/10.1016/0924-7963(92)90038-A.

    • Search Google Scholar
    • Export Citation

  • Ferrari, R., A. Mashayek, T. J. McDougall, M. Nikurashin, and J.-M. Campin, 2016: Turning ocean mixing upside down. J. Phys. Oceanogr., 46, 2239–2261, https://doi.org/10.1175/JPO-D-15-0244.1.

    • Search Google Scholar
    • Export Citation

  • Geyer, W. R., and P. MacCready, 2014: The estuarine circulation. Annu. Rev. Fluid Mech., 46, 175–197, https://doi.org/10.1146/annurev-fluid-010313-141302.

    • Search Google Scholar
    • Export Citation

  • Geyer, W. R., D. K. Ralston, and J.-L. Chen, 2020: Mechanisms of exchange flow in an estuary with a narrow, deep channel and wide, shallow shoals. J. Geophys. Res. Oceans, 125, e2020JC016092, https://doi.org/10.1029/2020JC016092.

    • Search Google Scholar
    • Export Citation

  • Gräwe, U., P. Holtermann, K. Klingbeil, and H. Burchard, 2015a: Advantages of vertically adaptive coordinates in numerical models of stratified shelf seas. Ocean Modell., 92, 56–68, https://doi.org/10.1016/j.ocemod.2015.05.008.

    • Search Google Scholar
    • Export Citation

  • Gräwe, U., M. Naumann, V. Mohrholz, and H. Burchard, 2015b: Anatomizing one of the largest saltwater inflows into the Baltic Sea in December 2014. J. Geophys. Res. Oceans, 120, 7676–7697, https://doi.org/10.1002/2015JC011269.

    • Search Google Scholar
    • Export Citation

  • Gräwe, U., G. Flöser, T. Gerkema, M. Duran-Matute, T. H. Badewien, E. Schulz, and H. Burchard, 2016: A numerical model for the entire Wadden Sea: Skill assessment and analysis of hydrodynamics. J. Geophys. Res. Oceans, 121, 5231–5251, https://doi.org/10.1002/2016JC011655.

    • Search Google Scholar
    • Export Citation

  • Groeskamp, S., S. M. Griffies, D. Iudicone, R. Marsh, A. J. G. Nurser, and J. D. Zika, 2019: The water mass transformation framework for ocean physics and biogeochemistry. Annu. Rev. Mar. Sci., 11, 271–305, https://doi.org/10.1146/annurev-marine-010318-095421.

    • Search Google Scholar
    • Export Citation

  • Hansen, D., and M. Rattray, 1965: Gravitational circulation in straits and estuaries. J. Mar. Res., 23, 104–122.

  • Henell, E., H. Burchard, U. Gräwe, and K. Klingbeil, 2023: Spatial composition of the diahaline overturning circulation in a fjord–type, non–tidal estuarine system. J. Geophys. Res. Oceans, 128, e2023JC019862, https://doi.org/10.1029/2023JC019862.

    • Search Google Scholar
    • Export Citation

  • Hetland, R. D., 2005: Relating river plume structure to vertical mixing. J. Phys. Oceanogr., 35, 1667–1688, https://doi.org/10.1175/JPO2774.1.

    • Search Google Scholar
    • Export Citation

  • Hieronymus, M., J. Nilsson, and J. Nycander, 2014: Water mass transformation in salinity–temperature space. J. Phys. Oceanogr., 44, 2547–2568, https://doi.org/10.1175/JPO-D-13-0257.1.

    • Search Google Scholar
    • Export Citation

  • Hofmeister, R., H. Burchard, and J.-M. Beckers, 2010: Non-uniform adaptive vertical grids for 3D numerical ocean models. Ocean Modell., 33, 70–86, https://doi.org/10.1016/j.ocemod.2009.12.003.

    • Search Google Scholar
    • Export Citation

  • Holtermann, P. L., L. Umlauf, T. Tanhua, O. Schmale, G. Rehder, and J. J. Waniek, 2012: The Baltic Sea Tracer Release Experiment: 1. Mixing rates. J. Geophys. Res., 117, C01021, https://doi.org/10.1029/2011JC007439.

    • Search Google Scholar
    • Export Citation

  • Holtermann, P. L., H. Burchard, U. Gräwe, K. Klingbeil, and L. Umlauf, 2014: Deep-water dynamics and boundary mixing in a nontidal stratified basin: A modeling study of the Baltic Sea. J. Geophys. Res. Oceans, 119, 1465–1487, https://doi.org/10.1002/2013JC009483.

    • Search Google Scholar
    • Export Citation

  • Kappenberg, J., and I. Grabemann, 2001: Variability of the mixing zones and estuarine turbidity maxima in the Elbe and Weser estuaries. Estuaries, 24, 699–706, https://doi.org/10.2307/1352878.

