95:
727:
1053:
511:
102:
diagram of the base state of the system. The flow under investigation represents a small perturbation away from this state. While the base state is parallel, the perturbation velocity has components in both
404:
519:
942:
772:
284:
334:
160:
935:
899:
850:
823:
800:
133:
417:
1152:
775:
86:, who derived the equation independently from each other in 1931. The third independent derivation, also in 1931, was made by B. Haurwitz.
1142:
72:
1124:
1081:
44:
68:
853:
722:{\displaystyle (U-c)^{2}\left({d^{2}{\tilde {\phi }} \over dz^{2}}-\alpha ^{2}{\tilde {\phi }}\right)+\left{\tilde {\phi }}=0,}
343:
24:
1147:
1048:{\displaystyle \alpha {\tilde {\phi }}={d{\tilde {\phi }} \over dz}=0\quad {\text{ at }}z=z_{1}{\text{ and }}z=z_{2}.}
112:
735:
28:
825:
is positive, then the flow is unstable, and the small perturbation introduced to the system is amplified in time.
1157:
168:
79:
60:
1162:
40:
292:
138:
1167:
1120:
1077:
904:
865:
871:
829:
83:
513:
for the flow, the following dimensional form of the Taylor–Goldstein equation is obtained:
36:
835:
808:
803:
785:
411:
118:
56:
32:
1136:
52:
779:
64:
852:
results in a flow which is always unstable. This instability is known as the
506:{\displaystyle u_{x}'=d{\tilde {\phi }}/dz,u_{z}'=-i\alpha {\tilde {\phi }}}
407:
99:
94:
48:
336:
is the unperturbed or basic flow. The perturbation velocity has the
108:
93:
16:
Ordinary differential equation used in the field of fluid dynamics
337:
51:
forces (e.g. gravity), for stably stratified fluids in the
399:{\displaystyle \mathbf {u} '\propto \exp(i\alpha (x-ct))}
67:. The Taylor–Goldstein equation is derived from the 2D
945:
907:
874:
864:
The relevant boundary conditions are, in case of the
838:
811:
788:
738:
522:
420:
346:
295:
171:
141:
121:
1047:
929:
893:
868:boundary conditions at the channel top and bottom
844:
817:
794:
766:
721:
505:
398:
328:
278:
154:
135:and a mean density gradient (with gradient-length
127:
1094:
1092:
767:{\displaystyle N={\sqrt {g \over L_{\rho }}}}
410:understood). Using this knowledge, and the
8:
1036:
1021:
1015:
1000:
971:
970:
964:
950:
949:
944:
918:
906:
885:
873:
837:
810:
787:
755:
745:
737:
699:
698:
684:
666:
659:
635:
607:
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600:
584:
564:
563:
557:
550:
539:
521:
492:
491:
470:
452:
441:
440:
425:
419:
348:
345:
294:
275:
172:
170:
146:
140:
120:
1064:
162:), for the perturbation velocity field
55:. Or, more generally, the dynamics of
1098:
279:{\displaystyle \mathbf {u} =\left,\,}
107:The equation is derived by solving a
7:
59:in the presence of a (continuous)
14:
1117:Wave interactions and fluid flows
349:
173:
999:
802:. If the imaginary part of the
1119:, Cambridge University Press,
976:
955:
704:
656:
644:
612:
569:
536:
523:
497:
446:
393:
390:
375:
366:
323:
308:
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296:
264:
246:
226:
208:
194:
188:
25:ordinary differential equation
1:
1076:, New York: Academic Press,
782:parameter of the problem is
78:The equation is named after
45:Kelvin–Helmholtz instability
1153:Equations of fluid dynamics
860:No-slip boundary conditions
854:Rayleigh–Taylor instability
1184:
1143:Atmospheric thermodynamics
329:{\displaystyle (U(z),0,0)}
29:geophysical fluid dynamics
155:{\displaystyle L_{\rho }}
115:, in presence of gravity
21:Taylor–Goldstein equation
930:{\displaystyle z=z_{2}:}
832:Brunt–Väisälä frequency
73:Boussinesq approximation
39:flows. It describes the
31:, and more generally in
894:{\displaystyle z=z_{1}}
776:Brunt–Väisälä frequency
35:, in presence of quasi-
1115:Craik, A.D.D. (1988),
1049:
931:
895:
846:
819:
796:
768:
723:
507:
400:
330:
280:
156:
129:
113:Navier–Stokes equation
104:
61:density stratification
53:dissipation-less limit
27:used in the fields of
1050:
932:
896:
847:
820:
797:
769:
724:
508:
401:
331:
281:
157:
130:
97:
1148:Atmospheric dynamics
1072:Kundu, P.J. (1990),
943:
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809:
786:
736:
520:
418:
344:
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842:
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503:
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396:
