655:
10,000 km), vertical wavelengths of 4–8 km, and upward group velocity. Similarly, westward-propagating mixed waves were also found in the
Atlantic Ocean by Weisberg et al. (1979) with periods of 31 days, horizontal wavelengths of 1200 km, vertical wavelengths of 1 km, and downward group velocity. Also, the vertically propagating gravity wave component was found in the stratosphere with periods of 35 hours, horizontal wavelengths of 2400 km, and vertical wavelengths of 5 km.
1102:
1990:
2011:
1091:
2000:
614:, the equatorially trapped Rossby wave and the mixed Rossby-gravity wave (which has some of the characteristics of the former two) . Equatorial gravity waves can be either westward- or eastward-propagating, and correspond to n=1 (same as for the equatorially trapped Rossby wave) on a dispersion relation diagram ("w-k" diagram). At
634:
As previously stated, the mixed Rossby-gravity waves are equatorially trapped waves unless the buoyancy frequency remains constant, introducing an additional vertical wave number to complement the zonal wave number and angular frequency. If this Brunt–Vaisala frequency does not change, then these
600:
654:
These vertically propagating mixed Rossby-gravity waves were first observed in the stratosphere as westward-propagating mixed waves by M. Yanai. They had the following characteristics: 4–5 days, horizontal wavenumbers of 4 (four waves circling the earth, corresponding to wavelengths of
238:
651: = 1 (gravity or Rossby waves) curves and would increase in the direction of increasing angular frequency. Typical group velocities for each component are the following: 1 cm/s for gravity waves and 2 mm/s for planetary (Rossby) waves.
626:), the solution appears to be a Rossby wave (hence the term Rossby-gravity waves). As mentioned earlier, the group velocity (or energy packet/dispersion) is always directed toward the east with a maximum for short waves (gravity waves).
412:
325:
438:
134:
115:
611:
420:
These three equations can be separated and solved using solutions in the form of zonally propagating waves, which are analogous to exponential solutions with a dependence on
342:
255:
607:
610:
is formulated in terms of ω, the angular frequency, the problem can be solved with three distinct solutions. These three solutions correspond to the
1730:
595:{\displaystyle {\begin{Bmatrix}u,v,\phi \end{Bmatrix}}={\begin{Bmatrix}{\hat {u}}(y),{\hat {v}}(y),{\hat {\phi }}(y)\end{Bmatrix}}e^{i(kx-\omega t)}}
1720:
779:
42:
The eastward speed of propagation of these waves can be derived for an inviscid slowly moving layer of fluid of uniform depth H. Because the
618: = 0 on a dispersion relation diagram, the mixed Rossby-gravity waves can be found where for large, positive zonal wave numbers (+
751:
Yanai, M. and T. Maruyama, 1966: Stratospheric wave disturbances propagating over the equatorial pacific. J. Met. Soc. Japan, 44, 291–194.
233:{\displaystyle {\frac {\partial \phi }{\partial t}}+c^{2}\left({\frac {\partial v}{\partial y}}+{\frac {\partial u}{\partial x}}\right)=0}
125:
the continuity equation (accounting for the effects of horizontal convergence and divergence and written with geopotential height):
58:
of the earth, 7.2921 × 10 rad/s, and θ is latitude) vanishes at 0 degrees latitude (equator), the “equatorial
1636:
34:. They always carry energy eastward, but their 'crests' and 'troughs' may propagate westward if their periods are long enough.
2003:
1051:
819:
77:
30:
does not remain constant). These waves have the same trapping scale as Kelvin waves, more commonly known as the equatorial
1283:
726:
Zhang, Dalin, 2008: Personal
Communication, “Waves in Rotating, Homogeneous Fluids,” University of Maryland, College Park.
772:
1173:
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2036:
1993:
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765:
1041:
31:
27:
1101:
1238:
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26:), meaning that they rapidly decay as their distance increases away from the equator (so long as the
1903:
1278:
1268:
1208:
844:
814:
1940:
1923:
1760:
1253:
1118:
1056:
1046:
939:
74:
is the distance from the equator and β is the variation of the
Coriolis parameter with latitude,
622:), the solution behaves like a gravity wave; but for large, negative zonal wave numbers (−
2041:
1935:
1873:
1300:
986:
669:
407:{\displaystyle {\frac {\partial v}{\partial t}}+u\beta y=-{\frac {\partial \phi }{\partial y}}}
320:{\displaystyle {\frac {\partial u}{\partial t}}-v\beta y=-{\frac {\partial \phi }{\partial x}}}
1768:
1750:
1258:
1153:
788:
1955:
1788:
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1213:
643:" dispersion diagram, the group velocity (energy) would be directed at right angles to the
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712:
International
Geophysics Series, Volume 30, Academic Press, 662 pp.
