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Magnetohydrodynamics

From Wikipedia, the free encyclopedia
Model of electrically conducting fluids
For the academic journal, seeMagnetohydrodynamics (journal).
The plasma making up the Sun can be modeled as an MHD system
Simulation of the Orszag–Tang MHD vortex problem, a well-known model problem for testing the transition to supersonic 2D MHD turbulence[1]
Electromagnetism
Solenoid

Magnetohydrodynamics (MHD; also calledmagneto-fluid dynamics orhydro­magnetics) is a model ofelectrically conductingfluids that treats all types ofcharged particles together as one continuous fluid. It is primarily concerned with the low-frequency, large-scale, magnetic behavior inplasmas andliquid metals and has applications in multiple fields includingspace physics,geophysics,astrophysics, andengineering.

The wordmagnetohydrodynamics is derived frommagneto- meaningmagnetic field,hydro- meaning water, anddynamics meaning movement. The field of MHD was initiated byHannes Alfvén, for which he received theNobel Prize in Physics in 1970.

History

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The MHD description of electrically conducting fluids was first developed byHannes Alfvén in a 1942 paper published inNature titled "Existence of Electromagnetic–Hydrodynamic Waves" which outlined his discovery of what are now referred to asAlfvén waves.[2][3] Alfvén initially referred to these waves as "electromagnetic–hydrodynamic waves"; however, in a later paper he noted, "As the term 'electromagnetic–hydrodynamic waves' is somewhat complicated, it may be convenient to call this phenomenon 'magneto–hydrodynamic' waves."[4]

Equations

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In MHD, motion in the fluid is described using linear combinations of the mean motions of the individualspecies: thecurrent densityJ{\displaystyle \mathbf {J} } and thecenter of mass velocityv{\displaystyle \mathbf {v} }. In a given fluid, each speciesσ{\displaystyle \sigma } has anumber densitynσ{\displaystyle n_{\sigma }}, massmσ{\displaystyle m_{\sigma }}, electric chargeqσ{\displaystyle q_{\sigma }}, and a mean velocityuσ{\displaystyle \mathbf {u} _{\sigma }}. The fluid's total mass density is thenρ=∑σmσnσ{\textstyle \rho =\sum _{\sigma }m_{\sigma }n_{\sigma }}, and the motion of the fluid can be described by the current density expressed asJ=∑σnσqσuσ{\displaystyle \mathbf {J} =\sum _{\sigma }n_{\sigma }q_{\sigma }\mathbf {u} _{\sigma }}and the center of mass velocity expressed as:v=1ρ∑σmσnσuσ.{\displaystyle \mathbf {v} ={\frac {1}{\rho }}\sum _{\sigma }m_{\sigma }n_{\sigma }\mathbf {u} _{\sigma }.}

MHD can be described by a set of equations consisting of acontinuity equation, an equation of motion (theCauchy momentum equation), anequation of state,Ampère's law,Faraday's law, andOhm's law. As with any fluid description to a kinetic system, aclosure approximation must be applied to the highest moment of the particle distribution equation. This is often accomplished with approximations to the heat flux through a condition ofadiabaticity orisothermality.

