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**Ocean Science**
An interactive open-access journal of the European Geosciences Union

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- Abstract
- Introduction
- Materials and methods
- Conceptual model
- Results
- Conclusions
- Data availability
- Appendix A: Analytical solutions of tidal hydrodynamics for a single tidal constituent
- Appendix B: Coefficients of the Godin's expansion
- Author contributions
- Competing interests
- Acknowledgements
- References
- Supplement

**Research article**
08 Aug 2018

**Research article** | 08 Aug 2018

Frictional interactions between tidal constituents in tide-dominated estuaries

^{1}Institute of Estuarine and Coastal Research, School of Marine Sciences, Sun Yat-sen University, Guangzhou 510275, China^{2}Department of Civil, Environmental and Mechanical Engineering, University of Trento, Trento, Italy^{3}Department of Water Management, Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, the Netherlands^{4}Centre for Marine and Environmental Research (CIMA), University of Algarve, Faro, Portugal

^{1}Institute of Estuarine and Coastal Research, School of Marine Sciences, Sun Yat-sen University, Guangzhou 510275, China^{2}Department of Civil, Environmental and Mechanical Engineering, University of Trento, Trento, Italy^{3}Department of Water Management, Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, the Netherlands^{4}Centre for Marine and Environmental Research (CIMA), University of Algarve, Faro, Portugal

**Correspondence**: Erwan Garel (egarel@ualg.pt)

**Correspondence**: Erwan Garel (egarel@ualg.pt)

Abstract

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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 (*M*_{2}, *S*_{2}, *N*_{2}, *O*_{1}, *K*_{1})
imposed independently at the estuary mouth. The analytical results appear to
agree very well with the observed tidal amplitudes and phases of the
different tidal constituents. The proposed method could be applicable to
other alluvial estuaries with a small tidal amplitude-to-depth ratio and
negligible river discharge.

How to cite

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How to cite.

Cai, H., Toffolon, M., Savenije, H. H. G., Yang, Q., and Garel, E.: Frictional interactions between tidal constituents in tide-dominated estuaries, Ocean Sci., 14, 769–782, https://doi.org/10.5194/os-14-769-2018, 2018.

1 Introduction

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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
(Schuttelaars et al., 2013; Winterwerp et al., 2013). Analytical models are invaluable
tools and have been developed to study the basic physics of tidal dynamics in
estuaries; for instance, to examine the sensitivity of tidal properties
(e.g., tidal damping or wave speed) to change in terms of external forcing
(e.g., spring–neap variations in amplitude) and geometry (e.g., depth or
channel length). However, most analytical solutions developed to date, which
make use of the linearized Saint-Venant equations, can only deal with one
predominant tidal constituent (e.g., *M*_{2}), which prevents consideration of
the nonlinear interactions between different tidal constituents. The
underlying problem is that the friction term in the momentum equation follows
a quadratic friction law, which causes nonlinear behavior, causing tidal
asymmetry as the tide propagates upstream. If the friction law were linear,
one would expect that the effective frictional effect for different tidal
constituents (e.g., *M*_{2} and *S*_{2}) could be computed independently (Pingree, 1983).

To explore the interaction between different constituents of the tidal flow,
the quadratic velocity $u\left|u\right|$ (where *u* is the velocity) is usually
approximated by a truncated series expansion, such as a Fourier expansion
(Proudman, 1953; Dronkers, 1964; Le Provost, 1973; Pingree, 1983; Fang, 1987; Inoue and Garrett, 2007).
If the tidal current is composed of one dominant constituent and a much
smaller second constituent, it has been shown by many researchers
(Jeffreys, 1970; Heaps, 1978; Prandle, 1997) that the weaker constituent is
acted on by up to 50 % more friction than acts on the dominant constituent.
However, this requires the assumption of a very small value of the ratio of
the magnitudes of the weaker and dominant constituents, which indicates that
this is only a first-order estimation. Later, some researchers extended
the analysis to improve the accuracy of estimates and to allow for more than
two constituents (Pingree, 1983; Fang, 1987; Inoue and Garrett, 2007).
Pingree (1983) investigated the interaction between *M*_{2} and *S*_{2}
tides, resulting in a second-order correction of the effective friction
coefficient acting on the predominant *M*_{2} tide and a fourth-order value for
the weaker *S*_{2} constituent of the tide. Fang (1987) derived exact
expressions of the coefficients of the Fourier expansion of $u\left|u\right|$ for two
tidal constituents but did not provide exact solutions for the case of three
or more constituents. Later, Inoue and Garrett (2007) used a novel approach to
determine the Fourier coefficients of $u\left|u\right|$, which allows the magnitude of
the effective friction coefficient to be determined for many tidal
constituents. For the general two-dimensional tidal wave propagation, the
expansion of quadratic bottom friction using a Fourier series was first
proposed by Le Provost (1973) and subsequently applied to spectral
models for regional tidal currents (Le Provost et al., 1981; Le Provost and Fornerino, 1985; Molines et al., 1989).
Building on the previous work by Le Provost (1973), the
importance of quadratic bottom friction in tidal propagation and damping was
discussed by Kabbaj and Le Provost (1980) and reviews of friction terms in models were
presented by Le Provost (1991).

In contrast, as noted by other researchers (Doodson, 1924; Dronkers, 1964; Godin, 1991, 1999), the quadratic
velocity $u\left|u\right|$ is, mathematically, an odd function, and it is possible to
approximate it by using a two- or three-term expression, such as
*α**u*+*β**u*^{3} or $\mathit{\alpha}u+\mathit{\beta}{u}^{\mathrm{3}}+\mathit{\xi}{u}^{\mathrm{5}}$, where *α*, *β* and
*ξ* are suitable numerical constants. The linear term *α**u* represents
the linear superposition of different constituents, while the nonlinear
interaction is attributed to a cubic term *β**u*^{3} and a fifth-order
term *ξ**u*^{5}. It is to be noted that such a method has the advantage of keeping
the hydrodynamic equations solvable in a closed form (Godin, 1991, 1999).

Previous studies explored the effect of frictional interaction between
different tidal constituents by quantifying a friction correction factor only
(e.g., Dronkers, 1964; Le Provost, 1973; Pingree, 1983; Fang, 1987; Godin, 1999; Inoue and Garrett, 2007).
In this study, for the first time, the mutual interactions between tidal
constituents in the frictional term were explored using a conceptual
analytical model. Specifically, a friction correction factor for each
constituent was defined by expanding the quadratic velocity using a Chebyshev
polynomials approach. The model has subsequently been applied to the Guadiana
and Guadalquivir estuaries in southern Iberia, for which cases the mutual
interaction between the predominant *M*_{2} tidal constituent and other tidal
constituents (e.g., *S*_{2}, *N*_{2}, *O*_{1}, *K*_{1}) is explored.

2 Materials and methods

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We are considering a semi-closed estuary that is forced by one predominant
tidal constituent (e.g., *M*_{2}) with the tidal frequency $\mathit{\omega}=\mathrm{2}\mathit{\pi}/T$,
where *T* is the tidal period. As the tidal wave propagates into the estuary,
it has a wave celerity of water level *c*_{A}, a wave celerity of
velocity *c*_{V}, an amplitude of tidal elevation *η*, a tidal velocity
amplitude *υ*, a phase of water level *ϕ*_{A}, and a phase of
velocity *ϕ*_{V}. The length of the estuary is indicated by *L*_{e}.

