The Yangtze River estuary, located on the east coast of China, extends ∼630 km from the Datong hydrological station (DT; the location of the tidal limit) to its mouth near the seaward end of the “South Branch” (Fig. ). Both tidal waves and river flow are the major sources of energy for the hydrodynamics along the Yangtze River estuary. Specifically, the estuary is a meso-tidal coastal area with a maximum and mean tidal range of 4.62 and 2.67 m near the estuary mouth, respectively. The tide has an irregular semi-diurnal character with average flood and ebb durations of 5 and 7.5 h, respectively . According to the observed data at DT (1950–2012), the annual mean river discharge is approximately 28 200 m3 s-1, and the monthly average river discharge reaches a maximum value of 49 500 m3 s-1 in July and a minimum value of 11 300 m3 s-1 in January. Unlike previous studies focusing on the tidal hydrodynamics near the estuary mouth, here we mainly concentrate on the tide–river dynamics in the mainstream of the Yangtze River estuary, extending from Tianshenggang gauging station to DT.

Monthly averaged hydrological data (including tidal range and water level) from the following six tidal gauging stations along the Yangtze River estuary were collected from the Yangtze Hydrology Bureau of the People's Republic of China for the period from 2003 to 2014: Tianshenggang (TSG); Jiangyin (JY), located 46 km upstream of TSG; Zhenjiang (ZJ), located 155 km upstream of TSG; Nanjing (NJ), located 236 km upstream of TSG; Maanshan (MAS), located 284 km upstream of TSG; and Wuhu (WH), located 330 km upstream of TSG. The tidal amplitude was defined as half of the tidal range either during the flood or the ebb period, and we determined the mean value by averaging the tidal amplitudes during flood and ebb periods. To correctly calculate the residual water level slope, measured water levels from the gauging stations were corrected to the Yellow Sea 1985 vertical datum of local mean sea level. Figure  illustrates the temporal variation of the monthly averaged tidal range H and residual water level Z‾ observed at the six gauging stations in addition to the monthly averaged river discharge observed at DT. In Fig. , we observe a strongly seasonal variation in H (except for TSG and JY) and Z‾ due to the strongly fluctuating river discharge. For the residual water level, we also note that the further upstream the station, the more evident the seasonal change is.

The dynamics of the residual water level can be derived from the one-dimensional momentum equation e.g.: 1∂U∂t+U∂U∂x+g∂Z∂x+gh2ρ∂ρ∂x+gU|U|K2h4/3=0, where U is the cross-sectional averaged velocity, Z is free surface elevation, h is water depth, g is the acceleration of gravity, t is time, ρ is water density, x is the longitudinal coordinate directed landward, and K is the Manning–Strickler friction coefficient. Assuming a periodic variation of flow velocity, the integration of Eq. () over a tidal cycle leads to an expression for the residual water level slope e.g.: 2∂Z‾∂x=-1K2U|U|h4/3‾-12g∂U2‾∂x-12ρ0h∂ρ∂x‾, where the over bars and the subscript zero indicate the tidal average and the value at the seaward boundary, respectively. As shown in Eq. (), the residual water level slope is caused by three contributions made by residual friction, advective acceleration and density effects that correspond to the three terms on the right-hand side of Eq. (). Note that the contribution from advective acceleration to the residual water level slope, 3∂Z‾adv∂x=-12g∂U2‾∂x, can be easily integrated to: 4Z‾adv=-12gU2‾-U02‾=-12Fr0‾U2‾U02‾-1h0‾, where we introduced the Froude number, Fr2‾=U2‾/(gh‾), computed with the averaged variables. In this case, the correction is local (not cumulative) and proportional to the flow depth via a coefficient that is negligible as long as the velocity does not change significantly, and Fr is small, as is common for most tidal flows. With regard to the contribution from the density effect, it has been shown by that the induced value of the residual water level only amounts to around 1.25 % of the estuary depth over the salt intrusion length. Hence, in this study we neglect the impact of density on the dynamics of residual water level.

Assuming a negligible advective acceleration influence and density effect, integration of Eq. () leads to an expression for the residual water level: 5Z‾(x)=-∫0x∂Z∂x‾=-∫0xU|U|K2h4/3‾, if we assume Z‾=0 at the estuary mouth (where x=0).