    • Search Google Scholar
    • Export Citation

  • Kappenberg, J., G. Schymura, and H.-U. Fanger, 1995: Sediment dynamics and estuarine circulation in the turbidity maximum of the Elbe River. Neth. J. Aquat. Ecol., 29, 229–237, https://doi.org/10.1007/BF02084220.

    • Search Google Scholar
    • Export Citation

  • Klingbeil, K., and E. Henell, 2023: A rigorous derivation of the water mass transformation framework, the relation between mixing and diasurface exchange flow, and links to recent theories in estuarine research. J. Phys. Oceanogr., 53, 2953–2968, https://doi.org/10.1175/JPO-D-23-0130.1.

    • Search Google Scholar
    • Export Citation

  • Klingbeil, K., M. Mohammadi-Aragh, U. Gräwe, and H. Burchard, 2014: Quantification of spurious dissipation and mixing—Discrete variance decay in a finite-volume framework. Ocean Modell., 81, 49–64, https://doi.org/10.1016/j.ocemod.2014.06.001.

    • Search Google Scholar
    • Export Citation

  • Klingbeil, K., F. Lemarié, L. Debreu, and H. Burchard, 2018: The numerics of hydrostatic structured-grid coastal ocean models: State of the art and future perspectives. Ocean Modell., 125, 80–105, https://doi.org/10.1016/j.ocemod.2018.01.007.

    • Search Google Scholar
    • Export Citation

  • Knudsen, M., 1900: Ein hydrographischer Lehrsatz. Ann. Hydrogr. Marit. Meteor., 28, 316–320.

  • Lacy, J. R., M. T. Stacey, J. R. Burau, and S. G. Monismith, 2003: Interaction of lateral baroclinic forcing and turbulence in an estuary. J. Geophys. Res., 108, 3089, https://doi.org/10.1029/2002JC001392.

    • Search Google Scholar
    • Export Citation

  • Lange, X., K. Klingbeil, and H. Burchard, 2020: Inversions of estuarine circulation are frequent in a weakly tidal estuary with variable wind forcing and seaward salinity fluctuations. J. Geophys. Res. Oceans, 125, e2019JC015789, https://doi.org/10.1029/2019JC015789.

    • Search Google Scholar
    • Export Citation

  • Lehmann, A., and K. Myrberg, 2008: Upwelling in the Baltic Sea—A review. J. Mar. Syst., 74 (Suppl.), S3–S12, https://doi.org/10.1016/j.jmarsys.2008.02.010.

    • Search Google Scholar
    • Export Citation

  • Lemagie, E. P., S. N. Giddings, P. MacCready, C. Seaton, and X. Wu, 2022: Measuring estuarine total exchange flow from discrete observations. J. Geophys. Res. Oceans, 127, e2022JC018960, https://doi.org/10.1029/2022JC018960.

    • Search Google Scholar
    • Export Citation

  • Li, X., M. Lorenz, K. Klingbeil, E. Chrysagi, U. Gräwe, J. Wu, and H. Burchard, 2022: Salinity mixing and diahaline exchange flow in a large multi-outlet estuary with islands. J. Phys. Oceanogr., 52, 2111–2127, https://doi.org/10.1175/JPO-D-21-0292.1.

    • Search Google Scholar
    • Export Citation

  • Lorenz, M., K. Klingbeil, P. MacCready, and H. Burchard, 2019: Numerical issues of the Total Exchange Flow (TEF) analysis framework for quantifying estuarine circulation. Ocean Sci., 15, 601–614, https://doi.org/10.5194/os-15-601-2019.

    • Search Google Scholar
    • Export Citation

  • MacCready, P., 2011: Calculating estuarine exchange flow using isohaline coordinates. J. Phys. Oceanogr., 41, 1116–1124, https://doi.org/10.1175/2011JPO4517.1.

    • Search Google Scholar
    • Export Citation

  • MacCready, P., and W. R. Geyer, 2001: Estuarine salt flux through an isohaline surface. J. Geophys. Res., 106, 11 629–11 637, https://doi.org/10.1029/2001JC900006.

    • Search Google Scholar
    • Export Citation

  • MacCready, P., and W. R. Geyer, 2024: Estuarine exchange flow in the Salish Sea. J. Geophys. Res. Oceans, 129, e2023JC020369, https://doi.org/10.1029/2023JC020369.

    • Search Google Scholar
    • Export Citation

  • MacCready, P., R. D. Hetland, and W. R. Geyer, 2002: Long-term isohaline salt balance in an estuary. Cont. Shelf Res., 22, 1591–1601, https://doi.org/10.1016/S0278-4343(02)00023-7.

    • Search Google Scholar
    • Export Citation

  • MacCready, P., W. R. Geyer, and H. Burchard, 2018: Estuarine exchange flow is related to mixing through the salinity variance budget. J. Phys. Oceanogr., 48, 1375–1384, https://doi.org/10.1175/JPO-D-17-0266.1.

    • Search Google Scholar
    • Export Citation

  • MacCready, P., and Coauthors, 2021: Estuarine circulation, mixing, and residence times in the Salish Sea. J. Geophys. Res. Oceans, 126, e2020JC016738, https://doi.org/10.1029/2020JC016738.

