plasma_plots.theory.kinetic

Kinetic dispersion relations of unmagnetized plasmas: Langmuir waves and Landau damping, ion-acoustic waves, beam-plasma, two-stream and bump-on-tail instabilities, and the Weibel instability.

Units

Electrostatic results are normalized to the reference electrons: time in 1/ω_pe, length in the Debye length λ_De and velocity in the thermal speed v_the = √(T_e/m_e), so that ω_pe = v_the/λ_De = 1. Densities are relative to the reference electron density, charges in e and masses in m_e. A Maxwellian species of thermal speed v_th = √(T/m) drifting at u enters through ζ = (ω − k u)/(√2 |k| v_th). The electromagnetic Weibel relation ([weibel()][weibel]) uses c instead: k in ω_pe/c and speeds in c.

All frequencies are complex for exp(i(kx − ωt)): a positive imaginary part is a growth rate, a negative one a damping rate. Functions take scalars or arrays that broadcast against each other, and return a Python complex for scalar input, else a complex numpy array.

Examples

>>> omega = langmuir(0.5)
>>> round(omega.real, 6), round(omega.imag, 6)
(1.415662, -0.153359)

Attributes

Name	Description
PROTON_ELECTRON_MASS_RATIO	No description.

Classes

Name	Description
Maxwellian	A drifting Maxwellian species, in the normalized units of this module.

Functions

Name	Description
beam_plasma_cold	Compute the cold beam-plasma frequencies: the roots of a quartic.
bohm_gross	Compute the Bohm–Gross frequency ω = √(1 + 3k²) of Langmuir waves.
bump_on_tail	Compute the most unstable kinetic root of a bump-on-tail distribution.
electrostatic_dielectric	Compute the electrostatic dielectric function ε(ω, k) = 1 + Σ_s χ_s of Maxwellian species.
ion_acoustic	Compute the kinetic ion-acoustic root of Maxwellian electrons and ions.
ion_acoustic_fluid	Compute the fluid ion-acoustic frequency with Boltzmann electrons and adiabatic ions.
landau_damping_weak	Compute the weak-damping approximation of Langmuir waves: Bohm–Gross plus Landau damping.
langmuir	Compute the exact kinetic Langmuir root: the least-damped zero of ε(ω, k) near Bohm–Gross.
maximum_growth	Find the wavenumber of maximum growth rate of a dispersion relation.
solve_dispersion	Find a complex root ω(k) of a dispersion relation for every wavenumber.
susceptibility	Compute the electrostatic susceptibility χ_s(ω, k) of one Maxwellian species.
two_stream	Compute the kinetic purely growing root of two warm counter-streaming electron beams.
two_stream_cold	Compute the cold symmetric two-stream frequencies in closed form.
weibel	Compute the purely growing (or damped) root of the electron Weibel instability.

PROTON_ELECTRON_MASS_RATIOattributemodule attribute#

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PROTON_ELECTRON_MASS_RATIO = 1836.15267343

Maxwellianclassdataclass#

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class Maxwellian(density: float = 1.0, charge: float = -1.0, mass: float = 1.0, thermal_speed: float = 1.0, drift: float = 0.0)

A drifting Maxwellian species, in the normalized units of this module.

f(v) = n/(√(2π) v_th) exp(−(v − u)²/(2 v_th²)). The fields may also be arrays that broadcast against the wavenumbers.

Attributes

Name	Type	Description
density	float	The density n_s, relative to the reference electron density. Default: 1.0.
charge	float	The charge q_s in units of e. Default: -1.0 (electrons).
mass	float	The mass m_s in units of m_e. Default: 1.0.
thermal_speed	float	The thermal speed v_th,s = √(T_s/m_s) in units of v_the, positive. Default: 1.0.
drift	float	The drift speed u_s in units of v_the. Default: 0.0.

Examples

>>> Maxwellian().plasma_frequency
1.0
>>> beam = Maxwellian(density=0.1, thermal_speed=0.5, drift=4.5)
>>> round(float(beam.plasma_frequency), 4)
0.3162

chargeattributeclass attributeinstance attribute#

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charge: float = -1.0

densityattributeclass attributeinstance attribute#

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density: float = 1.0

driftattributeclass attributeinstance attribute#

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drift: float = 0.0

massattributeclass attributeinstance attribute#

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mass: float = 1.0

thermal_speedattributeclass attributeinstance attribute#

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thermal_speed: float = 1.0

plasma_frequencyproperty#

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plasma_frequency

The plasma frequency ω_ps = √(n_s q_s²/m_s), in units of ω_pe.

ionsmethodclassmethod#

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def ions(mass_ratio=PROTON_ELECTRON_MASS_RATIO, temperature_ratio=1.0, charge=1.0, drift=0.0)

Create quasi-neutral ions for the reference electrons.

