2986:
131:. Waves that are asymmetric along the horizontal axis are called skewed waves. Asymmetry along the horizontal axis indicates that the wave crest deviates from the wave trough in terms of duration and height. Generally, skewed waves have a short and high wave crest and a long and flat wave trough. A skewed wave shape results in larger orbital velocities under the wave crest compared to smaller orbital velocities under the wave trough. For waves having the same velocity variance, the ones with higher skewness result in a larger net
3010:
2050:
3022:
2938:
2959:
901:, the skewness is maximum and the asymmetry is small and the waves have a skewed shape. For large Ursell numbers, the skewness approaches 0 and the asymmetry is maximum, resulting in an asymmetric wave shape. In this way, if the wave shape is known, the Ursell number can be predicted and consequently the size and direction of sediment transport at a certain location can be predicted.
2998:
100:
2039:
2948:
868:
973:
model with a focus on the deep ocean. SWAN is a nearshore model and mainly has coastal applications. Advantages of phase-averaged models are that they compute wave characteristics over a large domain, they are fast and they can be coupled to sediment transport models, which is an efficient tool to
143:
Waves that are asymmetric along the vertical axis are referred to as asymmetric waves. Wave asymmetry indicates the leaning forward or backward of the wave, with a steep front face and a gentle rear face. A steep front correlates with an upward tilt, a steep back is correlated with a downward tilt.
931:
as the high velocities under the crest are much more capable of moving large sediments. Beneath waves with high asymmetry, the change from onshore to offshore flow is more gradual than from offshore to onshore, where sediments are stirred up during peaks in offshore velocity and are transported
235:
956:
approach. With the phase-averaged approach, wave skewness and asymmetry are included based on parameterizations. Phase-averaged models incorporate the evolution of wave frequency and direction in space and time of the wave spectrum. Examples of these kinds of models are
913:
and swash zone. In the shoaling zone, the wave nonlinearity increases due to the decreasing depth and the sinusoidal waves approaching the coast will transform into skewed waves. As waves propagate further towards the coast, the wave shape becomes more asymmetric due to
334:
701:
230:
422:
Values for the skewness are positive with typical values between 0 and 1, where values of 1 indicate high skewness. Values for asymmetry are negative with typical values between -1.5 and 0, where values of -1.5 indicate high asymmetry.
111:) are waves having equal height and duration during the crest and the trough, and they can be mirrored in both the crest and the trough. Due to Non-linear effects, waves can transform from sinusoidal to a skewed and asymmetric shape.
1361:
144:
The duration and height of the wave-crest equal the duration and height of the wave-trough. An asymmetric wave shape results in a larger acceleration between trough and crest and a smaller acceleration between crest and trough.
926:
in shallow conditions, where it both affects the bedload transport as the suspended load transport. Skewed waves have higher flow velocities under the crest of the waves than under the trough, resulting in a net onshore
60:
and crest length. For practical engineering purposes, it is important to know the probability of these wave characteristics in seas and oceans at a given place and time. This knowledge is crucial for the prediction of
921:
Skewness and asymmetry are not only observed in the shape of the wave, but also in the orbital velocity profiles beneath the waves. The skewed and asymmetric velocity profiles have important implications for
36:
or a shallow area. As waves shoal in the nearshore zone, in addition to their wavelength and height changing, their asymmetry and skewness also change. Wave skewness and asymmetry are often implicated in
514:
872:
For small Ursell numbers, the skewness and asymmetry both approach zero and the waves have a sinusoidal shape, and thus waves having small Ursell numbers do not result in net sediment transport. For
248:
416:
863:{\displaystyle {\textrm {where}}\;B={\frac {0.857}{1+\exp({\frac {-0.471-\log(Ur)}{0.297}})}},\;{\textrm {and}}\;\psi =-90^{\circ }+90^{\circ }\tanh({\frac {0.8150}{Ur^{0.672}}})}
695:
647:
385:
2678:
899:
76:
data shows geographically coherent skewness fields in the ocean and from the data has been concluded that large values of skewness occur primarily in regions of large
157:
547:
2668:
1727:
962:
359:
595:
571:
2584:
435:, relates the skewness and asymmetry and quantifies the degree of sea surface elevation nonlinearity. Ruessink et al. defined the Ursell number as:
1999:
1767:
2231:
2121:
1720:
2826:
2253:
2141:
1213:
Elgar, Steve, and R. T. Guza. "Shoaling gravity waves: Comparisons between field observations, linear theory, and a nonlinear model."
2673:
1944:
32:, the two categories of nonlinearity are skewness and asymmetry. Wave skewness and asymmetry occur when waves encounter an opposing
2091:
440:
152:
Skewness (Sk) and asymmetry (As) are measures of the wave nonlinearity and can be described in terms of the following parameters:
2131:
2951:
2861:
1847:
2534:
2941:
1713:
1036:
966:
2976:
1989:
2049:
2186:
1605:
Ruessink, B. G.; Michallet, H.; Abreu, T.; Sancho, F.; A, D. A. Van der; Werf, J. J. Van der; Silva, P. A. (2011).
