List of Figures

View Image Figure 1:
Displacement versus time produced by two wave trains impacting on a prominence: (a) and (b) in the prominence apex; (c) and (d) in the NE leg; (e) and (f) in the SW leg (from Hershaw et al., 2011).
View Image Figure 2:
Time slices taken at three heights in a quiescent prominence. The bright sinusoidal patterns are caused by horizontal oscillations of the plasma with periods between 20 and 40 min. The orange lines denote oscillations with phases that approximately match. The slope of these lines implies an upward propagation speed of about 10 km s–1 (projected on the plane of the sky) (from Berger et al., 2008).
View Image Figure 3:
Period of prominence Doppler velocity oscillations as a function of the line-of-sight magnetic field strength. The top and bottom panels correspond to active region and non-active region prominences, respectively (from Harvey, 1969).
View Image Figure 4:
Temporal variation of the Doppler velocity along threads of a solar filament. Numbers on the right label the various threads. For each thread, the curves correspond to the Doppler velocity measured at different points along the thread (from Yi et al., 1991).
View Image Figure 5:
Displacement of two spines of a quiescent prominence (thin lines) and best fits using the function A [0]sin(2πt∕A [1 ] + A [2])exp (A[3]t) (thick lines). The fitted values of the parameters A [0] to A [3] are written at the bottom of the figure. Note that the values of A[3] displayed in the figure cannot be correct since they give a very strong amplification/damping that totally disagrees with the almost undamped behaviour of the thick lines (from Ning et al., 2009a).
View Image Figure 6:
Observed Doppler velocity (dots) and fitted function (continuous line) versus time at two different positions in a quiescent prominence. The period is 70 min in both positions and the damping time is 140 and 101 min, respectively. The function fitted to the observational data is of the form v0 cos(ωt + ϕ)exp (− t∕τ) (adapted from Molowny-Horas et al., 1999).
View Image Figure 7:
Results of fitting the function of Figure 6 to the Doppler velocity in the whole two-dimensional field of view. The spatial distribution of the fitted period and damping time is shown in the top panels, while that of the correlation coefficient and fitted amplitude is displayed in the bottom panels. The continuous white line (black in the top left panel) represents the approximate position of the prominence edge. The photosphere is slightly outside the image top (from Terradas et al., 2002).
View Image Figure 8:
Left: Time-frequency diagrams of the Doppler velocity at several aligned, equispaced points in a quiescent prominence. White/black correspond to large/small wavelet power. Right: Time variation of wavelet power from the diagrams on the left column for a period of 7.5 min (i.e., frequency around 2.2 mHz). The presence of power peaks suggests a finite duration of the perturbation, while the linear displacement of these peaks at the seven positions from t = 28 min to t = 42 min is an indication of a disturbance travelling with a group velocity vg ≥ 4.4 km s–1 (from Molowny-Horas et al., 1997).
View Image Figure 9:
(a) Time-space diagram of the 195 Å image intensity along the main axis of a filament. The vertical black stripes are caused by the lack of observational data. (b) Spatial distribution of the Fourier phase (gray coloured contours) (from Foullon et al., 2004).
View Image Figure 10:
Relative Fourier phase as a function of position along the slit for several sets of consecutive points with similar oscillatory period: (a) 10.0 min, (b) 7.5 min, (c) 12.0 min, and (d) 4.0 min. The variation of the phase in (a) is not linear and so this oscillatory feature is not interpreted as a true signal. Regarding (c) and (d), the phase varies linearly with position, but the number of points involved is too small to make a firm conclusion. Finally, the phase in (b) displays a very robust linear dependence with distance, so this is interpreted as a signature of wave propagation (from Molowny-Horas et al., 1997).
Watch/download Movie Figure 11: (avi-Movie; 3010 KB)
Movie: Hβ line center images of a quiescent prominence observed with the VTT of Sacramento Peak Observatory. Images have been coaligned and a persistent drift towards the left has been suppressed. The thick white line displays the prominence edge and the solar photosphere is at the top (from Terradas et al., 2002).
