Tidal asymmetry in a funnel-shaped estuary with mixed semidiurnal tides

Abstract

Different types of tidal asymmetry (see review of de Swart and Zimmerman Annu Rev Fluid Mech 41: 203–229, 2009) are examined in this study. We distinguish three types of tidal asymmetry: duration and magnitude differences between flood and ebb tidal flow, duration difference between the rising and falling tides. For waterborne substance transport, the first two asymmetries are important while the last one is not. In this study, we take the Huangmaohai Estuary (HE), Pearl River Delta, China as an example to examine the spatio-temporal variations of the tidal asymmetry in a mixed semidiurnal tidal regime and to explain them by investigating the associated mechanisms. The methodology defining the tidal duration asymmetry and velocity skewness, proposed by Nidzieko (J Geophys Res 115: C08006. doi: 10.1029/2009JC005864, 2010) and synthesized by Song et al. (J Geophys Res 116: C12007. doi: 10.1029/2011JC007270, 2011), is utilized here and referred to as tidal duration asymmetry (TDA) and flow velocity asymmetry (FVA), respectively. The methodology is further used to quantify the flow duration asymmetry (FDA). A positive asymmetry means a shorter duration of low water slack for FDA, a shorter duration of the rising tide for TDA, and a flood dominance for FVA and vice versa. The Regional Ocean Modeling System (ROMS) model is used to provide relatively long-term water elevation and velocity data and to conduct diagnostic experiments. In the HE, the main tidal constituents are diurnal tides K ₁, O ₁ and semidiurnal tides M ₂ and S ₂. The interaction among the diurnal and semidiurnal tides generates a negative tidal asymmetry, while the interactions among semidiurnal tides and their overtides or compound tides result in a positive tidal asymmetry. The competition among the above interactions determines the FDA and TDA, whereas for the FVA, aside from the interaction among different tidal constituents, an extra component, the residual flow, plays an important role. The results show that the FDA exhibits a predominant tendency of shorter duration of low water slack, favoring the landward transport of fine sediment. The FVA demonstrates prevailing ebb dominance in the study period, favoring the seaward transport of coarse sediment. This ebb dominance is primarily induced by the interaction among the residual flow and the tidal constituents. The external TDA in the ocean experiences distinct cyclic variations with positive asymmetry when semidiurnal tides dominate and negative asymmetry during the periods when diurnal tides dominate. The funnel shape of the HE is advantageous for the development of positive tidal asymmetry as the semidiurnal tides are more amplified than the diurnal tides. The effect of river flow can enhance the ebb dominance, while the baroclinic effect is more complex. The existence of channel and shoals favors the development of residual pattern with seaward flow (ebb dominance) in the channel and landward flow (flood dominance) at the shoal when the tides are strong (semidiurnal tides dominate) and the residual pattern with landward flow (flood dominance) in the channel and seaward flow (ebb dominance) at the shoal when the baroclinic effect is dominant (diurnal tides dominate).

Appendix: Derivation of tidal asymmetry through combinations of tidal constituents

For the FVA, the time series of the depth-averaged axial velocity at a Sta. is expressed as:

$$ u={\displaystyle \sum_{i=0}^M{u}_i=}{\displaystyle \sum_{i=0}^M{a}_i \cos}\left({\omega}_it-{\varphi}_i\right) $$

(18)

where a, ω, and φ are the velocity amplitude, frequency, and phase of the tidal constituents. The velocity skewness is

$$ {\gamma}_{FVA}=\frac{E\left[{u}^3\right]}{{\left(E\left[{u}^2\right]\right)}^{3/2}} $$

(19)

where E is the expectation of a variable.

$$ {u}^3={\left({\displaystyle \sum_{i=0}^M{u}_i}\right)}^3={\displaystyle \sum_{i=0}^M{u}_i^3+3{\displaystyle \sum_{\begin{array}{l}i,j=0\\ {}i\ne j\end{array}}^M{u}_i^2{u}_j+6{\displaystyle \sum_{\begin{array}{l}i,j,k=0\\ {}i\ne j\ne k\end{array}}^M{u}_i{u}_j{u}_k}}} $$

