On the frequency of rogue waves
Every surfer who's visited WA's North West would be aware of the 'King Waves Kill' sign.
Just out of Carnarvon, the sign warns tourists of rogue waves around the Quobba Blowholes. But what are rogue waves - sometimes called freak waves - and how frequent are they?
Mythical rogue waves have been documented for nearly two centuries, with the first observed in the Indian Ocean in 1826 by a French scientist and naval officer, Jules Dumont d'Urville, along with three colleagues. The 33m (108ft) waves were above what was thought plausible at that time - that waves could not exceed 9m (30ft) - and he was publicly ridiculed.
It wasn't until 1984 that the first rogue wave was recorded by scientific equipment on a platform in the North Sea's Gorm oil and gas field when a relatively low sea state produced an 11m wave.
However, it was the Draupner Wave in 1995 that grabbed the attention of scientists. The Draupner platform, also located in the North Sea, recorded a rogue wave of 25.6m (84ft) with a peak elevation (above sea level) of 18.5 metres.
Since then, there's been multiple attempts to re-create and produce rogue waves in a controlled environment, the main motivation being forecasts for such events.
In common literature a rogue/freak wave is labelled as a wave that is over 2x the Significant Wave Height, and this isn't as uncommon as you'd think.
Significant Wave Height is a statistical measure of the wave spectrum, that being the average wave height (from trough to crest) of the highest one-third of the waves. It was devised by Walter Munk during World War II as an estimation of wave height by a trained observer compared to a fixed point at sea.
When looking at an open ocean sea state there are waves from varying directions, sizes, and also with differing energies (periods).
On analysing the sea state over a period of time, the distribution of all the varying wave energy takes the form of a teardrop tipped on its side and cut in half length ways (see graph below).
This is known as a Rayleigh distribution and it's used in other applications such as analysing sound waves, wind speeds, as well as modelling noise variance in MRI's, among other examples.
The form of the distribution is such that wave height can be modelled as a function of the frequency of waves.
The combined dark blue and light blue areas in the above graph is what mathematicians call the Probability Density Function (PDF). Notice that its shape is more biased (weighted) towards the left, i.e smaller waves. This is because smaller waves are more likely than bigger waves, which can be confirmed by observing any sea state anywhere around the world.
The largest ⅓ of waves is coloured light blue, and the average of this area is where we get the value for Significant Wave Height.
If we look at a wave larger than the Significant Wave Height the probability of waves reaching this height becomes smaller and smaller, though not unexpected especially when the observation period increases and more waves are counted.
With such a distribution we can use statistics to predict the probability of waves being larger than, say, 10m Significant Wave Height.
- There's a 1/10 chance that a wave will be greater than 10.7m
- There's a 1/100 chance that a wave will be greater than 15.1m
- There's a 1/1,000 chance that a wave will be greater than 18.6m
- There's a 3.4/10,000 chance that a wave will be greater than 20m (2x Significant Wave Height)
And herein lies the answer to the question of observing a rogue wave. The greater the time and number of waves, the higher the chance and if we require this to be greater than 2x the Significant Wave Height then the odds of this occurring is 0.034% = 3.4 / 10,000 waves or 1 wave every 2,941 waves.
Let's try and put this into real terms. Surfers generally surf these highest ⅓ of waves with the smaller scraps left to go through to the keeper.
If looking at an East Coast beachbreak and the swell is fairly consistent you can probably expect to see 5-10 waves per minute across one spot, ranging from those sets to the in-betweeners. This equates to 300-600 waves per hour. So to reach the rogue wave threshold of 1 wave every 2,941 we're looking at this occurring once every 5-10 hours.
So even within a two hour session the chances of you seeing once of these rogue waves (double the size of the Significant Wave Height) is very low.
The 'King Waves Kill' warning sign for rock fisherman and tourists isn't for the waves discussed above, but moreso to signal that even though the ocean may look calm and benign, larger, long-period sets will arrive from the Indian Ocean, seemingly out of nowhere.
We as surfers understand they were generated some thousands of kilometres away, but for the layperson they are totally unaware. This is the main reason for the inordinate number of rock fisherman dying along the NSW coastline. It's not the large, stormy, and obviously dangerous swells, but the lully, long-period energy that catches fisherman off guard.
Beyond a rogue wave being classified as one greater than 2x the Significant Wave Height, the origins of such waves are still being debated.
The linear addition idea revolves around the simple constructive/destructive interference between interacting wave peaks, with the constructive phase causing these abnormally large waves.
The non-linear idea goes down the path of quantum theory and waves interacting and transferring energy between them instead of just passing through each other, conspiring to create a rogue wave.
Either way scientists are starting to get a better grasp on the initial sea states linked to rogue wave development in the open ocean and improving their forecasting for such conditions.