# Magnetohydrodynamics

The Mitsubishi experimental boat Yamato 1 with magnetohydrodynamic drive is now on display in front of the Maritime Museum in Kobe, Japan. Credit: Geofrog.
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“The word magnetohydrodynamics (MHD) is derived from magneto- meaning magnetic field, and hydro- meaning liquid, and -dynamics meaning movement.”[1]

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# Notation

Notation: let the symbol Def. indicate that a definition is following.

Notation: let the symbols between [ and ] be replacement for that portion of a quoted text.

# Universals

To help with definitions, their meanings and intents, there is the learning resource theory of definition.

Def. evidence that demonstrates that a concept is possible is called proof of concept.

The proof-of-concept structure consists of

1. background,
2. procedures,
3. findings, and
4. interpretation.[2]

The findings demonstrate a statistically systematic change from the status quo or the control group.

Def. "the study of the interaction of electrically conducting fluids with magnetic fields"[3] is called magnetohydrodynamics (MHD).

Def. "the fraction of a given part [or the whole] of [a stellar surface] covered by magnetic field"[4] is called the magnetic filling factor.

# Incompressible equations

"The incompressible MHD equations are

$\begin{array}{lcl} \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} & = & -\nabla p + \mathbf{B} \cdot \nabla \mathbf{B} + \nu \nabla^2 \mathbf{u} \\ \frac{\partial \mathbf{B}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{B} & = & \mathbf{B} \cdot \nabla \mathbf{u} + \eta \nabla^2 \mathbf{B} \\ \nabla \cdot \mathbf{u} & = & 0 \\ \nabla \cdot \mathbf{B} & = & 0. \end{array}$

where u, B, p represent the velocity, magnetic, and total pressure (thermal+magnetic) fields, $\nu$ and $\eta$ represent kinematic viscosity and magnetic diffusivity. The third equation is the incompressibility condition. In the above equation, the magnetic field is in Alfvén units (same as velocity units)."[5]

"The total magnetic field can be split into two parts: $\mathbf{B} = \mathbf{B_0} + \mathbf{b}$ (mean + fluctuations)."[5]

"The above equations in terms of Elsässer variables ($\mathbf{z}^{\pm} = \mathbf{u} \pm \mathbf{b}$) are"[5]

$\frac{\partial {\mathbf{z}^{\pm}}}{\partial t}\mp\left(\mathbf {B}_0\cdot{\mathbf \nabla}\right){\mathbf z^{\pm}} + \left({\mathbf z^{\mp}}\cdot{\mathbf \nabla}\right){\mathbf z^{\pm}} = -{\mathbf \nabla}p + \nu_+ \nabla^2 \mathbf{z}^{\pm} + \nu_- \nabla^2 \mathbf{z}^{\mp}$

where $\nu_\pm = \nu \pm \eta$. Nonlinear interactions occur between the Alfvénic fluctuations $z^{\mp}$.

"The important nondimensional parameters for MHD are"[5]

$\begin{array}{lcl} \text{Reynold number } Re & = & U L /\nu \\ \text{Magnetic Reynold number } Re_M & = & U L /\eta \\ \end{array}$

"The Magnetic Reynolds number is a dimensionless group that gives an estimate of the effects of magnetic advection to magnetic diffusion"[6].

$\begin{array}{lcl} \text{Magnetic Prandtl number } P_M & = & \nu / \eta. \end{array}$

"The magnetic Prandtl number is an important property of the fluid. Liquid metals have small magnetic Prandtl numbers, for example, liquid sodium's $P_M$ is around $10^{-5}$. But plasmas have large $P_M$."[5]

"The Reynolds number is the ratio of the nonlinear term $\mathbf{u} \cdot \nabla \mathbf{u}$ of the Navier-Stokes equation to the viscous term. While the magnetic Reynolds number is the ratio of the nonlinear term and the diffusive term of the induction equation."[5]

"In many practical situations, the Reynolds number $Re$ of the flow is quite large. For such flows typically the velocity and the magnetic fields are random. Such flows are called to exhibit MHD turbulence. Note that $Re_M$ need not be large for MHD turbulence. $Re_M$ plays an important role in dynamo (magnetic field generation) problem."[5]

"The mean magnetic field plays an important role in MHD turbulence, for example it can make the turbulence anisotropic; suppress the turbulence by decreasing energy cascade etc. The earlier MHD turbulence models assumed isotropy of turbulence, while the later models have studied anisotropic aspects. In the following discussions will summarize these models. More discussions on MHD turbulence can be found in Biskamp[7] and Verma[8]."[5]

# Isotropic turbulence

"Iroshnikov[9] and Kraichnan[10] formulated the first phenomenological theory of MHD turbulence. They argued that in the presence of a strong mean magnetic field, $z^+$ and $z^-$ wavepackets travel in opposite directions with the phase velocity of $B_0$, and interact weakly. The relevant time scale is Alfven time $(B_0 k)^{-1}$. As a results the energy spectra is

$E^u(k) \approx E^b(k) \approx A (\Pi V_A)^{1/2} k^{-3/2}.$

where $\Pi$ is the energy cascade rate."[5]

"Later Dobrowolny et al. [11]derived the following generalized formulas for the cascade rates of $z^{\pm}$ variables:

$\Pi^+ \approx \Pi^{-} \approx \tau^{\pm}_k E^{+}(k) E^{-}(k) k^4 \approx E^{+}(k) E^{-}(k) k^3 / B_0$

where $\tau^{\pm}$ are the interaction time scales of $z^{\pm}$ variables."[5]

