Magnetosonic wave

A magnetosonic wave (also called magnetoacoustic wave) is a linear magnetohydrodynamic (MHD) wave that is driven by both pressure (thermal and magnetic) and magnetic tension. There are two types of magnetosonic waves, the fast magnetosonic wave and the slow magnetosonic wave. Both fast and slow magnetosonic waves have been recently discovered in the solar corona,[1] which created an observational foundation for the novel technique for the coronal plasma diagnostics, coronal seismology.

Homogeneous plasma

In an ideal homogeneous plasma of infinite extent, and in the absence of gravity, the magnetosonic waves form, together with the Alfvén wave, the three basic linear MHD waves. Under the assumption of normal modes, namely that the linear perturbations of the physical quantities are of the form ${\displaystyle f_{1}={\tilde {f}}_{1}e^{i(k_{x}x+k_{y}y+k_{z}z-\omega t)}}$ (with ${\displaystyle {\tilde {f}}_{1}}$ the constant amplitude), a dispersion relation of the magnetosonic waves can be derived from the system of ideal MHD equations[2]:

${\displaystyle \omega ^{4}-k^{2}(v_{s}^{2}+v_{A}^{2})\omega ^{2}+k_{\parallel }k^{2}v_{s}^{2}v_{A}^{2}=0}$,

where ${\displaystyle v_{A}}$ is the Alfvén speed, ${\displaystyle v_{s}}$ is the sound speed, ${\displaystyle k={\sqrt {k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}}}$ is the magnitude of the wave vector and ${\displaystyle k_{\parallel }}$ is the component of the wave vector along the background magnetic field (which is straight and constant, because the plasma is assumed homogeneous).

This equation can be solved for the frequency ${\displaystyle \omega }$, yielding the frequencies of the fast and slow magnetosonic waves:

${\displaystyle \omega _{f,sl}^{2}={\frac {k^{2}(v_{s}^{2}+v_{A}^{2})}{2}}{\Biggl (}1\pm {\biggl (}1-{\frac {4k_{\parallel }^{2}v_{s}^{2}v_{A}^{2}}{k^{2}(v_{s}^{2}+v_{A}^{2})^{2}}}{\biggr )}^{1/2}{\Biggr )}}$.

It can be shown that ${\displaystyle \omega _{sl}\leq \omega _{A}\leq \omega _{f}}$ (with ${\displaystyle \omega _{A}=k_{\parallel }v_{A}}$ the Alfvén frequency), hence the name of "slow" and "fast" magnetosonic waves.

Limiting cases

Absent magnetic field

In the absence of a magnetic field, the whole MHD model reduces to the hydrodynamics (HD) model. In this case ${\displaystyle v_{A}=0}$, and hence ${\displaystyle \omega _{sl}^{2}=0}$ and ${\displaystyle \omega _{f}^{2}=k^{2}v_{s}^{2}}$. The slow wave thus disappears from the system, while the fast wave is just a sound wave, propagating isotropically.

Incompressible plasma

In case the plasma is incompressible, the sound speed ${\displaystyle v_{s}\to \infty }$ (this follows from the energy equation) and it can then be shown that ${\displaystyle \omega _{sl}^{2}=\omega _{A}^{2}}$ and ${\displaystyle \omega _{f}^{2}=\infty }$. The slow wave thus propagates with the Alfvén speed (although it remains different from an Alfvén wave in its nature), while the fast wave disappears from the system.

Cold plasma

Under the assumption that the background temperature ${\displaystyle T_{0}=0}$, it follows from the perfect gas law that the thermal pressure ${\displaystyle p_{0}=0}$ and thus that ${\displaystyle v_{s}=0}$. In this case, ${\displaystyle \omega _{sl}^{2}=0}$ and ${\displaystyle \omega _{f}^{2}=k^{2}v_{A}^{2}}$. Hence there are no slow waves in the system, and the fast waves propagate isotropically with the Alfvén speed.

Inhomogeneous plasma

In the case of an inhomogeneous plasma, i.e. a plasma where at least one of the background quantities is not constant, the MHD waves lose their defining nature and get mixed properties[3]. 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), the three MHD waves can still be clearly distinguished. But in general, the pure Alfvén, fast and slow magnetosonic waves don't exist and the waves in the plasma are coupled to each other in intricate ways.

See also

• Waves in plasmas
• Alfvén wave
• Ion acoustic wave
• Coronal seismology

References

1. Nakariakov, V.M.; Verwichte, E. (2005). "Coronal waves and oscillations". Living Rev. Sol. Phys. 2: 3.
2. Goossens, Marcel (2003). An Introduction to Plasma Astrophysics and Magnetohydrodynamics. Astrophysics and Space Science Library. 294. Dordrecht: Springer Netherlands. doi:10.1007/978-94-007-1076-4. ISBN 978-1-4020-1433-8.
3. Goossens, Marcel L.; Arregui, Inigo; Van Doorsselaere, Tom (2019-04-11). "Mixed Properties of MHD Waves in Non-uniform Plasmas". Frontiers in Astronomy and Space Sciences. 6: 20. doi:10.3389/fspas.2019.00020. ISSN 2296-987X.