## 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.

## Comments

love how the MSM refer to any boating mishap as being caused by a "rogue wave". A rogue wave to me is one that flips my boat in the garage it lives in 50m+ above sea level

Talk to Tom Hoye about rogue waves.... a triple-up at the Bears that almost took his life. Stu has the story I reckon.

How about rogue sets, had a set at Shipwreck one time 2-3 waves not sure how many as I was underwater for most of the first two, surfing 4-5' and a 10-12' set arrived to say hi, that was really spooky........

The Indo reefs sitting on deep water are a law unto themselves. I remember the first time I surfed Deserts, hiked in (yep, no passable road then) to be met with waist high waves dribbling across the reef. This sunburnt old timer sitting in a dirt floor hut just said "hold out....tides turning". Sure enough, double overhead freight trains within an hour. Then it stopped like someone had flicked a switch again. Literally had to paddle in. Just the sheer amount of tidal movement in those channels is enough to mess with wave energy. Crazy.

Classic Indo rogue set...

I love that clip. I watched that before my first ever Mentawai trip to get psyched up and mentally prepare myself for anything that may come. Makes me laugh and feel kind of sick at the same time. So good!

Ive been worked pretty hard at shipwreck, scared the heck out of me. Its a reasonably hectic combo sitting out there with an incoming tide and building swell. Way worse beat downs than similar size in the ments, bali & sumbawa from my experience

On the round Australia trip before leaving Carnarvon went to the tourist information and had a look at the weather forecast which was printed & bluetacked to the wall, looked like it was going to be solid.

Arrived at the blowholes as a wave closed out on the shelf & shot up the blowhole. Felt sick in the stomach & thought I was going to soil myself.

It was intimidating for a bloke from the Sunny Coast who hadnt surfed for 6 months.

Classic.

I’ve overheard quite a few conversations on the rocks involving an uneasy looking bloke staring wide -eyed at the lineup whilst his wife , who is already looking extremely uncomfortable sitting on the craggy limestone rocks , starts berating him along the lines of - “ We just drove three days to get here . You bloody well better be going surfing and bloody hurry up about doing it or else we’re driving out of this god forsaken, fly blown shithole right now.”

Thats the perfect combination of disaster awaiting to happen, true love also right........

Intersting..

Cape du couedic wave buoy.

April 21st this year and again on the 27th, both showing rogue wave spikes in the graph that qualify based on the 2x criteria.

Both times over 10 metres.

I took screenshots but dont remember seeing any irl...

Ride anything.

Unless you're right at the point of observation or very close to, the wave and energy would have dissipated and transformed, so you can't expect to see that same wave on the beach.

Hehe true, here's the screenies fwiw, i found another one in may that almost qualifies.

https://ibb.co/TYKC4K0

https://ibb.co/qjMbnGN

https://ibb.co/x694H6j

Wish i knew how to make the images show up and not some kooky links..

Ride anything.

i seen this rouge one once while watching the right from land that was like nothing ive ever seen before. i swear i was tripping, like did that just happen. its was on a 3mtr swell and was easy 30ft. that doesnt add up

Hows the cracker at 3.30pm yesterday just off Strahan.

Nice! Just falls 0.1m under the requirements but significant none the less.

https://en.wikipedia.org/wiki/File:Wea00800,1.jpg

Tanker in Bay of Biscay near 100 fathom line, 1940.

Another from the Bay of Biscay, this clip has a wave considerably bigger than the sea-state hit the ship at the start:

https://www.youtube.com/watch?v=uK_4V3zqAvg

Great vision.

Seems to be just a lone wave too, not part of a set.

Swimming at shipsterns taking water pics, trying to keep some what near the rock, on a inco 6 to maybe 8 ft day, and a 12 to 15 foot bomb comes thru! It really is miss a few heart beats stuff.....go out on a 10 to 12 foot day and the same thing happens!!!! Double or plus some my jaw drops, my body freezes for a coupla seconds, swim toward the shoulder , mick brennan wipes out, no photo opps now....im somewhat on the shoulder, and gently pulls me over the falls, flying deep down past stacks of boulders, down into the blackwater zone, well below thick a layer of airated frothy water that you cant swim through, its a long lonely wait for it to dissipate so you can swim back to the top.....mick had come up earlier further down towards the bay.