"Iroshnikov and Kraichnan's phenomenology follows once we choose $\tau^{\pm} \approx 1/(k V_A)$."[5]

"Marsch[12] chose the nonlinear time scale $T_{NL}^{\pm} \approx (k z_k^{\mp})^{-1}$ as the interaction time scale for the eddies and derived Kolmogorov-like energy spectrum for the Elsasser variables:

$E^{\pm}(k) = K^{\pm} (\Pi^{\pm})^{4/3} (\Pi^{\mp})^{-2/3} k^{-5/3}$

where $\Pi^+$ and $\Pi^-$ are the energy cascade rates of $z^+$ and $z^-$ respectively, and $K^{\pm}$ are constants."[5]

"Matthaeus and Zhou[13] attempted to combine the above two time scales by postulating the interaction time to be the harmonic mean of Alfven time and nonlinear time.[5]

"The main difference between the two competing phenomenologies (-3/2 and -5/3) is the chosen time scales for the interaction time. The main underlying assumption in that Iroshnikov and Kraichnan's phenomenology should work for strong mean magnetic field, whereas Marsh's phenomenology should work when the fluctuations dominate the mean magnetic field (strong turbulence)."[5]

# Anisotropic turbulence

"Mean magnetic field makes turbulence anisotropic. This aspect has been studied in last two decades. In the limit $\delta z^{\pm} \ll B_0$, Galtier et al.[14] showed using kinetic equations that

$E(k) \sim (\Pi B_0)^{1/2} k_{||}^{1/2} k_{\perp}^{-2}$

where $k_{||}$ and $k_{\perp}$ are components of the wavenumber parallel and perpendicular to mean magnetic field. The above limit is called the weak turbulence limit."[5]

"Under the strong turbulence limit, $\delta z^{\pm} \sim B_0$, Goldereich and Sridhar[15] argue that $k_{\perp} z_{k_{\perp}} \sim k_{||} B_0$ ("critical balanced state") which implies that"[5]

$\begin{array}{lcl} E(k) & \propto & k_{\perp}^{-5/3}; \\ k_{||} & \propto & k_{\perp}^{2/3} \end{array}$

"The above anisotropic turbulence phenomenology has been extended for large cross helicity MHD."[5]

# Stuart number

"The Stuart number (N, also known as magnetic interaction parameter) is a dimensionless number of fluids, i.e. gases or liquids."[16]

"It is defined as the ratio of electromagnetic to inertial forces, which gives an estimate of the relative importance of a magnetic field on a flow. The Stuart number is relevant for flows of conducting fluids, e.g. in fusion reactors, steel casters or plasmas.[17]"[16]

${\it N} = \frac{Ha^2}{Re} = \frac {B^2 \cdot L_{c} \cdot \sigma}{\rho U}$
• $Ha$ – Hartmann number
• $Re$ – Reynolds number
• $B$magnetic flux density
• $L_c$characteristic length
• $\sigma$electric conductivity
• $\mu$ – dynamic viscosity, sometimes denoted as $\eta$

# Electrohydrodynamics

Electrohydrodynamics (EHD), also known as electro-fluid-dynamics (EFD) or electrokinetics, is the study of the dynamics of electrically charged fluids.[18] It is the study of the motions of ionised particles or molecules and their interactions with electric fields and the surrounding fluid. The term may be considered to be synonymous with the rather elaborate electrostrictive hydrodynamics. EHD covers the following types of particle and fluid transport mechanisms: Electrophoresis, electrokinesis, dielectrophoresis, electro-osmosis, and electrorotation. In general, the phenomena relate to the direct conversion of electrical energy into kinetic energy, and vice versa."[19]

"In the first instance, shaped electrostatic fields create hydrostatic pressure (or motion) in dielectric media. When such media are fluids, a flow is produced. If the dielectric is a vacuum or a solid, no flow is produced. Such flow can be directed against the electrodes, generally to move the electrodes. In such case, the moving structure acts as an electric motor.”[19]

# Hartmann number

Hartmann number is the ratio of electromagnetic force to the viscous force first introduced by Hartmann.[20] It is defined by:

$H_a = BL\sqrt{\frac{\sigma} {\mu}}$

where

• $B$ is the magnetic field
• $L$ is the characteristic length scale
• $\sigma$ is the electrical conductivity
• $\mu$ is the viscosity[21]

# Computational magnetohydrodynamics

"Computational magnetohydrodynamics (CMHD) is a rapidly developing branch of magnetohydrodynamics that uses numerical methods and algorithms to solve and analyze problems that involve electrically conducting fluids. Most of the methods used in CMHD are borrowed from the well established techniques employed in Computational fluid dynamics. The complexity mainly arises due to the presence of a magnetic field and its coupling with the fluid. One of the important issues is to numerically maintain the $\nabla \cdot {\mathbf B} = 0$ (conservation of magnetic flux) condition, from Maxwell's equations, to avoid any unphysical effects."[22]

"Numerical resistivity is a problem in computer simulations of ideal magnetohydrodynamics (MHD). It is a form of numerical diffusion. In near-ideal MHD systems, the magnetic field can diffuse only very slowly through the plasma or fluid of the system; it is rate-limited by the resistivity of the fluid. In Eulerian simulations where the field is arbitrarily aligned compared to the simulation grid, the numerical diffusion rate takes the form similar to an additional resistivity, causing non-physical and sometimes bursty magnetic reconnection in the simulation. Numerical resistivity is a function of resolution, alignment of the magnetic field with the grid, and numerical method. In general, numerical resistivity will not behave isotropically, and there can be different effective numerical resistivities in different parts of the computational domain. For current (2005) simulations of the solar corona and inner heliosphere, this numerical effect can be several orders of magnitude larger than the physical resistivity of the plasma."[23]

