kinetic_alfven

Using the equation provided in Bellan [2012], this function calculates the numerical solution to the kinetic Alfvén dispersion relation presented by Hirose et al. [2004].

Parameters:

B (Quantity) – The magnetic field magnitude in units convertible to T.
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.).
k (Quantity) – Wavenumber in units convertible to rad / m. Either single valued or 1-D array of length \(N\).
n_i (Quantity) – Ion number density in units convertible to m^-3.
T_e (Quantity, keyword-only) – The electron temperature in units of K or eV.
T_i (Quantity, keyword-only) – The ion temperature in units of K or eV.
theta (Quantity) – The angle of propagation of the wave with respect to the magnetic field, \(\cos^{-1}(k_z / k)\), in units convertible to rad. Either single valued or 1-D array of size \(M\).
gamma_e (real number, keyword-only, default: 1) – The adiabatic index for electrons. The default value assumes that the electrons are able to equalize their temperature rapidly enough that the electrons are effectively isothermal.
gamma_i (real number, keyword-only, default: 3) – The adiabatic index for ions. The default value assumes that ion motion has only one degree of freedom, namely along magnetic field lines.
mass_numb (integer, keyword-only, optional) – The mass number corresponding to ion.
Z (real number, keyword-only, optional) – The charge number corresponding to ion.

Returns:

omega – A dictionary of computed wave frequencies in units rad / s. The dictionary contains a key for each: theta value provided. The value for each key will be an \(N × M\) array.

Return type:

Dict[str, Quantity]

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_e, gamma_i, or Z are not real numbers.
UnitTypeError – If applicable arguments do not have units convertible to the expected units.
ValueError – If any of B, k, n_i, T_e, or T_i is negative.
ValueError – If k is negative or zero.
ValueError – If ion is not of category ion or element.
ValueError – If B, n_i, T_e, or T_i are not single valued astropy.units.Quantity (i.e. an array).
ValueError – If k or theta are not single valued or a 1-D array.

Notes

Using the 2 × 2 matrix approach method from Bellan [2012], this function computes the corresponding wave frequencies in units of rad / s. This approach comes from Hasegawa and Uberoi [1982], Morales and Maggs [1997] and Lysak and Lotko [1996]; who argued that a 3 × 3 matrix that describes warm plasma waves can be represented as a 2 × 2 matrix because the compressional (i.e., fast) mode can be factored out. The result is that the determinant, when in the limit of \(ω ≫ k_{z}^{2} c^{2}_{\rm s}\), reduces to the kinetic Alfvén dispersion relation.

\[ω^2 = k_{\rm z}^2 v_{\rm A}^2 \left(1 + \frac{k_{\rm x}^2 c_{\rm s}^2}{ω_{\rm ci}^2} \right)\]

With \(c_{\rm s}\) being the wave speed and \(ω_{\rm ci}\) as the gyrofrequency of the respective ion. The regions in which this is valid are \(ω ≪ ω_{\rm ci}\) and \(ν_{\rm Te} ≫ \frac{ω}{k_{z}} ≫ ν_{\rm Ti}\), with \(ν_{\rm Ti}\) standing for the thermal speed of the respective ion. There is no restriction on propagation angle.

Examples

>>> import numpy as np
>>> import astropy.units as u
>>> from plasmapy.particles import Particle
>>> from plasmapy.dispersion.numerical import kinetic_alfven_
>>> inputs = {
...     "B": 8.3e-9 * u.T,
...     "ion": Particle("p+"),
...     "k": np.logspace(-7, -2, 2) * u.rad / u.m,
...     "n_i": 5 * u.m**-3,
...     "T_e": 1.6e6 * u.K,
...     "T_i": 4.0e5 * u.K,
...     "theta": 30 * u.deg,
...     "gamma_e": 3,
...     "gamma_i": 3,
...     "Z": 1,
... }
>>> kinetic_alfven(**inputs)
{30.0: <Quantity [1.24901116e+00, 3.45301796e+08] rad / s>}