    • Search Google Scholar
    • Export Citation

  • Marshall, J., D. Jamous, and J. Nilsson, 1999: Reconciling thermodynamic and dynamic methods of computation of water-mass transformation rates. Deep-Sea Res. I, 46, 545–572, https://doi.org/10.1016/S0967-0637(98)00082-X.

    • Search Google Scholar
    • Export Citation

  • McDougall, T. J., 1984: The relative roles of diapycnal and isopycnal mixing on subsurface water mass conversion. J. Phys. Oceanogr., 14, 1577–1589, https://doi.org/10.1175/1520-0485(1984)014<1577:TRRODA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • McDougall, T. J., and R. Ferrari, 2017: Abyssal upwelling and downwelling driven by near-boundary mixing. J. Phys. Oceanogr., 47, 261–283, https://doi.org/10.1175/JPO-D-16-0082.1.

    • Search Google Scholar
    • Export Citation

  • Pemberton, P., J. Nilsson, M. Hieronymus, and H. E. M. Meier, 2015: Arctic Ocean water mass transformation in S–T coordinates. J. Phys. Oceanogr., 45, 1025–1050, https://doi.org/10.1175/JPO-D-14-0197.1.

    • Search Google Scholar
    • Export Citation

  • Postma, H., and K. Kalle, 1955: Die Entstehung von Trübungszonen im Unterlauf der Flüsse, speziell im Hinblick auf die Verhältnisse in der Unterelbe. Dtsch. Hydrogr. Z., 8, 137–144, https://doi.org/10.1007/BF02019810.

    • Search Google Scholar
    • Export Citation

  • Reese, L., U. Gräwe, K. Klingbeil, X. Li, M. Lorenz, and H. Burchard, 2024: Local mixing determines spatial structure of diahaline exchange flow in a mesotidal estuary: A study of extreme runoff conditions. J. Phys. Oceanogr., 54, 3–27, https://doi.org/10.1175/JPO-D-23-0052.1.

    • Search Google Scholar
    • Export Citation

  • Reissmann, J. H., and Coauthors, 2009: Vertical mixing in the Baltic Sea and consequences for eutrophication—A review. Prog. Oceanogr., 82, 47–80, https://doi.org/10.1016/j.pocean.2007.10.004.

    • Search Google Scholar
    • Export Citation

  • Schulz, K., H. Burchard, V. Mohrholz, P. Holtermann, H. M. Schuttelaars, M. Becker, C. Maushake, and T. Gerkema, 2020: Intratidal and spatial variability over a slope in the EMS estuary: Robust along-channel SPM transport versus episodic events. Estuarine Coastal Shelf Sci., 243, 106902, https://doi.org/10.1016/j.ecss.2020.106902.

    • Search Google Scholar
    • Export Citation

  • Stevens, S. W., R. Pawlowicz, T. Tanhua, L. Gerke, W. A. Nesbitt, A. Drozdowski, J. Chassé, and D. W. R. Wallace, 2024: Deep inflow transport and dispersion in the Gulf of St. Lawrence revealed by a tracer release experiment. Commun. Earth Environ., 5, 338, https://doi.org/10.1038/s43247-024-01505-5.

    • Search Google Scholar
    • Export Citation

  • Umlauf, L., and H. Burchard, 2005: Second-order turbulence closure models for geophysical boundary layers. A review of recent work. Cont. Shelf Res., 25, 795–827, https://doi.org/10.1016/j.csr.2004.08.004.

    • Search Google Scholar
    • Export Citation

  • Umlauf, L., K. Bolding, and H. Burchard, 2005: GOTM – Scientific documentation, version 3.2. Baltic Sea Research Institute Tech. Rep. 63, 279 pp., https://www.io-warnemuende.de/files/forschung/meereswissenschaftliche-berichte/mebe63_2005-gotm.pdf.

  • Walin, G., 1977: A theoretical framework for the description of estuaries. Tellus, 29A, 128–136, https://doi.org/10.3402/tellusa.v29i2.11337.

    • Search Google Scholar
    • Export Citation

  • Wang, T., W. R. Geyer, and P. MacCready, 2017: Total exchange flow, entrainment, and diffusive salt flux in estuaries. J. Phys. Oceanogr., 47, 1205–1220, https://doi.org/10.1175/JPO-D-16-0258.1.

    • Search Google Scholar
    • Export Citation

  • Wüst, G., 1957: Ergebnisse eines hydrographisch-produktionsbiologischen Längsschnitts durch die Ostsee im Sommer 1956, I: Die Verteilung von Temperatur, Salzgehalt und Dichte. Kiel. Meeresforsch., 13, 163–185.

    • Search Google Scholar
    • Export Citation

  • Zika, J. D., M. H. England, and W. P. Sijp, 2012: The ocean circulation in thermohaline coordinates. J. Phys. Oceanogr., 42, 708–724, https://doi.org/10.1175/JPO-D-11-0139.1.

