27:
4057:. This is usually not the case in fusion, space and astrophysical plasmas. When this is not the case, or the interest is in smaller spatial scales, it may be necessary to use a kinetic model which properly accounts for the non-Maxwellian shape of the distribution function. However, because MHD is relatively simple and captures many of the important properties of plasma dynamics it is often qualitatively accurate and is therefore often the first model tried.
2831:
2803:
59:
35:
2741:. These can divide the fluid into magnetic domains, inside of which the currents are relatively weak. Current sheets in the solar corona are thought to be between a few meters and a few kilometers in thickness, which is quite thin compared to the magnetic domains (which are thousands to hundreds of thousands of kilometers across). Another example is in the Earth's
3231:
2730:
3938:
2300:
4453:
fueled by potassium-seeded coal combustion gas showed potential for more efficient energy conversion (the absence of solid moving parts allows operation at higher temperatures), but failed due to cost-prohibitive technical difficulties. One major engineering problem was the failure of the wall of the
4421:
or MHD propulsor is a method for propelling seagoing vessels using only electric and magnetic fields with no moving parts, using magnetohydrodynamics. The working principle involves electrification of the propellant (gas or water) which can then be directed by a magnetic field, pushing the vehicle in
4333:
derived from ideal MHD describes the equilibrium of axisymmetric toroidal plasma in a tokamak. In tokamak experiments, the equilibrium during each discharge is routinely calculated and reconstructed, which provides information on the shape and position of the plasma controlled by currents in external
4001:
Electron
Magnetohydrodynamics (EMHD) describes small scales plasmas when electron motion is much faster than the ion one. The main effects are changes in conservation laws, additional resistivity, importance of electron inertia. Many effects of Electron MHD are similar to effects of the Two fluid MHD
3804:
Extended MHD describes a class of phenomena in plasmas that are higher order than resistive MHD, but which can adequately be treated with a single fluid description. These include the effects of Hall physics, electron pressure gradients, finite Larmor Radii in the particle gyromotion, and electron
2710:
When the fluid cannot be considered as completely conductive, but the other conditions for ideal MHD are satisfied, it is possible to use an extended model called resistive MHD. This includes an extra term in Ohm's Law which models the collisional resistivity. Generally MHD computer simulations are
2093:
4093:
2417:
in the fluidâfor example, if a set of magnetic field lines are tied into a knot, then they will remain so as long as the fluid has negligible resistivity. This difficulty in reconnecting magnetic field lines makes it possible to store energy by moving the fluid or the source of the magnetic field.
2408:
which states that the bulk fluid and embedded magnetic field are constrained to move together such that one can be said to be "tied" or "frozen" to the other. Therefore, any two points that move with the bulk fluid velocity and lie on the same magnetic field line will continue to lie on the same
4052:
Another limitation of MHD (and fluid theories in general) is that they depend on the assumption that the plasma is strongly collisional (this is the first criterion listed above), so that the time scale of collisions is shorter than the other characteristic times in the system, and the particle
4483:
An important task in cancer research is developing more precise methods for delivery of medicine to affected areas. One method involves the binding of medicine to biologically compatible magnetic particles (such as ferrofluids), which are guided to the target via careful placement of permanent
4103:
Based on the MHD equations, Glatzmaier and Paul
Roberts have made a supercomputer model of the Earth's interior. After running the simulations for thousands of years in virtual time, the changes in Earth's magnetic field can be studied. The simulation results are in good agreement with the
1800:
3008:
2683:(from collisional resistivity) to be hundreds to thousands of years, much longer than the actual lifetime of a sunspotâso it would seem reasonable to ignore the resistivity. By contrast, a meter-sized volume of seawater has a magnetic diffusion time measured in milliseconds.
2699:, introducing small spatial scales into the system over which ideal MHD is broken and magnetic diffusion can occur quickly. When this happens, magnetic reconnection may occur in the plasma to release stored magnetic energy as waves, bulk mechanical acceleration of material,
995:. AlfvĂ©n initially referred to these waves as "electromagneticâhydrodynamic waves"; however, in a later paper he noted, "As the term 'electromagneticâhydrodynamic waves' is somewhat complicated, it may be convenient to call this phenomenon 'magnetoâhydrodynamic' waves."
4068:, a wide range of instabilities, chemical separation in space plasmas and electron runaway. In the case of ultra-high intensity laser interactions, the incredibly short timescales of energy deposition mean that hydrodynamic codes fail to capture the essential physics.
4186:
MHD forms the framework for understanding how populations of plasma interact within the local geospace environment. Researchers have developed global models using MHD to simulate phenomena within Earth's magnetosphere, such as the location of Earth's
2626:
1398:
4287:. That conservation would imply that as the mass concentrated in the center of the cloud to form the Sun, it would spin faster, much like a skater pulling their arms in. The high speed of rotation predicted by early theories would have flung the
3831:
1626:
1702:
1427:. As with any fluid description to a kinetic system, a closure approximation must be applied to highest moment of the particle distribution equation. This is often accomplished with approximations to the heat flux through a condition of
2184:
2706:
Magnetic reconnection in highly conductive systems is important because it concentrates energy in time and space, so that gentle forces applied to a plasma for long periods of time can cause violent explosions and bursts of radiation.
1920:
1312:
4433:, assigned his senior-year undergraduate students to develop a submarine with this new propulsion system. In the early 1990s, a foundation in Japan (Ship & Ocean Foundation (Minato-ku, Tokyo)) built an experimental boat, the
1993:
3794:). This diffusivity leads to a breaking in the magnetic topology; magnetic field lines can 'reconnect' when they collide. Usually this term is small and reconnections can be handled by thinking of them as not dissimilar to
2694:
exist that can increase the effective resistivity of the plasma by factors of more than 10. The enhanced resistivity is usually the result of the formation of small scale structure like current sheets or fine scale magnetic
1978:
4904:
Khabarova, O.; Malandraki, O.; Malova, H.; Kislov, R.; Greco, A.; Bruno, R.; Pezzi, O.; Servidio, S.; Li, Gang; Matthaeus, W.; Le Roux, J.; Engelbrecht, N. E.; Pecora, F.; Zelenyi, L.; Obridko, V.; Kuznetsov, V. (2021).
4104:
observations as the simulations have correctly predicted that the Earth's magnetic field flips every few hundred thousand years. During the flips, the magnetic field does not vanish altogetherâit just gets more complex.
2679:. This means that solutions to the ideal MHD equations are only applicable for a limited time for a region of a given size before diffusion becomes too important to ignore. One can estimate the diffusion time across a
3825:
term" in Ohm's law, a frequent simplification made in magnetic fusion theory. Hall-magnetohydrodynamics (HMHD) takes into account this electric field description of magnetohydrodynamics, and Ohm's law takes the form
1713:
3226:{\displaystyle {\frac {\omega }{k}}=\left({\tfrac {1}{2}}\left(v_{A}^{2}+v_{s}^{2}\right)\pm {\tfrac {1}{2}}{\sqrt {\left(v_{A}^{2}+v_{s}^{2}\right)^{2}-4v_{s}^{2}v_{A}^{2}\cos ^{2}\theta }}\right)^{\frac {1}{2}}}
4484:
magnets on the external body. Magnetohydrodynamic equations and finite element analysis are used to study the interaction between the magnetic fluid particles in the bloodstream and the external magnetic field.
1855:
2500:
3470:
3289:
is the ideal gas speed of sound. The plus branch corresponds to the fast-MHD wave mode and the minus branch corresponds to the slow-MHD wave mode. A summary of the properties of these waves is provided:
2997:
1242:
3284:
2932:
2553:
3757:
3649:
3501:
3340:
2647:
The resistivity due to these collisions is small. In particular, the typical magnetic diffusion times over any scale length present in the system must be longer than any time scale of interest.
5953:
Vencels, Juris; Jakovics, Andris; Geza, Vadims (2017). "Simulation of 3D MHD with free surface using Open-Source EOF-Library: levitating liquid metal in an alternating electromagnetic field".
3815:. As a result, the electron and ion momenta must be treated separately. This description is more closely tied to Maxwell's equations as an evolution equation for the electric field exists.
2172:
5838:
Conference Record of the 2000 IEEE Industry
Applications Conference. Thirty-Fifth IAS Annual Meeting and World Conference on Industrial Applications of Electrical Energy (Cat. No.00CH37129)
3371:
2857:
These modes have phase velocities that are independent of the magnitude of the wavevector, so they experience no dispersion. The phase velocity depends on the angle between the wave vector
2640:
The plasma is strongly collisional, so that the time scale of collisions is shorter than the other characteristic times in the system, and the particle distributions are therefore close to
3614:
5710:
1189:
616:
2561:
4242:. Most astrophysical systems are not in local thermal equilibrium, and therefore require an additional kinematic treatment to describe all the phenomena within the system (see
1323:
3704:
3559:
2453:
2367:
3933:{\displaystyle \mathbf {E} +\mathbf {v} \times \mathbf {B} \underbrace {-{\frac {1}{n_{e}e}}(\mathbf {J} \times \mathbf {B} )} _{\text{Hall current term}}=\eta \mathbf {J} ,}
589:
2131:
2323:
In view of the infinite conductivity, every motion (perpendicular to the field) of the liquid in relation to the lines of force is forbidden because it would give infinite
1550:
1528:
1056:
1030:
1160:
1133:
1106:
4544:
3795:
1558:
601:
1637:
2295:{\displaystyle \mathbf {J} \times \mathbf {B} ={\frac {\left(\mathbf {B} \cdot \nabla \right)\mathbf {B} }{\mu _{0}}}-\nabla \left({\frac {B^{2}}{2\mu _{0}}}\right),}
1483:
1076:
4368:
4038:
the (resistive) MHD equations can be reduced to a set of four closed scalar equations. This allows for, amongst other things, more efficient numerical calculations.
3969:
3407:
1506:
4143:. Researchers are attempting to learn more about this correlation to find out whether this method can be used as part of an early warning system for earthquakes.
3989:
3725:
3428:
1460:
4291:
apart before it could have formed. However, magnetohydrodynamic effects transfer the Sun's angular momentum into the outer solar system, slowing its rotation.
4089:. These eddies develop a magnetic field which boosts Earth's original magnetic fieldâa process which is self-sustaining and is called the geomagnetic dynamo.
1866:
1250:
2088:{\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}=\nabla \times (\mathbf {v} \times \mathbf {B} )+{\frac {\eta }{\mu _{0}}}\nabla ^{2}\mathbf {B} ,}
5620:
5741:
4081:
Beneath the Earth's mantle lies the core, which is made up of two parts: the solid inner core and liquid outer core. Both have significant quantities of
852:
621:
4340:
theory is known to govern the operational limits of tokamaks. For example, the ideal MHD kink modes provide hard limits on the achievable plasma beta (
4697:
26:
3995:. The most important difference is that in the absence of field line breaking, the magnetic field is tied to the electrons and not to the bulk fluid.
