An Analytical Solution for Investigating the Characteristics of Tidal Wave

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Article An Analytical Solution for Investigating the Characteristics of Tidal Wave and Surge Propagation Associated with Non-Tropical and Tropical Cyclones in the Humen Estuary, Pearl River

Zhuo Zhang 1,2, Fei Guo 1,2,* , Di Hu 1,2 and Dong Zhang 1,2

1 Key Laboratory of Virtual Geographic Environment, Nanjing Normal University, Ministry of Education, Nanjing 210023, China; [email protected] (Z.Z.); [email protected] (D.H.); [email protected] (D.Z.) 2 Jiangsu Center for Collaborative Innovation in Geographical Information Resource Development and Application, Nanjing 210023, China * Correspondence: [email protected]

Abstract: The Humen Estuary, one of the largest outlets of the Pearl River, is a long and wide tidal channel with a considerable tidal flow every year. Storm surges, always superposing spring tide, travel from the estuary and endanger the safety of people living around the river. However, little research has quantified the relationship between the hydraulic characteristics and the geometry features in this estuary. In this regard, an analytical model, combined with a numerical model, is applied to investigate the characteristics of tidal waves and surge propagations in the estuary. Given Citation: Zhang, Z.; Guo, F.; Hu, D.; the geometric, topographic, and tidal parameters at the mouth of the estuary, the tidal damping and Zhang, D. An Analytical Solution for wave celerity can be computed. The numerical results were used to calibrate and verify the analytical Investigating the Characteristics of model. The results indicate that the analytical model can describe the astronomical tidal dynamics Tidal Wave and Surge Propagation very well in correspondence with the numerical results. However, the analytical model cannot Associated with Non-Tropical and Tropical Cyclones in the Humen predict the tide well when a tropical cyclone-induced surge is superimposed on the astronomical Estuary, Pearl River. Water 2021, 13, tide. The reason is that this model does not take the wind stress and the pressure depression into 2375. https://doi.org/10.3390/ account. After reducing Manning’s coefficient, we found that the analytical results could be close to w13172375 the numerical results. Finally, we analyzed the characteristics of the tidal wave in the Humen Estuary using the analytical solution and its parameters. Academic Editors: Tony Wong and Vivek Srikrishnan Keywords: tidal damping; wave propagation; analytical model; surge; pearl river

Received: 1 June 2021 Accepted: 24 August 2021 Published: 29 August 2021 1. Introduction Tidal waves are a major hydrodynamic factor that influences the flooding, ecosystems, Publisher’s Note: MDPI stays neutral transport, and morphological evolution in estuaries. In contrast to tidal waves in deep with regard to jurisdictional claims in oceans, which are driven by astronomical forces with a high degree of predictability, waves published maps and institutional affil- iations. propagating along estuaries tend to adjust their amplitude, celerity, and shape during interactions with topography, friction, and other secondary factors such as wind stress and runoff. A tidal wave in a shallow estuary has the characteristics of amplification and damping, which alternate along the estuary. The tidal amplitude is influenced by two dominant processes: amplification due to convergent cross sections with a decrease Copyright: © 2021 by the authors. in depth in the landward direction and damping due to bottom friction. If the former Licensee MDPI, Basel, Switzerland. process dominates over the latter, the wave is amplified; otherwise, the wave is damped. This article is an open access article The celerity of propagation is also influenced by the processes of tidal damping and distributed under the terms and conditions of the Creative Commons amplification. If the tidal wave is damped, the celerity is decreased. Conversely, if the tidal Attribution (CC BY) license (https:// wave is amplified, the celerity is increased [1]. Moreover, the tidal wave can be strongly creativecommons.org/licenses/by/ distorted as it propagates into the estuarine system [2]. Therefore, tidal wave propagation 4.0/).

Water 2021, 13, 2375. https://doi.org/10.3390/w13172375 https://www.mdpi.com/journal/water Water 2021, 13, 2375 2 of 14