Parameters

Name	Type	Default	Description
mass_ratio	float	PROTON_ELECTRON_MASS_RATIO	m_i/m_e. Default: the proton mass ratio, 1836.15.
temperature_ratio	float	1.0	T_e/T_i. Default: 1.0.
charge	float	1.0	The charge number Z; the density is 1/Z. Default: 1.0.
drift	float	0.0	The drift speed in units of v_the. Default: 0.0.

Returns

Maxwellian

Ions with thermal speed √(T_i/m_i) = 1/√(mass_ratio · temperature_ratio) in units of v_the.

Examples

>>> ions = Maxwellian.ions(mass_ratio=100.0, temperature_ratio=4.0)
>>> ions.thermal_speed, round(float(ions.plasma_frequency), 4)
(0.05, 0.1)

beam_plasma_coldfunction#

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def beam_plasma_cold(k, beam_speed, beam_density=0.1, plasma_density=1.0, all_roots=False)

Compute the cold beam-plasma frequencies: the roots of a quartic.

A cold electron beam of density n_b at speed v_b through a cold plasma of density n_p (immobile ions): 1 = n_p/ω² + n_b/(ω − k v_b)², i.e. ω⁴ − 2kv_b ω³ + (k²v_b² − n_p − n_b) ω² + 2kv_b n_p ω − n_p k²v_b² = 0. For n_b ≪ n_p the maximum growth is γ ≈ (√3/2)(n_b/(2 n_p))^(1/3) √n_p at k v_b ≈ √n_p.

Parameters

Name	Type	Default	Description
k	float or array_like	required	The wavenumber in units of 1/λ_De (any length unit L works, with speeds in ω_pe L).
beam_speed	float or array_like	required	v_b.
beam_density	float or array_like	0.1	n_b. Default: 0.1.
plasma_density	float or array_like	1.0	n_p. Default: 1.0.
all_roots	bool	False	Return all four roots, sorted by real part. Default: False.

Returns

complex or numpy.ndarray

The root with the largest imaginary part where it is unstable, else the real root closest to the slow beam mode k v_b − √n_b. With all_roots, an extra last axis of length 4.

References

C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulation (IOP, 1991), sec. 5.10. R. J. Briggs, Electron-Stream Interaction with Plasmas (MIT Press, 1964).

Examples

>>> omega = beam_plasma_cold(1.0, beam_speed=1.0, beam_density=0.001)
>>> round(omega.real, 4), round(omega.imag, 4)
(0.9588, 0.066)

bohm_grossfunction#

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def bohm_gross(k)

Compute the Bohm–Gross frequency ω = √(1 + 3k²) of Langmuir waves.

The fluid (adiabatic, γ = 3) Langmuir frequency, the real part of langmuir() for k λ_De ≪ 1.

Parameters

Name	Type	Description
k	float or array_like	The wavenumber in units of 1/λ_De.

Returns

complex or numpy.ndarray

ω in units of ω_pe (real-valued).

References

D. Bohm and E. P. Gross, “Theory of plasma oscillations. A. Origin of medium-like behavior”, Phys. Rev. 75, 1851 (1949).

Examples

>>> bohm_gross(0.0)
(1+0j)
>>> round(bohm_gross(0.5).real, 4)
1.3229

bump_on_tailfunction#

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def bump_on_tail(k, beam_density=0.1, beam_speed=4.5, beam_thermal_speed=0.5, bulk_density=None)

Compute the most unstable kinetic root of a bump-on-tail distribution.

A Maxwellian bulk (density n_0, thermal speed 1, at rest) plus a Maxwellian beam (n_b, v_b, v_th,b) over immobile ions. Newton is started from the Bohm–Gross frequency and from phase speeds across the beam (k·v for v from v_b − 3 v_th,b to v_b + v_th,b), and the root with the largest imaginary part is returned: the growing Langmuir/beam mode where the bump makes the distribution unstable (phase speed on its rising flank), the least damped root found where not. For k < 0 the roots are those of |k| mirrored, ω → −conj ω.

