For the derivation of analytical solutions of the tidal hydrodynamics equations in estuaries, we require geometric functions to describe the estuary geometry, such as constant geometry e.g., a linear function e.g., a power function e.g., or an exponential function e.g.. Among these, the most common approach is to use an exponential function to describe the cross-sectional area, width, and depth in a tidally averaged scale. This method works very well in a tide-dominated estuary, which usually has a typical funnel shape. However, as opposed to what is generally done, the cross-sectional area and stream width do not converge to zero, but to constant river-dominated values. To better represent the geometry of such funnel–prismatic estuaries, we propose the following expressions to describe the longitudinal variation of cross-sectional area A‾ and stream width B‾ see also: A‾=Ar‾+(A0‾-Ar‾)exp⁡-xa,B‾=Br‾+(B0‾-Br‾)exp⁡-xb, where x is the distance (starting from the estuary mouth), A0‾ and B0‾ represent the cross-sectional area and stream width evaluated at the estuary mouth, Ar‾ and Br‾ represent the asymptotic riverine cross-sectional area and stream width (the overbar denotes the tidally averaged value), while a and b represent the convergence lengths of the cross-sectional area and stream width, respectively. This equation accounts for not only the exponential shape in the seaward part of the estuary, but also the nearly prismatic channel in the landward part. Assuming a near rectangular cross section, the tidally averaged depth is given by h‾=A‾/B‾.

Figure  illustrates the variation of the estuarine shape for different convergence lengths. In this approach, there is no need for an inflection point to cater for the transition from a funnel shape to a prismatic channel.

In a tidal river, we usually observe that the tidally averaged water level rises in landward direction e.g.. This residual water level increases with the fresh water discharge. In order to explore the underlying mechanism of this phenomenon and quantify the contributions of tide, river, and tide–river interaction to the increased residual water level, an analytical solution are invaluable tool since it provides direct insight into the tidal wave propagation under the influence of river discharge.

The density-induced pressure in the momentum equation is upstream directed and counteracted by a residual water level that equals 1.25 % of the mean water depth over the length of salt intrusion, having a significant influence on salt intrusion through gravitational circulation . In the Yangtze estuary the water level rise due to the density gradient is around 0.12 m (corresponding to an estuary depth of 9.5 m) over the salt intrusion length (approximately 50 km). Thus the density-induced slope is rather small (around 3.0×10-8) compared to the frictional dissipation induced by river discharge. Consequently, we neglect the effect of the density gradient on the mean water level profile in this paper.

It has been suggested by that the hydrodynamics in a tidal river is mainly determined by the four dimensionless parameters (see Table ), including the tidal amplitude to depth ratio ζ (representing the boundary condition in the seaward side), the estuary shape number γ (indicating the channel convergence), the friction number χ (representing the frictional dissipation), and the dimensionless river discharge φ (representing the effect of fresh water discharge). Note that in Table  η indicates the tidal amplitude, υ is the velocity amplitude, Ur is the river flow velocity, ω is the tidal frequency, g is the gravity acceleration, K is the Manning–Strickler friction coefficient, rS is the storage width ratio, and c0 is the classical wave celerity defined as c0=gh‾/rS. It is important to recognize that we use a new definition for the estuary shape number as suggested by to account for the asymptotic adjustment to the river cross section, the difference being a factor (1-Ar‾/A‾), which varies with distance although it remains close to unity in the most downstream reach of the estuary.

We use the analytical model for tidal dynamics proposed by , in which the solutions of the main tidal dynamics are obtained by means of solving a set of four implicit equations for the main dynamics, including tidal damping or amplification, wave celerity (or speed), velocity amplitude, and phase lag. The main dependent parameters are described by the following four variables (see Table ): δ represents the amplification number describing the damping (δ< 0) or amplified (δ> 0) rate of along-channel tidal amplitude, μ the velocity number indicating the ratio of actual velocity amplitude to that in a frictionless prismatic channel, λ the celerity number representing the classical wave celerity c0 scaled by the actual wave celerity (speed) c, and ε representing the phase lag between high water (HW) and high water slack (HWS) or between low water (LW) and low water slack (LWS). It is noted that 0≤ε≤π/2, where ε=0 indicates the tidal wave characterized by a standing wave, while ε=π/2 suggests a progressive wave. For a predominant tide (e.g. M2), the phase lag is determined by ε=π-(ϕZ-ϕU), in which ϕZ and ϕU represent the phase of water level and velocity, respectively .