Im not sure whats more frightening there rogue waves or great whites/orca threats.

At least you can somewhat prepare yourself for rogue waves, but when they appear and your towards or on the inside it is intimidating and you are at oceans mercy.

Likewise sitting out the back at MR Main Break, always bomb sets coming thru.

Seeing footage of the 2004 boxing day tsunami barreling towards a tourist village was a whole another perspective on rogue or king waves.....theyre very real.

mick b

This is really interesting and is starting to bring out the always lurking math geek in me.

The Rayleigh distribution has been found by statisticians to accurately model the distribution of the height of random waves in deepwater conditions with no influence from the bottom or from shores.

In shallow water conditions and/or the presence of a shore (e.g. a beach or a narrow seaway), the Rayleigh distribution requires some adjustment factors applied to it in order to retain an acceptable level of accuracy.

What might be interesting to people is if you have a large enough sample of deepwater wave heights (I see some small samples presented above) over a given time period in given unchanging sea-state conditions, you can use some very simple equations to get some interesting estimates.

For instance, let m be the mean (average) weight height in your sample. To start with, it can be shown that the significant wave height s is given by

s = 2*m*sqrt(2 / pi),

where:

- sqrt(x) us the square root of the number x,

- x*y is the number x multiplied by the number y,

- pi equals 3.14159265359... and

- x / y is the number x divided by the number y.

Also, the probability P(h) of observing a wave in the conditions in which the sample was taken exceeding a height of h can be shown to be given by

P(h) = 100*exp{ - 2*(h^2) / (s^2) } %,

where:

- x^2 equals the number x squared (multiplied by itself) and

- exp(x) = e^x, the exponential function of the number x.

Hence, given your mean wave height m in your sample (which I'm sure you can calculate), you can find P(h) for any wave height h. One interesting instance is the rogue wave height.

The height r of a rogue wave is defined to be twice the significant wave height:

r = 2*s.

Hence, by plugging in all the above, the probability P(r) of observing a wave in the conditions in which the sample was taken higher than a rogue wave is

P(r) = 100*exp{ -2*2*2 } = 0.033546262 %

which is what Craig got (but here without rounding). Taking the inverse, you could expect to see 1 in 2,981 waves higher than the rogue wave height, again what Craig got (here without rounding).

It is interesting to note that if we define the rogue wave height to be 2.5 times the significant wave height, these numbers blow out substantially: P(r) becomes 100*exp{ -2*2.5*2.5 } = 0.000372665 % and we would expect to see 1 in 268,337 waves larger than this new rogue wave height...not many...

Sorry bout the maths...

what if you slacken the definition of Rogue wave to 1.5 times the sig wave height?

how many kind of "bomb" sets could we expect?

That puts the chance at 1.11% = 1.11/100 = 1/90 which puts it in the range of one every nine to eighteen minutes if there are 5-10 waves per minute.

--Edited a couple of times as I was doing the maths on the fly.

Great maths GSCO, I've got a working spreadsheet with the probabilities etc and calculations for this article.

10/10 gsco, that was bloody great. I'm coming back to it in an hour after another coffee to see if I can get past the second line!

Surely there's a theoretical maximum wave height whereby stacking or quantum wizardry cannot exceed 2 or 3x the significant wave height of the primary swell. Secondary swells may contribute but my point being there'd be a relationship between the energy of a swell or combined swells and that their potential to amplify would be limited to the maximum combined energy of each swell. i.e a 2 meter swell might produce a 4 meter rogue wave every 2,981 waves but will never produce a 6 or 8 meter wave as the energy required simply doesn't exist in that sea state.

This might sound obvious but naturally I had the thought "does that mean 1 in every 2,000,000 waves will be a SUPER WAVE?" where's the ceiling

I tried to maths but it's been so long since uni my calculator has run out of battery....