Parameters

Name	Type	Default	Description
k	float or array_like	required	The wavenumber in units of 1/λ_De, nonzero.
beam_density	float or array_like	0.1	n_b. Default: 0.1.
beam_speed	float or array_like	4.5	v_b in units of v_the. Default: 4.5.
beam_thermal_speed	float or array_like	0.5	v_th,b in units of v_the, positive. Default: 0.5.
bulk_density	float or array_like	None	n_0. Default: 1 − n_b, so the total electron density is 1.

Returns

complex or numpy.ndarray

ω in units of ω_pe; nan if no guess converged.

References

T. M. O’Neil and J. H. Malmberg, “Transition of the dispersion roots from beam-type to Landau-type solutions”, Phys. Fluids 11, 1754 (1968).

Examples

>>> omega = bump_on_tail(0.3)
>>> round(omega.real, 4), round(omega.imag, 4)
(1.0012, 0.1981)

electrostatic_dielectricfunction#

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def electrostatic_dielectric(omega, k, species=None, derivative=0)

Compute the electrostatic dielectric function ε(ω, k) = 1 + Σ_s χ_s of Maxwellian species.

Its zeros are the electrostatic (Langmuir, ion-acoustic, beam) modes.

Parameters

Name	Type	Default	Description
omega	complex or array_like	required	The complex frequency in units of ω_pe.
k	float or array_like	required	The wavenumber in units of 1/λ_De, nonzero.
species	Maxwellian or sequence of Maxwellian	None	The mobile species. Default: the reference electrons, Maxwellian(), with immobile ions.
derivative	(0, 1)	0	0 for ε, 1 for ∂ε/∂ω. Default: 0.

Returns

complex or numpy.ndarray

ε or ∂ε/∂ω, broadcast over omega, k and the species fields.

See Also

susceptibility() : One species’ contribution χ_s.

Examples

>>> abs(electrostatic_dielectric(langmuir(0.3), 0.3)) < 1e-10
True

ion_acousticfunction#

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def ion_acoustic(k, temperature_ratio=10.0, mass_ratio=PROTON_ELECTRON_MASS_RATIO)

Compute the kinetic ion-acoustic root of Maxwellian electrons and ions.

The zero of ε = 1 + χ_e + χ_i with the reference electrons and ions of Maxwellian.ions(), damped by electron and ion Landau damping. It is continued from k λ_De = 0.01 and T_e/T_i = max(τ, 30), seeded with the fluid frequency and the weak damping γ/ω = −√(π/8) (√(m_e/m_i) + τ^(3/2) exp(−τ/2 − 3/2)) (Chen, eq. 7.144).

Parameters

Name	Type	Default	Description
k	float or array_like	required	The wavenumber in units of 1/λ_De, positive (ω(−k) = −conj ω(k)).
temperature_ratio	float or array_like	10.0	τ = T_e/T_i. Default: 10.0.
mass_ratio	float or array_like	PROTON_ELECTRON_MASS_RATIO	μ = m_i/m_e. Default: the proton mass ratio, 1836.15.

Returns

complex or numpy.ndarray

ω in units of ω_pe; nan where the root was lost.

See Also

ion_acoustic_fluid() : The fluid estimate.

References

F. F. Chen, Introduction to Plasma Physics and Controlled Fusion, 3rd ed. (Springer, 2016), sec. 7.6. B. D. Fried and R. W. Gould, “Longitudinal ion oscillations in a hot plasma”, Phys. Fluids 4, 139 (1961).

Examples

>>> omega = ion_acoustic(0.1, temperature_ratio=10.0, mass_ratio=100.0)
>>> round(omega.real, 5), round(omega.imag, 5)
(0.01178, -0.00076)

ion_acoustic_fluidfunction#

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def ion_acoustic_fluid(k, temperature_ratio=10.0, mass_ratio=PROTON_ELECTRON_MASS_RATIO, adiabatic_index=3.0)

Compute the fluid ion-acoustic frequency with Boltzmann electrons and adiabatic ions.

ω² = k² (T_e/(1 + k² λ_De²) + γ_i T_i)/m_i, i.e. in normalized units ω = k √((1/(1 + k²) + γ_i/τ)/μ) with τ = T_e/T_i and μ = m_i/m_e; for k λ_De ≪ 1, ω = k c_s with c_s = √((T_e + γ_i T_i)/m_i).

