6.1 Inelastic pitch-angle diffusion of ions in resonance with waves

The response of particles to turbulent electromagnetic wave fields has traditionally been described by the paradigm of inelastic pitch-angle diffusion. The corresponding Quasilinear Theory (QLT) was in parallel developed by several authors (Shapiro and ShevchenkoJump To The Next Citation Point, 1962Jump To The Next Citation Point, Rowlands et al.Jump To The Next Citation Point, 1966, and Kennel and EngelmannJump To The Next Citation Point), 1966Jump To The Next Citation Point), and then described in many other articles. Here we mainly refer to the excellent textbooks by Melrose and McPhedran (1991Jump To The Next Citation Point) and Stix (1992Jump To The Next Citation Point). QLT is quadratically non-linear in the coupling terms between the fluctuations of the VDFs and electromagnetic fields. But it is still linear (hence its name) in the sense that both kinds of fluctuations enter only linearly in the quadratic product terms of the perturbed second-order Vlasov equation. The wave properties, such as dispersion and polarization, are still evaluated from linear theory with slowly time-varying VDFs and power spectral density (PSD), implying weak wave growth or dissipation.

In QLT it is assumed that the electromagnetic wave fields can generally be Fourier-decomposed in plane waves with the frequency, ω = ωM (k ), and growth rate, γM (k), for a particular wave mode (index M) and a given real wave vector k, which is assumed here to be directed arbitrarily with respect to the constant background field, B0. The full dispersion equation for any linear plasma wave mode M in a multi-component plasma can for instance be found in Stix (1992Jump To The Next Citation Point). Mann et al. (1997) have studied in much detail the polarization properties of waves in a multi-component plasma. Ofman et al. (2005) recently discussed the possible observational implications of high-frequency Alfvén waves in a multi-ion corona. In the previous sections we have already given special, but for the solar wind particularly relevant, examples for dispersion relations in Equations (41View Equation) and (43View Equation).

QLT assumes the validity of the random-phase approximation, which ensures that no constructive interference occurs between the different waves modes, and thus these modes can be simply superposed linearly. Therefore, we can write the Fourier-transformed total electric field as a sum over the various modes as

∑ &tidle;E(k,ω ) = 2π δ [ω − ωM (k)]EM (k)eM (k). (46 ) M
Thus the Fourier components of the electric field vector can be expressed in terms of the wave amplitude, EM (k ), and the unimodular polarization vector, eM (k). The magnetic field, &tidle;BM (k), can through the induction equation be calculated as
B&tidle; (k) = ---c--k × &tidle;E (k). (47 ) M ωM (k) M
The wave growth rate, γM (k), or damping rate if it is negative, together with the real frequency, ωM (k), may be combined to a complex frequency, &tidle;ωM (k) = ωM (k) + iγM(k ). The spectral energy density of the magnetic field of mode M is given by ℬM (k ) = |B&tidle;M (k ) |2 ∕ (8 πV ), with arbitrarily large integration volume V, and evolves according to a simple exponential equation:
∂ ---ℬM (k) = 2γM (k)ℬM (k), (48 ) ∂t
which follows from the Fourier decomposition
∫ ∫ BM (x, t) = d3kB&tidle;M (k)eik⋅xe−i t0 dt′&tidle;ωM(k,t′), (49 )
where x is the spatial coordinate and t is the time. One has &tidle;B ∗(k) = B&tidle; (− k ) M M, since the magnetic field in Equation (49View Equation) must be real. Therefore, ℬM (k) = ℬM (− k) by definition. The asterisk indicates the complex conjugate number. It is often convenient to use the Doppler-shifted frequency denoted by a prime, ω ′M(k) = ωM (k ) − k∥Uj, as measured in a frame of reference moving with the bulk speed component, Uj, of species j along B0. The background electric field is taken to be zero, and the background plasma may be multi-component but is assumed to bear zero current and be quasi-neutral.

The quasilinear diffusion equation describes the evolution of the velocity distribution function, fj(v∥,v⊥,t), of any particle species j in an inertial frame of reference, in which the particles and waves are assumed to propagate. We will in this section of the paper assume that the VDF is normalised to a density of unity. The diffusion equation was originally derived by Shapiro and Shevchenko (1962) and is calculated in a transparent way by Stix (1992). It can generally be written as

∂ ∫ +∞ d3k ∑ 1 ∂ ( ∂ ) --fj(v∥,v⊥,t) = ----3- ℬˆM (k) ----- v ⊥νj,M (k;v∥,v⊥ )---fj(v ∥,v ⊥,t) , (50 ) ∂t −∞ (2π) M v⊥∂ α ∂α
where the pitch-angle gradient in the wave reference frame (defined by the phase speed vM(k ) = ωM(k )∕k∥, which for Alfvén waves would be equal to the Alfvén speed, VA) was introduced. It is given by the velocity derivative
∂-- -∂-- ( ) -∂-- ∂α = v⊥∂v ∥ − v∥ − vM(k) ∂v⊥ . (51 )
The magnetic field fluctuation spectrum is normalised to the background-field energy density. We find:
( ) ˆ ℬM-(k-)- k∥- 2 -------1-------- ℬM (k) = 8π B2 k ˆ 2. (52 ) 0 1− | k ⋅ eM(k ) |
The term in the denominator comes from the replacement of the electric field by the magnetic field. It turns out to be physically meaningful to introduce what we may call an ion-wave relaxation or scattering rate. It is defined as
2 +∑ ∞ || ||2 νj,M (k;v∥,v⊥ ) = π-Ω-j- δ(Vj (k, s) − v∥)|1(Js−1e+ + Js+1e−) + v∥-JseMz| . (53 ) | k∥ |s=− ∞ |2 M M v⊥ |
Note that this positive quantity has indeed the dimension of an inverse time or rate. Here we also introduced the s-order resonance speed and made use of the Bessel function (with index s):
ωM (k) − sΩj k⊥v⊥ Vj (k, s) = -------------,Js = Js(-----). (54 ) k∥ Ωj
The circular components of the wave polarization vector are defined as
e± (k) = eMx (k) ± i eMy (k). (55 ) M
The fundamental Equation (50View Equation) is quoted here without derivation as the starting point of our discussion on quasilinear diffusion. We have only slightly rewritten it in a form most appropriate for our subsequent purposes. Note that the famous quasilinear plateau in the VDF, to be discussed below, results for a vanishing pitch-angle gradient, meaning that ∂∕∂ α = 0, in which case wave absorption or emission ceases. The wave absorption coefficient can be calculated in QLT and is discussed below in Subsection 6.9.
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