According to the TEF analysis framework, the time-averaged volume transport per salinity class Q through a given cross-section A with salinity s greater than a given value S is obtained as 3Q(S)=∫A(s≥S)udA‾, where the bar ⋅‾ indicates time-averaging, and u is the along-channel current normal to the cross-section A. Changes in cross-sectional area due to tides are not taken into account.

From Eq. () the exchange profile of water transport per salinity class is obtained as 4q(S)=-dQ(S)dS, which verifies Q=∫s≥Sq(s)ds. Separating incoming and outgoing volume transports, it reads 5Qin=∫SminS0qds,   Qout=∫S0Smaxqds, where S0 is the dividing salinity which separates the inflow and outflow, and Smin and Smax are the minimum and maximum salinities in the cross-section. In general, Eq. () assumes the incoming and outgoing flows are arranged in two layers in the salinity space (not necessarily in the vertical coordinates as occurs with the classical estuarine circulation). also generalized the formulation for exchange flows with more than two layers in the salinity space.

Similarly to Eqs. () and (), the time-averaged transport of salt, Qs, reads 6Qs(S)=∫A(s≥S)sudA‾, where Qs=∫s≥Sqs(s)ds and 7Qins=∫SminS0qsds,   Qouts=∫S0Smaxqsds.

Based on quantities defined in Eqs. () and (), Kundsen-consistent salt concentrations for in- and outflows at a cross-section are 8sin=QinsQin,   sout=QoutsQout.

From Eq. () and the bulk mixing M≈sinsoutQr, considering the maximum possible mixing, i.e. Mmax=sin2Qr, the mixing completeness is defined as the ratio between both 9MC=M/Mmax=sout/sin.

Tidal current and salinity are assumed a superposition of the main semidiurnal constituent M2 and their most energetic overtide M4 as 10au(x,t)=ur(x)+ua(x)cos⁡(ωM2t)+ub(x)cos⁡(ωM4t+φb(x))10bs(x,t)=sr(x)+sa(x)cos⁡(ωM2t+ψa(x))+sb(x)cos⁡(ωM4t+ψb(x)) with x indicating the along-channel location of the cross-section, ur the residual current induced by the river flow, sr the residual or tidally-averaged salinity, ua and ub the current M2 and M4 amplitudes, φb the current M4 phase relative to that of the M2, sa and sb the salinity M2 and M4 amplitudes, and ψa and ψb the salinity M2 and M4 tidal phases relative to the current M2 phase. Residuals, amplitudes and phases are obtained (and prescribed) from field measurements (described below). Tidal periods are TM2=2π/ωM4 = 12.42 h and TM4=2π/ωM4 = 6.21 h.

The tidally-averaged (residual) salinity flux u⋅s‾ through a given cross-section at x is obtained from Eqs. () and () as 11u⋅s‾=ursr+uasa2cos⁡(ψa)+ubsb2cos⁡(φb-ψb).

Zero residual salinity flux, i.e. u⋅s‾=0, implies 12cos⁡(ψa)=-2ursruasa-ubsbuasacos⁡(φb-ψb).

This zero flux condition reduces the degrees of freedom of the problem to eight, to be determined from observations: ur, sr, ua, sa, ub, sb, φb and ψb. Considering the M2 tide only in Eqs. () and (), as in , the condition of zero residual salinity transport would reduce to cos⁡(ψa)=-2(ursr)/(uasa).

The Guadalquivir River Estuary is a coastal-plain estuary located in the south-western part of the Iberian Peninsula. The Guadalquivir estuary comprises the last 110 km of the Guadalquivir river, from head dam at the town of Alcalá del Río to Sanlúcar de Barrameda, where its waters flow into the Gulf of Cádiz in the Atlantic Ocean (Fig. ). The estuary is convergent with tidally-averaged cross-sections approximately decreasing exponentially from the mouth to the landward end according to 13A(x)=A0exp⁡(-x/a0), with A0 = 5800900 m2 and a0 = 605 km, where the subscript indicates the 95 % confidence interval in the last significant figure. Its mean depth in the thalweg, h≈ 7 m, is maintained by periodic dredging of the navigation channel . Tides are mesotidal (vertical tidal range at spring tides ∼ 3.5 m at the mouth) and semidiurnal, M2 being the most significant constituent. The estuary is flood-dominated as evidenced by the tidal phase differences between M2 and its first overtide M4, which accounts for (intra)tidal asymmetry . The tide propagation is dominated by friction and in the upper part by tidal wave reflection at the head dam of Alcalá del Río dam .