1454:
Ruessink, B. G.; Michallet, H.; Abreu, T.; Sancho, F.; A, D. A. Van der; Werf, J. J. Van der; Silva, P. A. (2011).
2086:
2721:
2126:
329:{\displaystyle As={\frac {\langle {\mathcal {H}}(\eta )^{3}\rangle }{\langle \eta ^{2}\rangle ^{\frac {3}{2}}}}}
2851:
2226:
2216:
2156:
1792:
1762:
1672:"On the parameterization of the free-stream non-linear wave orbital motion in nearshore morphodynamic models"
1417:"On the parameterization of the free-stream non-linear wave orbital motion in nearshore morphodynamic models"
2888:
2871:
2708:
2201:
2066:
2004:
1994:
1887:
550:
395:
77:
1267:
3042:
2883:
2821:
2248:
1934:
1671:
1521:"Parametrization of orbital velocity asymmetries of shoaling and breaking waves using bispectral analysis"
1416:
1228:
1073:"Parametrization of orbital velocity asymmetries of shoaling and breaking waves using bispectral analysis"
975:
1607:"Observations of velocities, sand concentrations, and fluxes under velocity-asymmetric oscillatory flows"
1520:
1456:"Observations of velocities, sand concentrations, and fluxes under velocity-asymmetric oscillatory flows"
1072:
2990:
2716:
2698:
2206:
2101:
1736:
1044:
66:
38:
25:
1606:
1455:
1160:
600:
The skewness and asymmetry at a certain location nearshore can be predicted from the Ursell number by:
653:
605:
2903:
2736:
2439:
2296:
2161:
1872:
1618:
1571:
1467:
1373:
1172:
1123:
953:
84:
17:
1110:
GĂłmez-Enri, J.; Gommenginger, C. P.; Srokosz, M. A.; Challenor, P. G.; Benveniste, J. (2007-06-01).
949:
366:
3009:
2898:
2783:
2778:
2504:
2176:
2136:
1852:
128:
42:
29:
2841:
2554:
2544:
2509:
2409:
2394:
2291:
1652:
1587:
1501:
1397:
1303:
1196:
1002:
928:
923:
132:
120:
108:
88:
936:
bar formation and provides a mechanism for the generation of three-dimensional features such as
225:{\displaystyle Sk={\frac {\langle \eta ^{3}\rangle }{\langle \eta ^{2}\rangle ^{\frac {3}{2}}}}}
1266:
Dugdale, Hannah L.; Macdonald, David W.; Pope, Lisa C.; Johnson, Paul J.; Burke, Terry (2008).
932:
onshore because of the sudden change in flow direction. The local sediment transport generates
2923:
2913:
2856:
2836:
2519:
2484:
2419:
2399:
2389:
2271:
1817:
1691:
1644:
1540:
1493:
1436:
1389:
1342:
1295:
1287:
1248:
1188:
1141:
1092:
987:
875:
388:
239:
1021:
Elgar, Steve, and R. T. Guza. "Observations of bispectra of shoaling surface gravity waves."
2878:
2846:
2816:
2625:
2610:
2479:
2414:
2306:
2221:
2151:
2076:
1857:
1827:
1757:
1752:
1683:
1634:
1626:
1579:
1532:
1483:
1475:
1428:
1381:
1334:
1279:
1240:
1180:
1131:
1084:
522:
234:
344:
2683:
2579:
2529:
2494:
2454:
2346:
2316:
2166:
2116:
2026:
1984:
1917:
1842:
1802:
1057:
970:
958:
33:
2038:
1622:
1575:
1471:
1377:
1176:
1127:
3026:
3014:
2793:
2788:
2693:
2688:
2524:
2464:
2459:
2191:
2081:
1902:
1837:
1812:
580:
556:
1558:
Ribberink, J. S.; Werf, J. J. van der; O'Donoghue, T.; Hassan, W. N. M. (2008-01-01).
