Wave overtopping is the time-averaged amount of water that is discharged (in liters per second) per structure length (in meters) by waves over a structure such as a breakwater, revetment or dike which has a crest height above still water level.
When waves break over a dike, it causes water to flow onto the land behind it. Excessive overtopping is undesirable because it can compromise the integrity of the structure or result in a safety hazard, particularly when the structure is in an area where people, infrastructure or vehicles are present, such as in the case of a dike fronting an esplanade or densely populated area.
Wave overtopping typically transpires during extreme weather events, such as intense storms, which often elevate water levels beyond average due to wind setup. These effects may be further intensified when the storm coincides with a high spring tide.
Excessive overtopping may cause damage to the inner slope of the dike, potentially leading to failure and inundation of the land behind the dike, or create water-related issues on the inside of the dike due to excess water pressure and inadequate drainage. The process is highly stochastic, and the amount of overtopping depends on factors including the freeboard, wave height, wave period, the geometry of the structure, and slope of the dike.
Overtopping factors and influences Overtopping can transpire through various combinations of water levels and wave heights, wherein a low water level accompanied by high waves may yield an equivalent overtopping outcome to that of a higher water level with lower waves. This phenomenon is inconsequential when water levels and wave heights exhibit correlation; however, it poses difficulties in river systems where these factors are uncorrelated. In such instances, a probabilistic calculation is necessary.
The freeboard is the height of the dike's crest above the still water level, which usually corresponds to the determining storm surge level or river water level. Overtopping is typically expressed in litres per second per metre of dike length (L/s/m), as an average value. Overtopping follows the cyclical nature of waves, resulting in a large amount of water flowing over a structure, followed by a period with no water. The official website of the EurOtop Manual, which is widely used in the design of coastal engineering structures, features a number of visualisations of wave overtopping.
In the case of overtopping at rubble-mound breakwaters, recent research using numerical models indicates that overtopping is strongly dependent on the slope angle. Since present design guidelines for non-breaking waves do not include the effect of the slope angle, modified guidelines have also been proposed. Whilst these observed slope effects are too large to be ignored, they still need to be verified by tests using physical models.
Overtopping behaviour is also influenced by the geometry and layout of different coastal structures. For example, seawalls (which are typically vertical, or near-vertical, as opposed to sloping breakwaters or revetments), are often situated behind natural beaches. Scour at the base of these structures during storms can have a direct impact on wave energy dissipation along their frontage, thus influencing wave overtopping. This phenomenon assumes critical importance when storms occur in such quick succession that the beach doesn't have sufficient time for sediments removed by the storm to be re-established. Experimental results show that, for near-vertical structures at the back of a beach, there is an increase in wave overtopping volume for a storm that starts from an eroded beach configuration, rather than a simple slope.
Calculation of overtopping
Wave overtopping predominantly depends on the respective heights of individual waves compared to the crest level of the coastal structure involved. This overtopping doesn't occur continuously; rather, it's a sporadic event that takes place when particularly high waves within a storm impact the structure.
The extent of wave overtopping is quantified by the volume of water that overflows onto the adjacent land. This can be measured either as the volume of water per wave for each unit length of the seawall, or as the average rate of overtopped water volume per unit length during the storm wave period.
Much research into overtopping has been carried out, ranging from laboratory experiments to full-scale testing and the use of simulators. In 1971, Jurjen Battjes developed a theoretically accurate equation for determining the average overtopping. However, the formula's complexity, involving error functions, has limited its widespread adoption in practical applications. Consequently, an alternative empirical relationship has been established:
in which is the dimensionless overtopping, and is the dimensionless freeboard:
in which: is the water depth is the freeboard is the overtopping discharge (in m³/s) is the significant wave height at the toe of the structure is the deep water wavelength is the inclination of the slope (of e.g. the breakwater or revetment) is the Iribarren number is a resistance term.
The values of and depend on the type of breaking wave, as shown in the table below:
{|class="wikitable" style="float:left;" ! Type of wave || Value for || Value for |- |breaking (plunging)|| 0.067 || 4.3 |- |non-breaking (surging)|| 0.2 || 2.3
The resistance term has a value between approximately 0.5 (for two layers of loosely dumped armourstone) and 1.0 (for a smooth slope). The effect of a berm and obliquely incident waves is also taken into account through the resistance term. This is determined in the same way as when calculating wave run-up. Special revetment blocks that reduce wave run-up (e.g., Hillblock, Quattroblock) also reduce wave overtopping. Since the governing overtopping is the boundary condition, this means that the use of such elements allows for a slightly lower flood barrier.
