View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CiteSeerX The role of meteorological focusing in generating rogue wave conditions M. A. Donelan1 and A. K. Magnusson2 1RSMAS, University of Miami, Miami , Florida, USA 2Norwegian Meteorological Institute, Berg, Norway Abstract. Rogue waves are believed to be the consequence of focusing of wave energy. While there are several ways that energy may be focused, we concen- trate here on the role of meteorological patterns in generating mixed sea condi- tions. We demonstrate the sharp increase in the probability of high wave crests for a given significant height when the sea is mixed, i.e., consists of wave trains arriving from very different directions. Then, using a full spectral wave predic- tion model forced by NCEP winds on the Atlantic Ocean, we illustrate the role of meteorological focusing in enhancing the probability of occurrence of rogue waves. Introduction peak, this is proportional to the steepness, π H λ p and the phase propagation speed c : The existence of “rogue waves” on the oceans is es- o tablished through the recorded encounters with ships or manned offshore platforms. Seafarers and offshore 2 ⎛⎞π H workers quickly gain a healthy respect for the sea and, Pc∝ ⎜⎟ (1) co⎜⎟ in particular, the energy containing surface gravity ⎝⎠λp waves that, at times, present a hazard to ships and struc- tures. The statistics of wave heights is well-known in a typical narrow banded (in frequency and direction) where H is trough to crest height and λp wavelength of wind driven sea. A wave exceeding the significant the spectral peak waves. height occurs roughly every 7.5 periods, while waves of Thus a rogue wave of double the significant height crest-to-trough extent more than twice the significant would produce a crest horizontal pressure of four times height (defined as “freak”, Kjeldsen, 1996) typically that of the significant height waves. If the rogue wave occur less frequently than one in three thousand waves. is breaking, the water velocity at the crest travels at the If the criterion for freakness is taken to be just 10% phase speed, co producing a further increase of a factor more, i.e., 2.2 H then Raleigh statistics appropriate to s of at least ten, based on the observation that the spectral narrow banded (linear) gaussian seas would suggest that peak waves do not break when π H0.3λ < . Thus a one in 17,000 is closer to the expectation for a very un- p usual (“freak”) wave. For typical storm wave periods of monster wave packing a four-fold wallop may be a haz- 15 seconds, this means that such a wave would occur on ard to ships and offshore structures, but, if it is also average every 70 hours at a given location. breaking, the forty fold increase above the design condi- Thus, the concept of “freakness” is clearly bound up tions will almost certainly produce significant damage. with the expectation of rarity, in the context of local conditions. But is a wave twice as big as the significant Focusing of wave energy height likely to damage a well-designed structure or ship operated in a standard sea-keeping fashion, or are The self-similarity of a wind-generated sea is well- there characteristics of these rogue waves over and established and the significant height H and peak pe- above mere height that render them particularly hazard- s ous? Taking the fluid drag on a fixed piling as an ex- riod Tp and peak wavelength λp are given by: ample of the damaging force per unit area (Pc ) that may be exerted by the crest of waves near the spectral 139 140 DONELAN AND MAGNUSSON 1.65 2 quite well described by the Rayleigh distribution U10 ⎛⎞Cp H0.2s = ⎜⎟ (Longuet-Higgins, 1952) based on the linear superposi- gU ⎝⎠10 tion of random wave components in a narrow-banded U C sea, although various higher order refinements (e.g. T2=×π p (2) p Tayfun, 1983) yield marginally better descriptions of the gU10 2 tail of the distribution. For illustrative purposes in con- 2 U ⎛⎞Cp sidering the additive effect (superposition) of mixed λπp =×2 ⎜⎟ gU⎝⎠10 seas on the statistics of high wave crests, the Rayleigh distribution is adequate. So that the steepness of the peak waves of significant It is important to note that the theoretical height dis- height, SS , is given by: tributions have been derived for the group envelope HS amplitude or equivalently, crest height above mean wa- −0.35 ter level. The theoretical height distribution for linear Hs ⎛⎞Cp SSH ==π 0.1⎜⎟ (3) seas is actually just that of twice the group envelope S λ U p10⎝⎠ magnitude. The group envelope can only be observed C at the crest or trough of the underlying