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Appendix A: Landau damping of ion acoustic waves in fluid theory
Appendix A.1: Elements of kinetic description
Within the twocomponent kinetic theory, Landau damping is well known. Here, we give a few details only in order to compare it with fluid description presented below in Sect. A.2. Small electrostatic perturbations of the form exp( − iωt + ikz) yield the dispersion equation (A.1)The dispersion Eq. (A.1) describes the plasma (Langmuir) and the IA oscillations, and ζ = v_{z}/v_{Tj}.
For the present case with electrons and ions and in the limit (A.2)the standard expansions may be used , and , where ω = ω_{r} + iγ is complex and its real part is assumed to be much larger than its imaginary part. The procedure yields the dispersion equation (A.3)We introduce ℑ △ (ω,k) and ℜ △ (ω,k) denoting the imaginary and real parts of Eq. (A.3), respectively. Setting the real part ℜ △ (ω,k) = 0 yields the approximate expression for the IA spectrum , . The approximate Landau damping of the wave is obtained from (A.4)Here, τ = T_{e}/T_{i}, and we have assumed massless electrons and singly charged ions, that is, the Landau damping is due to the ions only. Otherwise, the electron contribution to the Landau damping would appear in addition to the previously given Landau damping term, where z_{i} is the ion charge number.
Fig. A.1
Absolute value of the Landau damping of the IA wave (normalized to ω_{r}) in terms of τ = T_{e}/T_{i}. 

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In Fig. A.1, we give the absolute value of the approximate (dotted line) Landau damping (8) (normalized to ω_{r}) in terms of τ, together with the absolute value of the exact normalized Landau damping (full line) that may be obtained numerically from the general dispersion law (A.1). The nature of the dashed line will be given in Sect. A.2 below. It is seen that in the given range of τ the approximate kinetic expression (A.4) yields inaccurate values for the damping. For τ = 1, which is of interest for the solar corona, the exact Landau damping γ_{ex} = 0.394ω_{r} is larger by about a factor 2 than the approximate kinetic Landau damping γ_{app}.
For practical purposes, we note that the exact kinetic Landau damping may conveniently be expressed in terms of τ by the following polynomial (cf. Chen 1984): (A.5)The difference between γ_{ex} and γ_{app} follows from the fact that, in the limit τ → 1, the assumptions used to obtain the approximate kinetic damping become violated. One is thus supposed to use the rather inconvenient general dispersion Eq. (A.1), which contains an integral. Below, we show that this problem may be circumvented by using the twofluid theory and a modeled fluidLandau damping.
Appendix A.2: Twofluid description
We use the linearized momentum equation for the general species α in a quasineutral plasma (A.6)The term with μ_{L} is the one introduced for the first time by D’Angelo et al. (1979) to describe the Landau damping effect on the fast solar wind streams with a spatially varying ratio τ( =T_{e}/T_{i}) of the electron and ion temperatures. It is chosen in such a way to quantitatively describe the known properties of the Landau effect. These include the fact that the ratio d = δ/λ, between the attenuation length δ and the wavelength, is independent of the wavelength and the plasma density and dependent on τ in a prescribed way (Sessler & Pearson 1967; Andersen et al. 1968). These requirements appear to be fulfilled by (A.7)Here, , while d(τ) satisfies a curve that is such that the attenuation is strong at τ ≈ 1 and weak for higher values of τ. It turns out that a sufficiently good choice for d(τ) is
We observe that d(1) ≃ 0.4. We also use the ion continuity and in the massless electron limit from the electron momentum, we obtain just the Boltzmann distribution for electrons. This yields the dispersion equation for the Landaudamped IA wave (A.8)For the complex frequency ω = ω_{r} + iγ_{f}, we obtain (A.9)
This shows that the mode oscillation frequency ω_{r} is reduced as compared to the ideal plasma case, that is, without Landau damping. It is now appropriate to compare this fluidmodeled Landau damping γ_{f} with the exact kinetic damping given earlier. The normalized absolute fluidmodeled Landau damping  γ_{f}/ω_{r}  ≃ 1/(2πd) is presented by the dashed line in Fig. A.1. The difference between γ_{f} and γ_{ex} at τ = 1 is only 0.008. Clearly, the presented fluidmodeled Landau damping is, in fact, much more accurate than the approximate kinetic expression (A.4).
© ESO, 2013