in an estuary is an interesting but difficult issue to solve because of the nonlinear interaction resulting from the complex estuarine boundary lines and variably shallow topography. To understand the mechanism of tidal wave propagation along an estuary, vast efforts have been made to induce and solve the estuary hydrodynamic model. Generally, these works can be divided into two methods: analytical solutions and numerical simulations. With advances in computational technology, tidal wave propagation can be accurately simulated by numerical models [2–5]. Compared with analytical solutions, numerical models are better at providing high-resolution data under various controlling conditions, which can be comparable with the measurements in a real estuary. However, the numerical results should be further investigated and extracted through the analytical solution to obtain a deeper understanding of the effect by multiple controlling factors with the respect of tidal wave propagation. Specifically, to figure out the relationship between hydrodynamics and the geometry of the estuary, we need the analytical solution not only to fit the data but also to extract some important parameters such as velocity number, damping number, celerity number, and their relationships with the geometry parameter of the estuary. Thus, an analytical solution is still an irreplaceable instrument in analyzing the properties of tidal waves propagating in an estuary. Thus far, a range of analytical solutions based on 1D Saint–Venant equations have been derived by Hunt [6](1964), Ippen [7], and Prandle [8], which are all for a prismatic estuary and river. Meanwhile, the solutions for a more widely convergent estuary have been presented by Godin [9], Jay [10], Lanzoni and Seminara [11], and VanRijn [12]. Most of the above investigators linearized the equations by means of the perturbation method in an Eulerian framework. In contrast, Savenije [13] derived a relatively simple solution in a Lagrangian framework. Furthermore, Savenije [14] presented a fully explicit solution of the tidal equations by solving the set of four implicit equations derived from the Saint–Venant equations. With the advantage of considering the nonlinear effect of the bottom friction, Savenije’s analytical solution is constantly improved and widely used in analyzing the wave propagation on typical alluvial estuaries [15,16]. The Humen Estuary, one of the major accesses for a tidal wave from the South China Sea entering the Pearl River network, has a significant effect on flooding and inundation in Guangzhou city, which is the capital and economic center of Guangdong Province. However, limited measured data in the Humen Estuary restrict the understanding of tidal wave propagation along the Humen Estuary in the upstream direction. Although some previous numerical simulations have been performed [17,18], their results are not direct and clear enough to reflect the characteristics of how a certain controlling parameter affects other parameters. Another question is whether Savenije’s analytical solution can be applied to the situation under the interaction of tides and surges during typhoon landfall. If not, what are the changes in the hydrodynamic system and which parameters should be adjusted in the analytical solution? With the questions mentioned above, the objectives of the study are as follows: 1. to compose and validate a method of analysis that combines a classical analytical solution and the numerical results and is able to correctly reflect the characteristics of tide propagation in the Humen Estuary; 2. to check and verify a way to adapt the method of analysis to the context of tidal wave and surge propagation associated with tropical cyclones; and 3. to extract insight into the hydrodynamic features and factors in the Humen Estuary. Thus, we apply Savenije’s analytical solution to the Humen Estuary in conjunction with numerical simulation results to describe and analyze the characteristics of the tidal wave in the Humen Estuary, Pearl River. A series of tests are conducted to answer the above questions in this paper. The sections are organized as follows. First, some background information about the Humen Estuary in the Pearl River network is introduced in Section2, followed by a brief introduction of Savenije’s analytical solution in Section3. In Section4, the numerical model is verified. In Section5, calibration and verification for astronomic tides and storm surges are carried out. In Section6, the discussion focuses on the characteristics of tidal waves in the Humen Estuary. Finally, conclusions are drawn in Section7. Water 2021, 13, 2375 3 of 14

The sections are organized as follows. First, some background information about the Humen Estuary in the Pearl River network is introduced in Section 2, followed by a brief introduction of Savenije’s analytical solution in Section 3. In Section 4, the numerical model is verified. In Section 5, calibration and verification for astronomic tides and storm surges are carried out. In Section 6, the discussion focuses on the characteristics of tidal waves in the Humen Estuary. Finally, conclusions are drawn in Section 7.

Water 2021, 13, 2375 2. Study Areas 3 of 14 2.1. Overview

2. StudyThe Areas Pearl River network, located in South China, delivers a large amount of fresh 2.1.water Overview (ranging from 20,000 m3/s in the wet summer to 3600 m3/s in the dry winter) into theThe northern Pearl River South network, China located Sea through in South China,eight deliversoutlets a(Figure large amount 1). Four of fresh of the eight outlets water(Humen, (ranging Jiaomen, from 20,000 Hongqimen, m3/s in the wet and summer Hengmen) to 3600 m 3 enter/s in the Lingdingyang dry winter) into the Bay, which has a northerntrumpet-like South China shape Sea with through a width eight outlets of 5 km (Figure near1). the Four northern of the eight end outlets and (Humen, 35 km at the southern Jiaomen,end. Among Hongqimen, the four and outlets, Hengmen) the enter Humen Lingdingyang Estuary Bay, is the which largest has a mouth trumpet-like with a large amount shape with a width of 5 km near the northern end and 35 km at the southern end. Among theof fourtidal outlets, influx theand Humen outflux Estuary between is the the largest estuary mouth and with offshore. a large amountIt is obvious of tidal that the trumpet- influxlike andbay outfluxtends betweento amplify the estuarythe tide and amplitude offshore. It in is obviousconjunction that the with trumpet-like the convergent estuary baywhen tends the to amplifytidal wave the tide propagates amplitude in from conjunction the deep with sea the convergentinto the Humen estuary whenEstuary in the land- theward tidal direction. wave propagates Thus, fromwe focus the deep on seathe into characteristics the Humen Estuary of the in tidal the landwardwave along the Humen direction. Thus, we focus on the characteristics of the tidal wave along the Humen Estuary, specificallyEstuary, the specifically segment from the the segment mouth upstream from the to approximately mouth upstream 1.5 km downstream to approximately 1.5 km fromdownstream Huangpu Bridge. from Huangpu Bridge.

FigureFigure 1. The1. The location location of the of outlets the outlets of the Pearl of the River Pearl network. River network.