The key aspect of this method is to derive an analytical expression for tidal amplification or damping using the so-called “envelope method”, i.e. by subtracting the envelope curves at HW and LW for details see. In a Lagrangean reference frame, we assume that the velocity of a moving water particle V consists of a steady component Ur, generated by the fresh water discharge, and a time-dependent constituent Ut, introduced by the tidal flow: V=Ut-Ur=υsin⁡(ωt)-Q/A‾, where t is time and Q is the fresh water discharge (treated as a constant during the tidal wave propagation). Consequently, the velocity accounting for fresh water discharge at HW is given by VHW=υsin⁡(ε)-Ur=υsin⁡(ε)-φ, and similarly for LW: VLW=-υsin⁡(ε)-Ur=-υsin⁡(ε)+φ. Making use of Eqs. () and () and using the envelope method, the resulted damping equation, describing the tidal amplification or damping as a result of the balance between convergence (γθ) and friction (χμλΓ), is given by δ=μ2(γθ-χμλΓ)1+μ2β, where θ, β, and Γ account for the effect of river discharge. The expressions of θ and β are shown in Table , while Γ=1πp1-2p2φ+p3φ2(3+μ2λ2/φ2) is a friction factor obtained by using Chebyshev polynomials to represent the non-linear friction term in the momentum equation F=V|V|K2h‾4/3≈1K2h‾4/3πp0υ2+p1υV+p2V2+p3V3/υ, where pi (i=0, 1, 2, 3) represent the Chebyschev coefficients seep. 301, which are functions of φ through α=arccos⁡(-φ): p0=-7120sin⁡(2α)+124sin⁡(6α)-160sin⁡(8α),p1=76sin⁡(α)-730sin⁡(3α)-730sin⁡(5α)+110sin⁡(7α),p2=π-2α+13sin⁡(2α)+1930sin⁡(4α)-15sin⁡(6α),p3=43sin⁡(α)-23sin⁡(3α)+215sin⁡(5α). The coefficients p1, p2, and p3 quantify the contributions made by linear, quadratic, and cubic frictional interaction, respectively. In Fig. , it appears that the value of p0 is small with respect to the values of the other coefficients. We observe that the values of p1 and p2 increase with increasing φ until a maximum value is reached, after which p1 converges to 0 while p2 converges to -π. The value of p3 is decreased with φ and it reduces to 0 for φ<1. For φ≥1, p0=p1=p3=0 and p2=-π, so that Eq. () reduces to F=V2/(K2h‾4/3). If φ=0 (or Q=0), p0=p2=0, p1=16/15, and p3=32/15, so that Eq. () reduces to F=1615πυ2K2h‾4/3Vυ+2Vυ3. It is worth noting that the derived tidal damping Eq. () does account for the tidal asymmetry induced by the interaction between tide and river flow, since we described the velocity of a moving particle at HW and LW as a harmonic wave in combination with a river flow velocity, i.e. Eqs. () and ().

Apart from the damping Eq. (), the other three dimensionless equations are summarized as follows .

The scaling equation describes how the ratio of velocity amplitude to tidal amplitude depends on phase lag and wave speed (wave celerity): μ=sin⁡(ε)λ=cos⁡(ε)γ-δ,

The wave celerity (or speed) equation describes how the wave speed depends on the balance between convergence and tidal damping/amplification: λ2=1-δ(γ-δ),

The phase lag equation describes how the phase lag between HW and HWS depends on wave speed, convergence, and damping: tan⁡(ε)=λγ-δ.