# Dynamo

"[M]otions resulting from [a linear magnetohydrodynamic] instability act as a dynamo to sustain the magnetic field."[24] "Supersonic flows are initially generated by the Balbus-Hawley magnetic shear instability."[24]

"A plasma with local magnetohydrodynamic instabilities creates mechanical turbulence, motion, or shear (a dynamo) which in turn generates or sustains the local magnetic field."[25]

"Magnetic field generation by dynamo action is often studied within the theoretical framework of magnetohydrodynamics (MHD)."[26]

"For small values of the Hall parameter, the small-scale dynamo is more efficient, displaying faster growth and saturating at larger amplitudes of the magnetic field. For larger values of the Hall parameter, saturation of the magnetic field is reached at smaller amplitudes than in the MHD case."[26]

When this magnetohydrodynamic dynamo occurs between or within radiative layers, a radiative dynamo is operating.

"The kinematic approximation becomes invalid when the magnetic field becomes strong enough to affect the fluid motions. In that case the velocity field becomes affected by the Lorentz force, and so the induction equation is no longer linear in the magnetic field. In most cases this leads to a quenching of the amplitude of the dynamo. Such dynamos are sometimes also referred to as hydromagnetic dynamos. Virtually all dynamos in astrophysics and geophysics are hydromagnetic dynamos."[27]

"Numerical models are used to simulate fully nonlinear dynamos. A minimum of 5 equations are needed. They are as follows. The induction equation, see above. Maxwell's equation:"[27]

$\nabla \cdot \mathbf{B}=0$

"The (sometimes) Boussinesq conservation of mass:"[27]

$\nabla \cdot \mathbf{u} = 0$

"The (sometimes) Boussinesq conservation of momentum, also known as the Navier-Stokes equation:

$\frac{D\mathbf{u}}{Dt} = -\nabla p + \nu \nabla^2 \mathbf{u} + \rho^'\mathbf{g} + 2\mathbf{\Omega} \times \mathbf{u} + \mathbf{\Omega} \times \mathbf{\Omega} \times \mathbf{R} + \mathbf{J} \times \mathbf{B}$

where $\nu$ is the kinematic viscosity, $\rho^'$ is the density perturbation that provides buoyancy (for thermal convection $\rho^'=\alpha\Delta T$, $\Omega$ is the rotation rate of the Earth, and $\mathbf{J}$ is the electrical current density."[27]

"Finally, a transport equation, usually of heat (sometimes of light element concentration):

$\frac{\partial T}{\partial t} = \kappa \nabla^2 T +\epsilon$

where T is temperature, $\kappa=k/\rho c_p$ is the thermal diffusivity with k thermal conductivity, $c_p$ heat capacity, and $\rho$ density, and $\epsilon$ is an optional heat source."[27]

"Often the pressure is the dynamic pressure, with the hydrostatic pressure and centripetal potential removed. These equations are then non-dimensionalized, introducing the non-dimensional parameters,

$Ra=\frac{g\alpha T D^3}{\nu \kappa} , E=\frac{\nu}{\Omega D^2} , Pr=\frac{\nu}{\kappa} , Pm=\frac{\nu}{\eta}$

where Ra is the Rayleigh number, E the Ekman number, Pr and Pm the Prandtl and magnetic Prandtl number. Magnetic field scaling is often in Elsasser number units $B=\rho \Omega/\sigma$."[27]

# Antidynamo theorem

Def. “one of several results that restrict the type of magnetic fields that may be produced by dynamo action”[28] is called an antidynamo theorem.

Def. “no axisymmetric magnetic field can be maintained through a self-sustaining dynamo action by an axially symmetric current”[28][29] is called Cowling's antidynamo theorem.

“The impact of the known anti-dynamo theorems is that successful dynamos do not possess a high degree of symmetry.”[28]

# Plasma instabilities

This is a photo of the kink instability in action - the 3 by 25 cm pyrex tube at Aldermaston. Credit: Alan Sykes, UK Atomic Energy Authority.

"Plasma instabilities can be divided into two general groups:

1. hydrodynamic instabilities
2. kinetic instabilities."[30]

"Plasma instabilities are also categorised into different modes:[31]"[30]

 Mode (azimuthal wave number) Note Description Radial modes Description m=0 Sausage instability: displays harmonic variations of beam radius with distance along the beam axis n=0 Axial hollowing n=1 Standard sausaging n=2 Axial bunching m=1 Sinuous, kink or hose instability: represents transverse displacements of the beam cross-section without change in the form or in a beam characteristics other than the position of its center of mass m=2 Filamentation modes: growth leads towards the breakup of the beam into separate filaments. Gives an elliptic cross-section m=3 Gives a pyriform (pear-shaped) cross-section

"Ideal MHD instabilities driven by current or pressure gradients".[30]

"A kink instability, also oscillation or mode, is a class of magnetohydrodynamic instabilities which sometimes develop in a thin plasma column carrying a strong axial current. If a "kink" begins to develop in a column the magnetic forces on the inside of the kink become larger than those on the outside, which leads to growth of the perturbation.[32] As it develops at fixed areas in the plasma, kinks belong to the class of "absolute plasma instabilities", as opposed to convective processes."[33]

# Space plasma characteristics

"Space plasma pioneers Hannes Alfvén and Carl-Gunne Fälthammar divided the plasmas in the solar system into three different categories:"[34]

