-Tidal bores are fascinating phenomena that occurs in the estuary of many tidal rivers worldwide.
-Tidal bores have significant repercussions on ecology and morphodynamics of an estuary.
-Despite the interest, a predictive criterion for tidal bore occurrence is still missing.
-Tidal bore is assessed from a phenomenological standpoint, based on the results of a numerical study.
-The predictive criterion for bore occurrence performs well against both numerical and real data

Defina, Andrea

Viero, Daniele Pietro

English

Padua Research Archive

A quantitative criterion to predict the occurrence of tidal bores

Daniele Pietro Viero

Andrea Defina

Archivio istituzionale della ricerca - Università di Padova

Original Citation:A quantitative criterion to predict the occurrence of tidal boresPublisher:Published version:DOI:Terms of use:Open Access(Article begins on next page)This article is made available under terms and conditions applicable to Open Access Guidelines, as described athttp://www.unipd.it/download/file/fid/55401 (Italian only)Availability:This version is available at: 11577/3282761 since: 2018-11-06T11:49:46ZUniversità degli Studi di PadovaPadua Research Archive - Institutional RepositoryXXXVI Convegno Nazionale di Idraulica e Costruzioni Idrauliche Ancona, 12-14 Settembre 2018 A QUANTITATIVE CRITERION TO PREDICT THE OCCURRENCE  OF TIDAL BORES Daniele P. Viero 1* & Andrea Defina 1 (1) Dipartimento di Ingegneria Civile, Edile e Architettura, Università di Padova (Padova) *email: daniele.viero@unipd.it KEY POINTS  Tidal bores are fascinating phenomena that occurs in the estuary of many tidal rivers worldwide.  Tidal bores have significant repercussions on ecology and morphodynamics of an estuary.  Despite the interest, a predictive criterion for tidal bore occurrence is still missing.  Tidal bore is assessed from a phenomenological standpoint, based on the results of a numerical study.  The predictive criterion for bore occurrence performs well against both numerical and real data. 1 INTRODUCTION Tidal bores, one of the most fascinating and widely known phenomena observed in tidal rivers, are positive waves travelling upstream along the estuary of a river. Their occurrence is related to a relatively rapid rise of the tide, and often enhanced by the funnelling shape of the estuary. As the flooding tide advances upstream along an estuary, the swell due to the tide grows and its front steepens, thus promoting the formation of a sharp front wave, i.e., the tidal bore (Bartsch-Winkler & Lynch, 1988; Chanson 2011b). Tidal bores play a significant role on the ecology and morphodynamics of an estuary, as well as on the social activities that take place in these environments. Turbulent mixing and dispersion are enhanced at the passage of a tidal bore (Simpson et al., 2004; Koch & Chanson, 2009), and significant bed erosion and sediment resuspension take place (Khezri &Chanson, 2012; Keevil et al., 2015); the bed material is suspended, aerated, advected upstream with the bore, and redeposited on the retreat of the tide (Chanson et al., 2011; Lubin et al., 2010; Reungoat et al., 2014). This process has a positive and significant influence on the breeding of many small, estuarine invertebrates such as shrimps, molluscs and worms, which in turn feed several species of fish and provide important feeding grounds for wading birds and estuarine wildlife (Chanson, 2011a). Tidal bores provide opportunity for recreational activities such as surfing, thus acting as a major tourist attraction. Given their importance and appeal, tidal bores have long been studied theoretically, numerically, experimentally, and with field investigations. However, possibly because of the many mechanisms and conditions that determine whether a tidal bore forms or not (e.g., freshwater river flow velocity and depth, bed slope and friction, the funnelling shape of the estuary, see for example Shi et al., 2014), the prediction of bore occurrence through