1487:
1123:
1482:{\displaystyle {\begin{pmatrix}\omega ^{2}-v_{A}^{2}k^{2}-c_{s}^{2}k_{\perp }^{2}&0&-c_{s}^{2}k_{\parallel }k_{\perp }\\0&\omega ^{2}-v_{A}^{2}k_{\parallel }^{2}&0\\-c_{s}^{2}k_{\parallel }k_{\perp }&0&\omega ^{2}-c_{s}^{2}k_{\parallel }^{2}\end{pmatrix}}{\begin{pmatrix}v_{x1}\\v_{y1}\\v_{z1}\end{pmatrix}}=\mathbf {0} }
1762:
314:
3264:
In the case of an inhomogeneous fluids (that is, a fluid where at least one of the background quantities is not constant) the MHD waves lose their defining nature and get mixed properties. In some setups, such as the axisymmetric waves in a straight cylinder with a circular basis (one of the simplest
1062:
2993:
768:
2416:
2199:
2852:, so magnetic and thermal pressure perturbations have opposite signs. In other words, the two pressure perturbations reinforce one another in the fast mode, but oppose one another in the slow mode. As a result, the fast mode propagates at a faster speed than the slow mode.
2783:
507:
599:
2027:
399:
69:
and the equilibrium magnetic field and on the relative importance of fluid and magnetic processes in the medium. They only propagate with frequencies much smaller than the ion cyclotron or ion plasma frequencies of the medium, and they are
1572:
862:
186:
938:
944:
2873:
647:
2250:
2623:
3269:), the three MHD waves can still be clearly distinguished. But in general, the pure Alfvén and fast and slow magnetosonic waves don't exist, and the waves in the fluid are coupled to each other in intricate ways.
2038:
1840:
3237:, the fast and slow modes propagate as either a pure sound wave or pure Alfvén wave, with the fast mode identified with the larger of the two speeds and the slow mode identified with the smaller.
2532:
2640:
405:
101:
or anti-phase, respectively. This results in the phase velocity of any given fast mode always being greater than or equal to that of any slow mode in the same medium, among other differences.
3256:, the fast mode propagates as a longitudinal wave with phase velocity equal to the magnetosonic speed, and the slow mode propagates as a transverse wave with phase velocity approaching zero.
513:
2479:
1849:. This dispersion relation has three independent roots: one corresponding to the Alfvén wave and the other two corresponding to the magnetosonic modes. From the eigenvalue equation, the
2236:
as well as the equilibrium density, pressure, and magnetic field strength. From the roots of the magnetosonic dispersion relation, the associated phase velocities can be expressed as
135:
In an ideal, homogeneous, electrically conducting, magnetized fluid of infinite extent, there are two magnetosonic modes: the fast and slow modes. They form, together with the
1900:
320:
1757:{\displaystyle \left(\omega ^{2}-v_{A}^{2}k_{\parallel }^{2}\right)\left(\omega ^{4}-\omega ^{2}k^{2}c_{ms}^{2}+k^{2}k_{\parallel }^{2}v_{A}^{2}c_{s}^{2}\right)=0}
309:{\displaystyle \rho _{0}{\frac {\partial \mathbf {v} _{1}}{\partial t}}={\frac {(\nabla \times \mathbf {B} _{1})\times \mathbf {B} _{0}}{\mu _{0}}}-\nabla p_{1},}
795:
3050:-axis along the unperturbed magnetic field, these unit vectors correspond to those in the direction of increasing radial distance and increasing polar angle.
1057:{\displaystyle \mathbf {B} _{1}={\frac {(\mathbf {k} \cdot \mathbf {v} _{1})\mathbf {B} _{0}-(\mathbf {k} \cdot \mathbf {B} _{0})\mathbf {v} _{1}}{\omega }}.}
868:
2988:{\displaystyle \mathbf {v} _{g\pm }={\frac {d\omega }{d\mathbf {k} }}={\hat {k}}\,v_{\pm }+{\hat {\theta }}{\frac {\partial v_{\pm }}{\partial \theta }}}
763:{\displaystyle -\omega \rho _{0}\mathbf {v} _{1}={\frac {(\mathbf {k} \times \mathbf {B} _{1})\times \mathbf {B} _{0}}{\mu _{0}}}-\mathbf {k} p_{1}.}
2411:{\displaystyle ^{2}={\frac {\omega ^{2}}{k^{2}}}={\frac {1}{2}}\left(c_{ms}^{2}\pm {\sqrt {c_{ms}^{4}-4v_{A}^{2}c_{s}^{2}\cos ^{2}\theta }}\right)}
168:, the closed set of MHD equations consisting of the equation of motion, continuity equation, equation of state, and ideal induction equation (see
2549:
3508:
3471:
3405:
3350:
3488:
2194:{\displaystyle \omega ^{2}={\frac {k^{2}}{2}}\left(c_{ms}^{2}\pm {\sqrt {c_{ms}^{4}-4k_{\parallel }^{2}v_{A}^{2}c_{s}^{2}/k^{2}}}\right)}
65:
that acts to straighten bent magnetic field lines. The properties of magnetosonic waves are highly dependent on the angle between the
1773:
3380:
2778:{\displaystyle p_{m1}={\frac {v_{A}^{2}}{c_{s}^{2}}}\left(1-{\frac {c_{s}^{2}\cos ^{2}\theta }{v_{\pm }^{2}}}\right)p_{1}.}
2492:
502:{\displaystyle {\frac {\partial }{\partial t}}\left({\frac {p_{1}}{p_{0}}}-{\frac {\gamma \rho _{1}}{\rho _{0}}}\right)=0,}
1117:, respectively. The equation of motion after substituting for the perturbed quantities reduces to the eigenvalue equation
128:. They exist at frequencies far below the cyclotron and plasma frequencies of both ions and electrons in the medium (see
594:{\displaystyle {\frac {\partial \mathbf {B} _{1}}{\partial t}}=\nabla \times (\mathbf {v} _{1}\times \mathbf {B} _{0}),}
3043:
120:
Magnetosonic waves are a type of low-frequency wave present in electrically conducting, magnetized fluids, such as
3627:
2445:
1853:-component of the velocity perturbation decouples from the other two components giving the dispersion relation
46:
2204:
where the upper sign gives the fast magnetosonic mode and the lower sign gives the slow magnetosonic mode.