The climate in most of the Guadalquivir River basin is Mediterranean, characterized by hot, dry summers and mild, rainy winters. The basin is highly altered by human activities, with the discharge regime largely controlled by extensive upstream regulation and long-term land-use changes. Freshwater discharges inputs to the estuary from the Alcalá del Río dam are usually below Qr = 40 m3s-1, most often about Qr∼ 25 m3s-1. Salinity decreases from the mouth upstream due to freshwater input. The mesotidal conditions along with the relatively low values of Qr make the estuary tidally-energetic and well-mixed (partially-mixed near the mouth) in terms of salinity during low river flows. This is confirmed by the low values of the estuarine Richardson number (RiE < 0.08) and the potential energy anomaly . determined from an observational analysis that time correlation between tidal flow and salinity controls a substantial part of the salt transport. Modeling results by showed that the salt transport due to the density-driven flow interacts with that of the current-salinity correlation and that both transports are equally important in the Guadalquivir estuary. Under low river flow conditions, most of the observed suspended matter in the Guadalquivir estuary is due to the resuspension by tidal currents . The transport due to the M2 and M4 covariance of current velocity and suspended sediment explains the setting of Estuarine Turbidity Maxima in the Guadalquivir estuary .

Salinity and current data were recorded between 2008 and 2011 by a real time monitoring network, which was described in detail by and depicted in Fig. . Here only a brief description of the equipment is provided. Instrumentation was installed as close as possible to the navigation channel. Salinity data was recorded every 30 min in eight Conductivity-Temperature-Depth (CTD, denoted by γi in Fig. ) probes. The origin of the along-channel coordinate x was set at γ0, installed at the mouth of the estuary, and chosen chosen positive upstream. Table  shows the locations of the CTDs used in this study. Along-channel current data was obtained from six Acoustic Doppler Current Profilers (ADCPs) (αi in Fig. ), providing one data set every 15 min. Table  shows the kilometer points and geographic coordinates where the ADCPs were located in the Guadalquivir estuary.

Standard harmonic analysis was performed on the along-channel current and salinity time series using T_TIDE . Results are shown in Tables  and  . The time span for the harmonic analysis was 5 June 2008 through 5 December 2008. This time span was chosen according to the following three criteria. The interval must be larger than 28 d to separate the M2 from other semidiurnal constituents and to assure a zero residual net salt flux. And, finally, the chosen interval is that one with the fewer and smaller exceedances over 40 m3s-1 to assure that the data analysis corresponds with low river flow conditions.

Daily discharge data records at Alcalá del Río were provided by the Regional Water Management Agency (Red de seguimiento de la Confederación Hidrográfica del Guadalquivir, MAPAMA, station code 5072).

Volume transports and exchange profiles sorted by salinity classes are computed numerically using the analytical time series from Eqs. () and (), respectively. They are computed at different cross-sections along the Guadalquivir estuary indicated in Table  (and Fig. ). Tidal currents and salinity are obtained from Eqs. () and (), which include both the M2 and M4 constituents.

Values of ur, sr, ua, sa, ub, sb, φb and ψb (Eqs.  and ) at each cross-section in Fig.  are thus needed. They are obtained from Tables  and . From Eq. (), ψa is also computed from the other eight parameters at each cross-section. Differences between observed values of ψb (Table ) and those determined from Eq. () imposing zero residual salt flux are smaller than 12° at all cross-sections, i.e., {5.45°,11.20°,1.70°,5.54°,7.80°,2.16°}. Therefore, at the analysis scale, the estuary can be reasonably considered close to equilibrium conditions (zero residual salt flux).

Figure  shows the isohaline volume transport (solid lines) and the exchange profile (dotted lines), which are computed numerically using analytical time series according to Eqs. () and (), respectively, as a function of the salinity at cross-sections CSi. Overall, as the tide propagates upstream, i.e., towards estuarine parts of lower salinity and current, the maximum values of Q(S) become smaller. The largest volume transports Q(S) are observed at the CS1 and CS0 cross-sections, which are in the lower part of the estuary where the tidal currents are larger. The exchange profile q(S) per salinity class is structured in two layers at all locations, thereby showing a incoming transport of water per salinity class (qin) at higher salinity and an outgoing transport at lower salinity (qout). It is evident from Fig.  that incoming and outgoing transport vary more by salinity class in locations near the estuary mouth.

Figure  shows the Knudsen-bulk estimates along the main channel determined from the TEF analysis, Qin and Qout, determined integrating qin and qout (Eq. ), i.e. positive and negative transports, respectively. The results indicate that bulk along-channel exchange flow tends to decrease upstream, as expected. Incoming and outgoing water volume transports estimated here in the lower part of the estuary are about 10 % larger than volume transports inferred from the tidally-averaged horizontal current profiles shown by , who considered the density-driven flow only using an Eulerian approach at CS0 and CS1. This estimate difference from different approaches will be further discussed in Sect. . At the landward boundary at the head dam (CS6), the outgoing volume transport coincides with the freshwater discharge Qr = -25 m3s-1. Negligible values are obtained in the upper part of the estuary near the head dam. In the middle part of the estuary, incoming TEF bulk volume values below 150 m3s-1 are obtained. The largest net incoming water volume transport, viz. Qin≈ 300 m3s-1, is attained at the lower part of the estuary at CS1. The outgoing bulk value at CS1 is about 12-fold the normal river flow from the head dam at Alcalá del Río.