1244:
3036:
2963:
2811:
2731:
2620:
2539:
2514:
2449:
2379:
2286:
2181:
2058:
1979:
1939:
1912:
1822:
1772:
1687:
1591:
1536:
1432:
1401:
1283:
1268:"Reproductive skew and relatedness in social groups of European badgers, Meles meles"
1200:
1088:
933:
915:
1656:
1505:
1307:
99:
3021:
3002:
2958:
2918:
2866:
2806:
2757:
2635:
2630:
2605:
2589:
2564:
2281:
2171:
2111:
1897:
1807:
1782:
432:
2908:
2640:
2569:
2434:
2374:
2341:
2331:
2326:
2211:
2146:
2106:
2096:
2071:
1954:
1927:
1907:
1867:
1832:
997:
937:
53:
1338:
1322:
1161:"Modeling sediment transport beneath skewed asymmetric waves above a plane bed"
2726:
2574:
2549:
2444:
2424:
2351:
2336:
2321:
2311:
2276:
2196:
2016:
2011:
1974:
1969:
1964:
1862:
1583:
1385:
992:
574:
124:
62:
57:
1695:
1648:
1544:
1497:
1440:
1393:
1346:
1291:
1252:
1192:
1145:
1096:
238:
Skewness (top) and
Asymmetry (bottom) plotted against the Ursell number on a
2997:
2798:
2660:
2645:
2559:
2404:
2243:
2238:
2021:
1949:
1877:
1797:
1787:
1744:
1559:
910:
70:
49:
21:
1323:"Relationships involving third moments and bispectra of a harmonic process"
1299:
1112:"Measuring Global Ocean Wave Skewness by Retracking RA-2 Envisat Waveforms"
2893:
2615:
2474:
2366:
2356:
2301:
1777:
1630:
1560:"Sand motion induced by oscillatory flows: Sheet flow and vortex ripples"
1479:
1184:
1705:
1136:
1111:
918:
in the surf zone until the waves run up on the beach in the swash zone.
2762:
2752:
1922:
1892:
73:
2469:
1882:
1639:
1488:
46:
948:
Two different approaches exist to include wave shape in models: the
127:, skewness refers to a distortion or asymmetry that deviates from a
2831:
2650:
2429:
2384:
1159:
Ruessink, B. G.; Berg, T. J. J. van den; Rijn, L. C. van (2009).
2263:
1709:
1366:
Mathematical
Proceedings of the Cambridge Philosophical Society
1327:
IEEE Transactions on
Acoustics, Speech, and Signal Processing
1670:
Ruessink, B.G.; Ramaekers, G.; Van Rijn, L.C. (2012-07-01).
1415:
Ruessink, B.G.; Ramaekers, G.; Van Rijn, L.C. (2012-07-01).
509:{\displaystyle Ur={\frac {3}{8}}{\frac {H_{m0}k}{(kh)^{3}}}}
372:
269:
1362:"The long-wave paradox in the theory of gravity waves"
1229:"Transport rate under irregular sheet flow conditions"
909:
The nearshore zone is divided into the shoaling zone,
2974:
878:
704:
656:
608:
583:
559:
525:
443:
418:
the angle brackets indicate averaging over many waves
398:
369:
347:
251:
160:
103:
a) sinusoidal, b) skewed and c) asymmetric wave shape
2771:
2745:
2707:
2659:
2598:
2493:
2365:
2262:
2057:
1743:
1227:Dibajnia, Mohammad; Watanabe, Akira (1998-11-01).
893:
862:
689:
641:
589:
565:
541:
508:
410:
379:
353:
328:
224:
2679:North West Shelf Operational Oceanographic System
83:At the nearshore zone, skewness and asymmetry of
1037:"The effects of currents on wave nonlinearities"
2669:Deep-ocean Assessment and Reporting of Tsunamis
52:, in particular regarding the distribution of
1721:
1116:Journal of Atmospheric and Oceanic Technology
8:
944:Models including wave skewness and asymmetry
405:
399:
309:
295:
290:
264:
205:
191:
186:
173:
1728:
1714:
1706:
793:
785:
712:
1638:
1519:Doering, J.C.; Bowen, A.J. (1995-09-01).
1487:
1135:
1071:Doering, J.C.; Bowen, A.J. (1995-09-01).
877:
848:
835:
820:
807:
787:
786:
740:
719:
706:
705:
703:
655:
607:
582:
558:
530:
524:
497:
470:
463:
453:
442:
397:
371:
370:
368:
346:
312:
302:
284:
268:
267:
261:
250:
208:
198:
180:
170:
159:
233:
98:
2981:
1611:Journal of Geophysical Research: Oceans
1460:Journal of Geophysical Research: Oceans
1165:Journal of Geophysical Research: Oceans
1014:
361:is the zero-mean wave surface elevation
2000:one-dimensional Saint-Venant equations
1053:
1042:
411:{\displaystyle \langle \cdot \rangle }
7:
2947:
65:, which are a danger for ships and
2827:National Oceanographic Data Center
2254:World Ocean Circulation Experiment
2142:Global Ocean Data Analysis Project
20:refers to their deviations from a
14:
2674:Global Sea Level Observing System
3020:
3008:
2996:
2984:
2957:
2946:
2937:
2936:
2132:Geochemical Ocean Sections Study
2048:
2037:
1688:10.1016/j.coastaleng.2012.03.006
1433:10.1016/j.coastaleng.2012.03.006
1284:10.1111/j.1365-294X.2008.03708.x
690:{\displaystyle As=B*\sin(\psi )}
642:{\displaystyle Sk=B*\cos(\psi )}
2862:Ocean thermal energy conversion
2585:Vine–Matthews–Morley hypothesis
431:The Ursell number, named after
857:
832:
776:
767:
758:
737:
684:
678:
636:
630:
494:
484:
380:{\displaystyle {\mathcal {H}}}
281:
274:
1:
1245:10.1016/S0378-3839(98)00034-9
2122:El Niño–Southern Oscillation
2092:Craik–Leibovich vortex force
1848:Luke's variational principle
1537:10.1016/0378-3839(95)00007-X
1089:10.1016/0378-3839(95)00007-X
1035:van de Ven, Maartje (2018).