In Fig. , we see the contour plot displaying the main dependent parameters computed by solving the set of Eqs. (), (), (), and () over a wide range of estuary shapes (0<γ<4), and friction (0<χ<5) for given values of ζ=0.1, φ=0.5, rS=1.

Based on the assumptions of a negligible density effect and a periodic variation of velocity, the integral of the momentum equation over a tidal period yields the mean water level gradient with respect to distance see also: ∂z‾∂x=-F‾=-1K2h‾4/3πp0υ2+p2υV+p2V2+p3V3/υ‾, where z‾ is the mean water level or residual water level (see Fig. ). Substituting the total velocity V from Eq. () into the friction term F in Eq. () leads to three components contributing to the increase of mean water level: a tidal component Ft‾=1K2h‾4/3π12p2+p0υ2, a riverine component Fr‾=1K2h‾4/3πp2-p3φUr2, and tide–river interaction Ftr‾=1K2h‾4/3π-p1-32p3υUr.

Figure  shows the analytically computed gradient of the water surface over a wide range of river flow velocities (Ur=0–2 ms-1) and tidal velocity amplitudes (υ=0–2 ms-1) for given h‾=10 m and K=45 m1/3s-1. In general, we see that both river flow velocity and velocity amplitude trigger an increase of the water surface gradient and hence the mean tidal water level.

With the thus obtained water surface gradient ∂z‾/∂x, the mean water surface is given by z‾=∫0x∂z‾∂xdx=-∫0xF‾dx=-∫0xFt‾+Fr‾+Ftr‾dx. Equation () has been tested by comparing the analytical computations with numerical results and the good agreement suggests that it can well reproduce the correct mean water level profile along the estuary axis. For details, readers can refer to Sect. 5 of .

An iterative procedure is involved to determine the mean water surface because the analytical expression Eq. () contains two unknown variables, the velocity amplitude υ and the updated water depth expressed as hnew‾=h‾+z‾ (see Fig. ).

It was shown by that the quadratic velocity V|V| in the friction term can be linearized by means of adopting the first- and third-order terms as a function of non-dimensionlized velocity scaled by the maximum possible value of the velocity (i.e. υ+Ur in our case). Similar expressions as in Eqs. ()–() can be obtained by using Godin's approximation to the quadratic velocity V|V|, which are presented in Appendix . The Godin's approximation does perform well in the downstream part of the estuary, where the current is bi-directional. However, the approximation does not convergence to V2 in the river-dominated region (φ>1). Thus, we would prefer to use Dronkers' approximation to the friction term, which provides a consistent description for the whole estuary.

The dependent parameters δ, μ, λ, and ε represent the localized tidal dynamics since they depend on local (fixed position) values of the dimensionless tidal amplitude ζ, the shape of the estuary γ, the bottom friction χ, and the dimensionless river discharge φ. In order to correctly reproduce the main tidal hydrodynamics along the entire estuary axis, we adopt a multi-reach approach by subdividing the entire estuary into multiple reaches to account for the longitudinal variations of the cross sections (such as water depth and bottom friction). For given amplification number δ and tidal amplitude η0 at the seaward boundary of each reach, a tidal amplitude η1 at a distance Δx (e.g. 1 km) upstream can be calculated by a simple explicit integration of the amplification number: η1=η0+dηdxΔx=η0+η0ωδc0Δx.

Based on the computed η1 and the geometric feature (e.g. depth) of the next reach, the main tidal dynamics δ, μ, λ, and ϵ can be obtained by solving the set of Eqs. (), (), (), and (). Such a process can be repeated by moving the origin of axis for each reach, leading to the solutions for the entire estuary. In principle, the proposed method is valid for an arbitrary bed profile, even with strong longitudinal gradient of bed elevation. An example of MATLAB scripts is provided as supplement.