54:
39:
97:. The fast and slow modes are distinguished by magnetic and gas pressure oscillations that are either
3586:
3537:
3434:
169:
94:
2820:, so magnetic and thermal pressure perturbations have matching signs. Conversely, for the slow mode
3314:
622:
152:
140:
71:
2022:{\displaystyle \omega ^{4}-\omega ^{2}k^{2}c_{ms}^{2}+k^{2}k_{\parallel }^{2}v_{A}^{2}c_{s}^{2}=0}
394:{\displaystyle {\frac {\partial \rho _{1}}{\partial t}}+\rho _{0}\nabla \cdot \mathbf {v} _{1}=0,}
3604:
3309:
3282:
109:
3555:
3504:
3467:
3401:
3376:
3346:
3304:
2543:
129:
3594:
3545:
3496:
3442:
3338:
3294:
2214:
121:
62:
3500:
3337:. Astrophysics and Space Science Library. Vol. 294. Dordrecht: Springer Netherlands.
609:
1563:
857:{\displaystyle \rho _{1}=\rho _{0}{\frac {\mathbf {k} \cdot \mathbf {v} _{1}}{\omega }},}
105:
3590:
3541:
3438:
3281:
providing an observational foundation for the technique for coronal plasma diagnostics,
933:{\displaystyle p_{1}=\gamma p_{0}{\frac {\mathbf {k} \cdot \mathbf {v} _{1}}{\omega }},}
3299:
3064:
2856:
625:. Looking for a solution in the form of a superposition of plane waves which vary like
143:(MHD) waves. In this regime, magnetosonic waves are nondispersive at small amplitudes.
136:
86:
50:
3621:
3608:
3147:
3278:
3266:
125:
3146:
Under the assumption that the background temperature is zero, it follows from the
2032:
is the dispersion relation for the fast and slow magnetosonic modes. It has roots
604:
where equilibrium quantities have subscripts 0, perturbations have subscripts 1,
3029:
1523:
90:
58:
3342:
66:
3559:
3550:
3525:
780:, the remaining equations can be solved for perturbed quantities in terms of
2546:
perturbations associated with each mode. The magnetic pressure perturbation
162:
In an ideal electrically conducting fluid with a homogeneous magnetic field
20:
3574:
3423:"Atlas of the medium frequency waves in the ion-electron two-fluid plasma"
3245:
When the wavevector and the equilibrium magnetic field are perpendicular,
3599:
3524:
Goossens, Marcel L.; Arregui, Inigo; Van
Doorsselaere, Tom (2019-04-11).
98:
2618:{\displaystyle p_{m1}={\mathbf {B}}_{0}\cdot {\mathbf {B}}_{1}/\mu _{0}}
19:"Magnetoacoustic wave" redirects here. For sound-driven spin waves, see
2442:
The phase velocity of the fast mode is always greater than or equal to
27:
3447:
3422:
3230:
When the wavevector and the equilibrium magnetic field are parallel,
3187:, and the fast mode propagates isotropically with the Alfvén speed,
151:
The fast and slow magnetosonic waves are defined by a bi-quadratic
2542:. This is due to the differences in the signs of the thermal and
3277:
Both fast and slow magnetosonic waves have been observed in the
3097:. In this case, the slow mode propagates with the Alfvén speed,
1566:. Setting the determinant to zero gives the dispersion relation
42:
3335:
An
Introduction to Plasma Astrophysics and Magnetohydrodynamics
2625:
can be expressed in terms of the thermal pressure perturbation
1835:{\displaystyle \textstyle c_{ms}={\sqrt {v_{A}^{2}+c_{s}^{2}}}}
172:) linearized about a stationary equilibrium where the pressure
3218:. In this limit, the fast mode is sometimes referred to as a
2430:
of the fast mode and the lower sign gives the phase velocity
61:
of both the fluid and the magnetic field, as well as with an
3466:(Oxford Classics Series ed.). Oxford: Clarendon Press.
2217:
of the fast and slow magnetosonic waves depend on the angle
2481:, which is greater than or equal to that of the slow mode,
641:, the linearized equation of motion can be re-expressed as
3172:. In this case, the slow mode disappears from the system,
3464:
Cosmical
Magnetic Fields: Their Origin and Their Activity
1894:
for the Alfvén wave. The remaining bi-quadratic equation
155:
that can be derived from the linearized MHD equations.
3526:"Mixed Properties of MHD Waves in Non-uniform Plasmas"
3398:
3067:
fluid, the density and pressure perturbations vanish,
1777:
1413:
1132:
2876:
2643:
2552:
2527:{\displaystyle \geq {\frac {c_{ms}}{\sqrt {2}}}\geq }
2495:
2448:
2253:
2041:
1903:
1776:
1575:
1126:
947:
871:
798:
650:
516:
408:
323:
189:
3087:, resulting in the sound speed tending to infinity,
3160:, and, as a result, that the sound speed vanishes,
1067:Without loss of generality, we can assume that the
2987:
2867:of fast and slow magnetosonic waves is defined by
2777:
2617:
2526:
2473:
2410:
2193:
2021:
1834:
1756:
1481:
1056:
932:
856:
762:
593:
501:
393:
308:
3123:, and the fast mode disappears from the system,
2421:where the upper sign gives the phase velocity
3421:Huang, Y. C.; Lyu, L. H. (1 September 2019).