(20)

$$ {u}^2={\left({\displaystyle \sum_{i=0}^M{u}_i}\right)}^2={\displaystyle \sum_{i=0}^M{u}_i^2+2{\displaystyle \sum_{\begin{array}{l}i,j=0\\ {}i\ne j\end{array}}^M{u}_i{u}_j}} $$

(21)

and

$$ {u}_i^2=\frac{1}{2}{a}_i^2\left[1+ \cos \left(2{\omega}_it-2{\varphi}_i\right)\right] $$

(22‐1)

$$ {u}_i{u}_j=\frac{1}{2}{a}_i{a}_j\left\{ \cos \left[\left({\omega}_i+{\omega}_j\right)t-\left({\varphi}_i+{\varphi}_j\right)\right]+ \cos \left[\left({\omega}_i-{\omega}_j\right)t-\left({\varphi}_i-{\varphi}_j\right)\right]\right\} $$

(22‐2)

$$ {u}_i^3=\frac{1}{4}{a}_i^3\left[ \cos \left(3{\omega}_it-3{\varphi}_i\right)+3 \cos \left({\omega}_it-{\varphi}_i\right)\right] $$

(22‐3)

$$ {u}_i^2{u}_j=\frac{1}{4}{a}_i^2{a}_j\left\{ \cos \left[\left(2{\omega}_i+{\omega}_j\right)t-\left(2{\varphi}_i+{\varphi}_j\right)\right]+ \cos \left[\left(2{\omega}_i-{\omega}_j\right)t-\left(2{\varphi}_i-{\varphi}_j\right)\right]+2 \cos \left({\omega}_jt-{\varphi}_j\right)\right\} $$

(22‐4)

$$ \begin{array}{l}{u}_i{u}_j{u}_k=\frac{1}{4}{a}_i{a}_j{a}_k\left\{ \cos \left[\left({\omega}_i+{\omega}_j-{\omega}_k\right)t-\left({\varphi}_i+{\varphi}_j-{\varphi}_k\right)\right]+ \cos \left[\left({\omega}_i+{\omega}_j+{\omega}_k\right)t-\left({\varphi}_i+{\varphi}_j+{\varphi}_k\right)\right]\right.\hfill \\ {}\left.+ \cos \left[\left({\omega}_i-{\omega}_j-{\omega}_k\right)t-\left({\varphi}_i-{\varphi}_j-{\varphi}_k\right)\right]+ \cos \left[\left({\omega}_i-{\omega}_j+{\omega}_k\right)t-\left({\varphi}_i-{\varphi}_j+{\varphi}_k\right)\right]\right\}\hfill \end{array} $$

(22‐5)

Then:

$$ E\left[{u}_i{u}_j\right]=0 $$

(23‐1)

$$ \begin{array}{ccc}\hfill E\left[{u}_i^3\right]=\left\{\begin{array}{l}0\\ {}{a}_0^2 \cos \left({\varphi}_0\right)\end{array}\right.\hfill & \hfill \mathrm{when}\hfill & \hfill \begin{array}{l}{\omega}_i\ne 0\\ {}{\omega}_i=0\end{array}\hfill \end{array} $$

(23‐2)

$$ \begin{array}{ccc}\hfill E\left[{u}_i^2\right]=\left\{\begin{array}{c}\hfill \frac{1}{2}{a}_i^2\hfill \\ {}\hfill {a}_0^2\hfill \\ {}\hfill 0\hfill \end{array}\right.\hfill & \hfill \mathrm{when}\hfill & \hfill \begin{array}{l}{\omega}_i\ne 0\\ {}{\omega}_i=0\\ {} others\end{array}\hfill \end{array} $$

(23‐3)

$$ \begin{array}{ccc}\hfill E\left[{u}_i^2{u}_j\right]=\left\{\begin{array}{l}\frac{1}{4}{a}_i^2{a}_j \cos \left(2{\varphi}_i-{\varphi}_j\right)\\ {}\frac{1}{2}{a}_i^2{a}_0 \cos \left({\varphi}_0\right)\\ {}0\end{array}\right.\hfill & \hfill \mathrm{when}\hfill & \hfill \begin{array}{l}2{\omega}_i-{\omega}_j=0\\ {}{\omega}_i=0\\ {} others\end{array}\hfill \end{array} $$