The Yangtze River, which is the largest and longest river in the world, originates from the Tibetan Plateau and debouches into the East China Sea (Fig. ). Characterized by substantial fresh water discharges and meso-scale tides, the Yangtze estuary has a branched structure. Downstream from Xuliujing, the estuary is subdivided into the south branch and north branch divided by Chongming Island (see Fig. ). The south branch is the main channel conveying both fresh water discharge and sediment into the East China Sea, while the north branch is barely connected to the main channel and functions in isolation . Hence, in this paper we only consider the branched system downstream from the junction between the south branch and the north branch, which in our view functions as an entity for tidal hydrodynamics, so that we may treat it as a whole. Meanwhile, since we concentrate on the dominant tide–river interaction process in the Yangtze estuary, the influence of the net water, salt and sediment fluxes from the north branch into the south branch on the tide–river interaction is neglected.

The total length of the Yangtze estuary is around 600 km starting from the mouth, located at the Hengsha gauging station, up to the station of Datong, where the influence of tidal flow is vanishing. The estuary has a meso-scale tide with a maximum and mean tidal range of 4.62 and 2.67 m near the estuary mouth, respectively. The predominant tidal constituent in the Yangtze estuary is semi-diurnal, with averaged ebb and flood duration of 7.4 and 5 h near the estuary mouth, respectively . On the basis of observed data at Datong hydrological station from 1950 to 2012, the annual mean fresh water discharge is 28 200 m3s-1 and the monthly mean fresh water discharge reaches a maximum of 49 500 m3s-1 in July and a minimum of 11 300 m3s-1 in January. It has been suggested that the Canter–Cremers number (representing the ratio of the amount of fresh water to saline water entering the estuary during a tidal period) during a mean spring tide is around 0.1 during the dry season and about 0.24 during the wet season, which suggests a partially mixed salt intrusion in the south branch, where a well-mixed situation occurs during the dry season especially during the spring tide, when the Canter–Cremers number is less than 0.1 .

The topography used in this paper was obtained based on the navigation charts in 2007 having corrected to mean sea level of Huanghai 1985 datum. In Fig. , the geometric characteristics (i.e. the cross-sectional area, the stream width, the estuary depth) along the Yangtze estuary axis together with the best-fitting curves are shown. We see that both the cross-sectional area and stream width can be well represented by using functions of Eqs. () and (), which converge exponentially towards a constant cross section in the river part. The positions of the cross sections are presented in Fig.  as red line segments. It is noted that the conventional approach of using ordinary exponential functions (that converge to zero) can only be used if the estuary is subdivided into two reaches, i.e. a more strongly convergent channel in the seaward part and a more prismatic channel in the landward part of the estuary, with an inflection point at the position where the geometry switches from a funnel-shaped estuary to a more prismatic channel e.g.. The newly proposed Eqs. () and (), however, describe the shape of the entire estuary as an entity, using only one convergence scale, the convergence lengths a and b. From Fig. , we observe that the tidally averaged depth gradually increases until the position around x=245 km (between Jiangyin and Zhenjiang; see Fig. ), after which the depth decreases slightly towards a constant value. It should be noted the depth h‾ presented in Fig.  is the averaged depth relative to mean sea level, while the actual depth hnew‾ is reproduced by an iterative procedure described in Sect. 2.3.

The calibrated parameters that were obtained by fitting Eqs. () and () against observed geometry are presented in Table , where R2 is the coefficient of determination. The enhanced convergence length for cross-sectional area is 117 km, which is slightly larger than that for the width of 103 km.

To demonstrate the capability of the hydrodynamic model, the analytical solutions were compared with tidal amplitudes and residual water levels measured along the Yangtze estuary. The data were collected in February 2012 (6–26 February 2012, representing the dry season) and in August 2012 (10–26 August 2012, representing the flood season). In particular, the observed water levels at different gauging stations have been corrected and referenced to mean sea level of Huanghai 1985 datum. We determined the tidal amplitude by averaging the flood tidal amplitude and the ebb tidal amplitude. Figure  shows the observed tidal amplitude at the estuary mouth (Hengsha station) and fresh water discharge imposed at the upstream end (Datong station) for both the dry and flood season. Both measurements are tidally averaged values and cover a spring–neap cycle. From Fig. , we see a fluctuation of fresh water discharge during the dry season with a range between 14 850 and 15 900 m3s-1, while much larger values are observed during the flood season ranging between 46 500 and 59 000 m3s-1. We observe that the Yangtze estuary has an irregular semi-diurnal tide character, suggesting two tidal cycles within a day. The zigzag line in Fig.  has to do with the fact that the tidal amplitude is very different between the two tidal cycles within a day.

