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=Dam break flow=
  
Tidal bore dynamics
 
  
A tidal bore is a sudden elevation of the water surface that travels upstream an estuary with the incoming flood tide. This article describes the processes involved in this ultimate stage of tidal wave deformation and the modelling of these processes. For an introduction to the topic of tidal wave deformation the reader is referred to the article [[Tidal asymmetry and tidal basin morphodynamics]].
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This article discusses the often catastrophic flows that result from the failure of high dams that protect low-lying land. Dam break mechanisms are not dealt with; for this the reader is referred to the extensive literature existing on this subject, see for example  Zhang et al. (2016) <ref>Zhang, L., Peng, M., Chang, D. and Xu, Y. 2016. Dam failure and Risk Assessment. John Wiley and Sons, Singapore</ref> and Almog et al. (2011) <ref>Almog, E., Kelham, P and King, R. 2011. Modes of dam failure and monitoring and measuring techniques. Environmental Agency,UK https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/290819/scho0811buaw-e-e.pdf</ref>.
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==Introduction==
 
==Introduction==
  
[[Image:DordogneTidalBore.jpg|center|700px|thumb|Figure 1. Left image : Partially breaking tidal bore on the Dordogne River (Courtesy of Antony “YEP” Colas). Right image : Undular tidal bore on the Garonne River (Bonneton et al. 2011 <ref name=B11a>Bonneton, P, Parisot, J-P., Bonneton, N., Sottolichio, A., Castelle, B., Marieu, V., Pochon, N. and Van de Loock, J. 2011. Large amplitude undular tidal bore propagation in the Garonne River, France, Proceedings of the 21st ISOPE Conference: 870-874, ISBN 978-1-880653-96-8</ref>).]]
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[[image:Dijkdoorbraak1953.png|thumb|left|300px|Fig. 1. Sea dikes protecting low-lying polders in the Netherlands were breached during the extreme storm surge of 31 January 1953.]]
 
 
{|  style="border-collapse:collapse; font-size: 12px;  background:ivory;" cellpadding=5px align=right width=60%
 
|+ Table 1. Estuaries and rivers with substantial tidal bores (bore height of half a meter up to several meters). The maximum tidal range is the maximum range recorded at a tide gauge situated close to the location in the estuary where the tidal range is highest.
 
|- style="font-weight:bold;  font-size: 11px; text-align:center; background:lightblue"
 
! width="25% style=" border:1px solid blue;"| Estuary/river name
 
! width="5% style=" border:1px solid bleu;"| Maximum tidal range [m]
 
! width="20% style=" border:1px solid blue;"| Tide gauge location
 
! width="10% style=" border:1px solid blue;"| Country
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Shannon River
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 5.6
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Shannon
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Ireland
 
|-
 
| style="border:2px solid lightblue; font-size: 10px;  font-size: 10px; font-size: 10px; text-align:center"|  Humber,Trent
 
| style="border:2px solid lightblue; font-size: 10px;  font-size: 10px; text-align:center"| 8.3
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Hull
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center" rowspan="8"|  UK
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Great Ouse
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 7.4
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| King's Lynn
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Severn
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 12
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Portishead
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Dee
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 9.8
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Flint
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Mersey
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 10
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Liverpool
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Ribble
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 10.2
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Lytham
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Morecambe bay, Nith River, River Kent
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 10.9
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Morecambe
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| River Eden
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 10.3
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Silloth
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Seine
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 8.5
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Honfleur
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center" rowspan="4"|  France
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Canal de Carentan
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 7
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Carentan
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Baie du Mont Saint Michel – Sélune River
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 14
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Granville
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Garonne, Dordogne
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 6.3
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Podensac
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Pungue
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 7
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Beira
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Mozambique
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Qiantang
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 5
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Ganpu
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| China
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Indus
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 4
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Port Muhammad Bin Qasim
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Pakistan
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Hooghly
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 5.7
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Sagar Island
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| India
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Brahmaputra
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 5.7
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Chittagong
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Bangladesh
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Sittaung
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 6.3
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Moulmein
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Myanmar
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Batang Lupar River
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 5.6
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Kuching
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Sarawak
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Kampar River
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 5.3
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Pulo Muda
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Sumatra
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Hooghly
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 5.7
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Sagar Island
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| India
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Fly , Bamu, Turamu Rivers
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 4.2
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| estuary
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"|  West Papua
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Styx
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 6.4
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Mackay, Queensland
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center" rowspan="2"|  Australia
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Daly River
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 7.9
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| estuary
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Turnagain Arm, Knick Arm
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 7.9
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Anchorage
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"|  Alaska
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Bay of Fundy, Petitcodiac and Salmon rivers
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 16
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Truro
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Canada
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Colorado River
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 7.7
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| San Filipe
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Mexico
 
|-
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Amazon, Araguira, Guama, Capim and Mearim Rivers
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| 3.4
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Macapa
 
| style="border:2px solid lightblue; font-size: 10px; text-align:center"| Brazil
 
|}
 
 
 
 
 
When a tidal wave propagates upstream into an estuary, its shape is progressively distorted. In many estuaries the high-water (HW) tidal wave crest propagates faster up-estuary than the low-water (LW) tidal wave trough. HW propagates considerably faster than LW if the mean channel depth <math>D_0</math> is not much greater than the spring tidal range <math>2a</math> (same order of magnitude or a few times larger) and if the intertidal area is smaller than the tidal channel surface area. The tidal rise period is shortened and the tidal wave becomes steeper during propagation. If the tidal wave can propagate sufficiently far upstream the river without strong damping, the tidal wave front may become so steep that a water level jump – a so-called tidal bore, can form at the beginning of the rising tide.
 
 
 
Table 1 presents estuaries and tidal rivers in which significant tidal bores have been observed (see also Bartsch-Winkler and Lynch (1988) <ref> Bartsch-Winkler, S., and Lynch, D. K. 1988. Catalog of worldwide tidal bore occurrences and characteristics. US Government Printing Office</ref> and Colas (2017)<ref> Colas, A. 2017. Mascaret, prodige de la marée. Atlantica Editions</ref>). It appears that tidal bores generally occur for large tidal ranges (Tr0) at the estuary mouth. However, such a simple criterion is not sufficient to classify estuaries in terms of tidal bore occurrence. For instance, Furgerot (2014) <ref>Furgerot, L. (2014). Propriétés hydrodynamiques du mascaret et de son influence sur la dynamique sédimentaire: Une approche couplée en canal et in situ (estuaire de la Sée, Baie du Mont Saint Michel) (Doctoral dissertation, Caen)</ref> showed that in the Sée/Mont Saint Michel estuary, Tr0 must be larger than 10 m for tidal bore formation in the Sée River and, on the other hand, Bonneton et al. 2015 observed tidal bores in the Gironde/Garonne estuary even for Tr0 smaller than 2 m.
 
 
 
The estuarine shape (depth and width profiles) plays an important role in tidal bore formation. This is illustrated by the observation that engineering works (dredging, weirs) during the past century have weakened or even completely suppressed tidal bores in many rivers, for example in the rivers Seine, Loire, Charente and Petitcodiac. We will discuss in the following  general conditions for bore formation in terms of estuary geometry and flow conditions. 
 
 
 
Illustrations of tidal bores are shown in Fig. 1.
 
  
  
  
 +
Dam failure can lead to disastrous situations. Most dam break tragedies are related to the collapse of reservoir dams in mountain rivers. The failure of sea dikes can also cause major disasters, although in this case the level difference is not as large. A dramatic example is the failure of more than fifty sea dikes in the Netherlands during the extreme storm surge of 1953, see figure 1. Many of these dikes protected land lying below the average sea level, while the storm surge level exceeded more than five meters. Rapid flooding killed more than two thousand people who were unable to flee in time to safe places. The most common dam failure mechanisms are related to overtopping and seepage (also called piping or internal erosion)<ref name=F8>Froehlich, D.C. 2008. Embankment Dam Breach Parameters and Their Uncertainties. Journal of Hydraulic Engineering, ASCE 134: 1708-1721</ref>. In the case of the 1953 storm surge, overtopping and subsequent scour of the interior dike slope was the most important dike failure mechanism.
  
