The dispersion subgroup ofDocumentation Index
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WaveProperties_mod covers everything needed to characterise how electromagnetic waves propagate in a magnetised multi-species plasma: computing the Stix parameters S, D, P, R, L in both cold and warm limits; assembling the full cold or warm 3×3 dielectric tensor; and evaluating the squared refractive index η² and wave number κ for the four principal modes (R, L, O, X) plus free-space light. Cutoff-frequency helpers for the R and L modes are also provided.
stix_parameters
Computes the five cold-plasma Stix parameters from plasma densities and the background magnetic field magnitude. Internally the function derives plasma and cyclotron frequencies for electrons, H⁺, He⁺, and O⁺ from fundamental constants before assembling R, L, P, S, and D.
Reference: Stix, T. H. (1992). Waves in Plasmas. Springer Science & Business Media.
Parameters
Wave angular frequency [rad s⁻¹].
Electron number density [m⁻³].
H⁺ number density [m⁻³]. Pass
0 if hydrogen ions are absent.He⁺ number density [m⁻³]. Pass
0 if helium ions are absent.O⁺ number density [m⁻³]. Pass
0 if oxygen ions are absent.Background magnetic field magnitude [T].
Returns
Stix S parameter — arithmetic mean of R and L:
S = (R + L) / 2.Stix D parameter — half the difference of R and L:
D = (R − L) / 2.Stix P parameter — parallel (to B₀) dielectric response.
Stix R parameter — right-hand resonance factor; diverges at the electron cyclotron frequency.
Stix L parameter — left-hand resonance factor; diverges at the ion cyclotron frequencies.
Example
stix_parameters_warm
Applies finite-temperature (warm plasma) corrections to the cold Stix parameters using the warm dielectric tensor components computed by warm_dielectric_tensor. The correction is proportional to the dimensionless thermal parameter τ = (k_B T / m c²) × η².
Reference: Maxworth, A. and Gołkowski, M. (2017). Journal of Geophysical Research: Space Physics 122, 7323–7335.
Parameters
Cold-plasma Stix S parameter (from
stix_parameters).Cold-plasma Stix D parameter.
Cold-plasma Stix P parameter.
Electron temperature [eV].
Ion temperature [eV] (applied uniformly to H⁺, He⁺, and O⁺).
Warm-plasma refractive index η (dimensionless), used to form τ = (k_B T / m c²) × η².
Electron warm dielectric tensor components as a flat 9-element list
[K11, K12, ..., K33]
(output of warm_dielectric_tensor).H⁺ warm dielectric tensor components (same layout as
Ke).He⁺ warm dielectric tensor components.
O⁺ warm dielectric tensor components.
Returns
Temperature-corrected Stix S.
Temperature-corrected Stix D.
Temperature-corrected Stix P.
Temperature-corrected Stix R = S_warm + D_warm.
Temperature-corrected Stix L = S_warm − D_warm.
Example
cold_dielectric_tensor
Computes all nine components of the cold-plasma dielectric tensor K for a multi-species plasma. The tensor is Hermitian — K13 = K23 = K31 = K32 = 0 for a cold plasma in a uniform background field aligned with z, and K11 = K22 while K21 = −K12.
Reference: Maxworth, A. S., et al. (2020). Journal of Geophysical Research: Space Physics 125(4), e2019JA027154.
Parameters
Wave angular frequency [rad s⁻¹].
Electron cyclotron angular frequency [rad s⁻¹]. Defined as
q_e B₀ / m_e.Electron plasma angular frequency [rad s⁻¹].
H⁺ cyclotron angular frequency [rad s⁻¹].
H⁺ plasma angular frequency [rad s⁻¹].
He⁺ cyclotron angular frequency [rad s⁻¹].
He⁺ plasma angular frequency [rad s⁻¹].
O⁺ cyclotron angular frequency [rad s⁻¹].
O⁺ plasma angular frequency [rad s⁻¹].
Returns
Nine individual tensor components. For a cold plasma with B₀ aligned with ẑ the nonzero
independent components are K11 (= K22), K12 (purely imaginary), and K33.
All off-diagonal z-coupling terms (K13, K23, K31, K32) are identically zero.
Example
warm_dielectric_tensor
Computes warm-plasma dielectric tensor corrections for each ion species and electrons. The tensor components account for finite Larmor radius effects via the wave normal angle psi_arg. The function returns four lists, each a 9-element flat array indexed as [K11, K12, K13, K21, K22, K23, K31, K32, K33].
Reference: Maxworth & Gołkowski (2017). JGR: Space Physics 122, 7323–7335.
Parameters
Wave angular frequency [rad s⁻¹].
Electron cyclotron angular frequency [rad s⁻¹].
Electron plasma angular frequency [rad s⁻¹].
H⁺ cyclotron angular frequency [rad s⁻¹].
H⁺ plasma angular frequency [rad s⁻¹].
He⁺ cyclotron angular frequency [rad s⁻¹].
He⁺ plasma angular frequency [rad s⁻¹].
O⁺ cyclotron angular frequency [rad s⁻¹].
O⁺ plasma angular frequency [rad s⁻¹].
Wave normal angle [rad] with respect to B₀.
Returns
9-element list of electron warm-correction tensor components.
9-element list of H⁺ warm-correction tensor components.
9-element list of He⁺ warm-correction tensor components.
9-element list of O⁺ warm-correction tensor components.
Example
dispersion_R
Evaluates the R-mode (right-hand, whistler-branch) dispersion relation for a multi-species cold plasma. The R-mode is right-hand circularly polarised for parallel propagation and connects to the whistler mode at sub-electron-cyclotron frequencies.
