In
physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, magnetosonic waves, also known as magnetoacoustic waves, are low-frequency
compressive wave
In physics, mathematics, engineering, and related fields, a wave is a propagating dynamic disturbance (change from List of types of equilibrium, equilibrium) of one or more quantities. ''Periodic waves'' oscillate repeatedly about an equilibrium ...
s driven by mutual interaction between an
electrically conducting fluid and a
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 ...
. They are associated with
compression and
rarefaction
Rarefaction is the reduction of an item's density, the opposite of compression. Like compression, which can travel in waves (sound waves, for instance), rarefaction waves also exist in nature. A common rarefaction wave is the area of low relati ...
of both the fluid and the magnetic field, as well as with an
effective tension that acts to straighten bent magnetic field lines. The properties of magnetosonic waves are highly dependent on the angle between the
wavevector
In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
and the equilibrium magnetic field and on the relative importance of fluid and magnetic processes in the medium. They only propagate with frequencies much smaller than the ion cyclotron or ion plasma frequencies of the medium, and they are
nondispersive at small amplitudes.
There are two types of magnetosonic waves, fast magnetosonic waves and slow magnetosonic waves, which—together with
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
...
s—are the
normal mode
A normal mode of a dynamical system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies ...
s of ideal
magnetohydrodynamics. The fast and slow modes are distinguished by magnetic and gas pressure oscillations that are either
in-phase or anti-phase, respectively. This results in the phase velocity of any given fast mode always being greater than or equal to that of any slow mode in the same medium, among other differences.
Magnetosonic waves have been observed in the
Sun's corona and provide an observational foundation for
coronal seismology.
Characteristics
Magnetosonic waves are a type of low-frequency wave present in electrically conducting, magnetized fluids, such as
plasmas and
liquid metal
A liquid metal is a metal or a metal alloy which is liquid at or near room temperature.
The only stable liquid elemental metal at room temperature is Mercury (element), mercury (Hg), which is molten above −38.8 °C (234.3 K, −37.9  ...
s. They exist at frequencies far below the cyclotron and plasma frequencies of both ions and electrons in the medium (see ).
In an ideal, homogeneous, electrically conducting, magnetized fluid of infinite extent, there are two magnetosonic modes: the fast and slow modes. They form, together with the
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
...
, the three basic linear
magnetohydrodynamic (MHD) waves. In this regime, magnetosonic waves are nondispersive at small amplitudes.
Dispersion relation
The fast and slow magnetosonic waves are defined by a bi-quadratic
dispersion relation
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the ...
that can be derived from the linearized MHD equations.
Phase and group velocities
The
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, t ...
of the fast and slow magnetosonic waves depend on the angle between the wavevector and the equilibrium magnetic field as well as the equilibrium density, pressure, and magnetic field strength. From the roots of the magnetosonic dispersion relation, the associated phase velocities can be expressed as
where the upper sign gives the phase velocity of the fast mode and the lower sign gives the phase velocity of the slow mode.
The phase velocity of the fast mode is always greater than or equal to
, which is greater than or equal to that of the slow mode,
. This is due to the differences in the signs of the thermal and
magnetic pressure perturbations associated with each mode. The magnetic pressure perturbation
can be expressed in terms of the thermal pressure perturbation and phase velocity as
For the fast mode , so magnetic and thermal pressure perturbations have matching signs. Conversely, for the slow mode , so magnetic and thermal pressure perturbations have opposite signs. In other words, the two pressure perturbations reinforce one another in the fast mode, but oppose one another in the slow mode. As a result, the fast mode propagates at a faster speed than the slow mode.
The
group velocity
The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope (waves), envelope'' of the wave—propagates through space.
For example, if a stone is thro ...
of fast and slow magnetosonic waves is defined by
where and are local orthogonal
unit vector
In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
in the direction of and in the direction of increasing , respectively. In a
spherical coordinate system
In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are
* the radial distance along the line connecting the point to a fixed point ...
with a -axis along the unperturbed magnetic field, these unit vectors correspond to those in the direction of increasing radial distance and increasing polar angle.
Limiting cases
Incompressible fluid
In an
incompressible
Incompressible may refer to:
* Incompressible flow, in fluid mechanics
* incompressible vector field, in mathematics
* Incompressible surface, in mathematics
* Incompressible string, in computing
{{Disambig ...
fluid, the density and pressure perturbations vanish, and , resulting in the sound speed tending to infinity, . In this case, the slow mode propagates with the Alfvén speed, , and the fast mode disappears from the system, .
Cold limit
Under the assumption that the background temperature is zero, it follows from the
ideal gas law
The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
that the thermal pressure is also zero, , and, as a result, that the sound speed vanishes, . In this case, the slow mode disappears from the system, , and the fast mode propagates isotropically with the Alfvén speed, . In this limit, the fast mode is sometimes referred to as a ''compressional Alfvén wave''.
Parallel propagation
When the wavevector and the equilibrium magnetic field are parallel, , the fast and slow modes propagate as either a pure sound wave or pure Alfvén wave, with the fast mode identified with the larger of the two speeds and the slow mode identified with the smaller.
Perpendicular propagation
When the wavevector and the equilibrium magnetic field are perpendicular, , the fast mode propagates as a longitudinal wave with phase velocity equal to the magnetosonic speed, and the slow mode propagates as a transverse wave with phase velocity approaching zero.
Inhomogeneous fluid
In the case of an inhomogeneous fluids (that is, a fluid where at least one of the background quantities is not constant) the MHD waves lose their defining nature and get mixed properties. In some setups, such as the axisymmetric waves in a straight cylinder with a circular basis (one of the simplest models for a
coronal loop
In solar physics, a coronal loop is a well-defined arch-like structure in the Sun's Stellar atmosphere, atmosphere made up of relatively dense Plasma (physics), plasma confined and isolated from the surrounding medium by magnetic flux tubes. Co ...
), the three MHD waves can still be clearly distinguished. But in general, the pure Alfvén and fast and slow magnetosonic waves don't exist, and the waves in the fluid are coupled to each other in intricate ways.
Observations
Both fast and slow magnetosonic waves have been observed 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 ...
providing an observational foundation for the technique for coronal plasma diagnostics,
coronal seismology.
[{{cite journal, author=Nakariakov, V.M., author2=Verwichte, E., date=2005, title=Coronal waves and oscillations, url=http://www.livingreviews.org/lrsp-2005-3, journal=Living Rev. Sol. Phys., volume=2, issue=1, pages=3, doi=10.12942/lrsp-2005-3, bibcode=2005LRSP....2....3N, s2cid=123211890, doi-access=free]
See also
*
Waves in plasmas
In plasma physics, waves in plasmas are an interconnected set of particles and fields which propagate in a periodically repeating fashion. A plasma is a quasineutral, electrically conductive fluid. In the simplest case, it is composed of electro ...
*
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
...
*
Ion acoustic wave
*
Coronal seismology
*
Magnetogravity wave
References
Waves in plasmas