Maths & Oceans #2 : Mascaret, the tidal bore

During the equinox tides in September, the tidal bore returns to the spotlight. Let's outline a few details about this wave that travels up rivers, fascinating people all over the world (see videos of crowds of spectators in China (Hangzhou), sometimes showered by these impressive waves).

The origin of the French name remains unknown. It is interesting to note that in Gascon, the noun “mascaret” (tidal bore) refers to oxen with black, white, and gray spots1 . However, when it breaks in the river, this phenomenon does indeed appear with these colors on the surface due to the foam and sediments that are inevitably present (see Figure 12 ). In addition, the noise, which travels far inland, sounds remarkably like a herd of galloping cattle.

It is often said that the tidal bore only occurs once a year, during the equinox tides in September. In fact, it can occur throughout the year, but the wave is not always visible to the naked eye, and it breaks quite frequently: around 50 days a year in Gironde, a place known for this phenomenon, which serves as our guiding thread in this text.

The phenomenon can be observed in many places around the world. In France, we can mention its presence in Aquitaine, on the Dordogne and Garonne rivers, in Normandy, at Mont-Saint-Michel, and in Picardy, in the Baie de Somme.

Abroad, famous tidal bores include the Severn bore on the Severn River in the United Kingdom, the Guanchao on the Qiantang in China, and the Pororoca on the Amazon in Brazil. More recently, the Bono (nicknamed the “seven ghosts”) on the island of Sumatra has come to the forefront.

The tidal bore forms when the rising tide, which opposes the natural flow of the river, takes over and generates a wave known as a “tidal wave,” which travels up the river for tens of kilometers (for example, in Gironde, over 150 km and twice a day). Depending on conditions, this wave can break and create a wave that is surfed by enthusiasts: this is known as a tidal bore. At the mouth of the river, the shape of the delta, its width, and its slope are important parameters in the generation of the tidal bore. Inland, the depth, width of the riverbed, and meanders will also modify the propagation and breaking of the wave, as with an ocean wave. The wave may disappear and reappear with variations in the riverbed.

It is therefore understandable that by modifying the shape of the river3 , the tidal bore can be made to disappear, thus avoiding its inconveniences (destruction of banks, navigability, etc.). This is what was done on the Seine in the 1960s, which previously saw a four-meter wave breaking across its entire width (as shown in photos from that period in Quillebeuf, for example)4 .

Figure 2 [photographs below]: The tidal bore on the Seine at Quillebeuf, in the early 20th century5 .

The arrival time of the tidal bore can be predicted to within a few minutes thanks to knowledge of the tide times. The tidal coefficient (>90–100, for a good wave in Gironde) gives an idea of its size. However, it can be modified by the water level in the river (which varies depending on recent rainfall).

Here are some typical orders of magnitude (size/speed) for the tidal bores mentioned above (these are very approximate, as they depend on the event, and also, remember that size and speed vary along the river as the wave propagates):

• Dordogne: 1.5 m at a speed of approximately 15 km/h,
• Seine (before engineering modifications to the river): 3-4 m at approximately 20 km/h,
• Qiantang: 3 m on average (sometimes more in extreme circumstances) at approximately 25 km/h,
• Pororoca: up to 4 m and 25 km/h.

Let's return to the case of the Gironde: in its estuary, which collects the Garonne and Dordogne rivers, a wave can form across its entire width. Unable to rise higher than the tidal wave, the tidal bore stops near Génissac on the Dordogne and near La Réole on the Garonne. The best-known surfing spot (and the most publicized in France) is probably in Saint-Pardon, where there is a straight stretch of nearly 5 km. According to the regional press, there are 3,000 to 4,000 participants in September, but not all of them are in the water!

We also think it is worth saying a little more about the Bono and the seven ghosts (see Figure 36 ). This is a magnificent tidal bore on the Kampar River in Sumatra, where, from certain angles, seven fairly unique staggered waves can be seen very clearly (partly responsible for the nickname). Typically, it can reach a height of up to 2.5 m and a speed of around 20 km/h. The Bono was brought to light in 2011 by an explorer of the phenomenon, Antony Colas7 , who led a team of professional surfers there. They produced images, rare at the time, of well-formed, human-sized tubes being surfed on a river. A superb video is available at the following link: Link8 .

