Assessing the impacts of both natural (e.g. tidal forcing from the ocean) and human-induced changes (e.g. dredging for navigation and land reclamation) on estuarine morphology is particularly important for the protection and management of the estuarine environment. In this study, a novel analytical approach is proposed for the assessment of estuarine morphological evolution in terms of tidally averaged depth on the basis of the observed water levels along the estuary. The key lies in deriving a relationship between wave celerity and tidal damping or amplification. For given observed water levels at two gauging stations, it is possible to have a first estimation of both wave celerity (distance divided by tidal travelling time) and tidal damping or amplification rate (tidal range difference divided by distance), which can then be used to predict the morphological changes via an inverse analytical model for tidal hydrodynamics. The proposed method is applied to the Lingdingyang Bay of the Pearl River Estuary, located on the southern coast of China, to analyse the historical development of the tidal hydrodynamics and morphological evolution. The analytical results show surprisingly good correspondence with observed water depth and volume in this system. The merit of the proposed method is that it provides a simple approach for understanding the decadal evolution of the estuarine morphology through the use of observed water levels, which are usually available and can be easily measured.

Understanding the morphological changes in estuaries due to natural processes and human interventions is especially important with regard to sustainable water management and ecological impacts on the estuarine environment

In a frictionless, prismatic channel of a constant cross section (or an ideal estuary with no damping or amplification), the classical formula for tidal wave propagation can be described by

Previous studies have clearly demonstrated that the behaviour of both the tidal damping (or amplification) and wave speed strongly depends on the degree of channel convergence and on the intensity of friction

It is worth noting that considerable simplifications of geometry and forcing are made in order to derive generic analytical solutions of the governing Saint-Venant equations. Here, we restrict consideration to infinitely long tide-dominated estuaries with low tidal-amplitude-to-depth ratios and Froude numbers. The fundamental assumption is that the geometry (cross-sectional area, width and depth) of the channel can be described by a simple exponential function. We also assume that the velocity of river discharge is small compared to the tidal velocity. In this paper, we use the analytical relation of wave celerity, first proposed by

The Pearl River is the second-largest river in terms of water discharge in China

The paper is organized as follows. Section

The depth-averaged equations for the conservation of mass and momentum in a channel with a gradually varying cross section can be described by

To derive the analytical solution for the tidal hydrodynamics in convergent channels, it is assumed that the tidally averaged cross-sectional area

Considering a system that is forced by a sinusoidal tidal wave with period

Making use of these dimensionless parameters,

In addition, the solutions for the phases of elevation and velocity are given by

Contour plot of the dimensionless dependent parameters (

It is worth considering the solution in the special case of an ideal estuary (with no damping or amplification,

Figure

It is important to note that the analytical solutions of the linear model are local because they depend only on local (fixed position) quantities (i.e. the local tidal-amplitude-to-depth ratio

To address directly the correspondence between tidal dynamics and morphology, the celerity equation (Eq.

Contour plot of the celerity number

Estimation of the tidally averaged depth

Here, the wave celerity

In the inverse analytical model, it should be noted that the estuary is regarded as an ensemble system characterized by a spatially averaged water depth and a specific width convergence length. Based on the analytical expression of tidally averaged depth (Eq.

Contour plot of the estimated depth

Sketch of Lingdingyang Bay

The Pearl River (Fig.

In recent decades, intensive human activities (e.g. land reclamation, channel dredging, sand excavation and dam constructions) have substantially disturbed the natural morphological evolution of Lingdingyang Bay. In particular, land reclamation was usually done in areas shallower than approximately 0.5 m; thus, nearly 200 km

Bathymetric maps of Lingdingyang Bay with scales of

To explore the tidal hydrodynamics in Lingdingyang Bay, tidal water level records (see Fig.

Observed water levels (relative to mean sea level) in January for different periods:

Figure

Bathymetric maps of Lingdingyang Bay in 1965

Geometric characteristics observed in Lingdingyang Bay from 1965 to 2015.

In this paper, the observed wave celerity is derived from the travelling time of both high and low water levels (see Eq.

