Shocks and discontinuities (magnetohydrodynamics)
Encyclopedia
Shocks and discontinuities
are transition layers where the plasma
properties 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 magnetohydrodynamic (MHD) equations, assuming conservation of mass, momentum, energy and of
.
s for MHD. In the frame
moving with the shock/discontinuity, those jump conditions can be written:
where
, 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
.
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
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.
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.
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.
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
,
and the sonic speed
,
as follows:
where
is the Alfvén speed and 
is the angle between the incoming magnetic field
and the shock normal vector.
The normal component of the slow shock propagates with velocity
in the frame moving with the upstream plasma, that of the intermediate shock with velocity 
and that of the fast shock with velocity 
. 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.
Classification of discontinuities
Continuous functions are of utmost importance in mathematics and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there...
are transition layers where the plasma
Plasma (physics)
In physics and chemistry, plasma is a state of matter similar to gas in which a certain portion of the particles are ionized. Heating a gas may ionize its molecules or atoms , thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions...
properties 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 magnetohydrodynamic (MHD) equations, assuming conservation of mass, momentum, energy and of
.Rankine-Hugoniot jump conditions for MHD
The jump conditions across an MHD shock or discontinuity are referred as the Rankine-Hugoniot equationRankine-Hugoniot equation
The Rankine–Hugoniot conditions, also referred to as Rankine–Hugoniot jump conditions or Rankine–Hugoniot relations, relate to the behaviour of shock waves traveling normal to the prevailing flow. They are named in recognition of the work carried out by Scottish engineer and physicist William John...
s for MHD. In the frame
Frame of reference
A frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer.It may also refer to both an...
moving with the shock/discontinuity, those jump conditions can be written:
where
, v, p, B are the plasma densityDensity
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
, velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
, (thermal) pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
and magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
respectively. The subscripts t and n refer to the tangential and normal components
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 normal component of the vector...
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,.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
Magnetic pressure
Magnetic pressure is an energy density associated with the magnetic field. It is identical to any other physical pressure except that it is carried by the magnetic field rather than kinetic energy of the gas molecules. Interplay between magnetic pressure and ordinary gas pressure is important to...
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. 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.
Slow-mode and fast-mode shocks are compressive and are associated with an increase in entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
. 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 wave
An Alfvén wave, named after Hannes Alfvén, is a type of magnetohydrodynamic wave.-Definition:An Alfvén wave in a plasma is a low-frequency travelling oscillation of the ions and the magnetic field...
,
and the sonic speedSpeed of sound
The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....
,
as follows:where
is the Alfvén speed and is the angle between the incoming magnetic fieldMagnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
and the shock normal vector.
The normal component of the slow shock propagates with velocity
in the frame moving with the upstream plasma, that of the intermediate shock with velocity and that of the fast shock with velocity . The fast mode waves have higher phase velocitiesPhase velocity
The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity...
than the slow mode waves because the density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
and magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
are in phase, whereas the slow mode wave components are out of phase.
Example of shocks and discontinuities in space
- The Earth's bow shockBow shockA bow shock is the area between a magnetosphere and an ambient medium. For stars, this is typically the boundary between their stellar wind and the interstellar medium....
, which is the boundary where the solar windSolar windThe solar wind is a stream of charged particles ejected from the upper atmosphere of the Sun. It mostly consists of electrons and protons with energies usually between 1.5 and 10 keV. The stream of particles varies in temperature and speed over time...
's speed drops due to the presence of Earth's magnetosphereMagnetosphereA magnetosphere is formed when a stream of charged particles, such as the solar wind, interacts with and is deflected by the intrinsic magnetic field of a planet or similar body. Earth is surrounded by a magnetosphere, as are the other planets with intrinsic magnetic fields: Mercury, Jupiter,...
is a fast mode shock. The termination shock is a fast-mode shock due to the interaction of the solar wind with the interstellar mediumInterstellar mediumIn astronomy, the interstellar medium is the matter that exists in the space between the star systems in a galaxy. This matter includes gas in ionic, atomic, and molecular form, dust, and cosmic rays. It fills interstellar space and blends smoothly into the surrounding intergalactic space...
. - Magnetic reconnectionMagnetic reconnectionMagnetic reconnection is a physical process in highly conducting plasmas in which the magnetic topology is rearranged and magnetic energy is converted to kinetic energy, thermal energy, and particle acceleration...
can happen associated with a slow-mode shock (Petschek or fast magnetic reconnection) in the solar corona. - 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 remnantSupernova remnantA supernova remnant 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 material it sweeps up and shocks along the way.There are two...
) or due to the interaction of multiple CMECoronal mass ejectionA coronal mass ejection is a massive burst of solar wind, other light isotope plasma, and magnetic fields rising above the solar corona or being released into space....
driven shock waves. - The Earth's magnetopauseMagnetopauseThe 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 determined by the balance between the pressure of the...
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.
See also
- MagnetohydrodynamicsMagnetohydrodynamicsMagnetohydrodynamics is an academic discipline which studies the dynamics of electrically conducting fluids. Examples of such fluids include plasmas, liquid metals, and salt water or electrolytes...
- Shock waveShock waveA shock wave is a type of propagating disturbance. Like an ordinary wave, it carries energy and can propagate through a medium or in some cases in the absence of a material medium, through a field such as the electromagnetic field...
- Rankine-Hugoniot equationRankine-Hugoniot equationThe Rankine–Hugoniot conditions, also referred to as Rankine–Hugoniot jump conditions or Rankine–Hugoniot relations, relate to the behaviour of shock waves traveling normal to the prevailing flow. They are named in recognition of the work carried out by Scottish engineer and physicist William John...
- Alfvén waveAlfvén waveAn Alfvén wave, named after Hannes Alfvén, is a type of magnetohydrodynamic wave.-Definition:An Alfvén wave in a plasma is a low-frequency travelling oscillation of the ions and the magnetic field...
- Moreton waveMoreton waveA Moreton wave is the chromospheric signature of a large-scale solar coronal shock wave. Described as a kind of solar 'tsunami', they are generated by solar flares. They are named for American astronomer Gail Moreton, an observer at the Lockheed Solar Observatory in Burbank who spotted them in 1959...
- List of plasma (physics) applications articles