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magnetohydrodynamics Magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydro­magnetics) is the study of the magnetic properties and behaviour of electrically conducting fluids. Examples of such magneto­fluids include plasmas, liquid metals, ...
(MHD), shocks and discontinuities are transition layers where properties of a
plasma Plasma or plasm may refer to: Science * Plasma (physics), one of the four fundamental states of matter * Plasma (mineral), a green translucent silica mineral * Quark–gluon plasma, a state of matter in quantum chromodynamics Biology * Blood pla ...
change from one equilibrium state to another. The relation between the plasma properties on both sides of a shock or a discontinuity can be obtained from the conservative form of the MHD equations, assuming conservation of mass, momentum, energy and of \nabla \cdot \mathbf .


Rankine–Hugoniot jump conditions for MHD

The jump conditions across a time-independent MHD shock or discontinuity are referred as the Rankine–Hugoniot equations for MHD. In the
frame A frame is often a structural system that supports other components of a physical construction and/or steel frame that limits the construction's extent. Frame and FRAME may also refer to: Physical objects In building construction *Framing (con ...
moving with the shock/discontinuity, those jump conditions can be written: : \rho_1 v_ = \rho_2 v_, : B_ = B_, : \rho_1 v_^2+ p_1 + \frac=\rho_2 v_^2+ p_2 + \frac, : \rho_1 v_ \mathbf - \frac= \rho_2 v_ \mathbf - \frac, : \left(\frac\frac+\frac\right)\rho_1 v_+\frac-\frac=\left(\frac\frac+\frac\right)\rho_2 v_+\frac-\frac, : (\mathbf \times \mathbf)_ = (\mathbf \times \mathbf)_, where \rho, v, p, B are the plasma
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
,
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
, (thermal)
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
and
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
respectively. The subscripts t and n refer to the
tangential and normal components In mathematics, given a vector at a point on a curve, that vector can be decomposed uniquely as a sum of two vectors, one tangent to the curve, called the tangential component of the vector, and another one perpendicular to the curve, called the no ...
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


Contact and tangential discontinuities

Contact and tangential discontinuities are transition layers across which there is no particle transport. Thus, in the frame moving with the discontinuity, v_ = v_ =0. Contact discontinuities are discontinuities for which the thermal pressure, the magnetic field and the velocity are continuous. Only the mass density and temperature change. Tangential discontinuities are discontinuities for which the total pressure (sum of the thermal and
magnetic pressure In physics, magnetic pressure is an energy density associated with a magnetic field. In SI units, the energy density P_B of a magnetic field with strength B can be expressed as :P_B = \frac where \mu_0 is the vacuum permeability. Any magnetic fiel ...
s) is conserved. The normal component of the magnetic field is identically zero. The density, thermal pressure and tangential component of the magnetic field vector can be discontinuous across the layer.


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. 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 In thermodynamics, an isentropic process is an idealized thermodynamic process that is both adiabatic and reversible. The work transfers of the system are frictionless, and there is no net transfer of heat or matter. Such an idealized process ...
. All
thermodynamic Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of ther ...
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. Slow-mode and fast-mode shocks are compressive and are associated with an increase in
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
. Across slow-mode shock, the tangential component of the magnetic field decreases. Across fast-mode shock it increases. 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 Alfvén may refer to: People * Hannes Alfvén (1908–1995), Swedish plasma physicist and Nobel Prize in Physics laureate * Hugo Alfvén (1872–1960), Swedish composer, conductor, violinist, and painter * Marie Triepcke Krøyer Alfvén (1867–19 ...
, V_A and the sonic speed, c_s as follows: :a_^2 = \frac \left left(c_s^2 + V_A^2\right)-\sqrt\,\right : a_^2 = \frac \left left(c_s^2 + V_A^2\right)+\sqrt\,\right where V_ is the Alfvén speed and \theta_ is the angle between the incoming
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
and the shock normal vector. The normal component of the slow shock propagates with velocity a_ in the frame moving with the upstream plasma, that of the intermediate shock with velocity V_ and that of the fast shock with velocity a_. The fast mode waves have higher
phase velocities The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
than the slow mode waves because the
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
and
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
are in phase, whereas the slow mode wave components are out of phase.


