719:
500:
361:
787:
271:
473:
714:{\displaystyle \mathbf {F} (x,y,z)=\nabla \times g(x,y,z){\hat {\mathbf {z} }}+\nabla \times (\nabla \times h(x,y,z){\hat {\mathbf {z} }})+b_{x}(z){\hat {\mathbf {x} }}+b_{y}(z){\hat {\mathbf {y} }},}
165:
419:
110:
294:
727:
220:
478:
The poloidal–toroidal decomposition is unique if it is required that the average of the scalar fields Ψ and Φ vanishes on every sphere of radius
803:
430:
371:
993:
918:
Decomposition of solenoidal fields into poloidal fields, toroidal fields and the mean flow. Applications to the boussinesq-equations
930:, Lantz, S. R. and Fan, Y.; The Astrophysical Journal Supplement Series, Volume 121, Issue 1, Mar. 1999, pp. 247–264.
132:
494:, but a mean-field flow has to be included in this case. For example, every solenoidal vector field can be written as
388:
370:
is symmetric in that the curl of a toroidal field is poloidal, and the curl of a poloidal field is toroidal, known as
914:, Chandrasekhar, Subrahmanyan; International Series of Monographs on Physics, Oxford: Clarendon, 1961, p. 622.
78:
1026:
37:
29:
798:
491:
175:
33:
960:
924:, pp. 291–305; Lecture Notes in Mathematics, Springer Berlin/ Heidelberg, Vol. 1530/ 1992.
367:
981:
45:
211:
1004:
829:. International Series of Monographs on Physics. Oxford: Clarendon. See discussion on page 622.
356:{\displaystyle \mathbf {P} =\nabla \times (\nabla \times (\mathbf {r} \Phi (\mathbf {r} )))\,.}
989:
968:
945:
57:
41:
17:
964:
928:
938:
934:
1020:
951:
Backus, George (1986), "Poloidal and toroidal fields in geomagnetic field modeling",
927:
909:
824:
942:
191:
63:
782:{\displaystyle {\hat {\mathbf {x} }},{\hat {\mathbf {y} }},{\hat {\mathbf {z} }}}
21:
70:
972:
266:{\displaystyle \mathbf {T} =\nabla \times (\mathbf {r} \Psi (\mathbf {r} ))}
933:
Plane poloidal-toroidal decomposition of doubly periodic vector fields:
917:
865:
863:
382:
A toroidal vector field is tangential to spheres around the origin,
424:
while the curl of a poloidal field is tangential to those spheres
468:{\displaystyle \mathbf {r} \cdot (\nabla \times \mathbf {P} )=0.}
276:
and the poloidal field is derived from another scalar field Φ(
922:
The Navier–Stokes
Equations II — Theory and Numerical Methods
850:
848:
789:
denote the unit vectors in the coordinate directions.
160:{\displaystyle \mathbf {F} =\mathbf {T} +\mathbf {P} }
881:
869:
730:
503:
433:
391:
297:
223:
135:
81:
781:
713:
467:
413:
355:
265:
159:
104:
490:A poloidal–toroidal decomposition also exists in
414:{\displaystyle \mathbf {r} \cdot \mathbf {T} =0}
119:can be expressed as the sum of a toroidal field
8:
105:{\displaystyle \nabla \cdot \mathbf {F} =0,}
768:
766:
765:
751:
749:
748:
734:
732:
731:
729:
697:
695:
694:
679:
661:
659:
658:
643:
622:
620:
619:
566:
564:
563:
504:
502:
451:
434:
432:
400:
392:
390:
349:
335:
324:
298:
296:
252:
241:
224:
222:
190:). The toroidal field is obtained from a
152:
144:
136:
134:
88:
80:
911:Hydrodynamic and hydromagnetic stability
826:Hydrodynamic and hydromagnetic stability
815:
854:
839:
920:, Schmitt, B. J. and von Wahl, W; in
893:
7:
882:Backus, Parker & Constable 1996
870:Backus, Parker & Constable 1996
823:Subrahmanyan Chandrasekhar (1961).