In the adiabatic limit, that is, the assumption of anisotropic pressurep{\displaystyle p} and isotropic temperature, a fluid with anadiabatic indexγ{\displaystyle \gamma },electrical resistivityη{\displaystyle \eta }, magnetic fieldB{\displaystyle \mathbf {B} }, and electric fieldE{\displaystyle \mathbf {E} } can be described by the continuity equation∂ρ∂t+∇⋅(ρv)=0,{\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \left(\rho \mathbf {v} \right)=0,}the equation of stateddt(pργ)=0,{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {p}{\rho ^{\gamma }}}\right)=0,}the equation of motionρ(∂∂t+v⋅∇)v=J×B−∇p,{\displaystyle \rho \left({\frac {\partial }{\partial t}}+\mathbf {v} \cdot \nabla \right)\mathbf {v} =\mathbf {J} \times \mathbf {B} -\nabla p,}the low-frequency Ampère's lawμ0J=∇×B,{\displaystyle \mu _{0}\mathbf {J} =\nabla \times \mathbf {B} ,}Faraday's law∂B∂t=−∇×E,{\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}=-\nabla \times \mathbf {E} ,}and Ohm's lawE+v×B=ηJ.{\displaystyle \mathbf {E} +\mathbf {v} \times \mathbf {B} =\eta \mathbf {J} .}Taking the curl of this equation and using Ampère's law and Faraday's law results in theinduction equation,∂B∂t=∇×(v×B)+ημ0∇2B,{\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}=\nabla \times (\mathbf {v} \times \mathbf {B} )+{\frac {\eta }{\mu _{0}}}\nabla ^{2}\mathbf {B} ,}whereη/μ0{\displaystyle \eta /\mu _{0}} is themagnetic diffusivity.

In the equation of motion, theLorentz force termJ×B{\displaystyle \mathbf {J} \times \mathbf {B} } can be expanded using Ampère's law and avector calculus identity to giveJ×B=(B⋅∇)Bμ0−∇(B22μ0),{\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),}where the first term on the right hand side is themagnetic tension force and the second term is themagnetic pressure force.[5]

Ideal MHD

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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 infiniteeddy currents. Thus the matter of the liquid is "fastened" to the lines of force...

Hannes Alfvén, 1943[6]

The simplest form of MHD,ideal MHD, assumes that the resistive termηJ{\displaystyle \eta \mathbf {J} } inOhm's law is small relative to the other terms such that it can be taken to be equal to zero. This occurs in the limit of largemagnetic Reynolds numbers during whichmagnetic induction dominates overmagnetic diffusion at the velocity andlength scales under consideration.[5] Consequently, processes in ideal MHD that convert magnetic energy into kinetic energy, referred to asideal processes, cannot generateheat and raiseentropy.[7]: 6 

A fundamental concept underlying ideal MHD is thefrozen-in flux theorem 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 field line even as the points areadvected by fluid flows in the system.[8][7]: 25  The connection between the fluid and magnetic field fixes thetopology of the magnetic field in the fluid—for 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. The energy can then become available if the conditions for ideal MHD break down, allowingmagnetic reconnection that releases the stored energy from the magnetic field.

Ideal MHD equations

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In ideal MHD, the resistive termηJ{\displaystyle \eta \mathbf {J} } vanishes in Ohm's law giving the ideal Ohm's law,[5]E+v×B=0.{\displaystyle \mathbf {E} +\mathbf {v} \times \mathbf {B} =0.}Similarly, the magnetic diffusion termη∇2B/μ0{\displaystyle \eta \nabla ^{2}\mathbf {B} /\mu _{0}} in the induction equation vanishes giving the ideal induction equation,[7]: 23 ∂B∂t=∇×(v×B).{\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}=\nabla \times (\mathbf {v} \times \mathbf {B} ).}

Applicability of ideal MHD to plasmas

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Ideal MHD is only strictly applicable when:

  1. 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 toMaxwellian.
  2. 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.
  3. Interest in length scales much longer than the ionskin depth andLarmor radius perpendicular to the field, long enough along the field to ignoreLandau damping, and time scales much longer than the ion gyration time (system is smooth and slowly evolving).

Importance of resistivity

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In an imperfectly conducting fluid the magnetic field can generally move through the fluid following adiffusion law with the resistivity of the plasma serving as adiffusion constant. 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 asolar active region (from collisional resistivity) to be hundreds to thousands of years, much longer than the actual lifetime of a sunspot—so it would seem reasonable to ignore the resistivity. By contrast, a meter-sized volume of seawater has a magnetic diffusion time measured in milliseconds.

Even in physical systems[9]—which are large and conductive enough that simple estimates of theLundquist number suggest that the resistivity can be ignored—resistivity may still be important: manyinstabilities exist that can increase the effective resistivity of the plasma by factors of more than 109. The enhanced resistivity is usually the result of the formation of small scale structure like current sheets or fine scale magneticturbulence, 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,particle acceleration, and heat.