2775:. There are three MHD wave modes that can be derived from the linearized ideal-MHD equations for a fluid with a uniform and constant magnetic field:
1931:
631:
5062:
4128:, although a subsequent study indicates that this was little more than a sensor malfunction. On December 9, 2010, geoscientists announced that the
4426:
4271:
Previously, theories describing the formation of the Sun and planets could not explain how the Sun has 99.87% of the mass, yet only 0.54% of the
1795:{\displaystyle \rho \left({\frac {\partial }{\partial t}}+\mathbf {v} \cdot \nabla \right)\mathbf {v} =\mathbf {J} \times \mathbf {B} -\nabla p,}
5527:
5756:
3764:
The MHD oscillations will be damped if the fluid is not perfectly conducting but has a finite conductivity, or if viscous effects are present.
456:
5717:
4471:
Industrial MHD problems can be modeled using the open-source software EOF-Library. Two simulation examples are 3D MHD with a free surface for
5513:
5428:
4549:
471:
93:
4214:
have the potential to cause extensive damage to satellites and infrastructure, thus it is crucial that such events are detected early. The
4425:
The first prototype of this kind of propulsion was built and tested in 1965 by
Steward Way, a professor of mechanical engineering at the
6112:
5076:
4054:
2641:
38:
Simulation of the OrszagâTang MHD vortex problem, a well-known model problem for testing the transition to supersonic 2D MHD turbulence
5191:
Thomas, J. N.; Love, J. J.; Johnston, M. J. S. (April 2009). "On the reported magnetic precursor of the 1989 Loma Prieta earthquake".
4494:
1811:
3821:
In 1960, M. J. Lighthill criticized the applicability of ideal or resistive MHD theory for plasmas. It concerned the neglect of the "
2461:
351:
5853:
4777:
4576:
83:
5313:"Global Magnetohydrodynamic Simulations: Performance Quantification of Magnetopause Distances and Convection Potential Predictions"
3767:
MHD waves and oscillations are a popular tool for the remote diagnostics of laboratory and astrophysical plasmas, for example, the
3434:
2943:
4215:
4200:
266:
5166:
4732:
4529:
4371:
3242:
2696:
1194:
845:
611:
88:
20:
5795:
2888:
5010:
2508:
2389:
under consideration. Consequently, processes in ideal MHD that convert magnetic energy into kinetic energy, referred to as
6107:
4524:
2724:
1420:
626:
331:
5125:
4430:
3732:
3624:
3476:
3315:
491:
481:
466:
231:
98:
5112:
Fraser-Smith, Antony C.; Bernardi, A.; McGill, P. R.; Ladd, M. E.; Helliwell, R. A.; Villard Jr., O. G. (August 1990).
4330:
531:
221:
4121:
2373:
is small relative to the other terms such that it can be taken to be equal to zero. This occurs in the limit of large
784:
659:
556:
5869:
Kenjeres, S.; Hanjalic, K. (2000). "On the implementation of effects of
Lorentz force in turbulence closure models".
3002:
is the Alfvén speed. This branch corresponds to the shear Alfvén mode. Additionally the dispersion equation gives
2147:
989:
titled "Existence of
ElectromagneticâHydrodynamic Waves" which outlined his discovery of what are now referred to as
451:
3346:
4600:
4568:
4387:
4085:. The liquid outer core moves in the presence of the magnetic field and eddies are set up into the same due to the
3565:
284:
5609:
5100:
4514:
4461:, MHD is studied as a fluid pump for producing a continuous, nonpulsating flow in a complex microchannel design.
4418:
4097:
2378:
838:
799:
326:
316:
256:
251:
191:
5235:
2737:
In many MHD systems most of the electric current is compressed into thin nearly-two-dimensional ribbons termed
2374:
2175:
769:
649:
336:
4035:
2621:{\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}=\nabla \times (\mathbf {v} \times \mathbf {B} ).}
1165:
890:
774:
744:
176:
166:
161:
4394:. Accuracy improves with the size of the sensor. The sensor is capable of surviving in harsh environments.
1393:{\displaystyle \mathbf {v} ={\frac {1}{\rho }}\sum _{\sigma }m_{\sigma }n_{\sigma }\mathbf {u} _{\sigma }.}
6039:"The Behaviors of Ferro-Magnetic Nano-Particles In and Around Blood Vessels under Applied Magnetic Fields"
5042:
2405:
2306:
1486:
968:
596:
366:
141:
1003:
In MHD, motion in the fluid is described using linear combinations of the mean motions of the individual
4957:
Witalis, E.A. (1986). "Hall
Magnetohydrodynamics and Its Applications to Laboratory and Cosmic Plasma".
4391:
4231:
4180:
4140:
4061:
4007:
2712:
2700:
2419:
694:
381:
371:
321:
311:
58:
5481:
Kennel, C.F.; Arons, J.; Blandford, R.; Coroniti, F.; Israel, M.; Lanzerotti, L.; Lightman, A. (1985).
3660:
3515:
2671:
In an imperfectly conducting fluid the magnetic field can generally move through the fluid following a
2433:
2347:
5482:
5361:
4024:
MHD is also often used for collisionless plasmas. In that case the MHD equations are derived from the
6050:
6001:
5917:
5878:
5664:
5574:
5493:
5373:
5324:
5275:
5239:
5200:
5134:
4962:
4868:
4670:
4627:
4504:
4499:
4243:
4211:
2134:
819:
719:
684:
436:
301:
201:
186:
121:
5766:
4283:
such as the cloud of gas and dust from which the Sun was formed, mass and angular momentum are both
4092:
2101:
226:
5804:
4519:
4472:
4264:
may be the long-term effect of magnetic drag at the poles of the Sun, an MHD phenomenon due to the
4235:
4129:
4117:
2680:
1621:{\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \left(\rho \mathbf {v} \right)=0,}
1404:
901:
779:
759:
754:
561:
546:
431:
401:
296:
1533:
1511:
1039:
1013:
5906:"EOF-Library: Open-source Elmer FEM and OpenFOAM coupler for electromagnetics and fluid dynamics"
5781:
Operation of the
Thruster for Superconducting Electromagnetohydrodynamic Propu1sion Ship YAMATO 1
5692:
5590:
5564:
5519:
5434:
5293:
5265:
4978:
4926:
4886:
4837:
4643:
4465:
4294:
Breakdown of ideal MHD (in the form of magnetic reconnection) is known to be the likely cause of
3772:
2676:
2672:
2382:
1984:
1697:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {p}{\rho ^{\gamma }}}\right)=0,}
1432:
1408:
1138:
1111:
1084:
654:
394:
196:
156:
5408:
4817:
4318:
MHD describes a wide range of physical phenomena occurring in fusion plasmas in devices such as
2662:, and time scales much longer than the ion gyration time (system is smooth and slowly evolving).
1416:
5552:
5362:"Results from the LyonâFedderâMobarry global magnetospheric model for the electrojet challenge"
5254:"Enhanced ULF radiation observed by DEMETER two months around the strong 2010 Haiti earthquake"
4906:
4422:
the opposite direction. Although some working prototypes exist, MHD drives remain impractical.
6084:
6066:
6019:
5970:
5935:
5849:
5735:
5509:
5424:
5389:
5342:
5158:
5150:
4798:
4773:
4728:
4572:
3992:
2745:, where current sheets separate topologically distinct domains, isolating most of the Earth's
2651:
2414:
2310:
1428:
1412:
714:
4163:. Areas researched within space physics encompass a large number of topics, ranging from the
1468:
1061:
6074:
6058:
6009:
6005:
5962:
5925:
5886:
5841:
5682:
5672:
5582:
5501:
5416:
5381:
5332:
5283:
5208:
5142:
4970:
4918:
4876:
4829:
4678:
4635:
4539:
4411:
4383:
4347:
4337:
4284:
4272:
4239:
4218:(SWPC) runs MHD models to predict the arrival and impacts of space weather events at Earth.
2762:
2687:
2686:
Even in physical systemsâwhich are large and conductive enough that simple estimates of the
2413:
by fluid flows in the system. The connection between the fluid and magnetic field fixes the
2332:
1004:
984:
980:
964:
905:
897:
814:
729:
689:
679:
566:
521:
504:
421:
356:
126:
50:
4191:(the boundary between the Earth's magnetic field and the solar wind), the formation of the
3947:
3385:
1491:
5799:
5761:
5089:
4407:
4086:
4025:
2418:
The energy can then become available if the conditions for ideal MHD break down, allowing
1463:
1008:
749:
674:
669:
536:
411:
376:
271:
236:
136:
6014:
5990:"Numerical and experimental study of liquid metal stirring by rotating permanent magnets"
1983:
Taking the curl of this equation and using AmpĂšre's law and
Faraday's law results in the
789:
6054:
5921:
5882:
5836:
Fujisaki, Keisuke (Oct 2000). "In-mold electromagnetic stirring in continuous casting".
5668:
5578:
5497:
5377:
5328:
5279:
5204:
5138:
4966:
4872:
4674:
4631:
1915:{\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}=-\nabla \times \mathbf {E} ,}
1307:{\displaystyle \mathbf {J} =\sum _{\sigma }n_{\sigma }q_{\sigma }\mathbf {u} _{\sigma }}
6079:
6038:
4440:
4261:
4065:
3974:
3812:
3798:; this process has been shown to be important in the Earth-Solar magnetic interactions.
3710:
3413:
2659:
1445:
1079:
1033:
991:
944:
709:
704:
526:
416:
341:
291:
241:
214:
171:
146:
116:
109:
5890:
904:. It is primarily concerned with the low-frequency, large-scale, magnetic behavior in
6101:
5696:
5594:
5523:
5438:
4930:
4890:
4841:
4833:
4661:
FĂ€lthammar, Carl-Gunne (October 2007). "The discovery of magnetohydrodynamic waves".
4458:
4450:
4299:
4280:
4265:
4253:
4207:
4172:
4160:
4015:
4011:
2742:
2738:
2733:
Schematic view of the different current systems which shape the Earth's magnetosphere
2690:
suggest that the resistivity can be ignoredâresistivity may still be important: many
2655:
2141:
913:
824:
809:
794:
734:
446:
361:
346:
261:
246:
151:
5297:
5227:
4982:
958:
938:
5989:
5822:
5637:
5311:
Mukhopadhyay, Agnit; Jia, Xianzhe; Welling, Daniel T.; Liemohn, Michael W. (2021).
5113:
5047:
4647:
4403:
4341:
4276:
4227:
4192:
4188:
4156:
3822:
3768:
2386:
2324:
1244:, and the motion of the fluid can be described by the current density expressed as
921:
909:
804:
699:
664:
606:
541:
426:
306:
181:
2370:
1424:
461:
34:
5792:
5505:
4748:
Alfvén, Hannes (1943). "On the
Existence of Electromagnetic-Hydrodynamic Waves".
5930:
5905:
5457:
5385:
5017:
4682:
4534:
4323:
4302:
over a sunspot can store energy that is released suddenly as a burst of motion,
4295:
4113:
2691:
950:
925:
870:
724:
576:
406:
68:
6062:
5586:
5288:
5253:
5212:
4922:
4881:
4856:
4060:
Effects which are essentially kinetic and not captured by fluid models include
1973:{\displaystyle \mathbf {E} +\mathbf {v} \times \mathbf {B} =\eta \mathbf {J} .}
5746:
D.Titterton, J.Weston, Strapdown Inertial Navigation Technology, chapter 4.3.2
5420:
5337:
5312:
5252:
Athanasiou, M; Anagnostopoulos, G; Iliopoulos, A; Pavlos, G; David, K (2010).
