Volume 568, August 2014
|Number of page(s)||11|
|Published online||08 August 2014|
The eigen-value problem solver BVPSUITE seems new in the context of coronal seismology, hence a study validating its accuracy seems in order. To this end, we carry out a series of computations in both slab and cylindrical geometries, and compare the numerically derived dispersion curves with available analytic expectations. Given that these analytic expressions are derived in the limit of zero-beta MHD, we focus on the same situation accordingly.
We start with the slab geometry and consider the static case. The magnetic slab and the uniform equilibrium magnetic field are both aligned with the z-axis. The equilibrium density is structured in the x-direction. We consider only the two-dimensional propagation in the x-z plane. It then follows from the zero-beta MHD equations that the eigen-value problem for fast waves in the static case can be formulated as (e.g., Eq. (3) in Terradas et al. 2005) (A.1)together with the boundary conditions (A.2)where sausage and kink waves differ in their behavior at the slab axis (x = 0). For the symmetric Epstein profile it is possible to solve the dispersion relation analytically for the phase speed vph (Cooper et al. 2003; MacNamara & Roberts 2011). For kink waves, it reads (A.3)and for sausage waves, it reads (A.4)where ζ represents the density contrast ρ0/ρ∞. Figure A.1 compares the dispersion diagram computed with BVPSUITE (the asterisks) with the analytic expressions (the dashed red curves) for a representative density contrast ρ0/ρ∞ being 5. The internal and external Alfvén speed, vA0 and vA∞, are represented by the two horizontal dash-dotted lines. It can be seen that the analytic results are exactly reproduced. In fact, we have carried out a series of comparisons covering an extensive range of density ratios, and found exact agreement without exception.
Dependence on the longitudinal wavenumber k of the phase speed vph for a static cold slab with a symmetric Epstein density profile. The asterisks represent the results obtained with BVPSUITE by numerically solving the eigen-value problem, while the dashed red lines represent the analytic results, Eqs. (A.3) and (A.4). Here a density ratio ρ0/ρ∞ of 5 is adopted.
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We now move on to the cylindrical case, but still restrict ourselves to the static case for the moment. We solve the same eigen-value problem in the text, Eqs. (12) and (13), but now take the background flow to be zero. In addition, we take p to be infinity, corresponding to a discontinuous distribution of the equilibrium density. To facilitate the validation study, we focus on sausage waves given the availability of their analytic behavior both in the neighborhood of the cutoff wavenumber and for large wavenumbers. In the former, the dispersion behavior in terms of angular frequency ω can be expressed as (Eq. (53) in Vasheghani Farahani et al. 2014) (A.5)where Δω = ω − kcvA∞, Δk = k − kc, and kc is the cutoff wavenumber given by Eq. (3). For Eq. (A.5) to be valid, one nominally requires that Δk/kc ≪ 1. On the other hand, when ka ≫ 1, the dispersion relation Eq. (8b) in Edwin & Roberts (1983) can be shown to yield (A.6)where j1,l (l = 1,2,··· ) denotes the l-th zero of J1 and l denotes the infinite number of sausage branches. For the first branch we compute, j1,1 = 3.83.
Similar to Fig. A.1 but for sausage waves supported by static cylinders with discontinuous density distribution. The dashed red curves represent the analytic results in the vicinity of the cutoff wavenumber (Eq. (A.5)) and for big wavenumbers (Eq. (A.6)). Here two density ratios, 4 and 25, are adopted.
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Figure A.2 presents the dispersion curves expressing the phase speed vph as a function of longitudinal wavenumber k. For illustrative purposes, we present the results for two density contrasts, one large (ρ0/ρ∞ = 25) and the other rather mild (ρ0/ρ∞ = 4, inset). The results computed with BVPSUITE are given by the asterisks, and the analytic results are represented by the red dashed curves. The two horizontal dash-dotted lines represent the internal and external Alfvén speeds. Evidently, the numerical results excellently capture the cutoff wavenumber, and agree remarkably well with the analytic results for the appropriate wavenumber ranges. Actually, this can be said for all the tests we performed, which cover an extensive range of density ratios.
Our next validation study pertains to sausage waves supported by cold cylinders with flow. To this end we start with the comprehensive study by Goossens et al. (1992) where the sophisticated equilibrium configuration takes account of a background flow, and azimuthal components of the equilibrium velocity and magnetic field. By neglecting these azimuthal components and specializing to a piece-wise constant distribution for both the equilibrium density and flow speed, one finds that Eq. (18) in Goossens et al. (1992) simplifies to (A.7)where denotes the Fourier amplitude of the total pressure perturbation. Equation (A.7) is valid both inside and outside the cylinder, and m2 is defined as in which we have assumed that the ambient corona is static. For the simple configuration in question, Eq. (A.7) is analytically solvable in terms of Bessel function J0 (K0) inside (outside) the cylinder for trapped modes. A dispersion relation then follows from the continuity of the transverse Lagrangian displacement and total pressure perturbation at the cylinder boundary (see also, e.g., Terra-Homem et al. 2003), (A.8)where . Furthermore, the prime denotes the derivative of Bessel function with respect to its argument, e.g., with η = m∞a.
Dependence on the longitudinal wavenumber k of the phase speed vph for a cold cylinder with flow where the transverse distributions of both the equilibrium density and field-aligned flow adopt a step-function form. The asterisks represent the results obtained with BVPSUITE by numerically solving the eigen-value problem, while the dashed lines represent the solution to the analytically derived dispersion relation (Eq. (A.8)). The horizontal dash-dotted lines correspond to the internal and external Alfvén speeds. Here three different values of the internal flow speed U0 are examined for a density ratio ρ0/ρ∞ being 25.
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Figure A.3 presents the dependence of the phase speed vph on the axial wavenumber k for a representative density
contrast ρ0/ρ∞ being 25. For illustrative purposes, we examine three values of the internal flow speed U0, namely, 0 (red), 0.08 (green), and 0.16 (blue) times the external Alfvén speed vA∞. The horizontal dash-dotted lines represent the internal and external Alfvén speeds. The asterisks give the results from solving Eqs.(12) and (13) in the text with BVPSUITE, where both the density and flow speed profile steepnesses, p and u, are taken to be infinity. For comparison, the dashed curves represent the solutions to the algebraic dispersion relation, Eq. (A.8). One can see that the two sets of solutions agree with each other remarkably well. As a matter of fact, the agreement is found for all the tests we conducted, where we examined an extensive range of density contrasts and flow magnitudes.
In closing, we mention that at this stage of its development, the Matlab eigen-value problem solver BVPSUITE cannot find complex eigen-values, thereby limiting its use to trapped modes. Despite this, given that BVPSUITE is publicly available and easy to use with its friendly graphical user interface, this accurate code may find a wider application to problems encountered in coronal seismology.
© ESO, 2014
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