38:
2842:
2814:
70:
4068:. 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.
46:
2741:
2752:. 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
3242:
4104:
3949:
2311:
4464:
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
4432:
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
4344:
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
4012:
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
3815:
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
2721:
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
2104:
2428:
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.
2419:
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
4063:
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
4494:
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
4114:
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
1811:
3019:
2694:(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.
2710:, 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,
1006:. 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."
4079:, 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.
4197:
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
2637:
1409:
4298:. 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
3842:
1637:
1713:
1438:. 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
2195:
2717:
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.
1931:
1323:
4444:, 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
2004:
3805:). 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
2705:
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
1989:
4915:
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).
4115:
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.
2690:. 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
3836:
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
1724:
3237:{\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}}}
4495:
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.
1866:
2511:
3481:
3300:
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:
3008:
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3768:
3660:
3512:
3351:
2658:
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.
5964:
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".
3826:. 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.
2183:
5849:
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)
3382:
2868:
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
2651:
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
3625:
5721:
1200:
2572:
4253:. 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
1334:
3715:
3570:
2464:
2378:
3944:{\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} ,}
2142:
2334:
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
1561:
1539:
1067:
1041:
1171:
1144:
1117:
1569:
1648:
2306:{\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),}
1494:
1087:
4379:
4049:
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.
3980:
3418:
1517:
4154:. 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.
4000:
3736:
3439:
1471:
4302:
apart before it could have formed. However, magnetohydrodynamic effects transfer the Sun's angular momentum into the outer solar system, slowing its rotation.
627:
4100:. These eddies develop a magnetic field which boosts Earth's original magnetic fieldâa process which is self-sustaining and is called the geomagnetic dynamo.
1877:
1261:
2099:{\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}=\nabla \times (\mathbf {v} \times \mathbf {B} )+{\frac {\eta }{\mu _{0}}}\nabla ^{2}\mathbf {B} ,}
5631:
4092:
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
600:
4351:
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 (
4708:
4555:
4006:. 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.
3806:
2786:. 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:
1942:
612:
5073:
4139:, although a subsequent study indicates that this was little more than a sensor malfunction. On December 9, 2010, geoscientists announced that the
4282:
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
1806:{\displaystyle \rho \left({\frac {\partial }{\partial t}}+\mathbf {v} \cdot \nabla \right)\mathbf {v} =\mathbf {J} \times \mathbf {B} -\nabla p,}
5538:
5767:
3775:
The MHD oscillations will be damped if the fluid is not perfectly conducting but has a finite conductivity, or if viscous effects are present.
5728:
4482:
Industrial MHD problems can be modeled using the open-source software EOF-Library. Two simulation examples are 3D MHD with a free surface for
4225:
have the potential to cause extensive damage to satellites and infrastructure, thus it is crucial that such events are detected early. The
4436:
The first prototype of this kind of propulsion was built and tested in 1965 by
Steward Way, a professor of mechanical engineering at the
5752:
5087:
863:
632:
49:
Simulation of the OrszagâTang MHD vortex problem, a well-known model problem for testing the transition to supersonic 2D MHD turbulence
5202:
Thomas, J. N.; Love, J. J.; Johnston, M. J. S. (April 2009). "On the reported magnetic precursor of the 1989 Loma Prieta earthquake".
1822:
1431:
3832:
In 1960, M. J. Lighthill criticized the applicability of ideal or resistive MHD theory for plasmas. It concerned the neglect of the "
2472:
5324:"Global Magnetohydrodynamic Simulations: Performance Quantification of Magnetopause Distances and Convection Potential Predictions"
3778:
MHD waves and oscillations are a popular tool for the remote diagnostics of laboratory and astrophysical plasmas, for example, the
3445:
642:
2954:
4437:
467:
5524:
5439:
5177:
4560:
3253:
1205:
482:
477:
104:
17:
5806:
2899:
5021:
2519:
2400:
under consideration. Consequently, processes in ideal MHD that convert magnetic energy into kinetic energy, referred to as
492:
6123:
3743:
3635:
3487:
3326:
37:
5123:
Fraser-Smith, Antony C.; Bernardi, A.; McGill, P. R.; Ladd, M. E.; Helliwell, R. A.; Villard Jr., O. G. (August 1990).
4505:
4065:
2652:
2384:
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
5880:
Kenjeres, S.; Hanjalic, K. (2000). "On the implementation of effects of
Lorentz force in turbulence closure models".
5864:
4788:
4587:
3013:
is the Alfvén speed. This branch corresponds to the shear Alfvén mode. Additionally the dispersion equation gives
2158:
1000:
titled "Existence of
ElectromagneticâHydrodynamic Waves" which outlined his discovery of what are now referred to as
94:
3357:
362:
4611:
4096:. The liquid outer core moves in the presence of the magnetic field and eddies are set up into the same due to the
3576:
5620:
5111:
4472:, MHD is studied as a fluid pump for producing a continuous, nonpulsating flow in a complex microchannel design.
4226:
4211:
277:
4743:
4540:
4382:
2748:
In many MHD systems most of the electric current is compressed into thin nearly-two-dimensional ribbons termed
2707:
856:
622:
99:
31:
6118:
4535:
2735:
637:
342:
2632:{\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}=\nabla \times (\mathbf {v} \times \mathbf {B} ).}
1176:
5136:
4441:
4405:. Accuracy improves with the size of the sensor. The sensor is capable of surviving in harsh environments.
1404:{\displaystyle \mathbf {v} ={\frac {1}{\rho }}\sum _{\sigma }m_{\sigma }n_{\sigma }\mathbf {u} _{\sigma }.}
502:
242:
109:
6050:"The Behaviors of Ferro-Magnetic Nano-Particles In and Around Blood Vessels under Applied Magnetic Fields"
5053:
232:
4341:
4132:
1014:
In MHD, motion in the fluid is described using linear combinations of the mean motions of the individual
795:
670:
567:
542:
462:
4968:
Witalis, E.A. (1986). "Hall
Magnetohydrodynamics and Its Applications to Laboratory and Cosmic Plasma".
5492:
Kennel, C.F.; Arons, J.; Blandford, R.; Coroniti, F.; Israel, M.; Lanzerotti, L.; Lightman, A. (1985).
4579:
4398:
3671:
3526:
2682:
In an imperfectly conducting fluid the magnetic field can generally move through the fluid following a
2444:
2358:
295:
5493:
5372:
4035:
MHD is also often used for collisionless plasmas. In that case the MHD equations are derived from the
4525:
4429:
4108:
2389:
849:
810:
337:
327:
267:
262:
202:
5777:
4294:
such as the cloud of gas and dust from which the Sun was formed, mass and angular momentum are both
2112:
5246:
4275:
may be the long-term effect of magnetic drag at the poles of the Sun, an MHD phenomenon due to the
2385:
2186:
1632:{\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \left(\rho \mathbf {v} \right)=0,}
347:
1544:
1522:
1050:
1024:
780:
282:
5917:"EOF-Library: Open-source Elmer FEM and OpenFOAM coupler for electromagnetics and fluid dynamics"
5792:
Operation of the
Thruster for Superconducting Electromagnetohydrodynamic Propu1sion Ship YAMATO 1
4305:
Breakdown of ideal MHD (in the form of magnetic reconnection) is known to be the likely cause of
4046:
1708:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {p}{\rho ^{\gamma }}}\right)=0,}
1149:
1122:
1095:
901:
660:
187:
177:
172:
5419:
4828:
4329:
MHD describes a wide range of physical phenomena occurring in fusion plasmas in devices such as
2673:, and time scales much longer than the ion gyration time (system is smooth and slowly evolving).
785:
755:
5563:
5373:"Results from the LyonâFedderâMobarry global magnetospheric model for the electrojet challenge"
5265:"Enhanced ULF radiation observed by DEMETER two months around the strong 2010 Haiti earthquake"
4917:
4433:
the opposite direction. Although some working prototypes exist, MHD drives remain impractical.
2416:
2317:
1497:
979:
607:
377:
152:
2756:, where current sheets separate topologically distinct domains, isolating most of the Earth's
4402:
4242:
4191:
4174:. Areas researched within space physics encompass a large number of topics, ranging from the
4151:
4072:
4018:
2723:
2711:
2662:
2430:
1479:
1072:
705:
392:
382:
332:
322:
6016:
4358:
4229:(SWPC) runs MHD models to predict the arrival and impacts of space weather events at Earth.
2697:
Even in physical systemsâwhich are large and conductive enough that simple estimates of the
2424:
by fluid flows in the system. The connection between the fluid and magnetic field fixes the
6061:
6012:
5928:
5889:
5675:
5585:
5504:
5384:
5335:
5286:
5250:
5211:
5145:
4973:
4879:
4681:
4638:
4515:
4510:
4254:
4222:
4202:(the boundary between the Earth's magnetic field and the solar wind), the formation of the
3958:
3396:
2145:
1502:
830:
730:
695:
447:
312:
212:
197:
132:
5100:
2429:
The energy can then become available if the conditions for ideal MHD break down, allowing
69:
8:
6025:
6001:"Numerical and experimental study of liquid metal stirring by rotating permanent magnets"
5815:
4530:
4483:
4246:
4140:
4128:
2691:
1994:
Taking the curl of this equation and using AmpĂšre's law and
Faraday's law results in the
1415:
912:
790:
770:
765:
572:
557:
442:
412:
307:
237:
6065:
5932:
5893:
5847:
Fujisaki, Keisuke (Oct 2000). "In-mold electromagnetic stirring in continuous casting".