The geometry of a semi-closed estuary is shown in Fig. 1,
where *x* is the longitudinal coordinate, which is positive in the landward
direction, and *z* is the free surface elevation. The tidally averaged
cross-sectional area $\stackrel{\mathrm{\u203e}}{A}$ and width $\stackrel{\mathrm{\u203e}}{B}$ are assumed to
be exponentially convergent in the landward direction, as described by

$$\begin{array}{}\text{(1)}& {\displaystyle}& {\displaystyle}\stackrel{\mathrm{\u203e}}{A}=\stackrel{\mathrm{\u203e}}{{A}_{\mathrm{0}}}\mathrm{exp}(-x/a),\text{(2)}& {\displaystyle}& {\displaystyle}\stackrel{\mathrm{\u203e}}{B}=\stackrel{\mathrm{\u203e}}{{B}_{\mathrm{0}}}\mathrm{exp}(-x/b),\end{array}$$

where $\stackrel{\mathrm{\u203e}}{{A}_{\mathrm{0}}}$ and $\stackrel{\mathrm{\u203e}}{{B}_{\mathrm{0}}}$ are the respective values at the
estuary mouth (where *x*=0) and *a* and *b* are the convergence lengths of
cross-sectional area and width, respectively. We also assume a rectangular
cross section, from which it follows that the tidally averaged depth is given
by $\stackrel{\mathrm{\u203e}}{h}=\stackrel{\mathrm{\u203e}}{A}/\stackrel{\mathrm{\u203e}}{B}$. The possible influence of
storage area is described by the storage width ratio *r*_{S}, defined as the
ratio of the storage width *B*_{S} (width of the channel at averaged high water
level) to the tidally averaged width $\stackrel{\mathrm{\u203e}}{B}$ (i.e., ${r}_{\mathrm{S}}={B}_{\mathrm{S}}/\stackrel{\mathrm{\u203e}}{B}$).

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

$$\begin{array}{}\text{(3)}& {\displaystyle}{\displaystyle}{r}_{\mathrm{S}}{\displaystyle \frac{\partial h}{\partial t}}+u{\displaystyle \frac{\partial h}{\partial x}}+h{\displaystyle \frac{\partial u}{\partial x}}+{\displaystyle \frac{hu}{\stackrel{\mathrm{\u203e}}{B}}}{\displaystyle \frac{\mathrm{d}\stackrel{\mathrm{\u203e}}{B}}{\mathrm{d}x}}=\mathrm{0},\end{array}$$

where *t* is the time and *h* the instantaneous depth. Assuming negligible
density effects, the one-dimensional momentum equations can be cast as follows

$$\begin{array}{}\text{(4)}& {\displaystyle}{\displaystyle \frac{\partial u}{\partial t}}+u{\displaystyle \frac{\partial u}{\partial x}}+g{\displaystyle \frac{\partial z}{\partial x}}+{\displaystyle \frac{gu\left|u\right|}{{K}^{\mathrm{2}}{h}^{\mathrm{4}/\mathrm{3}}}}=\mathrm{0},\end{array}$$

where *g* is the acceleration due to gravity and *K* is the Manning–Strickler
friction coefficient.

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 Toffolon and Savenije (2011) to derive the linearized solution of the system of Eqs. (1) and (2). However, different from the standard linear solutions, we will retain the mutual interaction among different harmonics originating from the nonlinear frictional term, which contains two sources of nonlinearity: the quadratic velocity $u\left|u\right|$ and the variable depth in the denominator. While we neglect the latter factor, consistent with the assumption of small tidal amplitude, we will exploit Chebyshev polynomials to represent the harmonic interaction in the quadratic velocity (see Sect. 3.1). For clarity, we report here the linearized version of the momentum equation

$$\begin{array}{}\text{(5)}& {\displaystyle}{\displaystyle \frac{\partial u}{\partial t}}+g{\displaystyle \frac{\partial z}{\partial x}}+\mathit{\kappa}u\left|u\right|=\mathrm{0}\end{array}$$

and the friction coefficient

$$\begin{array}{}\text{(6)}& {\displaystyle}{\displaystyle}\mathit{\kappa}={\displaystyle \frac{g}{{K}^{\mathrm{2}}{\stackrel{\mathrm{\u203e}}{h}}^{\mathrm{4}/\mathrm{3}}}}.\end{array}$$

Toffolon and Savenije (2011) demonstrated that the tidal hydrodynamics in a
semi-closed estuary are controlled by a few dimensionless parameters that
depend on geometry and external forcing (for detailed information about
analytical solutions for tidal hydrodynamics, readers can refer to
Appendix Appendix A). They are defined in Table 1 and can be
interpreted as follows: *ζ*_{0} is the dimensionless tidal amplitude (the
subscript 0 indicating the seaward boundary condition); *γ* is the
estuary shape number (representing the effect of cross-sectional area
convergence); *χ*_{0} is the friction number (describing the role of the
frictional dissipation); ${L}_{\mathrm{e}}^{*}$ is the dimensionless estuary length. The
dimensional quantities used in the definition of the dimensionless parameters
are as follows: *η*_{0} is the tidal amplitude at the seaward boundary; ${c}_{\mathrm{0}}=\sqrt{g\stackrel{\mathrm{\u203e}}{h}/{r}_{\mathrm{S}}}$
is the frictionless wave celerity in a prismatic channel; ${L}_{\mathrm{0}}={c}_{\mathrm{0}}/\mathit{\omega}$
is the tidal length scale related to the frictionless tidal wave length by a factor 2*π*.

The main dependent dimensionless parameters are also presented in Table 1,
including the following: *ζ* is the actual tidal amplitude; *χ* is the
actual friction number; *μ* is the velocity number (the ratio of the actual
velocity amplitude to the frictionless value in a prismatic channel);
*λ*_{A} and *λ*_{V} are, respectively, the celerity for elevation and
velocity (the ratio between the frictionless wave celerity in a prismatic
channel and actual wave celerity); *δ*_{A} and *δ*_{V} are,
respectively, the amplification number for elevation and velocity (describing
the rate of increase, *δ*_{A} (or *δ*_{V})>0, or decrease,
*δ*_{A} (or *δ*_{V})<0, in the wave amplitudes along the estuary
axis); $\mathit{\varphi}={\mathit{\varphi}}_{\mathrm{V}}-{\mathit{\varphi}}_{\mathrm{A}}$ is the phase difference between
the phases of velocity and elevation.

It is important to remark that several nonlinear terms are present both in
the continuity and in the momentum equations (Parker, 1991), which are
responsible, for instance, for the internal generation of overtides (e.g.,
*M*_{4}). In this approximated approach, we disregard them and focus
exclusively on the mutual interaction among the external tidal constituents
mediated by the quadratic velocity dependence in the frictional term. In
fact, the nonlinear quadratic velocity term crucially affects the propagation
of the tidal waves associated with the different constituents that are
already present in the tidal forcing at the estuary mouth.