2.2. Shape of the Humen Estuary

The shapes of most alluvial estuaries are similar all over the world. The width and the area of the cross section decreases in the upstream direction, resulting in a convergent

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2.2. Shape of the Humen Estuary The shapes of most alluvial estuaries are similar all over the world. The width and the area of the cross section decreases in the upstream direction, resulting in a convergent (funnel-shaped) estuary. The main geometric parameters of the convergent estuary can be described by exponential functions along the estuary axis with the origin at the mouth: 푥 퐴 = 퐴 푒푥푝 (− ) (1) 푎 푥 퐵 = 퐵 푒푥푝 (− ) (2) 푏 푥 ℎ = ℎ 푒푥푝 (− ) (3) 푑

Water 2021, 13, 2375 where A, B, and h are the tidal average cross-sectional area, width, and4 of 14 depth; A0, B0, and h0 are the same variables at the estuary mouth; x is the distance from the estuary mouth; and a, b, and d are convergent lengths of the cross-sectional area, width, and depth. (funnel-shaped)The parameters estuary. of The Equations main geometric (1)–(3) parameters can be of obtained the convergent by regression estuary can be on the topographic data.described From by the exponential digital functionselevation along model the estuary (DEM) axis withof the the Pearl origin atRiver the mouth: Delta, 54 cross sections were extracted to obtain the cross-sectionalx area, width, and depth (Figure 2). Then, the A = A0exp(− ) (1) data were used to derive the convergent lengthsa a, b, and d listed in Table 1. Figure 3 shows x the regression lines plotted on Bsemilogarithmic= B exp(− ) coordinates. It shows (2)that the Humen Es- 0 b tuary can be divided into three sections: xthe downstream mouth section (x = 0–8.7 km), h = h exp(− ) (3) the middle section (x = 8.7–30.4 km),0 and thed upstream section (x = 30.4–36 km). The mouth sectionwhere isA, nearB, and ah areprismatic the tidal averageestuary, cross-sectional and the middle area, width, and and upstream depth; A0, Bsections0, and are the typical convergenth0 are the same estuary. variables at the estuary mouth; x is the distance from the estuary mouth; and a, b, and d are convergent lengths of the cross-sectional area, width, and depth. The parameters of Equations (1)–(3) can be obtained by regression on the topographic Tabledata. 1. FromShape the characteristics digital elevation of model the Humen (DEM) of Estuary. the Pearl River Delta, 54 cross sections were extracted to obtain the cross-sectional area, width, and depth (Figure2). Then, the dataSubsections were used to derive Range the convergent (km) lengths a, b A, and (km)d listed in Table1. B Figure (km)3 shows D (km) the regressionMouth lines plotted 0–8.7 on semilogarithmic coordinates. 166.7 It shows that 333.3 the Humen 333.3 Estuary can be divided into three sections: the downstream mouth section (x = 0–8.7 km), Middle 8.7–30.4 25 71.4 40.0 the middle section (x = 8.7–30.4 km), and the upstream section (x = 30.4–36 km). The mouth sectionUpstream is near a prismatic estuary,30.4–36 and the middle and 20 upstream sections are 8.3 the typical 208.3 convergent estuary.

FigureFigure 2. 2. TheThe deployment of the of cross the sectionscross sections along the Humenalong Estuary.the Humen Estuary.

Table 1. Shape characteristics of the Humen Estuary.

Subsections Range (km) A (km) B (km) D (km) Mouth 0–8.7 166.7 333.3 333.3 Middle 8.7–30.4 25 71.4 40.0 Upstream 30.4–36 20 8.3 208.3

WaterWater 20212021, 13, 13, ,2375 2375 5 of 14 5 of 14

2 FigureFigure 3. 3.The The regression regression lines lines for thefor cross-sectionalthe cross-sectional area A area(m ), A width (m2),B width(m), and B (m), depth andh (m) depth along h (m) along thethe estuary. estuary. 3. Analytical Solution 3. Analytical Solution The tidal dynamics in an estuary can be described by the following Saint–Venant equations: The tidal dynamics in an estuary can be described by the following Saint–Venant ∂U ∂U ∂d U|U| equations: + U + g + gI + gn2 = 0 (4) ∂t ∂x ∂x b d4/3 휕푈 휕푈 휕푑 푈|푈| ∂A ∂Q (4) + 푈 r ++푔 =+0푔퐼 + 푔푛 / = 0 (5) 휕푡 휕푥s ∂t ∂휕푥x 푑 where U is the tidal flow cross-sectional velocity,휕퐴 g is휕푄 the acceleration due to gravity, d is 푟 + = 0 (5) the flow depth, Ib is the bottom slope, n is the 휕푡 Manning휕푥 coefficient, and Q is the tidal flow discharge. rs is defined as the ratio between the storage width and the stream width, which iswhere often moreU is the than tidal unity flow in the cross-sectional river with shallow velocity, plains g [ 14 is] the (Savenije, acceleration 2008). due to gravity, d is the flowAfter depth, the scaling I is of the Equations bottom (4)slope, and n (5) is andthe underManning the coefficient, assumption and of a harmonicQ is the tidal flow wavedischarge. traveling 푟 fromis defined the estuary as the mouth ratio to thebetween upstream the reach, storage the followingwidth and dimensionless the stream width, parameter can be obtained: which is often more than unity in the riverc with shallow plains [14] (Savenije, 2008). γ = 0 (6) After the scaling of Equations (4) andωa (5) and under the assumption of a harmonic wave traveling from the estuary mouth toc the upstream reach, the following dimension- χ = r f 0 ζ (7) less parameter can be obtained: s ωh 1 vh = 푐 µ 훾 = (8) (6) rs ηc0 휔푎 c λ = 0 푐 (9) c 휒 = 푟푓 휁 (7) 1 dη c 휔ℎ δ = 0 (10) η dx ω 1 푣ℎ 휇 = (8) 푟 휂푐 푐 휆 = (9) 푐