  
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==Tidal bore formation in funnel-shaped estuaries==
 
  
 +
[[image:DamBreakFlowPrinciple.jpg|thumb|center|600px|Fig. 2. Left panel: Schematic representation of water retention behind a dam. Right panel: Positive downstream and negative upstream surges following instantaneous dam removal.]]
  
[[Image:BoreFormationCondition.jpg|left|400px|thumb|Figure 4. Estuarine classification in parameter space (<math>\epsilon, \phi</math>). Maximum surface elevation slope <math>A_{max}</math>, the white dashed line represents the <math>\epsilon_c</math> curve, namely the limit for tidal bore appearance following the criterion <math>A_{max} \ge 10^{-3}</math>. Each point in the plane (<math>\epsilon, \phi</math>) represents a numerical tidal wave solution for an idealized convergent estuary; this figure relies on 225 numerical simulations (Filippini et al., 2018)<ref></ref>.]]
 
  
 +
The consequences of dam failure have been studied for more than a century. It nevertheless remains a challenging topic due to the high non-linearity of the flood wave propagation. The problem can be tackled with numerical models, but rough estimates and insight into the tidal wave dynamics can be obtained with analytical solution methods.
  
The analysis of nonlinear tidal wave transformation in estuaries, in terms of tidal forcing at the estuary mouth and large-scale geometrical properties of the channel, has received considerable attention (see [[Tidal asymmetry and tidal basin morphodynamics]]). By contrast, the extreme nonlinear tidal-wave case where tidal bores form is much less studied.
 
  
[[Image:GaronneObservationOndularBoreDevelopment.jpg |center|800px|thumb|Figure 2. Tidal wave distortion and bore formation observed in the Gironde-Garonne estuary (Bonneton et al. 2015<ref name=B15></ref>) at spring tide. Left panel: Tide gauge locations on the estuary. Right top panel: Time series of tide elevation at 4 different locations, from the estuary mouth (Le Verdon) to the upper estuary (Podensac); Right bottom panel: Tidal bore illustrations at Podensac field site, located 126 km upstream the river mouth.]]
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==Frictionless dam-break flow: analytical solution==
  
[[Image:TidalBoreSimulation.jpg |center|600px|thumb| Figure 3. Tidal wave distortion and bore formation in a convergent estuary. Numerical simulations by Filippini et al. 2019 <ref name=F> Filippini, A. G., Arpaia, L., Bonneton, P., & Ricchiuto, M. 2019. Modeling analysis of tidal bore formation in convergent estuaries. European Journal of Mechanics-B/Fluids 73: 55-68</ref> Left panel: Rising tide elevation at different increasing times. Right panel: Zoom of  left panel.]]
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Analytical solutions relate to idealized situations as depicted in Fig. 2. The initial situation consists of an infinite reservoir with a water level that is <math> h_0 </math> higher than the horizontal ground level downstream of the dam. Dam break is simulated by instantaneous removal of the dam. This causes a positive surge in the positive <math> x </math>-direction and a negative surge in the negative <math> x </math>-direction. The floor is dry in front of the positive surge and the body of water behind the negative surge is undisturbed: water level =<math> h_0 </math> and current speed <math> u = 0 </math>.
  
The most intense tidal bores occur in long and shallow tide-dominated funnel-shaped estuaries (e.g. Qiantang, Severn, Kampar, Gironde/Garonne/Dordogne, …) . The formation of these tidal bores is governed by the progressive distortion of the tidal wave as it propagates up the estuary.  When the ebb-flood asymmetry (ebb duration longer than the flood and larger flood than ebb currents) is strong enough generally in the upper estuary (see tidal wave elevation in Fig. 2 at Bordeaux and Podensac), tidal bore can form (see Figs. 2 and 3).
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[[image:SouthForkDamFailure1889.jpg|thumb|left|300px|Fig. 3. The earth-filled South Fork Dam on Lake Conemaugh (Pennsylvania, US) collapsed in 1889, killing 2,209 people in downstream villages.]]
This extreme nonlinear deformation of a tidal wave occurs in a limited number of estuaries. To determine the conditions favourable to tidal bore occurrence, a scaling analysis can eb carried out for a schematic channel geometry. A usual schematization is based on the approximations <ref>Lanzoni, S. and Seminara, G. 1998. On tide propagation in convergent estuaries, J. Geophys. Res. 103: 30793–30812</ref><ref>Savenije, H.H.G. 2012. Salinity and Tides in Alluvial Estuaries, second ed., Salinity and Tides in Alluvial Estuaries, second ed., www.salinityandtides.com</ref><ref>Dronkers, J. 2017. Convergence of estuarine channels. Continental Shelf Res. 144: 120–133</ref>:
 
*Uniform channel depth <math>D</math>,
 
*Exponential convergence of the width <math>B(x)</math>.
 
The convergence length <math>L_b </math>, defined as <math>L_b=B(x)/dB/dx</math>, is assumed independent of the along-channel coordinate <math>x</math>.
 
The forcing tidal wave at the estuary mouth can be characterized by its tidal frequency <math>\omega=2 \pi / T</math> and its amplitude <math>A_0=Tr_0/2</math>, where <math>Tr_0</math> is the estuary mouth tidal range. Another important parameter which controls tide propagation in estuary is the dimensionless mean friction coefficient <math>c_D</math>. The tidal wave dynamics is then controlled, if we neglect fresh water discharge effects, by three dimensionless parameters:
 
*the nonlinearity parameter, <math>\epsilon=A_0/D</math>,
 
*the friction parameter, <math>\phi=c_D \sqrt{gD}/(\omega D)</math>,
 
*the convergence ratio, <math>\delta=\sqrt{gD}/(\omega L_b)</math>.
 
  
The formation of tidal bores is mainly governed by the progressive distortion of the tidal wave as it propagates up the estuary. This extreme nonlinear deformation of the tidal wave occurs under special conditions, in particular<ref name=B15>Bonneton, P., Bonneton, N., Parisot, J-P. and Castelle, B. 2015. Tidal bore dynamics in funnel-shaped estuaries.  J. Geophys. Res.: Ocean 120: 923-941. DOI: 10.1002/2014JC010267</ref> <ref name=R>Rousseaux, G., Mougenot, J. M., Chatellier, L., David, L. and Calluaud, D. 2016. A novel method to generate tidal-like bores in the laboratory. European Journal of Mechanics-B/Fluids 55:31-38</ref>:
 
*a large tidal amplitude <math>a</math>,
 
* a long, shallow and convergent channel.
 
  
River flow and river-bed slope also influence tidal bore formation.  River flow contributes to tidal wave deformation by enhancing the longitudinal velocity gradient and river-bed slope by upstream reduction of the water depth. However, river flow and river-bed slope also contribute to tidal wave damping and thus oppose tidal bore formation <ref> Horrevoets, A. C., Savenije, H. H. G., Schuurman, J. N. and Graas, S. 2004. The influence of river discharge on tidal damping in alluvial estuaries. Journal of Hydrology 294: 213-228</ref>. The latter effect usually dominates. Observations show that the tidal bore in the Garonne and Dordogne (France) is suppressed at high river runoff<ref name=B16>Bonneton, P., Filippini, A.G., Arpaia, L., Bonneton, N. and Ricchiuto, M.  2016. Conditions for tidal bore formation in convergent alluvial estuaries.  Estuarine, Coastal and Shelf Science. 172: 121-127</ref>. A similar effect is observed in the Daly estuary by Wolanski et al. (2006), who relate the occurrence of the tidal bore at low river discharge to the small water depth during such conditions. The opposite effect is reported for the Guamá-Capim river system near the mouth of the Amazon River, where tidal bores are observed only at high river discharges, in conjunction with high equinoctial tides<ref>Freitas, P.T.A., Silveira, O.F.M. and Asp, N.E. 2012. Tide distortion and attenuation in an Amazonian tidal river. Brazilian journal of oceanography, 60: 429-446</ref>. 
 