905:Impact on sediment transport
1360:Ursell, F. (October 1953).
1321:Elgar, S. (December 1987).
3059:
2187:Ocean dynamical thermostat
2035:
1339:10.1109/TASSP.1987.1165090
1215:Journal of fluid mechanics
1023:Journal of Fluid Mechanics
969:). WAVEWATCH3 is a global
2932:
2722:Ocean acoustic tomography
2535:Mohorovičić discontinuity
2127:General circulation model
1763:Benjamin–Feir instability
1584:10.1080/14685240802220009
1386:10.1017/S0305004100028887
597:is the mean water depth.
87:are the main drivers for
2852:Ocean surface topography
2227:Thermohaline circulation
2217:Subsurface ocean current
2157:Hydrothermal circulation
1990:Wave–current interaction
1768:Boussinesq approximation
894:{\displaystyle Ur\sim 1}
148:Mathematical description
24:shape. In the fields of
2889:Sea surface temperature
2872:Outline of oceanography
2067:Atmospheric circulation
2005:shallow water equations
1995:Waves and shallow water
1888:Significant wave height
551:significant wave height
78:significant wave height
2884:Sea surface microlayer
2249:Wind generated current
1052:Cite journal requires
1025:161.1 (1985): 425-448.
895:
864:
691:
643:
591:
567:
543:
542:{\displaystyle H_{m0}}
510:
412:
381:
355:
330:
243:
226:
104:
95:Skewness and asymmetry
2717:Deep scattering layer
2699:World Geodetic System
2207:Princeton Ocean Model
2087:Coriolis–Stokes force
1737:Physical oceanography
1564:Journal of Turbulence
896:
865:
692:
644:
592:
568:
544:
511:
413:
382:
356:
354:{\displaystyle \eta }
331:
237:
227:
107:Sinusoidal waves (or
102:
85:surface gravity waves
45:for the modelling of
26:physical oceanography
18:surface gravity waves
2737:Underwater acoustics
2297:Perigean spring tide
2162:Langmuir circulation
1873:Rossby-gravity waves
1631:10.1029/2010JC006443
1480:10.1029/2010JC006443
1185:10.1029/2009JC005416
876:
702:
654:
606:
581:
557:
523:
441:
396:
367:
345:
249:
158:
16:The nonlinearity of
2899:Science On a Sphere
2505:Convergent boundary
2177:Modular Ocean Model
2137:Geostrophic current
1853:Mild-slope equation
1676:Coastal Engineering
1623:2011JGRC..116.3004R
1576:2008JTurb...9...20R
1525:Coastal Engineering
1472:2011JGRC..116.3004R
1421:Coastal Engineering
1378:1953PCPS...49..685U
1233:Coastal Engineering
1177:2009JGRC..11411021R
1137:10.1175/JTECH2014.1
1128:2007JAtOT..24.1102G
1077:Coastal Engineering
940:and rhythmic bars.
129:normal distribution
67:offshore structures
43:coastal engineering
30:coastal engineering
2555:Seafloor spreading
2545:Outer trench swell
2510:Divergent boundary
2410:Continental margin
2395:Carbonate platform
2292:Lunitidal interval
1217:158 (1985): 47-70.
1003:Sediment transport
988:Infragravity waves
929:sediment transport
924:sediment transport
891:
860:
687:
639:
587:
563:
539:
506:
408:
377:
351:
326:
244:
222:
133:sediment transport
121:probability theory
105:
89:sediment transport
2972:
2971:
2964:Oceans portal
2924:World Ocean Atlas
2914:Underwater glider
2857:Ocean temperature
2520:Hydrothermal vent
2485:Submarine volcano
2420:Continental shelf
2400:Coastal geography
2390:Bathymetric chart
2272:Amphidromic point
1960:Wave nonlinearity
1818:Infragravity wave
1333:(12): 1725–1726.
1272:Molecular Ecology
952:approach and the
855:
790:
780:
774:
709:
590:{\displaystyle h}
566:{\displaystyle k}
504:
461:
389:Hilbert transform
324:
320:
240:logarithmic scale
220:
216:
39:ocean engineering
3050:
3025:
3024:
3013:
3012:
3001:
3000:
2989:
2988:
2987:
2980:
2962:
2961:
2950:
2949:
2940:
2939:
2879:Pelagic sediment
2817:Marine pollution
2611:Deep ocean water
2480:Submarine canyon
2415:Continental rise
2307:Rule of twelfths
2222:Sverdrup balance
2152:Humboldt Current
2077:Boundary current
2052:
2041:
1858:Radiation stress
1828:Iribarren number
1803:Equatorial waves
1758:Ballantine scale
1753:Airy wave theory
1730:
1723:
1716:
1707:
1700:
1699:
1667:
1661:
1660:
1642:
1602:
1596:
1595:
1555:
1549:
1548:
1516:
1510:
1509:
1491:
1451:
1445:
1444:
1412:
1406:
1405:
1357:
1351:
1350:
1318:
1312:
1311:
1278:(7): 1815–1827.