Classification of Magnetic Cosmic Plasmas

 Characteristic Space plasma density categories (Note that density does not refer to only particle density) Ideal comparison High density Medium Density Low Density Criterion λ << ρ λ << ρ << lc lc << λ lc << λD Examples Stellar interior Solar photosphere Solar chromosphere/corona Interstellar/intergalactic space Ionosphere above 70 km Magnetosphere during magnetic disturbance. Interplanetary space Single charges in a high vacuum Diffusion Isotropic Anisotropic Anisotropic and small No diffusion Conductivity Isotropic Anisotropic Not defined Not defined Electric field parallel to B in completely ionized gas Small Small Any value Any value Particle motion in plane perpendicular to B Almost straight path between collisions Circle between collisions Circle Circle Path of guiding centre parallel to B Straight path between collisions Straight path between collisions Oscillations (e.g. between mirror points) Oscillations (e.g. between mirror points) Debye Distance λD λD << lc λD << lc λD << lc λD >> lc Magnetohydrodynamics suitability Yes Approximately No No

λ=Mean free path. ρ= Larmor radius (gyroradius) of electron. λD=Debye length. lc=Characteristic length
Adapted From Cosmical Electrodynamics (2nd Ed. 1952) Alfvén and Fälthammar

# Magnetorotational instability

"Supersonic flows are initially generated by the Balbus-Hawley magnetic shear instability."[24]

“The magnetorotational instability or MRI is a fluid instability that arises when the angular velocity of a magnetized fluid decreases as the distance from the rotation center increases. It can also be known as the Velikhov-Chandrasekhar instability or Balbus-Hawley instability in the literature; not to be confused with the Velikhov instability which is the electrothermal instability. The MRI is of particular relevance in astrophysics where it is an important part of the dynamics in accretion discs. Balbus and Hawley were the first to realize the astrophysical importance of this instability and explain its physical mechanism.”[35]

“A rotating hydrodynamic fluid disc (e.g. a nonmagnetic accretion disc) will remain in a laminar flow state as long as the angular momentum (per unit mass) increases outwards. This is also known as the Rayleigh stability criterion:

$\frac{\partial(r^2\Omega)}{\partial r}>0$

where $\Omega$ is the angular velocity of a fluid element and $r$ is its distance to the rotation center. Instabilities leading ultimately to turbulence occur if a magnetic field is present and the angular velocity decreases with radius:

$\frac{\partial\Omega}{\partial r}<0$[35].

“The MRI was first noticed in a non-astrophysical context by Evgeny Velikhov in 1959 when considering the stability of Couette flow of an ideal hydromagnetic fluid.[36] His result was later generalized by S. Chandrasekhar in 1960.[37] This mechanism was proposed by Acheson & Hide (1973) to perhaps play a role in the context of the Earth's geodynamo problem.[38] The applicability of this mechanism to the problem of accretion disks was appreciated when S. A. Balbus and J. F. Hawley established that weak magnetic fields can substantially alter the stability character of accretion disks.[39][35]

“The observed accretion rates in astrophysical objects cannot be explained by a molecular viscosity, as the outward angular momentum transport in that case would not be enough to account for the inward flow of mass. The MRI provides a mechanism to account for the additional outward angular momentum transport. It is intrinsically a magnetohydrodynamic (MHD) phenomenon, with no hydrodynamic analog. Dynamo action is usually invoked to justify the existence of the magnetic field required for the instability to set in.[40][35]

# Resistive skin time

“The resistive skin time is a characteristic time of typical magnetohydrodynamic (MHD) phenomena.

The resistive skin time is defined as: $\tau_R=\frac{\mu_0 a^2}{\eta}$ where $\eta$ is the resistivity, $a$ is a typical radius of the device and $\mu_0$ is the magnetic permeability.”[41]

# Waves

"There are several distinct kinds of MHD modes which have quite different dispersive, polarisation, and propagation properties:

• Kink (or transverse) modes, which are oblique fast magnetoacoustic (also known as magnetosonic waves) guided by the plasma structure; the mode causes the displacement of the axis of the plasma structure. These modes are weakly compressible, but could nevertheless be observed with imaging instruments as periodic standing or propagating displacements of coronal structures, e.g. coronal loops. The frequency of transverse or "kink" modes is given by the following expression:
$\omega_{K}=\sqrt{\frac{2k_{z}B^{2}}{\mu(\rho_{i}+\rho_{e})}}$

For kink modes the parameter $m$ is equal to 1.

• Sausage modes, which are also oblique fast magnetoacoustic waves guided by the plasma structure; the mode causes expansions and contractions of the plasma structure, but does not displace its axis. These modes are compressible and cause significant variation of the absolute value of the magnetic field in the oscillating structure. The frequency of sausage modes is given by the following expression:
$\omega_{S}=\sqrt{\frac{k_{z}^{2}B^{2}}{\mu\rho_{e}}}$

For sausage modes the parameter $m$ is equal to 0.

• Longitudinal (or slow, or acoustic) modes, which are slow magnetoacoustic waves propagating mainly along the magnetic field in the plasma structure; these mode are essentially compressible. The magnetic field perturbation in these modes is negligible. The frequency of slow modes is given by the following expression:
$\omega_{L}=\sqrt{k^{2}_{z}\left ( \frac{C_{s}^{2}C_{A}^{2}}{C_{s}^{2}+C_{A}^{2}} \right )}$

Where we define $C_{s}$ as the sound speed and $C_{A}$ as the Alfvèn speed.