589:
580:
533:
445:
329:
315:
306:
246:
232:
82:
14:
980:Backus, George; Parker, Robert;
769:
752:
735:
698:
662:
623:
567:
505:
452:
435:
401:
393:
336:
325:
299:
253:
242:
225:
153:
145:
137:
89:
26:poloidal–toroidal decomposition
988:, Cambridge University Press,
935:Part 1. Fields with divergence
804:Chandrasekhar–Kendall function
773:
756:
739:
702:
691:
685:
666:
655:
649:
633:
627:
616:
598:
586:
571:
560:
542:
527:
509:
456:
442:
372:Chandrasekhar–Kendall function
346:
343:
340:
332:
321:
312:
260:
257:
249:
238:
20:, a topic in pure and applied
1:
288:), as a twice-iterated curl,
28:is a restricted form of the
986:Foundations of Geomagnetism
1043:
123:and poloidal vector field
55:
32:. It is often used in the
939:Part 2. Stokes equations
62:For a three-dimensional
38:solenoidal vector fields
973:10.1029/RG024i001p00075
486:Cartesian decomposition
30:Helmholtz decomposition
783:
715:
469:
415:
357:
267:
174:is a radial vector in
161:
106:
1003:Jones, Chris (2008),
953:Reviews of Geophysics
799:Toroidal and poloidal
784:
716:
492:Cartesian coordinates
470:
416:
358:
268:
176:spherical coordinates
162:
107:
56:Further information:
46:incompressible fluids
34:spherical coordinates
982:Constable, Catherine
728:
501:
431:
389:
295:
221:
210:), as the following
133:
79:
965:1986RvGeo..24...75B
779:
711:
465:
411:
353:
263:
157:
102:
776:
759:
742:
705:
669:
630:
574:
1034:
1012:
1011:
998:
975:
941:. G. D. McBain.
897:
891:
885:
879:
873:
867:
858:
852:
843:
837:
831:
830:
820:
788:
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712:
707:
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684:
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671:
670:
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660:
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632:
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621:
576:
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565:
508:
474:
472:
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466:
455:
438:
420:
418:
417:
412:
404:
396:
362:
360:
359:
354:
339:
328:
302:
272:
270:
269:
264:
256:
245:
228:
166:
164:
163:
158:
156:
148:
140:
111:
109:
108:
103:
92:
1042:
1041:
1037:
1036:
1035:
1033:
1032:
1031:
1027:Vector calculus
1017:
1016:
1009:
1002:
996:
979:
950:
906:
901:
900:
892:
888:
880:
876:
868:
861:
853:
846:
838:
834:
822:
821:
817:
812:
795:
726:
725:
675:
639:
499:
498:
488:
429:
428:
387:
386:
380:
293:
292:
219:
218:
131:
130:
77:
76:
60:
58:Vector operator
54:
42:magnetic fields
40:, for example,
18:vector calculus
12:
11:
5:
1040:
1038:
1030:
1029:
1019:
1018:
1015:
1014:
1000:
994:
977:
948:
931:
925:
915:
905:
902:
899:
898:
886:
884:, p. 179.
874:
872:, p. 178.
859:
844:
832:
814:
813:
811:
808:
807:
806:
801:
794:
791:
775:
771:
764:
758:
754:
747:
741:
737:
722:
721:
710:
704:
700:
693:
690:
687:
682:
678:
674:
668:
664:
657:
654:
651:
646:
642:
638:
635:
629:
625:
618:
615:
612:
609:
606:
603:
600:
597:
594:
591:
588:
585:
582:
579:
573:
569:
562:
559:
556:
553:
550:
547:
544:
541:
538:
535:
532:
529:
526:
523:
520:
517:
514:
511:
507:
487:
484:
476:
475:
464:
461:
458:
454:
450:
447:
444:
441:
437:
422:
421:
410:
407:
403:
399:
395:
379:
376:
364:
363:
352:
348:
345:
342:
338:
334:
331:
327:
323:
320:
317:
314:
311:
308:
305:
301:
274:
273:
262:
259:
255:
251:
248:
244:
240:
237:
234:
231:
227:
168:
167:
155:
151:
147:
143:
139:
113:
112:
101:
98:
95:
91:
87:
84:
53:
50:
13:
10:
9:
6:
4:
3:
2:
1039:
1028:
1025:
1024:
1022:
1008:
1007:
1006:Dynamo Theory
1001:
997:
995:0-521-41006-1
991:
987:
983:
978:
974:
970:
966:
962:
958:
954:
949:
947:
944:
940:
936:
932:
929:
926:
923:
919:
916:
913:
912:
908:
907:
903:
896:, p. 17.
