1

Tidal Bores Report for the course “Physics of Coastal Systems 2010”

Werner de Groot (3558835)

2

Table of contents

Abstract ............................................................................................................................................. 3

Introduction ...................................................................................................................................... 3

Results: a model for tidal bores ........................................................................................................ 4

Physical explanation ..................................................................................................................... 4

Mathematical description ............................................................................................................. 5

Different types of tidal bores .......................................................................................................... 12

Discussion: comparison to observations ........................................................................................ 12

Conclusion ....................................................................................................................................... 15

Cited material ................................................................................................................................. 15

3

Abstract

This report will provide the reader with an introduction to tidal bores. A tidal bore is a tidal phenomenon in which the incoming tide forms a large wave that travels up a river against the direction of the current. The introduction includes a discussion of the physical processes that are responsible for the generation of a tidal bore and a mathematical model that, among other things, quantifies the propagation velocity of a bore. Finally, the properties of tidal bores that were discussed in this report will be compared to recent observations, based on the publication “Undular tidal bore dynamics in the Daly Estuary, Nothern Australia” by Wolanski et al. (2004), which examines an undular bore observed in the Daly Estuary.

Introduction

A tidal bore is a phenomenon in which an incoming tidal wave travels up a river against the direction of the current and forms a steep and high front. It occurs when the flood tide gets confined to a narrow and shallow channel. This may not sound very exciting but it is, in fact, a fairly violent process in which two to three meters tide comes in all at once. To illustrate: a tidal bore on the Indus River apparently wiped out the entire fleet of Alexander the Great in 325 BC [1]. This might seem surprising at first – we usually regard tidal processes as being relatively smooth and quiet, so how can it produce these huge, fast waves? – but a simple model shows that, with a large enough tidal range and especially shallow waters, truly enormous waves can occur. In fact, a lot of videos and images of tsunamis circulating on the web are actually tidal bores! Nowadays, tidal bores can be accurately predicted and have become, more than anything else, a great opportunity for surfers and rafters.

Figure 1: A tidal bore on the Turnagain Arm in Canada, which can reach a height up to 2.5 meters, travelling with speeds up to 25 kilometers per hour. With the appropriate boundary conditions, the bore may travel as much as 80 kilometers. Source: http://www.photolib.noaa.gov/htmls/line1769.htm.

Tidal bores are extremely rare. Of all the rivers in the world, only a tiny subset have a large enough tidal range and the necessary relatively shallow waters to support them, and even then, often only during spring tide. The most famous tidal bores are probably those of the Seine River, France where it is called the Mascaret (which has been greatly reduced in magnitude recently due to man-made interventions), the Qiantang River, China where it is called the Black Dragon and those of the Severn River (see Abbott, 1956 for a discussion of the Severn River Bore) and the Humber, both in the United Kingdom.

The occurrence of tidal bores has a significant impact on river mouths and estuarine systems. Erosion and scour take place beneath the bore front, while suspended matters are carried upstream

4

by the large velocities caused by the bore. The existence of tidal bores, however, is based upon a truly fragile hydrodynamic balance which may easily be disturbed by changing the boundary conditions and the freshwater inflow. Man-made interventions led to the disappearance of several bores, with often a strong (negative) influence on the ecosystem, like in the case with the Mascaret of the Seine River [2]. Although these rivers gained considerably in safety in most cases (these rivers were generally regarded as very dangerous, as one can imagine) the elimination of several fish and bird species was also a direct consequence. Fish eat matter in suspension after the passage of the bore while birds fish directly behind the bore. Despite their impact on estuarine processes and their often lethal nature, little is known about the flow field, mixing and sediment motion beneath tidal bores. It is exactly because of this, that it is important to study tidal bores and to gain insight in the processes that play a role in the formation and propagation of these fascinating waves.

Results: a model for tidal bores

To investigate how a tidal bore is formed we consider a shallow river discharging in a wide and relatively deep bay, as is depicted in Figure 2. The basis of this explanation is the depth-averaged shallow water wave theory, in which we assume the wavelength of the waves to be much larger than the depth of the water. This is, of course, especially valid for the tidal waves we are considering. As is explained below, in the subsection Mathematical description, the fluid parcels in a tidal wave will

travel into the bay with a velocity , where is the gravitational acceleration, is the local water depth and is the undisturbed water depth (the water depth before the passage of the bore).