As a continuous probability density function defined for all nonnegative numbers, the Rayleigh distribution theoretically allows for any wave size, just with extremely low probabilities for extremely large waves. Of course this is unrealistic and impossible due to the physics.

Basically, the situation is that it's not correct to say that the distribution of wave heights "follows" a Rayleigh distribution.

It's more correct to say that in the range of sea-state and thus wave height conditions actually observed on planet earth, the Rayleigh distribution "accurately models" or is a "close approximation to" the distribution of wave heights. We all know that we will never observe a one million foot wave on planet earth in the usual range of sea-state conditions.

This is a similar situation to quantum mechanics vs Newtonian mechanics vs general relativity. On a human scale Newton's laws "accurately model" or are a "close approximation to" observable phenomenon, but they break down on the subatomic scale where quantum mechanics has been shown to be extremely accurate, and on the cosmological scale where Einstein's theory of general relativity is shown to be extremely accurate.

In general mathematical modelling is about finding "adequate" or "accurate enough" models of observable phenomenon on a scale at or in a range in which these phenomenon actually occur.

Yes I thought that would be the case, then it begs the question: just how far can a wave exceed the significant wave height in reality? variables aside, this data would exist, for example using Manly hydraulics lab data for recent east coast lows. This is more of a thought experiment and Saturday morning ponderings so I'm not gunna crunch numbers. But every time a big swell hits I find myself refreshing MHL every hour to check Hmax and Hsig. I love the phenomenon of big waves and it's basically why I'm here - I'll likely never paddle out in it.

This has probably been well covered in the literature when designing ships and offshore rigs or even just for navigation - they need to know what outliers to expect. Anyway cheers for your input.

oh and why we're here... I wonder if a million foot wave has occurred in the cosmos.... some planets are unfathomably large. Maybe there are 10,000ft methane waves peeling off a point on Jupiter right now

From https://cosmosmagazine.com/space/juno-sees-mega-waves-on-jupiter/:

"One wave was estimated to be 10 kilometres tall – a little less than the cruising altitude of commercial jets on Earth."

and if you read the whole article they go on to mention that Laird Hamilton was spotted out kite-foiling on them and making everyone else look very soft...

hahaha that's awesome. I wonder if rogue waves occur in atmospheric cloud waves

Yes well said, thanks GSCO.

Remember sitting at lances left, unsurfable lay day.

Out the port window a 2 wave 4 ft set reels through.

Spent the afternoon wake surfing behind the tender

Exact same thing happened to me at scar reef,3random 6footers rolled down the reef after lunch,I spent the arvo waiting in lake like conditions roasting!!!!

Yeh lances left is notorious for clean up sets.

Way Jambu in sth Sumatra

5-6ft long period energy

Caught everyone offgaurd

Rogue 12+ top to bottom

My go pro 4 made the footage look 4ft

That place is well known for it's rouge sets. Had a similar experience there but not quite to the same degree.

I read somewhere 100 ships a year disappear from rogue waves

Haven't looked into it, but wonder just how much of a role bathymetry plays. That being, do steeper bathy profiles increase the probability, whilst gradual/shallower profiles decrease?

From a location perspective, this might be like comparing West Cape at Yorkes to Seaford, on the Mid Coast.

Anecdotally, you'd think this might be the case - but perhaps it's just the way 'rogue' waves are experienced at deepwater locations (usually an arse-kicking), whilst shallower locations with more gradual bathy profiles attenuate/dissipate the wave energy before it shoals.

There has been some research and modelling on this question. For instance, in the "Design tools related to engineering" section of

https://www.sciencedirect.com/topics/engineering/rayleigh-distribution

it is stated that

"In shallow water, the Rayleigh distribution significantly underestimates the lower wave heights, and overestimates the highest. Several works deals with semiempirical adaptation to the Rayleigh distribution to allow for the effect of shallow water and breaking."

So it appears that Ben's intuition is correct.

The link then goes on the talk about some modifications that have been made to the Rayleigh distribution in order to handle this and provides some of the standard references in this area of enquiry. But it all gets a bit mathematical for the general reader.

