LandauDamp_mod: Electron Distributions and Damping

The LandauDamp_mod module ships five analytic electron phase-space distribution functions. Each takes a perpendicular velocity vperp and a parallel velocity vpar and returns the distribution value in units of s³ m⁻⁶. The distributions model different electron populations relevant to Landau damping of whistler-mode waves in the inner magnetosphere: a power-law outer-belt population (Bell and Bortnik), a loss-cone bi-Maxwellian, and two geomagnetic-activity-dependent hybrid models (Golden and Golden2) that interpolate between Bell and Bortnik depending on the L-shell position relative to the plasmapause.

from WPIT.LandauDamp_mod import (
    distribution_bell,
    distribution_bimaxwellian,
    distribution_bortnik,
    distribution_golden,
    distribution_golden2,
    landau_damping,
)

`distribution_bell`

Evaluates the Bell (2002) empirical power-law electron distribution fitted to outer radiation-belt observations. The distribution is expressed in centimetre–gram–second units internally and then converted to SI (s³ m⁻⁶).

f = distribution_bell(vperp, vpar)

vperp

float

Electron velocity component perpendicular to B₀, in m s⁻¹.

vpar

float

Electron velocity component parallel to B₀, in m s⁻¹.

float

Distribution function value in s³ m⁻⁶. The function uses fixed empirical constants a = 4.9e5, b = 8.3e14, c = 5.4e23 from Bell (2002), with a 10× scaling factor applied to the CGS-to-SI conversion.

Model reference: Bell, T. F. (2002). Electron distribution function fitted to outer radiation-belt observations. Journal of Geophysical Research.

import numpy as np
from WPIT.LandauDamp_mod import distribution_bell

vperp = 0.05 * 3e8   # 5% of c perpendicular [m/s]
vpar  = 0.02 * 3e8   # 2% of c parallel [m/s]
f_val = distribution_bell(vperp, vpar)
print(f"f(vperp, vpar) = {f_val:.4e} s^3/m^6")

`distribution_bortnik`

Evaluates the Bortnik et al. (2007) thermal electron power-law distribution. Internally the speed is computed in cm s⁻¹ and the result is converted to SI. The spectral index is determined by a1 = 0.755 and the amplitude by a0 = log10(2.14e7).

f = distribution_bortnik(vperp, vpar)

vperp

float

Perpendicular electron velocity, in m s⁻¹.

vpar

float

Parallel electron velocity, in m s⁻¹.

float

Distribution function value in s³ m⁻⁶. Uses a power-law exponent nu = 2*a1 + 2 = 3.51 and a mass-scaled amplitude mdot = 6.25e11 * me.

Model reference: Bortnik, J., Thorne, R. M., & Meredith, N. P. (2007). Modeling the propagation characteristics of chorus using CRRES suprathermal electron fluxes. Journal of Geophysical Research, 112(A8).

from WPIT.LandauDamp_mod import distribution_bortnik

f_val = distribution_bortnik(vperp, vpar)
print(f"Bortnik f = {f_val:.4e} s^3/m^6")

`distribution_bimaxwellian`

Evaluates a relativistic anisotropic bi-Maxwellian distribution with a loss-cone feature controlled by the parameter beta. The hot electron density is hardcoded as nh = 2e-3 m⁻³, the parallel thermal speed is 5% of c, and the perpendicular thermal speed is 3% of c. The loss-cone depth is set by beta = 0.01.

f = distribution_bimaxwellian(vperp, vpar)

vperp

float

Perpendicular electron velocity, in m s⁻¹.

vpar

float

Parallel electron velocity, in m s⁻¹.

float

Distribution function value in s³ m⁻⁶. The function converts (vperp, vpar) to relativistic four-momentum components (uper, upar) = (gamma*vper, gamma*vpar) before evaluating the Maxwellian exponents.

Internal parameters:

Symbol	Value	Meaning
`nh`	2×10⁻³ m⁻³	Hot electron number density
`Uthpar`	0.05 c	Parallel thermal speed
`Uthper`	0.03 c	Perpendicular thermal speed
`beta`	0.01	Loss-cone depth parameter

from WPIT.LandauDamp_mod import distribution_bimaxwellian

f_val = distribution_bimaxwellian(vperp, vpar)
print(f"Bi-Maxwellian f = {f_val:.4e} s^3/m^6")

`distribution_golden`

Evaluates the Golden et al. (2010) hybrid distribution, which interpolates in log-space between Bell and Bortnik using a sigmoid function centred on the plasmapause L-shell Lpp = 5.6 - 0.46 * Kpmax. The interpolation weight alpha = 5 controls how sharply the distribution transitions between the two models. This function requires explicit kpmax and Lmeas arguments.

f = distribution_golden(vperp, vpar, kpmax, Lmeas)

vperp

float

Perpendicular electron velocity, in m s⁻¹.

vpar

float

Parallel electron velocity, in m s⁻¹.

kpmax

float

Maximum Kp geomagnetic activity index used to locate the plasmapause. Typical range 0–9.

Lmeas

float

L-shell at the measurement point (dimensionless). Controls which side of the plasmapause the ray is on.

float

Interpolated hybrid distribution in s³ m⁻⁶. Inside the plasmasphere (Lmeas less than Lpp) the Bell model dominates; outside (Lmeas greater than Lpp) the Bortnik model dominates.

