def test_upper_hybrid_frequency():
r"""Test the upper_hybrid_frequency function in parameters.py."""
omega_uh = upper_hybrid_frequency(B, n_e=n_e)
omega_uh_hz = upper_hybrid_frequency(B, n_e=n_e, to_hz=True)
omega_ce = gyrofrequency(B, "e-")
omega_pe = plasma_frequency(n=n_e, particle="e-")
assert omega_ce.unit.is_equivalent(u.rad / u.s)
assert omega_pe.unit.is_equivalent(u.rad / u.s)
assert omega_uh.unit.is_equivalent(u.rad / u.s)
assert omega_uh_hz.unit.is_equivalent(u.Hz)
left_hand_side = omega_uh**2
right_hand_side = omega_ce**2 + omega_pe**2
assert np.isclose(left_hand_side.value, right_hand_side.value)
assert np.isclose(omega_uh_hz.value, 69385868857.90918)
with pytest.raises(ValueError):
upper_hybrid_frequency(5 * u.T, n_e=-1 * u.m**-3)
with pytest.warns(u.UnitsWarning):
assert upper_hybrid_frequency(1.2, 1.3) == upper_hybrid_frequency(
1.2 * u.T, 1.3 * u.m**-3)
with pytest.warns(u.UnitsWarning):
assert upper_hybrid_frequency(1.4 * u.T,
1.3) == upper_hybrid_frequency(
1.4, 1.3 * u.m**-3)
assert_can_handle_nparray(upper_hybrid_frequency)
def test_lower_hybrid_frequency():
r"""Test the lower_hybrid_frequency function in parameters.py."""
ion = "He-4 1+"
omega_ci = gyrofrequency(B, particle=ion)
omega_pi = plasma_frequency(n=n_i, particle=ion)
omega_ce = gyrofrequency(B)
omega_lh = lower_hybrid_frequency(B, n_i=n_i, ion=ion)
omega_lh_hz = lower_hybrid_frequency(B, n_i=n_i, ion=ion, to_hz=True)
assert omega_ci.unit.is_equivalent(u.rad / u.s)
assert omega_pi.unit.is_equivalent(u.rad / u.s)
assert omega_ce.unit.is_equivalent(u.rad / u.s)
assert omega_lh.unit.is_equivalent(u.rad / u.s)
left_hand_side = omega_lh ** -2
right_hand_side = (
1 / (omega_ci ** 2 + omega_pi ** 2) + omega_ci ** -1 * omega_ce ** -1
)
assert np.isclose(left_hand_side.value, right_hand_side.value)
assert np.isclose(omega_lh_hz.value, 299878691.3223296)
with pytest.raises(ValueError):
lower_hybrid_frequency(0.2 * u.T, n_i=5e19 * u.m ** -3, ion="asdfasd")
with pytest.raises(ValueError):
lower_hybrid_frequency(0.2 * u.T, n_i=-5e19 * u.m ** -3, ion="asdfasd")
with pytest.raises(ValueError):
lower_hybrid_frequency(np.nan * u.T, n_i=-5e19 * u.m ** -3, ion="asdfasd")
with pytest.warns(u.UnitsWarning):
assert lower_hybrid_frequency(1.3, 1e19) == lower_hybrid_frequency(
1.3 * u.T, 1e19 * u.m ** -3
)
assert_can_handle_nparray(lower_hybrid_frequency)
def spectral_density(
wavelengths: u.nm,
probe_wavelength: u.nm,
n: u.m**-3,
Te: u.K,
Ti: u.K,
efract: np.ndarray = None,
ifract: np.ndarray = None,
ion_species: Union[str, List[str], Particle, List[Particle]] = "H+",
electron_vel: u.m / u.s = None,
ion_vel: u.m / u.s = None,
probe_vec=np.array([1, 0, 0]),
scatter_vec=np.array([0, 1, 0]),
) -> Tuple[Union[np.floating, np.ndarray], np.ndarray]:
r"""
Calculate the spectral density function for Thomson scattering of a
probe laser beam by a multi-species Maxwellian plasma.
This function calculates the spectral density function for Thomson
scattering of a probe laser beam by a plasma consisting of one or more ion
species and a one or more thermal electron populations (the entire plasma
is assumed to be quasi-neutral)
.. math::
S(k,\omega) = \sum_e \frac{2\pi}{k}
\bigg |1 - \frac{\chi_e}{\epsilon} \bigg |^2
f_{e0,e} \bigg (\frac{\omega}{k} \bigg ) +
\sum_i \frac{2\pi Z_i}{k}
\bigg |\frac{\chi_e}{\epsilon} \bigg |^2 f_{i0,i}
\bigg ( \frac{\omega}{k} \bigg )
where :math:`\chi_e` is the electron component susceptibility of the
plasma and :math:`\epsilon = 1 + \sum_e \chi_e + \sum_i \chi_i` is the total
plasma dielectric function (with :math:`\chi_i` being the ion component
of the susceptibility), :math:`Z_i` is the charge of each ion, :math:`k`
is the scattering wavenumber, :math:`\omega` is the scattering frequency,
and :math:`f_{e0,e}` and :math:`f_{i0,i}` are the electron and ion velocity
distribution functions respectively. In this function the electron and ion
velocity distribution functions are assumed to be Maxwellian, making this
function equivalent to Eq. 3.4.6 in `Sheffield`_.
Parameters
----------
wavelengths : `~astropy.units.Quantity`
Array of wavelengths over which the spectral density function
will be calculated. (convertible to nm)
probe_wavelength : `~astropy.units.Quantity`
Wavelength of the probe laser. (convertible to nm)
n : `~astropy.units.Quantity`
Mean (0th order) density of all plasma components combined.
(convertible to cm^-3.)
Te : `~astropy.units.Quantity`, shape (Ne, )
Temperature of each electron component. Shape (Ne, ) must be equal to the
number of electron components Ne. (in K or convertible to eV)
Ti : `~astropy.units.Quantity`, shape (Ni, )
Temperature of each ion component. Shape (Ni, ) must be equal to the
number of ion components Ni. (in K or convertible to eV)
efract : array_like, shape (Ne, ), optional
An array-like object where each element represents the fraction (or ratio)
of the electron component number density to the total electron number density.
Must sum to 1.0. Default is a single electron component.
ifract : array_like, shape (Ni, ), optional
An array-like object where each element represents the fraction (or ratio)
of the ion component number density to the total ion number density.