  
Available field data suggest that tidal bores form in convergent estuaries characterized by large values of the convergence ratio <math\delta</math>.  Bonneton et al. (2016)<ref name=B16> Bonneton, P., Filippini, A.G., Arpaia, L., Bonneton, N. and Ricchiuto, M. 2016. Conditions for tidal bore formation in convergent alluvial estuaries. Estuarine, Coastal and Shelf Science. 172: 121-127</ref> and Filippini et al. (2018)<ref name=F></ref> have analysed tidal bore occurrence as a function of the dimensionless parameters <math>\epsilon</math> and <math>\phi</math> for a fixed value of the convergence ratio <math>\delta=2.4</math>. Fig. 4 presents the position of convergent estuaries in the parameter space (<math>\epsilon, \phi</math>). We can observe a clear separation between tidal-bore estuaries and non tidal-bore estuaries. Tidal bores occur when the nonlinearity parameter <math>\epsilon</math> is greater than a critical value, <math>\epsilon_c</math>, which is an increasing function of the friction parameter<math>\phi</math>. For small <math>/phi</math>-values (friction parameter <math>\phi \approx 15</math>), tidal bores can form for <math>\epsilon</math> greater than 0.2. By contrast, for large <math>\phi</math> values the tidal bore formation requires much larger nonlinearity parameters. These results show that bore formation is mainly controlled by the competition between two physical processes: (a) the nonlinear distortion of the tidal wave, which is favourable to bore inception; (b) the friction dissipation of the tidal wave, which is unfavourable to bore formation.
 
  
  
==Conditions for tidal bore formation==
 
  
More generally, the analysis of tidal bores observed in natural estuaries suggests the following conditions for tidal bore formation.
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The study of the dam break-flow was triggered by several dam failures in the 19th century, in particular the breach of the South Fork Dam in Pennsylvania (USA) in 1889, Fig. 3. Three years later Ritter <ref> Ritter, A. 1892. Die Fortpflanzung der Wasserwellen. Zeitschrift des Vereines Deutscher Ingenieure 36: 947-954 </ref> published an exact analytical solution for dam-break flow by assuming that frictional effects can be ignored. The derivation is given in box 1. According to this solution, the tip of the positive surge advances at high speed (supercritical flow) given by <math> u_f = 2 c_0</math>, where <math>c_0 = \sqrt {gh_0}</math> and <math> g </math> is the gravitational acceleration. The shape of the positive and negative wave is a concave-up parabola,
  
The first requirement for a tidal bore is sufficient length of the estuarine-tidal river system, i.e. the tide can penetrate far enough into the system. This does not depend so much on the convergence length <math>L_b</math>, but mainly on the average slope of the estuary-tidal river system and river flow. A steep slope and a large flow limit the penetration of the tide.
+
<math>h(x,t)= h_0, \; x<-c_0 t; \quad h(x,t)= \Large\frac{h_0}{9}\normalsize \; (2 - \Large\frac{x}{c_0 t}\normalsize)^2 , \, -c_0 t<x<2c_0 t; \quad h(x,t)= 0, \, x>2c_0 t ,  \qquad (1)</math>
  
The tidal bore develops in a section of the estuary-tidal river system where the nonlinearity parameter <math>\epsilon</math> is large (shallow depth, large tidal range) and where the tidal wave is already rather asymmetrical at the estuarine mouth. Sometimes the tidal bore is already forming close to the mouth of the estuary. This is the case of estuaries that have a large mouth bar, such as the Qiangtang and the Charente (before the large-scale interventions). But this is rather the exception. Most often the tidal bore is formed in a section of the estuary-tidal river system further upstream, in the tidal river or on the tidal flats at the head of the estuary (as for the Garonne, Seine, Sée, etc.). A low river flow (or low ebb tide) seems favorable to trigger the tidal bore. The convergence of the width plays a smaller role. If the longitudinal channel slope is weak, the tidal bore can travel over a long distance without much dissipation. A small channel slope is generally indicative of low-to-moderate current velocities and friction (steep slopes are generally related to strong friction generated by high current velocities and coarse bed sediments).
+
see Fig. 4. According to Eq. (1) and Box 1, the discharge through the breach per unit width is given by
  
The condition of a large tidal range and a strong tidal asymmetry depends on friction; one should therefore distinguish between muddy and non-silty estuaries. For estuaries where the solid fluvial flow is important and containing a large fraction of cohesive mud (mainly estuaries in the (sub) tropic regions, but also the Garonne), the estuarine channel bed is generally smooth so the friction is weak (small <math>\phi</math>); the tidal wave is not or hardly damped. In this case a small relative channel depth (large <math>\epsilon</math>) is the main condition to obtain a strong asymmetry of the tidal wave. Estuaries with low solid discharge or supplying sandy/coarse sediments have sandy or gravelly bottom channels in the mouth region that produces strong friction (large <math>\phi</math>) and strong tidal energy dissipation. In this case, the main condition for obtaining a large tidal range is a strong convergence of the width of the estuary (large <math>/delta</math>).
+
<math>q = \frac{8}{27} h_0 \sqrt{g h_0}. \qquad  (2) </math>  
Secondly, large river discharges induce strong tidal damping (Horrevoets et al. 2004)<ref> Horrevoets, A. C., Savenije, H. H. G., Schuurman, J. N. and Graas, S. 2004. The influence of river discharge on tidal damping in alluvial estuaries. Journal of Hydrology 294: 213-228</ref>, especially in the upper estuary, where tidal bores generally occur. Wolanski et al. 2006 showed that in the Daly estuary tidal bore occur at spring tides only for low river discharge. In the Garonne River, tidal bore formation is strongly controlled by river discharge (Bonneton et al. 2015, 2016). For large discharge (order of 1000 m3/s), tidal bores do not form. By contrast, for low discharge (order of 150 m3/s) tidal bore almost always occurs in the Garonne River and can even forms at neap tide. These findings was confirmed by Filippini et al. (2019)<ref name=F></ref> from numerical simulations. The opposite effect is reported for the Guamá-Capim river system near the mouth of the Amazon River, where tidal bores are observed only at high river discharges, in conjunction with high equinoctial tides<ref>Freitas, P.T.A., Silveira, O.F.M. and Asp, N.E. 2012. Tide distortion and attenuation in an Amazonian tidal river. Brazilian journal of oceanography, 60: 429-446</ref>. 
 
  
Variations of local river geometry and water depth can also significantly affect tidal bore dynamics (Bonneton et al. (2011)<ref name-B11a></ref> and Keevil et al. (2015)<ref>Keevil, C. E., Chanson, H. and Reungoat, D. 2015. Fluid flow and sediment entrainment in the Garonne River bore and tidal bore collision. Earth Surface Processes and Landforms 40: 1574-1586</ref>, Fig. 10 ??. Observations show that tidal bores often develop in shallow rivers that discharge from the higher upstream zone into a broad estuary. If the tidal wave has already acquired sufficient asymmetry when travelling through the estuary, a tidal bore develops when the tidal wave surges into the shallow river. The presence of a mouth bars (large <math>\epsilon</math>) is favourable to the development of a strong asymmetry of the tidal wave and tidal bore formation, see Fig. 5. On the contrary, the presence of local channel constrictions (large <math>\phi</math>) is unfavourable to tidal bore formation.  
+
Figure 4 also shows the shape of the dam break wave that was observed in laboratory experiments. For small values of <math> x / t </math>, the frictionless solution corresponds fairly well with the observations. However, the front zone is different: the shape is a bull nose, rather than a sharp edge. The front also advances more slowly; the speed of the front decreases with time and is closer to <math> u_f = c_0 </math> than to <math> u_f = 2 c_0 </math>. The reason for this difference is the neglect of frictional effects that are important in the thin fluid layer near the front.
  