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.

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 at least somewhat resistive because their computational grid introduces anumerical resistivity.

Structures in MHD systems

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Further information:Magnetosphere particle motion
Schematic view of the different current systems which shape the Earth's magnetosphere

In many MHD systems most of the electric current is compressed into thin nearly-two-dimensional ribbons termedcurrent sheets.[10] These can divide the fluid into magnetic domains, inside of which the currents are relatively weak. Current sheets inthe 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).[11] Another example is in the Earth'smagnetosphere, where current sheets separate topologically distinct domains, isolating most of the Earth'sionosphere from thesolar wind.

Waves

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See also:Waves in plasmas

The wave modes derived using the MHD equations are calledmagnetohydrodynamic waves orMHD waves. 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:

  • Alfvén waves
  • Slow magnetosonic waves
  • Fast magnetosonic waves
Phase velocity plotted with respect toθ
'"`UNIQ--postMath-0000001F-QINU`"'
vA >vs
'"`UNIQ--postMath-00000020-QINU`"'
vA <vs

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 vectork and the magnetic fieldB. An MHD wave propagating at an arbitrary angleθ with respect to the time independent or bulk fieldB0 will satisfy the dispersion relationωk=vAcos⁡θ{\displaystyle {\frac {\omega }{k}}=v_{\text{A}}\cos \theta }wherevA=B0μ0ρ{\displaystyle v_{\text{A}}={\frac {B_{0}}{\sqrt {\mu _{0}\rho }}}}is the Alfvén speed. This branch corresponds to the shear Alfvén mode. Additionally the dispersion equation givesωk=[12(vA2+vs2±(vA2+vs2)2−4vs2vA2cos2⁡θ)]1/2{\displaystyle {\frac {\omega }{k}}=\left[{\frac {1}{2}}\left(v_{\text{A}}^{2}+v_{\text{s}}^{2}\pm {\sqrt {\left(v_{\text{A}}^{2}+v_{\text{s}}^{2}\right)^{2}-4v_{\text{s}}^{2}v_{\text{A}}^{2}\cos ^{2}\theta }}\right)\right]^{1/2}}wherevs=γpρ{\displaystyle v_{\text{s}}={\sqrt {\frac {\gamma p}{\rho }}}}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:

ModeTypeLimiting phase speedsGroup
velocity		Direction of
energy flow
k∥B{\displaystyle \mathbf {k} \parallel \mathbf {B} }k⊥B{\displaystyle \mathbf {k} \perp \mathbf {B} }
Alfvén wavetransversal; incompressiblevA{\displaystyle v_{\text{A}}}0{\displaystyle 0}Bμ0ρ{\displaystyle {\frac {\mathbf {B} }{\sqrt {\mu _{0}\rho }}}}Q∥B{\displaystyle \mathbf {Q} \parallel \mathbf {B} }
Fast magnetosonic waveneither transversal nor
longitudinal; compressional		max(vA,vs){\displaystyle \max(v_{\text{A}},v_{\text{s}})}vA2+vs2{\displaystyle {\sqrt {v_{\text{A}}^{2}+v_{\text{s}}^{2}}}}equal to
phase velocity		approx.Q∥k{\displaystyle \mathbf {Q} \parallel \mathbf {k} }
Slow magnetosonic wavemin(vA,vs){\displaystyle \min(v_{\text{A}},v_{\text{s}})}0{\displaystyle 0}approx.Q∥B{\displaystyle \mathbf {Q} \parallel \mathbf {B} }

The MHD oscillations will be damped if the fluid is not perfectly conducting but has a finite conductivity, or if viscous effects are present.

MHD waves and oscillations are a popular tool for the remote diagnostics of laboratory and astrophysical plasmas, for example, thecorona of the Sun (Coronal seismology).