• Torsional (Alfvén or twist) modes are incompressible transverse perturbations of the magnetic field along certain individual magnetic surfaces. In contrast with kink modes, torsional modes cannot be observed with imaging instruments, as they do not cause the displacement of either the structure axis or its boundary.
$\omega_{A}=\sqrt{\frac{k_{z}^{2}B^{2}}{\mu\rho_{i}}}$"[42]

"Waves in plasmas are an interconnected set of particles and fields which propagates in a periodically repeating fashion. A plasma is a quasineutral, electrically conductive fluid. In the simplest case, it is composed of electrons and a single species of positive ions, but it may also contain multiple ion species including negative ions as well as neutral particles. Due to its electrical conductivity, a plasma couples to electric and magnetic fields. This complex of particles and fields supports a wide variety of waves."[43]

"Waves in plasmas can be classified as electromagnetic or electrostatic according to whether or not there is an oscillating magnetic field. Applying Faraday's law of induction to plane waves, we find $\mathbf{k}\times\tilde{\mathbf{E}}=\omega\tilde{\mathbf{B}}$, implying that an electrostatic wave must be purely longitudinal. An electromagnetic wave, in contrast, must have a transverse component, but may also be partially longitudinal."[43]

"Waves can be further classified by the oscillating species. In most plasmas of interest, the electron temperature is comparable to or larger than the ion temperature. This fact, coupled with the much smaller mass of the electron, implies that the electrons are much faster than the ions. An electron mode depends on the mass of the electrons, but the ions may be assumed to be infinitely massive, i.e. stationary. An ion mode depends on the ion mass, but the electrons are assumed to be massless and to redistribute themselves instantaneously according to the Boltzmann relation. Only rarely, e.g. in the lower hybrid oscillation, will a mode depend on both the electron and the ion mass."[43]

"The various modes can also be classified according to whether they propagate in an unmagnetized plasma or parallel, perpendicular, or oblique to the stationary magnetic field. Finally, for perpendicular electromagnetic electron waves, the perturbed electric field can be parallel or perpendicular to the stationary magnetic field."[43]

Summary of elementary plasma waves[43]
EM character oscillating species conditions dispersion relation name
electrostatic electrons $\vec B_0=0\ {\rm or}\ \vec k\|\vec B_0$ $\omega^2=\omega_p^2+3k^2v_{th}^2$ plasma oscillation (or Langmuir wave)
$\vec k\perp\vec B_0$ $\omega^2=\omega_p^2+\omega_c^2=\omega_h^2$ upper hybrid oscillation
ions $\vec B_0=0\ {\rm or}\ \vec k\|\vec B_0$ $\omega^2=k^2v_s^2=k^2\frac{\gamma_eKT_e+\gamma_iKT_i}{M}$ ion acoustic wave
$\vec k\perp\vec B_0$ (nearly) $\omega^2=\Omega_c^2+k^2v_s^2$ electrostatic ion cyclotron wave
$\vec k\perp\vec B_0$ (exactly) $\omega^2=[(\Omega_c\omega_c)^{-1}+\omega_i^{-2}]^{-1}$ lower hybrid oscillation
electromagnetic electrons $\vec B_0=0$ $\omega^2=\omega_p^2+k^2c^2$ light wave
$\vec k\perp\vec B_0,\ \vec E_1\|\vec B_0$ $\frac{c^2k^2}{\omega^2}=1-\frac{\omega_p^2}{\omega^2}$ O wave
$\vec k\perp\vec B_0,\ \vec E_1\perp\vec B_0$ $\frac{c^2k^2}{\omega^2}=1-\frac{\omega_p^2}{\omega^2}\, \frac{\omega^2-\omega_p^2}{\omega^2-\omega_h^2}$ X wave
$\vec k\|\vec B_0$ (right circ. pol.) $\frac{c^2k^2}{\omega^2}=1-\frac{\omega_p^2/\omega^2}{1-(\omega_c/\omega)}$ R wave (whistler mode)
$\vec k\|\vec B_0$ (left circ. pol.) $\frac{c^2k^2}{\omega^2}=1-\frac{\omega_p^2/\omega^2}{1+(\omega_c/\omega)}$ L wave
ions $\vec B_0=0$   none
$\vec k\|\vec B_0$ $\omega^2=k^2v_A^2$ Alfvén wave
$\vec k\perp\vec B_0$ $\frac{\omega^2}{k^2}=c^2\, \frac{v_s^2+v_A^2}{c^2+v_A^2}$ magnetosonic wave

$\omega$ - wave frequency, $k$ - wave number, $c$ - speed of light, $\omega_p$ - plasma frequency, $\omega_i$ - ion plasma frequency, $\omega_c$ - electron gyrofrequency, $\Omega_c$ - proton gyrofrequency, $\omega_h$ - upper hybrid frequency, $v_s$ - plasma "sound" speed, $v_A$ - plasma Alfven speed

# Rankine-Hugoniot jumps

"The jump conditions across an MHD shock or discontinuity are referred as the Rankine-Hugoniot equations for MHD. In the frame moving with the shock/discontinuity, those jump conditions can be written:

$\rho_1 v_{n1} = \rho_2 v_{n2},$
$B_{n1} = B_{n2},$
$\rho_1 v_{n1}^2+ p_1 + \frac{B_{t1}^2}{2 \mu_0}=\rho_2 v_{n2}^2+ p_2 + \frac{B_{t2}^2}{2 \mu_0},$
$\rho_1 v_{n1} \mathbf{v_{t1}} - \frac{\mathbf{B_{t1}}B_{n1}}{\mu_0}= \rho_2 v_{n2} \mathbf{v_{t2}} - \frac{\mathbf{B_{t2}}B_{n2}}{\mu_0},$
$\left(\frac{\gamma}{\gamma-1}\frac{p_1}{\rho_1}+\frac{v_1^2}{2}\right)\rho_1 v_{n1}+\frac{v_{n1} B_{t1}^2}{\mu_0}-\frac{B_{n1}(\mathbf{B_{t1}}\cdot \mathbf{v_{t1}})}{\mu_0}=\left(\frac{\gamma}{\gamma-1}\frac{p_2}{\rho_2}+\frac{v_2^2}{2}\right)\rho_2 v_{n2}+\frac{v_{n2} B_{t2}^2}{\mu_0}-\frac{B_{n2}(\mathbf{B_{t2}}\cdot \mathbf{v_{t2}})}{\mu_0},$
$(\mathbf{v} \times \mathbf{B})_{t1} = (\mathbf{v} \times \mathbf{B})_{t2},$

where $\rho$, v, p, B are the plasma density, velocity, (thermal) pressure and magnetic field respectively. The subscripts t and n refer to the tangential and normal components of a vector (with respect to the shock/discontinuity front). The subscripts 1 and 2 refer to the two states of the plasma on each side of the shock/discontinuity".[44]

# Shocks

"Shocks are transition layers across which there is a transport of particles. There are three types of shocks in MHD: slow-mode, intermediate and fast-mode shocks."[44]

"Intermediate shocks are non-compressive (meaning that the plasma density does not change across the shock). A special case of the intermediate shock is referred to as a rotational discontinuity. They are isentropic. All thermodynamic quantities are continuous across the shock, but the tangential component of the magnetic field can "rotate". Intermediate shocks in general however, unlike rotational discontinuities, can have a discontinuity in the pressure."[44]

"Slow-mode and fast-mode shocks are compressive and are associated with an increase in entropy. Across slow-mode shock, the tangential component of the magnetic field decreases. Across fast-mode shock it increases."[44]

"The type of shocks depend on the relative magnitude of the upstream velocity in the frame moving with the shock with respect to some characteristic speed. Those characteristic speeds, the slow and fast magnetosonic speeds, are related to the Alfvén speed, $v_A$ and the sonic speed, $c_s$ as follows:

$a_{\mathrm{slow}}^2 = \frac{1}{2} \left[\left(c_s^2 + V_A^2\right)-\sqrt{\left(c_s^2+V_A^2\right)^2-4c_s^2V_{A}^2 \cos^{2} \theta_{Bn}}\,\right],$
$a_{\mathrm{fast}}^2 = \frac{1}{2} \left[\left(c_s^2 + V_A^2\right)+\sqrt{\left(c_s^2+V_A^2\right)^2-4c_s^2V_{A}^2 \cos^{2} \theta_{Bn}}\,\right],$

where $V_{A}$ is the Alfvén speed and $\theta_{Bn}$ is the angle between the incoming magnetic field and the shock normal vector."[44]

"The normal component of the slow shock propagates with velocity $a_{\mathrm{slow} }$ in the frame moving with the upstream plasma, that of the intermediate shock with velocity $V_{An}$ and that of the fast shock with velocity $a_{\mathrm{fast}}$. The fast mode waves have higher phase velocities than the slow mode waves because the density and magnetic field are in phase, whereas the slow mode wave components are out of phase."[44]

# Magnetic tension

"The magnetic tension force is a restoring force (SI unit: Pa·m−1) that acts to straighten bent magnetic field lines. It equals:"[45]

$\frac{\left(\mathbf{B}\cdot\nabla\right)\mathbf{B}}{\mu_0}$

"It is analogous to rubber bands and their restoring force. The force is directed antiradially. Although magnetic tension is referred to as a force, it is actually a pressure gradient (Pa m−1) which is also a force density (N m−3)."[45]

"The magnetic pressure is the energy density of the magnetic field and it increases as magnetic field lines convene with each other. In contrast, magnetic tension force is determined by how much the magnetic pressure changes with distance. Magnetic tension forces also rely on vector current densities $\mathbf{J}$ and their interaction with the magnetic field $\mathbf{B}$. Plotting magnetic tension along adjacent field lines can give a picture as to their divergence and convergence with respect to each other as well as current densities $\mathbf{J}$."[45]

"Magnetic tension is particularly important in plasma physics and magnetohydrodynamics, where it controls dynamics of some systems and the shape of magnetized structures. In magnetohydrodynamics, the magnetic tension force can be derived from the momentum equation

$\rho\left(\frac{\partial}{\partial t}+ \mathbf{V}\cdot\nabla \right)\mathbf{V} = \mathbf{J}\times\mathbf{B} - \nabla p$

using the relation $\mu_0\mathbf{J}=\nabla\times\mathbf{B}$. The first term on the right hand side of the above equation represents electromagnetic forces and the second term represents pressure gradient forces."[45]

"[A] Taylor state is the minimum energy state of a plasma satisfying the constraint of conserving magnetic helicity.[46]"[47]

"Consider a closed, simply-connected, flux-conserving, perfectly conducting surface $S$ surrounding a plasma with negligible thermal energy ($\beta \rightarrow 0$)."[47]

"Since $\vec{B}.\vec{ds}=0$ on $S$. This implies that $\vec{A}_{||}=0$."[47]

"As discussed above, the plasma would relax towards a minimum energy state while conserving its magnetic helicity. Since the boundary is perfectly conducting, there cannot be any change in the associated flux. This implies $\delta \vec{B}.\vec{ds}=0$ and $\delta\vec{A}_{||}=0$ on $S$."[47]

"We formulate a variational problem of minimizing the plasma energy $W=\int d^3rB^2/2\mu_\circ$ while conserving magnetic helicity $K=\int d^3r\vec{A}.\vec{B}$."[47]

"The variational problem is $\delta W -\lambda \delta K = 0$.

