1-3 Appendix A

The elements of the diffusion matrix, $D$ , are given by

\begin{array}{r} (A1) & D_{α α} = \sum_{n = - \infty}^{\infty} \int_{0}^{\infty} k_{⊥} d k_{⊥} D_{α α}^{n k_{⊥}} \\ (A2) & D_{α p} = D_{p α} = \sum_{n = - \infty}^{\infty} \int_{0}^{\infty} k_{⊥} d k_{⊥} D_{α p}^{n k_{⊥}} \\ (A3) & D_{p p} = \sum_{n = - \infty}^{\infty} \int_{0}^{\infty} k_{⊥} d k_{⊥} D_{p p}^{n k_{⊥}} \end{array}

For each harmonic resonance number $n$ and $k_{⊥}$ , the pitch-angle diffusion coefficient $D_{α α}^{n k_{⊥}}$ is given by

\begin{matrix} (A4) & D_{α α}^{n k_{⊥}} = lim_{V \to \infty} \frac{q_{σ}^{2}}{4 π V} {{[\frac{n Ω_{σ} / (γ ω) - \sin^{2} α}{\cos α}]}^{2} \frac{| Θ_{n k} |^{2}}{| v_{∥} - \frac{\partial ω}{\partial k_{∥}} |} |}_{k_{∥}} \end{matrix}

The right-hand side of equation (A4) is evaluated at the resonant parallel wave number $k_{∥}$ and frequency $ω$ that satisfies the resonance condition

\begin{matrix} (A5) & ω - k_{∥} v_{∥} = n Ω_{σ} / γ \end{matrix}

The value of $Θ_{n k}$ is given by

\begin{matrix} (A6) & Θ_{n k} = \frac{v_{∥}}{v_{⊥}} E_{k, ∥} J_{n} + \frac{E_{k, L} J_{n - 1} + E_{k, R} J_{n + 1}}{\sqrt{2}} \end{matrix}

where $J_{n}$ is the n-th order Bessel function with argument $k_{⊥} p_{⊥} / (m_{σ} Ω_{σ})$ , and the Fourier transform of the wave electric field has been written in terms of its parallel $E_{k, ∥}$ , left-hand $E_{k, L}$ , and right-hand $E_{k, R}$ , components, with

\begin{array}{r} (A7) & E_{k, L} = \frac{1}{\sqrt{2}} (E_{k, x} + i E_{k, y}) \\ (A8) & E_{k, R} = \frac{1}{\sqrt{2}} (E_{k, x} - i E_{k, y}) \end{array}

From equations A7 and A8, we can easily solve for $E_{k, x}$ and $E_{k, y}$ :

\begin{matrix} (add) & E_{k, x} = \frac{1}{\sqrt{2}} (E_{k, L} + E_{k, R}), E_{k, y} = \frac{1}{i \sqrt{2}} (E_{k, L} - E_{k, R}) \end{matrix}

Using cold plasma theory, $E_{k, L}$ and $E_{k, R}$ can be rewritten in terms of $E_{k, ∥}$ , the refractive index $μ$ , the wave-normal angle $ψ$ , and the Stix parameters $R, L, S, D,$ and $P$ [Lyons, 1974b], to give

\begin{array}{r} (A9) & E_{k, L} = (\frac{μ^{2} \sin^{2} ψ - P}{μ^{2} \sin ψ \cos ψ}) (1 - \frac{D}{(μ^{2} - S)}) \frac{E_{k, ∥}}{\sqrt{2}} \\ (A10) & E_{k, R} = (\frac{μ^{2} \sin^{2} ψ - P}{μ^{2} \sin ψ \cos ψ}) (1 + \frac{D}{(μ^{2} - S)}) \frac{E_{k, ∥}}{\sqrt{2}} \end{array}

The Stix parameters are defined as:
$\begin{matrix} (add) & \begin{aligned} R & = 1 - \sum_{σ} \frac{ω_{p σ}^{2}}{ω (ω - Ω_{σ})} \\ L & = 1 - \sum_{σ} \frac{ω_{p σ}^{2}}{ω (ω + Ω_{σ})} \\ P & = 1 - \sum_{σ} \frac{ω_{p σ}^{2}}{ω^{2}} \\ S & = \frac{R + L}{2} \\ D & = \frac{R - L}{2} \end{aligned} \end{matrix}$

The Maxwell-Faraday equation in Fourier space is:

\begin{matrix} (add) & k \times E_{k} = ω B_{k} \to | E_{k, ⊥} | = c | B_{k} | \end{matrix}

and

\begin{matrix} (add) & | E_{k} |^{2} = | E_{k, ∥} |^{2} + | E_{k, ⊥} |^{2} = | E_{k, x} |^{2} + | E_{k, y} |^{2} \end{matrix}