Example of shocks and discontinuities in space

*The Earth's
bow shock In astrophysics, a bow shock occurs when the magnetosphere of an astrophysical object interacts with the nearby flowing ambient plasma such as the solar wind. For Earth and other magnetized planets, it is the boundary at which the speed of ...
, which is the boundary where the
solar wind The solar wind is a stream of charged particles released from the upper atmosphere of the Sun, called the corona. This plasma mostly consists of electrons, protons and alpha particles with kinetic energy between . The composition of the sola ...
's speed drops due to the presence of Earth's
magnetosphere In astronomy and planetary science, a magnetosphere is a region of space surrounding an astronomical object in which charged particles are affected by that object's magnetic field. It is created by a celestial body with an active interior dynam ...
is a fast mode shock. The
termination shock The heliosphere is the magnetosphere, astrosphere and outermost atmospheric layer of the Sun. It takes the shape of a vast, bubble-like region of space. In plasma physics terms, it is the cavity formed by the Sun in the surrounding interstell ...
is a fast-mode shock due to the interaction of the solar wind with the
interstellar medium In astronomy, the interstellar medium is the matter and radiation that exist in the space between the star systems in a galaxy. This matter includes gas in ionic, atomic, and molecular form, as well as dust and cosmic rays. It fills interstella ...
. *
Magnetic reconnection Magnetic reconnection is a physical process occurring in highly conducting plasmas in which the magnetic topology is rearranged and magnetic energy is converted to kinetic energy, thermal energy, and particle acceleration. Magnetic reconnectio ...
can happen associated with a slow-mode shock (Petschek or fast magnetic reconnection) in the
solar corona A corona ( coronas or coronae) is the outermost layer of a star's atmosphere. It consists of plasma. The Sun's corona lies above the chromosphere and extends millions of kilometres into outer space. It is most easily seen during a total solar e ...
. * The existence of intermediate shocks is still a matter of debate. They may form in MHD simulation, but their stability has not been proven. * Discontinuities (both contact and tangential) are observed in the solar wind, behind astrophysical shock waves (
supernova remnant A supernova remnant (SNR) is the structure resulting from the explosion of a star in a supernova. The supernova remnant is bounded by an expanding shock wave, and consists of ejected material expanding from the explosion, and the interstellar mat ...
) or due to the interaction of multiple CME driven shock waves. * The Earth's
magnetopause The magnetopause is the abrupt boundary between a magnetosphere and the surrounding plasma. For planetary science, the magnetopause is the boundary between the planet's magnetic field and the solar wind. The location of the magnetopause is d ...
is generally a tangential discontinuity. * Coronal Mass Ejections (CMEs) moving at super-Alfvénic speeds are able to drive fast-mode MHD shocks while propagating away from the Sun into the solar wind. Signatures of these shocks have been identified in both radio (as type II radio bursts) and ultraviolet (UV) spectra.S. Mancuso et al., ''UVCS/SOHO observations of a CME-driven shock: Consequences on ion heating mechanisms behind a coronal shock'', Astronomy and Astrophysics, 2002, v.383, p.267-274


See also

*
Alfvén wave In plasma physics, an Alfvén wave, named after Hannes Alfvén, is a type of plasma wave in which ions oscillate in response to a restoring force provided by an effective tension on the magnetic field lines. Definition An Alfvén wave is a ...
*
List of plasma physics articles This is a list of plasma physics topics. A * Ablation * Abradable coating * Abraham–Lorentz force * Absorption band * Accretion disk * Active galactic nucleus * Adiabatic invariant * ADITYA (tokamak) * Aeronomy * Afterglow plasma * Air ...
*
Magnetohydrodynamics Magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydro­magnetics) is the study of the magnetic properties and behaviour of electrically conducting fluids. Examples of such magneto­fluids include plasmas, liquid metals, ...
*
Moreton wave A Moreton wave, Solar Tsunami, or Moreton-Ramsey wave is the chromospheric signature of a large-scale solar corona shock wave. Described as a kind of solar "tsunami", they are generated by solar flares. They are named for American astronomer Ga ...
*
Rankine–Hugoniot conditions The Rankine–Hugoniot conditions, also referred to as Rankine–Hugoniot jump conditions or Rankine–Hugoniot relations, describe the relationship between the states on both sides of a shock wave or a combustion wave (deflagration or detonation) ...
*
Shock wave In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a med ...


References

The original research on MHD shock waves can be found in the following papers. * * * *Friedrichs, K. O. "Nonlinear wave motion in magnetohydrodynamics", ''Los Alamos Sci. Lab. Report LAMS-2105'' (Physics), written September 1954, distributed, March 1957. See also the somewhat emended and more available version of this report written jointly with H. Kranzer, Notes on magnetohydrodynamics, VIII, Nonlinear wave motion, AEC Computing and Applied Mathematics Center, Institute of Mathematical Sciences, New York University, Report No. NYO-6486 (1958). * * * * * Textbook references. *E. Priest, "''Solar magneto-hydrodynamics''"(chapter 5), Dordrecht, 1987. *T. Gombosi "''Physics of the Space Environment''" (chapter 6), Cambridge University Press, 1998. {{DEFAULTSORT:Shocks And Discontinuities (Magnetohydrodynamics) Plasma physics Space plasmas Shock waves Waves in plasmas