5679:
5589:
5508:
5388:
5339:
5290:
5215:
5149:
4977:
4883:
4685:
4642:
1926:{\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}=-\nabla \times \mathbf {E} ,}
1318:{\displaystyle \mathbf {J} =\sum _{\sigma }n_{\sigma }q_{\sigma }\mathbf {u} _{\sigma }}
6090:
6049:
5703:
5601:
5575:
5530:
5445:
5304:
5276:
4989:
4937:
4897:
4848:
4654:
4476:
3985:
3809:; this process has been shown to be important in the Earth-Solar magnetic interactions.
3783:
3721:
3424:
2687:
2683:
2393:
1995:
1456:
1443:
1419:
665:
405:
207:
167:
5901:
1427:
915:. It is primarily concerned with the low-frequency, large-scale, magnetic behavior in
487:
6095:
6077:
6030:
5981:
5946:
5860:
5746:
5707:
5605:
5534:
5520:
5449:
5435:
5400:
5353:
5169:
5161:
4941:
4901:
4852:
4844:
4809:
4784:
4739:
4672:
FĂ€lthammar, Carl-Gunne (October 2007). "The discovery of magnetohydrodynamic waves".
4583:
4003:
2744:
Schematic view of the different current systems which shape the Earth's magnetosphere
2701:
suggest that the resistivity can be ignoredâresistivity may still be important: many
2425:
2321:
1439:
1423:
725:
5308:
5238:
4993:
969:
949:
6085:
6069:
6020:
6000:
5973:
5936:
5897:
5852:
5693:
5683:
5648:
5593:
5512:
5427:
5392:
5343:
5322:
Mukhopadhyay, Agnit; Jia, Xianzhe; Welling, Daniel T.; Liemohn, Michael W. (2021).
5294:
5219:
5153:
5124:
4981:
4929:
4887:
4840:
4689:
4658:
4646:
4550:
4422:
4394:
4348:
4295:
4283:
4250:
4103:
2773:
2698:
1255:, and the motion of the fluid can be described by the current density expressed as
1015:
995:
916:
908:
825:
740:
700:
690:
577:
532:
515:
432:
367:
137:
61:
2343:
991:
975:
5810:
5803:
5772:
5516:
4759:
Alfvén, Hannes (1943). "On the
Existence of Electromagnetic-Hydrodynamic Waves".
4418:
4097:
4036:
1474:
1019:
760:
685:
680:
547:
422:
387:
247:
147:
5941:
5916:
5468:
5396:
5028:
4693:
4313:
over a sunspot can store energy that is released suddenly as a burst of motion,
961:
800:
6073:
5597:
5299:
5264:
5223:
4933:
4892:
4867:
4451:
4272:
4076:
4071:
Effects which are essentially kinetic and not captured by fluid models include
3823:
2670:
1984:{\displaystyle \mathbf {E} +\mathbf {v} \times \mathbf {B} =\eta \mathbf {J} .}
1090:
1044:
955:
720:
715:
537:
427:
352:
302:
252:
225:
182:
157:
127:
120:
5757:
D.Titterton, J.Weston, Strapdown Inertial Navigation Technology, chapter 4.3.2
5431:
5348:
5323:
5263:
Athanasiou, M; Anagnostopoulos, G; Iliopoulos, A; Pavlos, G; David, K (2010).
1002:
6112:
6081:
6034:
5985:
5950:
5814:, M. Mitchner and Charles H. Kruger, Jr., Mechanical Engineering Department,
5404:
5357:
5165:
4985:
4469:
4461:
4310:
4291:
4276:
4264:
4218:
4183:
4171:
4026:
4022:
2753:
2749:
2666:
2152:
990:
The MHD description of electrically conducting fluids was first developed by
924:
835:
820:
805:
745:
457:
372:
357:
272:
257:
162:
5856:
5157:
5074:"Earth's contrasting inner core rotation and magnetic field rotation linked"
3798:
Resistive MHD describes magnetized fluids with finite electron diffusivity (
6099:
5851:. Industry Applications Conference. Vol. 4. IEEE. pp. 2591â2598.
5833:
5173:
5058:
4954:
M. J. Lighthill, "Studies on MHD waves and other anisotropic wave motion,"
4414:
4352:
4287:
4238:
4203:
4199:
4167:
3833:
3779:
2397:
2381:
2335:
1435:
932:
920:
815:
710:
675:
617:
552:
472:
437:
317:
192:
2722:
at least somewhat resistive because their computational grid introduces a
5688:
5663:
5262:
4545:
4334:
4306:
4124:
2702:
936:
881:
735:
587:
417:
79:
5977:
2566:
in the induction equation vanishes giving the ideal induction equation,
5698:
4520:
4268:
4207:
4187:
4175:
4136:
2761:
2757:
2338:. Thus the matter of the liquid is "fastened" to the lines of force...
928:
452:
5818:. See Ch. 9 "Magnetohydrodynamic (MHD) Power Generation", pp. 214â230.
5125:"Low-Frequency Magnetic Field Measurements Near the Epicenter of the M
4813:
4805:
5999:
Dzelme, V.; Jakovics, A.; Vencels, J.; Köppen, D.; Baake, E. (2018).
4650:
4629:
Alfvén, H (1942). "Existence of Electromagnetic-Hydrodynamic Waves".
4318:
4299:
1450:
775:
750:
562:
84:
45:
5426:, Geophysical Monograph Series (1 ed.), Wiley, pp. 43â65,
5239:"Spacecraft Saw ULF Radio Emissions over Haiti before January Quake"
5580:
4446:
4393:
Magnetohydrodynamic sensors are used for precision measurements of
3822:
Two-fluid MHD describes plasmas that include a non-negligible Hall
2421:
527:
522:
142:
5422:, in Gannon, Jennifer L.; Swidinsky, Andrei; Xu, Zhonghua (eds.),
5281:
4486:
melting, and liquid metal stirring by rotating permanent magnets.
4479:
process of metals to suppress instabilities and control the flow.
4131:(ULF) activity. A remarkable example of this occurred before the
2841:
2813:
2669:
perpendicular to the field, long enough along the field to ignore
5501:
Unstable Current Systems and Plasma Instabilities in Astrophysics
5101:
NOVA | Magnetic Storm | What Drives Earth's Magnetic Field? | PBS
5054:"Why Earth's Inner and Outer Cores Rotate in Opposite Directions"
4330:
4260:
4014:
2409:
877:
497:
5564:"Relativistic fluid dynamics: Physics for many different scales"
5420:"Magnetohydrodynamic Models of B and Their Use in GIC Estimates"
1861:{\displaystyle \mu _{0}\mathbf {J} =\nabla \times \mathbf {B} ,}
5424:
Geomagnetically Induced Currents from the Sun to the Power Grid
4914:
4455:
4321:
when the main current sheet collapses, reconnecting the field.
4179:
4143:
satellite observed a dramatic increase in ULF radio waves over
2506:{\displaystyle \mathbf {E} +\mathbf {v} \times \mathbf {B} =0.}
582:
89:
5621:"Lecture Notes on Introduction to Plasma Astrophysics (Draft)"
5122:
5321:
4918:"Current Sheets, Plasmoids and Flux Ropes in the Heliosphere"
4314:
4163:
4144:
3476:{\displaystyle {\frac {\mathbf {B} }{\sqrt {\mu _{0}\rho }}}}
904:
5491:
4618:. Princeton University Department of Astrophysical Sciences.
3003:{\displaystyle v_{A}={\frac {B_{0}}{\sqrt {\mu _{0}\rho }}}}
41:
The plasma making up the Sun can be modeled as an MHD system
5998:
5371:
Wiltberger, M.; Lyon, J. G.; Goodrich, C. C. (2003-07-01).
4093:
2740:
2405:
1414:
MHD can be described by a set of equations consisting of a
6048:
Nacev, A.; Beni, C.; Bruno, O.; Shapiro, B. (2011-03-01).
5915:
Vencels, Juris; RĂ„back, Peter; GeĆŸa, Vadims (2019-01-01).
2778:
The wave modes derived using the MHD equations are called
4450:, which used a magnetohydrodynamic drive incorporating a
4279:
shape assumed by the extended magnetic field of the Sun.
2433:
that releases the stored energy from the magnetic field.
6005:
IOP Conference Series: Materials Science and Engineering
5494:"Perspectives on Space and Astrophysical Plasma Physics"
4709:"On the Existence of Electromagnetic-Hydrodynamic Waves"
2642:
1248:{\textstyle \rho =\sum _{\sigma }m_{\sigma }n_{\sigma }}
5827:
3290:{\displaystyle v_{s}={\sqrt {\frac {\gamma p}{\rho }}}}
5370:
3101:
3043:
2938:{\displaystyle {\frac {\omega }{k}}=v_{A}\cos \theta }
1449:
In the adiabatic limit, that is, the assumption of an
1208:
6047:
4361:
4245:(space between the planets), and possibly within the
3988:
3961:
3845:
3746:
3724:
3674:
3638:
3579:
3529:
3490:
3448:
3427:
3399:
3360:
3329:
3256:
3022:
2957:
2902:
2575:
2559:{\displaystyle \eta \nabla ^{2}\mathbf {B} /\mu _{0}}
2522:
2475:
2447:
2361:
2198:
2161:
2115:
2007:
1945:
1880:
1825:
1727:
1651:
1572:
1547:
1525:
1505:
1482:
1459:
1337:
1264:
1179:
1152:
1125:
1098:
1075:
1053:
1027:
5377:
Journal of Atmospheric and Solar-Terrestrial Physics
5022:"Reduced MHD equations for low aspect ratio plasmas"
4674:
Journal of Atmospheric and Solar-Terrestrial Physics
974:
meaning movement. The field of MHD was initiated by
5963:
4013:and the Hall MHD. EMHD is especially important for
3521:
neither transversal nor longitudinal; compressional
2884:with respect to the time independent or bulk field
2661:Interest in length scales much longer than the ion
2316:where the first term on the right hand side is the
5201:
4781:Magnetic Reconnection: MHD Theory and Applications
4373:
3994:
3974:
3943:
3763:{\displaystyle \mathbf {Q} \parallel \mathbf {B} }
3762:
3730:
3709:
3655:{\displaystyle \mathbf {Q} \parallel \mathbf {k} }
3654:
3619:
3564:
3507:{\displaystyle \mathbf {Q} \parallel \mathbf {B} }
3506:
3475:
3433:
3412:
3376:
3346:{\displaystyle \mathbf {k} \parallel \mathbf {B} }
3345:
3289:
3236:
3002:
2937:
2631:
2558:
2505:
2466:vanishes in Ohm's law giving the ideal Ohm's law,
2458:
2415:A fundamental concept underlying ideal MHD is the
2372:
2305:
2177:
2136:
2098:
1983:
1925:
1860:
1805:
1707:
1631:
1555:
1533:
1511:
1488:
1465:
1403:
1317:
1247:
1194:
1165:
1138:
1111:
1081:
1061:
1035:
923:and has applications in multiple fields including
5914:
4868:"Magnetic reconnection: MHD theory and modelling"
4556:Shocks and discontinuities (magnetohydrodynamics)
4465:primary-coal combustion chamber due to abrasion.
