Magneto-acoustic waves in compressible magnetically twisted flux tubes
Download the paper, Solar Physics, 2010.
Figure 1
Figure 1: The equilibrium configuration of the problem. The uniformly twisted magnetic flux tube is in an ambient straight and uniform magnetic field. At the boundary of the flux tube there is a jump in magnetic twist.
The general dispersion equation for linear MHD waves in a magnetically uniformly twisted compressible flux tube embedded in a compressible and uniform straight magnetic environment.
Figure 2a
Figure 2b
Figure 2a, 2b: The dimensionless phase speed (Vph) as function of the dimensionless wavenumber kr0 for cases (a) VAe > VAi (left panel) and (b) VAe < VAi (right panel) in an incompressible non-twisted flux tube. Solutions to the dispersion relation for the sausage (m = 0), kink (m = 1) and fluting (m > 1) modes are plotted. Comparison of our numerical solutions of the general dispersion equation, in the case of incompressible non-twisted magnetic flux tube limit with those obtained by Edwin and Roberts (1983) (see their Figure 2) shows an excellent agreement.
Figure 3a
Figure 3b
Figure 3a, 3b: The dimensionless phase speed (Vph) of the m = 0 (sausage) modes in a twisted tube (VAiφ = 0.1) as function of the dimensionless wavenumber kr0 for VAe > VAi and VAi > VAe are shown at the left and right panels, respectively. Surface and body modes are found. Note the change of character, from body to surface wave, at kr0 = 0.5 (left panel) and kr0 = 1 (right panel). The grey region corresponds to kα2 < 0, where only body waves are permitted. The dashed boundary curves correspond to the loci where kα2 ≡ 0 in the plots.
Figure 4a
Figure 4b
Figure 4a, 4b: The dimensionless phase speed (Vph) of the m = 1 fast kink modes in a magnetically twisted incompressible tube (VAiφ = 0.1) as function of the dimensionless wavenumber (kr0) VAe > VAi (left panel) and VAi > VAe (right panel). There are surface and body modes, which are both shown. Note the band of pure body kink modes accumulate around VAi as kr0 → ∞. Also, note that Vph → ∞ as kr0 → 0 (ie, thin tube approximation) for all of the kink modes.
Figure 5
Figure 5: The dimensionless phase speed (Vph) of the m=1 fast pseudomode in a twisted tube as function of the dimensionless wavenumber (kr0) for VAi > VAe for a series of a normalised magnetic twist: VAiφ = 0.0001, 0.001, 0.01 and 0.1.
Figure 6a
Figure 6b
Figure 6a, 6b: The dimensionless phase speed (Vph) as function of the dimensionless wavenumber (kr0) under (a) typical photospheric (ie, VAi > CSe > CSi > VAe) and (b) coronal conditions (ie, VAe > VAi > CSi > CSe) for an untwisted magnetic flux tube are shown in the (a) left and (b) right panels, respectively. Here (a) CSe = 0.75VAi, VAe = 0.25VAi, CSi = 0.5VAi for the photospheric case and (b) CSe = 0.25VAi, VAe = 2.5VAi, CSi = 0.5VAi for the solar corona one.
In the left panel three modes of the fast surface waves that are bounded within [CSi,CSe] and the infinite number of the slow body waves bounded within [CTi , CSi ] are plotted. The slow kink, sausage, etc. surface waves are very close to each other (just under CTi).
In the right panel sausage (m = 0), kink (m = 1) and fluting (m = 2) modes are shown. The infinite number of the fast body waves are bounded within [VAi,VAe] and the infinite number of the slow body waves are bounded within [CTi, CSi]. m = ij : the i refers to the mode (sausage, kink, etc.) and j refers to the j-th branch of the zeroes of eigenfunctions in the radial direction. Only three branches of each mode of the infinitely many slow and fast body waves are plotted.
Figure 7a
Figure 7b
Figure 7a, 7b: The dimensionless phase speed (Vph) of the sausage (m=0) and kink (m=1) modes as function of the dimensionless wavenumber (kr0) under photospheric conditions (ie, VAi > CSe > CSi > VAe) for a uniformly twisted intense magnetic flux tube (VAiφ = 0.1) are shown in the left and right panels, respectively. The sound and Alfvén speeds that characterise the equilibrium are the same as in Figure 6.
Figure 8a
Figure 8b
Figure 8a, 8b: Zoom in of the region where the fast sausage and kink modes are present (see Figure 7). The phase speeds of the sausage (left) and kink (right) wave modes are plotted for selected cases of magnetic twist: VAiφ=0.0, 0.001, 0.01 and 0.1.
Figure 9a
Figure 9c
Figure 9b
Figure 9d
Figure 9a, 9b, 9c, 9d: Zoom in of the region where slow kink surface (m=1), fluting (m=2) and body modes are present (see Figure 7). Only two branches (j=1, 2) of the infinitely many slow body waves are plotted. The modes are shown for a series of selected magnetic twist: VAiφ=0.0, 0.001, 0.01 and 0.1.
Figure 10a
Figure 10b
Figure 10c
Figure 10a, 10b, 10c: The normalised eigenfunctions ξr and PT of different branches (j=1..3) of the fluting mode (m=2) for VAiφ=0.1 for a series of non-dimensional phase speeds Vph=0.476, 0.482 and 0.488, and for a fixed dimensionless wavenumber kr0=3.0. These phase speeds (Vph) and wavenumber (kr0) correspond to the vertical dashed lines in Figure 9d.
Figure 11a
Figure 11c
Figure 11b
Figure 11d
Figure 11a, 11b, 11c, 11d: The dimensionless phase speed (Vph) of the fast and slow sausage surface mode m=0 and slow body sausage modes as function of the dimensionless wavenumber (kr0) illustrated for CSi = 0.5, CSe = 1.1, VAe = 0.75. This equilibrium case may correspond to a cooler evacuated flux tube embedded in a hotter, high-β magnetised plasma. Fast surface sausage and slow body waves are shown for four different values of magnetic twist VAiφ= 0, 0.05, 0.1, 0.2. Note the coupling of the fast mode to the Alfvén mode in the presence of a finite twist.
Figure 12a
Figure 12b
Figure 12a, 12b: The dimensionless phase speed (Vph) as function of dimensionless wavenumber (kr0) under coronal conditions (ie, VAe >VAiCSi >CSe) for a twisted (VAiφ = 0.1) intense and hot flux tube. In the left and right panels the sausage (m=0) and kink (m=1).
Figure 13a
Figure 13b
Figure 13a, 13b: Zoom in of the region where slow body sausage modes are present (see Figure 12). Only three branches (j = 1,...,3) of each modes of the infinitely many slow body waves are plotted. The slow body sausage modes are shown for selected cases of magnetic twist: VAiφ= 0.0 and 0.1, where the twist seems to have little effect on these modes.
Figure 14a
Figure 14c
Figure 14b
Figure 14d
Figure 14a, 14b, 14c, 14d: Zoom in of the region where slow body kink modes are present (see Figure 12). Only three branches (j=1,...,3) of each mode of the infinitely many slow body waves are plotted. The slow body kink modes are shown for selected cases of magnetic twist: VAiφ=0.0, 0.001, 0.01 and 0.1.