To correctly reproduce the residual water level profile in estuaries, an iterative procedure is required to properly calculate the friction term as presented in Eq. () as the velocity amplitude and water depth are unknown a priori. This is carried out using the analytical hydrodynamic model proposed by . In the analytical model, the fundamental assumption made for the geometry of the estuary is that the tidally averaged cross-sectional area (A‾) and width (B‾) can be approximated by the following exponential functions e.g.: 6A‾=Ar‾+(A0‾-Ar‾)exp⁡(-x/a),7B‾=Br‾+(B0‾-Br‾)exp⁡(-x/b), where A0‾ and B0‾ are the respective cross-sectional area and width at the estuary mouth, Ar‾ and Br‾ represent the respective asymptotic riverine cross-sectional area and width, and a and b denote the convergence lengths of the cross-sectional area and width, respectively. The tidally averaged depth (h‾) can be computed directly following the assumption of a mostly rectangular cross-section, i.e. h‾=A‾/B‾. The possible impact of the lateral storage areas (e.g. tidal flats or salt marshes) can be described by the storage width ratio rS=BS/B‾, defined as the ratio of the storage width BS and the tidally averaged stream width B‾.

Concentrating on a predominantly tidal constituent (e.g. M2) with frequency ω, it was shown by that tide–river dynamics are mainly determined by four externally defined, dimensionless parameters (see Table ), i.e. the dimensionless tidal amplitude ζ defined as the ratio of the tidal amplitude to the depth, the estuary shape number γ (describing the cross-sectional area convergence), the friction number χ (representing the role of frictional dissipation), and the dimensionless river discharge φ (indicating the influence of freshwater discharge Q imposed at the upstream boundary), where η is the tidal amplitude, υ is the velocity amplitude, Ur=Q/A‾ is the river flow velocity and c0 is the classical wave celerity in a prismatic frictionless channel, defined as c0=gh‾/rS.

The analytical solutions for the main tide–river dynamics can be obtained by solving a set of four implicit equations: the damping/amplification equation, 8δ=μ2γθ-χμλΓ1+μ2β; the phase lag equation, 9tan⁡ε=λγ-δ; the scaling equation, 10μ=sin⁡(ε)λ=cos⁡(ε)γ-δ; and the celerity equation, 11λ=1-δ(γ-δ).

The main dependent dimensionless parameters in Eqs. ()–() are presented in Table . They include the amplification/damping number δ, describing the rate of increase (δ>0) or decrease (δ<0) of the tidal wave amplitude along the channel axis; the velocity number μ, indicating the ratio of the actual velocity amplitude to the reference value in a prismatic frictionless channel; λ, the celerity number denoting the ratio of the classical wave celerity (c0) to the actual wave celerity (c), and ε, the phase lag between high water (HW) and high water slack (HWS) or between low water (LW) and low water slack (LWS), where ϕZ and ϕU are the water level phase and velocity, respectively. The unknown parameters θ, β and Γ in the damping/amplification equation account for the impact of river discharge, where θ and β are defined in Table , and Γ is given by 12Γ=1πp1-2p2φ+p3φ23+μ2λ2/φ2, which is a friction factor derived by means of a Chebyshev polynomials approach (Cai et al., 2016). In Eq. (), pi (i=0, 1, 2, 3) are the Chebyschev coefficients seep.301, which are functions of the dimensionless river discharge φ through α=arccos⁡(-φ): 13p0=-7120sin⁡2α+124sin⁡6α-160sin⁡8α,14p1=76sin⁡α-7303α-730sin⁡5α+110sin⁡7α,15p2=π-2α+13sin⁡2α+1930sin⁡4α-15sin⁡6α,16p3=43sin⁡α-23sin⁡3α+215sin⁡5α. The set of Eqs. ()–() can be regarded as a consistent analytical framework for understanding tide–river dynamics in estuaries. For more details about the computation procedure, readers can refer to .