Must sum to 1.0. Default is a single ion species.
ion_species : str or `~plasmapy.particles.Particle`, shape (Ni, ), optional
A list or single instance of `~plasmapy.particles.Particle`, or strings
convertible to `~plasmapy.particles.Particle`. Default is `'H+'`
corresponding to a single species of hydrogen ions.
electron_vel : `~astropy.units.Quantity`, shape (Ne, 3), optional
Velocity of each electron component in the rest frame. (convertible to m/s)
Defaults to a stationary plasma [0, 0, 0] m/s.
ion_vel : `~astropy.units.Quantity`, shape (Ni, 3), optional
Velocity vectors for each electron population in the rest frame
(convertible to m/s) Defaults zero drift
for all specified ion species.
probe_vec : float `~numpy.ndarray`, shape (3, )
Unit vector in the direction of the probe laser. Defaults to
[1, 0, 0].
scatter_vec : float `~numpy.ndarray`, shape (3, )
Unit vector pointing from the scattering volume to the detector.
Defaults to [0, 1, 0] which, along with the default `probe_vec`,
corresponds to a 90 degree scattering angle geometry.
Returns
-------
alpha : float
Mean scattering parameter, where `alpha` > 1 corresponds to collective
scattering and `alpha` < 1 indicates non-collective scattering. The
scattering parameter is calculated based on the total plasma density n.
Skw : `~astropy.units.Quantity`
Computed spectral density function over the input `wavelengths` array
with units of s/rad.
Notes
-----
For details, see "Plasma Scattering of Electromagnetic Radiation" by
Sheffield et al. `ISBN 978\\-0123748775`_. This code is a modified version
of the program described therein.
For a concise summary of the relevant physics, see Chapter 5 of Derek
Schaeffer's thesis, DOI: `10.5281/zenodo.3766933`_.
.. _`ISBN 978\\-0123748775`: https://www.sciencedirect.com/book/9780123748775/plasma-scattering-of-electromagnetic-radiation
.. _`10.5281/zenodo.3766933`: https://doi.org/10.5281/zenodo.3766933
.. _`Sheffield`: https://doi.org/10.1016/B978-0-12-374877-5.00003-8
"""
if efract is None:
efract = np.ones(1)
else:
efract = np.asarray(efract, dtype=np.float64)
if ifract is None:
ifract = np.ones(1)
else:
ifract = np.asarray(ifract, dtype=np.float64)
# If electon velocity is not specified, create an array corresponding
# to zero drift
if electron_vel is None:
electron_vel = np.zeros([efract.size, 3]) * u.m / u.s
# If ion drift velocity is not specified, create an array corresponding
# to zero drift
if ion_vel is None:
ion_vel = np.zeros([ifract.size, 3]) * u.m / u.s
# Condition ion_species
if isinstance(ion_species, (str, Particle)):
ion_species = [ion_species]
if len(ion_species) == 0:
raise ValueError("At least one ion species needs to be defined.")
for ii, ion in enumerate(ion_species):
if isinstance(ion, Particle):
continue
ion_species[ii] = Particle(ion)
# Condition Te
if Te.size == 1:
# If a single quantity is given, put it in an array so it's iterable
# If Te.size != len(efract), assume same temp. for all species
Te = np.repeat(Te, len(efract))
elif Te.size != len(efract):
raise ValueError(f"Got {Te.size} electron temperatures and expected "
f"{len(efract)}.")
# Condition Ti
if Ti.size == 1:
# If a single quantity is given, put it in an array so it's iterable
# If Ti.size != len(ion_species), assume same temp. for all species
Ti = [Ti.value] * len(ion_species) * Ti.unit
elif Ti.size != len(ion_species):
raise ValueError(f"Got {Ti.size} ion temperatures and expected "
f"{len(ion_species)}.")
# Make sure the sizes of ion_species, ifract, ion_vel, and Ti all match
if ((len(ion_species) != ifract.size) or (ion_vel.shape[0] != ifract.size)
or (Ti.size != ifract.size)):
raise ValueError(
f"Inconsistent number of species in ifract ({ifract}), "
f"ion_species ({len(ion_species)}), Ti ({Ti.size}), "
f"and/or ion_vel ({ion_vel.shape[0]}).")
# Make sure the sizes of efract, electron_vel, and Te all match
if (electron_vel.shape[0] != efract.size) or (Te.size != efract.size):
raise ValueError(
f"Inconsistent number of electron populations in efract ({efract.size}), "
f"Te ({Te.size}), or electron velocity ({electron_vel.shape[0]}).")
# Ensure unit vectors are normalized
probe_vec = probe_vec / np.linalg.norm(probe_vec)
scatter_vec = scatter_vec / np.linalg.norm(scatter_vec)
# Define some constants
C = const.c.si # speed of light
# Calculate plasma parameters
vTe = thermal_speed(Te, particle="e-")
vTi, ion_z = [], []
for T, ion in zip(Ti, ion_species):
vTi.append(thermal_speed(T, particle=ion).value)
ion_z.append(ion.integer_charge * u.dimensionless_unscaled)
vTi = vTi * vTe.unit
zbar = np.sum(ifract * ion_z)
ne = efract * n
ni = ifract * n / zbar # ne/zbar = sum(ni)
# wpe is calculated for the entire plasma (all electron populations combined)
wpe = plasma_frequency(n=n, particle="e-")
# Convert wavelengths to angular frequencies (electromagnetic waves, so
# phase speed is c)
ws = (2 * np.pi * u.rad * C / wavelengths).to(u.rad / u.s)
wl = (2 * np.pi * u.rad * C / probe_wavelength).to(u.rad / u.s)
# Compute the frequency shift (required by energy conservation)
w = ws - wl
# Compute the wavenumbers in the plasma
# See Sheffield Sec. 1.8.1 and Eqs. 5.4.1 and 5.4.2
ks = np.sqrt(ws**2 - wpe**2) / C
kl = np.sqrt(wl**2 - wpe**2) / C
# Compute the wavenumber shift (required by momentum conservation)
scattering_angle = np.arccos(np.dot(probe_vec, scatter_vec))
# Eq. 1.7.10 in Sheffield
k = np.sqrt(ks**2 + kl**2 - 2 * ks * kl * np.cos(scattering_angle))