4509:
4257:
4196:
4176:
4164:
4125:
2830:
2802:
2750:
2746:
917:
441:
6070:
6023:
5974:
5939:
5803:, M. Mitchner and Charles H. Kruger, Jr., Mechanical Engineering Department,
5393:
5346:
5154:
4974:
979:
The MHD description of electrically conducting fluids was first developed by
5845:
5146:
5063:"Earth's contrasting inner core rotation and magnetic field rotation linked"
4307:
4288:
3787:
Resistive MHD describes magnetized fluids with finite electron diffusivity (
1439:
764:
739:
551:
476:
73:
6088:
5840:. Industry Applications Conference. Vol. 4. IEEE. pp. 2591â2598.
5162:
4943:
M. J. Lighthill, "Studies on MHD waves and other anisotropic wave motion,"
2711:
at least somewhat resistive because their computational grid introduces a
5677:
5652:
4435:
2410:
516:
511:
131:
5966:
2555:
in the induction equation vanishes giving the ideal induction equation,
5687:
4319:
4249:
4003:
2398:
2327:. Thus the matter of the liquid is "fastened" to the lines of force...
866:
486:
5807:. See Ch. 9 "Magnetohydrodynamic (MHD) Power Generation", pp. 214â230.
5114:"Low-Frequency Magnetic Field Measurements Near the Epicenter of the M
4802:
4794:
5988:
Dzelme, V.; Jakovics, A.; Vencels, J.; Köppen, D.; Baake, E. (2018).
4639:
4618:
Alfvén, H (1942). "Existence of Electromagnetic-Hydrodynamic Waves".
4444:
4168:
571:
78:
5415:, Geophysical Monograph Series (1 ed.), Wiley, pp. 43â65,
5228:"Spacecraft Saw ULF Radio Emissions over Haiti before January Quake"
2729:
5569:
4382:
Magnetohydrodynamic sensors are used for precision measurements of
3811:
Two-fluid MHD describes plasmas that include a non-negligible Hall
5411:, in Gannon, Jennifer L.; Swidinsky, Andrei; Xu, Zhonghua (eds.),
5270:
4475:
melting, and liquid metal stirring by rotating permanent magnets.
4468:
process of metals to suppress instabilities and control the flow.
4303:
4152:
4133:
4120:(ULF) activity. A remarkable example of this occurred before the
4091:
2728:
2658:
perpendicular to the field, long enough along the field to ignore
893:
33:
25:
5490:
Unstable Current Systems and Plasma Instabilities in Astrophysics
5090:
NOVA | Magnetic Storm | What Drives Earth's Magnetic Field? | PBS
5043:"Why Earth's Inner and Outer Cores Rotate in Opposite Directions"
5553:"Relativistic fluid dynamics: Physics for many different scales"
5409:"Magnetohydrodynamic Models of B and Their Use in GIC Estimates"
4082:
2394:
1850:{\displaystyle \mu _{0}\mathbf {J} =\nabla \times \mathbf {B} ,}
5413:
Geomagnetically Induced Currents from the Sun to the Power Grid
4310:
when the main current sheet collapses, reconnecting the field.
4132:
satellite observed a dramatic increase in ULF radio waves over
2495:{\displaystyle \mathbf {E} +\mathbf {v} \times \mathbf {B} =0.}
5610:"Lecture Notes on Introduction to Plasma Astrophysics (Draft)"
4907:"Current Sheets, Plasmoids and Flux Ropes in the Heliosphere"
3465:{\displaystyle {\frac {\mathbf {B} }{\sqrt {\mu _{0}\rho }}}}
4607:. Princeton University Department of Astrophysical Sciences.
2992:{\displaystyle v_{A}={\frac {B_{0}}{\sqrt {\mu _{0}\rho }}}}
30:
The plasma making up the Sun can be modeled as an MHD system
5360:
Wiltberger, M.; Lyon, J. G.; Goodrich, C. C. (2003-07-01).
1403:
MHD can be described by a set of equations consisting of a
6037:
Nacev, A.; Beni, C.; Bruno, O.; Shapiro, B. (2011-03-01).
5904:
Vencels, Juris; RĂ„back, Peter; GeĆŸa, Vadims (2019-01-01).
2767:
The wave modes derived using the MHD equations are called
4439:, which used a magnetohydrodynamic drive incorporating a
4268:
shape assumed by the extended magnetic field of the Sun.
2422:
that releases the stored energy from the magnetic field.
5994:
IOP Conference Series: Materials Science and Engineering
5483:"Perspectives on Space and Astrophysical Plasma Physics"
4698:"On the Existence of Electromagnetic-Hydrodynamic Waves"
1237:{\textstyle \rho =\sum _{\sigma }m_{\sigma }n_{\sigma }}
3279:{\displaystyle v_{s}={\sqrt {\frac {\gamma p}{\rho }}}}
3090:
3032:
2927:{\displaystyle {\frac {\omega }{k}}=v_{A}\cos \theta }
1438:
In the adiabatic limit, that is, the assumption of an
1197:
4350:
4234:(space between the planets), and possibly within the
3977:
3950:
3834:
3735:
3713:
3663:
3627:
3568:
3518:
3479:
3437:
3416:
3388:
3349:
3318:
3245:
3011:
2946:
2891:
2564:
2548:{\displaystyle \eta \nabla ^{2}\mathbf {B} /\mu _{0}}
2511:
2464:
2436:
2350:
2187:
2150:
2104:
1996:
1934:
1869:
1814:
1716:
1640:
1561:
1536:
1514:
1494:
1471:
1448:
1326:
1253:
1168:
1141:
1114:
1087:
1064:
1042:
1016:
5366:
Journal of Atmospheric and Solar-Terrestrial Physics
5011:"Reduced MHD equations for low aspect ratio plasmas"
4663:
Journal of Atmospheric and Solar-Terrestrial Physics
963:
meaning movement. The field of MHD was initiated by
4002:and the Hall MHD. EMHD is especially important for
3510:
neither transversal nor longitudinal; compressional
2873:with respect to the time independent or bulk field
2650:Interest in length scales much longer than the ion
2305:where the first term on the right hand side is the
4770:Magnetic Reconnection: MHD Theory and Applications
4362:
3983:
3963:
3932:
3752:{\displaystyle \mathbf {Q} \parallel \mathbf {B} }
3751:
3719:
3698:
3644:{\displaystyle \mathbf {Q} \parallel \mathbf {k} }
3643:
3608:
3553:
3496:{\displaystyle \mathbf {Q} \parallel \mathbf {B} }
3495:
3464:
3422:
3401:
3365:
3335:{\displaystyle \mathbf {k} \parallel \mathbf {B} }
3334:
3278:
3225:
2991:
2926:
2620:
2547:
2494:
2455:vanishes in Ohm's law giving the ideal Ohm's law,
2447:
2404:A fundamental concept underlying ideal MHD is the
2361:
2294:
2166:
2125:
2087:
1972:
1914:
1849:
1794:
1696:
1620:
1544:
1522:
1500:
1477:
1454:
1392:
1306:
1236:
1183:
1154:
1127:
1100:
1070:
1050:
1024:
912:and has applications in multiple fields including
4857:"Magnetic reconnection: MHD theory and modelling"
4545:Shocks and discontinuities (magnetohydrodynamics)
4454:primary-coal combustion chamber due to abrasion.
2869:. An MHD wave propagating at an arbitrary angle
5101:Earth's Inconstant Magnetic Field â NASA Science
3664:
3519:
2675:with the resistivity of the plasma serving as a
4402:MHD is related to engineering problems such as
2321:
2167:{\displaystyle \mathbf {J} \times \mathbf {B} }
5819:Fundamentals and Applications of Microfluidics
5458:"What is Space Weather ? - Space Weather"
4772:(First ed.). Cambridge University Press.
3366:{\displaystyle \mathbf {k} \perp \mathbf {B} }
1317:and the center of mass velocity expressed as:
5653:"Solar Flares: Magnetohydrodynamic Processes"
4206:One prominent use of global MHD models is in
3609:{\displaystyle {\sqrt {v_{A}^{2}+v_{s}^{2}}}}
846:
8:
5871:International Journal of Heat and Fluid Flow
5193:Physics of the Earth and Planetary Interiors
4252:are caused by the Sun's magnetic fields, as
4136:in the month before the magnitude 7.0 M
4112:Some monitoring stations have reported that
2636:Ideal MHD is only strictly applicable when:
1552:can be described by the continuous equation
6043:Journal of Magnetism and Magnetic Materials
5651:Shibata, Kazunari; Magara, Tetsuya (2011).
5551:Andersson, Nils; Comer, Gregory L. (2021).
4565:Introduction to Modern Magnetohydrodynamics
1462:and isotropic temperature, a fluid with an
4855:Pontin, David I.; Priest, Eric R. (2022).
4763:
4761:
4759:
4718:
4716:
4714:
4260:is also governed by MHD. The differential
853:
839:
57:
41:
6078:
6013:
5929:
5686:
5676:
5568:
5336:
5317:Frontiers in Astronomy and Space Sciences
5287:
5269:
5258:Natural Hazards and Earth System Sciences
5016:. University of Wisconsin. Archived from
4880:
4727:. Cambridge: Cambridge University Press.
4349:
3976:
3955:
3949:
3922:
3910:
3896:
3888:
3873:
3863:
3857:
3851:
3843:
3835:
3833:
3744:
3736:
3734:
3712:
3687:
3674:
3662:
3636:
3628:
3626:
3598:
3593:
3580:
3575:
3569:
3567:
3542:
3529:
3517:
3488:
3480:
3478:
3450:
3440:
3438:
3436:
3415:
3393:
3387:
3358:
3350:
3348:
3327:
3319:
3317:
3259:
3250:
3244:
3212:
3193:
3183:
3178:
3168:
3163:
3147:
3136:
3131:
3118:
3113:
3101:
3089:
3075:
3070:
3057:
3052:
3031:
3012:
3010:
2977:
2966:
2960:
2951:
2945:
2909:
2892:
2890:
2607:
2599:
2571:
2565:
2563:
2539:
2530:
2525:
2519:
2510:
2481:
2473:
2465:
2463:
2440:
2435:
2354:
2349:
2276:
2262:
2256:
2238:
2228:
2212:
2204:
2196:
2188:
2186:
2174:can be expanded using AmpĂšre's law and a
2159:
2151:
2149:
2117:
2108:
2103:
2077:
2071:
2059:
2050:
2039:
2031:
2003:
1997:
1995:
1962:
1951:
1943:
1935:
1933:
1904:
1876:
1870:
1868:
1839:
1825:
1819:
1813:
1775:
1767:
1759:
1743:
1725:
1715:
1673:
1664:
1649:
1643:
1641:
1639:
1599:
1562:
1560:
1537:
1535:
1515:
1513:
1493:
1470:
1447:
1381:
1376:
1369:
1359:
1349:
1335:
1327:
1325:
1298:
1293:
1286:
1276:
1266:
1254:
1252:
1228:
1218:
1208:
1196:
1191:. The fluid's total mass density is then
1175:
1170:
1167:
1146:
1140:
1119:
1113:
1092:
1086:
1063:
1043:
1041:
1017:
1015:
5009:Kruger, S.E.; Hegna, C.C.; Callen, J.D.