To understand the importance of seasonal changes in river discharge on tide–river dynamics, we first explore the seasonal variation of the tidal damping rate and the residual water level slope (Fig. ). Here, the tidal damping rate (δH) and the residual water level slope (S) can be estimated for a reach of length Δx: 17δH=1η1+η2/2η2-η1Δx,18S=Z2‾-Z1‾Δx, where η1 and Z1‾ are the tidal amplitude and residual water level on the seaward side, respectively, and η2 and Z2‾ are the corresponding values at a distance Δx upstream, respectively.

The study period covers tide–river dynamics under both low and high flow conditions, where the monthly average river discharge observed at DT ranges from approximately 9174 to 61 400 m3 s-1 so that the nonlinear interaction between tidal wave and river flow varies considerably. It has been previously shown that river discharge primarily impacts the tidal damping rate and residual water level slope via the friction term . It can be clearly seen from Fig.  that both the tidal damping rate and residual water level slope vary strongly with river discharge. Remarkably, it appears that there is a threshold, corresponding to a critical value of river discharge, beyond which the relationship between the tidal damping rate and river discharge switches from negatively to positively correlated (Fig. a). This is particularly the case in the upper reach between the MAS and WH stations when the river discharge threshold is approximately 35 000 m3 s-1. The underlying mechanism will be elaborated upon further in Sect. 5.2. In Fig. b, it appears that the residual water level slope is linearly correlated with river discharge.

The main geometric characteristics (including the tidally averaged cross-sectional area, width and depth) used in this paper were extracted from a digital elevation model (DEM) produced from Yangtze River estuary navigation charts surveyed in 2007. The elevations have been corrected to the local mean sea level of the Yellow Sea 1985 vertical datum. Figure  displays the longitudinal geometric quantities along the Yangtze River estuary axis, in combination with the best-fitting curves reproduced by Eqs. () and (). Table  shows the calibrated geometric characteristics, where we observe a relatively large value of cross-sectional area convergence length (151 km), with a relatively small value for width (44 km), suggesting a fast transition from a funnel shaped reach to a prismatic reach in terms of width.

The analytical model was calibrated and verified against the observed tidal amplitude and residual water level along the Yangtze River estuary on the basis of the monthly averaged hydrological data during the 2003–2014 period. The adopted seaward tidal amplitude (at TSG) and upward river discharge (at DT) in the analytical model are displayed in Fig. . As the Yangtze River estuary features a typical semi-diurnal character, for the sake of simplification, we used a typical M2 tidal period (i.e. 12.42 h). The only calibrated parameter in the analytical model is the Manning–Strickler friction coefficient K. The storage width ratio rS was assumed to be rS=1. The calibrated value of K was 80 m1/3s-1 in the seaward region (x=0–32 km), whereas a smaller value of K=55 m1/3s-1 was used in the river-dominated region (x=52–450 km). Meanwhile, in order to avoid a discontinuous jump caused by the adoption of different friction coefficients, we adopted a friction coefficient of K=80–55 m1/3 s-1 (indicating a linear reduction of the friction coefficient) over the transitional reach (x=32–52 km). Figure  shows a comparison between the observed and computed tidal amplitude and residual water level at different gauging stations along the Yangtze River estuary for a wide range of tide and river discharge conditions. We observe that the correspondence between analytically computed results and observations is good with a high value for the coefficient of determination (R2>0.96), suggesting a reasonable performance of the analytical model for reproducing the first-order tide–river dynamics along the Yangtze River estuary. However, we note an overestimation of the analytically computed residual water level at upstream stations (i.e. MAS and WH) for values >5 m, which is likely due to the oversimplification of the geometry and flow characteristics (e.g. neglecting the M4 and M6 overtides) in the analytical model.

