The mean water level in estuaries rises in the landward direction due to a combination of the density gradient, the tidal asymmetry, and the backwater effect. This phenomenon is more prominent under an increase of the fresh water discharge, which strongly intensifies both the tidal asymmetry and the backwater effect. However, the interactions between tide and river flow and their individual contributions to the rise of the mean water level along the estuary are not yet completely understood. In this study, we adopt an analytical approach to describe the tidal wave propagation under the influence of substantial fresh water discharge, where the analytical solutions are obtained by solving a set of four implicit equations for the tidal damping, the velocity amplitude, the wave celerity, and the phase lag. The analytical model is used to quantify the contributions made by tide, river, and tide–river interaction to the water level slope along the estuary, which sheds new light on the generation of backwater due to tide–river interaction. Subsequently, the method is applied to the Yangtze estuary under a wide range of river discharge conditions where the influence of both tidal amplitude and fresh water discharge on the longitudinal variation of the mean tidal water level is explored. Analytical model results show that in the tide-dominated region the mean water level is mainly controlled by the tide–river interaction, while it is primarily determined by the river flow in the river-dominated region, which is in agreement with previous studies. Interestingly, we demonstrate that the effect of the tide alone is most important in the transitional zone, where the ratio of velocity amplitude to river flow velocity approaches unity. This has to do with the fact that the contribution of tidal flow, river flow, and tide–river interaction to the residual water level slope are all proportional to the square of the velocity scale. Finally, we show that, in combination with extreme-value theory (e.g. generalized extreme-value theory), the method may be used to obtain a first-order estimation of the frequency of extreme water levels relevant for water management and flood control. By presenting these analytical relations, we provide direct insight into the interaction between tide and river flow, which will be useful for the study of other estuaries that experience substantial river discharge in a tidal region.

It is of both theoretical and practical importance to understand the dynamics
of wave propagation under the backwater effect, for instance when a river is
backed up by an obstruction, such as a weir or a bridge, by a confluence with
a larger river, or by an ocean tide, resulting in a rise of the water level
upstream of the obstruction. Generally, the backwater effect can be
quantified by using the variation of the water level slope in the momentum
equation. Many researchers have explored the backwater effect in open
channels by disregarding one or more terms in the momentum equation (detailed
review can be found in

It has been suggested that the mean water surface of a tidal river is driven
by the fortnightly fluctuation due to the spring–neap changes in tidal
amplitude at the seaward side, but it also features a consistent increase in
the
landward direction, caused by the tide–river interaction

It was shown by

The current work is not just an application of a model to a case study, but an analysis that provides new analytical tools to assess the influence of fresh water discharge on water levels in estuaries. For the first time, we used a fully analytical approach to quantify the contributions made by different components (tide, river, and tide–river interaction) to the residual water level, which sheds new light on how backwaters are generated as a result of tide–river interaction. The method is subsequently used to estimate the frequency of extreme high water along the estuary, which is particularly useful for water management and flood control.

In the following section, the general methodology for describing the tidal wave propagation under riverine influence and contributions made by different frictional components (river, tide, tide–river interaction) to the rise of mean water level are presented. This is followed by an application to the Yangtze estuary where there is a notable influence of fresh water discharge on tidal dynamics (Sect. 3). We explored the response of the mean water level as a function of tidal forcing imposed at the mouth and the fresh water discharge from upstream. Subsequently, the method has been used to predict the envelopes of high water and low water in the Yangtze estuary. In particular, it is shown that the analytical model can be used to estimate the likelihood of extreme high water levels along the estuary for given probability of exceedance. Finally, conclusions are summarized in Sect. 4.

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

Figure

In a tidal river, we usually observe that the tidally averaged water level
rises in landward direction

The density-induced pressure in the momentum equation is upstream directed
and counteracted by a residual water level that equals 1.25

Variation of the estuarine shape (Eq.

It has been suggested by

Definitions of parameters used in the governing
Eqs. (

We use the analytical model for tidal dynamics proposed by

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

Variation of the Chebyshev coefficients

Apart from the damping Eq. (

The scaling equation describes how the ratio of velocity amplitude to tidal
amplitude depends on phase lag and wave speed (wave celerity):

The wave celerity (or speed) equation describes how the wave speed depends
on the balance between convergence and tidal damping/amplification:

The phase lag equation describes how the phase lag between HW and HWS
depends on wave speed, convergence, and damping:

In Fig.