2880:. An MHD wave propagating at an arbitrary angle
6110:
5879:
5112:Earth's Inconstant Magnetic Field â NASA Science
4058:
3675:
3530:
2686:with the resistivity of the plasma serving as a
4413:MHD is related to engineering problems such as
2178:{\displaystyle \mathbf {J} \times \mathbf {B} }
5830:Fundamentals and Applications of Microfluidics
5561:
5469:"What is Space Weather ? - Space Weather"
4783:(First ed.). Cambridge University Press.
4609:
4324:
3377:{\displaystyle \mathbf {k} \perp \mathbf {B} }
1328:and the center of mass velocity expressed as:
5664:"Solar Flares: Magnetohydrodynamic Processes"
5661:
5019:
4217:One prominent use of global MHD models is in
3620:{\displaystyle {\sqrt {v_{A}^{2}+v_{s}^{2}}}}
857:
5882:International Journal of Heat and Fluid Flow
5204:Physics of the Earth and Planetary Interiors
4865:
4778:
4263:are caused by the Sun's magnetic fields, as
4147:in the month before the magnitude 7.0 M
4123:Some monitoring stations have reported that
2729:
2677:
2647:Ideal MHD is only strictly applicable when:
1563:can be described by the continuous equation
6054:Journal of Magnetism and Magnetic Materials
5662:Shibata, Kazunari; Magara, Tetsuya (2011).
5562:Andersson, Nils; Comer, Gregory L. (2021).
5013:
4576:Introduction to Modern Magnetohydrodynamics
1473:and isotropic temperature, a fluid with an
5236:
4866:Pontin, David I.; Priest, Eric R. (2022).
4803:
4774:
4772:
4770:
4729:
4727:
4725:
4671:
4489:
4271:is also governed by MHD. The differential
864:
850:
68:
6089:
6024:
5940:
5697:
5687:
5579:
5347:
5328:Frontiers in Astronomy and Space Sciences
5298:
5280:
5269:Natural Hazards and Earth System Sciences
5027:. University of Wisconsin. Archived from
4891:
4738:. Cambridge: Cambridge University Press.
2185:can be expanded using AmpĂšre's law and a
1202:. The fluid's total mass density is then
5846:
5020:Kruger, S.E.; Hegna, C.C.; Callen, J.D.
4761:Arkiv för matematik, astronomi och fysik
4716:Arkiv för matematik, astronomi och fysik
4102:
2739:
44:
36:
5417:
4967:
4958:, London, vol. 252A, pp. 397â430, 1960.
4767:
4722:
4700:
4438:University of California, Santa Barbara
2803:Phase velocity plotted with respect to
2516:Similarly, the magnetic diffusion term
613:Electromagnetism and special relativity
27:Model of electrically conducting fluids
14:
6111:
5821:
5784:
5751:: CS1 maint: archived copy as title (
4826:
4758:
4733:
4706:
4628:
2436:
1195:{\displaystyle \mathbf {u} _{\sigma }}
5105:
5000:
4829:"Hydromagnetic stability of tokamaks"
4561:List of textbooks in electromagnetism
4127:are sometimes preceded by a spike in
2893:will satisfy the dispersion relation
2643:Applicability of ideal MHD to plasmas
633:Maxwell equations in curved spacetime
5790:Setsuo Takezawa et al. (March 1995)
5619:Kunz, Matthew W. (9 November 2020).
5618:
4779:Priest, Eric; Forbes, Terry (2000).
4458:, and could travel at 15 km/h.
5503:. Vol. 107. pp. 537â552.
5094:
4970:IEEE Transactions on Plasma Science
3982:is the electron number density and
24:
5828:Nguyen, N.T.; Wereley, S. (2006).
5006:W. Baumjohann and R. A. Treumann,
4567:
4506:Computational magnetohydrodynamics
2601:
2589:
2579:
2527:
2420:field line even as the points are
2355:, assumes that the resistive term
2260:
2231:
2079:
2033:
2021:
2011:
1909:
1894:
1884:
1844:
1794:
1762:
1742:
1738:
1661:
1655:
1596:
1584:
1576:
25:
6135:
5768:"Run Silent, Run Electromagnetic"
4440:. Way, on leave from his job at
4355:) and plasma current (set by the
3710:{\displaystyle \min(v_{A},v_{s})}
3565:{\displaystyle \max(v_{A},v_{s})}
2459:{\displaystyle \eta \mathbf {J} }
2441:In ideal MHD, the resistive term
2373:{\displaystyle \eta \mathbf {J} }
1069:. In a given fluid, each species
907:that treats all interpenetrating
5637:from the original on 2022-10-09.
4309:. The magnetic field in a solar
4212:geomagnetically induced currents
4162:The study of space plasmas near
4157:
3934:
3908:
3900:
3863:
3855:
3847:
3756:
3748:
3648:
3640:
3500:
3492:
3452:
3370:
3362:
3339:
3331:
2840:
2812:
2619:
2611:
2583:
2537:
2493:
2485:
2477:
2452:
2366:
2240:
2224:
2208:
2200:
2171:
2163:
2089:
2051:
2043:
2015:
1974:
1963:
1955:
1947:
1916:
1888:
1851:
1837:
1787:
1779:
1771:
1755:
1611:
1549:
1527:
1388:
1339:
1305:
1266:
1182:
1055:
1029:
944:magnetohydrodynamics
6041:
5992:
5957:
5908:
5873:
5840:
5796:
5760:
5714:
5668:Living Reviews in Solar Physics
5655:
5641:
5612:
5555:
5544:from the original on 2022-10-09
5485:
5461:
5411:
5364:
5315:
5237:KentuckyFC (December 9, 2010).
5230:
5195:
5183:from the original on 2022-10-09
5116:
5080:
5066:
5046:
4961:
4948:
4908:
4872:Living Reviews in Solar Physics
4859:
4232:
4227:Space Weather Prediction Center
4082:
2151:In the equation of motion, the
1816:the low-frequency AmpĂšre's law
6026:10.1088/1757-899X/424/1/012047
5418:Welling, Daniel (2019-09-25),
5010:, Imperial College Press, 1997
4820:
4797:
4752:
4736:Fundamentals of plasma physics
4665:
4622:
4603:
4541:Magnetohydrodynamic turbulence
4475:MHD can be implemented in the
4408:
4249:(space between the stars) and
4118:
4053:
3912:
3896:
3704:
3678:
3559:
3533:
2623:
2607:
2426:topology of the magnetic field
2137:{\displaystyle \eta /\mu _{0}}
2055:
2039:
32:Magnetohydrodynamics (journal)
30:For the academic journal, see
13:
1:
5902:10.1016/S0142-727X(00)00017-5
4804:Rosenbluth, M. (April 1956).
4597:
4536:Magnetohydrodynamic generator
4087:
4059:Importance of kinetic effects
3789:
2736:Magnetosphere particle motion
994:in a 1942 paper published in
638:Relativistic electromagnetism
5776:. 1966-09-23. Archived from
5568:Living Reviews in Relativity
5517:10.1007/978-94-009-6520-1_63
5137:Geophysical Research Letters
4734:Bellan, Paul Murray (2006).
4610:Philip Hopkins (July 2004).
4573:
4526:Lorentz force velocity meter
2327:
1556:{\displaystyle \mathbf {E} }
1534:{\displaystyle \mathbf {B} }
1062:{\displaystyle \mathbf {v} }
1036:{\displaystyle \mathbf {J} }
1009:
978:, for which he received the
7:
5942:10.1016/j.softx.2019.01.007
5397:10.1016/j.jastp.2003.08.003
5129:7.1 Loma Prieta Earthquake"
4694:10.1016/j.jastp.2006.08.021
4574:Galtier, Sebastien (2016).
4498:
4399:inertial navigation systems
4325:Magnetic confinement fusion
4133:1989 Loma Prieta earthquake
3391:transversal; incompressible
2320:and the second term is the
1166:{\displaystyle q_{\sigma }}
1139:{\displaystyle m_{\sigma }}
1112:{\displaystyle n_{\sigma }}
10:
6140:
6124:Plasma theory and modeling
6074:10.1016/j.jmmm.2010.09.008
5598:10.1007/s41114-021-00031-6
5300:10.5194/nhess-11-1091-2011
5224:10.1016/j.pepi.2008.11.014
5008:Basic Space Plasma Physics
4934:10.1007/s11214-021-00814-x
4893:10.1007/s41116-022-00032-9
4845:10.1088/0029-5515/18/1/010
4580:Cambridge University Press
4484:electromagnetic levitation
4417:, liquid-metal cooling of
4388:
2771:
2733:
2351:The simplest form of MHD,
985:
363:LiĂ©nardâWiechert potential
29:
5432:10.1002/9781119434412.ch3
5349:10.3389/fspas.2021.637197
4430:magnetohydrodynamic drive
3628:
3520:
3319:Direction of energy flow
3318:
3315:
3312:
3309:
3306:
2780:magnetohydrodynamic waves
2730:Structures in MHD systems
2678:Importance of resistivity
2386:magnetic Reynolds numbers
628:Mathematical descriptions
338:Electromagnetic radiation
328:Electromagnetic induction
268:Magnetic vector potential
263:Magnetic scalar potential
5247:Cambridge, Massachusetts
4986:10.1109/TPS.1986.4316632
4806:"Stability of the Pinch"
4425:casting (among others).