1 푑휂 푐 훿 = (10) 휂 푑푥 휔

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where c0 is the classical wave celerity of a frictionless progressive wave, ω is the frequency η of the wave, ζ = h is the dimensionless tidal amplitude, and f is the dimensionless friction factor deﬁned as " #−1 gn2 4 2 f = 1 − ζ (11) h1/3 3 Using the above dimensionless parameters in Equations (6)–(11), the scaling form of Equations (4) and (5) can be solved on the basis of the equations for the phase lag:

λ tan(e) = (12) γ − δ

sin(e) cos(e) µ = = (13) λ γ − δ µ2 δ = (γ − χµ2λ2) (14) µ2 + 1 λ2 = 1 − δ(γ − δ) (15) where e is the phase difference between high water and high water slack. It is clear that, for a progressive wave, e = 0.5π and that, for a standing wave, e = 0. The above equations should be applied to the following conditions: (1) the ratio between the tidal amplitude and the water depth is less than unity, (2) the runoff is much less than the tidal influx and outflux, and (3) the incident wave can be simplified as a harmonic wave. Equations (12)–(15) can be solved by an iteration process when γ and χ have been determined.

4. Numerical Model and Verification The Finite Volume Community Ocean Model (FVCOM), an unstructured-grid finite- volume, three-dimensional primitive equation coastal ocean model developed originally by Chen et al. [19] and upgraded by the UMASS-D/WHOI model development team [20,21], is applied to simulate the tide and surge propagation process in the Pearl River network. As shown in Figure4, the model domain covers the entire region of the Pearl River network, eight estuaries, and the nearshore. The model contains 58,564 nodes and 91,572 elements. Horizontally, the spatial resolution varies from 40 to 50 m within the river network and from 600 to 1000 m near the estuarine mouth and bay to 5 km at the offshore open boundary. To model the surge under various approaching cyclones in summer days, the Pearl River network model is nested at the open boundary with the South China Sea model, which is also based on the spherical coordinate version of the FVCOM. Vertically, the model contains five uniform sigma layers to capture the process of turbulence momentum transferring upwards from the bottom. The forces driving the model include tidal waves from the South China Sea and surges caused by meteorological forces such as storms, cyclones, and rainfalls. In this paper, we consider only tropical cyclone. The semi-empirical parametric cyclone model is integrated into the ocean model to drive the surge. The advantage of the parametric cyclone model is that it does not require as much meteorological data as the dynamic model to drive the model. The only data needed contain the track of the cyclone, the center pressure drop, the forward speed, and the maximum wind radius, most of which are available on weather forecast websites except for the maximum wind radius, which should be obtained using a statistical formula and then adjusted according to the surge results. Here, the cyclone No.1822 (Table2), well known as the Super Typhoon Mangosteen (2018), is used as an example to simulate the storm surge in conjunction with the astronomical tide. The numerical results are compared with the measurements and demonstrate good agreement in the high surge level between them in Figure5. WaterWater2021 2021,,13 13,, 2375 2375 77 of 14 14

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FigureFigure 4.4. TheThe domaindomain andand gridsgrids ofof thethe PearlPearl RiverRiver networknetwork model.model.

Table 2. The parameters of the Typhoon Mangosteen (two days before landfall).

Time Latitude (N) Longitude (E) Pressure (hPa) Maximum Wind Speed (m/s) 00:00 14/09 16 126.9 905 56.6 06:00 14/09 16.7 125.7 905 56.6 12:00 14/09 17.4 124.1 905 56.6 18:00 14/09 18 122.3 905 56.6 00:00 15/09 18 120.5 940 46.3 06:00 15/09 18.5 119.7 940 46.3 12:00 15/09 19.2 118.3 950 43.7 18:00 15/09 19.8 117 955 41.2 00:00 16/09 20.6 115.3 960 38.6 06:00 16/09(a) Nansha 21.7 Station 113.5 960 (b) Wanqingsha Station 38.6 12:00 16/09 22.2 111.6 970 33.4 Figure 18:004. The 16/09 domain and grids 22.7 of the Pearl 109.7River network model. 980 28.3

Figure 5. Verification of the numerical model in four stations: Nansha, Wanqinsha, Henmeng, and (a) Nansha Station (b) Wanqingsha Station Neigang. Figure 5. Cont.

(c) Hengmen Station (d) Neigang Station Figure 5. Verification of the numerical model in four stations: Nansha, Wanqinsha, Henmeng, and Neigang.

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Figure 4. The domain and grids of the Pearl River network model.