1263:
1257:
1256:
1224:
1218:
1211:
1205:
1204:
1156:
1150:
1149:
1139:
1122:(6): 1102–1116.
1107:
1101:
1100:
1068:
1062:
1061:
1055:
1050:
1048:
1040:
1032:
1026:
1019:
971:wave forecasting
900:
898:
897:
892:
869:
867:
866:
861:
856:
854:
853:
852:
836:
825:
824:
812:
811:
792:
791:
788:
781:
779:
775:
770:
741:
720:
711:
710:
707:
696:
694:
693:
688:
648:
646:
645:
640:
596:
594:
593:
588:
572:
570:
569:
564:
548:
546:
545:
540:
538:
537:
515:
513:
512:
507:
505:
503:
502:
501:
482:
478:
477:
464:
462:
454:
417:
415:
414:
409:
386:
384:
383:
378:
376:
375:
360:
358:
357:
352:
335:
333:
332:
327:
325:
323:
322:
321:
313:
307:
306:
293:
289:
288:
273:
272:
262:
231:
229:
228:
223:
221:
219:
218:
217:
209:
203:
202:
189:
185:
184:
171:
139:Asymmetric waves
3058:
3057:
3053:
3052:
3051:
3049:
3048:
3047:
3033:
3032:
3031:
3019:
3007:
2995:
2985:
2983:
2975:
2973:
2968:
2956:
2928:
2767:
2741:
2703:
2684:Sea-level curve
2655:
2594:
2580:Transform fault
2530:Mid-ocean ridge
2496:
2489:
2455:Oceanic plateau
2361:
2347:Tidal resonance
2317:Theory of tides
2258:
2167:Longshore drift
2117:Ekman transport
2053:
2047:
2046:
2045:
2044:
2043:
2042:
2033:
1985:Wave turbulence
1918:Trochoidal wave
1843:Longshore drift
1739:
1734:
1704:
1703:
1669:
1668:
1664:
1604:
1603:
1599:
1557:
1556:
1552:
1518:
1517:
1513:
1453:
1452:
1448:
1414:
1413:
1409:
1359:
1358:
1354:
1320:
1319:
1315:
1265:
1264:
1260:
1226:
1225:
1221:
1212:
1208:
1158:
1157:
1153:
1109:
1108:
1104:
1070:
1069:
1065:
1051:
1041:
1034:
1033:
1029:
1020:
1016:
1011:
984:
954:phase-resolving
946:
907:
874:
873:
844:
840:
816:
803:
742:
724:
700:
699:
652:
651:
604:
603:
579:
578:
555:
554:
526:
521:
520:
493:
483:
466:
465:
439:
438:
429:
394:
393:
365:
364:
343:
342:
308:
298:
294:
280:
263:
247:
246:
204:
194:
190:
176:
172:
156:
155:
150:
141:
117:
97:
12:
11:
5:
3056:
3054:
3046:
3045:
3035:
3034:
3030:
3029:
3017:
3005:
2993:
2991:Earth sciences
2970:
2969:
2967:
2966:
2954:
2944:
2933:
2930:
2929:
2927:
2926:
2921:
2916:
2911:
2906:
2904:Stratification
2901:
2896:
2891:
2886:
2881:
2876:
2875:
2874:
2864:
2859:
2854:
2849:
2844:
2839:
2834:
2829:
2824:
2819:
2814:
2809:
2804:
2796:
2794:Color of water
2791:
2789:Benthic lander
2786:
2781:
2775:
2773:
2769:
2768:
2766:
2765:
2760:
2755:
2749:
2747:
2743:
2742:
2740:
2739:
2734:
2729:
2724:
2719:
2713:
2711:
2705:
2704:
2702:
2701:
2696:
2694:Sea level rise
2691:
2689:Sea level drop
2686:
2681:
2676:
2671:
2665:
2663:
2657:
2656:
2654:
2653:
2648:
2643:
2638:
2633:
2628:
2623:
2618:
2613:
2608:
2602:
2600:
2596:
2595:
2593:
2592:
2587:
2582:
2577:
2572:
2567:
2562:
2557:
2552:
2547:
2542:
2537:
2532:
2527:
2525:Marine geology
2522:
2517:
2512:
2507:
2501:
2499:
2491:
2490:
2488:
2487:
2482:
2477:
2472:
2467:
2465:Passive margin
2462:
2460:Oceanic trench
2457:
2452:
2447:
2442:
2437:
2432:
2427:
2422:
2417:
2412:
2407:
2402:
2397:
2392:
2387:
2382:
2377:
2371:
2369:
2363:
2362:
2360:
2359:
2354:
2349:
2344:
2339:
2334:
2329:
2324:
2319:
2314:
2309:
2304:
2299:
2294:
2289:
2284:
2279:
2274:
2268:
2266:
2260:
2259:
2257:
2256:
2251:
2246:
2241:
2236:
2235:
2234:
2224:
2219:
2214:
2209:
2204:
2199:
2194:
2192:Ocean dynamics
2189:
2184:
2179:
2174:
2169:
2164:
2159:
2154:
2149:
2144:
2139:
2134:
2129:
2124:
2119:
2114:
2109:
2104:
2099:
2094:
2089:
2084:
2082:Coriolis force
2079:
2074:
2069:
2063:
2061:
2055:
2054:
2036:
2034:
2032:
2031:
2030:
2029:
2019:
2014:
2009:
2008:
2007:
2002:
1992:
1987:
1982:
1977:
1972:
1967:
1962:
1957:
1952:
1947:
1942:
1937:
1932:
1931:
1930:
1920:
1915:
1910:
1905:
1903:Stokes problem
1900:
1895:
1890:
1885:
1880:
1875:
1870:
1865:
1860:
1855:
1850:
1845:
1840:
1838:Kinematic wave
1835:
1830:
1825:
1820:
1815:
1810:
1805:
1800:
1795:
1790:
1785:
1780:
1775:
1770:
1765:
1760:
1755:
1749:
1747:
1741:
1740:
1735:
1733:
1732:
1725:
1718:
1710:
1702:
1701:
1662:
1597:
1550:
1531:(1–2): 15–33.
1511:
1446:
1407:
1372:(4): 685–694.
1352:
1313:
1258:
1239:(3): 167–183.
1219:
1206:
1151:
1102:
1083:(1–2): 15–33.
1063:
1054:|journal=
1027:
1013:
1012:
1010:
1007:
1006:
1005:
1000:
995:
990:
983:
980:
976:morphodynamics
950:phase-averaged
945:
942:
906:
903:
890:
887:
884:
881:
859:
851:
847:
843:
839:
834:
831:
828:
823:
819:
815:
810:
806:
802:
799:
796:
784:
778:
773:
769:
766:
763:
760:
757:
754:
751:
748:
745:
739:
736:
733:
730:
727:
723:
718:
715:
686:
683:
680:
677:
674:
671:
668:
665:
662:
659:
638:
635:
632:
629:
626:
623:
620:
617:
614:
611:
586:
562:
536:
533:
529:
500:
496:
492:
489:
486:
481:
476:
473:
469:
460:
457:
452:
449:
446:
428:
425:
420:
419:
407:
404:
401:
391:
374:
362:
350:
319:
316:
311:
305:
301:
297:
292:
287:
283:
279:
276:
271:
266:
260:
257:
254:
215:
212:
207:
201:
197:
193:
188:
183:
179:
175:
169:
166:
163:
149:
146:
140:
137:
116:
113:
96:
93:
13:
10:
9:
6:
4:
3:
2:
3055:
3044:
3043:Gravity waves
3041:
3040:
3038:
3028:
3023:
3018:
3016:
3011:
3006:
3004:
2999:
2994:
2992:
2982:
2978:
2965:
2960:
2955:
2953:
2945:
2943:
2935:
2934:
2931:
2925:
2922:
2920:
2917:
2915:
2912:
2910:
2907:
2905:
2902:
2900:
2897:
2895:
2892:
2890:
2887:
2885:
2882:
2880:
2877:
2873:
2870:
2869:
2868:
2865:
2863:
2860:
2858:
2855:
2853:
2850:
2848:
2845:
2843:
2840:
2838:
2835:
2833:
2830:
2828:
2825:
2823:
2820:
2818:
2815:
2813:
2812:Marine energy
2810:
2808:
2805:
2803:
2802:
2797:
2795:
2792:
2790:
2787:
2785:
2782:
2780:
2779:Acidification
2777:
2776:
2774:
2770:
2764:
2761:
2759:
2756:
2754:
2751:
2750:
2748:
2744:
2738:
2735:
2733:
2732:SOFAR channel
2730:
2728:
2725:
2723:
2720:
2718:
2715:
2714:
2712:
2710:
2706:
2700:
2697:
2695:
2692:
2690:
2687:
2685:
2682:
2680:
2677:
2675:
2672:
2670:
2667:
2666:
2664:
2662:
2658:
2652:
2649:
2647:
2644:
2642:
2639:
2637:
2634:
2632:
2629:
2627:
2624:
2622:
2619:
2617:
2614:
2612:
2609:
2607:
2604:
2603:
2601:
2597:
2591:
2588:
2586:
2583:
2581:
2578:
2576:
2573:
2571:
2568:
2566:
2563:
2561:
2558:
2556:
2553:
2551:
2548:
2546:
2543:
2541:
2540:Oceanic crust
2538:
2536:
2533:
2531:
2528:
2526:
2523:
2521:
2518:
2516:
2515:Fracture zone
2513:
2511:
2508:
2506:
2503:
2502:
2500:
2498:
2492:
2486:
2483:
2481:
2478:
2476:
2473:
2471:
2468:
2466:
2463:
2461:
2458:
2456:
2453:
2451:
2450:Oceanic basin