2767:
2187:vector calculus identity
6017:2018MS&E..424a2047D
5857:10.1109/IAS.2000.883188
5804:Partially Ionized Gases
5158:10.1029/GL017i009p01465
4707:Alfvén, Hannes (1943).
4616:www.astro.princeton.edu
4490:Magnetic drug targeting
4342:Grad-Shafranov equation
4267:theorized in 1919. The
4241:, including stars, the
3629:equal to phase velocity
2874:and the magnetic field
2796:Fast magnetosonic waves
2793:Slow magnetosonic waves
1718:the equation of motion
1489:{\displaystyle \gamma }
1082:{\displaystyle \sigma }
902:electrically conducting
178:Electrostatic induction
173:Electrostatic discharge
4972:. PS-14 (6): 842â848.
4956:Phil. Trans. Roy. Soc.
4375:
4374:{\displaystyle q>2}
4192:coronal mass ejections
4111:
4109:Earth's magnetic field
4029:, and plasma switches.
3996:
3976:
3945:
3764:
3732:
3711:
3666:Slow magnetosonic wave
3656:
3621:
3566:
3518:Fast magnetosonic wave
3508:
3477:
3435:
3414:
3378:
3347:
3291:
3238:
3004:
2939:
2745:
2633:
2560:
2507:
2460:
2417:frozen-in flux theorem
2374:
2340:
2318:magnetic tension force
2307:
2179:
2138:
2100:
1985:
1927:
1862:
1807:
1709:
1642:the equation of state
1633:
1557:
1535:
1513:
1498:electrical resistivity
1490:
1467:
1405:
1319:
1249:
1196:
1173:, and a mean velocity
1167:
1140:
1113:
1083:
1063:
1037:
980:Nobel Prize in Physics
968:
960:
948:
894:magneto-fluid dynamics
608:Electromagnetic tensor
50:
42:
4922:Space Science Reviews
4827:Wesson, J.A. (1978).
4442:Westinghouse Electric
4403:aerospace engineering
4376:
4243:interplanetary medium
4106:
4019:magnetic reconnection
3997:
3977:
3975:{\displaystyle n_{e}}
3946:
3765:
3733:
3712:
3657:
3622:
3567:
3509:
3478:
3436:
3415:
3413:{\displaystyle v_{A}}
3379:
3348:
3313:Limiting phase speeds
3292:
3239:
3005:
2940:
2743:
2734:Further information:
2724:numerical resistivity
2712:particle acceleration
2634:
2561:
2508:
2461:
2431:magnetic reconnection
2375:
2332:
2308:
2180:
2139:
2101:
1986:
1928:
1863:
1808:
1710:
1634:
1558:
1541:, and electric field
1536:
1514:
1512:{\displaystyle \eta }
1491:
1468:
1406:
1320:
1250:
1197:
1168:
1141:
1114:
1084:
1064:
1038:
911:together as a single
601:Covariant formulation
393:Synchrotron radiation
333:Electromagnetic pulse
323:Electromagnetic field
48:
40:
18:Magnetohydrodynamical
6119:Magnetohydrodynamics
5966:Magnetohydrodynamics
5780:on January 14, 2009.
5689:10.12942/lrsp-2011-6
5251:TechnologyReview.com
5062:. 19 September 2013.
5034:on 25 September 2015
4516:Electromagnetic pump
4511:Electrohydrodynamics
4462:MHD power generation
4359:
4255:Astrophysical plasma
4223:Intense solar storms
3986:
3959:
3843:
3744:
3722:
3672:
3636:
3577:
3527:
3488:
3446:
3425:
3397:
3358:
3327:
3254:
3020:
2955:
2900:
2573:
2520:
2473:
2445:
2396:at the velocity and
2359:
2196:
2159:
2146:magnetic diffusivity
2113:
2005:
1943:
1878:
1823:
1725:
1649:
1570:
1545:
1523:
1503:
1480:
1457:
1335:
1262:
1206:
1177:
1150:
1123:
1096:
1073:
1051:
1025:
898:hydromagnetics
886:magnetohydrodynamics
643:Stressâenergy tensor
568:Reluctance (complex)
313:Displacement current
6066:2011JMMM..323..651N
5978:10.22364/mhd.53.4.5
5933:2019SoftX...9...68V
5894:2000IJHFF..21..329K
5816:Stanford University
5680:2011LRSP....8....6S
5628:astro.princeton.edu
5590:2021LRR....24....3A
5509:1985IAUS..107..537K
5389:2003JASTP..65.1213W
5340:2021FrASS...8...45M
5291:2011NHESS..11.1091A
5216:2009PEPI..173..207T
5150:1990GeoRL..17.1465F
4978:1986ITPS...14..842W
4884:2022LRSP...19....1P
4686:2007JASTP..69.1604F
4643:1942Natur.150..405A
4531:Magnetic flow meter
4381:requirement of the
4247:interstellar medium
4208:auroral electrojets
4166:and throughout the
4129:ultra low frequency
4047:multiscale analysis
3614:
3596:
3199:
3184:
3152:
3134:
3091:
3073:
2692:solar active region
2437:Ideal MHD equations
1416:continuity equation
966:meaning water, and
558:Magnetomotive force
443:Electromotive force
413:Alternating current
348:Jefimenko equations
308:Cyclotron radiation
5809:2008-09-05 at the
5243:Physics arXiv Blog
4477:continuous casting
4415:plasma confinement
4395:angular velocities
4371:
4112:
4064:distributions are
3992:
3972:
3941:
3926:
3919:
3784:Coronal seismology
3760:
3728:
3707:
3652:
3617:
3600:
3582:
3562:
3504:
3473:
3431:
3410:
3374:
3343:
3287:
3234:
3185:
3170:
3138:
3120:
3110:
3077:
3059:
3052:
3000:
2935:
2746:
2688:diffusion constant
2629:
2556:
2503:
2456:
2404:, cannot generate
2394:magnetic diffusion
2390:magnetic induction
2370:
2303:
2175:
2134:
2096:
1996:induction equation
1981:
1923:
1858:
1803:
1705:
1629:
1553:
1531:
1509:
1486:
1463:
1420:equation of motion
1401:
1365:
1315:
1282:
1245:
1224:
1192:
1163:
1146:, electric charge
1136:
1109:
1079:
1059:
1033:
406:Electrical network
243:Gauss magnetic law
208:Static electricity
168:Electric potential
51:
43:
5526:978-90-277-1887-7
5441:978-1-119-43434-4
5383:(11): 1213â1222.
5076:. 7 October 2013.
4680:(14): 1604â1608.
4637:(3805): 405â406.
4612:"O-T Vortex Test"
4004:elementary charge
3995:{\displaystyle e}
3924:
3923:Hall current term
3894:
3869:
3867:
3773:
3772:
3731:{\displaystyle 0}
3615:
3471:
3470:
3434:{\displaystyle 0}
3285:
3284:
3231:
3216:
3109:
3051:
3031:
2998:
2997:
2911:
2596:
2322:magnetic pressure
2294:
2255:
2076:
2028:
1901:
1749:
1690:
1669:
1591:
1519:, magnetic field
1466:{\displaystyle p}
1424:equation of state
1356:
1354:
1273:
1215:
913:continuous medium
874:
873:
573:Reluctance (real)
543:Gyratorâcapacitor
488:Resonant cavities
378:Maxwell equations
16:(Redirected from
6131:
6104:
6103:
6093:
6045:
6039:
6038:
6028:
5996:
5990:
5989:
5961:
5955:
5954:
5944:
5912:
5906:
5905:
5877:
5871:
5870:
5844:
5838:
5837:
5825:
5819:
5800:
5794:
5788:
5782:
5781:
5764:
5758:
5756:
5750:
5742:
5740:
5739:
5733:
5727:. Archived from
5726:
5718:
5712:
5711:
5701:
5691:
5659:
5653:
5652:
5649:"Solar Activity"
5645:
5639:
5638:
5636:
5625:
5616:
5610:
5609:
5583:
5559:
5553:
5552:
5550:
5549:
5543:
5498:
5489:
5483:
5482:
5480:
5479:
5465:
5459:
5458:
5457:
5456:
5415:
5409:
5408:
5368:
5362:
5361:
5351:
5319:
5313:
5312:
5302:
5284:
5261:
5259:
5257:
5234:
5228:
5227:
5210:(3â4): 207â215.
5199:
5193:
5192:
5190:
5188:
5182:
5144:(9): 1465â1468.