[[Image:Seine-Qiangtang.jpg|center|700px|thumb|Figure 5. Many natural estuaries have large shoals in the mouth zone. These so-called mouth bars strongly enhance distortion of the tidal wave entering the estuary. Left image: Tidal bore in the Seine River around 1960, before the mouth bar was dredged. Right image: Tidal bore in the Qiangtang estuary, where a large mouth bar is still present (https://en.wikipedia.org/wiki/Qiantang_River ).]]
 
  
 +
[[image:DamBreakFlowDerivation.png|thumb|center|900px|Box 1. Derivation of the solution for frictionless dam break flow.]]
  
==Tidal bore characteristics==
 
  
A tidal bore can be schematically represented by a propagating transition between two streams of water depth <math>D_1</math> and <math>D_2</math>, with <math>D_1 < D_2</math> (see Fig. 6). Once such a hydraulic jump has developed, its flow velocities ahead and behind the jump can be derived from the mass and momentum balance equations (Fig. 6):
+
==Dam-break flow with friction==
  
<math>v_1=u_1 – c = - \sqrt{\Large \frac{g D_2 (D_2 + D_1)}{2 D_1}} \normalsize , \quad \quad (1) </math>  
+
[[image:DamBreakWaveProfiles.jpg|thumb|left|400px|Fig. 4. Wave profile after dam break. The red curve corresponds to the frictionless solution Eq. 1. The blue band represents data from different laboratory experiments performed at times <math>\small t \sim (40-80) \, \sqrt {h_0 / g}\normalsize </math> after removal of the dam <ref>Dressler, R. 1954. Comparison of theories and experiments for the hydraulic dam-break wave. Proc. Int. Assoc. Scientific Hydrology Assemblée Générale, Rome, Italy 3 (38)M 319–328</ref> <ref>Schoklitsch, A. 1917. Über Dambruchwellen. Kaiserliche Akademie der Wissenschaften, Wien, Mathematisch-Naturwissenschaftliche Klasse, Sitzungberichte IIa, 126: 1489–1514</ref><ref>Cavaillé,Y. 1965. Contribution à l’étude de l’écoulement variable accompagnant la vidange brusque d’une retenue. Publ. Scient. et Techn. du Ministère de l’Air, 410, Paris, France, 165</ref>. The frictionless solution only depends on <math>x/t</math>. Observed wave profiles also depend on time <math>t</math> because of the decreasing wave tip speed.]]
  
<math>v_2=u_2 – c = - \sqrt{\Large \frac{g D_1 (D_2 + D_1)}{2 D_2}} \normalsize , \quad \quad (2) </math>
 
  
where <math>c</math> is the bore celerity, <math>v</math> the velocity in the moving frame and <math>u</math> in the fixed frame. These equations show that the problem is entirely determined by the depth ratio <math>D_2 / D_1</math>, or equivalently by <math>\Delta D / D1</math>, where <math>\Delta D = D_2 – D_1</math> is the bore height. The Froude numbers around the jump, in the moving frame can be obtained from equations (1) and (2):
+
When friction terms are included in the flow equations, there is no exact analytical solution. It is generally assumed that the frictionless flow equations are approximately valid in a short period after dam break and for small values of <math>x / (c_0 t) </math>. In the front zone, where the water depth is small, the momentum balance is dominated by frictional momentum dissipation. If the inertia terms <math>\partial u / \partial t + u \partial u / \partial x </math> in the momentum equation are ignored, then the current in the front region is mainly determined by the balance of gravitational acceleration <math> g \partial h / \partial x </math> and shear stress <math>\tau</math>, the latter term being proportional to the square of the flow velocity<ref>Dressler, R.F. 1952. Hydraulic resistance effect upon the dambreak functions. J. Res. Natl. Bureau of Standards, 49(3): 217–225</ref><ref>Whitham, G.B. 1955. The effects of hydraulic resistance in the dam-break problem. Proc. Roy. Soc. of London, Serie A, 227: 399–407</ref><ref>Hogg, A.J. and Pritchard, D. 2004. The effects of hydraulic resistance on dam-break and other shallow inertial flows. J. Fluid Mech. 501: 179–212</ref><ref name=C09>Chanson, H. 2009. Application of the method of characteristics to the dam break wave problem. Journal of Hydraulic Research 47: 41–49</ref>. With such models the shape of the wave front is given by <math>h(s) \propto s^n, \; n=1/2</math>, where <math>s</math> is the distance measured from the wave front. The wave front speed <math>u_f</math> decreases with time. At large times  <math>t>>t_{\infty}</math>, the wave front speed varies approximately  as <math>u_f \approx c_0 \sqrt{t_{\infty}/t}</math> <ref name=C09></ref>. Here, <math>t_{\infty} = \sqrt{h_0/g}/(3 c_D)</math> and <math>c_D</math> is the friction coefficient (value in the range <math>\approx (1-5)\, 10^{-3}</math>). A more elaborate model of the frictional boundary layer near the wave front was studied by Nielsen (2018) <ref name=PN>Nielsen, P. 2018 Bed shear stress, surface shape and velocity field near the tips of dam-breaks, tsunami and wave runup velocity. Coastal Engineering 138: 126–131</ref>. According to this model, the wavefront is even more blunt, corresponding to <math> n = 1/4 </math>, which agrees well with laboratory experiments. This model also reveals a strong downward velocity component at the wavefront, which can contribute to the dislodging of particles from the sediment bed <ref name = PN> </ref>. This may explain the high near-bed of suspended concentrations associated with propagating surges observed in the field<ref>Khezri, N. and Chanson, H. 2012. Inception of bed load motion beneath a bore. Geomorphology 153: 39–47</ref>, see also the article [[Tidal bore dynamics]].
  
<math>F_1^2 = \Large \frac{v_1^2}{g D_1}\normalsize = \large \frac{D_2 (D_1+D_2)}{2 D_1^2} \normalsize  , \quad \quad (3) </math>
 
  
<math>F_2^2 = \Large \frac{v_2^2}{g D_2}\normalsize = \large \frac{D_1 (D_2+D_1)}{2 D_2^2} \normalsize . \quad \quad (4)</math>  
+
Data from numerous reservoir dam breaks that have occurred in the past have provided empirical formulas for the maximum discharge <math>Q_{max}</math> through the breach <ref>Froehlich, D.C. 2016. Predicting Peak Discharge from Gradually Breached Embankment Dam. Journal of Hydraulic Engineering, ASCE 04016041</ref><ref name=W> Wang, B., Chen, Y., Wu, C., Peng, Y., Song, J., Liu, W. and Liu, X. 2018. Empirical and semi-analytical models for predicting peak outflows caused by embankment dam failures. Journal of Hydrology 562: 692–702</ref>. From this data it was deduced that the width of the breach is mainly related to the volume of water <math>V</math> in the reservoir (volume above breach level). A reasonable estimate of the width is given by <ref name=F8></ref> <math> B \approx 0.3 V ^ {1/3} </math>. Empirical formulas for the maximum dam break discharge then yield <ref>Webby, M. G. 1996. Discussion of ‘Peak outflow from breached embankment dam.’ by D. C. Froehlich. J. Water Resour. Plann. Management 122:4(316), 316–317</ref> <ref name=W></ref>
  