2448:
2446:
2443:
2441:
2438:
2436:
2433:
2431:
2428:
2426:
2423:
2421:
2418:
2416:
2413:
2411:
2408:
2406:
2403:
2401:
2398:
2396:
2393:
2391:
2388:
2386:
2383:
2381:
2380:Abyssal plain
2378:
2376:
2373:
2372:
2370:
2368:
2364:
2358:
2355:
2353:
2350:
2348:
2345:
2343:
2340:
2338:
2335:
2333:
2330:
2328:
2325:
2323:
2320:
2318:
2315:
2313:
2310:
2308:
2305:
2303:
2300:
2298:
2295:
2293:
2290:
2288:
2287:Internal tide
2285:
2283:
2280:
2278:
2275:
2273:
2270:
2269:
2267:
2265:
2261:
2255:
2252:
2250:
2247:
2245:
2242:
2240:
2237:
2233:
2230:
2229:
2228:
2225:
2223:
2220:
2218:
2215:
2213:
2210:
2208:
2205:
2203:
2200:
2198:
2195:
2193:
2190:
2188:
2185:
2183:
2182:Ocean current
2180:
2178:
2175:
2173:
2170:
2168:
2165:
2163:
2160:
2158:
2155:
2153:
2150:
2148:
2145:
2143:
2140:
2138:
2135:
2133:
2130:
2128:
2125:
2123:
2120:
2118:
2115:
2113:
2110:
2108:
2105:
2103:
2100:
2098:
2095:
2093:
2090:
2088:
2085:
2083:
2080:
2078:
2075:
2073:
2070:
2068:
2065:
2064:
2062:
2060:
2056:
2051:
2040:
2028:
2025:
2024:
2023:
2020:
2018:
2015:
2013:
2010:
2006:
2003:
2001:
1998:
1997:
1996:
1993:
1991:
1988:
1986:
1983:
1981:
1980:Wave shoaling
1978:
1976:
1973:
1971:
1968:
1966:
1963:
1961:
1958:
1956:
1953:
1951:
1948:
1946:
1943:
1941:
1940:Ursell number
1938:
1936:
1933:
1929:
1926:
1925:
1924:
1921:
1919:
1916:
1914:
1911:
1909:
1906:
1904:
1901:
1899:
1896:
1894:
1891:
1889:
1886:
1884:
1881:
1879:
1876:
1874:
1871:
1869:
1866:
1864:
1861:
1859:
1856:
1854:
1851:
1849:
1846:
1844:
1841:
1839:
1836:
1834:
1831:
1829:
1826:
1824:
1823:Internal wave
1821:
1819:
1816:
1814:
1811:
1809:
1806:
1804:
1801:
1799:
1796:
1794:
1791:
1789:
1786:
1784:
1781:
1779:
1776:
1774:
1773:Breaking wave
1771:
1769:
1766:
1764:
1761:
1759:
1756:
1754:
1751:
1750:
1748:
1746:
1742:
1738:
1731:
1726:
1724:
1719:
1717:
1712:
1711:
1708:
1697:
1693:
1689:
1685:
1681:
1677:
1673:
1666:
1663:
1658:
1654:
1650:
1646:
1641:
1636:
1632:
1628:
1624:
1620:
1616:
1612:
1608:
1601:
1598:
1593:
1589:
1585:
1581:
1577:
1573:
1569:
1565:
1561:
1554:
1551:
1546:
1542:
1538:
1534:
1530:
1526:
1522:
1515:
1512:
1507:
1503:
1499:
1495:
1490:
1485:
1481:
1477:
1473:
1469:
1465:
1461:
1457:
1450:
1447:
1442:
1438:
1434:
1430:
1426:
1422:
1418:
1411:
1408:
1403:
1399:
1395:
1391:
1387:
1383:
1379:
1375:
1371:
1367:
1363:
1356:
1353:
1348:
1344:
1340:
1336:
1332:
1328:
1324:
1317:
1314:
1309:
1305:
1301:
1297:
1293:
1289:
1285:
1281:
1277:
1273:
1269:
1262:
1259:
1254:
1250:
1246:
1242:
1238:
1234:
1230:
1223:
1220:
1216:
1210:
1207:
1202:
1198:
1194:
1190:
1186:
1182:
1178:
1174:
1170:
1166:
1162:
1155:
1152:
1147:
1143:
1138:
1133:
1129:
1125:
1121:
1117:
1113:
1106:
1103:
1098:
1094:
1090:
1086:
1082:
1078:
1074:
1067:
1064:
1059:
1046:
1038:
1031:
1028:
1024:
1018:
1015:
1008:
1004:
1001:
999:
996:
994:
991:
989:
986:
985:
981:
979:
977:
972:
968:
964:
960:
955:
951:
943:
941:
939:
935:
930:
925:
919:
917:
916:wave breaking
912:
904:
902:
888:
885:
882:
879:
870:
849:
845:
841:
837:
829:
826:
821:
817:
813:
808:
804:
800:
797:
794:
782:
771:
764:
761:
755:
752:
749:
746:
743:
734:
731:
728:
725:
721:
716:
713:
697:
681:
675:
672:
669:
666:
663:
660:
657:
649:
633:
627:
624:
621:
618:
615:
612:
609:
601:
598:
584:
576:
573:is the local
560:
552:
549:is the local
534:
531:
527:
517:
498:
490:
487:
479:
474:
471:
467:
458:
455:
450:
447:
444:
436:
434:
427:Ursell number
426:
424:
402:
392:
390:
363:
348:
341:
340:
339:
336:
317:
314:
303:
299:
285:
277:
258:
255:
252:
241:
236:
232:
213:
210:
199:
195:
181:
177:
167:
164:
161:
153:
147:
145:
138:
136:
134:
130:
126:
122:
114:
112:
110:
101:
94:
92:
90:
86:
81:
79:
75:
72:
68:
64:
63:extreme waves
59:
55:
51:
48:
44:
40:
35:
31:
27:
23:
19:
2919:Water column
2867:Oceanography
2842:Observations
2837:Explorations
2807:Marginal sea
2800:
2758:OSTM/Jason-2
2590:Volcanic arc
2565:Slab suction
2282:Head of tide
2172:Loop Current
2112:Ekman spiral
1959:
1898:Stokes drift
1808:Gravity wave
1783:Cnoidal wave
1679:
1675:
1665:
1614:
1610:
1600:
1567:
1563:
1553:
1528:
1524:
1514:
1463:
1459:
1449:
1424:
1420:
1410:
1369:
1365:
1355:
1330:
1326:
1316:
1275:
1271:
1261:
1236:
1232:
1222:
1214:
1209:
1168:
1164:
1154:
1119:
1115:
1105:
1080:
1076:
1066:
1045:cite journal
1030:
1022:
1017:
965:) and SWAN (
947:
938:rip currents
920:
908:
871:
698:
650:
602:
599:
518:
437:
433:Fritz Ursell
430:
421:
337:
245:
154:
151:
142:
118:
115:Skewed waves
109:linear waves
106:
82:
74:Envisat RA-2
69:. Satellite
15:
2909:Thermocline
2626:Mesopelagic
2599:Ocean zones
2570:Slab window
2435:Hydrography
2375:Abyssal fan
2342:Tidal range
2332:Tidal power
2327:Tidal force
2212:Rip current
2147:Gulf Stream
2107:Ekman layer
2097:Downwelling
2072:Baroclinity
2059:Circulation
1955:Wave height
1945:Wave action
1928:megatsunami
1908:Stokes wave
1868:Rossby wave
1833:Kelvin wave
1813:Green's law
998:Wind stress
54:wave height
2847:Reanalysis
2746:Satellites
2727:Sofar bomb
2575:Subduction
2550:Ridge push
2445:Ocean bank
2425:Contourite
2352:Tide gauge
2337:Tidal race
2322:Tidal bore
2312:Slack tide
2277:Earth tide
2197:Ocean gyre
2017:Wind setup
2012:Wind fetch
1975:Wave setup
1970:Wave radar
1965:Wave power
1863:Rogue wave
1793:Dispersion
1009:References
993:Wind waves
959:WAVEWATCH3
575:wavenumber
338:In which:
125:statistics
58:wavelength
50:sea states
22:sinusoidal
2709:Acoustics
2661:Sea level
2560:Slab pull
2497:tectonics
2405:Cold seep
2367:Landforms
2244:Whirlpool
2239:Upwelling
2022:Wind wave
1950:Wave base
1878:Sea state
1798:Edge wave
1788:Cross sea
1696:0378-3839
1682:: 56–63.
1649:2156-2202
1640:2164/2592
1592:122442269
1545:0378-3839
1498:2156-2202
1489:2164/2592
1441:0378-3839
1427:: 56–63.
1402:121889662
1394:1469-8064
1347:0096-3518
1292:1365-294X
1253:0378-3839
1201:129854001
1193:2156-2202
1146:0739-0572
1097:0378-3839
934:nearshore
911:surf zone
886:∼
830:
822:∘
809:∘
801:−
795:ψ
756:
750:−
744:−
735:
682:ψ
676:
670:∗
634:ψ
628:
622:∗
406:⟩
403:⋅
400:⟨
349:η
310:⟩
300:η
296:⟨
291:⟩
278:η
265:⟨
206:⟩
196:η
192:⟨
187:⟩
178:η
174:⟨
71:altimeter
3037:Category
2942:Category
2894:Seawater
2621:Littoral
2616:Deep sea
2475:Seamount
2357:Tideline
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