The calibrated analytical model is subsequently used to explore the response of the main tide–river dynamics (represented by the damping/amplification number δ, the velocity number μ, the celerity number λ and the phase lag ε) to the seasonal variation of river discharge (Figs.  and S1–S3 in the Supplement). Figure a shows the spatial–temporal patterns of the damping number δ for the studied period (2003–2014), along with its minimum value δmin, which indicates the maximum tidal damping. The most noticeable feature of the spatial–temporal pattern of tidal damping is the seasonal variation with river discharge, which is clearly illustrated by the temporal variation of the tidal damping critical value δmin (see the red line in Fig. a, varying between 233 and 500 km in the Yangtze River estuary). To be more specific, in Fig. b, it can be observed that the critical value of tidal damping and its position along the estuary are negatively correlated with the corresponding river discharge. Generally, the critical value of tidal damping δmin reaches its minimum in December or January when the monthly average river discharge is minimum, and reaches its maximum in July with maximum river discharge throughout the year. Similarly, we observe that the position along the estuary with the critical δmin reaches its maximum value for minimum river discharge, and minimum value for maximum river discharge. Similar seasonal behaviour of velocity amplitude (denoted by the velocity number μ), wave celerity (denoted by the celerity number λ) and phase lag (denoted by ε) can also be reproduced using the calibrated analytical model (see Figs. S1–S3). In general, we observe a negatively correlated relation between μ, ε and Q, and a positively correlated relation between λ and Q.

For a typical tidal river, it is usually observed that the tidal range is reduced when the residual water level rises in the landward direction owing to the residual water level slope, which is mainly balanced by the residual frictional effect . To understand the underlying mechanism of tidal wave propagation under the influence of river discharge, we adopted a decomposition method that can be used to quantify the contributions of tide, river and tide–river interaction to the residual water level slope S see detailed derivation in, computed as: 19S=-1K2h‾4/3πp0υ2+p1υU+p2U2+p3U3/υ‾. Equation () can be decomposed into three components contributing to the increase of the residual water level: a tidal component, 20St=1K2h‾4/3π12p2+p0υ2; a riverine component, 21Sr=1K2h‾4/3πp2-p3φUr2; and tide–river interaction component, 22Str=1K2h‾4/3π-p1-32p3υUr.

Figure  shows the seasonal variation of the residual water level slope S, exhibiting a positively correlated relationship with the river discharge. It can be seen from Fig. a that the temporal variation of critical value Smax (maximum value) is quite similar to the tidal damping (see Fig. a), which suggests that the development of tide–river dynamics (e.g. tidal damping) is closely related to the residual water level slope. This indicates that both the maximum value of the residual water slope Smax (Fig. b) and its position along the estuary (Fig. a) are positively correlated with river discharge. To further understand the relative importance of tidal forcing alone, river flow alone and tide–river interactions, Eqs. ()–() were used to quantify the contributions made by both tidal and riverine forcing. The results of these separate components are presented in Fig. S4, where we observe that the main contribution lies in the riverine component Sr. In addition, we note that the position of the maximum riverine component Sr is landward of the corresponding maximum values of the other two contributions (St and Str), which is mainly due to the relatively larger residual frictional effect introduced by the riverine forcing.

To understand the main processes that control the development of a maximum tidal damping, we used the average values of the observed tidal amplitude at TSG and the river discharge at DT as model inputs and reproduced the main tide–river dynamics along the Yangtze River estuary. Figure  shows the longitudinal variation of the main tidal dynamics (δ, λ, μ and ε) in addition to the contributions made by both tidal and riverine forcing to the residual water level slope and the water depth for both the wet (Fig. a, c, e) and dry (Fig. b, d, f) seasons. The discontinuous jump occurring at x=42 km depends on the adoption of different friction coefficients in the analytical model. Apparently, the critical position of maximum tidal damping is closer to the sea side during the wet season (around x=305 km) than during the dry season (around x=410 km) owing to the river discharge flush. In addition, the position of maximum tidal damping (corresponding to the minimum value of the damping number δ, indicated by the dashed black line) is almost coincident with the maximum value of the celerity number λ and the minimum value of the velocity number μ. The slightly lagged responses of λ and μ to δ are due to nonlinear interaction between these main tide–river dynamics parameters, as described by the set of nonlinear equations, Eqs. ()–(). This also indicates the significantly nonlinear effect caused by estuary shape, bottom friction and river discharge as the tidal wave propagates upriver. The change in phase lag ε directly follows from the phase lag equation tan⁡(ε)=λ/(γ-δ) (see Eq. ). As can be seen from Fig. a–d, in general the phase lag ε is positively correlated with the damping number δ and the velocity number μ, whereas it is negatively correlated with the celerity number λ. Unlike tide-dominated estuaries with a negligible residual water level, the key parameter that determines the nonlinear relationship between the phase lag ε and the other variables (δ μ, λ) in tidal rivers lies in the water depth, which is controlled by the dynamics of the residual water level. The underlying mechanism generating the maximum tidal damping can be clearly shown in Fig. e and f, where we observe that the residual water level slope S and its dominant river component Sr increase to a maximum value near the critical position of maximum tidal damping, beyond which it is reduced. This means that the main tide–river dynamics are driven by the residual water level slope and that the critical position of maximum tidal damping is primarily controlled by the riverine forcing component. Furthermore, we also note that the maximum value of S corresponds to the local minimum h‾, which suggests a dominant impact of residual water level (and hence water depth) on tide–river dynamics in the Yangtze River estuary.