5133:
5120:
5114:
5109:
5103:
5098:
5092:
5091:
5084:
5078:
5077:
5070:
5064:
5063:
5050:
5044:
5043:
5041:
5039:
5033:
5026:
5017:
5011:
5004:
4998:
4997:
4965:
4959:
4952:
4946:
4945:
4912:
4906:
4905:
4895:
4863:
4857:
4856:
4824:
4818:
4817:
4801:
4795:
4794:
4776:
4765:
4764:
4756:
4750:
4749:
4731:
4720:
4719:
4713:
4704:
4698:
4697:
4669:
4663:
4662:
4651:10.1038/150405d0
4626:
4620:
4619:
4607:
4593:
4551:Plasma stability
4419:nuclear reactors
4380:
4378:
4377:
4372:
4284:angular momentum
4001:
3999:
3998:
3993:
3981:
3979:
3978:
3973:
3971:
3970:
3950:
3948:
3947:
3942:
3937:
3925:
3922:
3920:
3915:
3911:
3903:
3895:
3893:
3889:
3888:
3875:
3866:
3858:
3850:
3804:
3769:
3767:
3766:
3761:
3759:
3751:
3737:
3735:
3734:
3729:
3716:
3714:
3713:
3708:
3703:
3702:
3690:
3689:
3661:
3659:
3658:
3653:
3651:
3643:
3626:
3624:
3623:
3618:
3616:
3613:
3608:
3595:
3590:
3581:
3571:
3569:
3568:
3563:
3558:
3557:
3545:
3544:
3513:
3511:
3510:
3505:
3503:
3495:
3482:
3480:
3479:
3474:
3472:
3466:
3465:
3456:
3455:
3450:
3440:
3438:
3437:
3432:
3419:
3417:
3416:
3411:
3409:
3408:
3383:
3381:
3380:
3375:
3373:
3365:
3352:
3350:
3349:
3344:
3342:
3334:
3304:
3303:
3296:
3294:
3293:
3288:
3286:
3280:
3272:
3271:
3266:
3265:
3243:
3241:
3240:
3235:
3233:
3232:
3224:
3222:
3218:
3217:
3209:
3208:
3198:
3193:
3183:
3178:
3163:
3162:
3157:
3153:
3151:
3146:
3133:
3128:
3113:
3111:
3102:
3096:
3092:
3090:
3085:
3072:
3067:
3053:
3044:
3032:
3024:
3009:
3007:
3006:
3001:
2999:
2993:
2992:
2983:
2982:
2981:
2972:
2967:
2966:
2944:
2942:
2941:
2936:
2925:
2924:
2912:
2904:
2892:
2883:
2879:
2873:
2861:
2844:
2833:
2816:
2806:
2774:Waves in plasmas
2699:Lundquist number
2638:
2636:
2635:
2630:
2622:
2614:
2597:
2595:
2587:
2586:
2577:
2565:
2563:
2562:
2557:
2555:
2554:
2545:
2540:
2535:
2534:
2512:
2510:
2509:
2504:
2496:
2488:
2480:
2465:
2463:
2462:
2457:
2455:
2379:
2377:
2376:
2371:
2369:
2347:
2312:
2310:
2309:
2304:
2299:
2295:
2293:
2292:
2291:
2278:
2277:
2268:
2256:
2254:
2253:
2244:
2243:
2238:
2234:
2227:
2216:
2211:
2203:
2184:
2182:
2181:
2176:
2174:
2166:
2143:
2141:
2140:
2135:
2133:
2132:
2123:
2105:
2103:
2102:
2097:
2092:
2087:
2086:
2077:
2075:
2074:
2062:
2054:
2046:
2029:
2027:
2019:
2018:
2009:
1990:
1988:
1987:
1982:
1977:
1966:
1958:
1950:
1932:
1930:
1929:
1924:
1919:
1902:
1900:
1892:
1891:
1882:
1867:
1865:
1864:
1859:
1854:
1840:
1835:
1834:
1812:
1810:
1809:
1804:
1790:
1782:
1774:
1769:
1765:
1758:
1750:
1748:
1737:
1714:
1712:
1711:
1706:
1695:
1691:
1689:
1688:
1676:
1670:
1668:
1664:
1658:
1653:
1638:
1636:
1635:
1630:
1619:
1615:
1614:
1592:
1590:
1582:
1574:
1562:
1560:
1559:
1554:
1552:
1540:
1538:
1537:
1532:
1530:
1518:
1516:
1515:
1510:
1495:
1493:
1492:
1487:
1472:
1470:
1469:
1464:
1410:
1408:
1407:
1402:
1397:
1396:
1391:
1385:
1384:
1375:
1374:
1364:
1355:
1347:
1342:
1324:
1322:
1321:
1316:
1314:
1313:
1308:
1302:
1301:
1292:
1291:
1281:
1269:
1254:
1252:
1251:
1246:
1244:
1243:
1234:
1233:
1223:
1201:
1199:
1198:
1193:
1191:
1190:
1185:
1172:
1170:
1169:
1164:
1162:
1161:
1145:
1143:
1142:
1137:
1135:
1134:
1118:
1116:
1115:
1110:
1108:
1107:
1088:
1086:
1085:
1080:
1068:
1066:
1065:
1060:
1058:
1042:
1040:
1039:
1034:
1032:
946:is derived from
909:particle species
900:) is a model of
866:
859:
852:
533:Electric machine
516:Magnetic circuit
478:Parallel circuit
468:Network analysis
433:Electric current
368:London equations
213:Triboelectricity
203:Potential energy
72:
62:Electromagnetism
53:
52:
21:
6139:
6138:
6134:
6133:
6132:
6130:
6129:
6128:
6109:
6108:
6107:
6046:
6042:
5997:
5993:
5962:
5958:
5913:
5909:
5878:
5874:
5867:
5845:
5841:
5826:
5822:
5811:Wayback Machine
5801:
5797:
5789:
5785:
5766:
5765:
5761:
5744:
5743:
5737:
5735:
5731:
5724:
5722:"Archived copy"
5720:
5719:
5715:
5660:
5656:
5647:
5646:
5642:
5634:
5623:
5617:
5613:
5560:
5556:
5547:
5545:
5541:
5527:
5496:
5490:
5486:
5477:
5475:
5473:swe.ssa.esa.int
5467:
5466:
5462:
5454:
5452:
5442:
5416:
5412:
5369:
5365:
5320:
5316:
5255:
5253:
5235:
5231:
5200:
5196:
5186:
5184:
5180:
5131:
5128:
5121:
5117:
5110:
5106:
5099:
5095:
5086:
5085:
5081:
5072:
5071:
5067:
5052:
5051:
5047:
5037:
5035:
5031:
5024:
5018:
5014:
5005:
5001:
4966:
4962:
4953:
4949:
4913:
4909:
4864:
4860:
4825:
4821:
4802:
4798:
4791:
4777:
4768:
4757:
4753:
4746:
4732:
4723:
4711:
4705:
4701:
4670:
4666:
4627:
4623:
4608:
4604:
4600:
4590:
4570:
4568:Further reading
4565:
4501:
4492:
4423:electromagnetic
4411:
4391:
4360:
4357:
4356:
4327:
4237:MHD applies to
4235:
4160:
4152:2010 earthquake
4150:
4121:
4098:Coriolis effect
4090:
4085:
4061:
4056:
4037:Vlasov equation
3987:
3984:
3983:
3966:
3962:
3960:
3957:
3956:
3933:
3921:
3907:
3899:
3884:
3880:
3879:
3874:
3870:
3868:
3862:
3854:
3846:
3844:
3841:
3840:
3799:
3792:
3755:
3747:
3745:
3742:
3741:
3723:
3720:
3719:
3698:
3694:
3685:
3681:
3673:
3670:
3669:
3647:
3639:
3637:
3634:
3633:
3609:
3604:
3591:
3586:
3580:
3578:
3575:
3574:
3553:
3549:
3540:
3536:
3528:
3525:
3524:
3499:
3491:
3489:
3486:
3485:
3461:
3457:
3451:
3449:
3447:
3444:
3443:
3426:
3423:
3422:
3404:
3400:
3398:
3395:
3394:
3369:
3361:
3359:
3356:
3355:
3338:
3330:
3328:
3325:
3324:
3273:
3270:
3261:
3257:
3255:
3252:
3251:
3223:
3204:
3200:
3194:
3189:
3179:
3174:
3158:
3147:
3142:
3129:
3124:
3119:
3115:
3114:
3112:
3100:
3086:
3081:
3068:
3063:
3058:
3054:
3042:
3041:
3037:
3036:
3023:
3021:
3018:
3017:
2988:
2984:
2977:
2973:
2971:
2962:
2958:
2956:
2953:
2952:
2920:
2916:
2903:
2901:
2898:
2897:
2891:
2885:
2881:
2875:
2869:
2866:
2865:
2864:
2863:
2862:
2859:
2852:
2847:
2845:
2836:
2835:
2834:
2831:
2824:
2819:
2817:
2808:
2807:
2804:
2776:
2770:
2738:
2732:
2680:
2645:
2618:
2610:
2588:
2582:
2578:
2576:
2574:
2571:
2570:
2550:
2546:
2541:
2536:
2530:
2526:
2521:
2518:
2517:
2492:
2484:
2476:
2474:
2471:
2470:
2451:
2446:
2443:
2442:
2439:
2402:ideal processes
2392:dominates over
2365:
2360:
2357:
2356:
2349:
2342:
2330:
2287:
2283:
2279:
2273:
2269:
2267:
2263:
2249:
2245:
2239:
2223:
2222:
2218:
2217:
2215:
2207:
2199:
2197:
2194:
2193:
2170:
2162:
2160:
2157:
2156:
2128:
2124:
2119:
2114:
2111:
2110:
2088:
2082:
2078:
2070:
2066:
2061:
2050:
2042:
2020:
2014:
2010:
2008:
2006:
2003:
2002:
1973:
1962:
1954:
1946:
1944:
1941:
1940:
1915:
1893:
1887:
1883:
1881:
1879:
1876:
1875:
1850:
1836:
1830:
1826:
1824:
1821:
1820:
1786:
1778:
1770:
1754:
1741:
1736:
1735:
1731:
1726:
1723:
1722:
1684:
1680:
1675:
1671:
1660:
1659:
1654:
1652:
1650:
1647:
1646:
1610:
1606:
1602:
1583:
1575:
1573:
1571:
1568:
1567:
1548:
1546:
1543:
1542:
1526:
1524:
1521:
1520:
1504:
1501:
1500:
1481:
1478:
1477:
1475:adiabatic index
1458:
1455:
1454:
1392:
1387:
1386:
1380:
1376:
1370:
1366:
1360:
1346:
1338:
1336:
1333:
1332:
1309:
1304:
1303:
1297:
1293:
1287:
1283:
1277:
1265:
1263:
1260:
1259:
1239:
1235:
1229:
1225:
1219:
1207:
1204:
1203:
1186:
1181:
1180:
1178:
1175:
1174:
1157:
1153:
1151:
1148:
1147:
1130:
1126:
1124:
1121:
1120:
1103:
1099:
1097:
1094:
1093:
1074:
1071:
1070:
1054:
1052:
1049:
1048:
1028:
1026:
1023:
1022:
1020:current density
1012:
988:
870:
841:
840:
656:
648:
647:
603:
593:
592:
548:Induction motor
518:
508:
507:
423:Current density
408:
398:
397:
388:Poynting vector
298:
296:Electrodynamics
288:
287:
283:Right-hand rule
248:Magnetic dipole
238:BiotâSavart law
228:
218:
217:
153:Electric dipole
148:Electric charge
123:
35:
28:
23:
22:
15:
12:
11:
5:
6137:
6127:
6126:
6121:
6106:
6105:
6060:(6): 651â668.