It is evident from the TEF results shown in Fig.  that the exchange flow does not decrease continuously from the estuary mouth to the head. The along-channel variability of the bulk estimates (from Eq. ) are mainly due to changes in the along-channel currents and distribution of salinity. Cross-section CS1 is the closest to where the largest (averaged) along-channel salinity gradient and where the largest tidal currents are observed (Table ). The relative exchange flow mininum at cross-section CS2 is caused by a significant decrease of the M2 tidal current amplitude (Table ). A plausible source of variability could be due lateral variations in the along-channel current over the cross-section. Although instrumentation was installed as close as possible to the main channel of the estuary, the particular mooring location of each current meter may also affect tidal current amplitudes and, thus, TEF estimates. The exchange flow minimum at cross-section CS2 suggests that further upstream outflows are convergent and inflows are divergent, which can only be explained by a partial recirculation of the outflow towards the estuary head. Directly downstream of the minimum, outflow is divergent and inflow is convergent such that parts of the inflow is deflected back towards the mouth of the estuary. This mechanism has been described by as the efflux/reflux theory. In practice, this would imply longer residence times for conservative pollutants than in the absence of recirculation.

It should be noted that exchange flow in a well-mixed estuary does not mean that there is a distinct upstream flow of salty water near the bottom and a downstream flow of brackish water near the surface, even though the exchange profiles in salinity coordinates q(S) are structured in two layers (as in Fig. ). The exchange flow following the theory is formulated in salinity coordinates and means that the outflow Qout occurs at lower salinities than the inflow Qin. During flood a water parcel with a specific salinity passes through a transect, leaving an upstream flux contribution at a certain salinity class. Upstream of the transect, this water parcel exchanges salinity with other water parcels, such that during ebb it passes the transect at a different salinity, leaving a downstream contribution at this different salinity. Statistically, the flood flux happens at a higher salinity than the ebb flux, due to the lower salinities upstream, caused by the freshwater discharge from the upstream dam. This is why the fully cross-sectionally mixed idealized estuarine situation described in Eqs. () and () still results in an estuarine exchange flow, when formulated in salinity coordinates. Given that the Guadalquivir estuary is well-mixed during low river discharge, except for its partially-mixed lower part near the mouth, a vertically-structured exchange flow with persistent deep-water inflow and surface outflow might only develop in that lower region.

Figure  shows the along-channel Knudsen-consistent salt concentration estimates within inflows and outflows at each cross-section. Both curves resemble the (averaged) salinity along-channel distribution of salinity. The representative TEF bulk salinity values for the incoming transport are larger than those for the outgoing transport. Differences increase towards the estuary mouth from the head dam, where sin=sout=0. At CS0, which is the cross-section nearest to the mouth, the representative TEF bulk salinity value for inflows is 28 psu, whereas that for outflows at the same location is about 21 psu. Where the largest net incoming water volume transport occurs (CS1) these values are 16 and 24 psu, respectively.

The small relative differences between representative TEF bulk salinity values for inflows and outflows, i.e. 1-sout/sin (Fig. ), ensue from the high rates of mixing in the Guadalquivir estuary. The analysis of salt transport indicates that the mixing completeness, estimated from Eq. () (in %), is larger than 67 % at all cross-sections (Fig. ), thereby evidencing the poorly-stratified character of the Guadalquivir estuary during low river flows. The mixing completeness attains at the cross-section nearest to the mouth of the estuary. The net TEF exchange of variance upstream, at the tidal river part, is negligible due to vanishing salinities, such that the mixing is complete (100 %). Mixing completeness values in the lower and middle part of the Guadalquivir estuary, which are between 70 % and 75 %, are similar to the 75 % estimated near the mouth of the Elbe during low discharge conditions . The estimated values in the Guadalquivir are not far from those obtained in and idealized convergent V-shaped model estuary, viz. 64 % , which seems reasonable as the Guadalquivir estuary is a highly channelized estuary. While the mixing completeness can range between zero and 100 % in estuaries , the Guadalquivir estuary can be rated as of relatively high mixing completeness compared to other estuaries which typically at high discharge or neap-tide conditions are more stratified and therefore of lower mixing completeness (below 50 %).