Estimations of wave celerity

Note that both the wave celerity and the tidal damping (or amplification) rate reflect the imbalance between channel convergence and bottom friction

To investigate the underlying mechanism of such a spring–neap variability of wave celerity, we further rewrite the celerity equation (Eq.

Before an inverse analytical model can be used to predict the morphological changes in estuaries, it is required to calibrate and validate the model against observations well. Hence, the analytical model presented in Sect. 2.1 is used to reproduce the historic physical properties of the tidal wave (i.e. tidal damping or amplification rate and wave celerity) in Lingdingyang Bay. The estuarine system is subject to a harmonic tide at the estuary mouth (i.e. CW station). In order to calibrate and validate the analytical model, the tidal properties (including the tidal amplitude and phase) of the predominant

Longitudinal variations of the geometric characteristics (the tidally averaged cross-sectional area

Geometric characteristics of Lingdingyang Bay.

Calibrated parameters used for the analytical model and the evaluation of its performance using RMSE.

In Fig.

Comparison between analytically computed tidal amplitude

Subsequently, the analytically computed tidal characteristics of Lingdingyang Bay were used to describe how the tidal hydrodynamics are affected by the morphological evolution. In Fig.

Trajectories of the main dimensionless parameters as a function of estuary shape number

Spatially averaged dimensionless parameters in Lingdingyang Bay (0–58 km).

The successful reproduction of tidal hydrodynamics using an analytical model suggests a close relationship between tidal damping (or amplification) and wave celerity, which can be described by the celerity equation (Eq.

We adopted the width convergence length from topographic maps and estimated wave celerity and tidal damping (or amplification) from the observed water levels. Combining these parameters with the calibrated storage width ratio

Estimation of the tidally averaged depth

Figure

Comparison between analytically computed tidally averaged depth

Lingdingyang Bay is a typical funnel-shaped estuary, where the tidal dynamics are one of the main factors maintaining the stability state of estuarine morphology. Generally, the tides in the Pearl River Estuary are influenced by the geometry (i.e. cross-sectional variation) and river discharge

To better understand the response of the morphological evolution to tidal dynamics, we rewrite the equilibrium depth

It should be noted that several assumptions are made in order to derive the analytical solutions for tidal hydrodynamics. The fundamental assumption is that the tidal-amplitude-to-depth ratio and the Froude number are considerably lower than unity so that the linearized Saint-Venant equations can be used for the derivations. A second fundamental assumption is that both the cross-sectional area and width can be described by exponential functions, following Eqs. (

In this paper, a novel approach for estimating the tidally averaged depth was proposed to understand the morphological evolution based on the observed water levels at two stations (at least) in estuaries. The linear analytical hydrodynamics model proposed by

We introduce scaling in Eqs. (

When the non-linear continuity term

The real scales of velocity amplitude

For the case of a frictionless estuary with zero convergence, the classical wave celerity

Under the assumption that

Introducing the dimensionless parameters (Eqs.

Substituting the classical wave speed

The data and source codes used to reproduce the experiments presented in this paper are available from the authors upon request (caihy7@mail.sysu.edu.cn).

All authors contributed to the design and development of the work. The experiments were originally carried out by HC. PZ and SH carried out the data analysis. FL and HC prepared the paper with contributions from all co-authors. EG, PM and QY reviewed the paper.

The authors declare that they have no conflict of interest.

All the authors thank Mick van der Wegen and the other anonymous referee for their constructive comments and suggestions, which have greatly improved the quality of this paper.

This research has been supported by the National Key Research and Development Program of China (grant no. 2016YFC0402601), the National Natural Science Foundation of China (grant nos. 51979296, 51709287, 41706088 and 41476073), the Fundamental Research Funds for the Central Universities of China (grant no. 18lgpy29), the Water Resource Science and Technology Innovation Program of Guangdong Province (grant nos. 2016-20 and 2016-21), and the FCT research contract (grant no. IF/00661/2014/CP1234).

This paper was edited by Hubert H. G. Savenije and reviewed by Mick van der Wegen and one anonymous referee.