895:
890:
887:
883:
878:
875:
871:
866:
864:
860:
857:, p. 88.
856:
851:
849:
845:
842:, p. 87.
841:
836:
833:
828:
827:
819:
816:
809:
805:
802:
800:
797:
796:
792:
790:
762:
745:
708:
688:
680:
676:
672:
652:
644:
640:
636:
613:
610:
607:
604:
601:
595:
592:
583:
577:
557:
554:
551:
548:
545:
539:
536:
530:
524:
521:
518:
515:
512:
497:
496:
495:
493:
485:
483:
481:
462:
459:
448:
439:
427:
426:
425:
408:
405:
397:
385:
384:
383:
377:
375:
373:
369:
368:decomposition
350:
318:
309:
303:
291:
290:
289:
287:
283:
279:
235:
229:
217:
216:
215:
213:
209:
205:
201:
197:
193:
189:
185:
181:
177:
173:
149:
141:
129:
128:
127:
126:
122:
118:
99:
96:
93:
85:
75:
74:
73:
72:
68:
65:
59:
51:
49:
47:
43:
39:
35:
31:
27:
23:
19:
1005:
985:
956:
952:
921:
910:
889:
877:
835:
825:
818:
723:
489:
479:
477:
423:
381:
365:
285:
281:
277:
275:
207:
203:
199:
195:
192:scalar field
187:
183:
179:
171:
169:
124:
120:
116:
114:
66:
64:vector field
61:
36:analysis of
25:
15:
855:Backus 1986
840:Backus 1986
22:mathematics
959:: 75–109,
943:ANZIAM J.
904:References
894:Jones 2008
71:divergence
69:with zero
52:Definition
946:47 (2005)
774:^
757:^
740:^
703:^
667:^
628:^
593:×
590:∇
584:×
581:∇
572:^
537:×
534:∇
449:×
446:∇
440:⋅
398:⋅
330:Φ
319:×
316:∇
310:×
307:∇
247:Ψ
236:×
233:∇
86:⋅
83:∇
1021:Category
984:(1996),
793:See also
378:Geometry
961:Bibcode
992:
724:where
170:where
1010:(PDF)
810:Notes
366:This
115:this
990:ISBN
937:and
212:curl
44:and
24:, a
969:doi
16:In
1023::
967:,
957:24
955:,
862:^
847:^
482:.
463:0.
374:.
284:,
280:,
214:,
206:,
202:,
194:,
186:,
182:,
48:.
1013:.
999:.
976:.
971::
963::
770:z
763:,
753:y
746:,
736:x
709:,
699:y
692:)
689:z
686:(
681:y
677:b
673:+
663:x
656:)
653:z
650:(
645:x
641:b
637:+
634:)
624:z
617:)
614:z
611:,
608:y
605:,
602:x
599:(
596:h
587:(
578:+
568:z
561:)
558:z
555:,
552:y
549:,
546:x
543:(
540:g
531:=
528:)
525:z
522:,
519:y
516:,
513:x
510:(
506:F
480:r
460:=
457:)
453:P
443:(
436:r
409:0
406:=
402:T
394:r
351:.
347:)
344:)
341:)
337:r
333:(
326:r
322:(
313:(
304:=
300:P
286:φ
282:θ
278:r
261:)
258:)
254:r
250:(
243:r
239:(
230:=
226:T
208:φ
204:θ
200:r
198:(
196:Ψ
188:φ
184:θ
180:r
178:(
172:r
154:P
150:+
146:T
142:=
138:F
125:P
121:T
117:F
100:,
97:0
94:=
90:F
67:F
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