Something else I also found interesting is the "Ocean Energy" section of

https://www.sciencedirect.com/topics/earth-and-planetary-sciences/raylei...

but again it's a bit mathematical for the general reader.

There's some additional links that google will uncover for you.

Great stuff, thanks.

Going out wide for tuna off Sydney region the seas are often pretty lumpy just before the continental shelf dropoff but then smooth out as we head over the drop.

Strong currents seem to also be a major factor with rogue waves - the Agulhas current off South Africa pops up a lot in discussions.

Yep, massively.

The stories of the southward flowing Agulhas colliding with large Southern Ocean storms and spiking their amplitude and reducing the wave-length resulting in ships being lifted either end and snapped in half are horrific.

I was going to mention the Agulhas...

Craig, who would you have to know to pull up a synoptic of July 25/26 1909 for this area (if it exists)?

It is commonly regarded that the SS Waratah disappeared without a trace en route from Durban to Cape Town in a 'storm of exceptional violence'. Weather observations from other ships doing that passage on those days suggest a very strong northerly/NE wind preceding the front (think: a black nor'easter?), then a strong westerly wind shift with strong SW swell against the current, which battered many ships, causing others to come in days late.

Waratah and all souls on board disappeared, not one official bit of wreckage - though accounts have been posted of some debris washing ashore in Mossel Bay, and of some witnesses who claimed to see her roll over - remember, steamers took the inshore track (5 miles) or outer track (about 13 miles) so visibility from shore was possible. Waratah has all the elements of a gripping nautical horror story, from initial problems with her stability (resolved? or not?), possible shifting or explosive cargo; to passengers and crew being extremely uncomfortable during voyages with her long roll, sometimes hanging in the roll for minutes; to the ship's rats bailing the ship when stopped at Albany(!). One passenger had a prophetic nightmare, and disembarked after warning others, at Durban! The ship had no wireless, either, and was to be fitted with it upon arriving in London. The story really is Australia's "Titanic" - and more mysterious as the wreck has never been found.

It has been suggested that 20m waves are possible off this section of South Africa's coast with quite some regularity because of this current/swell/wind setup. Could Swellnet do a weather synoptic for this fateful day?

Seems plausible - the figure that always stuck with me as a kid was a 108ft wave sighted in the 1800s off the SA coast....

Wikipedia says "In 1826, French scientist and naval officer Captain Jules Dumont d'Urville reported waves as high as 108 ft (33 m) in the Indian Ocean with three colleagues as witnesses...."

I wish I could get a chart from back then but that's too far back in time sorry. Aghh.

I do love me my maths, but more as an interested observer than a practitioner. Thanks GSCO and Craig, always illuminating.

As a rule I always say to my mates that if it’s 4-5’ and you’re planning on being out there a couple of hours you need to be ready to deal with a 6-8’ pile driver somewhere in the mix. And that wouldn’t qualify for a rogue wave status.

Do you think they could introduce rogue waves into the mix at The Tub?

February 2007 ... cathedral rock Victoria ... 6-8 right on the maximum for me ... with a couple of locals ... set on the horizon ... omfg ... knew i was going to die ... looked up at the missus on the cliff and said goodbye .... not even going to guess how big but bigger than telegraph pole size ... two local guys stopped me panicking ... guided me to the deepest water and saved my ass ... had to be tsunami ....

Here’s a possible scenario: Two swells from similar directions ie SSW and SSE, resulting in set waves from the two different swells doubling up and forming a rogue wave. Any thoughts?

That's very common and is represented in the wave spectrum for that scenario.

IE the Average Wave Heights and Significant Wave Heights for such a sea state would be larger than if just one of those single swell trains were present. Hence a rogue wave would still have to be 2x above that SWH.

Cheers Craig. I’m beginning to realise that even though I’ve spent the last 33 years chasing waves, I actually know very little about how they work. It’s a fascinating subject.