    • Search Google Scholar
    • Export Citation
Save
Cover Journal of Physical Oceanography
Journal of Physical Oceanography

  • Broatch, E. M., and P. MacCready, 2022: Mixing in a salinity variance budget of the Salish Sea is controlled by river flow. J. Phys. Oceanogr., 52, 2305–2323, https://doi.org/10.1175/JPO-D-21-0227.1.

    • Search Google Scholar
    • Export Citation

  • Burchard, H., 2020: A universal law of estuarine mixing. J. Phys. Oceanogr., 50, 81–93, https://doi.org/10.1175/JPO-D-19-0014.1.

  • Burchard, H., and K. Bolding, 2002: GETM – A general estuarine transport model. Scientific documentation, European Commission Tech. Rep. EUR 20253 EN, 157 pp., https://op.europa.eu/en/publication-detail/-/publication/5506bf19-e076-4d4b-8648-dedd06efbb38.

  • Burchard, H., and H. Rennau, 2008: Comparative quantification of physically and numerically induced mixing in ocean models. Ocean Modell., 20, 293–311, https://doi.org/10.1016/j.ocemod.2007.10.003.

    • Search Google Scholar
    • Export Citation

  • Burchard, H., K. Bolding, and M. R. Villarreal, 1999: GOTM – A general ocean turbulence model. Theory, applications and test cases, European Commission Tech. Rep. EUR 18745 EN, 103 pp., https://op.europa.eu/en/publication-detail/-/publication/5b512e12-367d-11ea-ba6e-01aa75ed71a1/language-en.

  • Burchard, H., K. Bolding, and M. R. Villarreal, 2004: Three-dimensional modelling of estuarine turbidity maxima in a tidal estuary. Ocean Dyn., 54, 250–265, https://doi.org/10.1007/s10236-003-0073-4.

    • Search Google Scholar
    • Export Citation

  • Burchard, H., H. M. Schuttelaars, and D. K. Ralston, 2018a: Sediment trapping in estuaries. Annu. Rev. Mar. Sci., 10, 371–395, https://doi.org/10.1146/annurev-marine-010816-060535.

    • Search Google Scholar
    • Export Citation

  • Burchard, H., and Coauthors, 2018b: The Knudsen theorem and the Total Exchange Flow analysis framework applied to the Baltic Sea. Prog. Oceanogr., 165, 268–286, https://doi.org/10.1016/j.pocean.2018.04.004.

    • Search Google Scholar
    • Export Citation

  • Burchard, H., X. Lange, K. Klingbeil, and P. MacCready, 2019: Mixing estimates for estuaries. J. Phys. Oceanogr., 49, 631–648, https://doi.org/10.1175/JPO-D-18-0147.1.

    • Search Google Scholar
    • Export Citation

  • Burchard, H., U. Gräwe, K. Klingbeil, N. Koganti, X. Lange, and M. Lorenz, 2021: Effective diahaline diffusivities in estuaries. J. Adv. Model. Earth Syst., 13, e2020MS002307, https://doi.org/10.1029/2020MS002307.

    • Search Google Scholar
    • Export Citation

  • Chen, S.-N., W. R. Geyer, D. K. Ralston, and J. A. Lerczak, 2012: Estuarine exchange flow quantified with isohaline coordinates: Contrasting long and short estuaries. J. Phys. Oceanogr., 42, 748–763, https://doi.org/10.1175/JPO-D-11-086.1.

    • Search Google Scholar
    • Export Citation

  • Dickie, L. M., and R. W. Trites, 1983: The Gulf of St. Lawrence. Estuaries and Enclosed Seas, B. H. Ketchum, Ed., Ecosystems of the World, Vol. 26, Elsevier Science, 403–425.

  • Fennel, W., and M. Sturm, 1992: Dynamics of the western Baltic. J. Mar. Syst., 3, 183–205, https://doi.org/10.1016/0924-7963(92)90038-A.

    • Search Google Scholar
    • Export Citation

  • Ferrari, R., A. Mashayek, T. J. McDougall, M. Nikurashin, and J.-M. Campin, 2016: Turning ocean mixing upside down. J. Phys. Oceanogr., 46, 2239–2261, https://doi.org/10.1175/JPO-D-15-0244.1.

    • Search Google Scholar
    • Export Citation

  • Geyer, W. R., and P. MacCready, 2014: The estuarine circulation. Annu. Rev. Fluid Mech., 46, 175–197, https://doi.org/10.1146/annurev-fluid-010313-141302.

    • Search Google Scholar
    • Export Citation

  • Geyer, W. R., D. K. Ralston, and J.-L. Chen, 2020: Mechanisms of exchange flow in an estuary with a narrow, deep channel and wide, shallow shoals. J. Geophys. Res. Oceans, 125, e2020JC016092, https://doi.org/10.1029/2020JC016092.