Research for the EurOtop manual has provided much additional data, and based on this, the formula has been slightly modified to:
with a maximum of:
It turns out that this formula is also a perfect rational approximation of the original Battjes formula.
In certain applications, it may also be necessary to calculate individual overtopping quantities, i.e. the overtopping per wave. The volumes of individual overtopping waves are Weibull distributed. The overtopping volume per wave for a given probability of exceedance is given by:
in which is the probability of exceedance of the calculated volume, is the probability of overtopping waves, and is the crest height.
Calculation and measurement of overtopping at rock revetment crests In terms of revetments, the overtopping discussed in the EurOtop manual refers to the overtopping measured at the seaward edge of the revetment crest. The formulas above describe the wave overtopping occurring at the sea-side edge of the crest. In scenarios where the crest is impermeable (for example, a road surface or a clay layer), the volume of water overtopping the inland side of the crest would roughly equal that on the seaside. However, in the case of a rock armour breakwater with a more permeable crest, a large part of the overtopping water will seep into the crest, thus providing less overtopping on the inside of it. To analyse this effect, reduction coefficient can be used. This factor can be multiplied by 0.5 for a standard crest, with a width of about three rocks. This can result in a significant reduction in overtopping, and thus in the required crest height. If, behind the crest at a lower level, a permeable rock armour layer is installed with width , the amount of overtopping on the landside of this layer decreases still further. In that case, the reduction term (not to be confused with the reduction co-efficient ) can be multiplied by , in which is the crest width.
Berm breakwaters The circumstances surrounding overtopping at berm-type breakwaters differ slightly from those of dikes. Minor wave overtopping may occur as splashes from waves striking individual rocks. However, significant overtopping typically results in a horizontal flow across the crest, similar to what happens with dikes. The primary distinction lies in the wave heights used for designing these structures. Dikes rarely face wave heights exceeding 3 metres, while berm breakwaters are often designed to withstand wave heights of around 5 metres. This difference impacts the overtopping behaviour when dealing with smaller overtopping discharges.
Tolerable overtopping An understanding wave overtopping involves a combination of empirical data, physical modelling, and numerical simulations to predict and mitigate its impacts on coastal structures and safety. Traditionally, permissible average overtopping discharge has been utilised as a standard for designing coastal structures. It is necessary to restrict the average overtopping discharge to guarantee both the structural integrity of the structure, as well as the protection of individuals, vehicles, and properties situated behind it. Design handbooks often stipulate the thresholds for the maximum individual overtopping volumes, necessitating the examination of wave overtopping on a wave-per-wave basis. Often, to ensure a more dependable level of safety for pedestrians and vehicles, or to evaluate the stability of the inner slope of a revetment, it is necessary to consider the peak velocity and thickness of the overtopping flow.
The tolerable overtopping is the overtopping which the design accepts may occur during a design storm condition. It is dependent on a number of factors including the intended use of the dike or coastal structure, and the quality of the revetment. Tolerable overtopping volumes are site-specific and depend on various factors, including the size and usage of the receiving area, the dimensions and capacity of drainage ditches, damage versus inundation curves, and return period. For coastal defences safeguarding the lives and well-being of residents, workers, and recreational users, designers and overseeing authorities must also address the direct hazards posed by overtopping. This necessitates evaluating the level of hazard and its likelihood of occurrence, thereby enabling the development of suitable action plans to mitigate risks associated with overtopping events.
For rubble mound breakwaters (e.g., in harbour breakwaters) and a significant wave height greater than 5m on the outside, a heavy rubble mound revetment on the inside is required for overtopping of 10-30 L/s per metre. For overtopping of 5-20 L/s per metre, there is a high risk of damage to the crest.
{| class="wikitable" style="float:left;" ! Situation on the slope || (L/s per metre) |- |Quarry stone in waves > 5m, and some damage|| 1 |- |Quarry stone in waves > 5m, and some damage(*)|| 5 - 10 |- |Good grass cover between 1m and 3m|| 5 |- |Poor grass cover between 0.5m and 3m|| 0.1 |- |Poor grass cover < 1m|| 5 - 10 |- |Poor grass cover < 0.3 m|| Unlimited |- | colspan="2" |(*)and inner slope designed for overtopping