# Normal vector along k
k_vec = (scatter_vec - probe_vec) * u.dimensionless_unscaled
# Compute Doppler-shifted frequencies for both the ions and electrons
# Matmul is simultaneously conducting dot product over all wavelengths
# and ion components
w_e = w - np.matmul(electron_vel, np.outer(k, k_vec).T)
w_i = w - np.matmul(ion_vel, np.outer(k, k_vec).T)
# Compute the scattering parameter alpha
# expressed here using the fact that v_th/w_p = root(2) * Debye length
alpha = np.sqrt(2) * wpe / np.outer(k, vTe)
# Calculate the normalized phase velocities (Sec. 3.4.2 in Sheffield)
xe = (np.outer(1 / vTe, 1 / k) * w_e).to(u.dimensionless_unscaled)
xi = (np.outer(1 / vTi, 1 / k) * w_i).to(u.dimensionless_unscaled)
# Calculate the susceptibilities
chiE = np.zeros([efract.size, w.size], dtype=np.complex128)
for i, fract in enumerate(efract):
chiE[i, :] = permittivity_1D_Maxwellian(w_e[i, :], k, Te[i], ne[i],
"e-")
# Treatment of multiple species is an extension of the discussion in
# Sheffield Sec. 5.1
chiI = np.zeros([ifract.size, w.size], dtype=np.complex128)
for i, ion in enumerate(ion_species):
chiI[i, :] = permittivity_1D_Maxwellian(w_i[i, :],
k,
Ti[i],
ni[i],
ion,
z_mean=ion_z[i])
# Calculate the longitudinal dielectric function
epsilon = 1 + np.sum(chiE, axis=0) + np.sum(chiI, axis=0)
econtr = np.zeros([efract.size, w.size], dtype=np.complex128) * u.s / u.rad
for m in range(efract.size):
econtr[m, :] = efract[m] * (2 * np.sqrt(np.pi) / k / vTe[m] * np.power(
np.abs(1 - np.sum(chiE, axis=0) / epsilon), 2) *
np.exp(-xe[m, :]**2))
icontr = np.zeros([ifract.size, w.size], dtype=np.complex128) * u.s / u.rad
for m in range(ifract.size):
icontr[m, :] = ifract[m] * (
2 * np.sqrt(np.pi) * ion_z[m] / k / vTi[m] *
np.power(np.abs(np.sum(chiE, axis=0) / epsilon), 2) *
np.exp(-xi[m, :]**2))
# Recast as real: imaginary part is already zero
Skw = np.real(np.sum(econtr, axis=0) + np.sum(icontr, axis=0))
return np.mean(alpha), Skw
def test_plasma_frequency():
r"""Test the plasma_frequency function in parameters.py."""
assert plasma_frequency(n_e, "e-").unit.is_equivalent(u.rad / u.s)
assert plasma_frequency(n_e, "e-", to_hz=True).unit.is_equivalent(u.Hz)
assert np.isclose(plasma_frequency(1 * u.cm**-3, "e-").value,
5.64e4,
rtol=1e-2)
assert np.isclose(plasma_frequency(1 * u.cm**-3, particle="N").value,
3.53e2,
rtol=1e-1)
assert np.isclose(
plasma_frequency(1 * u.cm**-3, particle="N", to_hz=True).value,
56.19000195094519,
)
with pytest.raises(TypeError):
plasma_frequency(u.m**-3, "e-")
with pytest.raises(u.UnitTypeError):
plasma_frequency(5 * u.m**-2, "e-")
assert np.isnan(plasma_frequency(np.nan * u.m**-3, "e-"))
with pytest.warns(u.UnitsWarning):
assert plasma_frequency(1e19, "e-") == plasma_frequency(
1e19 * u.m**-3, "e-")
assert plasma_frequency(n_i,
particle="p").unit.is_equivalent(u.rad / u.s)
# Case where Z=1 is assumed
assert plasma_frequency(n_i,
particle="H-1+") == plasma_frequency(n_i,
particle="p")
assert np.isclose(plasma_frequency(mu * u.cm**-3, particle="p").value,
1.32e3,
rtol=1e-2)
with pytest.raises(ValueError):
plasma_frequency(n=5 * u.m**-3, particle="sdfas")
with pytest.warns(u.UnitsWarning):
plasma_freq_no_units = plasma_frequency(1e19, particle="p")
assert plasma_freq_no_units == plasma_frequency(1e19 * u.m**-3,
particle="p")
plasma_frequency(1e17 * u.cm**-3, particle="p")
# testing for user input z_mean
testMeth1 = plasma_frequency(1e17 * u.cm**-3, particle="p",
z_mean=0.8).si.value
testTrue1 = 333063562455.4028
errStr = f"plasma_frequency() gave {testMeth1}, should be {testTrue1}."
assert np.isclose(testMeth1, testTrue1, atol=0.0, rtol=1e-6), errStr
assert_can_handle_nparray(plasma_frequency)
def cold_plasma_permittivity_SDP(B: u.T, species, n, omega: u.rad / u.s):
r"""
Magnetized Cold Plasma Dielectric Permittivity Tensor Elements.
Elements (S, D, P) are given in the "Stix" frame, ie. with B // z.
The :math:`\exp(-i \omega t)` time-harmonic convention is assumed.
Parameters
----------
B : ~astropy.units.Quantity
Magnetic field magnitude in units convertible to tesla.
species : list of str
List of the plasma particle species
e.g.: ['e', 'D+'] or ['e', 'D+', 'He+'].
n : list of ~astropy.units.Quantity
`list` of species density in units convertible to per cubic meter
The order of the species densities should follow species.
omega : ~astropy.units.Quantity
Electromagnetic wave frequency in rad/s.
Returns
-------
sum : ~astropy.units.Quantity
S ("Sum") dielectric tensor element.
difference : ~astropy.units.Quantity
D ("Difference") dielectric tensor element.
plasma : ~astropy.units.Quantity
P ("Plasma") dielectric tensor element.
Notes
-----
The dielectric permittivity tensor is expressed in the Stix frame with
the :math:`\exp(-i \omega t)` time-harmonic convention as
:math:`\varepsilon = \varepsilon_0 A`, with :math:`A` being
.. math::
\varepsilon = \varepsilon_0 \left(\begin{matrix} S & -i D & 0 \\
+i D & S & 0 \\
0 & 0 & P \end{matrix}\right)
where:
.. math::
S = 1 - \sum_s \frac{\omega_{p,s}^2}{\omega^2 - \Omega_{c,s}^2}
D = \sum_s \frac{\Omega_{c,s}}{\omega}
\frac{\omega_{p,s}^2}{\omega^2 - \Omega_{c,s}^2}
P = 1 - \sum_s \frac{\omega_{p,s}^2}{\omega^2}
where :math:`\omega_{p,s}` is the plasma frequency and
:math:`\Omega_{c,s}` is the signed version of the cyclotron frequency
for the species :math:`s`.
References
----------
- T.H. Stix, Waves in Plasma, 1992.