Knowing that <math>D_2 > D_1</math>, it is straightforward to see that <math>F_1 > 1</math> and <math>F_2 < 1</math>. The jump correspond to the passage from a supercritical flow (<math>F_1 > 1</math>) to a subcritical flow (<math>F_2 < 1</math>). The bore intensity is characterized equally by <math>\Delta D / D_1</math> or by the Froude number <math>F_1</math>.
+
<math>Q_{max} \approx 0.04 \sqrt{g} \, V^{0.37} h_0^{1.4} \approx 0.15 h_0 \, B \, \sqrt{g h_0} \, V^{0.04} h_0^{-0.1}  ,  \qquad  (3)</math>  
  
[[Image:TidalBorePropagation.jpg|right|400px|thumb|Figure 6. Schematic representation of a hydraulic jump propagating with velocity <math>c</math>. In the frame moving with velocity <math>c</math> the hydraulic jump appears stationary; time derivatives are zero. The corresponding mass and momentum balance equations from which the propagation characteristics can be derived are indicated in the figure. The symbols used stand for: <math>u</math>= flow velocity in a fixed frame, <math>v</math>= flow velocity in the moving frame, <math>D</math>= water depth, <math>g</math>=gravitational acceleration; subscripts 1 and 2 indicate upstream and downstream conditions, respectively.]]
+
or
  
If <math>F_2 = 1</math> (the bore height <math> \Delta D</math> is zero) the bore velocity <math> c </math> is equal to
+
<math> q_{max} \approx  0.22 h_0 \, \sqrt{g h_0} ,  \qquad (4)</math>
  
<math>c_2 = u_2 + \sqrt{g D_2}</math>.  
+
where in the approximation for the maximum discharge per unit width we have considered a large reservoir volume (<math>V \approx 10^7 m^3</math>) and a water depth <math>h_0 \approx 15 m</math>. The empirical estimate (4) is about 75% of the estimate (2) given by the frictionless flow solution.
  
Another requirement for a tidal bore is therefore <math>c < c_2</math> or <math>F_2 < 1</math> and  <math>F_1 > 1</math>. Because <math>c_2</math> is the propagation speed of long-wave disturbances upstream of the bore, these disturbances (undulations) will catch up to the bore. A slowly propagating bore  (<math>F_2 << 1</math>) will thus grow faster and become higher than a fast propagating bore (<math>F_1, F_2 </math> close to 1) <ref>Dronkers, J.J. 1964. Tidal computations in rivers and coastal waters. North-Holland Publ. Co., 518 pp.</ref>.
 
  
It can be shown that shown that conservation of mass and momentum at the transition does not imply conservation of energy. From mass and momentum equations he obtained the following expression for the energy dissipation <ref>Lemoine, R. 1948. Sur les ondes positives de translation dans les canaux et sur le ressaut  ondulé de faible amplitude. Houille Blanche: 183-185</ref> <ref> Stoker, J.J. 1957. Water Waves. Interscience, New York</ref>:
+
==Numerical models==
  
<math>\Delta E = \rho g \Delta H = \rho g \Delta D + 0.5 \rho (v_2^2 -v_1^2) = \Large \frac{\rho g}{4} \sqrt{\frac{g (D_1+D_2)}{2 D_1 D_2}}\normalsize (D_2 – D_1)^3  , \quad \quad (5)</math>
+
The analytical methods for describing dam-break flow provide a qualitative picture and some first-order estimates for the wave profile and wave speed. They may only be used in situations that can be represented schematically by a simple prismatic channel. In most actual field situations, the dam-break flow must be modeled numerically. Detailed simulation of the forward and backward wave fronts requires 3D models of the Boussinesq type, taking into account vertical fluid accelerations<ref>Castro-Orgaz, O. and Chanson, H. 2017. Ritter’s dry-bed dam-break flows: positive and negative wave dynamics. Environ Fluid Mech (2017) 17:665–694</ref>. To correctly model the strong temporal and spatial gradients of the surging wave, a fine computational grid is required, with resulting long computer times. See also [[Tidal bore dynamics]].  
 
 
Two forms of energy dissipation can occur at the transition, leading to two different types of bores: undular and turbulent bores.
 
 
 
===Undular bores===
 
 
 
For <math>F_1</math> smaller than approximately 1.3, the bore transition is smooth and followed by a wave train (Fig. 1 right). The bore then consists of a mean jump between two water depths (see Fig. 6) on which secondary waves are superimposed. This type of bore is usually called an undular bore<ref name=C>Chanson, H. 2009. Current knowledge in hydraulic jumps and related phenomena. A survey of experimental results. European Journal of Mechanics-B/Fluids 28: 191-210</ref> and Favre (1935) <ref>Favre, H. 1935. Etude théorique et expérimentale des ondes de translation dans les canaux découverts (Theoretical and experimental study of travelling surges in open channels), Dunod, Paris</ref>  was the first to describe this phenomenon from laboratory experiments. That is why undular bores are sometimes referred to as Favre waves. This phenomenon is a dispersive wave process (http://www.coastalwiki.org/wiki/Dispersion_(waves)), which is also named “dispersive shock” in the mathematical community.
 
 
 
A simple model describing the secondary waves attached to the bore was proposed by Lemoine (1948) <ref name=L></ref>. He considers that secondary waves can be approximated by the linear theory. The propagation speed in the fixed frame is, according to linear [[shallow-water wave theory]],
 
 
 
<math> c_w = u_2 + \sqrt{gD_2 (1- \frac{k^2 D_2^2}{3})} , \quad \quad (6)</math>
 
 
 
where <math>k=2 \pi / \lambda</math> is the wave number and <math>\lambda</math> the wavelength. The secondary waves are stationary in a frame moving with the bore front, i.e.    <math>c_w  = c</math>. This equality provides an estimate of the wavelength:
 
 
 
<math>\lambda=\sqrt{2/3} \pi D_2 (F_1 – 1)^{-1/2} . \quad \quad (7)</math>
 
 
 
The secondary waves are stationary in the frame moving with the bore front, but in this frame wave energy is radiated behind the front. Equating the energy flux with the mean bore dissipation (Eq. 6) gives the secondary wave amplitude as a function of <math>F_1</math>:
 
 
 
<math>a_w=\frac{4}{3\sqrt{3}} D_2 (F_1 – 1) . \quad \quad (8)</math>
 
 
 
As this approach ignores wave-induced mass transport it is valid only for very small bores (<math>F_1</math> close to 1)<ref>Wilkinson, M.L. and Banner, M.L. 1977. 6th Australasian Hydraulics and Fluid Mechanics Conf. Adelaide, Australia, 5-9 December 1977</ref>. Nevertheless, equations (7) and (8) appear to give a correct estimate of the observed wavelength and amplitude of undular bore propagating in rectangular channels (<ref>Binnie, A. M., & Orkney, J. C. (1955). Experiments on the flow of water from a reservoir through an open horizontal channel II. The formation of hydraulic jumps. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 230(1181), 237-246</ref><ref>Chanson, H. (2010). Undular tidal bores: basic theory and free-surface characteristics. Journal of Hydraulic Engineering, 136(11), 940-944</ref>).
 