The idealized approach used in this study is simplified and does not take all physical processes affecting exchange flow in estuaries into account. The assumptions made to simplify the equations allow for the construction of a fast and understandable model, but they also limit its scope.

The analysis is restricted to normal, low river flows with zero residual salinity flux. Non-stationarity, which is not taken into account, influences TEF prior and after high-discharge events. Freshwater discharge variability is conditioned by extensive upstream regulation of the drainage basin and is characterized by seasonal, short-duration high-discharge events (days to a few weeks), reflecting the regulation effects of the river basin upstream of the Alcalá del Río dam.

The choice to consider only the most energetic constituent M2 and its main overtide M4 allows for a first reliable and approximate assessment of TEF in estuaries at low computational cost, particularly in cases where no complex high-resolution model is available, or as a benchmark prior to its implementation, highlighting the importance of the covariance between salinity and current. Additionally, this simplified scenario has allowed the effect on TEF of including M4, which accounts for tidal asymmetry, to be evaluated. However, it must be acknowledged that this simplification is a limitation that does not allow for an accurate estimation of TEF capable of capturing its spatio-temporal variability, which is controlled by tidal-fluvial interaction and complex bathymetry. In this regard, a precise estimation of TEF and its variability in the Guadalquivir estuary should be the subject of future research. To this end, other semidiurnal tides should be considered to capture the spring–neap modulation; diurnal constituents which generate diurnal inequality and also contribute to a semidiurnal–diurnal tidal asymmetry ; as well as their corresponding compound tides and overtides.

Although it is a common approximation in funneled estuaries, parameterizing the cross-sections using a decreasing exponential is an oversimplification of real bathymetric profiles. Besides, there are lateral and vertical variations of TEF in cross-sections that are not accounted for in this work and should be mentioned. Vertical salinity variations are negligible in an estuary such as the Guadalquivir that is mostly well-mixed, except perhaps in the lower reach near the mouth, where partial stratification may occur. Vertical variability in tidal currents is more significant and, in this regard, the approach adopted in this study may be overestimating total exchange flows. According to , within a monitored water column of ∼ 6 m, differences close to 50 % can occur between the M2 semimajor axis amplitude of the tidal ellipse at the surface and at the bottom. Vertical differences in the M2 tidal ellipse phase do not appear to be significant. Regarding lateral variations, with the available data it is difficult to quantify them. However, there is indirect evidence in the Guadalquivir estuary of their importance and, therefore, they should be included in future detailed studies with high-resolution numerical models to achieve a more accurate estimation of exchange flows. Lateral separation of the subtidal flow in estuaries occurs due to, e.g., tidal residual currents or lateral variations in the estuarine circulation . It is also well-known that local topographic features, such as bends, may have considerable influence on the lateral mixing driven by secondary flows and hence on the salt transport and exchange flows e.g..

Recently, studied the salt transport mechanisms in the North German Weser River Estuary using a detailed transport decomposition method applied to a high-resolution numerical model. These authors found that opposing flows often occur between the channel and adjacent shoals in this estuary. Outflow subtidal depth-averaged transport occurs on the shoals, whereas a net inflow occurs on the inner part of the channel. Also, in the bends of the Weser estuary, an outflow is found in the outer part of the bend, whereas in the center of the channel as well as the inner shoals, a residual inflow occurs. In the Guadalquivir estuary, provided indirect evidences of lateral separation of salt fluxes from analysis of observations near the thalweg. These authors noticed that during several months during low river-flows the 2 psu isohaline location (X2) did not show appreciable variations besides its a weak spring-neap variability. However, there was a persistent net salt influx on the inner part of the channel, thereby suggesting the presence of lateral variations over the cross-section and a compensating net salt outflux on the shoals.

Finally, TEF estimates of incoming and outgoing water volume transports in the lower part of the estuary obtained in this study are about 10 % larger than volume transports inferred from a tidally-averaged Eulerian approach by , who only considered the contribution of the density-driven flow to the exchange. The actual differences from these two different approaches are likely to be larger since the present study only considers the contribution of the M2 tidal constituent to TEF. As noted above, other constituents may potentially enhance tidal salt transport. observed that tidal salt transport is dominant, showing that Stokes transport and tidal pumping are at least one order of magnitude larger than those induced by vertically-sheared currents. Nevertheless, the water column structure in the lower part of the estuary is partially stratified rather than well-mixed, showing the residual horizontal currents an evident vertically-sheared profile . Semi-analytical model results by highlighted the importance of accounting for density-driven flow when estimating salt transport, particularly in this lower part of the estuary. They showed that this transport mechanism cannot be neglected. Salt transport due to density-driven flow interacts with that associated with current–salinity correlations, and both contributions are equally important in the lower part of the Guadalquivir Estuary.