effective criteria, that are not just qualitative, remains a challenge (Bonneton et al., 2016; Hoitink & Jay, 2016). In the present study, a wide series of numerical simulations are first carried out in a markedly simplified framework. Then, a phenomenological analysis of numerical results is used to shed light on the main processes and parameters controlling the formation of tidal bores (Viero & Defina, 2018). Finally, a criterion for predicting bore formation on the basis of external parameters is proposed. Besides well interpreting the results provided by numerical simulations, it also performs surprisingly well when compared with available data from real estuaries. 2 NUMERICAL RESULTS AND PHENOMENOLOGICAL ANALYSIS In this study, the problem is largely simplified by performing a wide series of numerical simulations in a rectangular channel of constant width, with a uniform subcritical flow forced downstream by rising the water level at a constant rate (Fig. 1). In the numerical simulations, the rate of downstream level rise resembles the D.P. Viero et al. – A quantitative criterion to predict the occurrence of tidal bores    maximum rising rates typical of semi-diurnal tides of real estuaries where tidal bores form (Friedrichs, 2010; Savenije, 2012; Bonneton et al., 2015). Most of simulations last less than 1-2 hours and, in few cases, extend up to 4 hours.  Figure 1. Schematic of the surge propagating upstream against a uniform flow of depth Y0 and velocity U0, with notations. xF  denotes the position of the foot of the front, i.e., the most upstream cross section where undisturbed water level is affected by the propagating tide; xU  denotes the position of the cross section where the flow reverse, i.e., the velocity is zero. Although the framework assumed in the present study to assess the formation and development of a tidal bore is extremely simple, yet the problem is still complex, and the solution is far from being trivial. From the results of numerical simulations, we identified three distinctive behaviours in which, respectively, a tidal bore forms, a tidal bore does not form, or a weak bore forms; the latter has a weakly steep front and, after the bore formed, its fate is to rapidly vanish.  We denote with “F ” the foot of the front, and with “U ” the cross-section where the flow reverses (see Fig. 1). According to the results of numerical simulations, when the section F travels upstream faster than the section U, no bore can form; otherwise, bores are found to form when U  meets F. When U  meets F, either the two sections align their velocities to move closely with the same speed (well-formed bore) or else, F speeds up and U  slows down, so that the just formed bore progressively reduces its height until it vanishes (weak bore). This first basic criterion, relying on the position and speed of sections U  and F, allows to univocally determine if, in a simulated scenario, a well-formed or a weak bore has formed, or not. 3 PREDICTIVE CRITERION FOR TIDAL BORE FORMATION For non-dimensionalization purposes, vertical lengths are scaled by the flow depth of the incoming uniform flow, Y0, horizontal lengths are scaled by the ratio Y0/s, where s = tanθ is the bottom slope, velocity is scaled by the small wave celerity c0 = (gY0)1/2, where g is gravity, and time is scaled by τ = Y0 /(c0 s) (see Viero et al., 2017). Non-dimensional variables are denoted with an asterisk (e.g., non-dimensional time is t∗ = t/τ). A criterion to predict the bore formation on the basis of external parameters is here proposed. First, we observe that the growth rate of downstream level controls, to some extent, the flow rate entering the channel from the sea, while the speed of the wave front determine the spreading of these volumes over the channel reach. When the growth rate of the downstream level is large compared to the speed of the wave front, the water from the sea is compressed within a short space and the front is pushed