However, natural estuary and river channels are non-rectangular and present most of the time a variable cross-section with a nearly trapezoidal shape and gently sloping banks. The propagation of undular bores over channels with variable cross-sections was studied by Treske (1994)<ref name=Tr>Treske, A. 1994. Undular bore (Favre-waves) in open channels - Experimental studies, J. Hydraulic Res. 32: 355-370</ref> in the laboratory and by Bonneton et al. (2015)<ref name=B15></ref> in the field. Both studies identified a transition around <math> F_t=1.15</math>. For <math>F_1 > F_t</math> the secondary wave field in the mid channel is very similar to the dispersive waves (Favre waves) described above, and follow the relations (7) and (8)<ref name=B15></ref>. For <math>F_1 < F_t</math>, the secondary wave wavelength in the whole channel is at least two to three times larger than in a rectangular channel for the same Froude numbers. It was shown that this new undular bore regime (Fig. A c,d and B b) differs significantly from classical dispersive undular bores in rectangular channels (i.e. Favre waves). Chassagne et al. (2019) <ref name=CF>Chassagne, R., Filippini, A., Ricchiuto, M. and Bonneton, P. 2019. Dispersive and dispersive-like bores in channels with sloping banks. Journal of Fluid Mechanics 870: 595-616. doi:10.1017/jfm.2019.287</ref> recently showed that this undular bore regime (named dispersive-like bore) is controlled by hydrostatic non-dispersive wave properties, with a dynamics similar to [[Infragravity waves|edge-waves]] in the near-shore. The transition between dispersive and dispersive-like bores is illustrated on Fig. 7 and 8. The dispersive-like bores are characterized by low wave steepness, which make them difficult to visually observe in the field. It is why such tidal bores are generally ignored and why tidal bore occurrence in the field is strongly underestimated <ref> Bonneton, P., Van de Loock, J., Parisot, J-P., Bonneton, N., Sottolichio, A., Detandt, G., Castelle, B., Marieu, V. and Pochon, N. 2011. On the occurrence of tidal bores – The Garonne River case. Journal of Coastal Research, SI 64: 11462-1466</ref>  and 2012 ??). 
 
 
 
[[Image:DispersiveBore.jpg|right|500px|thumb|Figure 7. Illustration of the two undular tidal bore regimes (Garonne River, Bonneton et al. 2015). Panels a, b: Dispersive regime (“mascaret” in French), F1=1.27. Panels c, d: Dispersive-like regime, Fr=1.08. Black line: elevation in the mid-channel; magenta line: elevation close to the river bank. Panel a:Two-dimensional (2-D) phase structure. Panel c: Quasi-1D phase structure.]]
 
 
 
[[Image:DispersiveUndularBores.jpg|left|300px|thumb|Figure 8. Numerical simulations of the two undular tidal bore regimes. Panel a: Dispersive regime, F1=1.20. Panel b: Dispersive-like regime, F1=1.05. From Chassagne et al. 2019<ref name=CF></ref>.]]
 
 
 
 
 
===Breaking bores===
 
For large Froude numbers, bores correspond to turbulent breaking fronts (Fig. 5), where the energy is dissipated by turbulent eddies<ref>Tu, J. and Fan, D. 2017. Flow and turbulence structure in a hypertidal estuary with the world's biggest tidal bore. Journal of Geophysical Research: Oceans, 122: 3417-3433</ref>. The transition between undular and turbulent bores is shown in Fig. 11.
 
Before arrival of the tidal bore the current velocity is generally low and in the ebb direction. At the arrival of the bore front the current velocity is almost instantaneously reversed into the flood direction and reaches a high value within a minute. Fig. 9 shows the sharp increase in the current velocity and suspended sediment concentration recorded in the megatidal Baie du Mont Saint Michel at spring tide when the tidal flood wave enters the tidal flat area and the small rivers at the head of the Baie <ref name=F> Furgerot, L., Mouazé, D., Tessier, B., Perez, L., Haquin, S., Weill, P., & Crave, A. (2016). Sediment transport induced by tidal bores. An estimation from suspended matter measurements in the Sée River (Mont-Saint-Michel Bay, northwestern France). Comptes Rendus Géoscience, 348(6), 432-441</ref>.  The sudden strong change of the current velocity has an important stirring effect on the bed sediments, causing a sharp increase of the suspended sediment concentration. Turbulent bores contribute significantly to upstream sediment transport<ref>Reungoat, D., Lubin, P., Leng, X. and Chanson, H. 2018. Coastal Engineering Journal 60: 484-498</ref>.
 
 
 
 
 
 
 
[[Image:TidalBoreSedimentSeeRiver_Furgerot.jpg|center|900px|thumb|Figure 9. Measurement of sediment transport by a tidal bore in the Sée River on 8 May 2012 (Furgerot et al, 2016 <ref name=F></ref>). Panel 1: Tidal bore entering the Sée channel at the head of the Baie du Mont Saint Michel. The monastery is visible at the background. Panel 2: Map of the inner part of the Baie du Mont Saint Michel. The measuring location in the Sée River is indicated by the red star. Panel 3: Temporal variation of the vertical distribution of suspended sediment concentration (top), current velocity (middle) and sediment flux just (bottom) before and after the arrival of the tidal bore. The current is ebb-directed with very low velocity just before arrival of the tidal bore and reverses to a flood-directed velocity simultaneously with the arrival of the tidal bore (middle panel). A very high short peak in the suspended sediment concentration near the bed occurs at the passage of the tidal bore front. High suspended sediment concentrations over the entire vertical (mainly fine silty sand) are recorded about 5 minutes later representing the sediment advected by the tidal bore (top panel). After a very high peak at the passage of the bore front the sediment flux is again high and in up-river direction some minutes later. The sediment flux is much lower during the remaining flood period (bottom panel) and similar to the up-river sediment flux when a tidal bore is absent. 
 
A video of the arrival of the tidal bore can be viewed by clicking [https://www.youtube.com/watch?v=waeS1MvhT9M here]. The photo at the right is a still from the video.]]
 
 
 
 
 
In many tidal rivers the bore propagation near the shallow channel banks differs from the bore propagation in the deeper middle part of the channel. The local Froude number is relatively lower at the deeper parts of the channel, where the bore has often an undulating character, while a higher breaking bore occurs in the shallower parts. This phenomenon is illustrated in Fig. 10 for the tidal bore in the Petitcodiac River.
 
 
 
[[Image:PetitcodiacRiver-KamparRiver.jpg|center|800px|thumb|Figure 10. Left image: Tidal bore in the Petitcodiac River, which is undular at the middle of the channel and breaking near the channel banks. Right image: Breaking tidal bore in the Kampar River, Sumatra.]]
 
 
 
 
 
 
 
==Modelling==
 
 
 
Tidal bore formation involves a large range of temporal and spatial scales, from the estuary to the turbulence scale. For this reason it is difficult to model these processes as a whole, both from physical models (laboratory experiments) and numerical approaches.
 
 
 
===Physical models===
 
 
 
[[Image:FlumeUndularBores.jpg|left|400px|thumb|Figure 11. Experimental study of undular bores (Favre-waves) in open rectangular channels (Treske 1994<name=Tr></ref>). Undular bores, Fr=1.2 and 1.3; transition to a turbulent bore, Fr=1.35; turbulent bore, Fr=1.45.]]
 
 
 
 
 
Due to the large range of scales, it is impossible to design a laboratory experiment in close similitude with natural tidal bores. However, leaving aside the tidal wave transformation and bore formation, the bore in itself (i.e. hydraulic jump in translation) can be studied in detail from flume experiments. The bore is commonly generated in a rectangular flume by using a fast-closing gate at the upstream end of the flume<ref name=Tr></ref><ref name=C></ref>. This method allows the study of the different bore regimes (see Fig. 11) and provides valuable insights in secondary wave structure and vortical motions.
 
 
 
To avoid the abrupt bore generation of the above method, Rousseaux et al. (2016)<ref name=R></ref> proposed a novel approach.  This method mimics the tidal asymmetry met in nature between the ebb and the flood.  Fig. 12 shows an example of a tidal-like bore generated with this method.
 
 
 
[[Image:FlumeUndularBoresDye.jpg|center|700px|thumb|Figure 12. Experimental study of tidal-like bore using a laser sheet and fluorescent dye (Rousseaux et al. 2016<ref name=R></ref>).]]
 
 
 
 
 
 
 
===Numerical models===
 
 
 
 
 
 
 
[[Image:BreakingBoreLES.jpg|left|400px|thumb|Figure 13. Large Eddy Simulation of turbulence generated by a weak breaking bore. Streamlines indicate recirculation structures under the bore (Lubin et al. 2010<ref name=L></ref>).]]
 
