The treatment of coronal loops as harmonic oscillators can be rigorously justified for particular equilibria. For example, neglecting gravity, gas pressure, resistivity and viscosity for simplicity and assuming linear perturbations about a static potential equilibrium, the momentum equation can be written as (A.1)where ξ is the plasma displacement vector, b is the magnetic field perturbation, ρ0 is the equilibrium density and B0 is the equilibrium magnetic field which we have assumed curl-free. Similarly, the time-integrated induction equation is (A.2)We consider a straight equilibrium magnetic field by setting B0 = B0ẑ (B0 is constant to give magnetic pressure balance in the equilibrium). Then, performing some algebra, the governing equations can be combined to give (A.3)where is the Alfvén speed and is a spatial differential operator defined by the equation above. In a medium with uniform Alfvén speed, incompressible solutions (with ∇·ξ⊥ = 0) to Eq. (A.3) are Alfvén waves, while the divergence of Eq. (A.3) gives the governing equation for fast waves in cold plasma. Equation (A.3) is also valid when the Alfvén speed is a function of position, as happens in equilibria supporting kink waves.
Standing wave solutions are obtained when ξ⊥ has separable time and spatial dependences, i.e. (A.4)Substituting this form into Eq. (A.3), we find (A.5)where X1 and X2 are the components of X in the chosen coordinate system and we have introduced −ω2 as the separation constant. It immediately follows that (A.6)which is the ordinary differential equation for a harmonic oscillator, while (A.7)implies that X1 and X2 are eigenfunctions of , each having eigenvalue −ω2. Oscillations appear for real ω, in which case Eq. (A.6) can be scaled so that T represents, e.g., the displacement of the loop apex from its equilibrium position. If ω is imaginary then the equilibrium can be unstable, but we exclude that possibility here for consistency with our observations. The final possibility for ω2 is that it could potentially be complex, which may introduce decay alongside oscillation, as discussed later.
The existence of suitable eigenfunctions of is expected on the basis that standing transverse oscillations of coronal loops are observed, including in the event presented in Sect. 2. We also provide mathematical justification by considering kink modes for the well-known magnetic cylinder model of Edwin & Roberts (1983). In this model, the equilibrium Alfvén speed is specified in cylindrical coordinates (r,θ,z) as (A.8)with vAe>vA0 so that the inner region of radius a acts as a waveguide for fast waves, giving rise to body modes. Since we have B0 constant for equilibrium magnetic-pressure balance, the Alfvén speed profile corresponds to a density enhancement inside the cylinder. We also assume that there are line-tied boundaries at z = 0 and z = L, which gives rise to standing waves with wave number kz, the fundamental harmonic having kz = π/L. The spatial eigenvalue Eq. (A.7) can be solved for this equilibrium using the approach described by Edwin & Roberts (1983), which yields Xr(r,θ,z) and Xθ(r,θ,z) with radial dependences in terms of Bessel functions. Meanwhile, the eigenvalue is determined by (A.9)where cp(akz) is the phase speed of the kink wave. In the thin-tube limit, kza ≪ 1, the waves become non-dispersive and cp reduces to the kink speed, (A.10)Since spatial eigenfunctions have been found and ω2 is real and positive, standing kink waves in the classic magnetic cylinder model are consistent with a harmonic oscillator view of coronal loops.
More generally, standing kink oscillations can decay over time due to physical damping by viscosity, resistivity etc. or due to transfer of energy from the kink mode by resonant absorption (Ruderman & Roberts 2002; Goossens et al. 2002). In these situations, the above approach of separation of variables can still be applied, leading to an oscillation equation of the form of Eq. (A.6). However, solution of the spatial operator equations now yields a complex frequency, ω = ωr + iωi. When this occurs it can be useful to recast the oscillation equation as a damped harmonic oscillator with real coefficients: (A.11)where ω0 is the frequency of the corresponding undamped oscillator and κ is the damping ratio. To see the correspondence, observe that harmonic solutions of this new equation with an eiωt time dependence also have complex frequency, with and ωi = ω0κ (we will consider underdamped solutions with 0 <κ< 1 so ωr is always real and ωi> 0 giving decay). This implies that the original oscillation equation with complex ω has the same oscillatory and decay behaviours as the damped oscillation equation with real coefficients set as and ω0κ = ωi, where ωr and ωi are obtained from the spatial eigenvalue equations.