Watch/download Movie Figure 12: (avi-Movie; 11684 KB)
Movie: Temporal evolution of the Doppler velocity in all points of the field of view of Figure 11 (from Terradas et al., 2002).
View Image Figure 13:
Two-dimensional Doppler velocity distribution at a given time in a quiescent prominence. The signal in the black rectangle can be fitted by a damped sinusoid with a high correlation coefficient (see Figure 7). Two paths (straight continuous and dashed lines) were selected. The continuous white line represents the approximate position of the prominence edge. The photosphere is slightly outside the image top (from Terradas et al., 2002).
View Image Figure 14:
Doppler velocity versus position and time along the solid path in Figure 13. Left: raw Doppler signal; middle: fitted exponentially damped sinusoid; right: Fourier phase associated to the 75 min periodicity (from Terradas et al., 2002).
View Image Figure 15:
Doppler velocity versus position and time along the dashed path in Figure 13. Left: raw Doppler signal; right: Fourier phase associated to the 75 min periodicity (from Terradas et al., 2002).
View Image Figure 16:
Fourier phase associated to a period around 75 min (that is, the one corresponding to the largest peak in the Fourier spectrum) for the rectangular region selected in Figure 13, both as a contour and as a surface plot. The selected paths are also displayed with continuous and dashed straight lines. Note that cuts of the Fourier phase along these two paths give rise to the Fourier phase displayed in Figures 14 and 15 (from Terradas et al., 2002).
View Image Figure 17:
Arrows represent the wavevector field computed from the gradient of the Fourier phase displayed in Figure 16, where the length of the arrows is proportional to the modulus of the wavevector. The phase velocity is shown with the help of different levels of grey and black and white colors (from Terradas et al., 2002).
Watch/download Movie Figure 18: (mpg-Movie; 7124 KB)
Movie: Hα line center images of a quiescent filament observed with the Swedish Solar Telescope in La Palma. The small-scale structures display the characteristic filament counter-streaming motions and undergo simultaneous transverse oscillations, detected as periodic Doppler variations (from Lin et al., 2003).
Watch/download Movie Figure 19: (mpg-Movie; 5825 KB)
Movie: Ca ii H line images taken with Hinode/SOT that shows ubiquitous continuous horizontal motions along the prominence threads at the top right of the image. These threads also oscillate up and down as they flow (from Okamoto et al., 2007).
View Image Figure 20:
Close-up view of a flowing thread displaying transverse oscillations. The measured flow speed is 39 km s–1, the amplitude of vertical oscillations is 900 km and the period is 174 s (from Okamoto et al., 2007).
View Image Figure 21:
Example of a prominence thread undergoing synchronous oscillations along its entire length (all images are shown in negative contrast). (a) The ends of the considered thread are marked by the two arrows. S1 to S5 indicate the locations used to make the height versus time plots shown in panels (b) to (f). (b) – (f) Height-time plots for the locations indicated in (a). Maximum and minimum amplitudes occur at nearly the same time for all locations (from Okamoto et al., 2007).
View Image Figure 22:
(a) and (b) Time-slice diagrams of the Hα line intensity and Doppler shift along a filament thread. (c) Data of panel (b) shown as a set of curves instead of as a contour plot. Each curve represents the Doppler velocity along the thread for a fixed time (frame). (d) and (e) Signals from panel (c) for some selected times (frames). (f) and (g) Power spectra of the Doppler shift along the thread for two times. Large peaks help identify the wavelength of propagating oscillations (from Lin et al., 2007).
View Image Figure 23:
Simple models of a prominence. (a) Mass suspended from an elastic string under the influence of gravity and the tension force. (b) Mass suspended from a taut string subject solely to the tension force (from Roberts, 1991). (c) Taut string with density ρc except for a central part with density ρp; gravity is also neglected (from Oliver et al., 1993). The size of the system (2ℓ in panels (a) and (b) and 2xc in panel (c)) is denoted by 2L in the text.
View Image Figure 24:
Spatial distribution of some normal modes of the string shown in Figure 23c. (a) Hybrid mode, (b) first internal even harmonic, (c) first internal odd harmonic, (d) first external odd harmonic. The spatial coordinate is given in units of L and the string of density ρp is in the range − 0.1 ≤ x ≤ 0.1. The wave speeds are cpro = 15 km s–1 and ccor = 166 km s–1, representative of prominence and coronal sound speeds, and the density ratio is ρp∕ρc = 11.25.