 
 
 
 
Recent approaches, based on the resolution of the general Navier Stokes equations in their multiphase form (water phase and air phase), allow a detailed description of bore structure, turbulence and air entrainment in a roller at the bore front<ref name=L>Lubin, P., Chanson, H. and Glockner, S. 2010. Large eddy simulation of turbulence generated by a weak breaking tidal bore. Environmental Fluid Mechanics 10: 587-602</ref><ref>Berchet, A., Simon, B., Beaudoin, A., Lubin, P., Rousseaux, G. and Huberson, S. 2018. Flow fields and particle trajectories beneath a tidal bore: A numerical study. International Journal of Sediment Research 33: 351-370</ref>. Figure 13 presents the simulation of recirculating structures under a breaking bore. Navier-Stokes approaches are dedicated to small scale bore processes that cannot be applied to tidal bores at the estuarine scales because of limited computation power. Such applications would require long-wave models, where small scale vorticity motions are parametrized and not directly resolved.
 
 
 
The most common long-wave model is the depth-averaged Saint Venant or Non Linear Shallow Water (NSW) model, which assumes hydrostatic pressure. This model gives a good description of the large-scale tidal wave transformation<ref name=S></ref>. The properties of breaking turbulent bores, are also quite well described by the non-dispersive NSW equations with jump conditions <ref>Chanson, H. 2012. Tidal bores, aegir, eagre, mascaret, pororoca: Theory and observations. World Scientific</ref>.
 
However, the onset of a tidal bore and its evolution upstream is controlled by non-hydrostatic dispersive mechanisms<ref name=P></ref>. If the onset of the tidal bore can be well described by classical weakly dispersive weakly nonlinear Boussinesq-type equations (<ref name=P>Peregrine, D. H. 1966. Calculations of the development of an undular bore. Journal of Fluid Mechanics 25: 321-330</ref>, see [https://en.wikipedia.org/wiki/Boussinesq_approximation_(water_waves) Boussinesq equations]), the subsequent nonlinear evolution, for high-intensity tidal bores, requires the use of the basic fully nonlinear Boussinesq equations, named Serre-Green Naghdi  (SGN) equations<ref name=B11c>Bonneton, P., Barthélemy, E., Chazel, F., Cienfuegos, R., Lannes, D., Marche, F. and Tissier, M. 2011. Recent advances in Serre–Green Naghdi modelling for wave transformation, breaking and runup processes. European Journal of Mechanics-B/Fluids 30: 589-597</ref><ref name=T>Tissier, M., Bonneton, P., Marche, F., Chazel, F., & Lannes, D. 2011. Nearshore dynamics of tsunami-like undular bores using a fully nonlinear Boussinesq model. Journal of Coastal Research SI 84: 603-607</ref><ref name=F></ref><ref name=C></ref>. This modelling approach allows an accurate description of both, the tidal bore formation at the estuarine scale (Fig. 4) and the bore structure at the local scale (Fig. 14).
 
 
 
[[Image:UndularBoreDevelopmentModel.jpg|right|500px|thumb|Figure 14. Serre-Green Naghdi  simulation of an undular bore. Comparisons between experimental data (at 6 gauges, Fr=1.104) from Soares-Frazao and Zech (2002) <ref>Soares-Frazão S. and Zech Y. 2002. Undular bores and secondary waves - Experiments and hybrid finite-volume modeling. Journal of Hydraulic Research, International Association of Hydraulic Engineering and Research (IAHR) 40: 33-43</ref> and model prediction (Tissier et al. 2011<ref name=T></ref>). ]]
 
 
 
 
 
==See also==
 
: [[Tidal asymmetry and tidal basin morphodynamics]]
 
: [[Ocean and shelf tides]]
 
: [[Morphology of estuaries]]
 
: [[Estuaries]]
 
 
 
 
 
==Further reading==
 
 
 
 
 
Dronkers, J.J. 1964. Tidal computations in rivers and coastal waters. North-Holland Publ. Co., 518 pp.
 
  
  
 
==References==
 
==References==
 
 
<references/>
 
<references/>
  
 
 
 
 
{{2Authors
 
|AuthorID1=15152
 
|AuthorFullName1= Philippe Bonneton
 
|AuthorName1= Bonneton P
 
|AuthorID2=120
 
|AuthorFullName2=Job Dronkers
 
|AuthorName2=Dronkers J
 
}}
 
  
  
 
[[Category:Physical coastal and marine processes]]
 
[[Category:Physical coastal and marine processes]]
[[Category:Estuaries and tidal rivers]]
 
 
[[Category:Hydrodynamics]]
 
[[Category:Hydrodynamics]]

Latest revision as of 21:13, 15 January 2020

Dam break flow

This article discusses the often catastrophic flows that result from the failure of high dams that protect low-lying land. Dam break mechanisms are not dealt with; for this the reader is referred to the extensive literature existing on this subject, see for example Zhang et al. (2016) [1] and Almog et al. (2011) [2].


Introduction

Fig. 1. Sea dikes protecting low-lying polders in the Netherlands were breached during the extreme storm surge of 31 January 1953.


Dam failure can lead to disastrous situations. Most dam break tragedies are related to the collapse of reservoir dams in mountain rivers. The failure of sea dikes can also cause major disasters, although in this case the level difference is not as large. A dramatic example is the failure of more than fifty sea dikes in the Netherlands during the extreme storm surge of 1953, see figure 1. Many of these dikes protected land lying below the average sea level, while the storm surge level exceeded more than five meters. Rapid flooding killed more than two thousand people who were unable to flee in time to safe places. The most common dam failure mechanisms are related to overtopping and seepage (also called piping or internal erosion)[3]. In the case of the 1953 storm surge, overtopping and subsequent scour of the interior dike slope was the most important dike failure mechanism.




Fig. 2. Left panel: Schematic representation of water retention behind a dam. Right panel: Positive downstream and negative upstream surges following instantaneous dam removal.


The consequences of dam failure have been studied for more than a century. It nevertheless remains a challenging topic due to the high non-linearity of the flood wave propagation. The problem can be tackled with numerical models, but rough estimates and insight into the tidal wave dynamics can be obtained with analytical solution methods.


Frictionless dam-break flow: analytical solution

Analytical solutions relate to idealized situations as depicted in Fig. 2. The initial situation consists of an infinite reservoir with a water level that is [math] h_0 [/math] higher than the horizontal ground level downstream of the dam. Dam break is simulated by instantaneous removal of the dam. This causes a positive surge in the positive [math] x [/math]-direction and a negative surge in the negative [math] x [/math]-direction. The floor is dry in front of the positive surge and the body of water behind the negative surge is undisturbed: water level =[math] h_0 [/math] and current speed [math] u = 0 [/math].

Fig. 3. The earth-filled South Fork Dam on Lake Conemaugh (Pennsylvania, US) collapsed in 1889, killing 2,209 people in downstream villages.




The study of the dam break-flow was triggered by several dam failures in the 19th century, in particular the breach of the South Fork Dam in Pennsylvania (USA) in 1889, Fig. 3. Three years later Ritter [4] published an exact analytical solution for dam-break flow by assuming that frictional effects can be ignored. The derivation is given in box 1. According to this solution, the tip of the positive surge advances at high speed (supercritical flow) given by [math] u_f = 2 c_0[/math], where [math]c_0 = \sqrt {gh_0}[/math] and [math] g [/math] is the gravitational acceleration. The shape of the positive and negative wave is a concave-up parabola,

[math]h(x,t)= h_0, \; x\lt -c_0 t; \quad h(x,t)= \Large\frac{h_0}{9}\normalsize \; (2 - \Large\frac{x}{c_0 t}\normalsize)^2 , \, -c_0 t\lt x\lt 2c_0 t; \quad h(x,t)= 0, \, x\gt 2c_0 t , \qquad (1)[/math]

see Fig. 4. According to Eq. (1) and Box 1, the discharge through the breach per unit width is given by

[math]q = \frac{8}{27} h_0 \sqrt{g h_0}. \qquad (2) [/math]

Figure 4 also shows the shape of the dam break wave that was observed in laboratory experiments. For small values of [math] x / t [/math], the frictionless solution corresponds fairly well with the observations. However, the front zone is different: the shape is a bull nose, rather than a sharp edge. The front also advances more slowly; the speed of the front decreases with time and is closer to [math] u_f = c_0 [/math] than to [math] u_f = 2 c_0 [/math]. The reason for this difference is the neglect of frictional effects that are important in the thin fluid layer near the front.