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 (Garel et al., 2009). Both the cross-sectional area and the
channel width are convergent and can be described by an exponential function,
with convergence lengths of *a*=31 km and *b*=38 km, respectively
(Fig. 2). The flow depth is generally between 4 and 8 m, with a
mean depth of about 5.5 m (Garel, 2017).
The tidal dynamics in the Guadiana estuary are derived from records obtained
using eight pressure transducers deployed for a period of 2 months (31 July
to 25 September 2015) approximately every 10 km along the estuary (from the
mouth to ∼70 km upstream). The data were collected during an extended
(months-long) period of drought with negligible river discharge (always
<20 m^{3} s^{−1} over the preceding 5 months). For each station, the amplitude and
phase of elevation of the tidal constituents were obtained from standard
harmonic analysis of the observed pressure records using the “t-tide” Matlab
toolbox (Pawlowicz et al., 2002). The harmonic results are displayed in
Table 2. Near the mouth, the largest diurnal (*K*_{1}), semidiurnal (*M*_{2})
and quarter-diurnal (*M*_{4}) frequencies are similar to those
previously reported at the same location based on pressure records taken over
∼9 months (see Garel and Ferreira, 2013). In particular, the value
$({\mathit{\eta}}_{{K}_{\mathrm{1}}}+{\mathit{\eta}}_{{O}_{\mathrm{1}}})/({\mathit{\eta}}_{{M}_{\mathrm{2}}}+{\mathit{\eta}}_{{S}_{\mathrm{2}}})$ is less than 0.1 at the sea
boundary, which indicates that the tide is dominantly semidiurnal.

The Guadalquivir estuary is located in southern Spain, at ∼100 km to
the east of the Guadiana River mouth. The estuary has a length of 103 km
starting from the mouth at Sanlúcar de Barrameda to the Alcalá del Río dam.
The geometry of the Guadalquivir estuary can be approximated by exponential
functions with a convergence length of *a*=60 km for the cross-sectional area
and *b*=66 km for the width (see Diez-Minguito et al., 2012). The flow depth
is more or less constant (7.1 m).

Tidal dynamics along the Guadalquivir estuary were analyzed by
Diez-Minguito et al. (2012) based on harmonic analyses of field measurements
collected from June to December 2008. The amplitude and phase of tidal
constituents near the mouth are highly similar to those at the entrance of
the Guadiana estuary (Table 2), producing a semidiurnal and
mesotidal signal with a mean spring tidal range of 3.5 m. In this paper, the
tidal observations of the Guadalquivir estuary are taken directly from
Diez-Minguito et al. (2012). The results apply to the low river discharge
conditions (<40 m^{3} s^{−1}) that usually predominate in the estuary.

3 Conceptual model

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Chebyshev polynomials can be used to approximate the quadratic dependence of the friction term on the velocity, $u\left|u\right|$. Adopting a two-term approximation, it is known that (Godin, 1991, 1999)

$$\begin{array}{}\text{(7)}& {\displaystyle}{\displaystyle}u\left|u\right|={\widehat{\mathit{\upsilon}}}^{\mathrm{2}}\left[\mathit{\alpha}\left({\displaystyle \frac{u}{\widehat{\mathit{\upsilon}}}}\right)+\mathit{\beta}{\left({\displaystyle \frac{u}{\widehat{\mathit{\upsilon}}}}\right)}^{\mathrm{3}}\right],\end{array}$$

where $\widehat{\mathit{\upsilon}}$ is the sum of the amplitudes of all the harmonic
constituents. The Chebyshev coefficients $\mathit{\alpha}=\mathrm{16}/\left(\mathrm{15}\mathit{\pi}\right)$ and $\mathit{\beta}=\mathrm{32}/\left(\mathrm{15}\mathit{\pi}\right)$
were determined by the expansion of cos (*n**x*) (*n*=1, 2, …) in
powers of cos (*x*) (Godin, 1991, 1999). It is important to note
that, unlike series developments (e.g., Fourier expansion), the Chebyshev
coefficients *α* and *β* vary with the number of terms that are used
in the development. Godin (1991) already showed that a two-term
approximation (such as Eq. 5) is adequate to satisfactorily account
for the friction.

For a single harmonic

$$\begin{array}{}\text{(8)}& {\displaystyle}{\displaystyle}u={\mathit{\upsilon}}_{\mathrm{1}}\mathrm{cos}\left({\mathit{\omega}}_{\mathrm{1}}t\right),\end{array}$$

where *υ*_{1} is the velocity amplitude and *ω*_{1} its frequency,
Eq. (5) can be expressed by exploiting standard trigonometric relations as

$$\begin{array}{}\text{(9)}& {\displaystyle}{\displaystyle}u\left|u\right|\cong {\mathit{\upsilon}}_{\mathrm{1}}^{\mathrm{2}}\left[{\displaystyle \frac{\mathrm{8}}{\mathrm{3}\mathit{\pi}}}\mathrm{cos}\left({\mathit{\omega}}_{\mathrm{1}}t\right)+{\displaystyle \frac{\mathrm{8}}{\mathrm{15}\mathit{\pi}}}\mathrm{cos}\left(\mathrm{3}{\mathit{\omega}}_{\mathrm{1}}t\right)\right].\end{array}$$

Focusing only on the original harmonic constituent leads to

$$\begin{array}{}\text{(10)}& {\displaystyle}{\displaystyle}u\left|u\right|\cong {\displaystyle \frac{\mathrm{8}}{\mathrm{3}\mathit{\pi}}}{\mathit{\upsilon}}_{\mathrm{1}}^{\mathrm{2}}\mathrm{cos}\left({\mathit{\omega}}_{\mathrm{1}}t\right),\end{array}$$

which coincides exactly with Lorentz's classical linearization (Lorentz, 1926) or a Fourier expansion of $u\left|u\right|$ (Proudman, 1953).

Considering a second tidal constituent, the velocity is given by

$$\begin{array}{ll}{\displaystyle}u& {\displaystyle}={\mathit{\upsilon}}_{\mathrm{1}}\mathrm{cos}\left({\mathit{\omega}}_{\mathrm{1}}t\right)+{\mathit{\upsilon}}_{\mathrm{2}}\mathrm{cos}\left({\mathit{\omega}}_{\mathrm{2}}t\right)\\ \text{(11)}& {\displaystyle}& {\displaystyle}=\widehat{\mathit{\upsilon}}\left[{\mathit{\epsilon}}_{\mathrm{1}}\mathrm{cos}\left({\mathit{\omega}}_{\mathrm{1}}t\right)+{\mathit{\epsilon}}_{\mathrm{2}}\mathrm{cos}\left({\mathit{\omega}}_{\mathrm{2}}t\right)\right],\end{array}$$

where *υ*_{2} and *ω*_{2} are the amplitude and frequency of the
second constituent, and ${\mathit{\epsilon}}_{\mathrm{1}}={\mathit{\upsilon}}_{\mathrm{1}}/\widehat{\mathit{\upsilon}}$ and
${\mathit{\epsilon}}_{\mathrm{2}}={\mathit{\upsilon}}_{\mathrm{2}}/\widehat{\mathit{\upsilon}}$ are the ratios of the
amplitudes to that of the maximum possible velocity
$\widehat{\mathit{\upsilon}}={\mathit{\upsilon}}_{\mathrm{1}}+{\mathit{\upsilon}}_{\mathrm{2}}$. Note that the possible phase lag
between the two constituents is neglected assuming a suitable time shift
(Inoue and Garrett, 2007). In this case, the truncated Chebyshev polynomials
approximation of $u\left|u\right|$ (focusing on two original tidal constituents) is
expressed as (see also Godin, 1999)