4750:Arkiv för matematik, astronomi och fysik
4705:Arkiv för matematik, astronomi och fysik
3292:
4947:, London, vol. 252A, pp. 397â430, 1960.
4591:
4427:University of California, Santa Barbara
2792:Phase velocity plotted with respect to
2505:Similarly, the magnetic diffusion term
602:Electromagnetism and special relativity
49:
16:Model of electrically conducting fluids
5740:: CS1 maint: archived copy as title (
5733:
1184:{\displaystyle \mathbf {u} _{\sigma }}
4818:"Hydromagnetic stability of tokamaks"
4550:List of textbooks in electromagnetism
4116:are sometimes preceded by a spike in
2882:will satisfy the dispersion relation
2632:Applicability of ideal MHD to plasmas
622:Maxwell equations in curved spacetime
7:
5779:Setsuo Takezawa et al. (March 1995)
5608:Kunz, Matthew W. (9 November 2020).
4768:Priest, Eric; Forbes, Terry (2000).
4447:, and could travel at 15 km/h.
5492:. Vol. 107. pp. 537â552.
4959:IEEE Transactions on Plasma Science
3971:is the electron number density and
5817:Nguyen, N.T.; Wereley, S. (2006).
4995:W. Baumjohann and R. A. Treumann,
4495:Computational magnetohydrodynamics
2590:
2578:
2568:
2516:
2409:field line even as the points are
2344:, assumes that the resistive term
2249:
2220:
2068:
2022:
2010:
2000:
1898:
1883:
1873:
1833:
1783:
1751:
1731:
1727:
1650:
1644:
1585:
1573:
1565:
14:
5757:"Run Silent, Run Electromagnetic"
4429:. Way, on leave from his job at
4344:) and plasma current (set by the
3699:{\displaystyle \min(v_{A},v_{s})}
3554:{\displaystyle \max(v_{A},v_{s})}
2448:{\displaystyle \eta \mathbf {J} }
2430:In ideal MHD, the resistive term
2362:{\displaystyle \eta \mathbf {J} }
1058:. In a given fluid, each species
896:that treats all interpenetrating
5626:from the original on 2022-10-09.
4298:. The magnetic field in a solar
4201:geomagnetically induced currents
4151:The study of space plasmas near
3923:
3897:
3889:
3852:
3844:
3836:
3745:
3737:
3637:
3629:
3489:
3481:
3441:
3359:
3351:
3328:
3320:
2829:
2801:
2608:
2600:
2572:
2526:
2482:
2474:
2466:
2441:
2355:
2229:
2213:
2197:
2189:
2160:
2152:
2078:
2040:
2032:
2004:
1963:
1952:
1944:
1936:
1905:
1877:
1840:
1826:
1776:
1768:
1760:
1744:
1600:
1538:
1516:
1377:
1328:
1294:
1255:
1171:
1044:
1018:
933:magnetohydrodynamics
5657:Living Reviews in Solar Physics
5533:from the original on 2022-10-09
5226:KentuckyFC (December 9, 2010).
5172:from the original on 2022-10-09
4861:Living Reviews in Solar Physics
4216:Space Weather Prediction Center
2140:In the equation of motion, the
1805:the low-frequency AmpĂšre's law
6015:10.1088/1757-899X/424/1/012047
5407:Welling, Daniel (2019-09-25),
4999:, Imperial College Press, 1997
4725:Fundamentals of plasma physics
4530:Magnetohydrodynamic turbulence
4464:MHD can be implemented in the
4238:(space between the stars) and
3901:
3885:
3693:
3667:
3548:
3522:
2612:
2596:
2415:topology of the magnetic field
2126:{\displaystyle \eta /\mu _{0}}
2044:
2028:
21:Magnetohydrodynamics (journal)
19:For the academic journal, see
1:
5891:10.1016/S0142-727X(00)00017-5
4793:Rosenbluth, M. (April 1956).
4525:Magnetohydrodynamic generator
4048:Importance of kinetic effects
2725:Magnetosphere particle motion
983:in a 1942 paper published in
627:Relativistic electromagnetism
5765:. 1966-09-23. Archived from
5557:Living Reviews in Relativity
5506:10.1007/978-94-009-6520-1_63
5126:Geophysical Research Letters
4723:Bellan, Paul Murray (2006).
4599:Philip Hopkins (July 2004).
4515:Lorentz force velocity meter
1545:{\displaystyle \mathbf {E} }
1523:{\displaystyle \mathbf {B} }
1051:{\displaystyle \mathbf {v} }
1025:{\displaystyle \mathbf {J} }
967:, for which he received the
5931:10.1016/j.softx.2019.01.007
5386:10.1016/j.jastp.2003.08.003
5118:7.1 Loma Prieta Earthquake"
4683:10.1016/j.jastp.2006.08.021
4563:Galtier, Sebastien (2016).
4388:inertial navigation systems
4314:Magnetic confinement fusion
4122:1989 Loma Prieta earthquake
3380:transversal; incompressible
2309:and the second term is the
1155:{\displaystyle q_{\sigma }}
1128:{\displaystyle m_{\sigma }}
1101:{\displaystyle n_{\sigma }}
6129:
6113:Plasma theory and modeling
6063:10.1016/j.jmmm.2010.09.008
5587:10.1007/s41114-021-00031-6
5289:10.5194/nhess-11-1091-2011
5213:10.1016/j.pepi.2008.11.014
4997:Basic Space Plasma Physics
4923:10.1007/s11214-021-00814-x
4882:10.1007/s41116-022-00032-9
4834:10.1088/0029-5515/18/1/010
4569:Cambridge University Press
4473:electromagnetic levitation
4406:, liquid-metal cooling of
2760:
2722:
2340:The simplest form of MHD,
352:LiĂ©nardâWiechert potential
18:
5421:10.1002/9781119434412.ch3
5338:10.3389/fspas.2021.637197
4419:magnetohydrodynamic drive
3617:
3509:
3308:Direction of energy flow
3307:
3304:
3301:
3298:
3295:
2769:magnetohydrodynamic waves
2719:Structures in MHD systems
2667:Importance of resistivity
2375:magnetic Reynolds numbers
617:Mathematical descriptions
327:Electromagnetic radiation
317:Electromagnetic induction
257:Magnetic vector potential
252:Magnetic scalar potential
5236:Cambridge, Massachusetts
4975:10.1109/TPS.1986.4316632
4795:"Stability of the Pinch"
4414:casting (among others).
2176:vector calculus identity
6006:2018MS&E..424a2047D
5846:10.1109/IAS.2000.883188
5793:Partially Ionized Gases
5147:10.1029/GL017i009p01465
4696:Alfvén, Hannes (1943).
4605:www.astro.princeton.edu
4479:Magnetic drug targeting
4331:Grad-Shafranov equation
4256:theorized in 1919. The
4230:, including stars, the
3618:equal to phase velocity
2863:and the magnetic field
2785:Fast magnetosonic waves
2782:Slow magnetosonic waves
1707:the equation of motion
1478:{\displaystyle \gamma }
1071:{\displaystyle \sigma }
891:electrically conducting
167:Electrostatic induction
162:Electrostatic discharge
4961:. PS-14 (6): 842â848.
4945:Phil. Trans. Roy. Soc.
4364:
4363:{\displaystyle q>2}
4181:coronal mass ejections
4100:
4098:Earth's magnetic field
4018:, and plasma switches.
3985:
3965:
3934:
3753:
3721:
3700:
3655:Slow magnetosonic wave
3645:
3610:
3555:
3507:Fast magnetosonic wave
3497:
3466:
3424:
3403:
3367:
3336:
3280:
3227:
2993:
2928:
2734:
2622:
2549:
2496:
2449:
2406:frozen-in flux theorem
2363:
2329:
2307:magnetic tension force
2296:
2168:
2127:
2089:
1974:
1916:
1851:
1796:
1698:
1631:the equation of state
1622:
1546:
1524:
1502:
1487:electrical resistivity
1479:
1456:
1394:
1308:
1238:
1185:
1162:, and a mean velocity
1156:
1129:
1102:
1072:
1052:
1026:
969:Nobel Prize in Physics
957:
949:
937:
883:magneto-fluid dynamics
597:Electromagnetic tensor
39:
31:
4911:Space Science Reviews
4816:Wesson, J.A. (1978).
4431:Westinghouse Electric
4392:aerospace engineering
4365:
4232:interplanetary medium
4095:
4008:magnetic reconnection
3986:
3966:
3964:{\displaystyle n_{e}}
3935:
3754:
3722:
3701:
3646:
3611:
3556:
3498:
3467:
3425:
3404:
3402:{\displaystyle v_{A}}
3368:
3337:
3302:Limiting phase speeds
3281:
3228:
2994:
2929:
2732:
2723:Further information:
2713:numerical resistivity
2701:particle acceleration
2623:
2550:
2497:
2450:
2420:magnetic reconnection
2364:
2297:
2169:
2128:
2090:
1975:
1917:
1852:
1797:
1699:
1623:
1547:
1530:, and electric field
1525:
1503:
1501:{\displaystyle \eta }
1480:
1457:
1395:
1309:
1239:
1186:
1157:
1130:
1103:
1073:
1053:
1027:
900:together as a single
590:Covariant formulation
382:Synchrotron radiation
322:Electromagnetic pulse
312:Electromagnetic field
37:
29:
6108:Magnetohydrodynamics
5955:Magnetohydrodynamics
5769:on January 14, 2009.
5678:10.12942/lrsp-2011-6
5240:TechnologyReview.com
5051:. 19 September 2013.