Water 2021, 13, 2375 8 of 14 (a) Nansha Station (b) Wanqingsha Station

FigureFigure 5. Verification 5. Verification of the ofnumerical the numerical model modelin four instations: four stations: Nansha, Nansha,Wanqinsha, Wanqinsha, Henmeng, Henmeng, and Neigang.and Neigang. where τ is the wind stress along the x-coordinate direction with a positive direction 5. Resultsfrom the mouth to the upstream. WeFrom used Equation the Humen (16), Estuarywe can find geometry that the presentedeffect of the in positive Table1 wind and stress the numerical term, which results oncaused 8–9 September a maximum 2018 surge, (about is equivalent one week to a beforereduction the in typhoonthe Manning landfall) coefficient: to calibrate the analytical model. The calibration process| | are as follows:| we| first divided the parameter 푈 푈 푈 푈 range into small subareas with푔 an푛 increment − τ of= 0.1푔푛 for rs and 0.001 for n with their(17) ranges listed in Table3. The range of rs and푑n were referred to푑 Savenije et al. (2005; 2008) [1,14]. Then, we used these parameters to obtain the tidal results. Lastly, we compare the analytical solutions with the numerical results and compute the푑 RMSE (root mean square error) for 푛 = 푛 − τ (18) every analytical solution corresponding to the parameters. 푔푈|푈| The one with the least RMSE is considered the best fit solution, and its parameters are the optimal ones. The calibration parameterswhere 푛 are is considered presented the in Tablereduced3. In version general, of the the original analytical Manning solution coefficient fits the n numericalunder resultthe verycyclone well surge (Figure scenarios.6), and the calibrated parameters are within the physically proper range. InTherefore, contrast, we in can the make upstream the analytical section, solution there appears close to the to benumerical more deviation result by tuning than in the otherManning’s two sections. coefficient The to reason be less. is The probably analytically that calculated the length tidal of thisamplitudes section before is too and short to matchafter a tuning more reasonablethe coefficient convergent are shown length. in Figure 8, and all of the parameters are listed in Table 3. The Manning coefficient considering the cyclone surge is significantly less than that only considering the astronomical tide. Some are even below the lower limit of the Table 3. Parameters used for analytical model. physically reasonable range. This means that the effect of the storm surge on the tidal damping is similar to the reduction in the bottomManning’s friction. Coefficient In other words,n (m−1/3 thes) wind that Storage Widthblew Ratio in the upstream direction before landfall of the cyclone counteracted part of the fric- Subsections Value Range Astronomical Considering rs tion when the tidal wave traveled upstream along the river. For an applicationValue of the Range solution in a surge scenario and a physicallySpring Tide reasonable estimateCyclone of Surge the Manning coefficient Mouth 1in the meantime, 1–2a new solution based 0.005 on Savenije’s analytical 0.005 model, which contains 0.017–0.06 the Middle 1.8wind stress term, 1–2should be derived in 0.031 the future. 0.018 0.017–0.06 Upstream 1.5 1–2 0.035 0.015 0.017–0.06

(a) (b)

FigureFigure 6. Comparison 6. Comparison of the of analyticalthe analytical results results for for (a ()a tidal) tidal amplitude amplitude and and ( (bb) )travel travel time, time, and and the the numerical numerical simulation simulation after calibration during 8–9 September 2018. The diamonds denotes the numerical simulation results and the same for the after calibrationfollowing figures. during 8–9 September 2018. The diamonds denotes the numerical simulation results and the same for the following ﬁgures.

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The model was further veriﬁed with the numerical results on 16–17 September 2018. The reason for choosing this period is that Super Typhoon Mangosteen approached and made landfall on the coast of Guangdong Province during this time. Thus, at ﬁrst, we only simulated the astronomical tide without exerting a wind force in the numerical model. Through this simulation, the analytical result was compared with the numerical result assuming no cyclone landfall. From Figure7, it can be seen that the correspondence with the numerical result is good for the tidal amplitude. However, the deviation for the travel time shows that the analytical model tends to overestimate the celerity compared with the numerical model. A similar problem concerning the distinct accuracies between the tidal Water 2021, 13, 2375 amplitude and the travel time was also encountered by others using Savenije’s analytical 10 of 14

model, who attributed this deviation to the effect of the storage width ratio rs at different tidal amplitudes (Cai et al., 2012).

(a) (b)

Figure 7. Comparison of the analytical results for ( a)) tidal tidal amplitude amplitude and and ( (b)) travel travel time, time, and and the the numerical numerical simulation simulation for verificationveriﬁcation under astronomical tide during 16–17 SeptemberSeptember 2018.2018.

Then, we simulated the typhoon-surge scenario by setting the cyclone model and by exerting the wind and pressure forces on the surface of the tidal ﬂow. The analytical result was compared with the numerical result. It is shown in Figure8 that the difference between them can reach a maximum of as much as 0.6 m. This result is unsurprising because the original scope of the analytical model does not include storm surges. With respect to the equation, a wind stress term should be supplemented in the momentum equation as follows:

∂U ∂U ∂d 2 U|U| + U + g + gIb + gn 4 − τW = 0 (16) ∂t ∂x ∂x d 3

where τW is the wind stress along the x-coordinate direction with a positive direction from Figurethe mouth 8. Comparison to the upstream. of the analytical tidal amplitude before (solid line) and after (dash line) tuning Manning’s coefficient, and numerical simulation for verification under tide and surge during 16–17 From Equation (16), we can find that the effect of the positive wind stress term, which September 2018. caused a maximum surge, is equivalent to a reduction in the Manning coefficient: Table 3. Parameters used for analytical model. 2 U|U| 2 U|U| gn 4 − τW = gnW 4 (17) Storage Width Ratio Manning’s Coefficient n (m−1/3s) Subsections Value Range d 3 d 3 rs Astronomical Spring Tide Considering Cyclone Surge Value Range s 4 Mouth 1 1–2 0.005 d 3 0.005 0.017–0.06 n = n2 − τ (18) Middle 1.8 1–2 0.031w W gU|U| 0.018 0.017–0.06 Upstream 1.5 1–2 0.035 0.015 0.017–0.06 where nw is considered the reduced version of the original Manning coefficient n under the 6.cyclone Discussion surge scenarios. 6.1. The Characteristics of the Tidal Wave Propagation in the Humen Estuary Savenije’s solution describes the variation of the four dimensionless parameters ϵ, μ, δ, and λ as a function of the shape number γ and the friction number χ, which is depicted in Figure 9. As shown in these figures, the solutions can be generally divided into two families. For the areas with a small shape number γ (γ < 2), all of the parameters depend on friction number χ. The phase lag ϵ decreases with an increase in χ, and the tidal wave is a mixture of the progressive wave and standing wave, indicating a riverine estuary. In contrast, for the areas with a large shape number γ (γ > 3.5), the friction number has little influence on the four dimensionless parameters. The phase lag ϵ increases with an increase of χ, and the tidal wave is much closer to the standing wave, indicating an oceanic estuary. In the intermediate area within 2 < γ < 3.5, the critical γ for transforming from the progressive wave to the standing wave increases with the increase in χ. The hydrodynamic analysis, as with the geometric analysis, indicates that there are three distinct segments for the mainstream of the Humen estuary. Figure 9 shows the

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(a) (b)

FigureWater 7.2021 Comparison, 13, 2375 of the analytical results for (a) tidal amplitude and (b) travel time, 10and of 14 the numerical simulation for verification under astronomical tide during 16–17 September 2018.

FigureFigure 8. Comparison 8. Comparison of the analytical of the tidal analytical amplitude beforetidal (solidamplitude line) and afterbefore (dash (solid line) line) and after (dash line) tuning tuningManning’s Manning’s coefficient, coefficient, and numerical and numerical simulation for simulation verification under for tide verification and surge during under tide and surge during 16–17 16–17 September 2018. September 2018. Therefore, we can make the analytical solution close to the numerical result by tuning Manning’s coefficient to be less. The analytically calculated tidal amplitudes before and after tuning theTable coefficient 3. Parameters are shown in Figureused8 ,for and analytical all of the parameters model. are listed in Table3. The Manning coefficient considering the cyclone surge is significantly less than Storage Width Ratiothat only considering the astronomical tide. Some are even belowManning’s the lower limitCoefficient of the n (m−1/3s) Subsections physically reasonableValue Range range. This means that the effect of the storm surge on the tidal rs damping is similar to the reductionAstronomical in the bottom friction. Spring In other Tide words, Considering the wind that Cyclone Surge Value Range Mouth 1 blew in the upstream 1–2 direction before landfall of 0.005 the cyclone counteracted part of the 0.005 0.017–0.06 friction when the tidal wave traveled upstream along the river. For an application of the Middle 1.8 solution in a surge 1–2 scenario and a physically reasonable 0.031 estimate of the Manning coefficient 0.018 0.017–0.06 Upstream 1.5 in the meantime, a 1–2 new solution based on Savenije’s 0.035 analytical model, which contains the 0.015 0.017–0.06 wind stress term, should be derived in the future. 6.6. Discussion Discussion 6.1. The Characteristics of the Tidal Wave Propagation in the Humen Estuary 6.1.Savenije’s The Characteristics solution describes theof variationthe Tidal of theWave four dimensionlessPropagation parameters in the eHumen, µ, Estuary δ, and λ as a function of the shape number γ and the friction number χ, which is depicted in FigureSavenije’s9. As shown insolution these figures, describes the solutions the can bevariation generally divided of the into four two dimensionless parameters ϵ, families.μ, δ, and For the λ areas as witha function a small shape of numberthe shapeγ (γ <2), number all of the parameters γ and dependthe friction number χ, which is de- on friction number χ. The phase lag e decreases with an increase in χ, and the tidal wave ispicted a mixture in of Figure the progressive 9. As wave shown and standing in these wave, figures, indicating the a riverine solutions estuary. can be generally divided into Intwo contrast, families. for the areas For with the a large areas shape with number a γsmall(γ > 3.5), shape the friction number number has γ (γ < 2), all of the parameters little influence on the four dimensionless parameters. The phase lag e increases with an increasedepend of χ, andon thefriction tidal wave number is much closer χ. toThe the standing phase wave, lag indicating ϵ decreases an oceanic with an increase in χ, and the estuary. In the intermediate area within 2 < γ < 3.5, the critical γ for transforming from the progressivetidal wave wave is to thea mixture standing wave of increasesthe progressive with the increase wave in χ. and standing wave, indicating a riverine estuary.The hydrodynamic In contrast, analysis, for as the with areas the geometric with analysis, a large indicates shape that number there are γ (γ > 3.5), the friction num- three distinct segments for the mainstream of the Humen estuary. Figure9 shows the parametersber has forlittle the three influence segments ofon the the estuary. four In thedimensionless mouth section (0–8.7 parameters. km), as a The phase lag ϵ increases resultwith of an the nearlyincrease prismatic of estuary,χ, and this the section tidal is featured wave by is the much riverine closer estuary dueto the standing wave, indicating to the small γ. Moreover, the friction in this section is also very small. Both the small γ andanχ oceanicmake the tidalestuary. wave in thisIn sectionthe intermediate travel as a series ofarea progressive within waves, 2 < with γ < 3.5, the critical γ for trans- smallforming amplification from of thethe wave progressive amplitude in the wave upstream to direction.the standing In the middle wave section increases with the increase in χ. The hydrodynamic analysis, as with the geometric analysis, indicates that there are three distinct segments for the mainstream of the Humen estuary. Figure 9 shows the