Box 1. Derivation of the solution for frictionless dam break flow.


Dam-break flow with friction

Fig. 4. Wave profile after dam break. The red curve corresponds to the frictionless solution Eq. 1. The blue band represents data from different laboratory experiments performed at times [math]\small t \sim (40-80) \, \sqrt {h_0 / g}\normalsize [/math] after removal of the dam [5] [6][7]. The frictionless solution only depends on [math]x/t[/math]. Observed wave profiles also depend on time [math]t[/math] because of the decreasing wave tip speed.


When friction terms are included in the flow equations, there is no exact analytical solution. It is generally assumed that the frictionless flow equations are approximately valid in a short period after dam break and for small values of [math]x / (c_0 t) [/math]. In the front zone, where the water depth is small, the momentum balance is dominated by frictional momentum dissipation. If the inertia terms [math]\partial u / \partial t + u \partial u / \partial x [/math] in the momentum equation are ignored, then the current in the front region is mainly determined by the balance of gravitational acceleration [math] g \partial h / \partial x [/math] and shear stress [math]\tau[/math], the latter term being proportional to the square of the flow velocity[8][9][10][11]. With such models the shape of the wave front is given by [math]h(s) \propto s^n, \; n=1/2[/math], where [math]s[/math] is the distance measured from the wave front. The wave front speed [math]u_f[/math] decreases with time. At large times [math]t\gt \gt t_{\infty}[/math], the wave front speed varies approximately as [math]u_f \approx c_0 \sqrt{t_{\infty}/t}[/math] [11]. Here, [math]t_{\infty} = \sqrt{h_0/g}/(3 c_D)[/math] and [math]c_D[/math] is the friction coefficient (value in the range [math]\approx (1-5)\, 10^{-3}[/math]). A more elaborate model of the frictional boundary layer near the wave front was studied by Nielsen (2018) [12]. According to this model, the wavefront is even more blunt, corresponding to [math] n = 1/4 [/math], which agrees well with laboratory experiments. This model also reveals a strong downward velocity component at the wavefront, which can contribute to the dislodging of particles from the sediment bed [12]. This may explain the high near-bed of suspended concentrations associated with propagating surges observed in the field[13], see also the article Tidal bore dynamics.


Data from numerous reservoir dam breaks that have occurred in the past have provided empirical formulas for the maximum discharge [math]Q_{max}[/math] through the breach [14][15]. From this data it was deduced that the width of the breach is mainly related to the volume of water [math]V[/math] in the reservoir (volume above breach level). A reasonable estimate of the width is given by [3] [math] B \approx 0.3 V ^ {1/3} [/math]. Empirical formulas for the maximum dam break discharge then yield [16] [15]

[math]Q_{max} \approx 0.04 \sqrt{g} \, V^{0.37} h_0^{1.4} \approx 0.15 h_0 \, B \, \sqrt{g h_0} \, V^{0.04} h_0^{-0.1} , \qquad (3)[/math]

or

[math] q_{max} \approx 0.22 h_0 \, \sqrt{g h_0} , \qquad (4)[/math]

where in the approximation for the maximum discharge per unit width we have considered a large reservoir volume ([math]V \approx 10^7 m^3[/math]) and a water depth [math]h_0 \approx 15 m[/math]. The empirical estimate (4) is about 75% of the estimate (2) given by the frictionless flow solution.


Numerical models

The analytical methods for describing dam-break flow provide a qualitative picture and some first-order estimates for the wave profile and wave speed. They may only be used in situations that can be represented schematically by a simple prismatic channel. In most actual field situations, the dam-break flow must be modeled numerically. Detailed simulation of the forward and backward wave fronts requires 3D models of the Boussinesq type, taking into account vertical fluid accelerations[17]. To correctly model the strong temporal and spatial gradients of the surging wave, a fine computational grid is required, with resulting long computer times. See also Tidal bore dynamics.


References

  1. Zhang, L., Peng, M., Chang, D. and Xu, Y. 2016. Dam failure and Risk Assessment. John Wiley and Sons, Singapore
  2. Almog, E., Kelham, P and King, R. 2011. Modes of dam failure and monitoring and measuring techniques. Environmental Agency,UK https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/290819/scho0811buaw-e-e.pdf
  3. 3.0 3.1 Froehlich, D.C. 2008. Embankment Dam Breach Parameters and Their Uncertainties. Journal of Hydraulic Engineering, ASCE 134: 1708-1721
  4. Ritter, A. 1892. Die Fortpflanzung der Wasserwellen. Zeitschrift des Vereines Deutscher Ingenieure 36: 947-954
  5. Dressler, R. 1954. Comparison of theories and experiments for the hydraulic dam-break wave. Proc. Int. Assoc. Scientific Hydrology Assemblée Générale, Rome, Italy 3 (38)M 319–328
  6. Schoklitsch, A. 1917. Über Dambruchwellen. Kaiserliche Akademie der Wissenschaften, Wien, Mathematisch-Naturwissenschaftliche Klasse, Sitzungberichte IIa, 126: 1489–1514
  7. Cavaillé,Y. 1965. Contribution à l’étude de l’écoulement variable accompagnant la vidange brusque d’une retenue. Publ. Scient. et Techn. du Ministère de l’Air, 410, Paris, France, 165
  8. Dressler, R.F. 1952. Hydraulic resistance effect upon the dambreak functions. J. Res. Natl. Bureau of Standards, 49(3): 217–225
  9. Whitham, G.B. 1955. The effects of hydraulic resistance in the dam-break problem. Proc. Roy. Soc. of London, Serie A, 227: 399–407
  10. Hogg, A.J. and Pritchard, D. 2004. The effects of hydraulic resistance on dam-break and other shallow inertial flows. J. Fluid Mech. 501: 179–212
  11. 11.0 11.1 Chanson, H. 2009. Application of the method of characteristics to the dam break wave problem. Journal of Hydraulic Research 47: 41–49
  12. 12.0 12.1 Nielsen, P. 2018 Bed shear stress, surface shape and velocity field near the tips of dam-breaks, tsunami and wave runup velocity. Coastal Engineering 138: 126–131 Cite error: Invalid <ref> tag; name "PN" defined multiple times with different content
  13. Khezri, N. and Chanson, H. 2012. Inception of bed load motion beneath a bore. Geomorphology 153: 39–47
  14. Froehlich, D.C. 2016. Predicting Peak Discharge from Gradually Breached Embankment Dam. Journal of Hydraulic Engineering, ASCE 04016041
  15. 15.0 15.1 Wang, B., Chen, Y., Wu, C., Peng, Y., Song, J., Liu, W. and Liu, X. 2018. Empirical and semi-analytical models for predicting peak outflows caused by embankment dam failures. Journal of Hydrology 562: 692–702
  16. Webby, M. G. 1996. Discussion of ‘Peak outflow from breached embankment dam.’ by D. C. Froehlich. J. Water Resour. Plann. Management 122:4(316), 316–317
  17. Castro-Orgaz, O. and Chanson, H. 2017. Ritter’s dry-bed dam-break flows: positive and negative wave dynamics. Environ Fluid Mech (2017) 17:665–694