View Image Figure 25:
Sketch of a prominence configuration based on the Kippenhahn and Schlüter (1957) model. In the text the system size (2Δ) and the prominence width (D) are denoted by 2L and 2x p, respectively. Except for the filed line curvature, this configuration is identical to that of Figure 27 (from Anzer, 2009).
View Image Figure 26:
Sketch of the prominence slab model with longitudinal magnetic field used by Joarder and Roberts (1992a). These authors assumed that the coronal environment in which the prominence is embedded extends infinitely in the x-direction. In this figure the width of the prominence is denoted by 2a, whereas in our text 2xp is used.
View Image Figure 27:
Schematic diagram of a prominence slab in a coronal environment. The magnetic field is perpendicular to the prominence axis and tied at the photosphere, represented by two rigid conducting walls at x = ± ℓ. Note that in the text the position of the photospheric walls is denoted by x = ±L (from Joarder and Roberts, 1992b).
View Image Figure 28:
Dispersion diagram of magnetoacoustic (a) kink modes and (b) sausage modes in the equilibrium model represented in Figure 27. Meaning of labels at the right of each curve: modes are identified as fundamental (f), first harmonic (1h), second harmonic (2h), …; internal or external (I or E); and fast or slow (F or S). Here ω and kz are the frequency and the wavenumber along the slab, while vAe is the coronal Alfvén speed and L is half the length of the supporting magnetic field. Parameter values used: v Ap = 28 km s–1, c sp = 15 km s–1, vAc = 315 km s–1, csc = 166 km s–1 (from Joarder and Roberts, 1992b).
View Image Figure 29:
Schematic diagram of a prominence slab in a coronal environment. The magnetic field makes an angle ϕ with the prominence axis and is tied at the photosphere, represented by two rigid, perfectly conducting walls (from Joarder and Roberts, 1993b).
View Image Figure 30:
Dispersion diagram of magnetoacoustic modes in the equilibrium structure of Figure 29. Meaning of labels at the right of each curve: modes are identified as string, fundamental (f), first harmonic (1h), second harmonic (2h), …; internal or external (I or E); and fast, Alfvén or slow (F, A or S) according to the mode’s nature for κ ≪ 1. Here ω and κ are the frequency and the wavenumber modulus, while vAe is the coronal Alfvén speed and ℓ is half the length of the supporting magnetic field. Parameter values used: vAp = 74 km s–1, csp = 15 km s–1, vAc = 828 km s–1, csc = 166 km s–1 (from Joarder and Roberts, 1993b).
View Image Figure 31:
Sketch of an infinitely long thread immersed in the solar corona (from Lin et al., 2009).
View Image Figure 32:
Sketch of an infinitely long thread immersed in the solar corona. The respective flow speeds in the thread and the corona are denoted by U p and U c (from Soler et al., 2008).
View Image Figure 33:
Period of the fundamental oscillatory modes of an infinitely long thread versus the mass flow, Up. The top, middle, and bottom panels correspond to the slow, kink, and thermal modes. Different line styles correspond to waves propagating in the absence of flow (dotted), parallel waves (solid), and anti-parallel waves (dashed). The wavenumber is given by k a = 10−2 z, which is consistent with the wavelength of observed propagating waves in prominences (Section 3.6); a is the thread radius (from Soler et al., 2008).
View Image Figure 34:
Sketch of an equilibrium model made of two infinitely long threads embedded in the solar corona (from Luna et al., 2008).
View Image Figure 35:
Fundamental normal modes of two parallel and infinitely long threads (Figure 34). Total pressure perturbation field (contour plot in arbitrary units) and transverse Lagrangian displacement vector field (arrows) in the xy-plane for the wave modes (a) S x, (b) A x, (c) S y, (d) A y, (e) Sz, and (f) Az for a separation between threads d = 4a and a longitudinal wavenumber −2 kza = 10, where a is the thread radius. The prominence thread boundaries are denoted by dotted circles (from Soler et al., 2009a).
