Shocks And Discontinuities (magnetohydrodynamics)
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magnetohydrodynamics In physics and engineering, magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydro­magnetics) is a model of electrically conducting fluids that treats all interpenetrating particle species together as a single Continuum ...
(MHD), shocks and discontinuities are transition layers where properties of a plasma 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 moving with the shock/discontinuity, those jump conditions can be written: \begin \rho_1 v_ = \rho_2 v_, \\ .2ex B_ = B_, \\ .2ex \rho_1 v_^2 + p_1 + \frac B_^2 = \rho_2 v_^2+ p_2 + \frac B_^2, \\ .2ex \rho_1 v_ \mathbf_ - \frac \mathbf_ B_ = \rho_2 v_ \mathbf_ - \frac \mathbf_ B_, \\ .2ex \displaystyle \left(\frac\frac + \frac\right) \rho_1 v_ + \frac \left - \right= \left(\frac\frac + \frac\right) \rho_2 v_ + \frac \left - \right \\ .2ex (\mathbf \times \mathbf)_ = (\mathbf \times \mathbf)_, \end where , , are the plasma
density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
,
velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
, (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 eve ...
and
magnetic field A magnetic field (sometimes called B-field) is a physical 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 ...
respectively. The subscripts \parallel and \perp 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


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 pressures) 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 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 is useful in eng ...
. 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 th ...
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, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...
. 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, V_A and the sonic speed, c_s as follows: \begin a_^2 &= \frac \left left(c_s^2 + V_A^2\right)-\sqrt\,\right \\ exa_^2 &= \frac \left left(c_s^2 + V_A^2\right)+\sqrt\,\right \end where V_ is the Alfvén speed and \theta_ is the angle between the incoming
magnetic field A magnetic field (sometimes called B-field) is a physical 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 ...
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 than the slow mode waves because the
density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
and
magnetic field A magnetic field (sometimes called B-field) is a physical 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 ...
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, which is the boundary where the
solar wind The solar wind is a stream of charged particles released from the Sun's outermost atmospheric layer, the Stellar corona, corona. This Plasma (physics), plasma mostly consists of electrons, protons and alpha particles with kinetic energy betwee ...
'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 Dynamo ...
is a fast mode shock. The termination shock is a fast-mode shock due to the interaction of the solar wind with the
interstellar medium The interstellar medium (ISM) is the matter and radiation that exists in the outer space, space between the star systems in a galaxy. This matter includes gas in ionic, atomic, and molecular form, as well as cosmic dust, dust and cosmic rays. It f ...
. *
Magnetic reconnection Magnetic reconnection is a physical process occurring in electrically conducting Plasma (physics), plasmas, in which the magnetic topology is rearranged and magnetic energy is converted to kinetic energy, thermal energy, and particle accelerati ...
can happen associated with a slow-mode shock (Petschek or fast magnetic reconnection) in the
solar corona In astronomy, a corona (: coronas or coronae) is the outermost layer of a star's Stellar atmosphere, atmosphere. It is a hot but relatively luminosity, dim region of Plasma (physics), plasma populated by intermittent coronal structures such as so ...
. * 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 (physics), plasma. For planetary science, the magnetopause is the boundary between the planet's magnetic field and the solar wind. The location of the ma ...
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 Magnetic tension force, effective tension on the magnetic field lines. Definition ...
*
Magnetohydrodynamics In physics and engineering, magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydro­magnetics) is a model of electrically conducting fluids that treats all interpenetrating particle species together as a single Continuum ...
* Moreton wave * Rankine–Hugoniot conditions *
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 me ...


References


Citations


General 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). * * * * *


Textbooks

* * {{DEFAULTSORT:Shocks And Discontinuities (Magnetohydrodynamics) Plasma phenomena Space plasmas Shock waves Waves in plasmas