Examples
--------
>>> from astropy import units as u
>>> from numpy import pi
>>> B = 2*u.T
>>> species = ['e', 'D+']
>>> n = [1e18*u.m**-3, 1e18*u.m**-3]
>>> omega = 3.7e9*(2*pi)*(u.rad/u.s)
>>> permittivity = S, D, P = cold_plasma_permittivity_SDP(B, species, n, omega)
>>> S
<Quantity 1.02422...>
>>> permittivity.sum # namedtuple-style access
<Quantity 1.02422...>
>>> D
<Quantity 0.39089...>
>>> P
<Quantity -4.8903...>
"""
S, D, P = 1, 0, 1
for s, n_s in zip(species, n):
omega_c = parameters.gyrofrequency(B=B, particle=s, signed=True)
omega_p = parameters.plasma_frequency(n=n_s, particle=s)
S += -(omega_p ** 2) / (omega ** 2 - omega_c ** 2)
D += omega_c / omega * omega_p ** 2 / (omega ** 2 - omega_c ** 2)
P += -(omega_p ** 2) / omega ** 2
return StixTensorElements(S, D, P)
def permittivity_1D_Maxwellian(
omega: u.rad / u.s,
kWave: u.rad / u.m,
T: u.K,
n: u.m ** -3,
particle,
z_mean: u.dimensionless_unscaled = None,
) -> u.dimensionless_unscaled:
r"""
The classical dielectric permittivity for a 1D Maxwellian plasma. This
function can calculate both the ion and electron permittivities. No
additional effects are considered (e.g. magnetic fields, relativistic
effects, strongly coupled regime, etc.)
Parameters
----------
omega : ~astropy.units.Quantity
The frequency in rad/s of the electromagnetic wave propagating
through the plasma.
kWave : ~astropy.units.Quantity
The corresponding wavenumber, in rad/m, of the electromagnetic wave
propagating through the plasma. This is often modulated by the
dispersion of the plasma or by relativistic effects. See em_wave.py
for ways to calculate this.
T : ~astropy.units.Quantity
The plasma temperature - this can be either the electron or the ion
temperature, but should be consistent with density and particle.
n : ~astropy.units.Quantity
The plasma density - this can be either the electron or the ion
density, but should be consistent with temperature and particle.
particle : str
The plasma particle species.
z_mean : str
The average ionization of the plasma. This is only required for
calculating the ion permittivity.
Returns
-------
chi : ~astropy.units.Quantity
The ion or the electron dielectric permittivity of the plasma.
This is a dimensionless quantity.
Notes
-----
The dielectric permittivities for a Maxwellian plasma are described
by the following equations [1]_
.. math::
\chi_e(k, \omega) = - \frac{\alpha_e^2}{2} Z'(x_e)
\chi_i(k, \omega) = - \frac{\alpha_i^2}{2}\frac{Z}{} Z'(x_i)
\alpha = \frac{\omega_p}{k v_{Th}}
x = \frac{\omega}{k v_{Th}}
:math:`chi_e` and :math:`chi_i` are the electron and ion permittivities
respectively. :math:`Z'` is the derivative of the plasma dispersion
function. :math:`\alpha` is the scattering parameter which delineates
the difference between the collective and non-collective Thomson
scattering regimes. :math:`x` is the dimensionless phase velocity
of the EM wave propagating through the plasma.
References
----------
.. [1] J. Sheffield, D. Froula, S. H. Glenzer, and N. C. Luhmann Jr,
Plasma scattering of electromagnetic radiation: theory and measurement
techniques. Chapter 5 Pg 106 (Academic press, 2010).
Example
-------
>>> from astropy import units as u
>>> from numpy import pi
>>> from astropy.constants import c
>>> T = 30 * 11600 * u.K
>>> n = 1e18 * u.cm**-3
>>> particle = 'Ne'
>>> z_mean = 8 * u.dimensionless_unscaled
>>> vTh = parameters.thermal_speed(T, particle, method="most_probable")
>>> omega = 5.635e14 * 2 * pi * u.rad / u.s
>>> kWave = omega / vTh
>>> permittivity_1D_Maxwellian(omega, kWave, T, n, particle, z_mean)
<Quantity -6.72809...e-08+5.76037...e-07j>
"""
# thermal velocity
vTh = parameters.thermal_speed(T=T, particle=particle, method="most_probable")
# plasma frequency
wp = parameters.plasma_frequency(n=n, particle=particle, z_mean=z_mean)
# scattering parameter alpha.
# explicitly removing factor of sqrt(2) to be consistent with Froula
alpha = np.sqrt(2) * (wp / (kWave * vTh)).to(u.dimensionless_unscaled)
# The dimensionless phase velocity of the propagating EM wave.
zeta = (omega / (kWave * vTh)).to(u.dimensionless_unscaled)
chi = alpha ** 2 * (-1 / 2) * plasma_dispersion_func_deriv(zeta.value)
return chi
def cold_plasma_permittivity_LRP(B: u.T, species, n, omega: u.rad / u.s):
r"""
Magnetized Cold Plasma Dielectric Permittivity Tensor Elements.
Elements (L, R, P) are given in the "rotating" basis, ie. in the basis
:math:`(\mathbf{u}_{+}, \mathbf{u}_{-}, \mathbf{u}_z)`,
where the tensor is diagonal and with B // z.
The :math:`\exp(-i \omega t)` time-harmonic convention is assumed.
Parameters
----------
B : ~astropy.units.Quantity
Magnetic field magnitude in units convertible to tesla.
species : list of str
The plasma particle species (e.g.: `['e', 'D+']` or
`['e', 'D+', 'He+']`.
n : list of ~astropy.units.Quantity
`list` of species density in units convertible to per cubic meter.
The order of the species densities should follow species.
omega : ~astropy.units.Quantity
Electromagnetic wave frequency in rad/s.
Returns
-------
left : ~astropy.units.Quantity
L ("Left") Left-handed circularly polarization tensor element.
right : ~astropy.units.Quantity
R ("Right") Right-handed circularly polarization tensor element.
plasma : ~astropy.units.Quantity
P ("Plasma") dielectric tensor element.
Notes
-----
In the rotating frame defined by
:math:`(\mathbf{u}_{+}, \mathbf{u}_{-}, \mathbf{u}_z)`
with :math:`\mathbf{u}_{\pm}=(\mathbf{u}_x \pm \mathbf{u}_y)/\sqrt{2}`,
the dielectric tensor takes a diagonal form with elements L, R, P with:
.. math::
L = 1 - \sum_s
\frac{\omega_{p,s}^2}{\omega\left(\omega - \Omega_{c,s}\right)}
R = 1 - \sum_s
\frac{\omega_{p,s}^2}{\omega\left(\omega + \Omega_{c,s}\right)}
P = 1 - \sum_s \frac{\omega_{p,s}^2}{\omega^2}
where :math:`\omega_{p,s}` is the plasma frequency and
:math:`\Omega_{c,s}` is the signed version of the cyclotron frequency
for the species :math:`s`.