View Image Figure 36:
Fundamental normal modes of two parallel and infinitely long threads (Figure 34). Left: Ratio of the frequency of the four collective kink-like modes, ω, to the frequency of the individual kink mode, ωk, as a function of the normalized distance between strand axes. Meaning of symbols: S x (solid line), A x (dotted line), S y (triangles), and A y (diamonds). Right: Ratio of the frequency of the two collective slow modes, ω, to the frequency of the individual slow mode, ωs for the Sz (solid line) and Az (dotted line) (from Soler et al., 2009a).
View Image Figure 37:
Cross-section of a two-thread model analogous to that of Figure 34 with the addition of mass flows along the cylinders (from Soler et al., 2009a).
View Image Figure 38:
Sketch of the thread equilibrium model used by Joarder et al. (1997), Díaz et al. (2001), and Díaz et al. (2003). The blue zone of length 2W represents the cold part of the flux tube, i.e., the prominence thread. The length of the magnetic structure is 2L and the thread thickness (equivalent to its diameter) is 2b. The magnetic field is uniform and parallel to the z-axis, and the whole configuration is invariant in the y-direction (from Díaz et al., 2001).
View Image Figure 39:
Kink mode normal velocity component across the axis of the Cartesian prominence thread depicted in Figure 38. Solutions are symmetric about the thread axis (x = 0) and so they are only shown for x ≥ 0. The length of magnetic field lines is 2L = 200,000 km. (a) In a very thick thread (with a “radius” of 10,000 km) the perturbation is essentially confined to the thread itself, i.e., to 0 ≤ x∕L ≤ 0.1. (b) In an actual thread (with a “radius” of 100 km) the velocity displays a large amplitude beyond the thread boundary, at x∕L = 0.001. This means that wave energy spreads into the surrounding coronal medium (from Díaz et al., 2001).
View Image Figure 40:
Normal velocity component (in arbitrary units) of the kink mode in the direction across the thread axis. The ratio of the thread “diameter” to the length of magnetic field lines is b∕L = 0.001, while the ratio of the thread length to the field lines length is W ∕L = 0.1. The solid, dotted, and dashed lines correspond to kyL = 0 (curve of Figure 39b), kyL = 3, and kyL = 20. All other parameter values are those of Figure 39. The thread boundary is marked by a vertical dashed line (from Díaz et al., 2003).
View Image Figure 41:
Sketch of the equilibrium configuration of a thread in a cylindrical coronal magnetic tube. The gray zone of length 2W represents the cold part of the flux tube, i.e., the prominence thread. The length of the magnetic structure is 2L and the thread radius is b. The magnetic field is uniform and parallel to the z-axis, and the whole configuration is invariant in the φ-direction (from Díaz et al., 2002).
View Image Figure 42:
Cut of the normal velocity component in Cartesian geometry (dashed line, i.e., curve of Figure 39b) and the radial velocity component in cylindrical geometry (dotted line) in the direction across the thread axis. These solutions correspond to the (fundamental) kink mode in a prominence thread with the parameter values used in Figure 40. The vertical long dashed line marks the thread boundary (from Díaz et al., 2002).
View Image Figure 43:
Sketch of a multi-thread equilibrium configuration. The grey zone represents the cold part of the magnetic tube, i.e., the prominence. The magnetic field is uniform and parallel to the z-axis, and the whole configuration is invariant in the y-direction (from Díaz et al., 2005).
View Image Figure 44:
Sketch of the density profile in the direction z = 0 of an inhomogeneous multi-thread system. The density values of the are normalized to the coronal value. Between and under the threads the dimensionless separation, 2c ∕L, and “diameter”, 2b∕L, are given (from Díaz et al., 2005).
View Image Figure 45:
Dimensionless frequency versus the dimensionless reference separation between threads in a multi-thread system. In this figure ca is the Alfvén speed in the corona (from Díaz et al., 2005).