upstream, thus promoting the formation of a bore. On the contrary, if the growth rate of the downstream level is not large, the small volumes of water from the sea are spread over a longer channel reach so that the free surface elevation gently reduces from the sea to the foot of the wave front and the bore does not form. Accordingly, given that during the early stage of the process the foot of the front moves with velocity a0 =c0 − U0, the non-dimensional ratio (dh∗/dt∗)/a0∗ is used to measure the strength of this formative mechanism. Second, a competing mechanisms has been recently highlighted by Viero et al. (2017), who showed that the fate of a positive surge, propagating upstream against a subcritical uniform flow, is to gradually reduce its height and velocity until vanishing at a distance LM, which is given approximately by D.P. Viero et al. – A quantitative criterion to predict the occurrence of tidal bores     200*12FFLM  (1) where F0 is the Froude number of the incoming uniform flow. Using the non-dimensional ratio 1/LM* to measure the intensity of this competing mechanisms, the prevalence of the formative factor over the competing factor is expressed by the ratio  **0** 1//MLadtdh (2) which must be greater than a threshold value, say a non-dimensional parameter α, in order for the bore to occur. Therefore, noting that a0∗ = 1 − F0, a predictive criterion for bore formation is readily obtained in the form    0200**211FFFdtdh   (3) The criterion given by Eq. (3) identifies the rising rate of the downstream level and the Froude number of the freshwater incoming flow as the two main controlling factors of the process. With α = 1, the condition (3) is found to compare well with the results of the numerical model (see Fig. 2a). If the cases in which a weak bore forms are excluded, the criterion has a 100% success rate. The two dashed lines of Fig. 2a are obtained from Eq. (3) with α = 1.2 (upper curve) and α = 0.65 (lower curve), and they enclose a region on the (F0, dh∗/dt∗) diagram that contains most of conditions for a weak bore to form. This occurrence strengthen the proposed criterion.  Figure 2. a) Comparison between numerical results and the theoretical criterion for tidal bore occurrence as given by Eq. (3). White and black circles denote conditions in which bores form and do not form, respectively; crosses denote conditions in which weak bores form. The solid curve is given by Eq. (3) with α = 1; the two dashed lines are given by Eq. (3) with α = 1.2 (upper curve), and α = 0.65 (lower curve). b) Data of real estuaries where bores do form (white circles) and do not form (black circles) are plotted in the (F0, 0 /0) diagram. The solid curve is given by Eq. (4) with α = 1; the two dashed lines are given by Eq. (4) with α = 1.2 (upper curve) and α = 0.65 (lower curve). 3.1 Application of the present criterion to real estuaries The framework within which the proposed criterion is sound, differs from that of real estuaries mainly because, in the present numerical simulations, the channel has not a funnel shape, and because water levels imposed at the sea do not vary gradually as for a real tide. However, although important, these aspects are not likely to affect dramatically the conclusions drawn above, at least from a qualitative point of view. D.P. Viero et al. – A quantitative criterion to predict the occurrence of tidal bores    Based on a scaling analysis of the one-dimensional shallow water equations, Bonneton et al. (2016) showed that the global tidal dynamics is governed by three dimensionless parameters, namely i) the dimensionless tidal amplitude 0 = A0 / D0, with A0 the tide amplitude and D0 a characteristic water depth; ii) the friction parameter 0 = Cf0 Lω0 / D0, with Cf0 a characteristic and constant friction coefficient, and Lω0 = (g D0)1/2 / ω0 the frictionless tidal-wave length scale (e.g., Friedrichs, 2010; Lanzoni & Seminara, 1998; Savenije, 2012), ω0 being the tidal angular frequency; iii) the