104:Magnetosonic waves have been observed in the
8:
108:and provide an observational foundation for
2474:{\displaystyle {\frac {c_{ms}}{\sqrt {2}}}}
77:There are two types of magnetosonic waves,
3366:
3364:
3362:
3598:
3549:
3530:Frontiers in Astronomy and Space Sciences
3446:
3375:. Cambridge: Cambridge University Press.
2968:
2958:
2947:
2946:
2937:
2932:
2921:
2920:
2909:
2895:
2883:
2878:
2875:
2766:
2749:
2744:
2727:
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2712:
2705:
2686:
2681:
2671:
2666:
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2648:
2642:
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2600:
2594:
2588:
2587:
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2551:
2505:
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2335:
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2143:
2133:
2128:
2112:
2104:
2098:
2089:
2081:
2061:
2055:
2046:
2040:
2007:
2002:
1992:
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1931:
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1454:
1437:
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1374:
1361:
1344:
1334:
1324:
1319:
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1235:
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1195:
1185:
1180:
1167:
1157:
1152:
1139:
1127:
1125:
1039:
1034:
1024:
1019:
1010:
998:
993:
983:
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954:
949:
946:
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839:
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822:
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803:
797:
751:
742:
731:
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715:
702:
697:
688:
682:
673:
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661:
649:
579:
574:
564:
559:
529:
524:
517:
515:
477:
466:
456:
445:
435:
429:
409:
407:
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358:
334:
324:
322:
297:
279:
268:
263:
250:
245:
232:
212:
207:
200:
194:
188:
3573:Nakariakov, V.M.; Verwichte, E. (2005).
3400:(2nd ed.). New York, NY: Springer.
3150:that the thermal pressure is also zero,
159:Derivation from linearized MHD equations
45:driven by mutual interaction between an
3493:Oxford Research Encyclopedia of Physics
3325:
16:Type of low-frequency compressive wave
170:Magnetohydrodynamics § Equations
7:
3501:10.1093/acrefore/9780190871994.013.7
130:Plasma parameters § Frequencies
3487:Nakariakov, V.M. (27 August 2020).
3038:and in the direction of increasing
2227:and the equilibrium magnetic field
2976:
2961:
549:
537:
520:
415:
411:
364:
342:
327:
290:
238:
220:
203:
14:
3575:"Coronal waves and oscillations"
2910:
2879:
2589:
2572:
1475:
1035:
1020:
1011:
994:
979:
970:
950:
911:
902:
835:
826:
743:
716:
698:
689:
669:
575:
560:
525:
372:
264:
246:
208:
3373:Fundamentals of plasma physics
2952:
2926:
1108:parallel and perpendicular to
1030:
1007:
989:
966:
708:
685:
585:
555:
256:
235:
180:are uniform and constant are:
1:
3371:Bellan, Paul Murray (2006).
3489:"Magnetohydrodynamic Waves"
3044:spherical coordinate system
53:. They are associated with
3644:
2209:Phase and group velocities
95:magnetohydrodynamics
18:
3343:10.1007/978-94-007-1076-4
3333:Goossens, Marcel (2003).
3241:Perpendicular propagation
3220:compressional Alfvén wave
139:, the three basic linear
3551:10.3389/fspas.2019.00020
3396:Somov, Boris V. (2012).
1080:and that the wavevector
1071:-axis is oriented along
2221:between the wavevector
1090:-plane with components
83:slow magnetosonic waves
79:fast magnetosonic waves
47:electrically conducting
3462:Parker, E. N. (1979).
2989:
2779:
2634:and phase velocity as
2619:
2528:
2475:
2412:
2195:
2023:
1836:
1758:
1483:
1058:
934:
858:
764:
635:and angular frequency
595:
503:
395:
310:
85:, which—together with
3579:Living Rev. Sol. Phys
3042:, respectively. In a
3028:are local orthogonal
2990:
2780:
2620:
2529:
2476:
2413:
2196:
2024:
1837:
1759:
1484:
1059:
935:
859:
765:
596:
504:
396:
311:
74:at small amplitudes.
36:magnetoacoustic waves
3600:10.12942/lrsp-2005-3
3226:Parallel propagation
3059:Incompressible fluid
3032:in the direction of
2874:
2641:
2550:
2493:
2446:
2251:
2039:
1901:
1774:
1573:
1124:
945:
869:
796:
648:
514:
406:
321:
187:
38:, are low-frequency
3591:2005LRSP....2....3N
3542:2019FrASS...6...20G
3439:2019PhPl...26i2102H
3315:Magnetogravity wave
3260:Inhomogeneous fluid
2754:
2722:
2691:
2676:
2384:
2369:
2348:
2325:
2168:
2153:
2138:
2117:
2094:
2012:
1997:
1982:
1954:
1828:
1810:
1742:
1727:
1712:
1684:
1623:
1608:
1399:
1384:
1329:
1302:
1287:
1230:
1205:
1190:
1162:
623:vacuum permeability
153:dispersion relation
147:Dispersion relation
141:magnetohydrodynamic
3427:Physics of Plasmas
3310:Coronal seismology
3283:coronal seismology
2985:
2788:For the fast mode
2775:
2740:
2708:
2677:
2662:
2615:
2524:
2471:
2439:of the slow mode.
