SlowMagnetosonicWave

class plasmapy.dispersion.analytical.mhd_waves_.SlowMagnetosonicWave( B: ~astropy.units.quantity.Quantity, density: (Unit("1 / m3"), Unit("kg / m3")), ion: str | int | ~numpy.integer | ~plasmapy.particles.particle_class.Particle | ~plasmapy.particles.particle_class.CustomParticle | ~astropy.units.quantity.Quantity, *, T: ~astropy.units.quantity.Quantity = <Quantity 0. K>, gamma: float = 1.6666666666666667, mass_numb: int | None = None, Z: float | None = None, )[source]

Bases: AbstractMHDWave

A class to represent slow magnetosonic waves.

Parameters:

B (Quantity) – The magnetic field magnitude in units convertible to T.
density (Quantity) – Either the ion number density \(n_i\) in units convertible to m^-3 or the total mass density \(ρ\) in units convertible to kg m^-3.
ion (particle-like) – Representation of the ion species (e.g., 'p+' for protons, 'D+' for deuterium, 'He-4 +1' for singly ionized helium-4, etc.). If no charge state information is provided, then the ions are assumed to be singly ionized.
T (Quantity, keyword-only, optional) – The plasma temperature in units of K or eV, which defaults to zero.
gamma (float or int, keyword-only, optional) – The adiabatic index for the plasma, which defaults to 3/5.
mass_numb (int, keyword-only, optional) – The mass number corresponding to ion.
Z (float or int, keyword-only, optional) – The charge number corresponding to ion.

Raises:

TypeError – If applicable arguments are not instances of Quantity or cannot be converted into one.
TypeError – If ion is not particle-like.
TypeError – If gamma or Z are not of type int or float.
TypeError – If mass_numb is not of type int.
UnitTypeError – If applicable arguments do not have units convertible to the expected units.
ValueError – If any of B, density, or T is negative.
ValueError – If ion is not of category ion or element.
ValueError – If B, density, or T are not single-valued astropy.units.Quantity (i.e. an array).

See also

AlfvenWave, FastMagnetosonicWave

Notes

Slow magnetosonic waves are also referred to as slow magnetoacoustic waves.

Examples

>>> import astropy.units as u
>>> from plasmapy.dispersion.analytical import SlowMagnetosonicWave
>>> slow = SlowMagnetosonicWave(1e-3 * u.T, 1e16 * u.m**-3, "p+", T=2.5e6 * u.K)
>>> slow.angular_frequency(1e-5 * u.rad / u.m, 0 * u.deg)
<Quantity 1.85454394 rad / s>
>>> slow.phase_velocity(1e-5 * u.rad / u.m, 0 * u.deg)
<Quantity 185454.39417735 m / s>
>>> slow.sound_speed
<Quantity 185454.39417735 m / s>

Attributes Summary

`alfven_speed`	The Alfvén speed of the plasma.
`beta`	The ratio of thermal pressure to magnetic pressure.
`magnetosonic_speed`	The magnetosonic speed of the plasma.
`sound_speed`	The sound speed of the plasma.

Methods Summary

`angular_frequency`(k, theta)	Calculate the angular frequency of slow magnetosonic waves.
`group_velocity`(k, theta)	Calculate the group velocities of slow magnetosonic waves.
`phase_velocity`(k, theta)	Calculate the phase velocities of magnetohydrodynamic waves.

Attributes Documentation

alfven_speed: The Alfvén speed of the plasma.

beta: The ratio of thermal pressure to magnetic pressure.

magnetosonic_speed

The magnetosonic speed of the plasma.

Defined as \(c_{ms} = \sqrt{v_A^2 + c_s^2}\) where \(v_A\) is the Alfvén speed and \(c_s\) is the sound speed.

sound_speed

The sound speed of the plasma.

Defined as \(c_s = \sqrt{γ k_B T / m_i}\) where \(gamma\) is the adiabatic index of the fluid, \(k_B\) is the Boltzmann constant, \(T\) is the temperature of the fluid, and \(m_i\) is the mass of the ion species in the fluid.

Methods Documentation

angular_frequency( k: Quantity, theta: Quantity, )[source]

Calculate the angular frequency of slow magnetosonic waves.

Parameters:

k (Quantity, single valued or 1-D array) – Wavenumber in units convertible to rad/m`. Either single valued or 1-D array of length \(N\).
theta (Quantity, single valued or 1-D array) – The angle of propagation of the wave with respect to the magnetic field, \(\cos^{-1}(k_z / k)\), in units must be convertible to radians. Either single valued or 1-D array of size \(M\).

