Figure 1: Energy density as a function of the radial position for the modes m=1, m=2 and m=3. In this plot the three modes have the same total energy E (energy density integrated over space). The loop length is L=50R and the density contrast is . The sausage mode is not represented since for these loop parameters this mode is not trapped. | |
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Figure 2: a) Example of kink perturbation and b) fluting (m=2) perturbation in the perpendicular plane to the loop axis. For this initial disturbance r_{0}=15R and a=2R. In the vertical direction the dependence is sin(kz). White colour corresponds to positive magnetic pressure perturbations while black colour represents negative values. The circular white region in the centre represents the loop. | |
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Figure 3: a) Displacement and b) trapped energy as a function of r_{0} and a for the m=1 mode. For these plots L=50R and . The behaviour of the displacement and energy with r_{0} and a is given by Eqs. (18) and (19) respectively in the limit . | |
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Figure 4: a) Displacement and b) energy as a function of the loop length. For all the cases the width of the perturbation is constant a=R. Solid line corresponds to r_{0}=10R, dashed to r_{0}=20R and dotted-dashed corresponds to r_{0}=40R. For these plots . | |
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Figure 5: a) Displacement at r=R and b) energy as a function of the density contrast. The perturbation is located at r_{0}=40R, a=3R. The solid line corresponds to a loop with L=50R while the dashed line corresponds to L=100R. | |
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Figure 6: Example of an azimuthally localised perturbation. The circular white region represents the loop. For this initial disturbance r_{0}=15R, a=2R and . This disturbance is decomposed in Fourier sinus series using Eq. (10). | |
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Figure 7: a) Energy of the Fourier components of the initial perturbation calculated using Eq. (21). For this plot r_{0}=40R, a=4R and (n=1, L=50R). b) Trapped energy of each eigenmode. Note that the vertical axis is in logarithmic scale. The difference in energy between consecutive modes increases with m. For example, the trapped energy for the m=1 is 4 10^{2} larger than the energy of the m=2. | |
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Figure 8: Example of longitudinally localised perturbation. The white area represents the loop. For this initial disturbance r_{0}=15R, a=2R, and z_{0}=L/4. This disturbance is decomposed in sinus series using Eq. (10) with m fixed (m=1). | |
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Figure 9: a) Energy of the Fourier components (calculated using Eq. (23) with m fixed) of the initial perturbation given by Eq. (22). The m=5 has the largest energy in the decomposition. Only the terms with m odd contribute to the sum. b) Trapped energy of each eigenmode (the vertical axis is in logarithmic scale). For this case r_{0}=40R, a=4R (L=50R), z_{0}=L/2 and . | |
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Figure 10: a) Energy of the Fourier components (calculated using Eq. (11) with m fixed) of the initial perturbation given by Eq. (22). The m=6 has the largest coefficient in the decomposition. b) Trapped energy of each eigenmode (the vertical axis is in logarithmic scale). Note that the fundamental mode has the largest energy (although the first harmonic has the maximum weight in the decomposition). For this case r_{0}=40R, a=4R (L=50R), z_{0}=L/4 and . | |
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