The extracted values of tidal amplitudes and residual water levels covering a spring–neap cycle from nine gauging stations along the Yangtze estuary (see their positions in Fig. ) have been used to calibrate the analytical model. Figure  shows the comparison of observed and computed tidal amplitude and residual water level at different gauging stations in the Yangtze estuary for both the dry and flood seasons. We see that the analytical results are in good agreement with observations, suggesting that the analytical model performs well and can correctly reproduce the main tidal dynamics in the Yangtze estuary. The scatter is mainly due to the fact that the simplified geometry adopted in the analytical model does not take account of the irregularities in the channel due to islands and fluctuations in the cross-sectional area. The calibrated friction coefficient K adopted in the seaward reach (0–245 km) is 75 m1/3s-1, which is realistic for a silt-mud part of the estuary, while K=55 m1/3s-1 in the landward reach (245–550 km) due to the fact that the sediment becomes coarse (sand) in the riverine part. For simplification, we used a constant storage width ratio rS of unit, indicating negligible influence of storage area on tidal dynamics. However, we note that the possible effect of bank storage area could be compensated by the adjustment of the friction coefficient.

Figure  shows the longitudinal variation of the mean water level under the influence of tide and river discharge at different tidal cycles for both dry season and flood season. We see that the development of the mean water level is closely related to the fresh water discharge and the tidal forcing at the estuary mouth. During the dry season when the river flow is small compared with the amplitude of tidal flow, we observe that the mean water level is mainly determined by the tidal forcing imposed at the estuary mouth (see Fig. a). Conversely, during the flood season when the river flow dominates, especially in the upstream reach of the estuary, we see the mean water level mainly depends on the fresh water discharge, although the tidal amplitude still has a strong influence on the mean water level variation in the seaward part where the tide flow dominates over the river flow (see Fig. b).

From the analysis presented in Sect. 2.3, it is suggested that the water level slope ∂z‾/∂x and the resulted residual water level z‾ is controlled by three parameters, i.e. the velocity amplitude, the river flow velocity, and the mean water depth. To illustrate the contributions made by both tidal and riverine forcings, we used the averaged values of F‾, Ft‾, Fr‾ and Ftr‾, which are computed at each tidal cycle during 6–26 February 2012 (representing the dry season) and during 10–26 August 2012 (representing the flood season), respectively. In Fig.  we see the longitudinal profiles for the tidal amplitude and its corresponding dimensionless damping rate (Fig. a and b), the longitudinal contributions of river and tide to the flow velocity (Fig. c and d) and the contributions of river and tide to the tidally averaged water level slope (Fig. e and f), for both the dry and the flood season. We observe that the tide–river interaction is the most dominant component in the seaward reach and its influence reduces to null until the critical position where the velocity amplitude is balanced by the river flow velocity (φ=1). We note that both p1 and p3 in Eq. () are equal to 0 when φ>1; thus, the tide–river component is negligible in the upstream reach of the estuary where the influence of river flow is dominant over the tidal flow. Interestingly, in the transitional zone where φ is close to 1, we see that all three components are crucial for the water level slope since they are proportional to the square of the velocity scale (see Eqs. –). With regard to the contribution made by tidal forcing, we observe that it increases to a maximum value near the critical position with φ=1, beyond which it reduces until zero is reached asymptotically. On the other hand, the riverine contribution is monotonously decreasing in the seaward direction. The jump observed around x=245 km has to do with the adoption of different friction coefficients in the analytical model. Meanwhile, a slightly negative contribution from tidal forcing is observed near the estuary mouth for the dry season case (see Fig. c), which is due to the positive value of the factor p2/ 2+p0 in Eq. ().

