Michel Olagnon IFREMER Brest, France

Extremes in Action - Symposium for Georg Lindgren 8th-9th June 2006

Michel OlagnonIFREMER

Brest, France

Looking forward to Rogue Waves


Outline

• What is a Rogue Wave ?

• Purpose of the study

• The long-term time-scale

• The medium-term time-scale

• The short-term time-scale

• Who can tell how rogue is the next wave going to be?


What is a rogue wave ?



Reconstructed water surface elevations over a 1000 m span, from T-30s (blue) to T (red) for the New Year Wave.



Georg tells us (Lindgren, 1970) that if it comes from the normal gaussian process, it is a wave that looks in retrospect like the autocorrelation function of the water surface elevation signal.

Sverre (Haver, 2000) states that it is a freak wave if it represents an outlier when seen in view of the population of events generated by a piecewise stationary and homogeneous second order model of the sea surface process.

Miguel and Al (Onorato & Osborne, 2005) tell us that according to the Schrödinger equation, it sucks energy from its neighbors and thus it is a freak invader from an outer statistical population.


It is nice to be able to recognize a Rogue Wave in the statistics after it occurred...

...For various reasons, a much nicer ability would be that of being successful when speculating that approaching waves are not rogue waves, or even that they are.

‘‘ When a woman at a party asks me what I do, I invariably say «I ’m just a speculator.» The encounter ’s over. The only worse conversation stopper is «I ’m just a statistician.» ’’

Victor Niederhoffer, The Education of a Speculator, Wiley, 1997


Purpose of the study

A wave is coming.

In order to predict its rogueness, should we use quasi-deterministically the non-linear Schrödinger equation or merely rely on the statistics derived from, for instance, Slepian processes ?


Discriminating questions:

1. Do we have more high waves than our conventional long-term statistical models predict ?

2. When we do have high waves, do other characteristics of the whole storm, of the sea state, or of the few previous waves look different from those of other storms, sea states, or sets of a few consecutive waves ?

3. Especially, do characteristics related to theoretical deterministic constructions of rogue waves exhibit statistical evidence of predictive power ?


Database:

20 years of data available from Frigg QP platform in the North Sea


Database:

1979-1989: mostly 3-hourly measurements, many time-series available.

1991-1999: mostly 20-minute statistics, only reduced parameters


Database:

Hmax and H1/3 retrieved preferably from the time-series when available (7%), from the statistics elsewhen.

For storms, missing zero-crossing period information was derived from T1/3 (9.4%) and drawn from the empirical H1/3-Tz distribution when no information at all was available (1.7%).

The final database consists of 265147 statistical records, it is thus equivalent to nearly 9 years of continuous measurements.


EKOFISK, operated by ConocoPhillips Laser measurements at the time of the ”Varg incident”

North Sea

Norway


Storm “freakiness”

We (Olagnon & Prevosto, 2005, Olagnon & Magnusson, 2004) try to investigate the widest time-scale: the whole storm.

Especially, the maximum wave expected in a storm is a more useful forecast to seafarers than the maximum wave in some particular 1- or 3-hour duration sea state of that storm.

It may thus appear natural to relate the maximum wave in a storm to the maximum predicted H1/3 in that whole storm rather than to the prevailing H1/3 at the precise instant of Hmax.



Storms are defined as durations > 12 hours with H1/3 > 5m



For each of the 187 identified storms, 1000 random simulations were made using the database statistical parameters and a Jonswap with gamma=3. Second order correction is then applied to all computed Hmax values.

Freakiness of a storm is defined as the quantile of that storm’s observed Hmax/ H1/3max in the corresponding distribution over the 187 actual storms (empirical) and over the 187000 simulated storms (2nd order theory).



QQ-plot of Hmax/ H1/3max = blue dots.

H1/3 = green dots

Hmax = red dots

Apart from a few ones, storms are less “freaky” than 2nd order theory would predict.



QQ-plot of Hmax/ H1/3max = blue dots.

Mean storm BFI = red dots

Benjamin-Feir instability at the time-scale of a storm can only be very weakly related to its “freakiness”.



Expectations based on experience rather than theory would be definitely too low: An explanation for freak waves ?


Medium term: the sea state time scaleFreaky sea states ?

Nerzic & Prevosto (98) proposed a Weibull-Stokes model for the distribution of maximum waves Hmax in a sea state, conditional to H1/3 and Tz of the sea state.

They used a 7% subset of the Frigg database, without any special emphasis on extremes, to derive their model.

We use the full database to study how the model performs with long-term extremes.


Distribution of maximum wave heights

The model applies scaling computed from the transformation between the linear and the non-linear second order models to the parameters of the (asymptotic limit) Gumbel law for the maximum wave in a given sea state


Distribution of maximum wave heights

Comparison of empirical distribution of Hmax with Nerzic & Prevosto model for H1/3>5 m.

No underestimation by model !Again, an appropriate transformation, limited to taking into account standard non-linearities up to second order, is sufficient to explain the observed extremes


Kurtosis and Benjamin-Feir instability

“When a similarity connection is achieved between two objects to 20 decimal places, the greater will move to the lesser”

A.E. Van Vogt, The World of Null-A, 1945

Even though conventional Hmax models seem acceptable for long-term distributions, it might be possible to predict when the extremes in the distribution are most likely to occur : at those times, the similarity between the actual world and the theoretical deterministic world of non-linear Schrödinger equation may be such that we can apply the rules of the latter for some limited time-space window. In that latter world, extremes are governed by Benjamin-Feir instability.


Kurtosis and Benjamin-Feir instability

Benjamin-Feir instability, i.e. the ratio of steepness to bandwidth, and signal kurtosis are strongly related (Mori & Janssen 2005)...

… but are kurtosis (BFI) excursions away from regular values the cause of freak waves, or a mere consequence of their observation ?

In other words, is kurtosis (BFI) a predictor or only a detector ?


Kurtosis and Hmax

Hmax/ H1/3 exhibitsa clear relationship to kurtosis...


Kurtosis and Hmax

…but if “kurtosis” is computed with removal of the largest wave’s time-duration, the relationship can no longer be seen.


What is there to be seen a few waves ahead ?

Instantaneous Benjamin-Feir instability index: nothing.

HH1/3BFI Index



Irregularity factor: nothing.

HH1/3Irr. Fact.



Steepness: let’s have a closer look.

HH1/3Steepness



HH1/3Steepness

CrestH1/3Steepness

NOTHING AGAIN !


Concluding remarks

• Extreme waves are not found more frequently than conventional long-term distribution models predict.

• When extremes are observed, no abnormal characteristic can be found in non-directional parameters at the time scale of the whole storm, of the sea state or of the set of a few consecutive waves. There is nothing more in rogue waves than what we can see in the statistics.

Knowing what he knows, Georg should thus be able to correctly assess the risk of a rogue wave if he wants to go enjoy his retirement on the beach, but...


Freak events do happen !

Please, Georg, don ’t go to a beach in Sicily !Thank you.

The death of Aeschylus was not of his own will; […]. Having come out of the place where he lived in Sicily, he sat under the sun. An eagle carrying a tortoise happened to fly above him. Mistaken by the whiteness of his bald head, it let the tortoise fall on to it, as it would have done to a stone, in order to break it and eat its flesh. The blow took his life away from the poet who first gave the most perfect form to

tragedy.

Valerius Maximus, Factorum ac dictorum memorabilium, IX 12, ca. 30 AD

Documents

Michel Olagnon IFREMER Brest, France