References
----------
- T.H. Stix, Waves in Plasma, 1992.
Examples
--------
>>> from astropy import units as u
>>> from numpy import pi
>>> B = 2*u.T
>>> species = ['e', 'D+']
>>> n = [1e18*u.m**-3, 1e18*u.m**-3]
>>> omega = 3.7e9*(2*pi)*(u.rad/u.s)
>>> L, R, P = permittivity = cold_plasma_permittivity_LRP(B, species, n, omega)
>>> L
<Quantity 0.63333...>
>>> permittivity.left # namedtuple-style access
<Quantity 0.63333...>
>>> R
<Quantity 1.41512...>
>>> P
<Quantity -4.8903...>
"""
L, R, P = 1, 1, 1
for s, n_s in zip(species, n):
omega_c = parameters.gyrofrequency(B=B, particle=s, signed=True)
omega_p = parameters.plasma_frequency(n=n_s, particle=s)
L += -(omega_p ** 2) / (omega * (omega - omega_c))
R += -(omega_p ** 2) / (omega * (omega + omega_c))
P += -(omega_p ** 2) / omega ** 2
return RotatingTensorElements(L, R, P)
def thermal_bremsstrahlung(
frequencies: u.Hz,
n_e: u.m ** -3,
T_e: u.K,
n_i: u.m ** -3 = None,
ion_species: Particle = "H+",
kmax: u.m = None,
) -> np.ndarray:
r"""
Calculate the bremsstrahlung emission spectrum for a Maxwellian plasma
in the Rayleigh-Jeans limit :math:`ℏ ω ≪ k_B T_e`
.. math::
\frac{dP}{dω} = \frac{8 \sqrt{2}}{3\sqrt{π}}
\bigg ( \frac{e^2}{4 π ε_0} \bigg )^3
\bigg ( m_e c^2 \bigg )^{-\frac{3}{2}}
\bigg ( 1 - \frac{ω_{pe}^2}{ω^2} \bigg )^\frac{1}{2}
\frac{Z_i^2 n_i n_e}{\sqrt(k_B T_e)}
E_1(y)
where :math:`E_1` is the exponential integral
.. math::
E_1 (y) = - \int_{-y}^∞ \frac{e^{-t}}{t}dt
and :math:`y` is the dimensionless argument
.. math::
y = \frac{1}{2} \frac{ω^2 m_e}{k_{max}^2 k_B T_e}
where :math:`k_{max}` is a maximum wavenumber approximated here as
:math:`k_{max} = 1/λ_B` where :math:`λ_B` is the electron
de Broglie wavelength.
Parameters
----------
frequencies : `~astropy.units.Quantity`
Array of frequencies over which the bremsstrahlung spectrum will be
calculated (convertible to Hz).
n_e : `~astropy.units.Quantity`
Electron number density in the plasma (convertible to m\ :sup:`-3`\ ).
T_e : `~astropy.units.Quantity`
Temperature of the electrons (in K or convertible to eV).
n_i : `~astropy.units.Quantity`, optional
Ion number density in the plasma (convertible to m\ :sup:`-3`\ ). Defaults
to the quasi-neutral condition :math:`n_i = n_e / Z`\ .
ion : `str` or `~plasmapy.particles.Particle`, optional
An instance of `~plasmapy.particles.Particle`, or a string
convertible to `~plasmapy.particles.Particle`.
kmax : `~astropy.units.Quantity`
Cutoff wavenumber (convertible to radians per meter). Defaults
to the inverse of the electron de Broglie wavelength.
Returns
-------
spectrum : `~astropy.units.Quantity`
Computed bremsstrahlung spectrum over the frequencies provided.
Notes
-----
For details, see "Radiation Processes in Plasmas" by
Bekefi. `ISBN 978\\-0471063506`_.
.. _`ISBN 978\\-0471063506`: https://ui.adsabs.harvard.edu/abs/1966rpp..book.....B/abstract
"""
# Default n_i is n_e/Z:
if n_i is None:
n_i = n_e / ion_species.integer_charge
# Default value of kmax is the electrom thermal de Broglie wavelength
if kmax is None:
kmax = (np.sqrt(const.m_e.si * const.k_B.si * T_e) / const.hbar.si).to(1 / u.m)
# Convert frequencies to angular frequencies
ω = (frequencies * 2 * np.pi * u.rad).to(u.rad / u.s)
# Calculate the electron plasma frequency
ω_pe = plasma_frequency(n=n_e, particle="e-")
# Check that all ω < wpe (this formula is only valid in this limit)
if np.min(ω) < ω_pe:
raise PhysicsError(
"Lowest frequency must be larger than the electron "
f"plasma frequency {ω_pe:.1e}, but min(ω) = {np.min(ω):.1e}"
)
# Check that the parameters given fall within the Rayleigh-Jeans limit
# hω << kT_e
rj_const = (
np.max(ω) * const.hbar.si / (2 * np.pi * u.rad * const.k_B.si * T_e)
).to(u.dimensionless_unscaled)
if rj_const.value > 0.1:
raise PhysicsError(
"Rayleigh-Jeans limit not satisfied: "
"hbar*ω/kT_e = {rj_const.value:.2e} > 0.1. "
"Try lower ω or higher T_e."
)
# Calculate the bremsstrahlung power spectral density in several steps
c1 = (
(8 / 3)
* np.sqrt(2 / np.pi)
* (const.e.si ** 2 / (4 * np.pi * const.eps0.si)) ** 3
* 1
/ (const.m_e.si * const.c.si ** 2) ** 1.5
)
Zi = ion_species.integer_charge
c2 = (
np.sqrt(1 - ω_pe ** 2 / ω ** 2)
* Zi ** 2
* n_i
* n_e
/ np.sqrt(const.k_B.si * T_e)
)
# Dimensionless argument for exponential integral
arg = 0.5 * ω ** 2 * const.m_e.si / (kmax ** 2 * const.k_B.si * T_e) / u.rad ** 2
# Remove units, get ndarray of values
arg = (arg.to(u.dimensionless_unscaled)).value
return c1 * c2 * exp1(arg)
def time_plasma_frequency(self):
plasma_frequency(1e19 * u.m**-3, particle='p', to_hz=True)
def hollweg_dispersion_solution(
*,
B: u.T,
ion: Union[str, Particle],
k: u.rad / u.m,
n_i: u.m**-3,
T_e: u.K,
T_i: u.K,
theta: u.deg,
gamma_e: Union[float, int] = 1,
gamma_i: Union[float, int] = 3,
z_mean: Union[float, int] = None,
):
# validate argument ion
if not isinstance(ion, Particle):
try:
ion = Particle(ion)
except TypeError:
raise TypeError(
f"For argument 'ion' expected type {Particle} but got {type(ion)}."