What are the results of mathematical research in interaction with physics on tidal bores?  

The phenomenon is still the subject of active research to understand its finer aspects, some of which are also linked to more abstract questions.

Tidal bores are associated with the transformation of the tidal wave into a “hydraulic jump,” somewhat like that of a kitchen sink [film]. To put it simply, it is a wave with a pronounced step at the front, which propagates at a constant speed without changing shape9 . However, and this is relatively recent, we are beginning to enrich the description of this jump by adding so-called “dispersive” effects that modify the shape of the wave10 .

For those who want to go further, one of the technical tools used to describe this dispersion is the Serre Green-Naghdi11  equations. However, we can provide an accessible intuition of these dispersive effects with the following diagram, which shows the evolution of the wave shape over time and space with typical magnitudes. Two curves are provided: one with a model without dispersion (black curve, “very simple”), and the other with a model containing dispersion (blue curve, where “oscillations” amplify over time).

As a real-life anecdote, tidal bore surfers will quickly recognize the dispersive wave train in nature: if the wave in front of you passes you by, the ones behind it won't be of much use to you! The train will move away without you; unfortunately, it's not like the Ohana in Lilo & Stitch!

You will also probably have noticed that, for both types of curves, there is a change in slope on both sides of the wave. In particular, the front of the wave tends towards an almost vertical slope: this is the subject of introductory mathematics courses on the subject (see also the blue curve) but also of current research (for example, in relation to the question of describing the phenomenon of wave breaking, whether at sea or in rivers).

To obtain Figure 4, scientists combined their expertise to propose a model adapted to tidal bores and innovative numerical methods that shed new light on this phenomenon. The simulations highlighted, for example, the influence of the type of tide, friction at the bottom, and the shape of the estuary12 . It is also interesting to note that the study of dispersive effects is not limited to tidal bores and hydrodynamics (the mathematics developed for these dispersion issues are used in various disciplines). It is found in contemporary fields such as fiber optic internet communications (long-distance light propagation), semiconductors (smartphones and computers), and plasma waves for nuclear fusion.

Tidal bores are a phenomenon that can be both magnificent and dangerous. Mathematics in interaction with other disciplines can help improve our understanding, prediction, and prevention of this natural hazard. This example is just one of many illustrations of the fruitful dialogue between mathematics and geophysics. Attempting to answer questions in geophysics leads to the creation of new and varied mathematical problems. Conversely, new mathematical methods make it possible to obtain more accurate results and, often, to better understand physics.

Acknowledgements

The author would like to express his warmest thanks to Philippe Bonneton for his insightful presentations on coastal hydrodynamics over many years. This article owes a great deal to him. Many thanks also to Antony Colas for providing us with photos of tidal bores in Gironde and Sumatra.

• 1Dictionnaire du béarnais et du gascon modernes. Simin Palay. Editions CNRS, 1980
• 2YEP 2006. Podensac, in Gironde
• 3Construction of dykes, modification of riverbanks, regular dredging of the riverbed at the mouth, etc.
• 4It should also be noted that the tidal bore was not responsible for the death of Victor Hugo's daughter Léopoldine on the Seine; the causes were more likely a gust of wind and an unbalanced boat.
• 5E. Mellet, postcard publisher in Harfleur (Seine-Maritime). 1920s (1927?); L ' Hernault, postcard publisher in Le Havre (Normandy). Date unknown but before 1963.
• 6 YEP, Rip Curl, Nate Lawrence, 2011 Bono campaign
• 7Mascaret, prodige de la marée. Antony Colas, YEP editions, Novembre 2017.
• 8YouTube/Rip Curl. 2011. Accessed September 18, 2025. Tubes at 2:05 and 4:05 (the latter by Tom Curren).
• 9In mathematics and fluid mechanics, in the theory of hyperbolic equations, we talk about “shock.”
• 10These are the two waves behind the first surfers, visible in Figure 1 and also in Figure 3. The shock is then referred to as a “dispersive shock.”
• 11Tissier et al. (2011) Nearshore Dynamics of Tsunami-like Undular Bores using a Fully Nonlinear Boussinesq Model. Journal of Coastal Research, pp. 603-607.
• 12See also other articles on this subject in the specialized press by P. Bonneton, D. Lannes, M. Ricchiuto, and their colleagues.