1814:
1804:
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635:
waves become vertically propagating solutions. On a typical "
629:
62:” approximation must be made. This approximation states that
1883:
1702:
1481:
1436:
1315:
428:
and the inclusion of structure functions that vary in the
110:{\displaystyle {\frac {\partial f}{\partial y}}=\beta }
481:
447:
16:
Equatorially trapped waves that carry energy eastwards
742:
Elsevier
Academic Press, Burlington, MA, pp. 394–400.
441:
345:
258:
137:
80:
333:
the V-momentum equation (meridional wind component):
594:
406:
319:
232:
109:
1731:North West Shelf Operational Oceanographic System
117:. With the inclusion of this approximation, the
2028:
647: = 0 (mixed Rossby-gravity waves) and
1721:Deep-ocean Assessment and Reporting of Tsunamis
246:the U-momentum equation (zonal wind component):
773:
630:Vertically propagating Rossby-gravity waves
780:
766:
22:are equatorially trapped waves (much like
787:
734:
732:
740:An Introduction to Dynamic Meteorology.
722:
720:
718:
2029:
1052:one-dimensional Saint-Venant equations
704:
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1999:
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13:
1879:National Oceanographic Data Center
1306:World Ocean Circulation Experiment
1194:Global Ocean Data Analysis Project
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92:
84:
14:
2053:
1726:Global Sea Level Observing System
612:equatorially trapped gravity wave
2009:
1998:
1989:
1988:
1184:Geochemical Ocean Sections Study
1100:
1089:
1914:Ocean thermal energy conversion
1637:Vine–Matthews–Morley hypothesis
745:
587:
569:
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538:
526:
520:
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121:become (neglecting friction):
1:
678:
37:
1174:El Niño–Southern Oscillation
1144:Craik–Leibovich vortex force
900:Luke's variational principle
7:
658:
66:is approximately equal to β
10:
2058:
1239:Ocean dynamical thermostat
1087:
710:Atmosphere-Ocean Dynamics,
1984:
1823:
1797:
1774:Ocean acoustic tomography
1759:
1711:
1650:
1587:Mohorovičić discontinuity
1545:
1417:
1314:
1179:General circulation model
1109:
815:Benjamin–Feir instability
795:
50: = 2Ω sin(
32:Rossby deformation radius
1904:Ocean surface topography
1279:Thermohaline circulation
1269:Subsurface ocean current
1209:Hydrothermal circulation
1042:Wave–current interaction
820:Boussinesq approximation
738:Holton, James R., 2004:
1941:Sea surface temperature
1924:Outline of oceanography
1119:Atmospheric circulation
1057:shallow water equations
1047:Waves and shallow water
940:Significant wave height
708:Gill, Adrian E., 1982:
28:Brunt–Vaisala frequency
1936:Sea surface microlayer
1301:Wind generated current
670:Equatorial Rossby wave
596:
408:
321:
234:
111:
2037:Physical oceanography
1769:Deep scattering layer
1751:World Geodetic System
1259:Princeton Ocean Model
1139:Coriolis–Stokes force
789:Physical oceanography
597:
409:
322:
235:
112:
1789:Underwater acoustics
1349:Perigean spring tide
1214:Langmuir circulation
925:Rossby-gravity waves
439:
343:
256:
135:
78:
20:Rossby-gravity waves
1951:Science On a Sphere
1557:Convergent boundary
1229:Modular Ocean Model
1189:Geostrophic current
905:Mild-slope equation
119:primitive equations
1607:Seafloor spreading
1597:Outer trench swell
1562:Divergent boundary
1462:Continental margin