$$\begin{array}{}\text{(12)}& {\displaystyle}{\displaystyle}u\left|u\right|\cong {\displaystyle \frac{\mathrm{8}}{\mathrm{3}\mathit{\pi}}}{\widehat{\mathit{\upsilon}}}^{\mathrm{2}}\left[{F}_{\mathrm{1}}{\mathit{\epsilon}}_{\mathrm{1}}\mathrm{cos}\left({\mathit{\omega}}_{\mathrm{1}}t\right)+{F}_{\mathrm{2}}{\mathit{\epsilon}}_{\mathrm{2}}\mathrm{cos}\left({\mathit{\omega}}_{\mathrm{2}}t\right)\right],\end{array}$$

with

$$\begin{array}{ll}{\displaystyle}{F}_{\mathrm{1}}& {\displaystyle}={\displaystyle \frac{\mathrm{3}\mathit{\pi}}{\mathrm{8}}}\left[\mathit{\alpha}+\mathit{\beta}\left({\displaystyle \frac{\mathrm{3}}{\mathrm{4}}}{\mathit{\epsilon}}_{\mathrm{1}}^{\mathrm{2}}+{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{\mathit{\epsilon}}_{\mathrm{2}}^{\mathrm{2}}\right)\right]={\displaystyle \frac{\mathrm{1}}{\mathrm{5}}}\left(\mathrm{2}+\mathrm{3}{\mathit{\epsilon}}_{\mathrm{1}}^{\mathrm{2}}+\mathrm{6}{\mathit{\epsilon}}_{\mathrm{2}}^{\mathrm{2}}\right)\\ \text{(13)}& {\displaystyle}& {\displaystyle}={\displaystyle \frac{\mathrm{1}}{\mathrm{5}}}\left(\mathrm{8}+\mathrm{9}{\mathit{\epsilon}}_{\mathrm{1}}^{\mathrm{2}}-\mathrm{12}{\mathit{\epsilon}}_{\mathrm{1}}\right),\end{array}$$

$$\begin{array}{ll}{\displaystyle}{F}_{\mathrm{2}}& {\displaystyle}={\displaystyle \frac{\mathrm{3}\mathit{\pi}}{\mathrm{8}}}\left[\mathit{\alpha}+\mathit{\beta}\left({\displaystyle \frac{\mathrm{3}}{\mathrm{4}}}{\mathit{\epsilon}}_{\mathrm{2}}^{\mathrm{2}}+{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{\mathit{\epsilon}}_{\mathrm{1}}^{\mathrm{2}}\right)\right]={\displaystyle \frac{\mathrm{1}}{\mathrm{5}}}\left(\mathrm{2}+\mathrm{3}{\mathit{\epsilon}}_{\mathrm{2}}^{\mathrm{2}}+\mathrm{6}{\mathit{\epsilon}}_{\mathrm{1}}^{\mathrm{2}}\right)\\ \text{(14)}& {\displaystyle}& {\displaystyle}={\displaystyle \frac{\mathrm{1}}{\mathrm{5}}}\left(\mathrm{5}+\mathrm{9}{\mathit{\epsilon}}_{\mathrm{1}}^{\mathrm{2}}-\mathrm{6}{\mathit{\epsilon}}_{\mathrm{1}}\right),\end{array}$$

where *F*_{1} and *F*_{2} represent the effective friction coefficients caused by
the nonlinear interactions between tidal constituents. The last equality in
Eqs. (10) and (10) is due to the fact that
${\mathit{\epsilon}}_{\mathrm{1}}+{\mathit{\epsilon}}_{\mathrm{2}}=\mathrm{1}$. It is worth noting that Eq. (9) is
a reasonable approximation only if the amplitude of the secondary constituent is
much smaller than that of the dominant one.

For illustration, approximations using Eqs. (5) and (9)
for a typical tidal current with ${\mathit{\epsilon}}_{\mathrm{1}}=\mathrm{3}/\mathrm{4}$ and ${\mathit{\epsilon}}_{\mathrm{2}}=\mathrm{1}/\mathrm{4}$
are displayed in Fig. 3 for the case of two tidal
constituents. It can be seen that the Chebyshev polynomials approximation
(Eq. 5) matches the nonlinear quadratic velocity well, while
Eq. (9), retaining only the original frequencies (*ω*_{1} and
*ω*_{2}), is still able to approximately capture the first-order trend of
the quadratic term.

It can be seen from Eqs. (10) and (10) that when
*ε*_{2}≪1 (hence, *ε*_{1}≃1 for the dominant tidal
constituent), *F*_{1}≃1, *F*_{2}≃1.6; thus, the weaker constituent
experiences proportionately 60 % more friction than the dominant constituent,
which is slightly larger than the classical result of 50 % more friction for
the weaker tidal constituent. Figure 4 shows the solutions of
effective friction coefficients *F*_{1} and *F*_{2} as a function of *ε*_{1}
for the case of two constituents. As expected, we see a
symmetric response of these coefficients in the function of *ε*_{1}
since ${\mathit{\epsilon}}_{\mathrm{1}}+{\mathit{\epsilon}}_{\mathrm{2}}=\mathrm{1}$. Specifically, we note that the
effective friction coefficient *F*_{1} reaches a minimum when
${\mathit{\epsilon}}_{\mathrm{1}}=\mathrm{2}/\mathrm{3}$, when the velocity amplitude of the dominant constituent
is twice as large as the weaker constituent.

Similarly, we are able to extend the same approach to the case of a generic
number *n* of astronomical tidal constituents (e.g., *K*_{1}, *O*_{1}, *M*_{2},
*S*_{2}, *N*_{2}):

$$\begin{array}{}\text{(15)}& {\displaystyle}{\displaystyle}u=\sum _{i=\mathrm{1}}^{n}{\mathit{\upsilon}}_{\mathrm{1}}\mathrm{cos}\left({\mathit{\omega}}_{i}t\right)=\widehat{\mathit{\upsilon}}\sum _{i=\mathrm{1}}^{n}{\mathit{\epsilon}}_{i}\mathrm{cos}\left({\mathit{\omega}}_{i}t\right),\end{array}$$

in which the subscript *i* represents the *i*th tidal constituent.
Considering only the original tidal constituents, the quadratic velocity can
be approximated as

$$\begin{array}{}\text{(16)}& {\displaystyle}{\displaystyle}u\left|u\right|\cong {\displaystyle \frac{\mathrm{8}}{\mathrm{3}\mathit{\pi}}}{\widehat{\mathit{\upsilon}}}^{\mathrm{2}}\sum _{i=\mathrm{1}}^{n}{F}_{i}{\mathit{\epsilon}}_{i}\mathrm{cos}\left({\mathit{\omega}}_{i}t\right),\end{array}$$

and the general expression for the effective friction coefficients of *j*th
tidal constituents is given by

$$\begin{array}{ll}{\displaystyle}{F}_{j}& {\displaystyle}={\displaystyle \frac{\mathrm{3}\mathit{\pi}}{\mathrm{8}}}\left\{\mathit{\alpha}+\mathit{\beta}\left[\sum _{i=\mathrm{1},i\ne j}^{n}{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{\mathit{\epsilon}}_{i}^{\mathrm{2}}-{\displaystyle \frac{\mathrm{3}}{\mathrm{4}}}{\mathit{\epsilon}}_{j}^{\mathrm{2}}\right]\right\}\\ \text{(17)}& {\displaystyle}& {\displaystyle}={\displaystyle \frac{\mathrm{1}}{\mathrm{5}}}\left(\mathrm{2}+\mathrm{3}{\mathit{\epsilon}}_{j}^{\mathrm{2}}+\sum _{i=\mathrm{1},i\ne j}^{n}\mathrm{6}{\mathit{\epsilon}}_{i}^{\mathrm{2}}\right).\end{array}$$

We provide the complete coefficients for the cases of one to three constituents in Appendix Appendix B.