Water 2021, 13, 2375 11 of 14

parameters for the three segments of the estuary. In the mouth section (0–8.7 km), as a result of the nearly prismatic estuary, this section is featured by the riverine estuary due to the small γ. Moreover, the friction in this section is also very small. Both the small γ and χ make the tidal wave in this section travel as a series of progressive waves, with small amplification of the wave amplitude in the upstream direction. In the middle section (8.7–30.4 km), γ is within the intermediate area and becomes small upstream while χ becomes large in the same direction. The sectional curve is crossed and divided into two parts by the ideal river curve. Above the ideal river curve, the effect of cross-sectional convergence dominates over the friction; thus, the tidal energy is accumulated and the tide is amplified in the downstream part of the middle section. Below the ideal river curve,

Water 2021, 13, 2375 the friction effect dominates over the cross-sectional convergence; thus, the tidal energy11 of 14 is dissipated and the tide is damped. The analysis corresponds with the result in the last section. In the upstream section (30.4–36 km), the large γ makes this section more similar to an oceanic estuary. The convergence effect is superior to friction according to the value (8.7–30.4of 훿; thus, km), theγ tideis withinis amplified. the intermediate The wave in area this and section becomes is typical small of upstream a standing while wave,χ becomeswhich is largenearly in independent the same direction. to the Thefriction. sectional In the curve process is crossed of calibration, and divided this section into two is partsinsensitive by the to ideal Manning’s river curve. coefficient. Above the ideal river curve, the effect of cross-sectional convergenceTo investigate dominates the overdifference the friction; made thus, by the the storm tidal energy surge iscompared accumulated with and the thenormal tide istidal amplified wave, we in thedepicted downstream the calibrated part of parameter the middle curves section. in the Below middle the idealsection river under curve, the thestorm friction surge. effect As expected, dominates the over curves the under cross-sectional the surge convergence;move slightly thus, toward the the tidal zone energy with isless dissipated friction from and the original tide is damped. curves under The analysis the astronomical corresponds tide. with This theis in result agreement in the with last section.the analysis In the of upstream the results section in the (30.4–36former section. km), the Previous large γ makes studies this have section indicated more that similar runoff to ancan oceanic increase estuary. the friction The convergencefor wave propagation effect is superior in an estuary; to friction however, according this study to the shows value that of δstorm; thus, surge the tide can is reduce amplified. the friction The wave for inthe this wave section propagation is typical while of a standingexaggerating wave, the which wave isamplitude. nearly independent to the friction. In the process of calibration, this section is insensitive to Manning’s coefficient.

(a) (b)

Figure 9. The relation between four parameters: (a) velocity number, (b) damping number, (c) celerity number, and (d) phase lag. The estuary shape number γ for different friction numbers χ is indicated by different line types. The bold lines indicate the parameters for the Humen Estuary under astronomical tide (red) and tide plus surge (blue), with the numbers indicating the distance from the estuary mouth in kilometers. The drawn line with the crosses represents the ideal estuary.

To investigate the difference made by the storm surge compared with the normal tidal wave, we depicted the calibrated parameter curves in the middle section under the storm surge. As expected, the curves under the surge move slightly toward the zone with less friction from the original curves under the astronomical tide. This is in agreement with the analysis of the results in the former section. Previous studies have indicated that runoff can increase the friction for wave propagation in an estuary; however, this study shows that storm surge can reduce the friction for the wave propagation while exaggerating the wave amplitude. Water 2021, 13, 2375 12 of 14

Figure 9. The relation between four parameters: (a) velocity number, (b) damping number, (c) celerity number, and (d) phase lag. The estuary shape number γ for different friction numbers χ is indicated by different line types. The bold lines indicate the parameters for the Humen Estuary under astronomical tide (red) and tide plus surge (blue), with the numbers Waterindicating2021, 13, 2375 the distance from the estuary mouth in kilometers. The drawn line with the crosses represents the ideal estuary. 12 of 14