After some algebra this leads to the following constraint for the minimum energy state $\nabla \times \vec{B} = \lambda \vec{B}$."[47]

# Solar dynamo

"The solar dynamo is the physical process that generates the Sun's magnetic field. The Sun is permeated by an overall dipole magnetic field, as are many other celestial bodies such as the Earth. The dipole field is produced by a circular electric current flowing deep within the star, following Ampère's law. The current is produced by shear (stretching of material) between different parts of the Sun that rotate at different rates, and the fact that the Sun itself is a very good electrical conductor (and therefore governed by the laws of magnetohydrodynamics)."[48]

"Any electrically conducting fluid can form a dynamo simply by shear within the fluid itself, because of a consequence of Lenz's law of induction: moving the fluid through a pre-existing magnetic field will induce electrical currents in the fluid that distort the pre-existing magnetic field. The direction of the distortion is such that the existing field lines tend to be dragged along with the fluid, like threads of dye embedded in taffy or syrup. If the flow has a strong shear component then the individual field lines are stretched by the flow, amplifying the existing magnetic field. Such systems are called MHD dynamos."[48]

"Depending on the structure of the flow, the dynamo may be self-exciting and stable, self-exciting and chaotic, or decaying. The Sun's dynamo is self-exciting: the direction of the field reverses itself about every 11 years, causing the sunspot cycle as ropes of magnetic field lines rise to the surface of the Sun and manifest as sunspots on the surface. The solar dynamo is thought to be located at the tachocline, a region inside the Sun that exhibits a very large shear profile."[48]

# Coronal cloud

In a coronal cloud are magnetohydrodynamic plasma flux tubes along magnetic field lines.[49]

# Interplanetary magnetic field

Notation: let the symbol IMF stand for interplanetary magnetic field.

"[T]he IMF switched from northward to southward while Mariner [10] was in the Mercurian magnetosphere."[50]

# Magnetosphere

“A magnetosphere is formed when a stream of charged particles, such as the solar wind, interacts with and is deflected by the magnetic field of a planet or similar body.”[51]

“Planets which generate magnetic fields in their interiors ... are surrounded by invisible magnetospheres. ... [I]n many respects, the magnetosphere of Venus is a scaled-down version of Earth’s. ... Earth’s magnetosphere is 10 times larger [than that of Venus]”[52]

# Mercury

The graph shows the relative strength of Mercury's magnetic field. Credit: .

"Despite its small size and slow 59-day-long rotation, Mercury has a significant, and apparently global, magnetic field. According to measurements taken by Mariner 10, it is about 1.1% as strong as the Earth's. The magnetic field strength at the Mercurian equator is about 300 nT.[53][54] Like that of Earth, Mercury's magnetic field is dipolar.[55] Unlike Earth, Mercury's poles are nearly aligned with the planet's spin axis.[56] Measurements from both the Mariner 10 and MESSENGER space probes have indicated that the strength and shape of the magnetic field are stable.[56]"[57]

"During its second flyby of the planet on October 6, 2008, MESSENGER discovered that Mercury's magnetic field can be extremely "leaky." The spacecraft encountered magnetic "tornadoes" – twisted bundles of magnetic fields connecting the planetary magnetic field to interplanetary space – that were up to 800 km wide or a third of the radius of the planet. These 'tornadoes' form when magnetic fields carried by the solar wind connect to Mercury's magnetic field. As the solar wind blows past Mercury's field, these joined magnetic fields are carried with it and twist up into vortex-like structures. These twisted magnetic flux tubes, technically known as flux transfer events, form open windows in the planet's magnetic shield through which the solar wind may enter and directly impact Mercury's surface.[58]"[57]

"The process of linking interplanetary and planetary magnetic fields, called magnetic reconnection, is common throughout the cosmos. It occurs in Earth's magnetic field, where it generates magnetic tornadoes as well. The MESSENGER observations show the reconnection rate is ten times higher at Mercury. Mercury's proximity to the Sun only accounts for about a third of the reconnection rate observed by MESSENGER.[58]"[57]

There are "observed MHD fluctuations in the magnetosphere"[50] of Mercury.

# Venus

"In 1967, Venera-4 found the Venusian magnetic field is much weaker than that of Earth. This magnetic field is induced by an interaction between the ionosphere and the solar wind,[59][60] ... Venus's small induced magnetosphere provides negligible protection to the atmosphere against cosmic radiation. This radiation may result in cloud-to-cloud lightning discharges.[61]"[62]

"The weak magnetosphere around Venus means the solar wind is interacting directly with the outer atmosphere of the planet. Here, ions of hydrogen and oxygen are being created by the dissociation of neutral molecules from ultraviolet radiation. The solar wind then supplies energy that gives some of these ions sufficient velocity to escape the planet's gravity field. This erosion process results in a steady loss of low-mass hydrogen, helium, and oxygen ions, while higher-mass molecules, such as carbon dioxide, are more likely to be retained."[62]

# Earth

This is a schematic diagram of Earth's interior. The outer core is the source of the geomagnetic field. Credit: NASA.
Reversals of Earth's magnetic field are diagrammed. Credit: NASA.
This is a schematic of the Birkeland currents and their connection to Earth's ionospheric current systems. Credit: Le, G., J. A. Slavin, and R. J. Strangeway.
Auroral-like Birkeland currents were created by scientist Kristian Birkeland in his terrella, featuring a magnetised anode globe in an evacuated chamber. Credit: Kristian Birkeland.
The complex self-constricting magnetic field lines and current paths in a Birkeland current that may develop in a plasma (Figure 15.3.2, Alfvén and Arrhenius, 1976)[63] Credit: .