)
if not (ion.is_ion or ion.is_category("element")):
raise ValueError(
"The particle passed for 'ion' must be an ion or element.")
# validate z_mean
if z_mean is None:
try:
z_mean = abs(ion.integer_charge)
except ChargeError:
z_mean = 1
else:
if not isinstance(z_mean, (int, np.integer, float, np.floating)):
raise TypeError(
f"Expected int or float for argument 'z_mean', but got {type(z_mean)}."
)
z_mean = abs(z_mean)
# validate arguments
for arg_name in ("B", "n_i", "T_e", "T_i"):
val = locals()[arg_name].squeeze()
if val.shape != ():
raise ValueError(
f"Argument '{arg_name}' must a single value and not an array of "
f"shape {val.shape}.")
locals()[arg_name] = val
# validate arguments
for arg_name in ("gamma_e", "gamma_i"):
if not isinstance(locals()[arg_name],
(int, np.integer, float, np.floating)):
raise TypeError(
f"Expected int or float for argument '{arg_name}', but got "
f"{type(locals()[arg_name])}.")
# validate argument k
k = k.squeeze()
if not (k.ndim == 0 or k.ndim == 1):
raise ValueError(
f"Argument 'k' needs to be a single valued or 1D array astropy Quantity,"
f" got array of shape {k.shape}.")
if np.any(k <= 0):
raise ValueError("Argument 'k' can not be a or have negative values.")
# validate argument theta
theta = theta.squeeze()
theta = theta.to(u.radian)
if not (theta.ndim == 0 or theta.ndim == 1):
raise ValueError(
f"Argument 'theta' needs to be a single valued or 1D array astropy "
f"Quantity, got array of shape {k.shape}.")
# Calc needed plasma parameters
n_e = z_mean * n_i
c_s = pfp.ion_sound_speed(
T_e=T_e,
T_i=T_i,
ion=ion,
n_e=n_e,
gamma_e=gamma_e,
gamma_i=gamma_i,
z_mean=z_mean,
)
v_A = pfp.Alfven_speed(B, n_i, ion=ion, z_mean=z_mean)
omega_ci = pfp.gyrofrequency(B=B, particle=ion, signed=False, Z=z_mean)
omega_pe = pfp.plasma_frequency(n=n_e, particle="e-")
# Parameters kx and kz
kz = np.cos(theta.value) * k
kx = np.sqrt(k**2 - kz**2)
# Bellan2012JGR beta param equation 3
beta = (c_s / v_A)**2
# Parameters D, F, sigma, and alpha to simplify equation 3
D = (c_s / omega_ci)**2
F = (c / omega_pe)**2
sigma = (kz * v_A)**2
alpha = (k * v_A)**2
# Polynomial coefficients: c3*x^3 + c2*x^2 + c1*x + c0 = 0
c3 = (F * kx**2 + 1) / sigma
c2 = -((alpha / sigma) * (1 + beta + F * kx**2) + D * kx**2 + 1)
c1 = alpha * (1 + 2 * beta + D * kx**2)
c0 = -beta * alpha * sigma
omega = {}
fast_mode = []
alfven_mode = []
acoustic_mode = []
# If a single k value is given
if np.isscalar(k.value) == True:
w = np.emath.sqrt(np.roots([c3.value, c2.value, c1.value, c0.value]))
fast_mode = np.max(w)
alfven_mode = np.median(w)
acoustic_mode = np.min(w)
# If mutliple k values are given
else:
# a0*x^3 + a1*x^2 + a2*x^3 + a3 = 0
for (a0, a1, a2, a3) in zip(c3, c2, c1, c0):
w = np.emath.sqrt(
np.roots([a0.value, a1.value, a2.value, a3.value]))
fast_mode.append(np.max(w))
alfven_mode.append(np.median(w))
acoustic_mode.append(np.min(w))
omega['fast_mode'] = fast_mode * u.rad / u.s
omega['alfven_mode'] = alfven_mode * u.rad / u.s
omega['acoustic_mode'] = acoustic_mode * u.rad / u.s
return omega
def hirose_dispersion_solution(
*,
B: u.T,
ion: Union[str, Particle],
k: u.rad / u.m,
n_i: u.m ** -3,
T_e: u.K,
T_i: u.K,
theta: u.deg,
gamma_e: Union[float, int] = 1,
gamma_i: Union[float, int] = 3,
z_mean: Union[float, int] = None,
):
# validate argument ion
if not isinstance(ion, Particle):
try:
ion = Particle(ion)
except TypeError:
raise TypeError(
f"For argument 'ion' expected type {Particle} but got {type(ion)}."
)
if not (ion.is_ion or ion.is_category("element")):
raise ValueError("The particle passed for 'ion' must be an ion or element.")
# validate z_mean
if z_mean is None:
try:
z_mean = abs(ion.integer_charge)
except ChargeError:
z_mean = 1
else:
if not isinstance(z_mean, (int, np.integer, float, np.floating)):
raise TypeError(
f"Expected int or float for argument 'z_mean', but got {type(z_mean)}."
)
z_mean = abs(z_mean)
# validate arguments
for arg_name in ("B", "n_i", "T_e", "T_i"):
val = locals()[arg_name].squeeze()
if val.shape != ():
raise ValueError(
f"Argument '{arg_name}' must a single value and not an array of "
f"shape {val.shape}."
)
locals()[arg_name] = val
# validate arguments
for arg_name in ("gamma_e", "gamma_i"):
if not isinstance(locals()[arg_name], (int, np.integer, float, np.floating)):
raise TypeError(
f"Expected int or float for argument '{arg_name}', but got "
f"{type(locals()[arg_name])}."
)
# validate argument k
k = k.squeeze()
if not (k.ndim == 0 or k.ndim == 1):
raise ValueError(
f"Argument 'k' needs to be a single valued or 1D array astropy Quantity,"
f" got array of shape {k.shape}."
)
if np.any(k <= 0):
raise ValueError("Argument 'k' can not be a or have negative values.")
# validate argument theta
theta = theta.squeeze()
theta = theta.to(u.radian)
if not (theta.ndim == 0 or theta.ndim == 1):
raise ValueError(
f"Argument 'theta' needs to be a single valued or 1D array astropy "
f"Quantity, got array of shape {k.shape}."