6040:
5991:
5972:(4): 643â652.
5956:
5907:
5888:(3): 329â337.
5872:
5865:
5839:
5820:
5795:
5783:
5759:
5713:
5654:
5640:
5611:
5554:
5525:
5484:
5460:
5440:
5410:
5363:
5314:
5229:
5194:
5126:
5115:
5104:
5093:
5079:
5065:
5045:
5012:
4999:
4960:
4947:
4907:
4858:
4833:Nuclear Fusion
4819:
4796:
4789:
4766:
4763:. 29B(2): 1â7.
4751:
4744:
4721:
4718:. 29B(2): 1â7.
4699:
4664:
4621:
4601:
4599:
4596:
4595:
4594:
4588:
4569:
4566:
4564:
4563:
4558:
4553:
4548:
4543:
4538:
4533:
4528:
4523:
4518:
4513:
4508:
4502:
4500:
4497:
4491:
4488:
4452:superconductor
4410:
4407:
4390:
4387:
4370:
4367:
4364:
4326:
4323:
4273:solar rotation
4234:
4231:
4159:
4156:
4148:
4120:
4117:
4089:
4086:
4084:
4081:
4077:Landau damping
4060:
4057:
4055:
4052:
4051:
4050:
4043:
4040:
4033:
4030:
4010:
4007:
3991:
3969:
3965:
3953:
3952:
3951:
3940:
3936:
3932:
3929:
3918:
3914:
3910:
3906:
3902:
3898:
3892:
3887:
3883:
3878:
3873:
3865:
3861:
3857:
3853:
3849:
3830:
3827:
3824:electric field
3820:
3817:
3813:
3810:
3796:
3791:
3788:
3771:
3770:
3758:
3754:
3750:
3738:
3727:
3717:
3706:
3701:
3697:
3693:
3688:
3684:
3680:
3677:
3667:
3663:
3662:
3650:
3646:
3642:
3630:
3627:
3612:
3607:
3603:
3599:
3594:
3589:
3585:
3572:
3561:
3556:
3552:
3548:
3543:
3539:
3535:
3532:
3522:
3519:
3515:
3514:
3502:
3498:
3494:
3483:
3469:
3464:
3460:
3454:
3441:
3430:
3420:
3407:
3403:
3392:
3389:
3385:
3384:
3372:
3368:
3364:
3353:
3341:
3337:
3333:
3321:
3320:
3317:
3316:Group velocity
3314:
3311:
3308:
3298:
3297:
3283:
3279:
3276:
3269:
3264:
3260:
3245:
3244:
3230:
3227:
3221:
3215:
3212:
3207:
3203:
3197:
3192:
3188:
3182:
3177:
3173:
3169:
3166:
3161:
3156:
3150:
3145:
3141:
3137:
3132:
3127:
3123:
3118:
3108:
3105:
3099:
3095:
3089:
3084:
3080:
3076:
3071:
3066:
3062:
3057:
3050:
3047:
3040:
3035:
3030:
3027:
3011:
3010:
2996:
2991:
2987:
2980:
2976:
2970:
2965:
2961:
2946:
2945:
2934:
2931:
2928:
2923:
2919:
2915:
2910:
2907:
2889:
2857:
2850:
2846:
2839:
2838:
2837:
2829:
2822:
2818:
2811:
2810:
2809:
2802:
2801:
2800:
2799:
2798:
2797:
2794:
2791:
2769:
2766:
2750:current sheets
2731:
2728:
2679:
2676:
2675:
2674:
2671:Landau damping
2659:
2656:
2644:
2641:
2640:
2639:
2628:
2625:
2621:
2617:
2613:
2609:
2606:
2603:
2600:
2594:
2591:
2585:
2581:
2553:
2549:
2544:
2539:
2533:
2529:
2525:
2514:
2513:
2502:
2499:
2495:
2491:
2487:
2483:
2479:
2454:
2450:
2438:
2435:
2368:
2364:
2331:
2329:
2326:
2314:
2313:
2302:
2298:
2290:
2286:
2282:
2276:
2272:
2266:
2262:
2259:
2252:
2248:
2242:
2237:
2233:
2230:
2226:
2221:
2214:
2210:
2206:
2202:
2173:
2169:
2165:
2131:
2127:
2122:
2118:
2107:
2106:
2095:
2091:
2085:
2081:
2073:
2069:
2065:
2060:
2057:
2053:
2049:
2045:
2041:
2038:
2035:
2032:
2026:
2023:
2017:
2013:
1992:
1991:
1980:
1976:
1972:
1969:
1965:
1961:
1957:
1953:
1949:
1936:and Ohm's law
1934:
1933:
1922:
1918:
1914:
1911:
1908:
1905:
1899:
1896:
1890:
1886:
1871:Faraday's law
1869:
1868:
1857:
1853:
1849:
1846:
1843:
1839:
1833:
1829:
1814:
1813:
1802:
1799:
1796:
1793:
1789:
1785:
1781:
1777:
1773:
1768:
1764:
1761:
1757:
1753:
1747:
1744:
1740:
1734:
1730:
1716:
1715:
1704:
1701:
1698:
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1687:
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1679:
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1667:
1663:
1657:
1640:
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1628:
1625:
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1618:
1613:
1609:
1605:
1601:
1598:
1595:
1589:
1586:
1581:
1578:
1551:
1529:
1508:
1485:
1462:
1412:
1411:
1400:
1395:
1390:
1383:
1379:
1373:
1369:
1363:
1359:
1353:
1350:
1345:
1341:
1326:
1325:
1312:
1307:
1300:
1296:
1290:
1286:
1280:
1276:
1272:
1268:
1242:
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1232:
1228:
1222:
1218:
1214:
1211:
1189:
1184:
1160:
1156:
1133:
1129:
1106:
1102:
1091:number density
1078:
1057:
1045:center of mass
1031:
1011:
1008:
987:
984:
956:magnetic field
892:; also called
872:
871:
869:
868:
861:
854:
846:
843:
842:
839:
838:
833:
828:
823:
818:
813:
808:
803:
798:
793:
788:
783:
778:
773:
768:
763:
758:
753:
748:
743:
738:
733:
728:
723:
718:
713:
708:
703:
698:
693:
688:
683:
678:
673:
668:
663:
657:
654:
653:
650:
649:
646:
645:
640:
635:
630:
625:
623:Four-potential
620:
615:
610:
604:
599:
598:
595:
594:
591:
590:
585:
580:
575:
570:
565:
560:
555:
550:
545:
540:
538:Electric motor
535:
530:
525:
519:
514:
513:
510:
509:
506:
505:
500:
495:
493:Series circuit
490:
485:
480:
475:
470:
465:
463:Kirchhoff laws
460:
455:
450:
445:
440:
435:
430:
428:Direct current
425:
420:
415:
409:
404:
403:
400:
399:
396:
395:
390:
385:
383:Maxwell tensor
380:
375:
370:
365:
360:
355:
353:Larmor formula
350:
345:
340:
335:
330:
325:
320:
315:
310:
305:
303:Bremsstrahlung
299:
294:
293:
290:
289:
286:
285:
280:
275:
270:
265:
260:
255:
253:Magnetic field
250:
245:
240:
235:
229:
226:Magnetostatics
224:
223:
220:
219:
216:
215:
210:
205:
200:
195:
190:
185:
180:
175:
170:
165:
160:
158:Electric field
155:
150:
145:
140:
135:
130:
128:Charge density
124:
121:Electrostatics
119:
118:
115:
114:
113:
112:
107:
102:
97:
92:
87:
82:
74:
73:
65:
64:
58:
57:
56:Articles about
26:
9:
6:
4:
3:
2:
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6125:
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6059:
6055:
6051:
6044:
6036:
6032:
6027:
6022:
6018:
6014:
6011:(1): 012047.