Understanding the complex behaviour of mean water level profile and its variation under external forcings (tide, river) is very important for water management to evaluate the influence of river floods, man-made structures (e.g. storm surge barriers, flood gates), and ecosystems protections. In particular, obtaining a first-order estimation of high water (h‾+η) and low water (h‾-η) levels is useful for flood control and in case problems arise with regard to fresh water withdrawal and navigation when sufficient data are not available to set up a detailed numerical model. In order to explore the response of high water and low water levels to the fresh water discharge, scenario simulation under given mean tidal amplitude (η0=1.3 m) and spring tidal amplitude (η0=2.3 m) were conducted. The results are shown in Fig. . In general, we see that both high water and low water levels increase in landward direction for different fresh water discharge conditions. Only during low flows do we see that the high water level reaches a maximum value; see Fig. a and b. This is illustrated in Fig. .

Figure  presents the case of extreme high water occurring near the transitional zone of the Yangtze estuary for a spring tide amplitude η0=2.3 m and a small fresh water discharge Q=10 000 m3s-1. The reason for this phenomenon lies in longitudinal variation of the depth, which has its maximum value near the transition zone. The larger depth causes less friction, which favours amplification. At higher discharges, the friction term gains prominence and the amplification disappears.

It is worth examining the likelihood of an extreme high water level (EHWL) as a function of the probability of exceedance along the estuary, since an EHWL is closely linked to flood control and planning of future engineering works (e.g. dam construction, channel deepening, confinement, or widening of channels). In this paper, we used the 3-parameter generalized extreme-value (GEV) distribution to interpret the probability distribution of EHWL. The method has been extensively used in a wide range of regional frequency analysis, such as annual floods, rainfall, wave height, and other natural extremes . For given positive random variable k, the cumulative distribution function of the GEV distribution is given by f(k;α1,α2,α3)=exp⁡-1+α3k-α1α2-1/α3, where α1, α2, and α3 represent shape, location, and scale of the distribution function, respectively. The critical value kr, which is defined as a value that is expected to be equalled or exceeded on average once every interval of time Tr (with probability of 1/Tr), can be computed by solving the equation of f(kr;α1,α2,α3)=1-1/Tr and is given by kr=-ln⁡(1-1/Tr)-α3α2-α2+α1α3α3.

In this paper, we first calculated the GEV distribution of maximum mean daily discharge at Datong gauging station based on the available historical record from 1947 to 2012 (see Fig. a). The three parameters were estimated by the method of maximum likelihood with α1=-0.114, α2=9400, and α3=54 300. From the fitted frequency distribution, we estimated the frequency of the mean daily discharge with a certain return period using Eq. (). Figure b shows the calculated flood discharge at Datong for 2, 5, 10, 20, 50, 100, 200, 500, and 1000 year return period. We assume a constant tidal amplitude of η0=2.3 m (corresponding to the mean spring tide) at the seaward boundary. Subsequently, the analytical model can be used to estimate the extreme high water levels along the estuary for floods of different return periods. Table  presents the resulting EHWL at different stations along the Yangtze estuary, which can be helpful in designing future engineering works to protect against extreme floods. It can be seen from Table  that the EHWL variations in the seaward reach (downstream from Jiangyin) is minor while significant changes occur in the upstream part of the estuary. This is due to the constant spring tidal amplitude imposed at the estuary mouth in the analytical model and thus the variations of EHWL are mainly controlled by the fresh water discharge.

One should be aware that the proposed analytical method only captures the first-order tide–river dynamics in estuaries since the model only accounts for the tidal asymmetry introduced by tide–river interaction while it neglects the tidal asymmetry caused by overtides (e.g. M4) and compound tides (e.g. MSf), which may have a significant impact on the residual water level in the central region of the estuary . For accurate prediction of high water and low water levels, it is required to further study the impact of higher-order terms (e.g. overtides and compound tides) on the generation of the residual water level.