What if just one wave is enough to cause disaster? (part 3): A Review of Abdalazeez et al. (2020)

The basics...

Understanding the occurrence of extreme coastal runup events — those in which individual waves reach unexpectedly high elevations on the beach — is essential for risk assessment in coastal zones. These events, often referred to as freak runups, are rare and difficult to predict using traditional statistical tools based on average wave conditions. Previous studies have been limited by short experimental records, simplified beach geometries, or assumptions of Gaussian statistics.

In their 2020 paper published in Water, Abdalazeez, Didenkulova, Dutykh, and Labart address this knowledge gap by investigating the statistical characteristics of wave runup on a composite beach using long-duration numerical simulations. Their work aims to explore how wave nonlinearity, wave breaking, and spectral bandwidth affect the statistical properties of runup and the likelihood of extreme inundation events.

The objective is twofold: (1) to quantify the effect of spectral properties and wave nonlinearity on the statistical distribution of runup heights, and (2) to assess whether a conditional Weibull distribution can provide a suitable statistical model for extreme runup values.

How was the problem tackled?

The study uses a one-dimensional nonlinear shallow water (NSW) model with shock-capturing finite volume methods. The model does not account for dispersion or overturning, but does allow for the formation of bore-type shocks, which the authors use as an approximation of wave breaking.

The beach profile consists of a composite geometry: a horizontal section of constant depth \( h_0 = 3.5\,\text{m} \) extends from \( x = 0 \) to \( x = 251.5\,\text{m} \), followed by a uniformly sloping section with a slope of \( \tan(\alpha) = \frac{1}{6} \) up to \( x = 291.5\,\text{m} \).

Wave forcing is applied at the offshore boundary using Gaussian random time series with a central frequency \( f_0 = 0.1\,\text{Hz} \) and two bandwidths:

  • Narrow-band: \( \Delta f / f_0 = 0.1 \)
  • Wide-band: \( \Delta f / f_0 = 0.4 \)

The significant wave height \( H_s \) is varied across five values corresponding to nonlinearity ratios:

\[ H_s / h_0 = 0.03,\ 0.06,\ 0.09,\ 0.11,\ 0.14 \]

This corresponds to \( H_s \in [0.105,\ 0.49]\,\text{m} \). For each combination of spectral width and wave height, the model was run for 1000 hours (approx. 360,000 waves), totalling 10 scenarios.

The Iribarren number is used to classify the breaking regime:

\[ \text{Ir} = \frac{\tan(\alpha)}{\sqrt{H_s / L}}, \quad L = \frac{g T_0^2}{2\pi} \approx 156.1\,\text{m} \]

This results in Iribarren numbers ranging from:

  • \( \text{Ir} \approx 4.3 \) for \( H_s = 0.105\,\text{m} \)
  • \( \text{Ir} \approx 1.87 \) for \( H_s = 0.49\,\text{m} \)

Therefore, the simulations cover a range \( \text{Ir} \in [1.87,\ 4.30] \), corresponding to surging and plunging wave breaking regimes.

Runup statistics were recorded at the shoreline (x = 291.5 m), and the significant runup height \( R_s \) was defined as the average of the highest third of individual runups. Events with \( R \geq 2 R_s \) were classified as freak runups.

Findings

The simulations reveal clear differences in runup statistics between narrow-band and wide-band wave fields, and across different levels of wave nonlinearity.

  • In narrow-band cases, runup distributions are narrower, with thinner tails. The runup values tend to stay closer to the mean. In other words, most values stay near the average, and you don’t get many big surprises.
  • In wide-band cases, runup distributions become wider and more asymmetric as nonlinearity increases. The tails become thicker, indicating a higher probability of extreme runup heights. In other words, there’s a higher chance of really big runup events, the kind you don’t expect often, but that can do real damage.

To really get what the authors measured, we need to talk about two stats terms (called statistical moments) that sound scarier than they actually are: skewness and kurtosis. Don’t worry, you won’t need a calculator for this.

Skewness is basically about whether your data leans to one side. If most of the values are low, but now and then a big one shows up, we say the distribution is skewed to the right. If the big values are rare and things mostly stick to the lower side, then it’s skewed to the left. In runup terms, negative skewness means you don’t get many massive waves — things tend to stay below the average. It’s like the ocean’s having a bit of a quiet day.

Kurtosis, on the other hand, tells you how “peaky” or flat the shape of the data is. If you’ve got a lot of average-sized waves but suddenly one hits you like a slap, that’s high kurtosis — most values are normal, but there’s a real chance of a surprise. If the values all stay pretty close to the middle and there aren’t any big shocks, that’s low kurtosis — a flatter shape, more predictable.