It is also worth examining the longitudinal and seasonal variations of the two controlling parameters represented by the estuary shape number γ and the friction number χ (see Fig. ), which are closely related to the strength of tidal damping δ. Remarkably, it is important to note that the effect of channel convergence (represented by γ) is stronger during the dry season (larger value of γ) than during the wet season. This indicates that river discharge also tends to reduce the channel convergence via the generation of the residual water level slope. In addition, we observe a switch of γ from positive to negative at x=290 km and x=394 km for the wet and dry seasons, respectively (Fig. a). The cause of the negative value for γ is that the cross-sectional area increases in the landward direction (hence dA‾/dx>0) owing to the increasing residual water level and depth. In Fig. b, we observe a larger value for the friction number χ during the dry season than during the wet season, which is mainly due to the relatively larger tidal amplitude during the dry season and the residual water level (hence the water depth) increasing with river discharge (see the definition of χ in Table ). Furthermore, it is also noted that χ asymptotically approaches zero with distance. This means that in the upstream part of the estuary tide–river dynamics are primarily determined by the geometric effect (i.e. the divergence of the cross-sectional area) and the residual frictional effect caused by riverine forcing Sr (see Eq. ).

Based on the analytical results, in Fig.  we display how the tidal damping δ, the residual water level slope S and the residual water level Z‾ develop as a function of river discharge Q for different positions in the upstream river-dominated region, where the maximum tidal damping occurs. Figure a shows the tidal damping at different positions with different river discharges. It also displays the critical value of river discharge corresponding to maximum tidal damping. As expected, more river discharge is required to change tidal damping from a negative gradient (indicating a strengthening damping with respect to river discharge Q) to a positive gradient (indicating a weakening damping with respect to river discharge Q) for the seaward positions where tide exerts more influence. The critical river discharge is approximately 34 000 m3 s-1 at x=470 km, and it gradually increases to 55 000 m3 s-1 at x=350 km. In Fig. a, we also note that beyond the critical value of maximum damping the δ appears to slightly increase until an asymptotic value is approached. Figure b and c show the relation between S, Z‾ and Q. It is noticeable that the curves for residual water level Z‾ appear as straight lines, corresponding to a consistent increase of the residual water level slope S with river discharge Q. Unlike the longitudinal variation of the maximum S value (see Fig. e, f), both S and Z‾ monotonously increase with Q. In addition, it can be seen from Fig. a that a threshold of approximately 15 000 m3 s-1 exists for the tidal damping with respect to the position along the estuary. At lower river discharges (Q<15 000 m3 s-1), the damping number δ tends to decrease (indicating a strengthening damping) in the landward direction, whereas opposite is seen at higher river discharges (Q>15 000 m3 s-1).

The underlying mechanism for achieving a critical river discharge for maximum tidal damping can be primarily attributed to the cumulative effect of the residual water level Z‾ altering the water depth and hence the channel convergence and effective friction, according to the definitions of the estuary shape number γ and the friction number χ in Table . Figure  presents these two controlling parameters (γ and χ) as a function of river discharge Q. It can be clearly seen in Fig. a that there is an apparent switch of the estuary shape number γ from positive (indicating a reduction of cross-sectional area in the landward direction) to negative (indicating an increase of cross-sectional area in the landward direction). In addition, more river discharge is required to achieve a switch in the estuary shape number γ for the seaward positions where tide exerts more influence. The main reason for such a switch is the consistent increase of the residual water level and, in turn, the water depth and cross-sectional area with river discharge. Conversely, the effective friction induced by tidal forcing (represented by χ) asymptotically approaches zero with the river discharge (see Fig. b), which suggests that the estuarine system is primarily controlled by the divergence of the cross-sectional area and the residual frictional effect caused by riverine forcing (represented by Sr in Eq. ) for high river discharge conditions. Additionally, we can conclude that the asymptotic behaviour of tidal damping δ with high river discharge (as shown in Fig. a) is due to the corresponding asymptotic behaviour of the estuary shape number γ and the friction number χ (and hence the residual frictional effect indicated by S as presented in Fig. b).