)
n_e = z_mean * n_i
c_s = pfp.ion_sound_speed(
T_e=T_e,
T_i=T_i,
ion=ion,
n_e=n_e,
gamma_e=gamma_e,
gamma_i=gamma_i,
z_mean=z_mean,
)
v_A = pfp.Alfven_speed(B, n_i, ion=ion, z_mean=z_mean)
omega_pi = pfp.plasma_frequency(n=n_i, particle=ion)
#Grid/vector creation for k?
#Parameters kz
kz = np.cos(theta.value) * k
#Parameters sigma, D, and F to simplify equation 3
A = (kz * v_A) ** 2
B = (k * c_s) ** 2
C = (k * v_A) ** 2
D = ((k * c) / omega_pi ) ** 2
#Polynomial coefficients where x in 'cx' represents the order of the term
c3 = 1
c2 = A * (1 + D) + B + C
c1 = A * (2 * B + C + B * D)
c0 = -B * A ** 2
[L1, L2, L3] = np.roots([c3, c2.value, c1.value, c0.value])
[omega1, omega2, omega3] = [np.emath.sqrt(L1), np.emath.sqrt(L2), np.emath.sqrt(L3)]
return omega1, omega2, omega3
def two_fluid_dispersion_solution(
*,
B: u.T,
ion: Union[str, Particle],
k: u.rad / u.m,
n_i: u.m**-3,
T_e: u.K,
T_i: u.K,
theta: u.deg,
gamma_e: Union[float, int] = 1,
gamma_i: Union[float, int] = 3,
z_mean: Union[float, int] = None,
):
r"""
Using the solution provided by Bellan 2012, calculate the analytical
solution to the two fluid, low-frequency (:math:`\omega/kc \ll 1`) dispersion
relation presented by Stringer 1963. This dispersion relation also
assummes a uniform magnetic field :math:`\mathbf{B_o}`, no D.C. electric
field :math:`\mathbf{E_o}=0`, and quasi-neutrality. For more information
see the **Notes** section below.
Parameters
----------
B : `~astropy.units.Quantity`
The magnetic field magnitude in units convertible to :math:`T`.
ion : `str` or `~plasmapy.particles.particle_class.Particle`
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.
k : `~astropy.units.Quantity`, single valued or 1-D array
Wavenumber in units convertible to :math:`rad / m`. Either single
valued or 1-D array of length :math:`N`.
n_i : `~astropy.units.Quantity`
Ion number density in units convertible to :math:`m^{-3}`.
T_e : `~astropy.units.Quantity`
The electron temperature in units of :math:`K` or :math:`eV`.
T_i : `~astropy.units.Quantity`
The ion temperature in units of :math:`K` or :math:`eV`.
theta : `~astropy.units.Quantity`, single valued or 1-D array
The angle of propagation of the wave with respect to the magnetic field,
:math:`\cos^{-1}(k_z / k)`, in units must be convertible to :math:`deg`.
Either single valued or 1-D array of size :math:`M`.
gamma_e : `float` or `int`, optional
The adiabatic index for electrons, which defaults to 1. This
value assumes that the electrons are able to equalize their
temperature rapidly enough that the electrons are effectively
isothermal.
gamma_i : `float` or `int`, optional
The adiabatic index for ions, which defaults to 3. This value
assumes that ion motion has only one degree of freedom, namely
along magnetic field lines.
z_mean : `float` or int, optional
The average ionization state (arithmetic mean) of the ``ion`` composing
the plasma. Will override any charge state defined by argument ``ion``.
Returns
-------
omega : Dict[str, `~astropy.units.Quantity`]
A dictionary of computed wave frequencies in units :math:`rad/s`. The
dictionary contains three keys: ``'fast_mode'`` for the fast mode,
``'alfven_mode'`` for the Alfvén mode, and ``'acoustic_mode'`` for the
ion-acoustic mode. The value for each key will be a :math:`N x M` array.
Raises
------
TypeError
If applicable arguments are not instances of `~astropy.units.Quantity` or
cannot be converted into one.
TypeError
If ``ion`` is not of type or convertible to `~plasmapy.particles.Particle`.
TypeError
If ``gamma_e``, ``gamma_i``, or``z_mean`` are not of type `int` or `float`.
~astropy.units.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.
Warns
-----
: `~plasmapy.utils.exceptions.PhysicsWarning`
When the computed wave frequencies violate the low-frequency
(:math:`\omega/kc \ll 1`) assumption of the dispersion relation.
Notes
-----
The complete dispersion equation presented by Springer 1963 [2]_ (equation 1
of Bellan 2012 [1]_) is:
.. math::
\left( \cos^2 \theta - Q \frac{\omega^2}{k^2 {v_A}^2} \right) &
\left[
\left( \cos^2 \theta - \frac{\omega^2}{k^2 {c_s}^2} \right)
- Q \frac{\omega^2}{k^2 {v_A}^2} \left(
1 - \frac{\omega^2}{k^2 {c_s}^2}
\right)
\right] \\
&= \left(1 - \frac{\omega^2}{k^2 {c_s}^2} \right)
\frac{\omega^2}{{\omega_{ci}}^2} \cos^2 \theta
where
.. math::
Q &= 1 + k^2 c^2/{\omega_{pe}}^2 \\
\cos \theta &= \frac{k_z}{k} \\
\mathbf{B_o} &= B_{o} \mathbf{\hat{z}}
:math:`\omega` is the wave frequency, :math:`k` is the wavenumber, :math:`v_A`
is the Alfvén velocity, :math:`c_s` is the sound speed, :math:`\omega_{ci}` is
the ion gyrofrequency, and :math:`\omega_{pe}` is the electron plasma frequency.
This relation does additionally assume low-frequency waves
:math:`\omega/kc \ll 1`, no D.C. electric field :math:`\mathbf{E_o}=0` and
quasi-neutrality.
Following section 5 of Bellan 2012 [1]_ the exact roots of the above dispersion
equation can be derived and expressed as one analytical solution (equation 38
of Bellan 2012 [1]_):
.. math::
\frac{\omega}{\omega_{ci}} = \sqrt{
2 \Lambda \sqrt{-\frac{P}{3}} \cos\left(
\frac{1}{3} \cos^{-1}\left(
\frac{3q}{2p} \sqrt{-\frac{3}{p}}
\right)
- \frac{2 \pi}{3}j
\right)
+ \frac{\Lambda A}{3}
}
where :math:`j = 0` represents the fast mode, :math:`j = 1` represents the
Alfvén mode, and :math:`j = 2` represents the acoustic mode. Additionally,
.. math::
p &= \frac{3B-A^2}{3} \; , \; q = \frac{9AB-2A^3-27C}{27} \\
A &= \frac{Q + Q^2 \beta + Q \alpha + \alpha \Lambda}{Q^2} \;
, \; B = \alpha \frac{1 + 2 Q \beta + \Lambda \beta}{Q^2} \;
, \; C = \frac{\alpha^2 \beta}{Q^2} \\
\alpha &= \cos^2 \theta \;
, \; \beta = \left( \frac{c_s}{v_A}\right)^2 \;
, \; \Lambda = \left( \frac{k v_{A}}{\omega_{ci}}\right)^2
References
----------
.. [1] PM Bellan, Improved basis set for low frequency plasma waves, 2012,
JGR, 117, A12219, doi: `10.1029/2012JA017856
<https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2012JA017856>`_.