    • Search Google Scholar
    • Export Citation

  • Gräwe, U., P. Holtermann, K. Klingbeil, and H. Burchard, 2015a: Advantages of vertically adaptive coordinates in numerical models of stratified shelf seas. Ocean Modell., 92, 56–68, https://doi.org/10.1016/j.ocemod.2015.05.008.

    • Search Google Scholar
    • Export Citation

  • Gräwe, U., M. Naumann, V. Mohrholz, and H. Burchard, 2015b: Anatomizing one of the largest saltwater inflows into the Baltic Sea in December 2014. J. Geophys. Res. Oceans, 120, 7676–7697, https://doi.org/10.1002/2015JC011269.

    • Search Google Scholar
    • Export Citation

  • Gräwe, U., G. Flöser, T. Gerkema, M. Duran-Matute, T. H. Badewien, E. Schulz, and H. Burchard, 2016: A numerical model for the entire Wadden Sea: Skill assessment and analysis of hydrodynamics. J. Geophys. Res. Oceans, 121, 5231–5251, https://doi.org/10.1002/2016JC011655.

    • Search Google Scholar
    • Export Citation

  • Groeskamp, S., S. M. Griffies, D. Iudicone, R. Marsh, A. J. G. Nurser, and J. D. Zika, 2019: The water mass transformation framework for ocean physics and biogeochemistry. Annu. Rev. Mar. Sci., 11, 271–305, https://doi.org/10.1146/annurev-marine-010318-095421.

    • Search Google Scholar
    • Export Citation

  • Hansen, D., and M. Rattray, 1965: Gravitational circulation in straits and estuaries. J. Mar. Res., 23, 104–122.

  • Henell, E., H. Burchard, U. Gräwe, and K. Klingbeil, 2023: Spatial composition of the diahaline overturning circulation in a fjord–type, non–tidal estuarine system. J. Geophys. Res. Oceans, 128, e2023JC019862, https://doi.org/10.1029/2023JC019862.

    • Search Google Scholar
    • Export Citation

  • Hetland, R. D., 2005: Relating river plume structure to vertical mixing. J. Phys. Oceanogr., 35, 1667–1688, https://doi.org/10.1175/JPO2774.1.

    • Search Google Scholar
    • Export Citation

  • Hieronymus, M., J. Nilsson, and J. Nycander, 2014: Water mass transformation in salinity–temperature space. J. Phys. Oceanogr., 44, 2547–2568, https://doi.org/10.1175/JPO-D-13-0257.1.

    • Search Google Scholar
    • Export Citation

  • Hofmeister, R., H. Burchard, and J.-M. Beckers, 2010: Non-uniform adaptive vertical grids for 3D numerical ocean models. Ocean Modell., 33, 70–86, https://doi.org/10.1016/j.ocemod.2009.12.003.

    • Search Google Scholar
    • Export Citation

  • Holtermann, P. L., L. Umlauf, T. Tanhua, O. Schmale, G. Rehder, and J. J. Waniek, 2012: The Baltic Sea Tracer Release Experiment: 1. Mixing rates. J. Geophys. Res., 117, C01021, https://doi.org/10.1029/2011JC007439.

    • Search Google Scholar
    • Export Citation

  • Holtermann, P. L., H. Burchard, U. Gräwe, K. Klingbeil, and L. Umlauf, 2014: Deep-water dynamics and boundary mixing in a nontidal stratified basin: A modeling study of the Baltic Sea. J. Geophys. Res. Oceans, 119, 1465–1487, https://doi.org/10.1002/2013JC009483.

    • Search Google Scholar
    • Export Citation

  • Kappenberg, J., and I. Grabemann, 2001: Variability of the mixing zones and estuarine turbidity maxima in the Elbe and Weser estuaries. Estuaries, 24, 699–706, https://doi.org/10.2307/1352878.

    • Search Google Scholar
    • Export Citation

  • Kappenberg, J., G. Schymura, and H.-U. Fanger, 1995: Sediment dynamics and estuarine circulation in the turbidity maximum of the Elbe River. Neth. J. Aquat. Ecol., 29, 229–237, https://doi.org/10.1007/BF02084220.

    • Search Google Scholar
    • Export Citation

  • Klingbeil, K., and E. Henell, 2023: A rigorous derivation of the water mass transformation framework, the relation between mixing and diasurface exchange flow, and links to recent theories in estuarine research. J. Phys. Oceanogr., 53, 2953–2968, https://doi.org/10.1175/JPO-D-23-0130.1.

    • Search Google Scholar
    • Export Citation

  • Klingbeil, K., M. Mohammadi-Aragh, U. Gräwe, and H. Burchard, 2014: Quantification of spurious dissipation and mixing—Discrete variance decay in a finite-volume framework. Ocean Modell., 81, 49–64, https://doi.org/10.1016/j.ocemod.2014.06.001.

