PlasmaBO

BOPBK(; N = 2)

Dispersion solver using the PBK matrix formulation.

N controls the truncation order of the cyclotron harmonic index used to build the dispersion matrix.

BOHH(; N = 2, J = 8)

Dispersion solver using the Hermite-Hankel (HH) matrix formulation.

N controls the truncation order of the cyclotron harmonic index. J controls the truncation order of the Hermite expansion used in the solver.

BOFluid()

Dispersion solver using the multi-fluid electromagnetic matrix formulation.

BOFluid()

Dispersion solver using the multi-fluid electromagnetic matrix formulation.

BOHH(; N = 2, J = 8)

Dispersion solver using the Hermite-Hankel (HH) matrix formulation.

N controls the truncation order of the cyclotron harmonic index. J controls the truncation order of the Hermite expansion used in the solver.

BOPBK(; N = 2)

Dispersion solver using the PBK matrix formulation.

N controls the truncation order of the cyclotron harmonic index used to build the dispersion matrix.

BranchPoint{T}
BranchPoint(k, ω)

Initial point specification for dispersion branch tracking.

Fields

• k: Wave vector at initial point
• ω: Frequency at initial point
• locator: Callable used to locate the seed eigenvalue at the initial point

Example

# Track unstable branch starting at k=0.03 with γ=0.1721
point1 = BranchPoint(0.03, 0.1721im)     # Auto: track by Im

# Track real frequency mode at k=0.5 with ωᵣ=2.5
point2 = BranchPoint(0.5, 2.5)  # Auto: track by Re

FluidSpecies(n, Tz, Tp = Tz; vdz=0.0, gamma_z=1.0, gamma_p=1.0, particle=:p, ...)

FluidSpecies{T}

Fluid species parameters for the multi-fluid dispersion relation solver.

Fields

• q: Charge
• m: Mass
• n: Number density (m⁻³)
• Tz: Parallel temperature (eV)
• Tp: Perpendicular temperature (eV)
• vdz: Parallel drift velocity (in units of c)
• gamma_z: Parallel polytrope exponent (default: 1.0 for isothermal)
• gamma_p: Perpendicular polytrope exponent (default: 1.0 for isothermal)

JPoleCoefficients{T}

J-pole approximation coefficients for the plasma dispersion function. Z(ζ) ≈ Σⱼ bⱼ/(ζ - cⱼ)

funAn(n, a, d, m, p)

Perpendicular integral for Jn²: ∫ Jn²(ay) exp(-(y-d)²) (y-d)^m y^p dy.

funBn(n, a, d, m, p)

Perpendicular integral for Jn·Jn': ∫ Jn(ay) Jn'(ay) exp(-(y-d)²) (y-d)^m y^p dy.

funCn(n, a, d, m, p)

Perpendicular integral for (Jn')²: ∫ [Jn'(ay)]² exp(-(y-d)²) (y-d)^m y^p dy.

funIn(n)

Normalization integral I_n = ∫ v^n exp(-v²) dv / √π.

Returns Γ((n+1)/2)/√π for even n, 0 for odd n.

get_jpole_coefficients(J=8)

Get J-pole approximation coefficients for the plasma dispersion function.

Supported values: J ∈ {4, 6, 8, 10, 12, 16, 20, 24, 28, 32}

hermite_H(n::Int, x)

Compute physicists' Hermite polynomial H_n(x) using recurrence relation.

Recurrence Relations

• H_0(x) = 1
• H_1(x) = 2x
• H{n+1}(x) = 2x*Hn(x) - 2n*H_{n-1}(x)

Examples

hermite_H(0, 1.5)  # Returns 1.0
hermite_H(1, 1.5)  # Returns 3.0 (= 2*1.5)
hermite_H(2, 1.5)  # Returns 7.0 (= 2*1.5*3.0 - 2*1*1.0)

hermite_a0_to_a(a0lm)

Convert normalized Hermite basis coefficients: a0lm -> alm

Transforms expansion:

f(z,x) = Σ_{l,m} a0_{l,m} * ρ_l(z) * u_m(x)

to:

f(z,x) = Σ_{l,m} a_{l,m} * g_l(z) * h_m(x)

where:

• ρl, um are normalized Hermite functions
• g_l(z) ∝ z^l * exp(-z²/2)
• h_m(x) ∝ x^m * exp(-x²/2)

hermite_coefficients_matrix(nmax)

Compute coefficient matrix for expanding Hermite polynomials in power basis given the maximum order nmax.

Returns matrix cHn[n+1, k+1] such that:

H_n(x) / √(2^n n! √π) = Σ_k cHn[n+1, k+1] * x^k / √(2^{n-k} n! √π)

This is used to convert between normalized Hermite basis and power-law basis.

Returns

• cHn: (nmax+1) × (nmax+1) coefficient matrix

hermite_expansion(
    fv::AbstractMatrix{T},
    vz, vx, vtz, vtp;
    Nz = 16, Nx = 16,
    dz = zero(T), dx = zero(T)
) where {T}

Compute Hermite expansion coefficients from gridded distribution function.

Expands arbitrary 2D distribution f(vparallel, vperp) as:

f(vz, vx) = Σ_{l=0}^{Nz} Σ_{m=0}^{Nx} a_{l,m} * ρ_l(vz) * u_m(vx)

where ρl and um are normalized Hermite basis functions. Nz, Nx are the maximum parallel and perpendicular Hermite indices, respectively.

Arguments

• fv: Distribution function values on (vz, vx) grid
• vz: Parallel velocity grid (1D or matches fv size)
• vx: Perpendicular velocity grid (1D or matches fv size)
• vtz: Parallel thermal velocity (default: 1.0)
• vtp: Perpendicular thermal velocity (default: 1.0)
• dz: Parallel drift velocity (default: 0.0)
• dx: Perpendicular drift velocity (default: 0.0)

Returns

Named tuple with:

• alm: (Nz+1) × (Nx+1) coefficient matrix in power-law basis
• a0lm: (Nz+1) × (Nx+1) coefficient matrix in Hermite basis

Algorithm

1. Compute a0_{l,m} via numerical integration:

   a0_{l,m} = ∫∫ f(z,x) ρ_l(z) u_m(x) dz dx * (2/(Lz*Lx))

2. Normalize by distribution integral

3. Convert to power-law basis: a{l,m} = hermitea0toa(a0_{l,m})

solve_dispersion_matrix(params, kx, kz; N=2, J=8)

Solve the kinetic dispersion relation using the matrix eigenvalue method.

Returns all eigenfrequencies ω(k) for the given wave vector (kx, kz).

This method transforms the dispersion relation into a matrix eigenvalue problem using J-pole approximation for the plasma dispersion function, allowing simultaneous computation of all wave modes.

• kx: Perpendicular wave vector component (m⁻¹)
• kz: Parallel wave vector component (m⁻¹)
• J: Number of poles for Z-function approximation (default: 8)

See also: get_jpole_coefficients

track(solution, point)

Track a single dispersion branch across parameter space from initial point.

The algorithm:

1. Start at the k-point nearest to the initial point
2. Track bidirectionally using interpolation to predict ω at each k
3. Select nearest eigenvalue to prediction

Example

initial = BranchPoint(0.03, 0.1721im)
k_branch, ω_branch = track(solution, initial)

See also: BranchPoint