Parameters

Name	Type	Default	Description
k	float or array_like	required	The wavenumber in units of 1/λ_De.
temperature_ratio	float or array_like	10.0	τ = T_e/T_i. Default: 10.0.
mass_ratio	float or array_like	PROTON_ELECTRON_MASS_RATIO	μ = m_i/m_e. Default: the proton mass ratio, 1836.15.
adiabatic_index	float	3.0	γ_i of the ions. Default: 3.0 (one-dimensional adiabatic compression).

Returns

complex or numpy.ndarray

ω in units of ω_pe (real-valued).

References

F. F. Chen, Introduction to Plasma Physics and Controlled Fusion, 3rd ed. (Springer, 2016), sec. 4.6.

Examples

>>> round(
...     ion_acoustic_fluid(0.1, temperature_ratio=10.0, mass_ratio=100.0).real,
...     5,
... )
0.01136

landau_damping_weakfunction#

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def landau_damping_weak(k)

Compute the weak-damping approximation of Langmuir waves: Bohm–Gross plus Landau damping.

ω = √(1 + 3k²) − i √(π/8) k⁻³ exp(−1/(2k²) − 3/2), the textbook expansion for k λ_De ≪ 1 (Chen, eq. 7.133). It overestimates the damping at moderate k (by ~60 % at k = 0.3); use langmuir() for the exact root.

Parameters

Name	Type	Description
k	float or array_like	The wavenumber in units of 1/λ_De, nonzero.

Returns

complex or numpy.ndarray

ω in units of ω_pe.

References

F. F. Chen, Introduction to Plasma Physics and Controlled Fusion, 3rd ed. (Springer, 2016), sec. 7.5.

Examples

>>> omega = landau_damping_weak(0.2)
>>> round(omega.real, 4), round(omega.imag, 7)
(1.0583, -6.51e-05)

langmuirfunction#

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def langmuir(k)

Compute the exact kinetic Langmuir root: the least-damped zero of ε(ω, k) near Bohm–Gross.

Electrons (Maxwellian, v_th = 1) with immobile ions. The root is continued in k from k λ_De = 0.1, where the weak-damping approximation is accurate.

Parameters

Name	Type	Description
k	float or array_like	The wavenumber in units of 1/λ_De.

Returns

complex or numpy.ndarray

ω in units of ω_pe, with ω(−k) = ω(k) and ω(0) = 1; nan where the root was lost.

See Also

landau_damping_weak() : The small-k approximation.

References

L. D. Landau, “On the vibrations of the electronic plasma”, J. Phys. USSR 10, 25 (1946). J. Canosa, “Numerical solution of Landau’s dispersion equation”, J. Comput. Phys. 13, 158 (1973).

Examples

>>> omega = langmuir([0.3, 0.4, 0.5, 1.0])
>>> np.round(omega, 4)
array([1.1598-0.0126j, 1.2851-0.0661j, 1.4157-0.1534j, 2.0459-0.8513j])

maximum_growthfunction#

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def maximum_growth(function, k_range, samples=64, tol=1e-08)

Find the wavenumber of maximum growth rate of a dispersion relation.

function is sampled at samples points across k_range, and the largest Im ω is refined by golden-section search between the neighbours of the best sample.

Parameters

Name	Type	Default	Description
function	callable	required	function(k) returning the complex ω for an array of wavenumbers, e.g. bump_on_tail() or lambda k: two_stream(k, 3.0, 0.3).
k_range	(float, float)	required	The interval of wavenumbers searched.
samples	int	64	The number of samples of the initial scan. Default: 64.
tol	float	1e-08	The absolute tolerance in k of the refinement. Default: 1e-8.

Returns

tuple of (float, complex)

The wavenumber of maximum growth and ω there.

Raises

ValueError

If function returns no finite value on the scan.

Examples

>>> k, omega = maximum_growth(lambda k: two_stream_cold(k, 1.0), (0.01, 1.4))
>>> round(k, 5), round(omega.imag, 6)  # √(3/8), √0.5/2
(0.61237, 0.353553)

solve_dispersionfunction#

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def solve_dispersion(function, k, guess, derivative=None, continuation=True, tol=1e-11, maxiter=60)

Find a complex root ω(k) of a dispersion relation for every wavenumber.