Returns:

omega – An \(N × M\) array of computed wave frequencies in units rad/s.

Return type:

Quantity

Raises:

UnitTypeError – If applicable arguments do not have units convertible to the expected units.
ValueError – If k is negative or zero.
ValueError – If k or theta are not single valued or a 1-D array.

Warns:

PhysicsWarning – When the computed wave frequencies violate the low-frequency (\(ω ≪ ω_c,ω_p\)) assumption of the dispersion relation.

Notes

The angular frequency \(ω\) of a slow magnetosonic wave is given by the equation

\[ω^2 = \frac{k^2}{2} \left(c_{ms}^2 - \sqrt{c_{ms}^4 - 4 v_A^2 c_s^2 \cos^2 θ}\right)\]

where \(k\) is the wavenumber, \(v_A\) is the Alfvén speed, \(c_s\) is the ideal sound speed, \(c_{ms} = \sqrt{v_A^2 + c_s^2}\) is the magnetosonic speed, and \(θ\) is the angle between the wavevector and the equilibrium magnetic field.

Examples

>>> import astropy.units as u
>>> from plasmapy.dispersion.analytical import SlowMagnetosonicWave
>>> slow = SlowMagnetosonicWave(1e-3 * u.T, 1e16 * u.m**-3, "p+", T=2.5e6 * u.K)
>>> slow.angular_frequency(1e-5 * u.rad / u.m, 0 * u.deg)
<Quantity 1.85454394 rad / s>
>>> slow.angular_frequency([1e-5, 2e-4] * (u.rad / u.m), 0 * u.deg)
<Quantity [ 1.85454394, 37.09087884] rad / s>
>>> slow.angular_frequency(1e-5 * u.rad / u.m, [0, 45, 90] * u.deg)
<Quantity [1.85454394, 1.07839372, 0.        ] rad / s>
>>> slow.angular_frequency([1e-5, 2e-4] * (u.rad / u.m), [0, 45, 90] * u.deg)
<Quantity [[ 1.85454394,  1.07839372,  0.        ],
           [37.09087884, 21.56787445,  0.        ]] rad / s>

group_velocity( k: Quantity, theta: Quantity, )[source]

Calculate the group velocities of slow magnetosonic waves.

Parameters:

k (Quantity) – Wavenumber in units convertible to rad/m`. Either single valued or 1-D array of length \(N\).
theta (Quantity) – The angle of propagation of the wave with respect to the magnetic field, \(\cos^{-1}(k_z / k)\), in units convertible to radians. Either single valued or 1-D array of size \(M\).

Returns:

group_velocity – An array of group_velocities in units m/s with shape \(2 × N × M\). The first dimension maps to the two coordinate arrays in the direction of k and in the direction of increasing theta, the second dimension maps to the k array, and the third dimension maps to the theta array.

Return type:

Quantity of shape (2, N, M)

Raises:

UnitTypeError – If applicable arguments do not have units convertible to the expected units.
ValueError – If k is negative or zero.
ValueError – If k or theta are not single valued or a 1-D array.

Warns:

PhysicsWarning – When the computed wave frequencies violate the low-frequency (\(ω ≪ ω_c,ω_p\)) assumption of the dispersion relation.

Notes

The group velocity \(\mathbf{v}_g\) is given by

\[\]

mathbf{v}_g = frac{dω}{dmathbf{k}}

= hat{mathbf{k}} v_{ph}

hat{mathbf{θ}} frac{∂ v_{ph}}{∂θ}

where \(ω\) is the angular frequency, \(\mathbf{k}\) is the wavevector, \(θ\) is the angle between \(\mathbf{k}\) and the unperturbed magnetic field, and \(v_{ph}\) is the phase velocity.

phase_velocity( k: Quantity, theta: Quantity, ) → Quantity

Calculate the phase velocities of magnetohydrodynamic waves.

Parameters:

k (Quantity) – Wavenumber in units convertible to rad/m`. Either single valued or 1-D array of size \(N\).
theta (Quantity) – The angle of propagation of the wave with respect to the magnetic field, \(\cos^{-1}(k_z / k)\), in units convertible to radians. Either single valued or 1-D array of size \(M\).

Returns:

phase_velocity – An \(N × M\) array of computed phase velocities in units of m/s.

Return type:

Quantity

Raises:

UnitTypeError – If applicable arguments do not have units convertible to the expected units.
ValueError – If k is negative or zero.
ValueError – If k or theta are not single valued or a 1-D array.

Warns:

PhysicsWarning – When the computed wave frequencies violate the low-frequency (\(ω ≪ ω_c,ω_p\)) assumption of the dispersion relation.