1447:Carbonate platform
1344:Lunitidal interval
608:frequency relation
592:
555:
467:
404:
317:
230:
107:
44:Coriolis parameter
2024:
2023:
2016:Oceans portal
1976:World Ocean Atlas
1966:Underwater glider
1909:Ocean temperature
1572:Hydrothermal vent
1537:Submarine volcano
1472:Continental shelf
1452:Coastal geography
1442:Bathymetric chart
1324:Amphidromic point
1012:Wave nonlinearity
870:Infragravity wave
541:
517:
493:
402:
364:
315:
277:
217:
194:
156:
99:
54:) where Ω is the
2049:
2014:
2013:
2002:
2001:
1992:
1991:
1931:Pelagic sediment
1869:Marine pollution
1663:Deep ocean water
1532:Submarine canyon
1467:Continental rise
1359:Rule of twelfths
1274:Sverdrup balance
1204:Humboldt Current
1129:Boundary current
1104:
1093:
910:Radiation stress
880:Iribarren number
855:Equatorial waves
810:Ballantine scale
805:Airy wave theory
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56:angular velocity
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2020:
2008:
1980:
1819:
1793:
1755:
1736:Sea-level curve
1707:
1646:
1632:Transform fault
1582:Mid-ocean ridge
1548:
1541:
1507:Oceanic plateau
1413:
1399:Tidal resonance
1369:Theory of tides
1310:
1219:Longshore drift
1169:Ekman transport
1105:
1099:
1098:
1097:
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1095:
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1085:
1037:Wave turbulence
970:Trochoidal wave
895:Longshore drift
791:
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1741:Sea level drop
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1577:Marine geology
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1512:Oceanic trench
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955:Stokes problem
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947:
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890:Kinematic wave
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2042:Gravity waves
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1864:Marine energy
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1592:Oceanic crust
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1345:
1342:
1340:
1339:Internal tide
1337:
1335:
1332:
1330:
1327:
1325:
1322:
1321:
1319:
1317:
1313:
1307:
1304:
1302:
1299:
1297:
1294:
1292:
1289:
1285:
1282:
1281:
1280:
1277:
1275:
1272:
1270:
1267:
1265:
1262:
1260:
1257:
1255:
1252:
1250:
1247:
1245:
1242:
1240:
1237:
1235:
1234:Ocean current
1232:
1230:
1227:
1225:
1222:
1220:
1217:
1215:
1212:
1210:
1207:
1205:
1202:
1200:
1197:
1195:
1192:
1190:
1187:
1185:
1182:
1180:
1177:
1175:
1172:
1170:
1167:
1165:
1162:
1160:
1157:
1155:
1152:
1150:
1147:
1145:
1142:
1140:
1137:
1135:
1132:
1130:
1127:
1125:
1122:
1120:
1117:
1116:
1114:
1112:
1108:
1103:
1092:
1080:
1077:
1076:
1075:
1072:
1070:
1067:
1065:
1062:
1058:
1055:
1053:
1050:
1049:
1048:
1045:
1043:
1040:
1038:
1035:
1033:
1032:Wave shoaling
1030:
1028:
1025:
1023:
1020:
1018:
1015:
1013:
1010:
1008:
1005:
1003:
1000:
998:
995:
993:
992:Ursell number
990:
988:
985:
981:
978:
977:
976:
973:
971:
968:
966:
963:
961:
958:
956:
953:
951:
948:
946:
943:
941:
938:
936:
933:
931:
928:
926:
923:
921:
918:
916:
913:
911:
908:
906:
903:
901:
898:
896:
893:
891:
888:
886:
883:
881:
878:
876:
875:Internal wave
873:
871:
868:
866:
863:
861:
858:
856:
853:
851:
848:
846:
843:
841:
838:
836:
833:
831:
828:
826:
825:Breaking wave
823:
821:
818:
816:
813:
811:
808:
806:
803:
802:
800:
798:
794:
790:
783:
778:
776:
771:
769:
764:
763:
760:
748:
741:
735:
733:
723:
721:
719:
711:
705:
703:
701:
699:
697:
695:
693:
691:
689:
684:
676:
671:
668:
666:
663:
662:
656:
652:
650:
646:
642:
638:
627:
625:
621:
617:
613:
609:
584:
581:
578:
575:
572:
566:
562:
556:
547:
535:
529:
523:
511:
505:
499:
487:
478:
473:
468:
462:
459:
456:
453:
450:
444:
435:
434:
433:
431:
427:
423:
398:
390:
381:
378:
375:
372:
369:
366:
360:
352:
339:
338:
337:
336:
332:
331:
311:
303:
294:
291:
288:
285:
282:
279:
273:
265:
252:
251:
250:
249:
245:
244:
227:
224:
220:
213:
205:
196:
190:
182:
172:
166:
162:
158:
152:
144:
131:
130:
129:
128:
124:
123:
122:
120:
104:
101:
95:
87:
73:
69:
65:
61:
57:
53:
49:
45:
35:
33:
29:
25:
21:
1971:Water column
1919:Oceanography
1894:Observations
1889:Explorations
1859:Marginal sea
1852:
1810:OSTM/Jason-2
1642:Volcanic arc
1617:Slab suction
1334:Head of tide
1224:Loop Current
1164:Ekman spiral
950:Stokes drift
924:
860:Gravity wave
835:Cnoidal wave
747:
739:
709:
674:
653:
648:
644:
640:
636:
633:
623:
619:
615:
605:
432:-direction:
429:
425:
421:
419:
71:
67:
63:
51:
47:
41:
24:Kelvin waves
19:
18:
1961:Thermocline
1678:Mesopelagic
1651:Ocean zones
1622:Slab window
1487:Hydrography
1427:Abyssal fan
1394:Tidal range
1384:Tidal power
1379:Tidal force
1264:Rip current
1199:Gulf Stream
1159:Ekman layer
1149:Downwelling
1124:Baroclinity
1111:Circulation
1007:Wave height
997:Wave action
980:megatsunami
960:Stokes wave
920:Rossby wave
885:Kelvin wave
865:Green's law
665:Rossby wave
2031:Categories
1899:Reanalysis
1798:Satellites
1779:Sofar bomb
1627:Subduction
1602:Ridge push
1497:Ocean bank
1477:Contourite
1404:Tide gauge
1389:Tidal race
1374:Tidal bore
1364:Slack tide
1329:Earth tide
1249:Ocean gyre
1069:Wind setup
1064:Wind fetch
1027:Wave setup
1022:Wave radar
1017:Wave power
915:Rogue wave
845:Dispersion
679:References
60:beta plane
38:Derivation
1761:Acoustics
1713:Sea level
1612:Slab pull
1549:tectonics
1457:Cold seep
1419:Landforms
1296:Whirlpool
1291:Upwelling
1074:Wind wave
1002:Wave base
930:Sea state
850:Edge wave
840:Cross sea
606:Once the
582:ω
579:−
539:^
536:ϕ
515:^
491:^
463:ϕ
396:∂
391:ϕ
388:∂
382:−
373:β
358:∂
350:∂
309:∂
304:ϕ
301:∂
295:−
286:β
280:−
271:∂
263:∂
211:∂
203:∂
188:∂
180:∂
150:∂
145:ϕ
142:∂
105:β
93:∂
85:∂
1994:Category
1946:Seawater
1673:Littoral
1668:Deep sea
1527:Seamount
1409:Tideline
1354:Rip tide
1284:shutdown
1254:Overflow
987:Undertow
830:Clapotis
659:See also
70:, where
2004:Commons
1874:Mooring
1824:Related
1815:Jason-3
1805:Jason-1
1688:Pelagic
1683:Oceanic
1658:Benthic
975:Tsunami
945:Soliton
1693:Photic
1522:Seabed
935:Seiche
1884:Ocean
1853:Alvin
1703:Swash
1547:Plate
1492:Knoll
1482:Guyot
1437:Atoll
1316:Tides
1079:model
965:Swell
797:Waves
1851:DSV
1836:Argo
1698:Surf
1154:Eddy
424:and
2033::
731:^
717:^
687:^
781:e
774:t
767:v
649:n
645:n
641:k
639:,
637:m
624:k
620:k
616:n
602:.
588:)
585:t
576:x
573:k
570:(
567:i
563:e
557:}
551:)
548:y
545:(
530:,
527:)
524:y
521:(
512:v
506:,
503:)
500:y
497:(
488:u
479:{
474:=
469:}
460:,
457:v
454:,
451:u
445:{
430:y
426:t
422:x
414:.
399:y
379:=
376:y
370:u
367:+
361:t
353:v
312:x
292:=
289:y
283:v
274:t
266:u
228:0
225:=
221:)
214:x
206:u
197:+
191:y
183:v
173:(
167:2
163:c
159:+
153:t
102:=
96:y
88:f
72:y
68:y
64:f
52:θ
48:f
46:(
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