6.2. Comparison with Other Estuaries 6.2.The Comparison Humen withEstuary Other is Estuariesso complicated that it can be divided into three sections. Any sectionThe demonstrates Humen Estuary its distinct is so complicated features compared that itcan with be the divided others. into To threebetter sections. understand Any thesection property, demonstrates we compared its distinct them featureswith other compared typical estuaries, with theothers. which Tohave better been understand analyzed bythe Savenije property, et al. we [14] compared (2008). themAs shown with otherin Figure typical 10, estuaries,the feature which of the have mouth been section analyzed (0– 8.7by km) Savenije in the et Humen al. [14] estuary (2008). can As shownbe comparable in Figure with 10, the Hau feature Estuariy, of the mouthespecially section the upstream(0–8.7 km) section in the (57–160 Humen km). estuary They can are be all comparable riverine estuaries with the with Hau small Estuariy, shape especially numbers. the Thus,upstream they have section a riverine (57–160 character km). They with are a all long riverine convergence estuaries length with and small a high shape phase numbers. lag. TheThus, difference they have is athat riverine the mouth character section with of a the long Humen convergence Estuary length has less and friction a high phasethan the lag. HauThe Estuary. difference Therefore, is that the the mouth tide sectionis amplified of the for Humen the former Estuary and has damped less friction for the than latter, the and Hau theEstuary. celerity Therefore, in the mouth the section tide is ampliﬁedof the Humen for theEstuary former is larger and damped than that for of thethe latter,Hau estuary. and the Thecelerity middle in thesection mouth (8.7–30.4 section km) of of the the Humen Humen Estuary Estuary is is larger comparable than that with of thethe HauSchelde estuary. Es- tuary,The middle with close section shape (8.7–30.4 numbers km) located of the inHumen the intermediate Estuary iszone. comparable Both of their with curves the Schelde cross theEstuary, ideal estuary with close curve, shape which numbers indicates located a transition in theintermediate point between zone. tidal Both damping of their and curves amplification.cross the idealThe difference estuary curve, is that which the depth indicates of the a transitionHumen Estuary point betweenis decreased tidal in damping the up- streamand ampliﬁcation. direction and Thethat differenceit is the opposite is that thefor the depth Schelde of the Estuary. Humen Moreover, Estuary is the decreased length of in thethe middle upstream Humen direction estuary and is that much it is less the than opposite that for of the the Schelde Schelde Estuary, Estuary. but Moreover, the curve the lengthlength is of similar. the middle This means Humen that estuary the parameter is much lessvariation than thatrate of the middle Schelde Humen Estuary, Estuary but the iscurve greater length than isthat similar. of the This Schelde means Estuary. that the parameter variation rate of the middle Humen Estuary is greater than that of the Schelde Estuary.

(a) (b)

FigureFigure 10. 10. PositioningPositioning of of the the Humen Humen (red (red line), line), Schelde (yellow squares), andand HauHau (pink(pink triangles)triangles) Estuaries Estuaries in in the the damping damping number (a) and celerity number (b) diagrams, with the numbers at the inflection points indicating the distance from number (a) and celerity number (b) diagrams, with the numbers at the inﬂection points indicating the distance from the the estuary mouth (in kilometers). The other lines are the same as those in Figure 9. estuary mouth (in kilometers). The other lines are the same as those in Figure9.

7.7. Conclusions Conclusions InIn this this study, study, Savenije’s Savenije’s solution solution was was applied applied for for the the first ﬁrst time time to to the the Humen Humen Estuary. Estuary. DueDue to to the the scarcity scarcity and and low low resolution resolution of of the the measured measured data, data, we we used used the the numerical numerical results results toto calibrate calibrate and and verify verify the the analytical analytical model. model. The The merit merit of using of using an analytical an analytical model model is that is itthat can itprovide can provide direct insight direct insightinto relationships into relationships among the among tidal theproperties, tidal properties, such as velocity such as amplitude,velocity amplitude, the tidal damping the tidal rate, damping and wave rate, celerity; and wave the geometry celerity; the indicator; geometry friction; indicator; and thefriction; tidal forces. and the This tidal analytical forces. This model analytical demonstrated model demonstratedgood accuracy good when accuracy it was applied when it towas an estuary applied where to an estuary the tide where dominates the tide in dominatescomparison in with comparison the river with discharge. the river The discharge. results indicateThe results that indicatethe analytical that the model analytical can predict model the can astronomical predict the astronomicaltide well in the tide Humen well in Estuarythe Humen after calibration. Estuary after However, calibration. it cannot However, predict it the cannot tide well predict when the a tropical tide well cyclone- when a inducedtropical surge cyclone-induced is superimposed surge onto is superimposed the astronomical onto tide. the The astronomical reason may tide. lie Thein the reason fact may lie in the fact that Savenije’s solution does not take the wind forces into account. After reducing Manning’s coefﬁcient, we found that the analytical result could be close to the numerical results. This means that the loss of the wind forcing can partly be compensated by adjusting the friction. Water 2021, 13, 2375 13 of 14

By analyzing the tidal wave propagation along the Humen Estuary, we found that the characteristics of the three sections—the mouth section (0–8.7 km), the middle section (8.7–30.4 km), and the upstream section (30.4–36 km)—are not alike at all. The mouth section is a typical riverine estuary with a nearly constant cross section. The tidal wave traveling there is in the form of a progressive wave and with a nearly classical celerity in a frictionless state. In contrast, the upstream section is a typical oceanic estuary with a short convergence length. The tidal wave there is in the form of a standing wave with a nearly infinite celerity. The middle section is in the intermediate zone of the two solution families. The tidal wave is first amplified and then damped, with a transition point that is the location of the maximum tidal amplitude. Finally, a comparison was conducted between the Humen Estuary and other estuaries.

Author Contributions: Conceptualization, Z.Z. and F.G.; methodology, Z.Z.; validation, Z.Z.; formal analysis, D.Z.; investigation, Z.Z.; resources, D.H.; data curation, D.H.; writing—original draft preparation, Z.Z.; writing—review and editing, F.G.; visualization, D.Z.; supervision, D.Z.; project administration, F.G.; funding acquisition, F.G. All authors have read and agreed to the published version of the manuscript. Funding: This paper was supported by the National Key R&D Program of China (grant No. 2018YFB0505500 and 2018YFB0505502) and by the National Natural Science Foundation of China (grant No. 41771421, 41771447, and 41571386). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: No report any data. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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