Figure 2: A river with a large discharge (velocities ) is connected to a large bay which is subject to tidal forcing.

Physical explanation 1 The variations in the free surface due to the tides travel as waves towards the river, as is shown

below in Figure 3.1. The discharging river, however, makes it impossible for these waves to propagate further than the mouth of the river: the velocity of the incoming waves (about ) is less than the outbound river flow (about ). The wave is essentially detained at the mouth of the river and the tides cannot travel any further. Since there is still a net inflow due to the tides at the seaward boundary of the bay and the discharge of the river, the water level in the bay will start to rise. Because of the increase in water depth , the potential propagation velocity of the wave, , will increase as well. This is depicted in Figure 3.2, in which we see that the velocity at which the wave wants to travel up the river has increased to a value of , but is still insufficient to overcome the outbound river flow. The water level in the bay will, however, continue to rise until the wave velocity is high enough to finally overcome the flow velocity of the river. When this happens, the wave will move into the river as a bore. All the water parcels beneath the bore will move forward

1 See the lecture given by Clive Dorman for justification of the claims made in this paragraph.

5

(a bore is a very shallow water wave, as opposed to pure shallow water waves in which water parcels will move in elliptical orbits, eventually returning to their original position) andthe tidal bore will thus, simply, reverse the flow in the river, as is illustrated in Figure 3.3.

The formation of a tidal bore can also be explained by considering a wave travelling up the channel that gets steeper with time and eventually breaks forming a bore front. This is described in the next section.

Figure 3: A cartoon of the main process causing tidal bores. Tidal bores need to have a sufficiently large water level change to overcome the outbound river water speed.

Mathematical description As was mentioned earlier, we will use the depth-averaged shallow water wave theory. For

simplicity we will regard the case of a broad rectangular channel of constant inclination and work with depth and horizontal velocity as basic variables, as is depicted in Figure 4. We assume conditions to be uniform in the -direction, thereby making it essentially a two-dimensional model. The inclination of the channel will give rise to the strong outbound flow velocity which is needed for the bore to exist, as was discussed in the subsection Physical explanation above.

With the volume flux equal to we see that equations of motion (the depth-averaged shallow water equations in which friction is neglected) can be expressed as

6

where

denotes the material derivative. We have disregarded the inclination of the

channel for the moment for ease of understanding. From these equations we immediately see that we cannot use linear dynamics to model the behavior of a tidal bore. Linearizing the equations gives:

where is the constant average water depth. This will lead to waves

with constant velocity , which means that all disturbances will move with the same velocity

at any fixed point in the channel: any two waves are always separated by the same time interval! Bore formation (which requires that disturbances near the head of the wave are overtaken by the later disturbances carrying a greater depth, as is explained in the next paragraph) is not within the scope of the linear theory. The tidal bore must therefore be a non-linear phenomenon. In hindsight this was what one would expect, since gradients on the bore front must be extremely large.

Figure 4: Illustration of the channel we will consider in this subsection. The -axis is chosen perpendicular to the bottom of the river.

It can be shown (see Whitham, 1974) that fluid parcels in waves modeled with the non-linear depth averaged shallow water equation propagate with the shallow water celerity, which depends on the local depth,

This means that the crest of the wave moves faster than the trough, resulting in a steepening of the wave. It is immediately clear that any part of the wave where the propagation velocity is a decreasing function of will ultimately break to give a tripled value solution for . The breaking starts at time when the wave first develops an infinite slope. This corresponds to the green curve in Figure 5 below.

Of course, this is not what actually happens. When breaking occurs, the approximations that were made to derive the model will (at least partially) cease to be valid. The model no longer describes the physical phenomenon observed. In most cases, we have to formulate an improved theory that does not depend on the approximations mentioned earlier. However, these theories often prove to be too

7

complicated to solve analytically. It turns out, as we shall see, that our simple equations can be maintained by allowing discontinuities into the solution: there is then a single-valued solution with a simple jump discontinuity to replace the multivalued continuous solution.

When, for instance, friction is ignored, like was done using the depth-averaged shallow water equations, the equations of fluid motion have ‘unphysical’ solutions since, as the gradients become extremely steep (just before breaking), the effects of viscosity are no longer negligible. These effects can be included to give an improved theory and waves no longer break in that theory. There is a thin region, a shock wave much like the shock wave caused by breaking the sound barrier, in which friction is crucially important; outside the shock wave, friction may still be neglected. This shock region is idealized into a discontinuity in the ‘extended’ frictionless theory.