convergence ratio δ0 = Lω0 / Lb0 where Lb0 = −B/(dB/dx) is the convergence length, with B the channel width. Bore formation is weakly affected by the convergence ratio δ0, possibly because most of real estuaries all have approximately the same δ0 (Bonneton et al., 2016). In order to compare the prediction of our criterion with the available data for real estuaries, we rewrite condition (3) using the parameters suggested by Bonneton et al. (2015), i.e., 0 and 0     20 00001 14F FFk   (4) where k is a calibration factor. The predictive criterion (4), with k = 2.1, has been tested in predicting the occurrence of tidal bores in real estuaries for which data are available in the literature (Fig. 2b). The available experimental data do not distinguish strong from weak bores; however, it is interesting to observe that points pertaining to estuaries where very strong bores are observed, locate well above the plotted curves. Despite the obvious limitations of the theoretical and numerical frameworks used in this study, the predictions are found to compare favourably with field data, suggesting that the key features controlling the formation of tidal bores are retained in the proposed theory. REFERENCES Bartsch-Winkler, S. & Lynch, D. K. Catalog of worldwide tidal bore occurrences and characteristics. U.S. Geological Survey Circular 1022, 1–17, 1988. Bonneton, P., Bonneton, N., Parisot, J.-P. & Castelle, B. Tidal bore dynamics in funnel-shaped estuaries. J. Geophys. Res. Oceans 120, 923–941, 2015. Bonneton, P., Filippini, A. G., Arpaia, L., Bonneton, N. & Ricchiuto, M. Conditions for tidal bore formation in convergent alluvial estuaries. Estuar. Coast. Shelf Sci. 172, 121–127, 2016. Chanson, H. Current knowledge in tidal bores and their environmental, ecological and cultural impacts. Environ. Fluid. Mech. 11, 77–98, 2011a. Chanson, H. Tidal Bores, Aegir, Eagre, Mascaret, Pororoca – Theory and Observations. World Scientific Publishing, 2011b. Chanson, H., Reungoat, D., Simon, B. & Lubin, P. High-frequency turbulence and suspended sediment concentration measurements in the Garonne River tidal bore. Estuar. Coast. Shelf Sci. 95, 298–306, 2011. Friedrichs, C. T. Barotropic tides in channelized estuaries. In A. Valle-Levinson (Ed.), Contemporary issues in estuarine physics, Chapter 3, pp. 27–61. New York: Cambridge University Press, 2010. Hoitink, A. F. J. & Jay, D. A. Tidal river dynamics: Implications for deltas. Rev. Geophys. 54, 240–272, 2016. Keevil, C. E., Chanson, H. & Reungoat, D. Fluid flow and sediment entrainment in the Garonne River bore and tidal bore collision. Earth Surf. Process. Landforms 40, 1574–1586, 2015. Khezri, N. & Chanson, H. Inception of bed load motion beneath a bore. Geomorphology 153–154, 39–47, 2012. Koch, C., and Chanson, H. Turbulence measurements in positive surges and bores, J. Hydraul. Res., 47, 29–40, 2009. Lanzoni, S., and Seminara, G. On tide propagation in convergent estuaries, J. Geophys. Res. Oceans, 103, 30793–30812, 1998. Lubin, P., Chanson, H. & Glockner, S. Large Eddy Simulation of turbulence generated by a weak breaking tidal bore. Environ. Fluid. Mech. 10, 587–602, 2010. Reungoat, D., H. Chanson, & B. Caplain. Sediment processes and flow reversal in the undular tidal bore of the Garonne River (France). Environ. Fluid Mech. 14, 591–616, 2014. Savenije, H. H. G. Salinity and Tides in Alluvial Estuaries, 2nd completely revised edition, 2012, salinityandtides.com. Shi, J., Tong, C., Yan, Y. & Luo, X. Influence of varying shape and depth on the generation of tidal bores. Environ. Earth Sci. 72, 2489–2496, 2014. Simpson, J. H., Fisher, N. & Wiles, P. Reynolds stress and TKE production in an estuary with a tidal bore. Estuar. Coast. Shelf Sci. 60, 619–627, 2004. Viero, D. P. & Defina, A. A look into the mechanisms of tidal bore formation. Submitted to Water Resour. Res., 2018. Viero, D. P., Peruzzo, P.  & Defina, A. Positive surge propagation in sloping channels. Water 9, 518, 2017. 