Thus, $E_{k, ∥}$ can be written as a function of the magnitude of the Fourier transform of the wave magnetic field at each $k$ , namely $| B_{k} |$ :

\begin{matrix} (A11) & | E_{k, ∥} |^{2} = \frac{c^{2} | B_{k} |^{2}}{μ^{2}} \cdot {[{(\frac{R - L}{2 (μ^{2} - S)})}^{2} {(\frac{μ^{2} \sin^{2} ψ - P}{μ^{2} \sin ψ \cos ψ})}^{2} + {(\frac{P}{μ^{2} \sin ψ})}^{2}]}^{- 1} \end{matrix}

Combining these results gives

\begin{matrix} (A12) & | Θ_{n k} |^{2} = \frac{c^{2} | B_{k} |^{2}}{μ^{2}} | Φ_{n, k} |^{2} \end{matrix}

with

\begin{aligned} (A13) & | Φ_{n, k} |^{2} & = \frac{{[(\frac{μ^{2} - L}{μ^{2} - S}) J_{n + 1} + (\frac{μ^{2} - R}{μ^{2} - S}) J_{n - 1}] (\frac{μ^{2} \sin^{2} ψ - P}{2 μ^{2}}) + \cot α \sin ψ \cos ψ J_{n}}^{2}}{{(\frac{R - L}{2 (μ^{2} - S)})}^{2} {(\frac{P - μ^{2} \sin^{2} ψ}{μ^{2}})}^{2} + {(\frac{P \cos ψ}{μ^{2}})}^{2}} \end{aligned}

We assume that the wave magnetic field can be described by [Lyons, 1974b]

\begin{matrix} (A14) & | B_{k} |^{2} = \frac{V}{N (ω)} B^{2} (ω) g (ψ) \end{matrix}

with

\begin{matrix} (A15) & N (ω) = \frac{1}{2 π^{2}} \int_{X_{min}}^{X_{max}} g (X) | J (\frac{k_{⊥}, k_{∥}}{ω, X}) | k_{⊥} d X \end{matrix}

Here $B^{2} (ω)$ is the wave magnetic field intensity squared per unit frequency and $g (X)$ gives the variation of wave magnetic field energy with wave normal angle,

\begin{matrix} (add) & \begin{array}{r} g (X) = {\begin{cases} \exp [- {(\frac{X - X_{m}}{X_{w}})}^{2}] & X_{min} \leq X \leq X_{max} \\ 0 & otherwise, w i t h X = \tan (ψ) \end{cases} \end{array} \end{matrix}

Using the cold plasma dispersion relation $D (k, ω, X) = 0$ , and the Jacobian $J (\frac{k_{∥}, k_{⊥}}{ω, X})$ , A15 can be rewritten to obtain the equation (see 1-3 Appendix B):

\begin{matrix} (add) & N (ω) = \frac{1}{2 π^{2}} \int_{X_{min}}^{X_{max}} \frac{g (X) k^{2}}{(1 + X^{2})^{\frac{3}{2}}} \frac{\partial D}{\partial ω} {[\frac{\partial D}{\partial k}]}^{- 1} X d X \end{matrix}

Combining equations (A1)–(A3), (A4), and (A12) and transforming the integrals from $k_{⊥}$ to $X$ leads to the final form of the diffusion coefficients' expression:

\begin{aligned} D_{α α} & = \sum_{n = n_{l}}^{n_{h}} \int_{X_{min}}^{X_{max}} X d X D_{α α}^{n X} \\ D_{α p} & = D_{p α} = \sum_{n = n_{l}}^{n_{h}} \int_{X_{min}}^{X_{max}} X d X D_{α p}^{n X} \\ D_{p p} & = \sum_{n = n_{l}}^{n_{h}} \int_{X_{min}}^{X_{max}} X d X D_{p p}^{n X} \end{aligned}

with

\begin{aligned} D_{α α}^{n X} & = \sum_{i} \frac{q_{σ}^{2} ω_{i}^{2}}{4 π (1 + X^{2}) N (ω_{i})} {[\frac{n Ω_{σ} / (γ ω_{i}) - \sin^{2} α}{\cos α}]}^{2} \cdot {B^{2} (ω_{i}) g (X) \frac{| Φ_{n, k} |^{2}}{| v_{∥} - \frac{\partial ω}{\partial k_{∥}} |} |}_{k_{| | i}} \\ D_{α p}^{n X} & = D_{α α}^{n X} {[\frac{\sin α \cos α}{n Ω_{σ} / (γ ω_{i}) - \sin^{2} α}]}_{k_{| | i}} \\ D_{p p}^{n X} & = D_{α α}^{n X} {[\frac{\sin α \cos α}{n Ω_{σ} / (γ ω_{i}) - \sin^{2} α}]}_{k_{| | i}}^{2} \end{aligned}