Although the current analytical model can reproduce the first-order tide–river dynamics well, it also has some limitations. The fundamental assumption is that the tidal wave can be described by a combination of a steady residual term (generated by the river discharge) and a time-dependent harmonic wave (introduced by the tidal flow). Thus, the proposed model can only capture the tidal asymmetry caused by tide–river interaction while it neglects the tidal asymmetry introduced by astronomical tides (e.g. nonlinear interactions among K1, O1 and M2), overtides (e.g. M4) and compound tides (e.g. MSf). Consequently, the proposed analytical method is preferably applied to tidal rivers with a predominant tidal constituent (e.g. M2 or K1).

It is assumed that both the tidally averaged cross-sectional area and channel width can be approximated by exponential functions following Eqs. ()–(). However, this is not a restrictive assumption as the model can, in principle, be applied to an arbitrary estuarine shape (i.e. bed elevation and channel width), as long as the variation of the cross-section is gradual. The proposed model can also be used to quantify the spring–neap variability of the tide–river dynamics based on daily averaged tidal amplitude and river discharge conditions see example in. However, the model cannot be used to explore the tide–river dynamics within a tidal cycle as it is based on a tidally averaged scale. This means that it may not be applicable to the cases with rapidly varying river discharge.

Knowledge of the development and evolution of tide–river dynamics that determine the behaviour of tidal damping and residual water level slope under external forcing (e.g. tidal and riverine flow) and geometry changes (e.g. deepening and land reclamation) are essential for improving the sustainable water management in estuaries. Adopting the method proposed in this study, one can evaluate the influence of human intervention on the estuarine system (such as large-scale sand excavation, dredging for navigational channels or freshwater withdrawal), on flood control structures (e.g. storm surge barriers, flood gates) and on the aquatic environment (e.g. such as salt intrusion and the related water quality). For instance, explored how the freshwater regulation of the Three Gorges Dam (the world's largest hydroelectric station in terms of installed power capacity) may affect the alteration of the tidal limit in the Yangtze River estuary using the analytical model proposed in this paper. It was shown that the largest change of the tidal limit of around 75 km occurred in October owing to the substantial increase in freshwater discharge. When combined with ecological or salt intrusion models, the analytical approach presented in this study is particularly useful for a quick computation of the longitudinal distribution of the salinity e.g.. Using salinity as a general predictor, it is possible to assess the potential impacts of human intervention on the aquatic ecosystem health in general (e.g., water quality, water utilisation and agricultural development in the estuarine area).

As tide propagates into an estuary, it is distorted and becomes asymmetric due to significant nonlinear interactions with geometry and river flow. Tidal asymmetry is regarded as one of the most important mechanisms generating residual sediment transport e.g.. Although the current analytical method can only deal with tide–river interaction for a single predominant tidal constituent (e.g. M2), the model does capture the major tidal asymmetry induced by geometric effect and riverine flow (e.g. the tidal asymmetry caused by the residual river flow) and can reproduce the seasonal behaviour of tidal damping and residual water level slope well. It was shown by that the erosion and deposition patterns along an estuary are strongly related to the shape of the residual water level profile, which we have shown to be linked to the tide–river dynamics and the geometry of the estuary. The successful reproduction of the seasonal behaviour of tide–river dynamics and residual water level slope in the Yangtze River estuary suggests that the proposed analytical approach can be used as a tool for detecting the evolution of estuarine morphology under various external forcing conditions. However, further studies are required to quantify the relationship between the residual water level slope and the estuarine morphology.