5023:on 25 September 2015
4505:Electromagnetic pump
4500:Electrohydrodynamics
4451:MHD power generation
4348:
4244:Astrophysical plasma
4212:Intense solar storms
3975:
3948:
3832:
3733:
3711:
3661:
3625:
3566:
3516:
3477:
3435:
3414:
3386:
3347:
3316:
3243:
3009:
2944:
2889:
2562:
2509:
2462:
2434:
2385:at the velocity and
2348:
2185:
2148:
2135:magnetic diffusivity
2102:
1994:
1932:
1867:
1812:
1714:
1638:
1559:
1534:
1512:
1492:
1469:
1446:
1324:
1251:
1195:
1166:
1139:
1112:
1085:
1062:
1040:
1014:
887:hydromagnetics
875:magnetohydrodynamics
632:Stressâenergy tensor
557:Reluctance (complex)
302:Displacement current
6055:2011JMMM..323..651N
5967:10.22364/mhd.53.4.5
5922:2019SoftX...9...68V
5883:2000IJHFF..21..329K
5805:Stanford University
5669:2011LRSP....8....6S
5617:astro.princeton.edu
5579:2021LRR....24....3A
5498:1985IAUS..107..537K
5378:2003JASTP..65.1213W
5329:2021FrASS...8...45M
5280:2011NHESS..11.1091A
5205:2009PEPI..173..207T
5139:1990GeoRL..17.1465F
4967:1986ITPS...14..842W
4873:2022LRSP...19....1P
4675:2007JASTP..69.1604F
4632:1942Natur.150..405A
4520:Magnetic flow meter
4370:requirement of the
4236:interstellar medium
4197:auroral electrojets
4155:and throughout the
4118:ultra low frequency
4036:multiscale analysis
3603:
3585:
3188:
3173:
3141:
3123:
3080:
3062:
2681:solar active region
2426:Ideal MHD equations
1405:continuity equation
955:meaning water, and
547:Magnetomotive force
432:Electromotive force
402:Alternating current
337:Jefimenko equations
297:Cyclotron radiation
5798:2008-09-05 at the
5232:Physics arXiv Blog
4466:continuous casting
4404:plasma confinement
4384:angular velocities
4360:
4101:
4053:distributions are
3981:
3961:
3930:
3915:
3908:
3773:Coronal seismology
3749:
3717:
3696:
3641:
3606:
3589:
3571:
3551:
3493:
3462:
3420:
3399:
3363:
3332:
3276:
3223:
3174:
3159:
3127:
3109:
3099:
3066:
3048:
3041:
2989:
2924:
2735:
2677:diffusion constant
2618:
2545:
2492:
2445:
2393:, cannot generate
2383:magnetic diffusion
2379:magnetic induction
2359:
2292:
2164:
2123:
2085:
1985:induction equation
1970:
1912:
1847:
1792:
1694:
1618:
1542:
1520:
1498:
1475:
1452:
1409:equation of motion
1390:
1354:
1304:
1271:
1234:
1213:
1181:
1152:
1135:, electric charge
1125:
1098:
1068:
1048:
1022:
395:Electrical network
232:Gauss magnetic law
197:Static electricity
157:Electric potential
40:
32:
5515:978-90-277-1887-7
5430:978-1-119-43434-4
5372:(11): 1213â1222.
5065:. 7 October 2013.
4669:(14): 1604â1608.
4626:(3805): 405â406.
4601:"O-T Vortex Test"
3993:elementary charge
3984:{\displaystyle e}
3913:
3912:Hall current term
3883:
3858:
3856:
3762:
3761:
3720:{\displaystyle 0}
3604:
3460:
3459:
3423:{\displaystyle 0}
3274:
3273:
3220:
3205:
3098:
3040:
3020:
2987:
2986:
2900:
2585:
2311:magnetic pressure
2283:
2244:
2065:
2017:
1890:
1738:
1679:
1658:
1580:
1508:, magnetic field
1455:{\displaystyle p}
1413:equation of state
1345:
1343:
1262:
1204:
902:continuous medium
863:
862:
562:Reluctance (real)
532:Gyratorâcapacitor
477:Resonant cavities
367:Maxwell equations
6120:
6093:
6092:
6082:
6034:
6028:
6027:
6017:
5985:
5979:
5978:
5950:
5944:
5943:
5933:
5901:
5895:
5894:
5866:
5860:
5859:
5833:
5827:
5826:
5814:
5808:
5789:
5783:
5777:
5771:
5770:
5753:
5747:
5745:
5739:
5731:
5729:
5728:
5722:
5716:. Archived from
5715:
5707:
5701:
5700:
5690:
5680:
5648:
5642:
5641:
5638:"Solar Activity"
5634:
5628:
5627:
5625:
5614:
5605:
5599:
5598:
5572:
5548:
5542:
5541:
5539:
5538:
5532:
5487:
5478:
5472:
5471:
5469:
5468:
5454:
5448:
5447:
5446:
5445:
5404:
5398:
5397:
5357:
5351:
5350:
5340:
5308:
5302:
5301:
5291:
5273:
5250:
5248:
5246:
5223:
5217:
5216:
5199:(3â4): 207â215.
5188:
5182:
5181:
5179:
5177:
5171:
5133:(9): 1465â1468.
5122:
5109:
5103:
5098:
5092:
5087:
5081:
5080:
5073:
5067:
5066:
5059:
5053:
5052:
5039:
5033:
5032:
5030:
5028:
5022:
5015:
5006:
5000:
4993:
4987:
4986:
4954:
4948:
4941:
4935:
4934:
4901:
4895:
4894:
4884:
4852:
4846:
4845:
4813:
4807:
4806:
4790:
4784:
4783:
4765:
4754:
4753:
4745:
4739:
4738:
4720:
4709:
4708:
4702:
4693:
4687:
4686:
4658:
4652:
4651:
4640:10.1038/150405d0
4615:
4609:
4608:
4596:
4582:
4540:Plasma stability
4408:nuclear reactors
4369:
4367:
4366:
4361:
4273:angular momentum
3990:
3988:
3987:
3982:
3970:
3968:
3967:
3962:
3960:
3959:
3939:
3937:
3936:
3931:
3926:
3914:
3911:
3909:
3904:
3900:
3892:
3884:
3882:
3878:
3877:
3864:
3855:
3847:
3839:
3793:
3758:
3756:
3755:
3750:
3748:
3740:
3726:
3724:
3723:
3718:
3705:
3703:
3702:
3697:
3692:
3691:
3679:
3678:
3650:
3648:
3647:
3642:
3640:
3632:
3615:
3613:
3612:
3607:
3605:
3602:
3597:
3584:
3579:
3570:
3560:
3558:
3557:
3552:
3547:
3546:
3534:
3533:
3502:
3500:
3499:
3494:
3492:
3484:
3471:
3469:
3468:
3463:
3461:
3455:
3454:
3445:
3444:
3439:
3429:
3427:
3426:
3421:
3408:
3406:
3405:
3400:
3398:
3397:
3372:
3370:
3369:
3364:
3362:
3354:
3341:
3339:
3338:
3333:
3331:
3323:
3293:
3285:
3283:
3282:
3277:
3275:
3269:
3261:
3260:
3255:
3254:
3232:
3230:
3229:
3224:
3222:
3221:
3213:
3211:
3207:
3206:
3198:
3197:
3187:
3182:
3172:
3167:
3152:
3151:
3146:
3142:
3140:
3135:
3122:
3117:
3102:
3100:
3091:
3085:
3081:
3079:
3074:
3061:
3056:
3042:
3033:
3021:
3013:
2998:
2996:
2995:
2990:
2988:
2982:
2981:
2972:
2971:
2970:
2961:
2956:
2955:
2933:
2931:
2930:
2925:
2914:
2913:
2901:
2893:
2881:
2872:
2868:
2862:
2850:
2833:
2822:
2805:
2795:
2763:Waves in plasmas
2688:Lundquist number
2627:
2625:
2624:
2619:
2611:
2603:
2586:
2584:
2576:
2575:
2566:
2554:
2552:
2551:
2546:
2544:
2543:
2534:
2529:
2524:
2523:
2501:
2499:
2498:
2493:
2485:
2477:
2469:
2454:
2452:
2451:
2446:
2444:
2368:
2366:
2365:
2360:
2358:
2336:
2301:
2299:
2298:
2293:
2288:
2284:
2282:
2281:
2280:
2267:
2266:
2257:
2245:
2243:
2242:
2233:
2232:
2227:
2223:
2216:
2205:
2200:
2192:
2173:
2171:
2170:
2165:
2163:
2155:
2132:
2130:
2129:
2124:
2122:
2121:
2112:
2094:
2092:
2091:
2086:
2081:
2076:
2075:
2066:
2064:
2063:
2051:
2043:
2035:
2018:
2016:
2008:
2007:
1998:
1979:
1977:
1976:
1971:
1966:
1955:
1947:
1939:
1921:
1919:
1918:
1913:
1908:
1891:
1889:
1881:
1880:
1871:
1856:
1854:
1853:
1848:
1843:
1829:
1824:
1823:
1801:
1799:
1798:
1793:
1779:
1771:
1763:
1758:
1754:
1747:
1739:
1737:
1726:
1703:
1701:
1700:
1695:
1684:
1680:
1678:
1677:
1665:
1659:
1657:
1653:
1647:
1642:
1627:
1625:
1624:
1619:
1608:
1604:
1603:
1581:
1579:
1571:
1563:
1551:
1549:
1548:
1543:
1541:
1529:
1527:
1526:
1521:
1519:
1507:
1505:
1504:
1499:
1484:
1482:
1481:
1476:
1461:
1459:
1458:
1453:
1399:
1397:
1396:
1391:
1386:
1385:
1380:
1374:
1373:
1364:
1363:
1353:
1344:
1336:
1331:
1313:
1311:
1310:
1305:
1303:
1302:
1297:
1291:
1290:
1281:
1280:
1270:
1258:
1243:
1241:
1240:
1235:
1233:
1232:
1223:
1222:
1212:
1190:
1188:
1187:
1182:
1180:
1179:
1174:
1161:
1159:
1158:
1153:
1151:
1150:
1134:
1132:
1131:
1126:
1124:
1123:
1107:
1105:
1104:
1099:
1097:
1096:
1077:
1075:
1074:
1069:
1057:
1055:
1054:
1049:
1047:
1031:
1029:
1028:
1023:
1021:
935:is derived from
898:particle species
889:) is a model of
855:
848:
841:
522:Electric machine
505:Magnetic circuit
467:Parallel circuit
457:Network analysis
422:Electric current
357:London equations
202:Triboelectricity
192:Potential energy
61:
51:Electromagnetism
42:
6128:
6127:
6123:
6122:
6121:
6119:
6118:
6117:
6098:
6097:
6096:
6036:
6035:
6031:
5987:
5986:
5982:
5952:
5951:
5947:
5903:
5902:
5898:
5868:
5867:
5863:
5856:
5835:
5834:
5830:
5816:
5815:
5811:
5800:Wayback Machine
5790:
5786:
5778:
5774:
5755:
5754:
5750:
5732:
5726:
5724:
5720:
5713:
5711:"Archived copy"
5709:
5708:
5704:
5650:
5649:
5645:
5636:
5635:
5631:
5623:
5612:
5607:
5606:
5602:
5550:
5549:
5545:
5536:
5534:
5530:
5516:
5485:
5480:
5479:
5475:
5466:
5464:
5462:swe.ssa.esa.int
5456:
5455:
5451:
5443:
5441:
5431:
5406:
5405:
5401:
5359:
5358:
5354:
5310:
5309:
5305:
5251:
5244:
5242:
5225:
5224:
5220:
5190:
5189:
5185:
5175:
5173:
5169:
5120:
5117:
5111:
5110:
5106:
5099:
5095:
5088:
5084:
5075:
5074:
5070:
5061:
5060:
5056:
5041:
5040:
5036:
5026:
5024:
5020:
5013:
5008:
5007:
5003:
4994:
4990:
4956:
4955:
4951:
4942:
4938:
4903:
4902:
4898:
4854:
4853:
4849:
4815:
4814:
4810:
4792:
4791:
4787:
4780:
4767:
4766:
4757:
4747:
4746:
4742:
4735:
4722:
4721:
4712:
4700:
4695:
4694:
4690:
4660:
4659:
4655:
4617:
4616:
4612:
4598:
4597:
4593:
4589:
4579:
4562:
4559:
4557:Further reading
4554:
4490:
4481:
4412:electromagnetic
4400:
4380:
4346:
4345:
4316:
4226:MHD applies to
4224:
4149:
4141:2010 earthquake
4139:
4110:
4087:Coriolis effect
4079:
4074:
4050:
4045:
4026:Vlasov equation
3973:
3972:
3951:
3946:
3945:
3869:
3868:
3859:
3830:
3829:
3788:
3781:
3731:
3730:
3709:
3708:
3683:
3670:
3659:
3658:
3623:
3622:
3564:
3563:
3538:
3525:
3514:
3513:
3475:
3474:
3446:
3433:
3432:
3412:
3411:
3389:
3384:
3383:
3345:
3344:
3314:
3313:
3262:
3246:
3241:
3240:
3189:
3108:
3104:
3103:
3047:
3043:
3030:
3026:
3025:
3007:
3006:
2973:
2962:
2947:
2942:
2941:
2905:
2887:
2886:
2880:
2874:
2870:
2864:
2858:
2855:
2854:
2853:
2852:
2851:
2848:
2841:
2836:
2834:
2825:
2824:
2823:
2820:
2813:
2808:
2806:
2797:
2796:
2793:
2765:
2759:
2727:
2721:
2669:
2634:
2577:
2567:
2560:
2559:
2535:
2515:
2507:
2506:
2460:
2459:
2432:
2431:
2428:
2391:ideal processes
2381:dominates over
2346:
2345:
2338:
2331:
2319:
2272:
2268:
2258:
2252:
2234:
2211:
2207:
2206:
2183:
2182:
2146:
2145:
2113:
2100:
2099:
2067:
2055:
2009:
1999:
1992:
1991:
1930:
1929:
1882:
1872:
1865:
1864:
1815:
1810:
1809:
1730:
1724:
1720:
1712:
1711:
1669:
1660:
1648:
1636:
1635:
1595:
1591:
1572:
1564:
1557:
1556:
1532:
1531:
1510:
1509:
1490:
1489:
1467:
1466:
1464:adiabatic index
1444:
1443:
1375:
1365:
1355:
1322:
1321:
1292:
1282:
1272:
1249:
1248:
1224:
1214:
1193:
1192:
1169:
1164:
1163:
1142:
1137:
1136:
1115:
1110:
1109:
1088:
1083:
1082:
1060:
1059:
1038:
1037:
1012:
1011:
1009:current density
1001:
977:
859:
830:
829:
645:
637:
636:
592:
582:
581:
537:Induction motor
507:
497:
496:
412:Current density
397:
387:
386:
377:Poynting vector
287:
285:Electrodynamics
277:
276:
272:Right-hand rule
237:Magnetic dipole
227:BiotâSavart law
217:
207:
206:
142:Electric dipole
137:Electric charge
112:
24:
17:
12:
11:
5:
6126:
6124:
6116:
6115:
6110:
6100:
6099:
6095:
6094:
6049:(6): 651â668.