Damped Newton iteration in ω, with the analytic derivative when one is given and a central difference otherwise. With continuation, the wavenumbers are solved in the given order and each root seeds the next (linearly extrapolated from the last two), which follows one branch along a k-grid; otherwise every k starts from guess and all are solved at once.

Parameters

Name	Type	Default	Description
function	callable	required	function(omega, k), analytic in ω, whose zero is sought, e.g. lambda w, k: electrostatic_dielectric(w, k, species). It must broadcast over arrays.
k	float or array_like	required	The wavenumbers; one-dimensional with continuation, else any shape.
guess	complex or array_like	required	The starting frequency: a scalar for the first k with continuation, else anything that broadcasts against k.
derivative	callable	None	derivative(omega, k), the ω-derivative of function. Default: a central difference.
continuation	bool	True	Continue the root along k. Default: True.
tol	float	1e-11	Converged once the Newton step is below tol · |ω|. Default: 1e-11.
maxiter	int	60	The maximum number of Newton steps per wavenumber. Default: 60.

Returns

complex or numpy.ndarray

ω for every k, nan where Newton did not converge (no exception is raised); with continuation the next wavenumber is then seeded by the last converged root.

Raises

ValueError

With continuation, if k has more than one dimension or guess is not a scalar.

Examples

The Langmuir branch followed from k = 0.2 to 0.5:

>>> k = np.linspace(0.2, 0.5, 7)
>>> omega = solve_dispersion(electrostatic_dielectric, k, guess=1.06)
>>> round(complex(omega[-1]).real, 4), round(complex(omega[-1]).imag, 4)
(1.4157, -0.1534)

susceptibilityfunction#

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def susceptibility(omega, k, species, derivative=0)

Compute the electrostatic susceptibility χ_s(ω, k) of one Maxwellian species.

χ_s = (ω_ps²/(k² v_th,s²)) [1 + ζ_s Z(ζ_s)], with ζ_s = (ω − k u_s)/(√2 |k| v_th,s) and Z the plasma dispersion function (Landau’s continuation, valid for any complex ω). For a negative k the drift enters as k u_s, so χ_s(ω, −k) is χ_s(ω, k) with the drift reversed.

Parameters

Name	Type	Default	Description
omega	complex or array_like	required	The complex frequency in units of ω_pe.
k	float or array_like	required	The wavenumber in units of 1/λ_De, nonzero.
species	Maxwellian	required	The species.
derivative	(0, 1)	0	0 for χ_s, 1 for ∂χ_s/∂ω. Default: 0.

Returns

complex or numpy.ndarray

χ_s or ∂χ_s/∂ω, broadcast over omega, k and the species fields.

Raises

ValueError

If derivative is not 0 or 1.

References

B. D. Fried and S. D. Conte, The Plasma Dispersion Function (Academic Press, 1961). T. H. Stix, Waves in Plasmas (AIP, 1992), ch. 8.

Examples

The cold limit χ → −ω_p²/ω²:

>>> round(susceptibility(10.0, 0.01, Maxwellian()).real, 6)
-0.01

two_streamfunction#

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def two_stream(k, beam_speed, thermal_speed, beam_density=0.5)

Compute the kinetic purely growing root of two warm counter-streaming electron beams.

Two Maxwellian electron beams of density n_b and thermal speed v_t at ±v_b over immobile ions. By symmetry ε(iγ, k) is real, and the two-stream instability is purely growing, ω = iγ: it exists where ε(0, k) < 0, and γ is the largest zero of ε(iγ, k), found by a scan and bisection. Where the beams are stable, the least damped root reached by Newton from phase speeds between 0 and v_b is returned instead, with Re ω ≥ 0 (−conj ω is a root too): past the marginal wavenumber the growing root iγ meets its mirror −iγ near γ = 0 and turns into a weakly damped oscillating pair, so γ(k) stays continuous. For v_t → 0 the growth approaches two_stream_cold().

Parameters

Name	Type	Default	Description
k	float or array_like	required	The wavenumber in units of 1/λ_De (ω(−k) = ω(k)).
beam_speed	float or array_like	required	v_b in units of v_the.
thermal_speed	float or array_like	required	v_t of each beam in units of v_the, positive.
beam_density	float or array_like	0.5	n_b, each beam’s density. Default: 0.5, so the total density is 1.