Extensions

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Resistive
Resistive MHD describes magnetized fluids with finite electron diffusivity (η ≠ 0). 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 toshocks; this process has been shown to be important in the Earth-Solar magnetic interactions.
Extended
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 inertia.
Two-fluid
Two-fluid MHD describes plasmas that include a non-negligible Hallelectric field. 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.
Hall
In 1960, M. J. Lighthill criticized the applicability of ideal or resistive MHD theory for plasmas.[12] It concerned the neglect of the "Hall current 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 formE+v×B−1nee(J×B)⏟Hall current term=ηJ,{\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} ,} wherene{\displaystyle n_{e}} is the electron number density ande{\displaystyle e} is theelementary charge. 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.[13]
Electron MHD
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 and the Hall MHD. EMHD is especially important forz-pinch,magnetic reconnection,ion thrusters,neutron stars, and plasma switches.
Collisionless
MHD is also often used for collisionless plasmas. In that case the MHD equations are derived from theVlasov equation.[14]
Reduced
By using amultiscale analysis 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.[15]

Limitations

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Importance of kinetic effects

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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 distributions areMaxwellian. 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.

Effects which are essentially kinetic and not captured by fluid models includedouble layers,Landau damping, 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.

Applications

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Geophysics

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Beneath the Earth's mantle lies the core, which is made up of two parts: the solid inner core and liquid outer core.[16][17] Both have significant quantities ofiron. The liquid outer core moves in the presence of the magnetic field and eddies are set up into the same due to theCoriolis effect.[18] These eddies develop a magnetic field which boosts Earth's original magnetic field—a process which is self-sustaining and is called the geomagnetic dynamo.[19]

Reversals ofEarth's magnetic field

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 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 altogether—it just gets more complex.[20]

Earthquakes

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Some monitoring stations have reported thatearthquakes are sometimes preceded by a spike inultra low frequency (ULF) activity. A remarkable example of this occurred before the1989 Loma Prieta earthquake inCalifornia,[21] although a subsequent study indicates that this was little more than a sensor malfunction.[22] On December 9, 2010, geoscientists announced that theDEMETER satellite observed a dramatic increase in ULF radio waves overHaiti in the month before the magnitude 7.0 Mw2010 earthquake.[23] 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.

Space physics

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The study of space plasmas nearEarth and throughout theSolar System is known asspace physics. Areas researched within space physics encompass a large number of topics, ranging from theionosphere toauroras, Earth'smagnetosphere, theSolar wind, andcoronal mass ejections.

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'smagnetopause[24] (the boundary between the Earth's magnetic field and the solar wind), the formation of thering current,auroral electrojets,[25] andgeomagnetically induced currents.[26]

One prominent use of global MHD models is inspace weather forecasting.Intense solar storms have the potential to cause extensive damage to satellites[27] and infrastructure, thus it is crucial that such events are detected early. TheSpace Weather Prediction Center (SWPC) runs MHD models to predict the arrival and impacts of space weather events at Earth.

Astrophysics

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MHD applies toastrophysics, including stars, theinterplanetary medium (space between the planets), and possibly within theinterstellar medium (space between the stars) andjets.[28] Most astrophysical systems are not in local thermal equilibrium, and therefore require an additional kinematic treatment to describe all the phenomena within the system (seeAstrophysical plasma).[29][30]

Sunspots are caused by the Sun's magnetic fields, asJoseph Larmor theorized in 1919. Thesolar wind, predicted byEugene Parker, is also described by MHD. The differentialsolar rotation may be the long-term effect of magnetic drag at the poles of the Sun, an MHD phenomenon due to theParker spiral shape assumed by the extended magnetic field of the Sun.

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 theangular momentum in theSolar System. In aclosed system such as the cloud of gas and dust from which the Sun was formed, mass and angular momentum are bothconserved. 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 theproto-Sun apart before it could have formed. However, magnetohydrodynamic effects transfer the Sun's angular momentum into the outer solar system, slowing its rotation.