For a single tidal constituent *u*=*υ*_{1}cos (*ω*_{1}*t*), the
quadratic velocity term $u\left|u\right|$ is often approximated by adopting Lorentz's
linearization equation (Eq. 8), and thus the friction term in
Eq. (3) becomes

$$\begin{array}{}\text{(18)}& {\displaystyle}{\displaystyle}\mathit{\kappa}u\left|u\right|=\left(\mathit{\kappa}{\displaystyle \frac{\mathrm{8}}{\mathrm{3}\mathit{\pi}}}{\mathit{\upsilon}}_{\mathrm{1}}\right)u=ru,\end{array}$$

which is the “standard” case for a monochromatic wave, i.e., when we only
deal with a predominant tidal constituent (e.g., *M*_{2}).

For illustration of the method, we consider a tidal current that is composed
of one dominant constituent (e.g., *M*_{2} with velocity *u*_{1}) and a weaker
constituent (e.g., *S*_{2} with velocity *u*_{2}), which is a simple but
important example in estuaries, i.e., $u={u}_{\mathrm{1}}+{u}_{\mathrm{2}}$. In this case, the combination of Eq. (3) and the Chebyshev polynomials expansion
of $u\left|u\right|$ (Eq. 9) yields

$$\begin{array}{}\text{(19)}& {\displaystyle}{\displaystyle \frac{\partial {u}_{\mathrm{1}}}{\partial t}}+{\displaystyle \frac{\partial {u}_{\mathrm{2}}}{\partial t}}+g{\displaystyle \frac{\partial {z}_{\mathrm{1}}}{\partial x}}+g{\displaystyle \frac{\partial {z}_{\mathrm{2}}}{\partial x}}+\mathit{\kappa}{\displaystyle \frac{\mathrm{8}}{\mathrm{3}\mathit{\pi}}}\widehat{\mathit{\upsilon}}\left({F}_{\mathrm{1}}{u}_{\mathrm{1}}+{F}_{\mathrm{2}}{u}_{\mathrm{2}}\right)=\mathrm{0},\end{array}$$

where *z*_{1} is the free surface elevation for the dominant constituent and
*z*_{2} for the secondary constituent. Exploiting the linearity of
Eq. (13), we can solve the two problems independently. As a result, we
see that the actual friction term that is felt in Eq. (13) is
different from that which would be felt by the single constituent alone (Eq. 12).

Introducing a general form of the linearized momentum equation for the
generic *i*th constituent

$$\begin{array}{}\text{(20)}& {\displaystyle}{\displaystyle \frac{\partial {u}_{i}}{\partial t}}+g{\displaystyle \frac{\partial {z}_{i}}{\partial x}}+{f}_{i}{r}_{i}{u}_{i}=\mathrm{0},\end{array}$$

with

$$\begin{array}{}\text{(21)}& {\displaystyle}{\displaystyle}{r}_{i}=\mathit{\kappa}{\displaystyle \frac{\mathrm{8}}{\mathrm{3}\mathit{\pi}}}{\mathit{\upsilon}}_{i},\end{array}$$

as in the standard case, we see that the effective friction term contains a correction factor

$$\begin{array}{}\text{(22)}& {\displaystyle}{\displaystyle}{f}_{i}={\displaystyle \frac{{F}_{i}}{{\mathit{\epsilon}}_{i}}},\end{array}$$

through the coefficient *F*_{i}. Since the ratio *ε*_{i} can be quite
small for a weaker constituent, the friction actually felt can be significantly stronger.

4 Results

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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 *f*_{n}
defined in Eq. (16) if the other (weaker) constituents are treated
in the same way as the predominant constituent. As a result, the friction
number *χ*_{n} for each tidal constituent can be modified as

$$\begin{array}{}\text{(23)}& {\displaystyle}{\displaystyle}{\mathit{\chi}}_{n}={f}_{n}\mathit{\chi},\end{array}$$

where *χ* is the friction number (see definition in Table 1)
experienced if only a single tidal constituent is considered.

We note that the modified friction number *χ*_{n} in Eq. (17)
contains the friction coefficient *K*. In many applications, *K* is
calibrated separately for each tidal constituent to account for the different
friction exerted due to the combined tide, either changing *K* directly or
through calibration of the different correction friction factors *f*_{n}
(see, e.g., Cai et al., 2015, 2016). The current study aims at avoiding
the need to adjust *K* individually, so that only a single value of *K* needs
to be calibrated, based on the physical consideration that friction mostly
depends on bottom roughness, and the other factors (tide interaction) are to
be correctly modeled.

With a hydrodynamic model for a single constituent (see
Appendix Appendix A), an iterative procedure can be designed to study the
propagation of the different constituents by calibrating a single value of
the Manning–Strickler friction parameter *K*. The flowchart illustrating the
computation process is presented in Fig. 5. Initially, we
assume the friction correction factor *f*_{i}=1 for each tidal constituent and
compute the first tentative values of velocity amplitude *υ*_{i} along
the channel using the hydrodynamic model. This allows defining $\widehat{\mathit{\upsilon}}$
and, hence, *ε*_{i}. Taking into account the
frictional interaction between tidal constituents, the revised *f*_{i} is
calculated using Eqs. (12) and (16). Subsequently, using
the updated *f*_{i}, the new velocity amplitude *υ*_{i} along the channel
can be computed using the hydrodynamic model. This process is repeated until
the result is stable. In this paper, two examples of Matlab scripts are
provided together with the observed tidal data in the Guadiana and
Guadalquivir estuaries (see Supplement).

It is worth stressing that the single constituents are not calibrated
independently, as was done in previous analyses (e.g., Cai et al., 2015).
Conversely, only a single friction parameter, *K*, is calibrated or estimated
based on the physical knowledge of the system (bed roughness). This feature
represents a major advantage of the proposed method because the frictional
interaction is modeled in mechanistic terms using Eq. (16).

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 *η* and
associated phase of elevation *ϕ*_{A}.

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^{2}, see Garel, 2017). The Manning–Strickler
friction coefficient (*K*=42 m^{1∕3} s^{−1}) was determined by
calibrating the model outputs (obtained using the iterative procedure
presented in Sect. 4.2) with observations. It can be seen from
Fig. 6 that the computed tidal amplitude and phase of elevation are
in good agreement with the observed values for different tidal constituents
in the Guadiana estuary. The *N*_{2} amplitude is slightly overestimated in the
central part of the estuary, which may suggest that the harmonic analysis has
some difficulties in resolving this constituent in relation to the length of
the considered time series (54 days). In support, the *N*_{2} amplitude
(0.16 m) from a longer time series (85 days) collected in 2017 at 58 km from the
mouth matches the model output better, while results for other constituents
are similar in 2015 and 2017 (Erwan Garel, personal communication, 2017). Otherwise, the
correspondence is poorest for the semidiurnal constituents at the most
upstream station, owing to the truncation of the lowest water levels by a sill
located about 65 km from the river mouth (Garel, 2017).
Table 3 displays the mean friction correction coefficient *f* obtained
from the iterative procedure to account for the nonlinear interaction between
different tidal constituents. In particular, the mean friction correction
factors *f* for the minor constituents *S*_{2}, *N*_{2}, *O*_{1} and *K*_{1}
are 4.6, 8.1, 41.1 and 49.8, respectively.