"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 iron. The liquid outer core moves in the presence of the magnetic field and eddies are set up into the same due to the Coriolis effect. These eddies develop a magnetic field which boosts Earth's original magnetic field - a process which is self-sustaining and is called as the geomagnetic dynamo.[64]"[1]

"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 thousands of years. During the flips, the magnetic field doesn't vanish altogether - it just gets more complicated.[65]"[1]

"The Atmospheric dynamo is a pattern of electrical currents that are set up in the Earth's ionosphere by multiple effects, mostly the Sun's solar wind, but also the tides of the Moon and Sun.[66][67] The currents flow in circuits between the poles and the equator, but they are not well understood.[68]"[69]

"Magnetite typically carries the dominant magnetic signature in rocks, and so it has been a critical tool in paleomagnetism, a science important in discovering and understanding plate tectonics and as historic data for magnetohydrodynamics and other scientific fields. The relationships between magnetite and other iron-rich oxide minerals such as ilmenite, hematite, and ulvospinel have been much studied, as the complicated reactions between these minerals and oxygen influence how and when magnetite preserves records of the Earth's magnetic field."[70]

The schematic at right describes the Birkeland or field-aligned currents and the ionospheric current systems they connect to.[71]

"A Birkeland current is a set of currents which flow along geomagnetic field lines connecting the Earth’s magnetosphere to the Earth's high latitude ionosphere. Lately, the term Birkeland currents has been expanded by some authors to include magnetic field aligned currents in space plasmas in general. In the Earth’s magnetosphere, the currents are driven by the solar wind and interplanetary magnetic field and by bulk motions of plasma through the magnetosphere (convection which is indirectly driven by the interplanetary environment). The strength of the Birkeland currents changes with activity in the magnetosphere (e.g. during substorms). Small scale variations in the upward current sheets (downward flowing electrons) accelerate magnetospheric electrons which, when they reach the upper atmosphere, create the Aurora Borealis and Australis. In the high latitude ionosphere (or auroral zones), the Birkeland currents close through the region of the auroral electrojet, which flows perpendicular to the local magnetic field in the ionosphere. The Birkeland currents occur in two pairs of field-aligned current sheets. One pair extends from noon through the dusk sector to the midnight sector. The other pair extends from noon through the dawn sector to the midnight sector. The sheet on the high latitude side of the auroral zone is referred to as the Region 1 current sheet and the sheet on the low latitude side is referred to as the Region 2 current sheet."[72]

"A reason why Birkeland currents are particularly interesting is that, in the plasma forced to carry them, they cause a number of plasma physical processes to occur (waves, instabilities, fine structure formation). These in turn lead to consequences such as acceleration of charged particles, both positive and negative, and element separation (such as preferential ejection of oxygen ions). Both of these classes of phenomena should have a general astrophysical interest far beyond that of understanding the space environment of our own Earth."[73]

"Auroral Birkeland currents carry about 100,000 amperes during quiet times[74] and more than 1 million amperes during geomagnetically disturbed times.[75] Birkeland had estimated currents "at heights of several hundred kilometres, and strengths of up to a million amperes" in 1908.[76] The ionospheric currents which connect the field-aligned currents heat up the upper atmosphere due to the finite conductivity of the ionosphere. The heat (also known as Joule heat) is transferred from the ionospheric plasma to the gas of the upper atmosphere which rises and increases drag on low-altitude satellites."[72]

"Birkeland currents can also be created in the laboratory with multi-terawatt pulsed power generators. The resulting cross-section pattern indicates a hollow beam of electrons in the form of a circle of vortices, a formation called the diocotron instability[77] (similar to, but different from, the Kelvin-Helmholtz instability), that subsequently leads to filamentation. Such vortices can be seen in aurora as "auroral curls".[78]"[72]

"Birkeland currents are also one of a class of plasma phenomena called a z-pinch, so named because the azimuthal magnetic fields produced by the current pinches the current into a filamentary cable. This can also twist, producing a helical pinch that spirals like a twisted or braided rope, and this most closely corresponds to a Birkeland current. Pairs of parallel Birkeland currents will also interact due to Ampère's force law: parallel Birkeland currents moving in the same direction will attract each other with an electromagnetic force inversely proportional to their distance apart whilst parallel Birkeland currents moving in opposite directions will repel each other. There is also a short-range circular component to the force between two Birkeland currents that is opposite to the longer-range parallel forces.[79]"[72]

"Electrons moving along a Birkeland current may be accelerated by a plasma double layer. If the resulting electrons approach relativistic velocities (i.e. if they approach the speed of light) they may subsequently produce a Bennett pinch, which in a magnetic field causes the electrons to spiral and emit synchrotron radiation that may include radio, optical (i.e. visible light), x-rays, and gamma rays."[72]

"[T]he currents there are imagined as having come into existence mainly as a secondary effect of the electric corpuscles from the sun drawn in out of space, and thus far come under the second of the possibilities mentioned above". And p. 105, "Fig. 50a represents those in which the current-directions at the storm-centre are directed westwards, and 50b those in which the currents move eastwards"[76].

"[C]urrents there [in the aurora] are imagined as having come into existence mainly as a secondary effect of the electric corpuscles from the sun drawn in out of space,"[76]

"A U.S. Navy satellite 1963-38C, launched in 1963 and carrying a magnetometer above the ionosphere.[80] ... In 1966 Alfred Zmuda, J.H. Martin, and F.T.Heuring[81] analysed the satellite magnetometer results and reported their findings of magnetic disturbance in the aurora"[72]

# Mars

Mars has a magnetosphere.[59]

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