Breakdown of ideal MHD (in the form of magnetic reconnection) is known to be the likely cause ofsolar flares. The magnetic field in a solaractive region over a sunspot can store energy that is released suddenly as a burst of motion,X-rays, andradiation when the main current sheet collapses, reconnecting the field.[31][32]

Magnetic confinement fusion

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MHD describes a wide range of physical phenomena occurring in fusion plasmas in devices such astokamaks orstellarators.

TheGrad-Shafranov equation 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 coils.

MHD stability 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 (Troyon limit) and plasma current (set by theq>2{\displaystyle q>2} requirement of thesafety factor).

In a tokamak, instabilities also emerge from resistive MHD. For instance,tearing modes are instabilities arising within the framework of non-ideal MHD.[33] This is an active field of research, since these instabilities are the starting point for disruptions.[34]

Sensors

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Magnetohydrodynamic sensors are used for precision measurements ofangular velocities ininertial navigation systems such as inaerospace engineering. Accuracy improves with the size of the sensor. The sensor is capable of surviving in harsh environments.[35]

Engineering

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MHD is related to engineering problems such asplasma confinement, liquid-metal cooling ofnuclear reactors, andelectromagnetic casting (among others).

Amagnetohydrodynamic drive 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 the opposite direction. Although some working prototypes exist, MHD drives remain impractical.

The first prototype of this kind of propulsion was built and tested in 1965 by Steward Way, a professor of mechanical engineering at theUniversity of California, Santa Barbara. Way, on leave from his job atWestinghouse Electric, assigned his senior-year undergraduate students to develop a submarine with this new propulsion system.[36] In the early 1990s, a foundation in Japan (Ship & Ocean Foundation (Minato-ku, Tokyo)) built an experimental boat, theYamato-1, which used a magnetohydrodynamic drive incorporating asuperconductor cooled byliquid helium, and could travel at 15 km/h.[37]

MHD power generation 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.[38] One major engineering problem was the failure of the wall of the primary-coal combustion chamber due to abrasion.

Inmicrofluidics, MHD is studied as a fluid pump for producing a continuous, nonpulsating flow in a complex microchannel design.[39]

MHD can be implemented in thecontinuous casting process of metals to suppress instabilities and control the flow.[40][41]

Industrial MHD problems can be modeled using the open-source software EOF-Library.[42] Two simulation examples are 3D MHD with a free surface forelectromagnetic levitation melting,[43] and liquid metal stirring by rotating permanent magnets.[44]

Magnetic drug targeting

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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 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.[45]