6029:
5980:
5961:(4): 643â652.
5945:
5896:
5877:(3): 329â337.
5861:
5854:
5828:
5809:
5784:
5772:
5748:
5702:
5643:
5629:
5600:
5543:
5514:
5473:
5449:
5429:
5399:
5352:
5303:
5218:
5183:
5115:
5104:
5093:
5082:
5068:
5054:
5034:
5001:
4988:
4949:
4936:
4896:
4847:
4822:Nuclear Fusion
4808:
4785:
4778:
4755:
4752:. 29B(2): 1â7.
4740:
4733:
4710:
4707:. 29B(2): 1â7.
4688:
4653:
4610:
4590:
4588:
4585:
4584:
4583:
4577:
4558:
4555:
4553:
4552:
4547:
4542:
4537:
4532:
4527:
4522:
4517:
4512:
4507:
4502:
4497:
4491:
4489:
4486:
4480:
4477:
4441:superconductor
4399:
4396:
4379:
4376:
4359:
4356:
4353:
4315:
4312:
4262:solar rotation
4223:
4220:
4148:
4145:
4137:
4109:
4106:
4078:
4075:
4073:
4070:
4066:Landau damping
4049:
4046:
4044:
4041:
4040:
4039:
4032:
4029:
4022:
4019:
3999:
3996:
3980:
3958:
3954:
3942:
3941:
3940:
3929:
3925:
3921:
3918:
3907:
3903:
3899:
3895:
3891:
3887:
3881:
3876:
3872:
3867:
3862:
3854:
3850:
3846:
3842:
3838:
3819:
3816:
3813:electric field
3809:
3806:
3802:
3799:
3785:
3780:
3777:
3760:
3759:
3747:
3743:
3739:
3727:
3716:
3706:
3695:
3690:
3686:
3682:
3677:
3673:
3669:
3666:
3656:
3652:
3651:
3639:
3635:
3631:
3619:
3616:
3601:
3596:
3592:
3588:
3583:
3578:
3574:
3561:
3550:
3545:
3541:
3537:
3532:
3528:
3524:
3521:
3511:
3508:
3504:
3503:
3491:
3487:
3483:
3472:
3458:
3453:
3449:
3443:
3430:
3419:
3409:
3396:
3392:
3381:
3378:
3374:
3373:
3361:
3357:
3353:
3342:
3330:
3326:
3322:
3310:
3309:
3306:
3305:Group velocity
3303:
3300:
3297:
3287:
3286:
3272:
3268:
3265:
3258:
3253:
3249:
3234:
3233:
3219:
3216:
3210:
3204:
3201:
3196:
3192:
3186:
3181:
3177:
3171:
3166:
3162:
3158:
3155:
3150:
3145:
3139:
3134:
3130:
3126:
3121:
3116:
3112:
3107:
3097:
3094:
3088:
3084:
3078:
3073:
3069:
3065:
3060:
3055:
3051:
3046:
3039:
3036:
3029:
3024:
3019:
3016:
3000:
2999:
2985:
2980:
2976:
2969:
2965:
2959:
2954:
2950:
2935:
2934:
2923:
2920:
2917:
2912:
2908:
2904:
2899:
2896:
2878:
2846:
2839:
2835:
2828:
2827:
2826:
2818:
2811:
2807:
2800:
2799:
2798:
2791:
2790:
2789:
2788:
2787:
2786:
2783:
2780:
2758:
2755:
2739:current sheets
2720:
2717:
2668:
2665:
2664:
2663:
2660:Landau damping
2648:
2645:
2633:
2630:
2629:
2628:
2617:
2614:
2610:
2606:
2602:
2598:
2595:
2592:
2589:
2583:
2580:
2574:
2570:
2542:
2538:
2533:
2528:
2522:
2518:
2514:
2503:
2502:
2491:
2488:
2484:
2480:
2476:
2472:
2468:
2443:
2439:
2427:
2424:
2357:
2353:
2320:
2318:
2315:
2303:
2302:
2291:
2287:
2279:
2275:
2271:
2265:
2261:
2255:
2251:
2248:
2241:
2237:
2231:
2226:
2222:
2219:
2215:
2210:
2203:
2199:
2195:
2191:
2162:
2158:
2154:
2120:
2116:
2111:
2107:
2096:
2095:
2084:
2080:
2074:
2070:
2062:
2058:
2054:
2049:
2046:
2042:
2038:
2034:
2030:
2027:
2024:
2021:
2015:
2012:
2006:
2002:
1981:
1980:
1969:
1965:
1961:
1958:
1954:
1950:
1946:
1942:
1938:
1925:and Ohm's law
1923:
1922:
1911:
1907:
1903:
1900:
1897:
1894:
1888:
1885:
1879:
1875:
1860:Faraday's law
1858:
1857:
1846:
1842:
1838:
1835:
1832:
1828:
1822:
1818:
1803:
1802:
1791:
1788:
1785:
1782:
1778:
1774:
1770:
1766:
1762:
1757:
1753:
1750:
1746:
1742:
1736:
1733:
1729:
1723:
1719:
1705:
1704:
1693:
1690:
1687:
1683:
1676:
1672:
1668:
1663:
1656:
1652:
1646:
1629:
1628:
1617:
1614:
1611:
1607:
1602:
1598:
1594:
1590:
1587:
1584:
1578:
1575:
1570:
1567:
1540:
1518:
1497:
1474:
1451:
1401:
1400:
1389:
1384:
1379:
1372:
1368:
1362:
1358:
1352:
1348:
1342:
1339:
1334:
1330:
1315:
1314:
1301:
1296:
1289:
1285:
1279:
1275:
1269:
1265:
1261:
1257:
1231:
1227:
1221:
1217:
1211:
1207:
1203:
1200:
1178:
1173:
1149:
1145:
1122:
1118:
1095:
1091:
1080:number density
1067:
1046:
1034:center of mass
1020:
1000:
997:
976:
973:
945:magnetic field
881:; also called
861:
860:
858:
857:
850:
843:
835:
832:
831:
828:
827:
822:
817:
812:
807:
802:
797:
792:
787:
782:
777:
772:
767:
762:
757:
752:
747:
742:
737:
732:
727:
722:
717:
712:
707:
702:
697:
692:
687:
682:
677:
672:
667:
662:
657:
652:
646:
643:
642:
639:
638:
635:
634:
629:
624:
619:
614:
612:Four-potential
609:
604:
599:
593:
588:
587:
584:
583:
580:
579:
574:
569:
564:
559:
554:
549:
544:
539:
534:
529:
527:Electric motor
524:
519:
514:
508:
503:
502:
499:
498:
495:
494:
489:
484:
482:Series circuit
479:
474:
469:
464:
459:
454:
452:Kirchhoff laws
449:
444:
439:
434:
429:
424:
419:
417:Direct current
414:
409:
404:
398:
393:
392:
389:
388:
385:
384:
379:
374:
372:Maxwell tensor
369:
364:
359:
354:
349:
344:
342:Larmor formula
339:
334:
329:
324:
319:
314:
309:
304:
299:
294:
292:Bremsstrahlung
288:
283:
282:
279:
278:
275:
274:
269:
264:
259:
254:
249:
244:
242:Magnetic field
239:
234:
229:
224:
218:
215:Magnetostatics
213:
212:
209:
208:
205:
204:
199:
194:
189:
184:
179:
174:
169:
164:
159:
154:
149:
147:Electric field
144:
139:
134:
129:
124:
119:
117:Charge density
113:
110:Electrostatics
108:
107:
104:
103:
102:
101:
96:
91:
86:
81:
76:
71:
63:
62:
54:
53:
47:
46:
45:Articles about
15:
13:
10:
9:
6:
4:
3:
2:
6125:
6114:
6111:
6109:
6106:
6105:
6103:
6090:
6086:
6081:
6076:
6072:
6068:
6064:
6060:
6056:
6052:
6048:
6044:
6040:
6033:
6030:
6025:
6021:
6016:
6011:
6007:
6003:
6000:(1): 012047.