Figure 5: The breaking of a wave governed by the depth-averaged shallow water equations. The different colored curves represent the wave at different times.

Of course, we need to take care that the discontinuity still adheres to the physical laws imposed on the system: we must insist on conservation of volume and momentum. Both the multivalued curve and the discontinuous curve must, for instance, satisfy the conservation of volume. Therefore, the integral must be the same for both curves and, hence, the discontinuity must cut of areas of equal size (see Figure 6).

Figure 6: Idealization of the shock region into a discontinuity. Both shaded areas are of equal size to ensure the conservation of volume.

The equations of motion, based on the depth-averaged shallow water equations, can be written in local form to give

8

where subscripts denote derivatives and and . The pressure gradient is reduced because gravity is no longer perpendicular to the bottom topography, but an extra forcing term appears because gravity has a component parallel to the -coordinate. Furthermore, quadratic bottom friction is included which works in the positive -direction, against the direction of the river current. The constant is a friction coefficient which shall be discussed in more detail in

the remainder of this section. The undisturbed (steady) flow has constant values , (see Figure 4) from which

follows that, to smallest order, . The wave front will, in this case, have constant velocity

. To analyze the behavior of the solution of the system near the wave front, we let the disturbance occur at time and introduce the variable . We formulate an expansion of the solutions of the equations in the form

where the functions are the th -derivatives of the water depth at the bore front ( ):

which depend only on time , since any direct -dependence of the equations was eliminated by introducing and expanding around . Obviously, the same applies to the functions . Reducing the two equations of motion (in successive powers of ) to

(1) (2)

(3)

(4)

gives

From the equations (1) and (2) we conclude that the propagation speed is:

(5)

which is equal to the velocity of a shallow water wavelet in a rectangular channel (when friction is ignored) superimposed on the background flow, much like one would expect. We also see that

The combined second and third order equations for and become, using these results,

and after use of equation (4) it follows that

9

Since a tidal bore is a wave propagating upstream, in the positive -direction, we know that

(remember ) and the last equation reduces to

This wave will (for negative ) break if

since then the value of will increase without a bound (remember that the quantity is the value of the derivative at the wave front) and the wave will develop a discontinuity. To apply the wave front results we use the maximum rate of change in the tidal variation in the bay to determine . We assume that the tidal variation at the mouth of the river is

with

, the tidal period. Since , the maximum value of is .

In the discontinuity analysis, the initial value of at the wave front is

and therefore we choose [5]

The bore condition thus becomes:

or either

or

where

and where is the traditional quadratic drag coefficient.

For rivers, the local Froude number is reasonably small, so the right-hand side of the two equations directly above can be approximated by the first factor. For typical values of , and this leads to values of 30 meters for . The effect of a narrowing

channel (and other factors, e.g. a complex bottom topography) bring the value down considerably although high tides and rapidly changing bottom topography are still necessary. That is why only few rivers have bores. For the Severn River, including bottom topography and the fact that the river decreases in width upstream, Abbott (1956) [6] finds that the required tidal range . The spring tides have an average range of 12 while the neap tides have an average range of 7 . Abbott therefore predicts that bore formation should occur for about 4 days around the times of the highest tides, which corresponds nicely to observations.

It should be noted that, although this model correctly predicts the occurrence of a tidal bore, some of the quantative values it predicts (for instance ) are nothing more than crude estimates. A better approximation for is, however, supplied in the next paragraph.

10

Of course, we need to take care that the discontinuity that ultimately develops still adheres to the physical laws imposed on the system: we must insist on conservation of volume and momentum. Take for instance the conservation of volume:

(where we refer to Figure 7 below for the meaning of the different symbols). When we switch to the (primed) coordinate system travelling with the bore:

we immediately see that the conservation of volume yields:

Figure 7: A discontinuity propagating upstream (in the positive -direction), travelling with velocity against the river current ( ). We have again neglected the inclination of the channel for ease of understanding.