Model reference: Golden, D. I., Spasojevic, M., Li, W., & Nishimura, Y. (2010). Statistical modeling of plasmaspheric hiss amplitude using the Solar Wind and geomagnetic activity as drivers. Journal of Geophysical Research.

from WPIT.LandauDamp_mod import distribution_golden

kpmax = 4.0    # activity level
Lmeas = 4.5    # measurement L-shell
f_val = distribution_golden(vperp, vpar, kpmax, Lmeas)
print(f"Golden f = {f_val:.4e} s^3/m^6")

`distribution_golden2`

Convenience wrapper around distribution_golden with fixed internal parameters kpmax = 5 and Lmeas = 2. Designed for quick use in landau_damping when no explicit geomagnetic context is needed.

f = distribution_golden2(vperp, vpar)

vperp

float

Perpendicular electron velocity, in m s⁻¹.

vpar

float

Parallel electron velocity, in m s⁻¹.

float

Hybrid distribution value in s³ m⁻⁶, computed at kpmax = 5, Lmeas = 2 (deep inside the plasmapause for strong activity).

distribution_golden2 is the version selected by landau_damping(ray_file, distr='Golden'). For activity- and L-shell-dependent results, call distribution_golden directly.

`landau_damping`

Computes the spatial Landau damping rate at every point along a ray path and accumulates it into a normalised wave-power profile. The function reads the raw ray file, resolves the wave vector and magnetic field projections at each step, selects the distribution function, evaluates the imaginary hot-plasma dispersion integral numerically (summing over resonance orders m = −2 to +2), and writes the resulting power time-series to <ray_file>_damping.csv. A preview plot of normalised wave power versus time is shown on screen.

landau_damping(ray_file, distr)

ray_file

str

Path to the input ray file in the whitespace-delimited format read by read_input_ray. The damping CSV is written to the same directory as ray_file + '_damping.csv'.

distr

str

Electron distribution to use. Accepted values: 'Bell', 'Bortnik', 'Bimaxw', 'Golden'. Any other string causes a warning and no distribution is set.

(side-effect)

file

Writes <ray_file>_damping.csv with two columns: time (seconds) and damp (normalised wave power, initialised to 1 at t=0 and decaying as the wave propagates).

Internal steps:

Load ray data via read_input_ray.
At each time step, decompose the refractive index vector into components parallel and perpendicular to the local B field.
Call spatialdamping(f, kperp, kpar, w, mres, wch, qh, mh, qs, Ns, ms, nus, B0) to obtain the imaginary wave number ki.
Project ki along the group velocity direction (ki_along_vg).
Accumulate: magnitude[i] = magnitude[i-1] * exp(-dist * ki), where dist is the spatial step length.

Only the Landau resonance (m=0) and first four cyclotron harmonics (m = ±1, ±2) are included in the damping sum. Higher harmonics are negligible for typical whistler-mode waves. Points where the refractive index magnitude is zero (evanescent regions) are skipped.

import WPIT.LandauDamp_mod as ld

# Compute damping using Bell distribution
ld.landau_damping("path/to/ray.ray", distr="Bell")
# → saves path/to/ray.ray_damping.csv
# → displays normalised wave power plot

# Switch to Bortnik distribution for outer belt
ld.landau_damping("path/to/ray.ray", distr="Bortnik")

Comparing distribution functions

The following snippet evaluates all five distributions over a grid and compares their shapes.

import numpy as np
import matplotlib.pyplot as plt
from WPIT.LandauDamp_mod import (
    distribution_bell, distribution_bortnik,
    distribution_bimaxwellian, distribution_golden, distribution_golden2
)

c = 3e8   # speed of light [m/s]
vpar  = 0.0
vperp_arr = np.linspace(0.005 * c, 0.15 * c, 300)

labels = ['Bell', 'Bortnik', 'Bi-Maxwellian', 'Golden (Kp=4,L=4.5)', 'Golden2']
funcs  = [
    lambda vp: distribution_bell(vp, vpar),
    lambda vp: distribution_bortnik(vp, vpar),
    lambda vp: distribution_bimaxwellian(vp, vpar),
    lambda vp: distribution_golden(vp, vpar, kpmax=4.0, Lmeas=4.5),
    lambda vp: distribution_golden2(vp, vpar),
]

fig, ax = plt.subplots()
for label, func in zip(labels, funcs):
    f_arr = np.array([func(vp) for vp in vperp_arr])
    ax.semilogy(vperp_arr / c, f_arr, label=label)

ax.set_xlabel("v_perp / c")
ax.set_ylabel("f  [s³ m⁻⁶]")
ax.legend()
ax.set_title("Electron distributions at v_par = 0")
plt.tight_layout()
plt.show()

Environment Module

Wave Properties Module

WPI Module

Landau Damping Module

LandauDamp_mod: Electron Distributions and Damping

`distribution_bell`

`distribution_bortnik`

`distribution_bimaxwellian`

`distribution_golden`

`distribution_golden2`

`landau_damping`

Comparing distribution functions

Build docs developers (and LLMs) love

Environment Module

Wave Properties Module

WPI Module

Landau Damping Module

Documentation Index

​distribution_bell

​distribution_bortnik

​distribution_bimaxwellian

​distribution_golden

​distribution_golden2

​landau_damping

​Comparing distribution functions

Build docs developers (and LLMs) love

`distribution_bell`

`distribution_bortnik`

`distribution_bimaxwellian`

`distribution_golden`

`distribution_golden2`

`landau_damping`

Comparing distribution functions