To understand the tidal dynamics between different tidal constituents along
the Guadiana estuary, the longitudinal variations in the tidal
damping/amplification number *δ*_{A} and celerity number *λ*_{A} (see
their definitions in Table 1) are shown in Fig. 7
where similar minor constituents in semidiurnal (*S*_{2}, *N*_{2}) and diurnal
(*O*_{1}, *K*_{1}) bands behave more or less the same. As shown in
Fig. 7a, the minor constituents *S*_{2}, *N*_{2}, *O*_{1} and *K*_{1}
experience more friction compared with the predominant *M*_{2} tide.
Interestingly, we observe a stronger damping (*δ*_{A}<0) of semidiurnal
constituents (*S*_{2}, *N*_{2}) than of diurnal constituents (*O*_{1}, *K*_{1})
in the seaward part of the estuary (around *x*=0–40 km) although the
amplitudes of the diurnal constituents are less than those of the semidiurnal
ones. In contrast, the amplification (*δ*_{A}>0) of semidiurnal
constituents (*S*_{2}, *N*_{2}) is more apparent than those of diurnal
constituents (*O*_{1}, *K*_{1}) in the landward part of the estuary. For the wave
celerity, as expected the dominant *M*_{2} tide travels faster
(smaller *λ*_{A}) than minor tidal constituents. In addition, we observe that the
wave celerity of semidiurnal tidal constituents is larger than those of
diurnal constituents in the seaward reach (around *x*=0–30 km), while it is
the opposite in the landward reach, which suggests a complex relation between
tidal damping/amplification and wave celerity due to the combined impacts of
channel convergence, bottom friction and reflected wave. It is important to
note that a standing wave pattern with celerity approaching infinity is
produced near the sill due to the superimposition of the incident and
reflected waves (see also Garel and Cai, 2018).

For the Guadalquivir estuary, the geometry can be approximated as a
converging estuary with a width convergence length of *b*=65.5 km and a
constant stream depth of about 7.1 m. A linear reduction of the storage width
ratio of 1.5–1 was adopted over the reach of 0–103 km. The observed tidal
amplitudes and phases are best reproduced by using the model for *K*=46 m^{1∕3} s^{−1}
(see Fig. 8). In general, the observed
tidal properties (tidal amplitude and phase) of different constituents are
well reproduced. The enhanced frictional coefficient *f* for the minor
constituents *S*_{2}, *N*_{2}, *O*_{1} and *K*_{1} are 5.4, 9.7, 40.7 and 43.7,
respectively (Table 3).

Figure 9 shows the longitudinal variations in tidal
damping/amplification and wave celerity for the Guadalquivir estuary, which
are similar to those in the Guadiana estuary. In general, we observe that the
dominant *M*_{2} tide experiences less friction than other secondary
semidiurnal tidal constituents although it travels at more or less the same
speed in the seaward reach (*x*=0–35 km). Unlike the Guadiana estuary, the
damping experienced by the secondary semidiurnal tides is less than that of
diurnal constituents near the estuary mouth (around *x*=0–7 km;
Fig. 9a), while the wave celerity is consistently larger in the
seaward reach (*x*=0–38 km; Fig. 9b). Similar to the Guadiana
estuary, we observe that the tidal damping for the secondary semidiurnal
tides is stronger than that of diurnal constituents in the central parts of
the estuary (around *x*=7–52 km), whereas their amplifications are larger in
the landward part of the estuary although their wave speeds are less.

In particular, the tidal damping along the first half of these two estuaries
is mainly due to the damping of the dominant *M*_{2} wave owning to the fact
that the impact of bottom friction dominates over the channel convergence.
Along the upper reach, enhanced morphological convergence and reflection
effects (that reduce the overall friction experienced by the propagating
wave) result in the overall amplification of the tidal wave. For more details
of the tidal hydrodynamics in these two estuaries, readers can refer to
Garel and Cai (2018) for the Guadiana estuary and Diez-Minguito et al. (2012)
for the Guadalquivir estuary.

In order to clarify the behavior of different tidal constituents, we present
Fig. 10 showing the longitudinal variations in estuary shape
number *γ* (representing the channel convergence) and friction number *χ*_{n}
(representing the bottom friction), two major factors determining
the tidal hydrodynamics, in both estuaries. Note that the variable estuary
shape number *γ* observed in the Guadalquivir estuary is due to the
adoption of a variable storage width ratio *r*_{S} in the analytical model. On
the one hand, the estuary shape numbers for diurnal tides are approximately
twice larger than those for semidiurnal tides (Fig. 10a and d)
due to the tidal frequency differences (see definition
of *γ* in Table 1). On the other hand, the effective friction
experienced by the diurnal tides is much larger than those of the semidiurnal
tides due to the mutual interaction between different tidal constituents
(Fig. 10b and e, see also Table 3). However, the
propagation of different tidal constituents mainly depends on the imbalance
between channel convergence and friction, except for those reaches where wave
reflection matters (generally close to the head). In particular, in the
seaward reach the tidal damping for each tidal constituent can be
approximately estimated by ${\mathit{\delta}}_{\mathrm{A}}=\mathit{\gamma}/\mathrm{2}-{\mathit{\chi}}_{n}\mathit{\mu}\mathrm{cos}\left(\mathit{\varphi}\right)/\left(\mathrm{2}{\mathit{\lambda}}_{\mathrm{A}}\right)$
(see Eq. 20 by Cai et al., 2012). While
the channel convergence effect (represented by *γ*∕2) is much stronger
for diurnal tides than for semidiurnal tides, the frictional effect
(represented by *χ*_{n}*μ*cos (*ϕ*)∕(2*λ*_{A})) is only slightly larger
(Fig. 10c and f). Hence, diurnal tides generally experience
relatively less damping in the seaward reach (Figs. 7a
and 9a). In the case of the Guadalquivir estuary, diurnal tides are
more damped than semidiurnal tides very near the estuary mouth (*x*=0–7 km).
For the second (landward) half of the estuary, the lower amplification
experienced by diurnal tides is mainly due to the wave reflection from the
closed end (see Garel and Cai, 2018).

The importance of mutual interaction between different tidal constituents is
illustrated with the iteratively refined model implemented in both case
studies (Figs. 7 and 9). For comparison,
Fig. 11 shows the analytically computed damping/amplification
number *δ*_{A} and celerity number *λ*_{A} without considering mutual
interaction (by setting *f*_{n}=1 in the model). In this case, the damping
experienced by both secondary diurnal and semidiurnal tides is apparently
underestimated due to the unrealistic friction adopted in the model
(Fig. 11a and c; see also Figs. 7a and 9a). Similarly, the computed wave celerities for secondary tidal
constituents are apparently overestimated due to the underestimated bottom
friction (Fig. 11b and d; see also Figs. 7b
and 9b). To correctly reproduce the main features of
different tidal waves, it is required to use the iteratively refined model
proposed in this study.