6010:
6006:
6002:
5995:
5987:
5983:
5979:
5975:
5971:
5967:
5960:
5952:
5948:
5943:
5938:
5934:
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5926:
5922:
5918:
5911:
5903:
5899:
5895:
5891:
5887:
5883:
5876:
5868:
5866:0-7803-6401-5
5862:
5858:
5854:
5850:
5843:
5835:
5831:
5824:
5817:
5813:
5812:
5808:
5805:
5799:
5793:
5787:
5779:
5775:
5774:
5769:
5763:
5754:
5748:
5734:on 2014-08-20
5730:
5723:
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5709:
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5695:
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5406:
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5359:
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5337:
5333:
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5325:
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5310:
5306:
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5283:
5278:
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5225:
5221:
5217:
5213:
5209:
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5198:
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5159:
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5147:
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5097:
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5023:
5016:
5009:
5003:
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4991:
4987:
4983:
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4957:
4951:
4943:
4939:
4935:
4931:
4927:
4923:
4919:
4911:
4903:
4899:
4894:
4889:
4885:
4881:
4877:
4873:
4869:
4862:
4854:
4850:
4846:
4842:
4838:
4834:
4830:
4823:
4815:
4811:
4807:
4800:
4792:
4790:0-521-48179-1
4786:
4782:
4775:
4773:
4771:
4762:
4755:
4747:
4741:
4737:
4730:
4728:
4726:
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4710:
4703:
4695:
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4683:
4679:
4675:
4668:
4660:
4656:
4652:
4648:
4644:
4640:
4636:
4632:
4625:
4617:
4613:
4606:
4602:
4591:
4589:9781107158658
4585:
4581:
4577:
4572:
4571:
4562:
4559:
4557:
4554:
4552:
4549:
4547:
4544:
4542:
4539:
4537:
4534:
4532:
4529:
4527:
4524:
4522:
4519:
4517:
4514:
4512:
4509:
4507:
4504:
4503:
4496:
4487:
4485:
4480:
4478:
4473:
4471:
4470:microfluidics
4466:
4463:
4459:
4457:
4456:liquid helium
4453:
4449:
4448:
4443:
4439:
4434:
4431:
4426:
4424:
4420:
4416:
4406:
4404:
4400:
4396:
4386:
4384:
4383:safety factor
4368:
4365:
4362:
4354:
4350:
4349:MHD stability
4346:
4343:
4338:
4336:
4332:
4322:
4320:
4316:
4312:
4311:active region
4308:
4303:
4301:
4297:
4293:
4292:closed system
4289:
4285:
4280:
4278:
4277:Parker spiral
4274:
4270:
4266:
4265:Joseph Larmor
4262:
4258:
4256:
4252:
4248:
4244:
4240:
4230:
4228:
4224:
4221:forecasting.
4220:
4219:space weather
4215:
4213:
4209:
4205:
4201:
4195:
4193:
4189:
4185:
4184:magnetosphere
4181:
4177:
4173:
4172:space physics
4169:
4165:
4158:Space Physics
4155:
4153:
4146:
4142:
4138:
4134:
4130:
4126:
4116:
4110:
4107:Reversals of
4105:
4101:
4099:
4095:
4080:
4078:
4074:
4073:double layers
4069:
4067:
4048:
4044:
4041:
4038:
4034:
4032:Collisionless
4031:
4028:
4027:neutron stars
4024:
4023:ion thrusters
4020:
4016:
4011:
4008:
4005:
3989:
3967:
3963:
3954:
3938:
3930:
3927:
3916:
3904:
3890:
3885:
3881:
3876:
3871:
3859:
3851:
3839:
3838:
3835:
3831:
3828:
3825:
3821:
3818:
3814:
3811:
3808:
3802:
3797:
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3793:
3787:
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3781:
3776:
3752:
3739:
3725:
3718:
3699:
3695:
3691:
3686:
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3668:
3665:
3664:
3644:
3631:
3610:
3605:
3601:
3597:
3592:
3587:
3583:
3573:
3554:
3550:
3546:
3541:
3537:
3523:
3517:
3516:
3496:
3484:
3467:
3462:
3458:
3442:
3428:
3421:
3405:
3401:
3393:
3390:
3387:
3386:
3366:
3354:
3335:
3323:
3322:
3305:
3302:
3281:
3277:
3274:
3267:
3262:
3258:
3250:
3249:
3248:
3228:
3225:
3219:
3213:
3210:
3205:
3201:
3195:
3190:
3186:
3180:
3175:
3171:
3167:
3164:
3159:
3154:
3148:
3143:
3139:
3135:
3130:
3125:
3121:
3116:
3106:
3103:
3097:
3093:
3087:
3082:
3078:
3074:
3069:
3064:
3060:
3055:
3048:
3045:
3038:
3033:
3028:
3025:
3016:
3015:
3014:
2994:
2989:
2985:
2978:
2974:
2968:
2963:
2959:
2951:
2950:
2949:
2932:
2929:
2926:
2921:
2917:
2913:
2908:
2905:
2896:
2895:
2894:
2888:
2878:
2872:
2860:
2853:
2843:
2832:
2825:
2815:
2795:
2792:
2789:
2788:
2787:
2785:
2781:
2775:
2765:
2763:
2759:
2755:
2754:magnetosphere
2751:
2742:
2737:
2727:
2725:
2719:
2715:
2713:
2709:
2704:
2703:instabilities
2700:
2695:
2693:
2689:
2685:
2684:diffusion law
2672:
2668:
2667:Larmor radius
2664:
2660:
2657:
2654:
2650:
2649:
2648:
2626:
2615:
2604:
2598:
2592:
2569:
2568:
2567:
2551:
2547:
2542:
2531:
2523:
2500:
2497:
2489:
2481:
2469:
2468:
2467:
2448:
2434:
2432:
2427:
2423:
2418:
2413:
2411:
2407:
2403:
2399:
2398:length scales
2395:
2391:
2388:during which
2387:
2383:
2362:
2354:
2348:
2345:
2344:Hannes Alfvén
2339:
2337:
2336:eddy currents
2325:
2323:
2319:
2300:
2296:
2288:
2284:
2280:
2274:
2270:
2264:
2257:
2250:
2246:
2235:
2228:
2219:
2212:
2204:
2192:
2191:
2190:
2188:
2167:
2154:
2153:Lorentz force
2149:
2147:
2129:
2125:
2120:
2116:
2093:
2083:
2071:
2067:
2063:
2058:
2047:
2036:
2030:
2024:
2001:
2000:
1999:
1997:
1978:
1970:
1967:
1959:
1951:
1939:
1938:
1937:
1920:
1912:
1906:
1903:
1897:
1874:
1873:
1872:
1855:
1847:
1841:
1831:
1827:
1819:
1818:
1817:
1800:
1797:
1791:
1783:
1775:
1766:
1759:
1751:
1745:
1732:
1728:
1721:
1720:
1719:
1702:
1699:
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1692:
1685:
1681:
1677:
1672:
1665:
1645:
1644:
1643:
1626:
1623:
1620:
1616:
1607:
1603:
1599:
1593:
1587:
1579:
1566:
1565:
1564:
1506:
1499:
1483:
1476:
1460:
1452:
1447:
1445:
1444:isothermality
1441:
1437:
1433:
1432:Faraday's law
1429:
1425:
1421:
1417:
1398:
1393:
1381:
1377:
1371:
1367:
1361:
1357:
1351:
1348:
1343:
1331:
1330:
1329:
1310:
1298:
1294:
1288:
1284:
1278:
1274:
1270:
1258:
1257:
1256:
1240:
1236:
1230:
1226:
1220:
1216:
1212:
1209:
1187:
1158:
1154:
1131:
1127:
1104:
1100:
1092:
1076:
1046:
1021:
1017:
1007:
1005:
1004:
999:
998:
993:
992:Hannes Alfvén
983:
981:
977:
976:Hannes Alfvén
973:
972:
971:
965:
964:
963:
957:
953:
952:
951:
945:
940:
938:
934:
930:
926:
925:space physics
922:
921:liquid metals
918:
914:
910:
906:
903:
899:
895:
891:
887:
883:
879:
867:
862:
860:
855:
853:
848:
847:
845:
844:
837:
834:
832:
829:
827:
824:
822:
819:
817:
814:
812:
809:
807:
804:
802:
799:
797:
794:
792:
789:
787:
784:
782:
779:
777:
774:
772:
769:
767:
764:
762:
759:
757:
754:
752:
749:
747:
744:
742:
739:
737:
734:
732:
729:
727:
724:
722:
719:
717:
714:
712:
709:
707:
704:
702:
699:
697:
694:
692:
689:
687:
684:
682:
679:
677:
674:
672:
669:
667:
664:
662:
659:
658:
652:
651:
644:
641:
639:
636:
634:
631:
629:
626:
624:
621:
619:
616:
614:
611:
609:
606:
605:
602:
597:
596:
589:
586:
584:
581:
579:
576:
574:
571:
569:
566:
564:
561:
559:
556:
554:
551:
549:
546:
544:
541:
539:
536:
534:
531:
529:
526:
524:
521:
520:
517:
512:
511:
504:
501:
499:
496:
494:
491:
489:
486:
484:
481:
479:
476:
474:
471:
469:
466:
464:
461:
459:
458:Joule heating
456:
454:
451:
449:
446:
444:
441:
439:
436:
434:
431:
429:
426:
424:
421:
419:
416:
414:
411:
410:
407:
402:
401:
394:
391:
389:
386:
384:
381:
379:
376:
374:
373:Lorentz force
371:
369:
366:
364:
361:
359:
356:
354:
351:
349:
346:
344:
341:
339:
336:
334:
331:
329:
326:
324:
321:
319:
316:
314:
311:
309:
306:
304:
301:
300:
297:
292:
291:
284:
281:
279:
276:
274:
273:Magnetization
271:
269:
266:
264:
261:
259:
258:Magnetic flux
256:
254:
251:
249:
246:
244:
241:
239:
236:
234:
231:
230:
227:
222:
221:
214:
211:
209:
206:
204:
201:
199:
196:
194:
191:
189:
186:
184:
181:
179:
176:
174:
171:
169:
166:
164:
163:Electric flux
161:
159:
156:
154:
151:
149:
146:
144:
141:
139:
136:
134:
131:
129:
126:
125:
122:
117:
116:
111:
108:
106:
103:
101:
100:Computational
98:
96:
93:
91:
88:
86:
83:
81:
78:
77:
76:
75:
71:
67:
66:
63:
60:
59:
55:
54:
47:
39:
33:
19:
6057:
6053:
6043:
6008:
6004:
5994:
5969:
5965:
5959:
5924:
5920:
5910:
5885:
5881:
5875:
5848:
5842:
5834:Artech House
5829:
5823:
5802:
5798:
5791:
5786:
5778:the original
5771:
5762:
5736:. Retrieved
5729:the original
5716:
5671:
5667:
5657:
5643:
5627:
5614:
5571:
5567:
5557:
5546:. Retrieved
5500:
5487:
5476:. Retrieved
5472:
5463:
5453:, retrieved
5423:
5413:
5380:
5376:
5366:
5331:
5327:
5317:
5272:
5268:
5256:December 18,
5254:. Retrieved
5242:
5232:
5207:
5203:
5197:
5187:December 18,
5185:. Retrieved
5141:
5135:
5118:
5107:
5096:
5082:
5068:
5059:Live Science
5057:
5048:
5036:. Retrieved
5029:the original
5015:
5007:
5002:
4969:
4963:
4955:
4950:
4925:
4921:
4910:
4875:
4871:
4861:
4836:
4832:
4822:
4799:
4780:
4760:
4754:
4735:
4715:
4702:
4677:
4673:
4667:
4634:
4630:
4624:
4615:
4605:
4575:
4493:
4481:
4474:
4467:
4460:
4445:
4435:
4427:
4412:
4392:
4353:Troyon limit
4347:
4339:
4335:stellarators
4328:
4307:solar flares
4304:
4288:Solar System
4281:
4259:
4239:astrophysics
4236:
4233:Astrophysics
4216:
4204:ring current
4200:magnetopause
4196:
4170:is known as
4168:Solar System
4161:
4122:
4113:
4091:
4083:Applications
4070:
4062:
4009:Electron MHD
3834:Hall current
3800:
3782:of the Sun (
3777:
3774:
3299:
3246:
3012:
2947:
2886:
2876:
2870:
2867:
2855:
2848:
2827:
2820:
2790:Alfvén waves
2783:
2779:
2777:
2747:
2720:
2716:
2714:, and heat.