Reference: Swanson, D. G. (2012). Plasma Waves. Elsevier.
Parameters
Wave angular frequency [rad s⁻¹].
Electron plasma angular frequency [rad s⁻¹].
Electron cyclotron angular frequency [rad s⁻¹].
H⁺ plasma angular frequency [rad s⁻¹].
H⁺ cyclotron angular frequency [rad s⁻¹].
He⁺ plasma angular frequency [rad s⁻¹]. Pass
0 if absent.He⁺ cyclotron angular frequency [rad s⁻¹]. Pass
0 if absent.O⁺ plasma angular frequency [rad s⁻¹]. Pass
0 if absent.O⁺ cyclotron angular frequency [rad s⁻¹]. Pass
0 if absent.Returns
Squared refractive index η² = n² for the R mode (dimensionless).
Positive values indicate a propagating wave; negative values indicate evanescence.
Wave number κ = η ω / c [m⁻¹]. Computed as
sqrt(ω² η² / c²).Example
The R-mode becomes evanescent (η² less than 0) below the R-mode cutoff frequency and above
the electron cyclotron frequency. Use
cutoff_R to find the lower propagation boundary.dispersion_L
Evaluates the L-mode (left-hand circularly polarised, parallel propagation) dispersion relation for a multi-species cold plasma. The L mode resonates at the ion cyclotron frequencies and is relevant for EMIC wave modelling.
Reference: Swanson, D. G. (2012). Plasma Waves. Elsevier.
Parameters
Wave angular frequency [rad s⁻¹].
Electron plasma angular frequency [rad s⁻¹].
Electron cyclotron angular frequency [rad s⁻¹].
H⁺ plasma angular frequency [rad s⁻¹].
H⁺ cyclotron angular frequency [rad s⁻¹].
He⁺ plasma angular frequency [rad s⁻¹]. Pass
0 if absent.He⁺ cyclotron angular frequency [rad s⁻¹]. Pass
0 if absent.O⁺ plasma angular frequency [rad s⁻¹]. Pass
0 if absent.O⁺ cyclotron angular frequency [rad s⁻¹]. Pass
0 if absent.Returns
Squared refractive index η² for the L mode (dimensionless).
Wave number κ [m⁻¹].
Example
The L-mode resonates (η² → ∞) at each ion cyclotron frequency. The wave is evanescent
below the L-mode cutoff. Use
cutoff_L to locate this boundary.dispersion_O
Evaluates the O-mode (ordinary mode, electric field polarised parallel to B₀) dispersion relation. The O-mode is independent of the magnetic field; it propagates when ω exceeds the total plasma frequency.
Reference: Swanson, D. G. (2012). Plasma Waves. Elsevier.
Parameters
Wave angular frequency [rad s⁻¹].
Electron plasma angular frequency [rad s⁻¹].
H⁺ plasma angular frequency [rad s⁻¹]. Pass
0 if absent.He⁺ plasma angular frequency [rad s⁻¹]. Pass
0 if absent.O⁺ plasma angular frequency [rad s⁻¹]. Pass
0 if absent.Returns
Squared refractive index η² = 1 − Σ ωₚₛ² / ω² (dimensionless).
Wave number κ [m⁻¹].
Example
dispersion_X
Evaluates the X-mode (extra-ordinary mode, perpendicular propagation) dispersion relation using the lower hybrid resonance frequency wlh as input. The X-mode exists in two distinct pass bands separated by the upper hybrid resonance and the plasma frequency.
Reference: Swanson, D. G. (2012). Plasma Waves. Elsevier.
Parameters
Wave angular frequency [rad s⁻¹].
Electron plasma angular frequency [rad s⁻¹].
Lower hybrid resonance angular frequency [rad s⁻¹].
Returns
Squared refractive index η² for the X mode (dimensionless).
Wave number κ [m⁻¹].
Example
dispersion_light
Returns the squared refractive index and wave number for a free-space electromagnetic (light) wave. This is equivalent to the O-mode for an electron-only plasma with η² = 1 − ωₚₑ²/ω². The function accepts wce for interface consistency but does not use it in the calculation.
Reference: Swanson, D. G. (2012). Plasma Waves. Elsevier.
Parameters
Wave angular frequency [rad s⁻¹].
Electron plasma angular frequency [rad s⁻¹].
Electron cyclotron angular frequency [rad s⁻¹]. Accepted for interface consistency
but not used in the calculation.
Returns
Squared refractive index η² = (ω² − ωₚₑ²) / ω² (dimensionless).
Wave number κ = sqrt((ω² − ωₚₑ²) / c²) [m⁻¹].
Example
cutoff_L
Computes the L-mode cutoff angular frequency, below which the L mode is evanescent.
The formula is omega_L = 0.5 * (−wce + sqrt(wce² + 4 * wpe²)).
Reference: Swanson, D. G. (2012). Plasma Waves. Elsevier.
Parameters
Electron cyclotron angular frequency [rad s⁻¹].
Electron plasma angular frequency [rad s⁻¹].
Returns
L-mode cutoff angular frequency [rad s⁻¹].
Example
cutoff_R
Computes the R-mode cutoff angular frequency, below which the R mode is evanescent.
The formula is omega_R = 0.5 * (wce + sqrt(wce² + 4 * wpe²)).
Reference: Swanson, D. G. (2012). Plasma Waves. Elsevier.
Parameters
Electron cyclotron angular frequency [rad s⁻¹].
Electron plasma angular frequency [rad s⁻¹].
Returns
R-mode cutoff angular frequency [rad s⁻¹].
Example
By construction, omega_R is greater than omega_L when wpe is positive. The whistler mode
propagates between the L-cutoff and wce on the R-mode branch.