    • Search Google Scholar
    • Export Citation

  • Klingbeil, K., F. Lemarié, L. Debreu, and H. Burchard, 2018: The numerics of hydrostatic structured-grid coastal ocean models: State of the art and future perspectives. Ocean Modell., 125, 80–105, https://doi.org/10.1016/j.ocemod.2018.01.007.

    • Search Google Scholar
    • Export Citation

  • Knudsen, M., 1900: Ein hydrographischer Lehrsatz. Ann. Hydrogr. Marit. Meteor., 28, 316–320.

  • Lacy, J. R., M. T. Stacey, J. R. Burau, and S. G. Monismith, 2003: Interaction of lateral baroclinic forcing and turbulence in an estuary. J. Geophys. Res., 108, 3089, https://doi.org/10.1029/2002JC001392.

    • Search Google Scholar
    • Export Citation

  • Lange, X., K. Klingbeil, and H. Burchard, 2020: Inversions of estuarine circulation are frequent in a weakly tidal estuary with variable wind forcing and seaward salinity fluctuations. J. Geophys. Res. Oceans, 125, e2019JC015789, https://doi.org/10.1029/2019JC015789.

    • Search Google Scholar
    • Export Citation

  • Lehmann, A., and K. Myrberg, 2008: Upwelling in the Baltic Sea—A review. J. Mar. Syst., 74 (Suppl.), S3–S12, https://doi.org/10.1016/j.jmarsys.2008.02.010.

    • Search Google Scholar
    • Export Citation

  • Lemagie, E. P., S. N. Giddings, P. MacCready, C. Seaton, and X. Wu, 2022: Measuring estuarine total exchange flow from discrete observations. J. Geophys. Res. Oceans, 127, e2022JC018960, https://doi.org/10.1029/2022JC018960.

    • Search Google Scholar
    • Export Citation

  • Li, X., M. Lorenz, K. Klingbeil, E. Chrysagi, U. Gräwe, J. Wu, and H. Burchard, 2022: Salinity mixing and diahaline exchange flow in a large multi-outlet estuary with islands. J. Phys. Oceanogr., 52, 2111–2127, https://doi.org/10.1175/JPO-D-21-0292.1.

    • Search Google Scholar
    • Export Citation

  • Lorenz, M., K. Klingbeil, P. MacCready, and H. Burchard, 2019: Numerical issues of the Total Exchange Flow (TEF) analysis framework for quantifying estuarine circulation. Ocean Sci., 15, 601–614, https://doi.org/10.5194/os-15-601-2019.

    • Search Google Scholar
    • Export Citation

  • MacCready, P., 2011: Calculating estuarine exchange flow using isohaline coordinates. J. Phys. Oceanogr., 41, 1116–1124, https://doi.org/10.1175/2011JPO4517.1.

    • Search Google Scholar
    • Export Citation

  • MacCready, P., and W. R. Geyer, 2001: Estuarine salt flux through an isohaline surface. J. Geophys. Res., 106, 11 629–11 637, https://doi.org/10.1029/2001JC900006.

    • Search Google Scholar
    • Export Citation

  • MacCready, P., and W. R. Geyer, 2024: Estuarine exchange flow in the Salish Sea. J. Geophys. Res. Oceans, 129, e2023JC020369, https://doi.org/10.1029/2023JC020369.

    • Search Google Scholar
    • Export Citation

  • MacCready, P., R. D. Hetland, and W. R. Geyer, 2002: Long-term isohaline salt balance in an estuary. Cont. Shelf Res., 22, 1591–1601, https://doi.org/10.1016/S0278-4343(02)00023-7.

    • Search Google Scholar
    • Export Citation

  • MacCready, P., W. R. Geyer, and H. Burchard, 2018: Estuarine exchange flow is related to mixing through the salinity variance budget. J. Phys. Oceanogr., 48, 1375–1384, https://doi.org/10.1175/JPO-D-17-0266.1.

    • Search Google Scholar
    • Export Citation

  • MacCready, P., and Coauthors, 2021: Estuarine circulation, mixing, and residence times in the Salish Sea. J. Geophys. Res. Oceans, 126, e2020JC016738, https://doi.org/10.1029/2020JC016738.

    • Search Google Scholar
    • Export Citation

  • Marshall, J., D. Jamous, and J. Nilsson, 1999: Reconciling thermodynamic and dynamic methods of computation of water-mass transformation rates. Deep-Sea Res. I, 46, 545–572, https://doi.org/10.1016/S0967-0637(98)00082-X.

    • Search Google Scholar
    • Export Citation

  • McDougall, T. J., 1984: The relative roles of diapycnal and isopycnal mixing on subsurface water mass conversion. J. Phys. Oceanogr., 14, 1577–1589, https://doi.org/10.1175/1520-0485(1984)014<1577:TRRODA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • McDougall, T. J., and R. Ferrari, 2017: Abyssal upwelling and downwelling driven by near-boundary mixing. J. Phys. Oceanogr., 47, 261–283, https://doi.org/10.1175/JPO-D-16-0082.1.