Returns

complex or numpy.ndarray

ω in units of ω_pe: iγ where unstable; nan where stable and Newton found no root.

See Also

two_stream_cold() : The cold limit.

References

T. H. Stix, Waves in Plasmas (AIP, 1992), sec. 8.11. C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulation (IOP, 1991), sec. 5.10.

Examples

>>> omega = two_stream(0.2, beam_speed=3.0, thermal_speed=0.3)
>>> round(omega.real, 6), round(omega.imag, 4)
(0.0, 0.3491)

two_stream_coldfunction#

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def two_stream_cold(k, beam_speed, beam_density=0.5, all_roots=False)

Compute the cold symmetric two-stream frequencies in closed form.

Two cold electron beams of density n_b each at ±v_b over immobile ions: 1 = n_b/(ω − k v_b)² + n_b/(ω + k v_b)², a quadratic in ω² with the roots ω² = k²v_b² + n_b ± √(4 n_b k² v_b² + n_b²). The “slow” root ω² < 0, i.e. the purely growing ω = iγ, exists for k² v_b² < 2 n_b; the maximum growth is γ = √n_b/2 at k² v_b² = 3 n_b/4.

Parameters

Name	Type	Default	Description
k	float or array_like	required	The wavenumber in units of 1/λ_De (any length unit L works, with speeds in ω_pe L).
beam_speed	float or array_like	required	v_b, each beam’s speed.
beam_density	float or array_like	0.5	n_b, each beam’s density (its ω_pb²). Default: 0.5, so the total density is 1.
all_roots	bool	False	Return all four roots instead of the slow one. Default: False.

Returns

complex or numpy.ndarray

The slow root √(ω₋²) with Im ω ≥ 0: iγ where unstable, the real slow beam mode where stable. With all_roots, an extra last axis of length 4: (√(ω₊²), −√(ω₊²), √(ω₋²), −√(ω₋²)).

References

C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulation (IOP, 1991), sec. 5.10.

Examples

>>> omega = two_stream_cold(np.sqrt(3 / 8), beam_speed=1.0)
>>> round(omega.real, 6), round(omega.imag, 6)
(0.0, 0.353553)

weibelfunction#

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def weibel(k, anisotropy, parallel_thermal_speed)

Compute the purely growing (or damped) root of the electron Weibel instability.

Transverse electromagnetic waves along k ∥ z in bi-Maxwellian electrons (T⊥ across k, T∥ along k) with immobile ions:

ω² − k²c² − ω_pe² + ω_pe² A [1 + ζ Z(ζ)] = 0, A = T⊥/T∥, ζ = ω/(√2 k v_∥),

with v_∥ = √(T∥/m_e). It reduces to light waves ω² = ω_pe² + k²c² for T → 0. The root is ω = iγ, with γ > 0 exactly for k²c² < (A − 1) ω_pe² and γ < 0 (a damped, non-oscillating mode) beyond; γ is found by bisection of the real function D(iγ), which decreases in γ. Near the cutoff γ ≈ √(2/π) k v_∥ (A − 1 − k²c²/ω_pe²)/A.

Units: unlike the rest of this module, k in ω_pe/c, ω in ω_pe and speeds in c.

Parameters

Name	Type	Description
k	float or array_like	The wavenumber in units of ω_pe/c (ω(−k) = ω(k), ω(0) = 0).
anisotropy	float or array_like	A = T⊥/T∥, positive; unstable for A > 1.
parallel_thermal_speed	float or array_like	v_∥ = √(T∥/m_e) in units of c, positive.

Returns

complex or numpy.ndarray

ω = iγ in units of ω_pe.

References

E. S. Weibel, “Spontaneously growing transverse waves in a plasma due to an anisotropic velocity distribution”, Phys. Rev. Lett. 2, 83 (1959). R. C. Davidson, D. A. Hammer, I. Haber and C. E. Wagner, “Nonlinear development of electromagnetic instabilities in anisotropic plasmas”, Phys. Fluids 15, 317 (1972). N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics (McGraw-Hill, 1973), sec. 9.10.

Examples

>>> round(weibel(1.0, anisotropy=4.0, parallel_thermal_speed=0.1).imag, 4)
0.061
>>> weibel(
...     np.sqrt(3.0), anisotropy=4.0, parallel_thermal_speed=0.1
... ).imag < 1e-12
True