5 Conclusions

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In this study, we provide insight into the mutual interactions between one
predominant (e.g., *M*_{2}) and other tidal constituents in estuaries and the
role of quadratic friction on tidal wave propagation. An analytical method
exploiting Chebyshev polynomials was developed to quantify the effective
friction experienced by different tidal constituents. Based on linearization
of the quadratic friction, a conceptual model has been used to explore the
nonlinear interaction of different tidal constituents, which enables them to
be treated independently by means of an iterative procedure. Thus, an
analytical hydrodynamic model for a single tidal constituent can be used to
reproduce the correct wave behavior for different tidal constituents. In
particular, it was shown that a correction of the friction term needs to be
used to correctly reproduce the tidal dynamics for minor tidal constituents.
The application to the Guadiana and Guadalquivir estuaries shows that the
conceptual model can interpret the nonlinear interaction reasonably well when
combined with an analytical model for tidal hydrodynamics.

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.

Data availability

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Data availability.

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

Appendix A: Analytical solutions of tidal hydrodynamics for a single tidal constituent

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In this paper, analytical solutions for a semi-closed estuary proposed by Toffolon and Savenije (2011) were used to reproduce the longitudinal tidal dynamics along the estuary axis. The solution makes use of the parameters that are defined in Table 1.

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

$$\begin{array}{}\text{(A1)}& {\displaystyle}& {\displaystyle}\mathit{\eta}={\mathit{\zeta}}_{\mathrm{0}}\stackrel{\mathrm{\u203e}}{{h}_{\mathrm{0}}}\left|{A}^{*}\right|,\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathit{\upsilon}={r}_{\mathrm{S}}{\mathit{\zeta}}_{\mathrm{0}}{c}_{\mathrm{0}}\left|{V}^{*}\right|,\text{(A2)}& {\displaystyle}& {\displaystyle}\mathrm{tan}\left({\mathit{\varphi}}_{\mathrm{A}}\right)={\displaystyle \frac{\mathrm{\Im}\left({A}^{*}\right)}{\mathrm{\Re}\left({A}^{*}\right)}},\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathrm{tan}\left({\mathit{\varphi}}_{\mathrm{V}}\right)={\displaystyle \frac{\mathrm{\Im}\left({V}^{*}\right)}{\mathrm{\Re}\left({V}^{*}\right)}},\end{array}$$

where ℜ and ℑ are the real and imaginary parts of the corresponding
term, and *A*^{*} and *V*^{*} are unknown complex functions varying along the
dimensionless coordinate ${x}^{*}=x/{L}_{\mathrm{0}}$:

$$\begin{array}{}\text{(A3)}& {\displaystyle}& {\displaystyle}{A}^{*}={a}_{\mathrm{1}}^{*}\mathrm{exp}\left({w}_{\mathrm{1}}^{*}{x}^{*}\right)+{a}_{\mathrm{2}}^{*}\mathrm{exp}\left({w}_{\mathrm{2}}^{*}{x}^{*}\right),\text{(A4)}& {\displaystyle}& {\displaystyle}{V}^{*}={v}_{\mathrm{1}}^{*}\mathrm{exp}\left({w}_{\mathrm{1}}^{*}{x}^{*}\right)+{v}_{\mathrm{2}}^{*}\mathrm{exp}\left({w}_{\mathrm{2}}^{*}{x}^{*}\right).\end{array}$$

For a tidal channel with a closed end, the analytical solutions for the unknown variables in Eqs. (A1) and (A1) are listed in Table A1, where Λ is a complex variable, defined as

$$\begin{array}{}\text{(A5)}& {\displaystyle}{\displaystyle}\mathrm{\Lambda}=\sqrt{{\mathit{\gamma}}^{\mathrm{2}}/\mathrm{4}-\mathrm{1}+i\widehat{\mathit{\chi}}},\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\widehat{\mathit{\chi}}={\displaystyle \frac{\mathrm{8}}{\mathrm{3}\mathit{\pi}}}\mathit{\mu}\mathit{\chi},\end{array}$$

where the coefficient 8∕(3*π*) stems from the adoption of Lorentz's
linearization when considering only one single predominant tidal constituent
(e.g., *M*_{2}).

Since the friction parameter $\widehat{\mathit{\chi}}$ depends on the unknown value
of *μ* (or *υ*), an iterative procedure was used to determine the
correct wave behavior. In addition, to account for the longitudinal variation in the cross section (e.g., estuary depth), a multi-reach technique was
adopted by subdividing the entire estuary into multiple sub-reaches; the
solutions were obtained by solving a set of linear equations with internal
boundary conditions at the junction of the sub-reaches satisfying the
continuity condition (see details in Toffolon and Savenije, 2011).

For given computed values of *A*^{*} and *V*^{*}, the dependent parameters
defined in Table 1 can be computed using the following equations:

$$\begin{array}{}\text{(A6)}& {\displaystyle}& {\displaystyle}\mathit{\mu}=\left|{V}^{*}\right|,\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathit{\varphi}={\mathit{\varphi}}_{\mathrm{V}}-{\mathit{\varphi}}_{\mathrm{A}},\text{(A7)}& {\displaystyle}& {\displaystyle}{\mathit{\delta}}_{\mathrm{A}}=\mathrm{\Re}\left({\displaystyle \frac{\mathrm{1}}{{A}^{*}}}{\displaystyle \frac{{\mathrm{d}}^{{A}^{*}}}{{\mathrm{d}}^{{x}^{*}}}}\right),\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{\mathit{\delta}}_{\mathrm{V}}=\mathrm{\Re}\left({\displaystyle \frac{\mathrm{1}}{{V}^{*}}}{\displaystyle \frac{{\mathrm{d}}^{{V}^{*}}}{{\mathrm{d}}^{{x}^{*}}}}\right),\text{(A8)}& {\displaystyle}& {\displaystyle}{\mathit{\lambda}}_{\mathrm{A}}=\left|\mathrm{\Im}\left({\displaystyle \frac{\mathrm{1}}{{A}^{*}}}{\displaystyle \frac{{\mathrm{d}}^{{A}^{*}}}{{\mathrm{d}}^{{x}^{*}}}}\right)\right|,\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{\mathit{\lambda}}_{\mathrm{V}}=\left|\mathrm{\Im}\left({\displaystyle \frac{\mathrm{1}}{{V}^{*}}}{\displaystyle \frac{{\mathrm{d}}^{{V}^{*}}}{{\mathrm{d}}^{{x}^{*}}}}\right)\right|.\end{array}$$

Appendix B: Coefficients of the Godin's expansion

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The following trigonometric equation

$$\begin{array}{}\text{(B1)}& {\displaystyle}{\displaystyle}{\mathrm{cos}}^{\mathrm{3}}\left({\mathit{\omega}}_{\mathrm{1}}t\right)={\displaystyle \frac{\mathrm{3}}{\mathrm{4}}}\mathrm{cos}\left({\mathit{\omega}}_{\mathrm{1}}t\right)+{\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}\mathrm{cos}\left(\mathrm{3}{\mathit{\omega}}_{\mathrm{1}}t\right),\end{array}$$

is used to convert the third-order terms of Eq. (5) to the harmonic constituents. For a single harmonic, it follows that

$$\begin{array}{}\text{(B2)}& {\displaystyle}{\displaystyle}u\left|u\right|={\mathit{\upsilon}}_{\mathrm{1}}^{\mathrm{2}}\left[\left(\mathit{\alpha}+{\displaystyle \frac{\mathrm{3}}{\mathrm{4}}}\mathit{\beta}\right)\mathrm{cos}\left({\mathit{\omega}}_{\mathrm{1}}t\right)+{\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}\mathit{\beta}\mathrm{cos}\left(\mathrm{3}{\mathit{\omega}}_{\mathrm{1}}\right)\right].\end{array}$$