2408:
2370:
2355:
2331:
2308:
2191:
2154:
2139:
2124:
2100:
2077:
2019:
1998:
1983:
1968:
1937:
1847:magnetosonic speed
1832:
1831:
1814:
1796:
1754:
1728:
1713:
1698:
1667:
1609:
1594:
1479:
1465:
1402:
1385:
1370:
1315:
1288:
1273:
1216:
1191:
1176:
1148:
1054:
930:
854:
773:And assuming that
760:
591:
499:
391:
306:
160:
110:coronal seismology
32:magnetosonic waves
3510:978-0-19-087199-4
3473:978-0-19-882996-6
3448:10.1063/1.5110991
3407:978-1-4614-4283-7
3352:978-1-4020-1433-8
3305:Ion acoustic wave
2983:
2955:
2929:
2915:
2755:
2692:
2544:magnetic pressure
2519:
2518:
2469:
2468:
2401:
2301:
2288:
2184:
2070:
1829:
1049:
925:
849:
737:
544:
483:
451:
422:
349:
285:
227:
158:
63:effective tension
3635:
3628:Waves in plasmas
3613:
3612:
3602:
3570:
3564:
3563:
3553:
3521:
3515:
3514:
3484:
3478:
3477:
3459:
3453:
3452:
3450:
3418:
3412:
3411:
3393:
3387:
3386:
3368:
3357:
3356:
3330:
3295:Waves in plasmas
3255:
3236:
3217:
3216:
3215:
3199:
3198:
3186:
3184:
3183:
3171:
3159:
3137:
3135:
3134:
3122:
3121:
3120:
3109:
3108:
3096:
3086:
3076:
3049:
3041:
3037:
3027:
3026:
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3024:
3019:
3012:
3011:
3010:
3009:
3004:
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2991:
2986:
2984:
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2974:
2973:
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2959:
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2948:
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2757:
2756:
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2614:
2613:
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2576:
2575:
2565:
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2541:
2533:
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2530:
2525:
2520:
2514:
2513:
2512:
2500:
2489:
2480:
2478:
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2450:
2438:
2429:
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2403:
2402:
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2368:
2363:
2347:
2342:
2330:
2324:
2319:
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2294:
2289:
2287:
2286:
2277:
2276:
2267:
2262:
2261:
2247:
2235:
2226:
2220:
2215:phase velocities
2200:
2198:
2197:
2192:
2190:
2186:
2185:
2183:
2182:
2173:
2167:
2162:
2152:
2147:
2137:
2132:
2116:
2111:
2099:
2093:
2088:
2071:
2066:
2065:
2056:
2051:
2050:
2028:
2026:
2025:
2020:
2011:
2006:
1996:
1991:
1981:
1976:
1967:
1966:
1953:
1948:
1936:
1935:
1926:
1925:
1913:
1912:
1893:
1892:
1891:
1881:
1880:
1867:
1866:
1841:
1839:
1838:
1833:
1830:
1827:
1822:
1809:
1804:
1795:
1790:
1789:
1763:
1761:
1760:
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1747:
1743:
1741:
1736:
1726:
1721:
1711:
1706:
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1678:
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1665:
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1643:
1642:
1628:
1624:
1622:
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1607:
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1589:
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1560:
1559:
1521:
1520:
1519:
1488:
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1480:
1478:
1470:
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1462:
1461:
1445:
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1428:
1427:
1407:
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1398:
1393:
1383:
1378:
1366:
1365:
1349:
1348:
1339:
1338:
1328:
1323:
1301:
1296:
1286:
1281:
1269:
1268:
1250:
1249:
1240:
1239:
1229:
1224:
1204:
1199:
1189:
1184:
1172:
1171:
1161:
1156:
1144:
1143:
1116:
1107:
1098:
1089:
1085:
1079:
1070:
1063:
1061:
1060:
1055:
1050:
1045:
1044:
1043:
1038:
1029:
1028:
1023:
1014:
1003:
1002:
997:
988:
987:
982:
973:
964:
959:
958:
953:
939:
937:
936:
931:
926:
921:
920:
919:
914:
905:
899:
897:
896:
881:
880:
863:
861:
860:
855:
850:
845:
844:
843:
838:
829:
823:
821:
820:
808:
807:
788:
779:
769:
767:
766:
761:
756:
755:
746:
738:
736:
735:
726:
725:
724:
719:
707:
706:
701:
692:
683:
678:
677:
672:
666:
665:
640:
634:
629:with wavevector
628:
620:
607:
600:
598:
597:
592:
584:
583:
578:
569:
568:
563:
545:
543:
535:
534:
533:
528:
518:
508:
506:
505:
500:
489:
485:
484:
482:
481:
472:
471:
470:
457:
452:
450:
449:
440:
439:
430:
423:
421:
410:
400:
398:
397:
392:
381:
380:
375:
363:
362:
350:
348:
340:
339:
338:
325:
315:
313:
312:
307:
302:
301:
286:
284:
283:
274:
273:
272:
267:
255:
254:
249:
233:
228:
226:
218:
217:
216:
211:
201:
199:
198:
179:
175:
167:
34:, also known as
3643:
3642:
3638:
3637:
3636:
3634:
3633:
3632:
3618:
3617:
3616:
3572:
3571:
3567:
3523:
3522:
3518:
3511:
3486:
3485:
3481:
3474:
3461:
3460:
3456:
3420:
3419:
3415:
3408:
3395:
3394:
3390:
3383:
3370:
3369:
3360:
3353:
3332:
3331:
3327:
3323:
3291:
3275:
3262:
3246:
3243:
3231:
3228:
3214:
3209:
3208:
3207:
3197:
3194:
3193:
3192:
3188:
3182:
3179:
3178:
3177:
3173:
3169:
3161:
3157:
3151:
3144:
3133:
3130:
3129:
3128:
3124:
3119:
3116:
3115:
3114:
3107:
3104:
3103:
3102:
3098:
3094:
3088:
3084:
3078:
3074:
3068:
3061:
3056:
3047:
3039:
3033:
3020:
3017:
3016:
3015:
3014:
3005:
3002:
3001:
3000:
2999:
2975:
2964:
2960:
2933:
2905:
2897:
2877:
2872:
2871:
2865:
2859:
2844:
2839:
2838:
2837:
2830:
2827:
2826:
2825:
2821:
2812:
2807:
2806:
2805:
2798:
2795:
2794:
2793:
2789:
2762:
2723:
2707:
2698:
2694:
2644:
2639:
2638:
2632:
2626:
2605:
2586:
2569:
2553:
2548:
2547:
2540:
2534:
2501:
2491:
2490:
2488:
2482:
2451:
2444:
2443:
2437:
2431:
2428:
2422:
2385:
2307:
2303:
2278:
2268:
2254:
2249:
2248:
2246:
2240:
2234:
2228:
2222:
2218:
2211:
2206:
2174:
2076:
2072:
2057:
2042:
2037:
2036:
1958:
1927:
1917:
1904:
1899:
1898:
1890:
1887:
1886:
1885:
1879:
1874:
1873:
1872:
1865:
1860:
1859:
1858:
1854:
1852:
1778:
1772:
1771:
1688:
1657:
1647:
1634:
1633:
1629:
1581:
1580:
1576:
1571:
1570:
1558:
1552:
1546:
1544:
1542:
1535:
1527:
1518:
1511:
1505:
1503:
1501:
1493:
1464:
1463:
1450:
1447:
1446:
1433:
1430:
1429:
1416:
1409:
1401:
1400:
1357:
1355:
1350:
1340:
1330:
1309:
1308:
1303:
1260:
1258:
1252:
1251:
1241:
1231:
1211:
1206:
1163:
1135:
1128:
1122:
1121:
1115:
1109:
1106:
1100:
1097:
1091:
1087:
1081:
1078:
1072:
1068:
1033:
1018:
992:
977:
965:
948:
943:
942:
909:
900:
888:
872:
867:
866:
833:
824:
812:
799:
794:
793:
787:
781:
774:
747:
727:
714:
696:
684:
667:
657:
646:
645:
636:
630:
626:
619:
613:
610:adiabatic index
605:
573:
558:
536:
523:
519:
512:
511:
473:
462:
458:
441:
431:
428:
424:
414:
404:
403:
370:
354:
341:
330:
326:
319:
318:
293:
275:
262:
244:
234:
219:
206:
202:
190:
185:
184:
177:
173:
163:
149:
118:
116:Characteristics
24:
17:
12:
11:
5:
3641:
3639:
3631:
3630:
3620:
3619:
3615:
3614:
3565:
3516:
3509:
3479:
3472:
3454:
3413:
3406:
3388:
3381:
3358:
3351:
3324:
3322:
3319:
3318:
3317:
3312:
3307:
3302:
3297:
3290:
3287:
3274:
3271:
3261:
3258:
3242:
3239:
3227:
3224:
3210:
3195:
3180:
3165:
3155:
3143:
3140:
3131:
3117:
3105:
3092:
3082:
3072:
3065:incompressible
3060:
3057:
3055:
3054:Limiting cases
3052:
2996:
2995:
2981:
2978:
2971:
2967:
2963:
2954:
2951:
2945:
2940:
2936:
2928:
2925:
2919:
2912:
2908:
2903:
2900:
2894:
2889:
2886:
2881:
2863:
2857:group velocity
2840:
2828:
2808:
2796:
2786:
2785:
2774:
2769:
2765:
2760:
2752:
2747:
2743:
2738:
2735:
2730:
2726:
2720:
2715:
2711:
2704:
2701:
2697:
2689:
2684:
2680:
2674:
2669:
2665:
2659:
2654:
2651:
2647:
2630:
2612:
2608:
2603:
2597:
2591:
2585:
2580:
2574:
2568:
2563:
2560:
2556:
2538:
2523:
2517:
2511:
2508:
2504:
2498:
2486:
2467:
2461:
2458:
2454:
2435:
2426:
2419:
2418:
2406:
2400:
2397:
2392:
2388:
2382:
2377:
2373:
2367:
2362:
2358:
2354:
2351:
2346:
2341:
2338:
2334:
2328:
2323:
2318:
2315:
2311:
2306:
2300:
2297:
2292:
2285:
2281:
2275:
2271:
2265:
2260:
2256:
2244:
2232:
2210:
2207:
2202:
2201:
2189:
2181:
2177:
2172:
2166:
2161:
2157:
2151:
2146:
2142:
2136:
2131:
2127:
2123:
2120:
2115:
2110:
2107:
2103:
2097:
2092:
2087:
2084:
2080:
2075:
2069:
2064:
2060:
2054:
2049:
2045:
2030:
2029:
2018:
2015:
2010:
2005:
2001:
1995:
1990:
1986:
1980:
1975:
1971:
1965:
1961:
1957:
1952:
1947:
1944:
1940:
1934:
1930:
1924:
1920:
1916:
1911:
1907:
1888:
1875:
1861:
1850:
1843:
1842:
1826:
1821:
1817:
1813:
1808:
1803:
1799:
1793:
1788:
1785:
1781:
1765:
1764:
1753:
1750:
1746:
1740:
1735:
1731:
1725:
1720:
1716:
1710:
1705:
1701:
1695:
1691:
1687:
1682:
1677:
1674:
1670:
1664:
1660:
1654:
1650:
1646:
1641:
1637:
1632:
1627:
1621:
1616:
1612:
1606:
1601:
1597:
1593:
1588:
1584:
1579:
1556:
1550:
1540:
1531:
1516:
1509:
1497:
1490:
1489:
1477:
1473:
1468:
1460:
1457:
1453:
1449:
1448:
1443:
1440:
1436:
1432:
1431:
1426:
1423:
1419:
1415:
1414:
1412:
1405:
1397:
1392:
1388:
1382:
1377:
1373:
1369:
1364:
1360:
1356:
1354:
1351:
1347:
1343:
1337:
1333:
1327:
1322:
1318:
1314:
1311:
1310:
1307:
1304:
1300:
1295:
1291:
1285:
1280:
1276:
1272:
1267:
1263:
1259:
1257:
1254:
1253:
1248:
1244:
1238:
1234:
1228:
1223:
1219:
1215:
1212:
1210:
1207:
1203:
1198:
1194:
1188:
1183:
1179:
1175:
1170:
1166:
1160:
1155:
1151:
1147:
1142:
1138:
1134:
1133:
1131:
1113:
1104:
1095:
1076:
1065:
1064:
1053:
1048:
1042:
1037:
1032:
1027:
1022:
1017:
1013:
1009:
1006:
1001:
996:
991:
986:
981:
976:
972:
968:
962:
957:
952:
940:
929:
924:
918:
913:
908:
904:
895:
891:
887:
884:
879:
875:
864:
853:
848:
842:
837:
832:
828:
819:
815:
811:
806:
802:
785:
771:
770:
759:
754:
750:
745:
741:
734:
730:
723:
718:
713:
710:
705:
700:
695:
691:
687:
681:
676:
671:
664:
660:
656:
653:
617:
602:
601:
590:
587:
582:
577:
572:
567:
562:
557:
554:
551:
548:
542:
539:
532:
527:
522:
509:
498:
495:
492:
488:
480:
476:
469:
465:
461:
455:
448:
444:
438:
434:
427:
420:
417:
413:
401:
390:
387:
384:
379:
374:
369:
366:
361:
357:
353:
347:
344:
337:
333:
329:
316:
305:
300:
296:
292:
289:
282:
278:
271:
266:
261:
258:
253:
248:
243:
240:
237:
231:
225:
222:
215:
210:
205:
197:
193:
157:
148:
145:
117:
114:
51:magnetic field
15:
13:
10:
9:
6:
4:
3:
2:
3640:
3629:
3626:
3625:
3623:
3610:
3606:
3601:
3596:
3592:
3588:
3584:
3580:
3576:
3569:
3566:
3561:
3557:
3552:
3547:
3543:
3539:
3535:
3531:
3527:
3520:
3517:
3512:
3506:
3502:
3498:
3494:
3490:
3483:
3480:
3475:
3469:
3465:
3458:
3455:
3449:
3444:
3440:
3436:
3432:
3428:
3424:
3417:
3414:
3409:
3403:
3399:
3392:
3389:
3384:
3378:
3374:
3367:
3365:
3363:
3359:
3354:
3348:
3344:
3340:
3336:
3329:
3326:
3320:
3316:
3313:
3311:
3308:
3306:
3303:
3301:
3298:
3296:
3293:
3292:
3288:
3286:
3284:
3280:
3272:
3270:
3268:
3265:models for a
3259:
3257:
3253:
3249:
3240:
3238:
3234:
3225:
3223:
3221:
3213:
3206:
3203:
3191:
3176:
3168:
3164:
3154:
3149:
3148:ideal gas law
3141:
3139:
3127:
3113:
3101:
3091:
3081:
3071:
3066:
3058:
3053:
3051:
3045:
3036:
3031:
3023:
3008:
2979:
2969:
2965:
2949:
2943:
2938:
2934:
2923:
2917:
2906:
2901:
2898:
2892:
2887:
2884:
2870:
2869:
2868:
2862:
2858:
2853:
2850:
2843:
2836:
2824:
2818:
2811:
2804:
2792:
2772:
2767:
2763:
2758:
2750:
2745:
2741:
2736:
2733:
2728:
2724:
2718:
2713:
2709:
2702:
2699:
2695:
2687:
2682:
2678:
2672:
2667:
2663:
2657:
2652:
2649:
2645:
2637:
2636:
2635:
2629:
2610:
2606:
2601:
2595:
2583:
2578:
2566:
2561:
2558:
2554:
2545:
2537:
2521:
2515:
2509:
2506:
2502:
2496:
2485:
2465:
2459:
2456:
2452:
2440:
2434:
2425:
2404:
2398:
2395:
2390:
2386:
2380:
2375:
2371:
2365:
2360:
2356:
2352:
2349:
2344:
2339:
2336:
2332:
2326:
2321:
2316:
2313:
2309:
2304:
2298:
2295:
2290:
2283:
2279:
2273:
2269:
2263:
2258:
2255:
2243:
2239:
2238:
2237:
2231:
2225:
2216:
2208:
2205:
2187:
2179:
2175:
2170:
2164:
2159:
2155:
2149:
2144:
2140:
2134:
2129:
2125:
2121:
2118:
2113:
2108:
2105:
2101:
2095:
2090:
2085:
2082:
2078:
2073:
2067:
2062:
2058:
2052:
2047:
2043:
2035:
2034:
2033:
2016:
2013:
2008:
2003:
1999:
1993:
1988:
1984:
1978:
1973:
1969:
1963:
1959:
1955:
1950:
1945:
1942:
1938:
1932:
1928:
1922:
1918:
1914:
1909:
1905:
1897:
1896:
1895:
1884:
1878:
1871:
1864:
1857:
1848:
1824:
1819:
1815:
1811:
1806:
1801:
1797:
1791:
1786:
1783:
1779:
1770:
1769:
1768:
1751:
1748:
1744:
1738:
1733:
1729:
1723:
1718:
1714:
1708:
1703:
1699:
1693:
1689:
1685:
1680:
1675:
1672:
1668:
1662:
1658:
1652:
1648:
1644:
1639:
1635:
1630:
1625:
1619:
1614:
1610:
1604:
1599:
1595:
1591:
1586:
1582:
1577:
1569:
1568:
1567:
1565:
1555:
1549:
1539:
1534:
1530:
1525:
1515:
1508:
1500:
1496:
1471:
1466:
1458:
1455:
1451:
1441:
1438:
1434:
1424:
1421:
1417:
1410:
1403:
1395:
1390:
1386:
1380:
1375:
1371:
1367:
1362:
1358:
1352:
1345:
1341:
1335:
1331:
1325:
1320:
1316:
1312:
1305:
1298:
1293:
1289:
1283:
1278:
1274:
1270:
1265:
1261:
1255:
1246:
1242:
1236:
1232:
1226:
1221:
1217:
1213:
1208:
1201:
1196:
1192:
1186:
1181:
1177:
1173:
1168:
1164:
1158:
1153:
1149:
1145:
1140:
1136:
1129:
1120:
1119:
1118:
1112:
1103:
1094:
1084:
1075:
1051:
1046:
1040:
1025:
1015:
1004:
999:
984:
974:
960:
955:
941:
927:
922:
916:
906:
893:
889:
885:
882:
877:
873:
865:
851:
846:
840:
830:
817:
813:
809:
804:
800:
792:
791:
790:
784:
777:
757:
752:
748:
739:
732:
728:
721:
711:
703:
693:
679:
674:
662:
658:
654:
651:
644:
643:
642:
639:
633:
624:
616:
611:
588:
580:
570:
565:
552:
546:
540:
530:
510:
496:
493:
490:
486:
478:
474:
467:
463:
459:
453:
446:
442:
436:
432:
425:
418:
402:
388:
385:
382:
377:
367:
359:
355:
351:
345:
335:
331:
317:
303:
298:
294:
287:
280:
276:
269:
259:
251:
241:
229:
223:
213:
195:
191:
183:
182:
181:
171:
166:
156:
154:
146:
144:
142:
138:
133:
131:
127:
126:liquid metals
123:
115:
113:
111:
107:
102:
100:
96:
92:
88:
84:
80:
75:
73:
72:nondispersive
68:
64:
60:
56:
52:
48:
44:
41:
37:
33:
29:
22:
3582:
3578:
3568:
3533:
3529:
3519:
3492:
3482:
3463:
3457:
3430:
3426:
3416:
3397:
3391:
3372:
3334:
3328:
3279:solar corona
3276:
3273:Observations
3267:coronal loop
3263:
3251:
3247:
3244:
3232:
3229:
3219:
3211:
3204:
3201:
3189:
3174:
3166:
3162:
3152:
3145:
3125:
3111:
3099:
3089:
3079:
3069:
3062:
3034:
3021:
3006:
2997:
2860:
2854:
2848:
2841:
2834:
2822:
2816:
2809:
2802:
2790:
2787:
2627:
2535:
2483:
2441:
2432:
2423:
2420:
2241:
2229:
2223:
2212:
2203:
2031:
1882:
1876:
1869:
1862:
1855:
1846:
1844:
1766:
1564:Alfvén speed
1553:
1547:
1537:
1532:
1528:
1513:
1506:
1498:
1494:
1491:
1110:
1101:
1092:
1086:lies in the
1082:
1073:
1066:
782:
775:
772:
637:
631:
614:
603:
176:and density
164:
161:
150:
134:
119:
106:Sun's corona
103:
91:normal modes
87:Alfvén waves
82:
78:
76:
49:fluid and a
35:
31:
25:
3300:Alfvén wave
3030:unit vector
1524:sound speed
137:Alfvén wave
59:rarefaction
55:compression
40:compressive
3382:0521528003
3321:References
3142:Cold limit
67:wavevector
3609:123211890
3560:2296-987X
3235:→ 0
3136:→ ∞
2980:θ
2977:∂
2970:±
2962:∂
2953:^
2950:θ
2939:±
2927:^
2902:ω
2888:±
2746:±
2737:θ
2734:
2703:−
2607:μ
2584:⋅
2522:≥
2497:≥
2399:θ
2396:
2350:−
2327:±
2270:ω
2130:∥
2119:−
2096:±
2044:ω
1974:∥
1919:ω
1915:−
1906:ω
1704:∥
1649:ω
1645:−
1636:ω
1615:∥
1592:−
1583:ω
1391:∥
1368:−
1359:ω
1346:⊥
1336:∥
1313:−
1294:∥
1271:−
1262:ω
1247:⊥
1237:∥
1214:−
1197:⊥
1174:−
1146:−
1137:ω
1047:ω
1016:⋅
1005:−
975:⋅
923:ω
907:⋅
886:γ
847:ω
831:⋅
814:ρ
801:ρ
740:−
729:μ
712:×
694:×
659:ρ
655:ω
652:−
571:×
553:×
550:∇
538:∂
521:∂
475:ρ
464:ρ
460:γ
454:−
416:∂
412:∂
368:⋅
365:∇
356:ρ
343:∂
332:ρ
328:∂
291:∇
288:−
277:μ
260:×
242:×
239:∇
221:∂
204:∂
192:ρ
93:of ideal
89:—are the
21:Spin wave
3622:Category
3585:(1): 3.
3289:See also
3250:→
2864:g ±
99:in-phase
3587:Bibcode
3538:Bibcode
3435:Bibcode
3046:with a
2829:−
2436:−
1889:∥
1845:is the
1562:is the
1545:√
1522:is the
1507:γp
1504:√
1105:⊥
1096:∥
621:is the
608:is the
122:plasmas
28:physics
3607:
3558:
3536:: 20.
3507:
3470:
3404:
3379:
3349:
3252:π
3248:θ
3233:θ
3070:ρ
3063:In an
3040:θ
3022:θ
2998:where
2849:θ
2817:θ
2245:±
2219:θ
1856:ω
1767:where
1554:ρ
1548:μ
1514:ρ
1492:where
776:ω
638:ω
615:μ
612:, and
606:γ
178:ρ
3605:S2CID
3433:(9).