    • Search Google Scholar
    • Export Citation

  • Pemberton, P., J. Nilsson, M. Hieronymus, and H. E. M. Meier, 2015: Arctic Ocean water mass transformation in S–T coordinates. J. Phys. Oceanogr., 45, 1025–1050, https://doi.org/10.1175/JPO-D-14-0197.1.

    • Search Google Scholar
    • Export Citation

  • Postma, H., and K. Kalle, 1955: Die Entstehung von Trübungszonen im Unterlauf der Flüsse, speziell im Hinblick auf die Verhältnisse in der Unterelbe. Dtsch. Hydrogr. Z., 8, 137–144, https://doi.org/10.1007/BF02019810.

    • Search Google Scholar
    • Export Citation

  • Reese, L., U. Gräwe, K. Klingbeil, X. Li, M. Lorenz, and H. Burchard, 2024: Local mixing determines spatial structure of diahaline exchange flow in a mesotidal estuary: A study of extreme runoff conditions. J. Phys. Oceanogr., 54, 3–27, https://doi.org/10.1175/JPO-D-23-0052.1.

    • Search Google Scholar
    • Export Citation

  • Reissmann, J. H., and Coauthors, 2009: Vertical mixing in the Baltic Sea and consequences for eutrophication—A review. Prog. Oceanogr., 82, 47–80, https://doi.org/10.1016/j.pocean.2007.10.004.

    • Search Google Scholar
    • Export Citation

  • Schulz, K., H. Burchard, V. Mohrholz, P. Holtermann, H. M. Schuttelaars, M. Becker, C. Maushake, and T. Gerkema, 2020: Intratidal and spatial variability over a slope in the EMS estuary: Robust along-channel SPM transport versus episodic events. Estuarine Coastal Shelf Sci., 243, 106902, https://doi.org/10.1016/j.ecss.2020.106902.

    • Search Google Scholar
    • Export Citation

  • Stevens, S. W., R. Pawlowicz, T. Tanhua, L. Gerke, W. A. Nesbitt, A. Drozdowski, J. Chassé, and D. W. R. Wallace, 2024: Deep inflow transport and dispersion in the Gulf of St. Lawrence revealed by a tracer release experiment. Commun. Earth Environ., 5, 338, https://doi.org/10.1038/s43247-024-01505-5.

    • Search Google Scholar
    • Export Citation

  • Umlauf, L., and H. Burchard, 2005: Second-order turbulence closure models for geophysical boundary layers. A review of recent work. Cont. Shelf Res., 25, 795–827, https://doi.org/10.1016/j.csr.2004.08.004.

    • Search Google Scholar
    • Export Citation

  • Umlauf, L., K. Bolding, and H. Burchard, 2005: GOTM – Scientific documentation, version 3.2. Baltic Sea Research Institute Tech. Rep. 63, 279 pp., https://www.io-warnemuende.de/files/forschung/meereswissenschaftliche-berichte/mebe63_2005-gotm.pdf.

  • Walin, G., 1977: A theoretical framework for the description of estuaries. Tellus, 29A, 128–136, https://doi.org/10.3402/tellusa.v29i2.11337.

    • Search Google Scholar
    • Export Citation

  • Wang, T., W. R. Geyer, and P. MacCready, 2017: Total exchange flow, entrainment, and diffusive salt flux in estuaries. J. Phys. Oceanogr., 47, 1205–1220, https://doi.org/10.1175/JPO-D-16-0258.1.

    • Search Google Scholar
    • Export Citation

  • Wüst, G., 1957: Ergebnisse eines hydrographisch-produktionsbiologischen Längsschnitts durch die Ostsee im Sommer 1956, I: Die Verteilung von Temperatur, Salzgehalt und Dichte. Kiel. Meeresforsch., 13, 163–185.

