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Appendix A: Alfvén wave propagation in a one-dimensional, ideal, β = 0 plasma

	Fig. A.1 a) Transverse perturbations vz for ideal (η = ν = 0) β = 0 plasma. b) Longitudinal perturbations vy for ideal β = 0 plasma. In both subfigures, t = 1000τA and dotted line represents y = vAt.
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Consider a cold (β = 0), ideal (η = ν = 0) plasma in our one-dimensional system (Sect. 3). Driving this system with boundary condition (31) generates an Alfvén wave at y = 0; this can be seen in Fig. A.1a. The Alfvén wave (v_z) propagates in the direction of increasing/positive y with amplitude A = 0.01 and there is no damping. The time of the snapshot can be read directly from the y-axis using the relation t = y / v_A. The ramp-up over the first four wavelengths is clearly visible.

Figure A.1b shows the longitudinal motions (v_y) in the system. Here, v_y is driven by the nonlinear terms of Eq. (29) (~B_z∂B_z / ∂y) and again there is no damping. In this paper, we call this second-order nonlinear effect the ponderomotive effect, and thus these longitudinal motions, which propagate at the Alfvén speed, are driven by the ponderomotive force (Alfvén wave-pressure gradients). At early times, these longitudinal motions are governed by a simplified version of Eq. (29): (A.1)Assuming boundary conditions (31) in our ideal, cold plasma and that v_x, v_y are initially zero, Eq. (A.1) has an analytical solution of the form: (A.2)which is valid for 0 < y < v_At. Note that these perturbations have an amplitude of and are always positive for an Alfvén wave propagating in the positive y-direction (due to the constant of integration, and thus due to the boundary conditions). Finally, note that the frequency of these longitudinal motions is twice that of the driving frequency, and that the motions do not grow with time. Interestingly, the ramp-up to maximum amplitude now occurs over eight wavelengths, i.e. twice that of the driven wave.

Appendix B: Derivation of Eqs. (38)–(41)

Equation (29) governs the driven, longitudinal motions in the system, and in 1D (∂ / ∂x = 0) has the form: (B.1)

Let us assume that our linear Alfvén wave can be represented as:

where A is the amplitude of our wave, ω is the driving frequency, k = k_R + ik_I is our wavenumber, with complex conjugate k ^∗  = k_R − ik_I. Thus, from Eq. (7) (with N₃ = 0) we have:

(B.2)

The right-hand-side of Eq. (B.1) has two contributions, which can be calculated using our expression for b_z. The first term has the form:

(B.3)

and the second has the form:

(B.4)

The form of Eqs. (B.2) − (B.4) lead to Eqs. (38)−(41).

Equations (B.2) − (B.4) can also be derived explicitly from our choice of v_z, i.e.:

where as before k = k_R + ik_I. Thus (from Eq. (7)) b_z has the form: (B.5)

Now we can explicitly determine the two contributions:

(B.6)

and the second has the form:

(B.7)

Equations (B.5) − (B.7) give equivalent solutions to Eqs. (B.2) − (B.4), and thus also lead to Eqs. (38) − (41).

Note that in both derivations, we have assumed an infinite harmonic solution for v_z, whereas our numerical simulations are for driven harmonic wavetrains. Thus, our solutions are only valid for c_st < y < v_At.

© ESO, 2011