326:
276:
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125:
105:
1101:, pp. 27–28)
1024:
1003:
991:
979:
958:
845:{\displaystyle N}
818:{\displaystyle c}
795:{\displaystyle c}
762:
761:
707:
691:
615:
591:
572:
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449:
128:{\displaystyle g}
1175:
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849:
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843:
830:purely imaginary
824:
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1074:Fluid Mechanics
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1023: and
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518:
517:
416:
415:
414:representation
347:
342:
341:
340:-like solution
291:
290:
238:
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184:
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167:
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142:
137:
136:
117:
116:
111:version of the
92:
69:Euler equations
17:
12:
11:
5:
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1165:
1160:
1158:Fluid dynamics
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412:streamfunction
395:
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175:
149:
145:
124:
91:
88:
57:internal waves
33:fluid dynamics
15:
13:
10:
9:
6:
4:
3:
2:
1180:
1169:
1166:
1164:
1161:
1159:
1156:
1154:
1151:
1149:
1146:
1144:
1141:
1140:
1138:
1128:
1126:0-521-36829-4
1122:
1118:
1113:
1112:
1108:
1100:
1095:
1093:
1089:
1085:
1083:0-12-178253-0
1079:
1075:
1068:
1065:
1058:
1042:
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1033:
1029:
1026:
1016:
1012:
1008:
1005:
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993:
987:
984:
973:
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961:
952:
946:
939:
938:
937:
924:
919:
915:
911:
908:
886:
882:
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875:
867:
859:
857:
855:
839:
831:
826:
812:
805:
789:
781:
777:
756:
752:
748:
742:
739:
716:
713:
710:
701:
694:
685:
681:
677:
672:
667:
663:
653:
650:
647:
641:
636:
632:
627:
623:
619:
609:
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597:
593:
585:
581:
577:
566:
558:
554:
546:
540:
532:
529:
526:
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515:
514:
494:
488:
485:
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479:
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467:
463:
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409:
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96:
89:
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81:
76:
74:
70:
66:
62:
58:
54:
50:
47:, subject to
46:
42:
38:
34:
30:
26:
22:
1163:Oceanography
1116:
1073:
1067:
863:
828:Note that a
827:
774:denotes the
731:
288:
106:
84:S. Goldstein
77:
71:, using the
20:
18:
1099:Craik (1988
103:directions.
90:Formulation
80:G.I. Taylor
1137:Categories
1109:References
804:wave speed
780:eigenvalue
109:linearized
65:shear flow
977:~
974:ϕ
956:~
953:ϕ
947:α
757:ρ
705:~
702:ϕ
651:−
642:−
613:~
610:ϕ
598:α
594:−
570:~
567:ϕ
530:−
498:~
495:ϕ
489:α
483:−
447:~
444:ϕ
408:real part
382:−
373:α
364:
358:∝
148:ρ
100:schematic
1168:Buoyancy
476:′
431:′
354:′
243:′
205:′
49:buoyancy
41:dynamics
866:no-slip
43:of the
1123:
1080:
778:. The
732:where
289:where
23:is an
1059:Notes
1121:ISBN
1078:ISBN
901:and
338:wave
82:and
63:and
19:The
361:exp
1139::
1091:^
856:.
98:A
75:.
37:2D
1043:.
1038:2
1034:z
1030:=
1027:z
1017:1
1013:z
1009:=
1006:z
997:0
994:=
988:z
985:d
968:d
962:=
925::
920:2
916:z
912:=
909:z
887:1
883:z
879:=
876:z
840:N
813:c
790:c
753:L
749:g
743:=
740:N
717:,
714:0
711:=
695:]
686:2
682:z
678:d
673:U
668:2
664:d
657:)
654:c
648:U
645:(
637:2
633:N
628:[
624:+
620:)
602:2
586:2
582:z
578:d
559:2
555:d
547:(
541:2
537:)
533:c
527:U
524:(
486:i
480:=
472:z
468:u
464:,
461:z
458:d
454:/
438:d
435:=
427:x
423:u
406:(
394:)
391:)
388:t
385:c
379:x
376:(
370:i
367:(
350:u
324:)
321:0
318:,
315:0
312:,
309:)
306:z
303:(
300:U
297:(
273:,
269:]
265:)
262:t
259:,
256:z
253:,
250:x
247:(
240:w
236:,
233:0
230:,
227:)
224:t
221:,
218:z
215:,
212:x
209:(
202:u
198:+
195:)
192:z
189:(
186:U
182:[
178:=
174:u
144:L
123:g
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
