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Alfvén Reflection and Reverberation in the Solar Atmosphere

Abstract

Magneto-atmospheres with Alfvén speed [a] that increases monotonically with height are often used to model the solar atmosphere, at least out to several solar radii. A common example involves a uniform vertical or inclined magnetic field in an isothermal atmosphere, for which the Alfvén speed is exponential. We address the issue of internal reflection in such atmospheres, both for time-harmonic and for transient waves. It is found that a mathematical boundary condition may be devised that corresponds to perfect absorption at infinity, and, using this, that many atmospheres where a(x) is analytic and unbounded present no internal reflection of harmonic Alfvén waves. However, except for certain special cases, such solutions are accompanied by a wake, which may be thought of as a kind of reflection. For the initial-value problem where a harmonic source is suddenly switched on (and optionally off), there is also an associated transient that normally decays with time as \(\mathcal{O}(t^{-1})\) or \(\mathcal{O}(t^{-1}\ln t)\), depending on the phase of the driver. Unlike the steady-state harmonic solutions, the transient does reflect weakly. Alfvén waves in the solar corona driven by a finite-duration train of p-modes are expected to leave such transients.

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Notes

  1. 1.

    A quadratic wave-energy equation is easily constructed from the linearised momentum and induction equations: \(\partial\mathcal{E}/\partial{t} + \partial{\mathcal{F}}/\partial {x}=0\), where \(\mathcal{E}=\frac{1}{2}\rho v^{2} + \frac{1}{2}b^{2}\) is the energy density, \(\mathcal{F}=-Bbv\) is the wave-energy flux, v=∂ξ/∂t is the plasma velocity, and b=−B∂ξ/∂x is the magnetic field perturbation. Energy fluxes may be attributed to solutions of the wave equation using the formula for \(\mathcal{F}\), and hence reflection coefficients may be calculated if these solutions can be split into upward- and downward-propagating parts. If ξ and b are being modelled as complex quantities, then we should write \(\mathcal {E}=\frac{1}{2}\rho|v|^{2} + \frac{1}{2}|b|^{2}\) and \(\mathcal{F}=-B \operatorname {Re}(bv^{*})\).

  2. 2.

    Alternatively, keep r real but insert a weak frictional force [−γ∂ξ/∂t] on the right-hand side of Equation (7), with 0<γω. Then the upgoing solution is \(H_{0}^{(2)}(\omega r\sqrt{1+\mathrm{i} \gamma /\omega})\). This places the argument of the Hankel function below the cut for negative r.

  3. 3.

    All formulae used in this section relating to Bessel functions may be found in Olver and Maximon (2010), particularly the asymptotic formulae 10.17.5 – 6 and the analytic continuations 10.11.5 and 10.11.7 – 8.

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Correspondence to P. S. Cally.

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Cally, P.S. Alfvén Reflection and Reverberation in the Solar Atmosphere. Sol Phys 280, 33–50 (2012). https://doi.org/10.1007/s11207-012-0052-3

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Keywords

  • Waves, Alfvén
  • Waves, magnetohydrodynamic