(23‐4)

$$ \begin{array}{ccc}\hfill E\left[{u}_i{u}_j{u}_k\right]=\left\{\begin{array}{l}\frac{1}{4}{a}_i{a}_j{a}_k \cos \left({\varphi}_i+{\varphi}_j-{\varphi}_k\right)\\ {}0\end{array}\right.\hfill & \hfill \mathrm{when}\hfill & \hfill \begin{array}{l}{\omega}_i+{\omega}_j-{\omega}_k=0\\ {} others\end{array}\hfill \end{array} $$

(23‐5)

Thus:

$$ {\gamma}_{FVa}=\frac{{\displaystyle \sum_{\omega_i+{\omega}_j={\omega}_k}\frac{3}{2}{a}_i{a}_j{a}_k \cos \left({\varphi}_i+{\varphi}_j-{\varphi}_k\right)+{\displaystyle \sum_{2{\omega}_i={\omega}_j}\frac{3}{4}{a}_i^2{a}_j \cos \left(2{\varphi}_i-{\varphi}_j\right)}+{\displaystyle \sum_{i=1}\frac{3}{2}{a}_i^2{a}_0 \cos \left({\varphi}_0\right)+{a}_0^3 \cos }}\left({\varphi}_0\right)}{{\left({a}_0^2+\frac{1}{2}{\displaystyle \sum_{i=1}^N{a}_i^2}\right)}^{3/2}} $$

(24)

N is the total number of the tidal constituents which satisfies the relationship of ω _i  + ω _j  = ω _k or 2ω _i  = ω _j .

For the TDA, the water elevation at a Sta. is expressed as:

$$ \eta ={\displaystyle \sum_{i=0}^M{\eta}_i=}{\displaystyle \sum_{i=0}^M{a}_i \cos}\left({\omega}_it-{\varphi}_i\right) $$

(25)

where a, ω, and φ are the water surface elevation’s amplitude, frequency, and phase of the tidal constituents. The skewness is

$$ \gamma =\frac{E\left[{\varsigma}^3\right]}{{\left(E\left[{\varsigma}^2\right]\right)}^{3/2}} $$

(26)

where ζ is the acceleration of water elevation, $\zeta =\frac{d\eta }{dt}$.

$$ {\gamma}_{TDA}=\frac{E\left[{\zeta}^3\right]}{{\left(E\left[{\zeta}^2\right]\right)}^{3/2}}=\frac{{\displaystyle \sum_{\omega_i+{\omega}_j={\omega}_k}\frac{3}{2}{a}_i{\omega}_i{a}_j{\omega}_j{a}_k{\omega}_k \sin \left({\varphi}_i+{\varphi}_j-{\varphi}_k\right)+{\displaystyle \sum_{2{\omega}_i={\omega}_j}\frac{3}{4}{a}_{i^2}{\omega}_{i^2}{a}_j{\omega}_j \sin }}\left(2{\varphi}_i-{\varphi}_j\right)}{{\left(\frac{1}{2}{\displaystyle \sum_{i=1}^N{a}_i^2{\omega}_i^2}\right)}^{3/2}} $$

(27)

For the FDA, the skewness calculation is applied to the acceleration of flow velocity and the similar result is obtained:

$$ {\gamma}_{FDA}=\frac{E\left[{\chi}^3\right]}{{\left(E\left[{\chi}^2\right]\right)}^{3/2}}=\frac{{\displaystyle \sum_{\omega_i+{\omega}_j={\omega}_k}\frac{3}{2}{a}_i{\omega}_i{a}_j{\omega}_j{a}_k{\omega}_k \sin \left({\varphi}_i+{\varphi}_j-{\varphi}_k\right)+{\displaystyle \sum_{2{\omega}_i={\omega}_j}\frac{3}{4}{a}_{i^2}{\omega}_{{}_i}^2{a}_j{\omega}_j \sin }}\left(2{\varphi}_i-{\varphi}_j\right)}{{\left(\frac{1}{2}{\displaystyle \sum_{i=1}^N{a}_i^2{\omega}_i^2}\right)}^{3/2}} $$

(28)

where $\chi =\frac{du}{dt}$ and a and ω and φ are the velocity amplitude, frequency, and phase of the tidal constituents.