Original Citation:A quantitative criterion to predict the occurrence of tidal boresPublisher:Published version:DOI:Terms of use:Open Access(Article begins on next page)This article is made available under terms and conditions applicable to Open Access Guidelines, as described athttp://www.unipd.it/download/file/fid/55401 (Italian only)Availability:This version is available at: 11577/3282759 since: 2018-11-06T11:35:02ZUniversità degli Studi di PadovaPadua Research Archive - Institutional RepositoryA quantitative criterion to predict the occurrence of tidalboresD. P. Viero & A. DefinaUniversity of Padova, ItalyABSTRACTTidal bores are positive waves travelling upstreamalong the estuary of many rivers worldwide, with im-portant implications on ecology, morphology, and so-cial activities. A predictive criterion for tidal bore for-mation is proposed and applied to real estuaries.1 INTRODUCTIONTidal bores, one of the most fascinating and widelyknown phenomena observed in tidal rivers, are posi-tive waves travelling upstream along the estuary of ariver. Their occurrence is related to a relatively rapidrise of the tide, and often enhanced by the funnellingshape of the estuary. As the flooding tide advancesupstream along an estuary, the swell due to the tidegrows and its front steepens, thus promoting the for-mation of a sharp front wave, i.e., the tidal bore [1, 2].Tidal bores play a significant role on the ecologyand morphodynamics of an estuary, as well as on thesocial activities that take place in this environment.Turbulent mixing and dispersion are enhanced at thepassage of a tidal bore [3], and significant bed erosionand sediment resuspension take place; the bed ma-terial is suspended, aerated, advected upstream withthe bore, and redeposited on the retreat of the tide[4]. This process has a positive and significant influ-ence on the breeding of many small, estuarine inver-tebrates such as shrimps, molluscs and worms, whichin turn feed several species of fish and provide impor-tant feeding grounds for wading birds and estuarinewildlife [2]. Tidal bores provide opportunity also forrecreational activities such as surfing, thus acting as amajor tourist attraction.Given its importance and appeal, tidal bore haslong been studied theoretically, numerically, experi-mentally, and with field investigations. However, pos-sibly because of the many mechanisms and conditionsthat determine whether a tidal bore forms or not (e.g.,freshwater river flow velocity and depth, bed slopeand friction, the funnelling shape of the estuary, etc.;see, for example, [5]), the prediction of bore occur-rence through effective criteria, that are not just qual-itative, remains a challenge [6].In this study, a phenomenological analysis of nu-merical results is carried out to shed light on the mainprocesses and parameters controlling the formation oftidal bore [7]. The problem is largely simplified byperforming a wide series of numerical simulations [8]in a rectangular channel of constant width, with a uni-form subcritical flow forced downstream by rising thewater level at a constant rate (Fig. 1). In the numericalsimulations, the rate of downstream level rise resem-bles the maximum rising rates typical of semi-diurnaltides of real estuaries where tidal bores form [9].aUU0Y Y0xFxhθxUU=0∆YUFFigure 1: Schematic of the surge propagating upstream against auniform flow of depthY0 and velocityU0, with notations.∆Yis the bore front height,xF denotes the position of the foot ofthe front, i.e., the most upstream cross section where undisturbedwater level is affected by the propagating tide;xU denotes theposition of the cross section where the flow reverse, i.e., thve-locity is zero.Although the framework assumed in the presentstudy to assess the formation and development of atidal bore is extremely simple, yet the problem is stillcomplex, and solutions are far from being trivial.From the results of numerical simulations, we iden-tify three distinctive behaviours, in which a tidal boreforms, a tidal bore does not form, and a weak boreforms; the latter has a weakly steep front and, afterthe bore formed, its fate is to rapidly vanish. With ref-erence to Fig. 1, we denote with “F ” the foot of thefront, and with “U ” the cross-section where the flowreverses. When the sectionF travels upstream fasterthan the sectionU , no bore can form; otherwise,bores were found to form whenU meetsF . WhenU meetsF , either the two sections align their ve-locities to move closely with a common speed (well-formed bore) or, alternatively,F speeds up andUslows down, so that the bore just formed progressivelyreduces its height until it