So in short, skewness tells you which way things lean, and kurtosis tells you if the story comes with wild twists or just keeps things steady.

For narrow-band cases, the runup distribution is skewed to the left (negative skewness), meaning that most runup values are smaller, with fewer large extremes. As wave nonlinearity increases (i.e., as \( H_s / h_0 \) becomes larger), this negative skewness becomes more pronounced. The kurtosis — which describes how heavy or thin the tails of the distribution are — does not follow a consistent trend; it increases in some cases and decreases in others. In wide-band cases, both skewness and kurtosis also fluctuate as nonlinearity increases, but without a clear or monotonic pattern, indicating a more complex response of the runup statistics to wave conditions.

The number of freak runups (defined as \( R / R_s \geq 2 \)) is found to be larger in wide-band simulations than in narrow-band ones. In cases with low nonlinearity (e.g. \( H_s / h_0 = 0.03 \)), some freak runups occur in both spectral cases. As nonlinearity increases, especially for \( H_s / h_0 = 0.09 \) and above, freak runups disappear in narrow-band cases and become rare in wide-band cases.

The authors propose a conditional Weibull distribution to describe the tail of the runup distribution for \( R \geq 0.7 R_s \). The parameters of this distribution — shape \( k \), scale \( \lambda \), and threshold \( s \) — are fitted using maximum likelihood estimation. The model provides a good fit across the different simulation scenarios.

Summary of Key Findings

Aspect Narrow-band Waves Wide-band Waves
Runup distribution shape Narrower, thin tails
Values stay closer to the mean		 Wider, thicker tails
Higher variability, more spread
Skewness Negative and becomes more negative as nonlinearity increases
Runups tend to be lower than average		 Fluctuates with nonlinearity, no clear trend
Behaviour is more complex
Kurtosis Changes non-monotonically with nonlinearity
No consistent increase or decrease		 Also non-monotonic
Indicates more complex tail behaviour
Freak runup occurrence
(\( R / R_s \geq 2 \))		 Rare or disappears as \( H_s / h_0 \) increases More frequent than in narrow-band
But becomes rare at high nonlinearity
Effect of wave nonlinearity Suppresses extreme runup
Skewness becomes more negative		 Modifies shape of distribution
But not in a predictable (monotonic) way
Extreme runup modelling Conditional Weibull distribution fitted for \( R \geq 0.7 R_s \); works well across all scenarios

Discussion

The results show that both wave nonlinearity and spectral bandwidth significantly influence the statistical characteristics of wave runup. Narrow-band wave fields tend to produce more freak waves offshore, but these do not necessarily lead to freak runups at the coast. In contrast, wide-band wave fields are associated with higher runup variability and a greater likelihood of extreme runup values.

Wave breaking, as characterised by the Iribarren number, plays a filtering role. As nonlinearity increases and more plunging breakers occur (lower Ir), the number of freak runups decreases sharply. This suggests that wave breaking limits the ability of the wave field to produce extreme shoreline excursions.

The authors do not perform spectral decomposition of the runup time series, and no specific analysis of long-period oscillations is presented. The study does not address the possibility of low-frequency energy components arising from wave generation, reflection, or nonlinear interactions. The potential influence of spurious long waves or reflection patterns in the numerical domain is not discussed.

The slope of \( \tan(\alpha) = \frac{1}{6} \) is relatively steep compared to natural beaches, which may limit the generalisability of the findings. Nonetheless, the composite beach model is well suited to test the influence of slope and nonlinearity in a controlled framework.

Conclusion

Abdalazeez et al. (2020) present a thorough numerical investigation into the statistical behaviour of wave runup on a composite beach under random wave forcing. Their long-duration simulations provide rare insight into the tails of runup distributions, especially in relation to wave breaking and spectral bandwidth.

The study shows that extreme runups (freak runups) are more likely under wide-band wave conditions, particularly in low to moderate nonlinearity cases. As nonlinearity increases and wave breaking intensifies, these extremes become less frequent.

The use of a conditional Weibull distribution is demonstrated to be effective for modelling the upper tail of runup heights, offering a practical tool for coastal hazard assessment.

While the simulations are comprehensive, future work could benefit from investigating the spectral content of the runup signal, assessing possible low-frequency components, and testing milder slopes. This would help to extend the applicability of the findings to a broader range of real-world beach conditions.

Reference

Abdalazeez, A., Didenkulova, I., Dutykh, D., & Labart, C. (2020). Extreme Inundation Statistics on a Composite Beach. Water, 12(6), 1573. https://doi.org/10.3390/w12061573

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