.. [2] TE Stringer, Low-frequency waves in an unbounded plasma, 1963, JNE,
Part C, doi: `10.1088/0368-3281/5/2/304
<https://doi.org/10.1088/0368-3281/5/2/304>`_
Examples
--------
>>> from astropy import units as u
>>> from plasmapy.dispersion import two_fluid_dispersion
>>> inputs = {
... "k": 0.01 * u.rad / u.m,
... "theta": 30 * u.deg,
... "B": 8.3e-9 * u.T,
... "n_i": 5e6 * u.m ** -3,
... "T_e": 1.6e6 * u.K,
... "T_i": 4.0e5 * u.K,
... "ion": "p+",
... }
>>> omegas = two_fluid_dispersion_solution(**inputs)
>>> omegas
{'fast_mode': <Quantity 1520.57... rad / s>,
'alfven_mode': <Quantity 1261.75... rad / s>,
'acoustic_mode': <Quantity 0.688152... rad / s>}
>>> inputs = {
... "k": [1e-7, 2e-7] * u.rad / u.m,
... "theta": [10, 20] * u.deg,
... "B": 8.3e-9 * u.T,
... "n_i": 5e6 * u.m ** -3,
... "T_e": 1.6e6 * u.K,
... "T_i": 4.0e5 * u.K,
... "ion": "He+",
... }
>>> omegas = two_fluid_dispersion_solution(**inputs)
>>> omegas['fast_mode']
<Quantity [[0.00767..., 0.00779... ],
[0.01534..., 0.01558...]] rad / s>
"""
# validate argument ion
if not isinstance(ion, Particle):
try:
ion = Particle(ion)
except TypeError:
raise TypeError(
f"For argument 'ion' expected type {Particle} but got {type(ion)}."
)
if not (ion.is_ion or ion.is_category("element")):
raise ValueError(
f"The particle passed for 'ion' must be an ion or element.")
# validate z_mean
if z_mean is None:
try:
z_mean = abs(ion.integer_charge)
except ChargeError:
z_mean = 1
else:
if not isinstance(z_mean, (int, np.integer, float, np.floating)):
raise TypeError(
f"Expected int or float for argument 'z_mean', but got {type(z_mean)}."
)
z_mean = abs(z_mean)
# validate arguments
for arg_name in ("B", "n_i", "T_e", "T_i"):
val = locals()[arg_name].squeeze()
if val.shape != ():
raise ValueError(
f"Argument '{arg_name}' must a single value and not an array of "
f"shape {val.shape}.")
locals()[arg_name] = val
# validate arguments
for arg_name in ("gamma_e", "gamma_i"):
if not isinstance(locals()[arg_name],
(int, np.integer, float, np.floating)):
raise TypeError(
f"Expected int or float for argument '{arg_name}', but got "
f"{type(locals()[arg_name])}.")
# validate argument k
k = k.squeeze()
if not (k.ndim == 0 or k.ndim == 1):
raise ValueError(
f"Argument 'k' needs to be a single valued or 1D array astropy Quantity,"
f" got array of shape {k.shape}.")
if np.any(k <= 0):
raise ValueError(f"Argument 'k' can not be a or have negative values.")
# validate argument theta
theta = theta.squeeze()
theta = theta.to(u.radian)
if not (theta.ndim == 0 or theta.ndim == 1):
raise ValueError(
f"Argument 'theta' needs to be a single valued or 1D array astropy "
f"Quantity, got array of shape {k.shape}.")
# Calc needed plasma parameters
n_e = z_mean * n_i
with warnings.catch_warnings():
warnings.simplefilter("ignore", category=PhysicsWarning)
c_s = pfp.ion_sound_speed(
T_e=T_e,
T_i=T_i,
ion=ion,
n_e=n_e,
gamma_e=gamma_e,
gamma_i=gamma_i,
z_mean=z_mean,
)
v_A = pfp.Alfven_speed(B, n_i, ion=ion, z_mean=z_mean)
omega_ci = pfp.gyrofrequency(B=B, particle=ion, signed=False, Z=z_mean)
omega_pe = pfp.plasma_frequency(n=n_e, particle="e-")
# Bellan2012JGR params equation 32
alpha = np.cos(theta.value)**2
beta = (c_s / v_A).to(u.dimensionless_unscaled).value**2
alphav, kv = np.meshgrid(alpha, k.value) # create grid
Lambda = (kv * v_A.value / omega_ci.value)**2
# Bellan2012JGR params equation 2
Q = 1 + (kv * c.value / omega_pe.value)**2
# Bellan2012JGR params equation 35
A = ((1 + alphav) / Q) + beta + (alphav * Lambda / Q**2)
B = alphav * (1 + 2 * Q * beta + Lambda * beta) / Q**2
C = beta * (alphav / Q)**2
# Bellan2012JGR params equation 36
p = (3 * B - A**2) / 3
q = (9 * A * B - 2 * A**3 - 27 * C) / 27
# Bellan2012JGR params equation 38
R = 2 * Lambda * np.emath.sqrt(-p / 3)
S = 3 * q / (2 * p) * np.emath.sqrt(-3 / p)
T = Lambda * A / 3
omega = {}
for ind, key in enumerate(("fast_mode", "alfven_mode", "acoustic_mode")):
# The solution corresponding to equation 38
w = omega_ci * np.emath.sqrt(
R * np.cos(1 / 3 * np.emath.arccos(S) - 2 * np.pi / 3 * ind) + T)
omega[key] = w.squeeze()
# check for violation of dispersion relation assumptions
# (i.e. low-frequency, w/kc << 0.1)
wkc_max = np.max(w.value / (kv * c.value))
if wkc_max > 0.1:
warnings.warn(
f"The {key} calculation produced a high-frequency wave (w/kc == "
f"{wkc_max:.3f}), which violates the low-frequency (w/kc << 1) "
f"assumption of the dispersion relation.",
PhysicsWarning,
)
return omega