For two harmonic constituents, the Chebyshev polynomials approximation of $u\left|u\right|$ is expressed as

$$\begin{array}{ll}{\displaystyle}u\left|u\right|& {\displaystyle}={\mathit{\upsilon}}_{\mathrm{1}}^{\mathrm{2}}\left\{\mathit{\alpha}\left[{\mathit{\epsilon}}_{\mathrm{1}}\mathrm{cos}\left({\mathit{\omega}}_{\mathrm{1}}t\right)+{\mathit{\epsilon}}_{\mathrm{2}}\mathrm{cos}\left({\mathit{\omega}}_{\mathrm{2}}t\right)\right]\right.\\ \text{(B3)}& {\displaystyle}& {\displaystyle}\left.+\mathit{\beta}{\left[{\mathit{\epsilon}}_{\mathrm{1}}\mathrm{cos}\left({\mathit{\omega}}_{\mathrm{1}}t\right)+{\mathit{\epsilon}}_{\mathrm{2}}\mathrm{cos}\left({\mathit{\omega}}_{\mathrm{2}}t\right)\right]}^{\mathrm{3}}\right\}.\end{array}$$

In Eq. (B3), the cubic term can be expanded as

$$\begin{array}{ll}{\displaystyle}\left[{\mathit{\epsilon}}_{\mathrm{1}}\mathrm{cos}\right.& {\displaystyle}{\left.\left({\mathit{\omega}}_{\mathrm{1}}t\right)+{\mathit{\epsilon}}_{\mathrm{2}}\mathrm{cos}\left({\mathit{\omega}}_{\mathrm{2}}t\right)\right]}^{\mathrm{3}}={\mathit{\epsilon}}_{\mathrm{1}}^{\mathrm{3}}{\mathrm{cos}}^{\mathrm{3}}\left({\mathit{\omega}}_{\mathrm{1}}t\right)\\ {\displaystyle}& {\displaystyle}+\mathrm{3}{\mathit{\epsilon}}_{\mathrm{1}}{\mathit{\epsilon}}_{\mathrm{2}}^{\mathrm{2}}\mathrm{cos}\left({\mathit{\omega}}_{\mathrm{1}}t\right){\mathrm{cos}}^{\mathrm{2}}\left({\mathit{\omega}}_{\mathrm{2}}t\right)\\ \text{(B4)}& {\displaystyle}& {\displaystyle}+\mathrm{3}{\mathit{\epsilon}}_{\mathrm{2}}{\mathit{\epsilon}}_{\mathrm{1}}^{\mathrm{2}}\mathrm{cos}\left({\mathit{\omega}}_{\mathrm{2}}t\right){\mathrm{cos}}^{\mathrm{2}}\left({\mathit{\omega}}_{\mathrm{1}}t\right)+{\mathit{\epsilon}}_{\mathrm{2}}^{\mathrm{3}}{\mathrm{cos}}^{\mathrm{3}}\left({\mathit{\omega}}_{\mathrm{2}}t\right).\end{array}$$

Making use of the trigonometric equations to expand the power of the cosine
functions (e.g., cos ^{3}(*ω*_{1}*t*) and cos ^{2}(*ω*_{1}*t*)) and
extracting only the harmonic terms with frequencies *ω*_{1} and *ω*_{2},
Eq. (B3) can be reduced to Eq. (9).

For the case of many constituents, here we only provide the exact coefficients for *n*=3:

$$\begin{array}{ll}{\displaystyle}{F}_{\mathrm{1}}& {\displaystyle}={\displaystyle \frac{\mathrm{3}\mathit{\pi}}{\mathrm{8}}}\left[\mathit{\alpha}+\mathit{\beta}\left({\displaystyle \frac{\mathrm{3}}{\mathrm{4}}}{\mathit{\epsilon}}_{\mathrm{1}}^{\mathrm{2}}+{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{\mathit{\epsilon}}_{\mathrm{2}}^{\mathrm{2}}+{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{\mathit{\epsilon}}_{\mathrm{3}}^{\mathrm{2}}\right)\right]\\ \text{(B5)}& {\displaystyle}& {\displaystyle}={\displaystyle \frac{\mathrm{1}}{\mathrm{5}}}\left(\mathrm{2}+\mathrm{3}{\mathit{\epsilon}}_{\mathrm{1}}^{\mathrm{2}}+\mathrm{6}{\mathit{\epsilon}}_{\mathrm{2}}^{\mathrm{2}}+\mathrm{6}{\mathit{\epsilon}}_{\mathrm{3}}^{\mathrm{2}}\right),\end{array}$$

$$\begin{array}{ll}{\displaystyle}{F}_{\mathrm{2}}& {\displaystyle}={\displaystyle \frac{\mathrm{3}\mathit{\pi}}{\mathrm{8}}}\left[\mathit{\alpha}+\mathit{\beta}\left({\displaystyle \frac{\mathrm{3}}{\mathrm{4}}}{\mathit{\epsilon}}_{\mathrm{2}}^{\mathrm{2}}+{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{\mathit{\epsilon}}_{\mathrm{1}}^{\mathrm{2}}+{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{\mathit{\epsilon}}_{\mathrm{3}}^{\mathrm{2}}\right)\right]\\ \text{(B6)}& {\displaystyle}& {\displaystyle}={\displaystyle \frac{\mathrm{1}}{\mathrm{5}}}\left(\mathrm{2}+\mathrm{3}{\mathit{\epsilon}}_{\mathrm{2}}^{\mathrm{2}}+\mathrm{6}{\mathit{\epsilon}}_{\mathrm{1}}^{\mathrm{2}}+\mathrm{6}{\mathit{\epsilon}}_{\mathrm{3}}^{\mathrm{2}}\right),\end{array}$$

$$\begin{array}{ll}{\displaystyle}{F}_{\mathrm{3}}& {\displaystyle}={\displaystyle \frac{\mathrm{3}\mathit{\pi}}{\mathrm{8}}}\left[\mathit{\alpha}+\mathit{\beta}\left({\displaystyle \frac{\mathrm{3}}{\mathrm{4}}}{\mathit{\epsilon}}_{\mathrm{3}}^{\mathrm{2}}+{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{\mathit{\epsilon}}_{\mathrm{1}}^{\mathrm{2}}+{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{\mathit{\epsilon}}_{\mathrm{2}}^{\mathrm{2}}\right)\right]\\ \text{(B7)}& {\displaystyle}& {\displaystyle}={\displaystyle \frac{\mathrm{1}}{\mathrm{5}}}\left(\mathrm{2}+\mathrm{3}{\mathit{\epsilon}}_{\mathrm{3}}^{\mathrm{2}}+\mathrm{6}{\mathit{\epsilon}}_{\mathrm{1}}^{\mathrm{2}}+\mathrm{6}{\mathit{\epsilon}}_{\mathrm{2}}^{\mathrm{2}}\right).\end{array}$$

Equations (B3) to (B3) reduce to Eqs. (10) and (10)
when *ε*_{3}=0 (i.e., *υ*_{3}=0).

Supplement

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Supplement.

The supplement related to this article is available online at: https://doi.org/10.5194/os-14-769-2018-supplement.

Author contributions

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Author contributions.

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.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

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

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