5999:
5995:
5991:
5984:
5981:
5976:
5972:
5968:
5964:
5960:
5956:
5949:
5946:
5941:
5937:
5932:
5927:
5923:
5919:
5915:
5911:
5907:
5900:
5897:
5892:
5888:
5884:
5880:
5876:
5872:
5865:
5862:
5857:
5855:0-7803-6401-5
5851:
5847:
5843:
5839:
5832:
5829:
5824:
5820:
5813:
5810:
5806:
5802:
5801:
5797:
5794:
5788:
5785:
5782:
5776:
5773:
5768:
5764:
5763:
5758:
5752:
5749:
5743:
5737:
5723:on 2014-08-20
5719:
5712:
5706:
5703:
5698:
5694:
5689:
5684:
5679:
5674:
5670:
5666:
5662:
5658:
5654:
5647:
5644:
5639:
5633:
5630:
5622:
5618:
5611:
5604:
5601:
5596:
5592:
5588:
5584:
5580:
5576:
5571:
5566:
5562:
5558:
5554:
5547:
5544:
5529:
5525:
5521:
5517:
5511:
5507:
5503:
5499:
5495:
5491:
5484:
5477:
5474:
5463:
5459:
5453:
5450:
5440:
5436:
5432:
5426:
5422:
5418:
5414:
5410:
5403:
5400:
5395:
5391:
5387:
5383:
5379:
5375:
5371:
5367:
5363:
5356:
5353:
5348:
5344:
5339:
5334:
5330:
5326:
5322:
5318:
5314:
5307:
5304:
5299:
5295:
5290:
5285:
5281:
5277:
5272:
5267:
5263:
5259:
5255:
5241:
5237:
5233:
5229:
5222:
5219:
5214:
5210:
5206:
5202:
5198:
5194:
5187:
5184:
5168:
5164:
5160:
5156:
5152:
5148:
5144:
5140:
5136:
5132:
5128:
5127:
5119:
5108:
5105:
5102:
5097:
5094:
5091:
5086:
5083:
5078:
5072:
5069:
5064:
5058:
5055:
5050:
5049:
5044:
5038:
5035:
5019:
5012:
5005:
5002:
4998:
4992:
4989:
4984:
4980:
4976:
4972:
4968:
4964:
4960:
4953:
4950:
4946:
4940:
4937:
4932:
4928:
4924:
4920:
4916:
4912:
4908:
4900:
4897:
4892:
4888:
4883:
4878:
4874:
4870:
4866:
4862:
4858:
4851:
4848:
4843:
4839:
4835:
4831:
4827:
4823:
4819:
4812:
4809:
4804:
4800:
4796:
4789:
4786:
4781:
4779:0-521-48179-1
4775:
4771:
4764:
4762:
4760:
4756:
4751:
4744:
4741:
4736:
4730:
4726:
4719:
4717:
4715:
4711:
4706:
4699:
4692:
4689:
4684:
4680:
4676:
4672:
4668:
4664:
4657:
4654:
4649:
4645:
4641:
4637:
4633:
4629:
4625:
4621:
4614:
4611:
4606:
4602:
4595:
4592:
4586:
4580:
4578:9781107158658
4574:
4570:
4566:
4561:
4560:
4556:
4551:
4548:
4546:
4543:
4541:
4538:
4536:
4533:
4531:
4528:
4526:
4523:
4521:
4518:
4516:
4513:
4511:
4508:
4506:
4503:
4501:
4498:
4496:
4493:
4492:
4487:
4485:
4478:
4476:
4474:
4469:
4467:
4462:
4460:
4459:microfluidics
4455:
4452:
4448:
4446:
4445:liquid helium
4442:
4438:
4437:
4432:
4428:
4423:
4420:
4415:
4413:
4409:
4405:
4397:
4395:
4393:
4389:
4385:
4377:
4375:
4373:
4372:safety factor
4357:
4354:
4351:
4343:
4339:
4338:MHD stability
4335:
4332:
4327:
4325:
4321:
4313:
4311:
4309:
4305:
4301:
4300:active region
4297:
4292:
4290:
4286:
4282:
4281:closed system
4278:
4274:
4269:
4267:
4266:Parker spiral
4263:
4259:
4255:
4254:Joseph Larmor
4251:
4247:
4245:
4241:
4237:
4233:
4229:
4221:
4219:
4217:
4213:
4210:forecasting.
4209:
4208:space weather
4204:
4202:
4198:
4194:
4190:
4184:
4182:
4178:
4174:
4173:magnetosphere
4170:
4166:
4162:
4161:space physics
4158:
4154:
4147:Space Physics
4146:
4144:
4142:
4135:
4131:
4127:
4123:
4119:
4115:
4107:
4105:
4099:
4096:Reversals of
4094:
4090:
4088:
4084:
4076:
4071:
4069:
4067:
4063:
4062:double layers
4058:
4056:
4047:
4042:
4037:
4033:
4030:
4027:
4023:
4021:Collisionless
4020:
4017:
4016:neutron stars
4013:
4012:ion thrusters
4009:
4005:
4000:
3997:
3994:
3978:
3956:
3952:
3943:
3927:
3919:
3916:
3905:
3893:
3879:
3874:
3870:
3865:
3860:
3848:
3840:
3828:
3827:
3824:
3820:
3817:
3814:
3810:
3807:
3803:
3800:
3797:
3791:
3786:
3783:
3782:
3778:
3776:
3774:
3770:
3765:
3741:
3728:
3714:
3707:
3688:
3684:
3680:
3675:
3671:
3657:
3654:
3653:
3633:
3620:
3599:
3594:
3590:
3586:
3581:
3576:
3572:
3562:
3543:
3539:
3535:
3530:
3526:
3512:
3506:
3505:
3485:
3473:
3456:
3451:
3447:
3431:
3417:
3410:
3394:
3390:
3382:
3379:
3376:
3375:
3355:
3343:
3324:
3312:
3311:
3294:
3291:
3270:
3266:
3263:
3256:
3251:
3247:
3239:
3238:
3237:
3217:
3214:
3208:
3202:
3199:
3194:
3190:
3184:
3179:
3175:
3169:
3164:
3160:
3156:
3153:
3148:
3143:
3137:
3132:
3128:
3124:
3119:
3114:
3110:
3105:
3095:
3092:
3086:
3082:
3076:
3071:
3067:
3063:
3058:
3053:
3049:
3044:
3037:
3034:
3027:
3022:
3017:
3014:
3005:
3004:
3003:
2983:
2978:
2974:
2967:
2963:
2957:
2952:
2948:
2940:
2939:
2938:
2921:
2918:
2915:
2910:
2906:
2902:
2897:
2894:
2885:
2884:
2883:
2877:
2867:
2861:
2849:
2842:
2832:
2821:
2814:
2804:
2784:
2781:
2778:
2777:
2776:
2774:
2770:
2764:
2756:
2754:
2752:
2748:
2744:
2743:magnetosphere
2740:
2731:
2726:
2718:
2716:
2714:
2708:
2704:
2702:
2698:
2693:
2692:instabilities
2689:
2684:
2682:
2678:
2674:
2673:diffusion law
2666:
2661:
2657:
2656:Larmor radius
2653:
2649:
2646:
2643:
2639:
2638:
2637:
2631:
2615:
2604:
2593:
2587:
2581:
2558:
2557:
2556:
2540:
2536:
2531:
2520:
2512:
2489:
2486:
2478:
2470:
2458:
2457:
2456:
2437:
2425:
2423:
2421:
2416:
2412:
2407:
2402:
2400:
2396:
2392:
2388:
2387:length scales
2384:
2380:
2377:during which
2376:
2372:
2351:
2343:
2337:
2334:
2333:Hannes Alfvén
2328:
2326:
2325:eddy currents
2316:
2314:
2312:
2308:
2289:
2285:
2277:
2273:
2269:
2263:
2259:
2253:
2246:
2239:
2235:
2224:
2217:
2208:
2201:
2193:
2181:
2180:
2179:
2177:
2156:
2143:
2142:Lorentz force
2138:
2136:
2118:
2114:
2109:
2105:
2082:
2072:
2060:
2056:
2052:
2047:
2036:
2025:
2019:
2013:
1990:
1989:
1988:
1986:
1967:
1959:
1956:
1948:
1940:
1928:
1927:
1926:
1909:
1901:
1895:
1892:
1886:
1863:
1862:
1861:
1844:
1836:
1830:
1820:
1816:
1808:
1807:
1806:
1789:
1786:
1780:
1772:
1764:
1755:
1748:
1740:
1734:
1721:
1717:
1710:
1709:
1708:
1691:
1688:
1685:
1681:
1674:
1670:
1666:
1661:
1654:
1634:
1633:
1632:
1615:
1612:
1609:
1605:
1596:
1592:
1588:
1582:
1576:
1568:
1555:
1554:
1553:
1495:
1488:
1472:
1465:
1449:
1441:
1436:
1434:
1433:isothermality
1430:
1426:
1422:
1421:Faraday's law
1418:
1414:
1410:
1406:
1387:
1382:
1370:
1366:
1360:
1356:
1350:
1346:
1340:
1337:
1332:
1320:
1319:
1318:
1299:
1287:
1283:
1277:
1273:
1267:
1263:
1259:
1247:
1246:
1245:
1229:
1225:
1219:
1215:
1209:
1205:
1201:
1198:
1176:
1147:
1143:
1120:
1116:
1093:
1089:
1081:
1065:
1035:
1010:
1006:
998:
996:
994:
993:
988:
987:
982:
981:Hannes Alfvén
974:
972:
970:
966:
965:Hannes Alfvén
962:
961:
960:
954:
953:
952:
946:
942:
941:
940:
934:
929:
927:
923:
919:
915:
914:space physics
911:
910:liquid metals
907:
903:
899:
895:
892:
888:
884:
880:
876:
872:
868:
856:
851:
849:
844:
842:
837:
836:
834:
833:
826:
823:
821:
818:
816:
813:
811:
808:
806:
803:
801:
798:
796:
793:
791:
788:
786:
783:
781:
778:
776:
773:
771:
768:
766:
763:
761:
758:
756:
753:
751:
748:
746:
743:
741:
738:
736:
733:
731:
728:
726:
723:
721:
718:
716:
713:
711:
708:
706:
703:
701:
698:
696:
693:
691:
688:
686:
683:
681:
678:
676:
673:
671:
668:
666:
663:
661:
658:
656:
653:
651:
648:
647:
641:
640:
633:
630:
628:
625:
623:
620:
618:
615:
613:
610:
608:
605:
603:
600:
598:
595:
594:
591:
586:
585:
578:
575:
573:
570:
568:
565:
563:
560:
558:
555:
553:
550:
548:
545:
543:
540:
538:
535:
533:
530:
528:
525:
523:
520:
518:
515:
513:
510:
509:
506:
501:
500:
493:
490:
488:
485:
483:
480:
478:
475:
473:
470:
468:
465:
463:
460:
458:
455:
453:
450:
448:
447:Joule heating
445:
443:
440:
438:
435:
433:
430:
428:
425:
423:
420:
418:
415:
413:
410:
408:
405:
403:
400:
399:
396:
391:
390:
383:
380:
378:
375:
373:
370:
368:
365:
363:
362:Lorentz force
360:
358:
355:
353:
350:
348:
345:
343:
340:
338:
335:
333:
330:
328:
325:
323:
320:
318:
315:
313:
310:
308:
305:
303:
300:
298:
295:
293:
290:
289:
286:
281:
280:
273:
270:
268:
265:
263:
262:Magnetization
260:
258:
255:
253:
250:
248:
247:Magnetic flux
245:
243:
240:
238:
235:
233:
230:
228:
225:
223:
220:
219:
216:
211:
210:
203:
200:
198:
195:
193:
190:
188:
185:
183:
180:
178:
175:
173:
170:
168:
165:
163:
160:
158:
155:
153:
152:Electric flux
150:
148:
145:
143:
140:
138:
135:
133:
130:
128:
125:
123:
120:
118:
115:
114:
111:
106:
105:
100:
97:
95:
92:
90:
89:Computational
87:
85:
82:
80:
77:
75:
72:
70:
67:
66:
65:
64:
60:
56:
55:
52:
48:
44:
43:
36:
28:
22:
6046:
6042:
6032:
5997:
5993:
5983:
5958:
5954:
5948:
5913:
5909:
5899:
5874:
5870:
5864:
5837:
5831:
5823:Artech House
5818:
5812:
5791:
5787:
5780:
5775:
5767:the original
5760:
5751:
5725:. Retrieved
5718:the original
5705:
5660:
5656:
5646:
5632:
5616:
5603:
5560:
5556:
5546:
5535:. Retrieved
5489:
5476:
5465:. Retrieved
5461:
5452:
5442:, retrieved
5412:
5402:
5369:
5365:
5355:
5320:
5316:
5306:
5261:
5257:
5245:December 18,
5243:. Retrieved
5231:
5221:
5196:
5192:
5186:
5176:December 18,
5174:. Retrieved
5130:
5124:
5107:
5096:
5085:
5071:
5057:
5048:Live Science
5046:
5037:
5025:. Retrieved
5018:the original
5004:
4996:
4991:
4958:
4952:
4944:
4939:
4914:
4910:
4899:
4864:
4860:
4850:
4825:
4821:
4811:
4788:
4769:
4749:
4743:
4724:
4704:
4691:
4666:
4662:
4656:
4623:
4619:
4613:
4604:
4594:
4564:
4482:
4470:
4463:
4456:
4449:
4434:
4424:
4416:
4401:
4381:
4342:Troyon limit
4336:
4328:
4324:stellarators
4317:
4296:solar flares
4293:
4277:Solar System
4270:
4248:
4228:astrophysics
4225:
4222:Astrophysics
4205:
4193:ring current
4189:magnetopause
4185:
4159:is known as
4157:Solar System
4150:
4111:
4102:
4080:
4072:Applications
4059:
4051:
3998:Electron MHD
3823:Hall current
3789:
3771:of the Sun (
3766:
3763:
3288:
3235:
3001:
2936:
2875:
2865:
2859:
2856:
2844:
2837:
2816:
2809:
2779:Alfvén waves
2772:
2768:
2766:
2736:
2709:
2705:
2703:, and heat.