Yeah I don't think surfers surf the highest 1/3 or Hsig waves, that's probably more like what you report as the size of the swell, say you call it a 1-3ft swell the Hsig is probably more representative of all the 1-3ft waves you see and the 2-3ft'ers are the 1/10 that actually break good enough to ride (we tend to ignore anything smaller we see). FWIW, some Hawaiians did a study on it more than a decade ago, someone here can make a link, look for Pat Caldwell and An Empirical Method for Estimating Surf Heights from Deepwater Significant Wave Heights...

https://www.researchgate.net/publication/253901043_An_Empirical_Method_f...

Two experiences to share:

Watching the Molokai Channel during Cat 5 hurricane Iniki 1992 (strongest ever to hit Hawaiian Islands) when 3 massive waves (> twice the height of the rest, waves that were running down the reef in front of us at Hawaiian 20’) rolled down the channel. The first one engulfed a steel ship, she never surfaced. 3 crew I was told. Can still see those 3 mountains in my minds eye.

Inbound to Bangkok from London after dawn Boxing Day 2004, waiting for the beautiful Thai coastline to appear, a sight I had enjoyed many times, I was confused then horrified to watch the tsunamis roll thru and then bounce around the Islands. Can still see that one too.

Waves of horror that you cannot unsee.

Wow, what type of work where you doing?

Pilot. Visiting a surfing mate for Iniki during a couple of weeks off. Operating for the second on descent into Bangkok.

Normally a great view from the front seats, that day a clear view of unfolding devastation.

That's super heavy and unimaginable. Seeing the whole ocean slop around like that between islands etc.

Woh, that must've been heavy

Thanks for all the comments, insights and equations fellas.

Attached is a photo of a rogue in the Mentawais shot a long way away from our boat a few years back:

https://ibb.co/89RbpmQ

Sorry u may have to cut and paste in browser or just click on.

Swell had backed right off so we were surfing a random peak @ 2-4 ft max and I had just caught one and was paddling out when the ocean around me started doing weird things and was sucking me back towards where I had flicked off.

I'm in the trough on the other side of the shaven headed one and about to watch my mates get destroyed by this cobra head while Im scrambling for the shoulder.

Didn't see em resurface for at least 30 metres!

Thanks Craig, interesting stuff but more worried about the rogue surfer these days especially at Manly

(Lenny)

And then we have this...

Have a look at a book by Susan Casey called "The Wave" great read about rogue waves

Also watched a doco called Rogue Waves, I thought it was about rogue waves but it was about a guy from Tassie with a real interesting story

At sea we've used a similar frequency for estimating encounters with rogue waves - in bad weather have worked off rule of thumb of a rogue wave every 2000th wave or 1 every 5-6 hours, every 2nd watch or so.

On rogue wave heights, closer to home the 1998 Sydney Hobart race had incredible footage of boats dealing with huge seas south-east of Gabo Island, with anecdotes of 30m rogues common. One of the skippers in the race, Peter Joubert of the boat Kingurra, did an analysis of helicopter altimeter data after the race and found evidence of a wave measured at 140 feet (43m) and one of the highest ever recorded, if not highest. Kingurra was hit hard in the race, knocked down beyond the horizontal by a rogue wave, and lost a man overboard who was rescued ok. The helicopter that did the rescue was measuring its height with altimeter - at one point whilst hovering at 100ft it was confronted with a wall of water heading towards it - it quickly went up 50ft and the wave passed 10ft below it. Joubert worked with the heli pilot after the race and analysed the altimeter data and its reliability - he well qualified for the task as a Professor of Mechanical Eng, and published this analysis in the literature. I have a copy of the paper and can email it in, not sure can post it in here.

Bathymetry and ocean currents off Gabo definitely played a part, with the south flowing current of the EAC running at 4-5 knots at the time of the race and colliding with the SW storm swell, generated by storm force sou'wester that was measured gusting 92 knots at the Prom. South of Gabo is also the Bass Canyon where bathymetry drops out from 100m to around 3000m in short distance. This bathymetery and the EAC are factors in a general view that Eastern Bass Strait, when it blows up, is the roughest part of the Strait.

Thanks for those quantifications, my old boss, experienced sailor, was a navigator in that race, thought they were all going to die...