Analytical solutions of the four dependent
dimensionless variables –

Contour plot of the water surface gradient

Sketch of the water levels in a tidal river

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

Figure

Location of the study area

With the thus obtained water surface gradient

An iterative procedure is involved to determine the mean water
surface because the analytical expression Eq. (

It was shown by

The dependent parameters

Semi-logarithmic plot of the geometric characteristics
(the cross-sectional area

Based on the computed

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.

Observations of tidal amplitude at the estuary mouth
(Hengsha station) and fresh water discharge at the upstream boundary (Datong
station) during the dry

The total length of the Yangtze estuary is around 600

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.

Comparison between analytically computed tidal
amplitude

Longitudinal variation of the mean water level along
the Yangtze estuary axis as a function of time for the dry season

The geometric characteristics of the Yangtze estuary.

The calibrated parameters that were obtained by fitting Eqs. (

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

Longitudinal variation of the tidal amplitude and
corresponding damping number

Longitudinal variation of the high water level

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.

Figure

From the analysis presented in Sect. 2.3, it is suggested that the water
level slope

Shape of the Yangtze estuary

The return values of EHWL (m) at different positions along the Yangtze estuary.

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
(

The fitted GEV distribution against observed maximum
mean daily discharge

Figure

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

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.

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. M

To investigate the impact of tide–river dynamics on the behaviour of mean
water level profile in estuaries, an analytical approach was used to explore
the response of residual water level to the two dominant forcings, i.e. tide
and river flow. The analytical model allows for quantifying the contributions
made by tide and river forcings to the rise of the mean water level along the
estuary by making use of the Dronkers' Chebyshev polynomials approximation to
the friction term. The distinguishing feature of the present approach is that
it allows for analytical prediction of tidally averaged mean water level and
tidal amplitude for given inputs of tidal forcing at the estuary mouth,
geometry, and fresh water discharge, while the previous studies adopted a
linear regression model to estimate the sub-tidal water level and usually
required long-term time series of water level or velocity

The analytical model requires certain assumptions on the geometry and flow
characteristics. The fundamental assumption is that the funnel–prismatic
shape of a typical tidal river can be described by Eqs. (

Despite the fact that the analytical model requires a certain number of assumptions and thus the results are not as accurate as those of a fully non-linear numerical model, there are some important advantages in using a simplified analytical approach, as compared to numerical models. First of all, the analytical models are completely transparent, allowing direct assessment of the influence of individual variables and parameters on the resulting mean water level. In addition, analytical methods are fast and efficient so that wide ranges of input parameters can be considered. Furthermore, they are more appropriate in data-poor (or ungauged) estuaries since only a minimum amount of (geometrical) data is required. Finally, they provide direct insight into cause–effect relations, which is not as straightforward in numerical models.

The hydrodynamics model has been used to reproduce the main dynamics in the Yangtze estuary, which shows good correspondence with observed data. The model is subsequently used to explore the longitudinal variation of mean water level under a wide range of tidal amplitude and fresh water discharge conditions. It is shown that both tidal amplitude and fresh water discharge tend to rise the mean water level along the Yangtze estuary as a result of the non-linear frictional dissipation. Specifically, the mean water level is influenced primarily by the tide–river interaction in tide-dominated region, while it is mainly controlled by the river flow in the upstream part of the estuary. The contribution made by pure tidal influence only becomes important in the transitional zone, where the river flow velocity to tidal velocity amplitude ratio approximately equals 1. Finally, we also demonstrate that the proposed method can be used to predict the envelopes of high water and low water, which is very useful when assessing the potential influence of intensified extreme river floods and human interventions (e.g. dredging for navigational channel or fresh water withdrawal along the estuary) on along-channel water levels. More importantly, the analytical approach in combination with extreme-value theory can be used to estimate the extreme high water level frequency distribution and the likelihood of various extreme values as a function of return period, which makes the proposed method a useful tool for water management (e.g. flood control measures).

Longitudinal variation of the tidal amplitude and
corresponding damping number

Nomenclature.

Continued.

Substituting the total velocity

It should be noted that the main dynamics (e.g. the velocity amplitude

Making use of the friction factor

Based on the obtained hydrodynamics along the estuary, it follows directly
from Eq. (

Figure

The authors would like to thank Maximiliano Sassi and the other anonymous referee for their constructive comments and suggestions, which have substantially improved this paper. This research was financially supported by National Natural Science Foundation of China with the reference no. 41476073. Edited by: E. Zehe