When different tidal constituents propagate along an estuary, they interact
because of the presence of nonlinear terms in the hydrodynamic equations. In
particular, due to the quadratic velocity in the friction term, the effective
friction experienced by both the predominant and the minor tidal constituents
is enhanced. We explore the underlying mechanism with a simple conceptual
model by utilizing Chebyshev polynomials, enabling the effect of the
velocities of the tidal constituents to be summed in the friction term and,
hence, the linearized hydrodynamic equations to be solved analytically in a
closed form. An analytical model is adopted for each single tidal constituent
with a correction factor to adjust the linearized friction term, accounting
for the mutual interactions between the different tidal constituents by means
of an iterative procedure. The proposed method is applied to the Guadiana
(southern Portugal–Spain border) and Guadalquivir (Spain) estuaries for
different tidal constituents (

Numerous studies have been conducted in recent decades to model tidal wave
propagation along an estuary since an understanding of tidal dynamics is
essential for exploring the influence of human-induced (such as dredging for
navigational channels) or natural (such as global sea level rises)
interventions on estuarine environments

Geometry of a semi-closed estuary and basic notation (after

To explore the interaction between different constituents of the tidal flow,
the quadratic velocity

In contrast, as noted by other researchers

Previous studies explored the effect of frictional interaction between
different tidal constituents by quantifying a friction correction factor only

We are considering a semi-closed estuary that is forced by one predominant
tidal constituent (e.g.,

The geometry of a semi-closed estuary is shown in Fig.

With the above assumptions, the one-dimensional continuity equation reads

In order to obtain an analytical solution, we assume a negligible river
discharge and that the tidal amplitude is small with respect to the mean
depth and follow

Definitions of dimensionless parameters.

The main dependent dimensionless parameters are also presented in Table

It is important to remark that several nonlinear terms are present both in
the continuity and in the momentum equations

Both the Guadiana and Guadalquivir estuaries are located in the southwest part of the Iberian Peninsula. These systems are good candidates for the application of a 1-D hydrodynamic model of tidal propagation. Both estuaries feature a simple geometry, consisting of a single, narrow and moderately deep channel with relatively smooth bathymetric variations. Moreover, their tidal prism exceeds their average freshwater inputs by several orders of magnitude due to strong regulation by dams. Under these usual, low river discharge conditions, both estuaries are well-mixed, and the water circulation is mainly driven by tides.

The Guadiana estuary, at the southern border between Spain and Portugal,
connects the Guadiana River to the Gulf of Cádiz. Tidal water level
oscillations are observed along the channel as far as a weir 78 km upstream
of the river mouth

The Guadalquivir estuary is located in southern Spain, at

Tidally averaged depth (m, black dots), width (m, blue dots) and
cross-sectional area (m

Tidal elevation amplitudes (m) and phases (

Tidal dynamics along the Guadalquivir estuary were analyzed by

Chebyshev polynomials can be used to approximate the quadratic dependence
of the friction term on the velocity,

For a single harmonic

Considering a second tidal constituent, the velocity is given by

For illustration, approximations using Eqs. (

Approximation to the quadratic velocity

It can be seen from Eqs. (

Computed effective friction coefficients

Similarly, we are able to extend the same approach to the case of a generic
number

For a single tidal constituent

For illustration of the method, we consider a tidal current that is composed
of one dominant constituent (e.g.,

Introducing a general form of the linearized momentum equation for the
generic

If there are many tidal constituents, then the friction experienced by one is
affected by the others. As suggested by our conceptual model, the mutual
effects can be incorporated by using the friction correction factor

We note that the modified friction number

Computation process for tidal properties of different constituents in an estuary.

With a hydrodynamic model for a single constituent (see
Appendix

It is worth stressing that the single constituents are not calibrated
independently, as was done in previous analyses

Tidal constituents

In this study, the analytical model for a semi-closed estuary presented in
Sect. 2.1 was applied to the Guadiana and Guadalquivir estuaries to
reproduce the correct tidal behavior for different tidal constituents. The
analytical results were compared with observed tidal amplitude

The morphology of the Guadiana estuary was represented in the model with a
constant depth (5.5 m), an exponentially converging width (length scale,
38 km) and a constant storage ratio of 1 representative of the limited salt
marsh areas (about 20 km

Mean correction friction factor

Longitudinal variations in tidal damping/amplification number

To understand the tidal dynamics between different tidal constituents along
the Guadiana estuary, the longitudinal variations in the tidal
damping/amplification number

For the Guadalquivir estuary, the geometry can be approximated as a
converging estuary with a width convergence length of

Figure

In particular, the tidal damping along the first half of these two estuaries
is mainly due to the damping of the dominant

Tidal constituents

Longitudinal variations in tidal damping/amplification number

In order to clarify the behavior of different tidal constituents, we present
Fig.

The importance of mutual interaction between different tidal constituents is
illustrated with the iteratively refined model implemented in both case
studies (Figs.

Longitudinal variations in estuary shape number

Longitudinal variations in damping/amplification number

In this study, we provide insight into the mutual interactions between one
predominant (e.g.,

A crucial feature of the proposed approach is the deterministic description of the mutual frictional interaction among tidal constituents, which avoids the need of an independent calibration of the friction parameter for the single constituent. In this respect, further work is required to explore whether a reliable value of the friction coefficient estimated through this method can be parameterized based on observations of the bottom roughness of the estuary.

The data and source codes used to reproduce the experiments presented in this paper are available from the authors upon request (egarel@ualg.pt).

In this paper, analytical solutions for a semi-closed estuary proposed by

The analytical solutions for the tidal wave amplitudes and phases are given by

Since the friction parameter

For given computed values of

Analytical expressions for unknown complex variables for the case of a closed estuary.

The following trigonometric equation

For the case of many constituents, here we only provide the exact coefficients for

The supplement related to this article is available online at:

HC and EG conceived the study and wrote the draft of the paper. MT, HHGS and QY contributed to the improvement of the paper. All authors reviewed the paper.

The authors declare that they have no conflict of interest.

We acknowledge the financial support from the National Key R & D of China (grant no. 2016YFC0402600), from the National Natural Science Foundation of China (grant no. 51709287), from the Basic Research Program of Sun Yat-Sen University (grant no. 17lgzd12), and from the Water Resource Science and Technology Innovation Program of Guangdong Province (grant no. 2016-20). The work of Erwan Garel was supported by FCT research contract IF/00661/2014/CP1234. Edited by: John M. Huthnance Reviewed by: David Bowers and Job Dronkers