Momentum is conserved as well, providing us with the equation

where the terms are (1) the rate of increase of momentum per unit length, , in the interval (2) the net transport of momentum across and and (3) the net total pressure force acting across and . The latter term may require some additional explanation. Since the pressure satisfies a hydrostatic law:

we obtain

where is the atmospheric pressure. The total contribution to the perturbed pressure integrated over a cross-section of the channel is hence

with the width of the channel. This is the origin of the pressure term in the momentum balance above. (One might wonder why we have chosen to give the expression in integral form instead of in

11

local form. The answer, however, is simple: the equations in local form are not suited to deal with discontinuities.) We thus must impose that:

where is the position of the discontinuity and, naturally, : the propagation speed of the flood wave. Furthermore,

Since is bounded on each of the intervals and separately, we can take the

limit as and to remove the integrals from the consideration

When we again switch to the coordinate system travelling with the bore front we can eliminate (since )

where

, as defined above. Multiplying this equation with

gives

We thus obtain

or, since we are considering a bore moving in the positive -direction,

This is a better approximation than the one obtained earlier. If we consider only small bores, for which , we see that the expression for obtained in this section becomes equivalent to equation (5):

12

The tidal bore is a phenomenon in which a wave travels against the direction of the river current. Nowhere in the explanation, however, the derivation of the conditions for bore formation and the velocity of such a wave depends crucially on the sign of river velocity. The same phenomenon is thus expected to occur also for a river flowing in the opposite direction and even for a motionless river. These types of waves have been given different names, and shall therefore not be discussed in this text. It is, however, interesting to mention that for a river current moving in the positive -direction, the existence of a bore does not depend on the magnitude of tidal variations near the mouth of the river but instead depends on the magnitude of the Froude number.

Different types of tidal bores

A tidal bore may take on various forms, ranging from a single breaking, yet stable, wave front called a turbulent bore or breaking bore to so-called undular tidal bores, which consist of a large but smooth wave front followed by a train of secondary waves.

A turbulent bore is a continuously breaking wave, in which energy is released as kinetic energy (see Figure 1 for a picture of a turbulent bore). Since the particle velocity in front of the bore and inside the bore have opposite directions, both pointed towards to bore front like was discussed in the subsection Physical explanation, turbulent motions which act to resuspend sediment result. Turbulent bores generally occur when the Froude number, defined as seen by an observer moving with the bore front, has a value that is larger than [2].

In less extreme cases ( ), the bore is only a series of non-breaking waves and is called an undular bore. In the case of an undular bore, turbulent mixing is almost absent and, hence, it is expected that these types of bores hardly induce sediment resuspension at all. The secondary waves following the bore front are generally referred to as lee waves. A picture of a undular bore is shown in Figure 8 and on the title page of this report.

Figure 8: An undular bore being enjoyed by surfers. Note the absence of a turbulent, breaking wave front.

Discussion: comparison to observations

In the article by Wolanski et al. [7] an observation of an undular bore in the Daly Estuary (in Northern Australia) was discussed. The Daly Estuary is an especially shallow estuary with an average maximum depth of about and has a relatively large tidal range of about . These are, obviously, favorable conditions for the formation of a tidal bore. The field data concerning undular bore dynamics is the first of its kind, making this a great opportunity to compare our theoretical studies above to actual observations. When an undular bore was observed, the water was smooth upstream of the bore and, in addition, in the leading wave. Although the waves were mainly traveling along-channel, some waves near river beds were travelling in the across-channel direction.

13

On September 9, 2002 (spring tide), a tidal bore was observed that propagated upstream of the Daly Estuary at a speed of about . Between the two measuring stations used (which were placed approximately apart) the amplitude of the leading wave was reduced from to , possibly due to the diffraction of energy along the wave crest toward the breaking wave at the shoreline, fluidization of the bed or progressive wave reflection from rising bed and shoals. The along-channel flow velocity changed abruptly from about (oriented downriver) ahead of the bore to (oriented up-river) after the passage of the bore. The two dominant lee waves (the first two secondary waves) generated current fluctuations of about . This is illustrated in Figure 9, in which the passage of the bore front (the large increase in water depth at 7:30) and the following Lee waves with an amplitude of about half of the amplitude of the frontal wave can easily be observed.