2685:
2670:
2635:
2504:
2429:
2403:
2390:
2341:
2339:
2330:
2322:
2304:
2139:
2097:
1982:
1924:
1859:
1804:
1706:
1630:
1437:
1429:adiabaticity
1417:AmpĂšre's Law
1402:
1316:
1002:
992:Alfvén waves
990:
985:
978:
956:
948:
936:
932:
930:
922:astrophysics
886:
882:
878:
874:
864:
607:Four-current
542:Linear motor
427:Electrolysis
307:Eddy current
267:Permeability
187:Polarization
182:Permittivity
5688:2433/153022
5264:(4): 1091.
5077:"Geodynamo"
4535:Molten salt
4398:Engineering
4390:such as in
4114:earthquakes
4108:Earthquakes
4043:Limitations
4034:By using a
3377:Alfvén wave
926:engineering
871:engineering
577:Transformer
407:Capacitance
332:Faraday law
127:Coulomb law
69:Electricity
6102:Categories
5727:2014-08-19
5570:2008.12069
5537:2019-07-22
5467:2023-03-10
5444:2023-03-10
4828:: 87â132.
4734:0521528003
4587:References
4510:Ferrofluid
4443:cooled by
4258:solar wind
4177:Solar wind
4171:, Earth's
4165:ionosphere
4126:California
4077:Geophysics
4055:Maxwellian
3779:Extensions
2761:See also:
2751:solar wind
2747:ionosphere
2697:turbulence
2652:skin depth
2642:Maxwellian
2397:and raise
2335:, 1943
918:geophysics
644:Scientists
492:Waveguides
472:Resistance
442:Inductance
222:AmpĂšre law
6071:0304-8853
6024:1757-899X
5975:0024-998X
5940:2352-7110
5916:: 68â72.
5910:SoftwareX
5697:122217405
5595:235631174
5524:117512943
5439:204194812
5394:1364-6826
5347:2296-987X
5271:1012.1533
5155:0094-8276
4931:231592434
4891:248673571
4842:122227433
4308:radiation
4289:proto-Sun
4285:conserved
3920:η
3906:⏟
3894:×
3861:−
3849:×
3808:Two-fluid
3784:Resistive
3742:∥
3634:∥
3486:∥
3457:ρ
3448:μ
3356:⊥
3325:∥
3271:ρ
3264:γ
3203:θ
3200:
3154:−
3087:±
3015:ω
2984:ρ
2975:μ
2922:θ
2919:
2895:ω
2773:MHD waves
2749:from the
2605:×
2594:×
2591:∇
2579:∂
2569:∂
2537:μ
2517:∇
2513:η
2479:×
2438:η
2371:Ohm's law
2352:η
2342:ideal MHD
2317:Ideal MHD
2274:μ
2250:∇
2247:−
2236:μ
2221:∇
2218:⋅
2194:×
2157:×
2115:μ
2106:η
2069:∇
2057:μ
2053:η
2037:×
2026:×
2023:∇
2011:∂
2001:∂
1960:η
1949:×
1902:×
1899:∇
1896:−
1884:∂
1874:∂
1837:×
1834:∇
1817:μ
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1781:−
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1752:∇
1749:⋅
1732:∂
1728:∂
1718:ρ
1675:γ
1671:ρ
1597:ρ
1589:⋅
1586:∇
1574:∂
1569:ρ
1566:∂
1496:η
1473:γ
1442:pressure
1440:isotropic
1425:Ohm's law
1383:σ
1371:σ
1361:σ
1351:σ
1347:∑
1341:ρ
1300:σ
1288:σ
1278:σ
1268:σ
1264:∑
1230:σ
1220:σ
1210:σ
1206:∑
1199:ρ
1177:σ
1148:σ
1121:σ
1094:σ
1066:σ
1036:velocity
999:Equations
971:in 1970.
931:The word
800:Steinmetz
730:Kirchhoff
715:Jefimenko
710:Hopkinson
695:Helmholtz
690:Heaviside
552:Permeance
437:Impedance
177:Insulator
172:Gauss law
122:Conductor
99:Phenomena
94:Textbooks
74:Magnetism
6089:21278859
5796:Archived
5736:cite web
5663:(1): 6.
5621:Archived
5563:(1): 3.
5528:Archived
5298:53456663
5167:Archived
5027:27 April
4983:31433317
4867:(1): 1.
4488:See also
4436:Yamato-1
4320:tokamaks
4250:Sunspots
3805:inertia.
3801:Extended
3729:approx.
3621:approx.
2411:advected
2178:to give
1032:and the
959:dynamics
943:meaning
939:magneto-
825:Wiechert
780:Poynting
670:Einstein
517:DC motor
512:AC motor
347:Lenz law
132:Electret
6080:3029028
6051:Bibcode
6002:Bibcode
5918:Bibcode
5879:Bibcode
5665:Bibcode
5575:Bibcode
5494:Bibcode
5374:Bibcode
5325:Bibcode
5276:Bibcode
5201:Bibcode
5163:1795290
5135:Bibcode
4963:Bibcode
4869:Bibcode
4803:4329910
4671:Bibcode
4648:4072220
4628:Bibcode
4378:Sensors
4334:coils.
4279:. In a
4275:in the
4169:auroras
4130:DEMETER
4031:Reduced
4004:z-pinch
3991:is the
2399:entropy
2313:force.
2133:is the
1108:, mass
1005:species
975:History
906:plasmas
867:physics
810:Thomson
785:Ritchie
775:Poisson
760:Neumann
755:Maxwell
750:Lorentz
745:Liénard
675:Faraday
660:Coulomb
487:Voltage
462:Ohm law
84:History
6087:
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4620:Nature
4575:
4410:, and
4306:, and
4304:X-rays
4199:, and
4179:, and
4175:, the
3944:where
3796:shocks
3769:corona
3236:where
2937:where
2098:where
1423:, and
1078:has a
1007:: the
986:Nature
951:hydro-
924:, and
894:fluids
795:Singer
790:Savart
770:Ărsted
735:Larmor
725:Kelvin
680:Fizeau
650:AmpĂšre
572:Stator
79:Optics
5721:(PDF)
5714:(PDF)
5693:S2CID
5624:(PDF)
5613:(PDF)
5591:S2CID
5565:arXiv
5531:(PDF)
5520:S2CID
5486:(PDF)
5435:S2CID
5294:S2CID
5266:arXiv
5170:(PDF)
5121:(PDF)
5021:(PDF)
5014:(PDF)
4979:S2CID
4927:S2CID
4917:(3).
4887:S2CID
4838:S2CID
4701:(PDF)
4644:S2CID
4153:Earth
4134:Haiti
2843:<
2815:>
2757:Waves
2144:term
1411:, an
1407:, an
820:Weber
815:Volta
805:Tesla
720:Joule
705:Hertz
700:Henry
685:Gauss
567:Rotor
6085:PMID
6067:ISSN
6020:ISSN
5971:ISSN
5936:ISSN
5850:ISBN
5762:Time
5742:link
5510:ISBN
5425:ISBN
5390:ISSN
5343:ISSN
5247:2010
5178:2010
5159:OCLC
5151:ISSN
5029:2015
4799:OSTI
4774:ISBN
4729:ISBN
4573:ISBN
4355:>
4329:The
4240:jets
4083:iron
3818:Hall
3299:Type
3296:Mode
2654:and
2395:heat
908:and
869:and
740:Lenz
665:Davy
655:Biot
6075:PMC
6059:doi
6047:323
6010:doi
5998:424
5963:doi
5926:doi
5887:doi
5842:doi
5683:hdl
5673:doi
5583:doi
5502:doi
5417:doi
5382:doi
5333:doi
5284:doi
5209:doi
5197:173
5143:doi
4971:doi
4919:doi
4915:217
4877:doi
4830:doi
4679:doi
4636:doi
4624:150
4457:In
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4322:or
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3792:â 0
3775:).
3665:min
3520:max
3191:cos
2916:cos
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765:Ohm
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