We conclude by noting that the approach described above is fairly general and it can be applied to analyses that make fewer simplifying assumptions. For example, gas pressure was in fact included by Edwin & Roberts (1983), while magnetic curvature, density stratification, flux tube expansion and non-circular cross-sections have all been considered since then (see Ruderman 2003, 2015; Van Doorsselaere et al. 2004; Andries et al. 2005; Ruderman et al. 2008). In each case, suitable spatial eigenfunctions were obtained and a dispersion relation giving ω was derived, which determines the real coefficients in the corresponding damped oscillation equation.
Our model equation produces all three types of solution discussed in Sect. 3. Writing Eq. (2) in dimensionless form, (B.1)where with τ = 2π/ω, and with D an appropriate length scale such as the final displacement. This equation is equivalent to a pair of coupled first-order ordinary differential equations, (B.2)which are easily integrated using fourth-order Runge-Kutta. Setting the damping parameter as κ = 0.1 produces decay similar to the motivating example. Solutions of the three types shown in Fig. 4 can then be obtained using different driving functions for . Impulsively excited oscillations (Fig. 4a) result if the system is driven using a step function, e.g. (B.3)where is the time of the collapse. Examples of gradual displacement (Fig. 4b) can be produced using (B.4)with Δ ≳ 1. In fact, Eq. (B.4) can produce any of the three types of response, depending on the value of Δ, and converges to Eq. (B.3) for Δ → 0. Nonetheless, the clearest examples of oscillation during collapse (Fig. 4c) are produced when the change in equilibrium starts sharply, e.g. under (B.5)with Δ = 1.
It was noted in Sect. 3 that the greatest amplitude oscillations are obtained when the change in equilibrium position is initially sharp. Physical justification was given in Sect. 3, but a quantitative demonstration is also provided here. Using the non-dimensionalised model of Eq. (B.1) with damping turned off (κ = 0), we consider a test problem in which the equilibrium position accelerates from rest to motion at a constant speed.
Assuming the equilibrium position accelerates at a constant rate during a switch-on interval equivalent to δ periods of oscillation, (C.1)Here, the normalising length has been set to the distance that the equilibrium position moves per period in the constant speed phase, . The specified driver produces oscillations superimposed on a never-ending collapse, and the amplitude of the oscillations is readily measured from the long term solution. Figure C.1 plots the normalised amplitude for a range of switch-on times, δ, which is the only free parameter in the dimensionless model. In this test problem, dimensional amplitude is limited only by the maximum rate of collapse, with amplitudes capped at 0.16 times the maximum displacement per period. Nodes in Fig. C.1 show cases where acceleration of the equilibrium position resonates with the loop period in such a way that long-term oscillations are not produced. Most significantly, the plot confirms that the largest amplitude oscillations are excited when the switch-on time is a small fraction of the oscillation period.
Examining the motion of loop L3 in Fig. 2, the inferred equilibrium position shifts by approximately 12 Mm during the loop’s first period of oscillation. Multiplying this distance by 0.16 gives an estimate of 2 Mm for the maximum amplitude that can be excited by the corresponding rate of contraction. That estimate is very close to the actual amplitude seen for L3. Based on these values and the fall-off of amplitude with increasing switch-on time (Fig. C.1) we suggest that a quarter of L3’s period (approximately 40 s) is a reasonable upper limit to place on the energy-release switch-on time for the SOL2012-03-09 flare.
Amplitude of oscillation as a function of the energy-release switch-on time, for a test problem in which the equilibrium position accelerates from rest to motion at a constant speed. The switch-on time (duration of the acceleration) is normalised to the period of oscillation, and the amplitude of oscillation is normalised to the equilibrium displacement per period after the acceleration.
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