With the measured initial water depth , the water depth after the passage of the bore , the initial river flow velocity and the river flow after the bore has passed we obtain the surge celerity:

which is in good agreement with the observed .2 The information obtained by measurements and the calculation of the bore front velocity furthermore enables us to determine the Froude number, defined as seen by an observer moving with the bore front:

which confirms that an undular tidal bore is expected. The bore generated, as one would expect in the case of undular bores, no direct pronounced

vertical velocities or any resulting sediment resuspension, neither did it have any effect on the net sediment budget. This was, however, somewhat unexpected in light of observations of (stationary) undular bores in laboratory studies, where a so-called recirculation bubble was observed directly beneath the bore front [8]. These recirculation bubbles arguably contribute to sediment resuspension due to the resultant strong vertical motions. The observed undular bore did not, however, generate any measurable recirculation bubble. This may be due to slight differences in bottom friction: the large natural roughness and bed forms of the Daly estuary might significantly perturb the flow field close to the bottom and dampen possible vertical motions. Another reason for the absence of the recirculation bubble was speculated to be the difference between the Froude number of the observed bore ( ) and the Froude number of the stationary bore in the laboratory ) which may lead to a small difference in dynamics. The undular bore will, regardless of the absence of the recirculation bubble, still contribute measurably to sediment dynamics. The bore broke along the bank where it resuspended the fine sediment. This erosive power contributes to the channel widening and migration that is clearly observed in the Daly estuary.

As can be seen from Figure 9, accelerating tidal currents resuspended sediment not directly but after 2 minutes after the passage of the bore. The time lag between the passage of the bore front and the sediment resuspension is due to the time it takes for sediment to diffuse upward: although the tidal currents are strong enough for sediment resuspension, the absence of vertical motions makes it difficult for the sediment to reach the measurement equipment placed and above the bottom of the estuary. The main process that causes the sediment to eventually reach the SSC-sensors is diffusion, which was calculated to need about 2 minutes to transport sediment upward, which is in good agreement with the observations.

2 Note that Wolanski et al. do not include the term in the expression for the surge celerity. They

therefore arrive at a different and, in fact, less accurate prediction for the propagation velocity of the bore: .

14

About 20 minutes after the passage of the bore, rapid fluctuations of the along-channel velocity of were measured at intervals, superimposed on a tidal current of (see Figure 10). The cross-channel velocity also showed similar fluctuations, suggesting the presence of eddies embedded within the main tidal flow. The entire event, which lasted about 3 minutes, is still unexplained, but is speculated to be due to lateral shear instabilities.

Figure 9: Time series during the passage of the bore of (1) the water level, (2) the velocity at from the bottom, (3) the vertical velocity at from the bottom and (4) the suspended sediment concentration at and from the bottom.

15

Figure 10: Rapid fluctuations ( ) in the along-channel velocity occur about 20 minutes after the passage of the bore at 7:30. The phenomenon has yet to be explained.

Conclusion

In this report, a model was developed to describe the origin and the propagation of tidal bores in a rectangular channel. Of course, we do not expect the model results to be directly applicable to a narrowing channel with a complex bathymetry which is, in addition, subject to effects of wind stress and stratification. The section Discussion: comparison to observations, however, suggests that the model developed (in particular the propagation velocity) is in close agreement with reality. Even quantities, such as the required tidal range are not entirely off. The required tidal range for the Daly Estuary is, for instance,

for , and . This is, indeed, of the same order as the

observed tidal range .

Cited material

[1] Arrian (1976). The Campaigns of Alexander, vol. 2, pp. 156 – 161. [2] Koch, C. and Chanson, H. (2008). Turbulent mixing beneath an undular bore front. Journal of

Coastal Research 24(4):999-1007. [3] Lecture given by Clive Dorman on tidal bores on the Severn River during the SDSU Geological

Sciences Seminar (http://www.scivee.tv/node/7928). [4] Simonsen, H.J. (2003) Numerical modeling of tidal bores with emphasis on the Hangzhou Bore.

(Master thesis). [5] Whitham, G.B. Linear and nonlinear waves. ISBN 0-471-94090-9. [6] Abbott, M.R. (1956) A theory of the propagation of bores in channels and rivers. Proceedings of

the Cambridge Philosophical Society, 52:344-362. [7] Wolanski, E. et al. (2004) Undular tidal bore dynamics in the Daly Estuary, Northern Australia.

Estuarine, coastal and shelf science, 60:629-636. [8] Chanson, H. (1995) Flow characteristics of undular hydraulic jumps. Comparison with near-critical

flows. Report Ch. 45/95, Department of civil engineering, University of Queensland, Australia, pp. 202.