2696:
2681:
2646:
2515:
2440:
2414:
2401:
2352:
2350:
2341:
2333:
2315:
2150:
2108:
1993:
1935:
1870:
1815:
1717:
1641:
1448:
1440:adiabaticity
1428:AmpĂšre's Law
1413:
1327:
1013:
1003:Alfvén waves
1001:
996:
989:
967:
959:
947:
943:
941:
933:astrophysics
897:
893:
889:
885:
875:
618:Four-current
553:Linear motor
438:Electrolysis
318:Eddy current
278:Permeability
198:Polarization
193:Permittivity
5699:2433/153022
5275:(4): 1091.
5088:"Geodynamo"
4546:Molten salt
4409:Engineering
4401:such as in
4125:earthquakes
4119:Earthquakes
4054:Limitations
4045:By using a
3388:Alfvén wave
937:engineering
882:engineering
588:Transformer
418:Capacitance
343:Faraday law
138:Coulomb law
80:Electricity
6113:Categories
5738:2014-08-19
5581:2008.12069
5548:2019-07-22
5478:2023-03-10
5455:2023-03-10
4839:: 87â132.
4745:0521528003
4598:References
4521:Ferrofluid
4454:cooled by
4269:solar wind
4188:Solar wind
4182:, Earth's
4176:ionosphere
4137:California
4088:Geophysics
4066:Maxwellian
3790:Extensions
2772:See also:
2762:solar wind
2758:ionosphere
2708:turbulence
2663:skin depth
2653:Maxwellian
2408:and raise
2346:, 1943
929:geophysics
655:Scientists
503:Waveguides
483:Resistance
453:Inductance
233:AmpĂšre law
6082:0304-8853
6035:1757-899X
5986:0024-998X
5951:2352-7110
5927:: 68â72.
5921:SoftwareX
5708:122217405
5606:235631174
5535:117512943
5450:204194812
5405:1364-6826
5358:2296-987X
5282:1012.1533
5166:0094-8276
4942:231592434
4902:248673571
4853:122227433
4319:radiation
4300:proto-Sun
4296:conserved
3931:η
3917:⏟
3905:×
3872:−
3860:×
3819:Two-fluid
3795:Resistive
3753:∥
3645:∥
3497:∥
3468:ρ
3459:μ
3367:⊥
3336:∥
3282:ρ
3275:γ
3214:θ
3211:
3165:−
3098:±
3026:ω
2995:ρ
2986:μ
2933:θ
2930:
2906:ω
2784:MHD waves
2760:from the
2616:×
2605:×
2602:∇
2590:∂
2580:∂
2548:μ
2528:∇
2524:η
2490:×
2449:η
2382:Ohm's law
2363:η
2353:ideal MHD
2328:Ideal MHD
2285:μ
2261:∇
2258:−
2247:μ
2232:∇
2229:⋅
2205:×
2168:×
2126:μ
2117:η
2080:∇
2068:μ
2064:η
2048:×
2037:×
2034:∇
2022:∂
2012:∂
1971:η
1960:×
1913:×
1910:∇
1907:−
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1760:⋅
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1608:ρ
1600:⋅
1597:∇
1585:∂
1580:ρ
1577:∂
1507:η
1484:γ
1453:pressure
1451:isotropic
1436:Ohm's law
1394:σ
1382:σ
1372:σ
1362:σ
1358:∑
1352:ρ
1311:σ
1299:σ
1289:σ
1279:σ
1275:∑
1241:σ
1231:σ
1221:σ
1217:∑
1210:ρ
1188:σ
1159:σ
1132:σ
1105:σ
1077:σ
1047:velocity
1010:Equations
982:in 1970.
942:The word
811:Steinmetz
741:Kirchhoff
726:Jefimenko
721:Hopkinson
706:Helmholtz
701:Heaviside
563:Permeance
448:Impedance
188:Insulator
183:Gauss law
133:Conductor
110:Phenomena
105:Textbooks
85:Magnetism
6100:21278859
5807:Archived
5747:cite web
5674:(1): 6.
5632:Archived
5574:(1): 3.
5539:Archived
5309:53456663
5178:Archived
5038:27 April
4994:31433317
4878:(1): 1.
4499:See also
4447:Yamato-1
4331:tokamaks
4261:Sunspots
3816:inertia.
3812:Extended
3740:approx.
3632:approx.
2422:advected
2189:to give
1043:and the
970:dynamics
954:meaning
950:magneto-
836:Wiechert
791:Poynting
681:Einstein
528:DC motor
523:AC motor
358:Lenz law
143:Electret
6091:3029028
6062:Bibcode
6013:Bibcode
5929:Bibcode
5890:Bibcode
5676:Bibcode
5586:Bibcode
5505:Bibcode
5385:Bibcode
5336:Bibcode
5287:Bibcode
5212:Bibcode
5174:1795290
5146:Bibcode
4974:Bibcode
4880:Bibcode
4814:4329910
4682:Bibcode
4659:4072220
4639:Bibcode
4389:Sensors
4345:coils.
4290:. In a
4286:in the
4180:auroras
4141:DEMETER
4042:Reduced
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4002:is the
2410:entropy
2324:force.
2144:is the
1119:, mass
1016:species
986:History
917:plasmas
878:physics
821:Thomson
796:Ritchie
786:Poisson
771:Neumann
766:Maxwell
761:Lorentz
756:Liénard
686:Faraday
671:Coulomb
498:Voltage
473:Ohm law
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4421:, and
4317:, and
4315:X-rays
4210:, and
4190:, and
4186:, the
3955:where
3807:shocks
3780:corona
3247:where
2948:where
2109:where
1434:, and
1089:has a
1018:: the
997:Nature
962:hydro-
935:, and
905:fluids
806:Singer
801:Savart
781:Ărsted
746:Larmor
736:Kelvin
691:Fizeau
661:AmpĂšre
583:Stator
90:Optics
5732:(PDF)
5725:(PDF)
5704:S2CID
5635:(PDF)
5624:(PDF)
5602:S2CID
5576:arXiv
5542:(PDF)
5531:S2CID
5497:(PDF)
5446:S2CID
5305:S2CID
5277:arXiv
5181:(PDF)
5132:(PDF)
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4938:S2CID
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4655:S2CID
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2854:<
2826:>
2768:Waves
2155:term
1422:, an
1418:, an
831:Weber
826:Volta
816:Tesla
731:Joule
716:Hertz
711:Henry
696:Gauss
578:Rotor
6096:PMID
6078:ISSN
6031:ISSN
5982:ISSN
5947:ISSN
5861:ISBN
5773:Time
5753:link
5521:ISBN
5436:ISBN
5401:ISSN
5354:ISSN
5258:2010
5189:2010
5170:OCLC
5162:ISSN
5040:2015
4810:OSTI
4785:ISBN
4740:ISBN
4584:ISBN
4366:>
4340:The
4251:jets
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3829:Hall
3310:Type
3307:Mode
2665:and
2406:heat
919:and
880:and
751:Lenz
676:Davy
666:Biot
6086:PMC
6070:doi
6058:323
6021:doi
6009:424
5974:doi
5937:doi
5898:doi
5853:doi
5694:hdl
5684:doi
5594:doi
5513:doi
5428:doi
5393:doi
5344:doi
5295:doi
5220:doi
5208:173
5154:doi
4982:doi
4930:doi
4926:217
4888:doi
4841:doi
4690:doi
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