See also

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Further reading

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References

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  1. ^Philip Hopkins (July 2004)."O-T Vortex Test".www.astro.princeton.edu. Princeton University Department of Astrophysical Sciences.
  2. ^Alfvén, H (1942). "Existence of Electromagnetic-Hydrodynamic Waves".Nature.150 (3805):405–406.Bibcode:1942Natur.150..405A.doi:10.1038/150405d0.S2CID 4072220.
  3. ^Fälthammar, Carl-Gunne (October 2007). "The discovery of magnetohydrodynamic waves".Journal of Atmospheric and Solar-Terrestrial Physics.69 (14):1604–1608.Bibcode:2007JASTP..69.1604F.doi:10.1016/j.jastp.2006.08.021.
  4. ^Alfvén, Hannes (1943)."On the Existence of Electromagnetic-Hydrodynamic Waves"(PDF).Arkiv för matematik, astronomi och fysik. 29B(2):1–7.
  5. ^abcBellan, Paul Murray (2006).Fundamentals of plasma physics. Cambridge: Cambridge University Press.ISBN 0-521-52800-3.
  6. ^Alfvén, Hannes (1943). "On the Existence of Electromagnetic-Hydrodynamic Waves".Arkiv för matematik, astronomi och fysik. 29B(2):1–7.
  7. ^abcPriest, Eric; Forbes, Terry (2000).Magnetic Reconnection: MHD Theory and Applications (First ed.). Cambridge University Press.ISBN 0-521-48179-1.
  8. ^Rosenbluth, M. (April 1956)."Stability of the Pinch".OSTI 4329910.
  9. ^Wesson, J.A. (1978)."Hydromagnetic stability of tokamaks".Nuclear Fusion.18:87–132.doi:10.1088/0029-5515/18/1/010.S2CID 122227433.
  10. ^Pontin, David I.; Priest, Eric R. (2022)."Magnetic reconnection: MHD theory and modelling".Living Reviews in Solar Physics.19 (1): 1.Bibcode:2022LRSP...19....1P.doi:10.1007/s41116-022-00032-9.S2CID 248673571.
  11. ^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)."Current Sheets, Plasmoids and Flux Ropes in the Heliosphere".Space Science Reviews.217 (3) 38.doi:10.1007/s11214-021-00814-x.S2CID 231592434.
  12. ^M. J. Lighthill, "Studies on MHD waves and other anisotropic wave motion,"Phil. Trans. Roy. Soc., London, vol. 252A, pp. 397–430, 1960.
  13. ^Witalis, E.A. (1986). "Hall Magnetohydrodynamics and Its Applications to Laboratory and Cosmic Plasma".IEEE Transactions on Plasma Science. PS-14 (6):842–848.Bibcode:1986ITPS...14..842W.doi:10.1109/TPS.1986.4316632.S2CID 31433317.
  14. ^W. Baumjohann and R. A. Treumann,Basic Space Plasma Physics, Imperial College Press, 1997
  15. ^Kruger, S.E.; Hegna, C.C.; Callen, J.D."Reduced MHD equations for low aspect ratio plasmas"(PDF). University of Wisconsin. Archived fromthe original(PDF) on 25 September 2015. Retrieved27 April 2015.
  16. ^"Why Earth's Inner and Outer Cores Rotate in Opposite Directions".Live Science. 19 September 2013.
  17. ^"Earth's contrasting inner core rotation and magnetic field rotation linked". 7 October 2013.
  18. ^"Geodynamo".
  19. ^NOVA | Magnetic Storm | What Drives Earth's Magnetic Field? | PBS
  20. ^"Earth's Inconstant Magnetic Field – NASA Science". Archived fromthe original on 2022-11-01. Retrieved2017-07-12.
  21. ^Fraser-Smith, Antony C.; Bernardi, A.; McGill, P. R.; Ladd, M. E.; Helliwell, R. A.; Villard Jr., O. G. (August 1990)."Low-Frequency Magnetic Field Measurements Near the Epicenter of the Ms 7.1 Loma Prieta Earthquake"(PDF).Geophysical Research Letters.17 (9):1465–1468.Bibcode:1990GeoRL..17.1465F.doi:10.1029/GL017i009p01465.ISSN 0094-8276.OCLC 1795290.Archived(PDF) from the original on 2022-10-09. RetrievedDecember 18, 2010.
  22. ^Thomas, J. N.; Love, J. J.; Johnston, M. J. S. (April 2009). "On the reported magnetic precursor of the 1989 Loma Prieta earthquake".Physics of the Earth and Planetary Interiors.173 (3–4):207–215.Bibcode:2009PEPI..173..207T.doi:10.1016/j.pepi.2008.11.014.
  23. ^KentuckyFC (December 9, 2010)."Spacecraft Saw ULF Radio Emissions over Haiti before January Quake".Physics arXiv Blog.Cambridge, Massachusetts:TechnologyReview.com. RetrievedDecember 18, 2010.Athanasiou, M; Anagnostopoulos, G; Iliopoulos, A; Pavlos, G; David, K (2010)."Enhanced ULF radiation observed by DEMETER two months around the strong 2010 Haiti earthquake".Natural Hazards and Earth System Sciences.11 (4): 1091.arXiv:1012.1533.Bibcode:2011NHESS..11.1091A.doi:10.5194/nhess-11-1091-2011.S2CID 53456663.
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