wave making it where p is the wave age. impossible in a dispersive wave field to measure the U10 actual height of a given wave from time series at a These relationships are taken from observations of point. Furthermore in most cases rogue waves are dan- fetch-limited waves (Donelan et al., 1985). Breaking of gerous not because their heights are extraordinary, but the peak waves in a wind generated spectrum occurs for rather because their crest heights are. Nonlinear distor- SS> 0.3. Consequently breaking of these peak waves tions to large waves tend to steepen the crests and flat- will occur when their height exceeds the significant ten the troughs, with little change in the total height. height by this ratio: Consequently, for both theoretical and practical reasons 0.35 it is appropriate to discuss the probability of rogue H ⎛⎞Cp waves in terms of crest heights above mean water level B > 3 (4) ⎜⎟ rather than crest to trough heights. Where we refer to HUS10⎝⎠ height, H this is just double the crest envelope, 2A. In For full development, corresponding to wave age of a wind-sea of mean variance of elevation about the 1.2, the spectral peak waves break when H exceeds 3 mean sea level, σ 2 , the probability of occurrence H , while in strongly forced storm conditions (for S pA( ) of wave crests of height A is wave age of 0.3, say) breaking of the spectral peak waves will occur at more modest heights of 2HS . Wave energy may be focused in deep water by their AA⎧⎫− 2 pA()= 22 exp⎨⎬ (5) interaction with current gradients or simply through the σσ⎩⎭2 propagation of energy from different areas where the meteorological forcing yields waves propagating to- so that the probability of exceedance of any crest height, wards a common destination. This may be through dif- A is ferent meteorological disturbances in the same ocean basin or even within a single storm where the curvature 2 ∞ ⎧⎫−A of the streamlines generates waves propagating from pAdAexp() = ⎨⎬ (6) ∫A 2 one sector of the storm to another producing a mixed ⎩⎭2σ sea of recent swell and local wind sea. This circum- stance is common and can lead to hazardous conditions This is graphed in Figure 1 against the ratio of the ex- with enhanced probability of rogue waves and of break- ceedance of crest height A to σ . The vertical lines on ing rogue waves. the figure (eq. 4) indicate the heights that must be ex- ceeded for breaking of the spectral peak waves to occur. At full development (wave-age = 1.2) this probability is The enhanced probability of rogue waves in essentially zero (10−9 ) , whereas in the teeth of an in- mixed seas tense storm with wave age of 0.3 the probability rises to −4 The statistics of occurrence of wave crest heights (or, 510× or about one breaker every eight hours for 15 equivalently, the group envelope) in a random sea is second peak waves. METEOROLOGICAL FOCUSING 141 22 AA12⎧⎫ A 1 A 2 pA,A()12=−−22 exp⎨⎬ 2 2 (7) σσ12⎩⎭22 σ 1 σ 2 Under the constraint of linear superposition, this is also the probability of occurrence of crest height in the mixed sea, AAAm12= + . Thus, the probability of mixed sea crest height, Am exceeding nσ may readily be computed from the integral of eq. 7: ∞∞ p()A,A12 dAdA 12 (Donelan, 2005b). The ∫∫AnAσ − 11 highest probabilities of extreme waves occur for the case of equal energy components (σ12= σ ) of the mixed sea and these are shown in Figure 1a. It is clear that the intersection of two trains produces an enhance- ment of the probability of extreme crests. The probabil- ity of exceedance of twice the standard deviation in a mixed sea is 2.5 times higher than for a pure wind sea and the enhancement increases with A/σ (Figure 1b). Furthermore these enhanced probabilities of higher crests raises the probability of high waves being also breakers and hence far more dangerous. In these mixed seas, the probability of the spectral peak waves breaking is still negligible for mature seas, but increases to 1/550 Figure 1. (a) the probability of exceedance of crest height for wave age of 0.3. For 15 second waves this trans- in relation to the ratio of height to the standard deviation of lates into a mean time between giant breakers of 2.5 surface elevation for both wind seas (lower curve) and hours. Thus, the maturity of the storms has a direct mixed seas (upper curve). The vertical dashed lines indi- bearing of the danger of the seas, i.e., breaking or non- cate the heights above which breaking of the (spectral) breaking giant waves, but the extreme crest heights are peak waves will occur for the wave ages indicated at the largely dependent on the mixture of focused seas.