vanishes (weak bore). Thisfirst basic criterion allows to univocally determine if,in a simulated scenario, a well-formed or a weak borehas formed, or not.More importantly, a further criterion is proposed inorder to predict the formation of tidal bore on the ba-sis of external parameters. First, we observe that thegrowth rate of downstream level controls, to some ex-tent, the flow rate (and thus the volume of water) en-tering the channel from the sea, whereas the speed ofthe wave front determine the spreading of this vol-ume over the channel reach. When the growth rate ofthe downstream level is large compared to the speedof the wave front, the water from the sea is com-pressed within a short space and the front is pushedupstream, this way promoting the formation of a bore.In the opposite case, the small volume of water fromthe sea is spread over a longer channel reach so thatthe free surface elevation gently reduces from the seato the foot of the wave front and the bore does notform. Accordingly, given that during the early stageof the process the foot of the front moves with ve-locity a0 =√gY0 − U0, the non-dimensional ratio(dh/dt)/a0s is used to measure the strength of thisfirst formative mechanism (g is gravity, Y0 and U0are respectively the uniform flow depth and velocity,s = tan θ is the bottom slope, andt denotes time; adimensional analysis is carried out in [10, 11]).Secondly, a competing mechanisms has been re-cently highlighted by [11], who showed that the fateof a positive surge propagating upstream against asubcritical uniform flow is to gradually reduce itsheight and velocity until vanishing at a distanceLM ,which is given byLM =Y0s2F01− F 20(1)whereF0 is the Froude number of the incoming uni-form flow. Using the non-dimensional ratioY0/LMsto measure the intensity of this competing mecha-nisms, the prevalence of the formative over the com-peting mechanism is expressed by the ratio[dh/dta0s]/[Y0LMs](2)which must be greater than a threshold value, sayα,in order for the bore to occur. The predictive criterionfor bore formation is thus obtained in the form1c0sdhdt≥ α(1− F0)(1− F 20 )2F0(3)which identifies the rising rate of the downstreamlevel and the Froude number of the freshwater incom-ing flow as the two main controlling factors of the pro-cess.Suitably rearranged in terms of significant non-dimensional parameters according to [6], the criterion(3) is tested in predicting the occurrence of tidal boresin real estuaries for which sufficient data are availablein the literature. Despite the obvious limitations of thetheoretical framework used in this study, the predic-tions are found to compare favourably with field data,suggesting that the key features controlling the forma-tion of tidal bores are retained in the proposed theory.REFERENCES[1] Bartsch-Winkler, S. and Lynch, D.K., 1988. Cat-alog of worldwide tidal bore occurrences andcharacteristics. U.S. Geological Survey Circular,1022: 1-17.[2] Chanson, H., 2011. Current knowledge in tidalbores and their environmental, ecological and cul-tural impacts. Environmental Fluid Mechanics,11: 77-98.[3] Simpson, J.H., Fisher, N. and Wiles, P., 2004.Reynolds stress and TKE production in an estu-ary with a tidal bore. Estuarine, Coastal and ShelfScience, 60: 619-627.[4] Reungoat, D., Chanson, H. and Caplain, B.(2014). Sediment processes and flow reversalin the undular tidal bore of the Garonne River(France). Environ. Fluid Mech. 14: 591-616.[5] Shi, J., Tong, C., Yan, Y. and Luo, X., 2014. Influ-ence of varying shape and depth on the generationof tidal bores. Environmental Earth Sciences, 72:2489-2496.[6] Bonneton, P., Filippini, A.G., Arpaia, L., Bon-neton, N. and Ricchiuto, M., 2016. Conditions fortidal bore formation in convergent alluvial estu-aries. Estuarine, Coastal and Shelf Science, 172:121-127.[7] Viero, D.P. and Defina, A., 2018. A look into themechanisms of tidal bore formation. Water Re-sources Research, in review.[8] Defina, A. and Viero, D.P. (2010). Open channelflow through a linear contraction, Physics of Flu-ids, 22: 036602.[9] Savenije, H.H.G., 2012. Salinity and Tides in Al-luvial Estuaries, 2nd completely revised edition.salinityandtides.com.[10] Bonneton, P., Bonneton, N., Parisot, J.-P. andCastelle, B., 2015. Tidal bore dynamics in funnel-shaped estuaries. Journal of Geophysical Re-search, Oceans, 120: 923-941.[11] Viero, D.P., Peruzzo, P. and Defina, A., 2017.Positive surge propagation in sloping channels.Water, 9: 518.

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A quantitative criterion to predict the occurrence of tidal bores

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