    • Search Google Scholar
    • Export Citation

  • Zika, J. D., M. H. England, and W. P. Sijp, 2012: The ocean circulation in thermohaline coordinates. J. Phys. Oceanogr., 42, 708–724, https://doi.org/10.1175/JPO-D-11-0139.1.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Sketch explaining the connection between exchange flow and diahaline mixing. (a) Dependence of the diahaline volume transport per unit horizontal area udia,z on the divergence of horizontal transport Ux, for simplicity only shown for the x direction, (b) two-layer flow in x–z coordinates, (c) two-layer flow in x–S coordinates, and (d) three-layer flow in x–z coordinates. The inner part of the isohalines is shown as bold red lines, and the outer part is shown as bold blue lines. Inflowing transports (>0) are shown as red arrows, and outflowing transports (<0) are shown as blue arrows. In (b)–(d), the water volume is marked by light blue shading.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Locations and bathymetries of the two estuaries investigated in the present study. (a) Map of Europe with an outline of the map area of (b) indicated; (b) Map of northern Germany with the complete model domains for the two estuaries; white lines indicate the open boundaries; (c) Bathymetry of the lower Elbe estuary, with a white line marking the open boundary and a red line marking the Cuxhaven transect; and (d) Bathymetry of the Warnow estuary, with a red line marking the Warnow transect.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Mixing and exchange flow results for the Elbe estuary for (a),(c),(e),(g) low (September 2012) and (b),(d),(f),(h) high (July 2013) runoff situations. The average discharge Qr for each month is given in the axis titles. (a),(b) Left-hand side (black line) and right-hand side (solid gray line) of the mixing–exchange flow relation, see (9), taken along the Elbe estuary; dashed gray lines represent the right-hand side of relation (9) corrected by the storage term ΔVest/ΔT (ΔT denoting the averaging period); the Cuxhaven transect [see (e)–(h) for its location] at 139 km is indicated as dashed red line; (c),(d) TEF q(S) = −∂Q/∂S across the Cuxhaven transect; the amounts of Qin (red) and Qout (blue) are indicated as numbers and graphically as shaded areas. The dividing salinity is marked as a thin dashed line; (e),(f) diahaline mixing per salinity class, m on the sdiv isohaline of the Cuxhaven transect (bold red line); the mean relative isohaline heights above ground of σ = −0.8 (solid), σ = −0.5 (dash–dotted), and σ = −0.2 (dotted); (g),(h) as in (e) and (f), but for the diahaline volume transport per unit horizontal area udia,z across the sdiv isohaline of the Cuxhaven transect; incoming (positive) transports are marked in red, and outflowing (negative) transports are marked in blue.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Mixing and exchange flow results for the Warnow estuary during the year 2014. (a) TEF across the Warnow transect shown in (c)–(f) averaged over the entire year; (b) temporally resolved TEF; local mixing per salinity class m on the dividing salinity isohalines for (c) sdiv,low and (d) sdiv,high; and diahaline volume transport per unit horizontal area udia,z on the dividing salinity isohalines for (e) sdiv,low and (f) sdiv,high. In (a) and (b), the horizontal lines indicate the two dividing salinities.

Fig. 1.

Sketch explaining the connection between exchange flow and diahaline mixing. (a) Dependence of the diahaline volume transport per unit horizontal area udia,z on the divergence of horizontal transport Ux, for simplicity only shown for the x direction, (b) two-layer flow in x–z coordinates, (c) two-layer flow in x–S coordinates, and (d) three-layer flow in x–z coordinates. The inner part of the isohalines is shown as bold red lines, and the outer part is shown as bold blue lines. Inflowing transports (>0) are shown as red arrows, and outflowing transports (<0) are shown as blue arrows. In (b)–(d), the water volume is marked by light blue shading.
Fig. 2.

Locations and bathymetries of the two estuaries investigated in the present study. (a) Map of Europe with an outline of the map area of (b) indicated; (b) Map of northern Germany with the complete model domains for the two estuaries; white lines indicate the open boundaries; (c) Bathymetry of the lower Elbe estuary, with a white line marking the open boundary and a red line marking the Cuxhaven transect; and (d) Bathymetry of the Warnow estuary, with a red line marking the Warnow transect.
Fig. 3.

Mixing and exchange flow results for the Elbe estuary for (a),(c),(e),(g) low (September 2012) and (b),(d),(f),(h) high (July 2013) runoff situations. The average discharge Qr for each month is given in the axis titles. (a),(b) Left-hand side (black line) and right-hand side (solid gray line) of the mixing–exchange flow relation, see (9), taken along the Elbe estuary; dashed gray lines represent the right-hand side of relation (9) corrected by the storage term ΔVest/ΔT (ΔT denoting the averaging period); the Cuxhaven transect [see (e)–(h) for its location] at 139 km is indicated as dashed red line; (c),(d) TEF q(S) = −∂Q/∂S across the Cuxhaven transect; the amounts of Qin (red) and Qout (blue) are indicated as numbers and graphically as shaded areas. The dividing salinity is marked as a thin dashed line; (e),(f) diahaline mixing per salinity class, m on the sdiv isohaline of the Cuxhaven transect (bold red line); the mean relative isohaline heights above ground of σ = −0.8 (solid), σ = −0.5 (dash–dotted), and σ = −0.2 (dotted); (g),(h) as in (e) and (f), but for the diahaline volume transport per unit horizontal area udia,z across the sdiv isohaline of the Cuxhaven transect; incoming (positive) transports are marked in red, and outflowing (negative) transports are marked in blue.
Fig. 4.

Mixing and exchange flow results for the Warnow estuary during the year 2014. (a) TEF across the Warnow transect shown in (c)–(f) averaged over the entire year; (b) temporally resolved TEF; local mixing per salinity class m on the dividing salinity isohalines for (c) sdiv,low and (d) sdiv,high; and diahaline volume transport per unit horizontal area udia,z on the dividing salinity isohalines for (e) sdiv,low and (f) sdiv,high. In (a) and (b), the horizontal lines indicate the two dividing salinities.
All Time Past Year Past 30 Days
Abstract Views 8 8 8
Full Text Views 688 688 688
PDF Downloads 352 352 352