AlfvenWave

class plasmapy.dispersion.analytical.mhd_waves_.AlfvenWave(B: ~typing.Annotated[~astropy.units.quantity.Quantity, Unit("T")], 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: ~typing.Annotated[~astropy.units.quantity.Quantity, Unit("K")] = <Quantity 0. K>, gamma: float = 1.6666666666666667, mass_numb: int | None = None, Z: float | None = None)

Bases: AbstractMHDWave

A class to represent magnetohydrodynamic Alfvén 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, default: 5/3) – The adiabatic index for the plasma.

• 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

FastMagnetosonicWave, SlowMagnetosonicWave

Examples

>>> import astropy.units as u
>>> from plasmapy.dispersion.analytical import AlfvenWave
>>> alfven = AlfvenWave(1e-3 * u.T, 1e16 * u.m**-3, "p+")
>>> alfven.angular_frequency(1e-5 * u.rad / u.m, 0 * u.deg)
<Quantity 2.18060973 rad / s>
>>> alfven.phase_velocity(1e-5 * u.rad / u.m, 0 * u.deg)
<Quantity 218060.97295233 m / s>
>>> alfven.alfven_speed
<Quantity 218060.97295233 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 magnetohydrodynamic Alfvén waves.
group_velocity(k, theta)	Calculate the group velocities of magnetohydrodynamic Alfvén 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: Annotated[Quantity, Unit('rad / m')],

theta: Annotated[Quantity, Unit('rad')],

)

Calculate the angular frequency of magnetohydrodynamic Alfvén 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 magnetohydrodynamic Alfvén wave is given by

\[ω = k v_A \cosθ\]

where \(k\) is the wavenumber, \(v_A\) is the Alfvén 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 AlfvenWave
>>> alfven = AlfvenWave(1e-3 * u.T, 1e16 * u.m**-3, "p+")
>>> alfven.angular_frequency(1e-5 * u.rad / u.m, 0 * u.deg)
<Quantity 2.18060973 rad / s>
>>> alfven.angular_frequency([1e-5, 2e-4] * (u.rad / u.m), 0 * u.deg)
<Quantity [ 2.18060973, 43.61219459] rad / s>
>>> alfven.angular_frequency(1e-5 * u.rad / u.m, [0, 45, 90] * u.deg)
<Quantity [2.18060973e+00, 1.54192393e+00, 1.33523836e-16] rad / s>
>>> alfven.angular_frequency([1e-5, 2e-4] * (u.rad / u.m), [0, 45, 90] * u.deg)
<Quantity [[2.18060973e+00, 1.54192393e+00, 1.33523836e-16],
           [4.36121946e+01, 3.08384785e+01, 2.67047673e-15]] rad / s>

group_velocity(

k: Annotated[Quantity, Unit('rad / m')],

theta: Annotated[Quantity, Unit('rad')],

)

Calculate the group velocities of magnetohydrodynamic Alfvén 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) – 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{B}} v_{A}

where \(\hat{\mathbf{B}}\) is the unit vector in the direction of the unperturbed magnetic field and \(v_A\) is the Alfvén speed.

phase_velocity(

k: Annotated[Quantity, Unit('rad / m')],

theta: Annotated[Quantity, Unit('rad')],

) → Annotated[Quantity, Unit('m / s')]

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.