2833:<
2801:>
43:waves
3556:ISSN
3505:ISBN
3468:ISBN
3402:ISBN
3377:ISBN
3347:ISBN
3077:and
3013:and
2855:The
2847:cos
2815:cos
2213:The
1526:and
1099:and
124:and
81:and
57:and
3595:doi
3546:doi
3497:doi
3443:doi
3339:doi
3185:= 0
3170:= 0
3158:= 0
3095:→ ∞
3085:= 0
3075:= 0
2725:cos
2387:cos
778:≠ 0
627:exp
132:).
26:In
3624::
3603:.
3593:.
3581:.
3577:.
3554:.
3544:.
3532:.
3528:.
3503:.
3495:.
3491:.
3441:.
3431:26
3429:.
3425:.
3361:^
3345:.
3285:.
3254:/2
3222:.
3200:=
3181:sl
3138:.
3110:=
3106:sl
1868:=
1536:=
1502:=
1088:xz
789::
112:.
30:,
3611:.
3597::
3589::
3583:2
3562:.
3548::
3540::
3534:6
3513:.
3499::
3476:.
3451:.
3445::
3437::
3410:.
3385:.
3355:.
3341::
3212:A
3205:v
3202:k
3196:f
3190:ω
3175:ω
3167:s
3163:c
3156:0
3153:p
3132:f
3126:ω
3118:A
3112:ω
3100:ω
3093:s
3090:c
3083:1
3080:p
3073:1
3048:z
3035:k
3018:∧
3007:k
3003:∧
2966:v
2944:+
2935:v
2924:k
2918:=
2911:k
2907:d
2899:d
2893:=
2885:g
2880:v
2861:v
2842:s
2835:c
2823:v
2810:s
2803:c
2797:+
2791:v
2773:.
2768:1
2764:p
2759:)
2751:2
2742:v
2729:2
2719:2
2714:s
2710:c
2700:1
2696:(
2688:2
2683:s
2679:c
2673:2
2668:A
2664:v
2658:=
2653:1
2650:m
2646:p
2631:1
2628:p
2611:0
2602:/
2596:1
2590:B
2579:0
2573:B
2567:=
2562:1
2559:m
2555:p
2539:-
2536:v
2516:2
2510:s
2507:m
2503:c
2487:+
2484:v
2466:2
2460:s
2457:m
2453:c
2433:v
2427:+
2424:v
2405:)
2391:2
2381:2
2376:s
2372:c
2366:2
2361:A
2357:v
2353:4
2345:4
2340:s
2337:m
2333:c
2322:2
2317:s
2314:m
2310:c
2305:(
2299:2
2296:1
2291:=
2284:2
2280:k
2274:2
2264:=
2259:2
2242:v
2233:0
2230:B
2224:k
2188:)
2180:2
2176:k
2171:/
2165:2
2160:s
2156:c
2150:2
2145:A
2141:v
2135:2
2126:k
2122:4
2114:4
2109:s
2106:m
2102:c
2091:2
2086:s
2083:m
2079:c
2074:(
2068:2
2063:2
2059:k
2053:=
2048:2
2017:0
2014:=
2009:2
2004:s
2000:c
1994:2
1989:A
1985:v
1979:2
1970:k
1964:2
1960:k
1956:+
1951:2
1946:s
1943:m
1939:c
1933:2
1929:k
1923:2
1910:4
1883:k
1877:A
1870:v
1863:A
1851:y
1825:2
1820:s
1816:c
1812:+
1807:2
1802:A
1798:v
1792:=
1787:s
1784:m
1780:c
1752:0
1749:=
1745:)
1739:2
1734:s
1730:c
1724:2
1719:A
1715:v
1709:2
1700:k
1694:2
1690:k
1686:+
1681:2
1676:s
1673:m
1669:c
1663:2
1659:k
1653:2
1640:4
1631:(
1626:)
1620:2
1611:k
1605:2
1600:A
1596:v
1587:2
1578:(
1557:0
1551:0
1543:/
1541:0
1538:B
1533:A
1529:v
1517:0
1512:/
1510:0
1499:s
1495:c
1476:0
1472:=
1467:)
1459:1
1456:z
1452:v
1442:1
1439:y
1435:v
1425:1
1422:x
1418:v
1411:(
1404:)
1396:2
1387:k
1381:2
1376:s
1372:c
1363:2
1353:0
1342:k
1332:k
1326:2
1321:s
1317:c
1306:0
1299:2
1290:k
1284:2
1279:A
1275:v
1266:2
1256:0
1243:k
1233:k
1227:2
1222:s
1218:c
1209:0
1202:2
1193:k
1187:2
1182:s
1178:c
1169:2
1165:k
1159:2
1154:A
1150:v
1141:2
1130:(
1114:0
1111:B
1102:k
1093:k
1083:k
1077:0
1074:B
1069:z
1052:.
1041:1
1036:v
1031:)
1026:0
1021:B
1012:k
1008:(
1000:0
995:B
990:)
985:1
980:v
971:k
967:(
961:=
956:1
951:B
928:,
917:1
912:v
903:k
894:0
890:p
883:=
878:1
874:p
852:,
841:1
836:v
827:k
818:0
810:=
805:1
786:1
783:v
758:.
753:1
749:p
744:k
733:0
722:0
717:B
709:)
704:1
699:B
690:k
686:(
680:=
675:1
670:v
663:0
632:k
618:0
589:,
586:)
581:0
576:B
566:1
561:v
556:(
547:=
541:t
531:1
526:B
497:,
494:0
491:=
487:)
479:0
468:1
447:0
443:p
437:1
433:p
426:(
419:t
389:,
386:0
383:=
378:1
373:v
360:0
352:+
346:t
336:1
304:,
299:1
295:p
281:0
270:0
265:B
257:)
252:1
247:B
236:(
230:=
224:t
214:1
209:v
196:0
174:p
165:B
23:.
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