View Image Figure 46:
Wave damping by thermal effects in a uniform, infinitely long thread (Figure 32). Period (left), damping time (center), and ratio of the damping time to the period (right) versus the flow velocity for the fundamental oscillatory modes. The upper, middle, and lower panels correspond to the slow, fast kink, and thermal modes, respectively. Different line styles represent parallel waves (solid line), anti-parallel waves (dashed line), and solutions in the absence of flow (dotted line) (from Soler et al., 2008).
View Image Figure 47:
Wave damping by thermal effects in a two-thread system. Left: Ratio of the damping time to the period versus the distance between the thread axes of the Sx (solid line), Ax (dotted line), Sy (triangles), and Ay (diamonds) kink-like modes. Right: The same for the Sz (solid line) and A z (dotted line) slow wave modes (from Soler et al., 2009a).
View Image Figure 48:
Wave damping by ion-neutral effects in a uniform medium. (a) – (c) Ratio of the damping time to the period, τd∕P, versus the wavenumber, k, corresponding to the Alfvén wave, fast wave and slow wave, respectively. (d) Damping time, τ d, of the thermal wave versus the wavenumber, k. The different linestyles represent the following abundances: ξHeii = 0% (solid line), ξHeii = 10% (dotted line), and ξHeii = 20% (dashed line). In all computations, &tidle;μH = 0.8 and δHe = 0.1. The results for ξHe ii = 10% and δHe = 0.5 are plotted by means of symbols for comparison. The shaded regions correspond to the range of typically observed wavelengths of prominence oscillations. In all the figures the angle between the wavevector and the magnetic field is π∕4 (from Soler et al., 2010b).
View Image Figure 49:
Wave damping by ion-neutral effects in an infinitely long prominence thread. Dimensionless phase speed (left panels) and ratio of the damping time to the period (right panels) as a function of k a z for Alfvén waves (top panels), kink waves (middle panels), and slow waves (bottom panels). The different linestyles represent different ionization degrees: &tidle;μp = 0.5 (dotted), &tidle;μp = 0.6 (dashed), &tidle;μp = 0.8 (solid), and &tidle;μp = 0.95 (dash-dotted). Symbols are the approximate solution given by Equation (36) in Soler et al. (2009c) for μ&tidle;p = 0.8. The shaded zones correspond to the range of typically observed wavelengths of prominence oscillations. The Alfvén speed in the thread, vAf, the kink speed, ck, and the cusp speed in the thread, cTp, have been used to compute the dimensionless phase speed (adapted from Soler et al., 2009c).
View Image Figure 50:
Model used by Arregui et al. (2008b) to represent a radially non-uniform filament fine structure of mean radius a and transverse inhomogeneity length scale l.
View Image Figure 51:
Wave damping by Alfvén resonant absorption in an infinitely long prominence thread. Damping time over period for fast kink waves in filament threads with radius a = 100 km. (a) As a function of the density contrast, with l∕a = 0.2 and for two wavelengths. (b) As a function of the wavelength, with l∕a = 0.2, for two density contrasts. (c) As a function of the transverse inhomogeneity length-scale, for two combinations of wavelength and density contrast. In all plots solid lines correspond to analytical solutions given by Equation (30View Equation), with F = 2∕π (from Arregui et al., 2008b).
View Image Figure 52:
Wave damping by Alfvén and slow resonances in an infinitely long prominence thread. Kink mode ratio of the damping time to the period, τ ∕P d, as a function of the dimensionless wavenumber, kza, for l∕a = 0.2. The solid line is the full numerical solution. The symbols and the dashed line are the results of the thin boundary approximation for the Alfvén and slow resonances, i.e., the two terms in Equation (31View Equation). The shaded region represents the range of typically observed values for the wavelengths in prominence oscillations (from Soler et al., 2009e).
View Image Figure 53:
Wave damping by ion-neutral effects in an infinitely long cylindrical prominence thread. Ratio of the damping time to the period of the kink mode as a function of kza for a thread without transitional layer, i.e., l∕a = 0. (a) Results for a = 100 km and different ionization degrees: &tidle;μp = 0.5 (dotted line), &tidle;μp = 0.6 (dashed line), &tidle;μp = 0.8 (solid line), and &tidle;μp = 0.95 (dash-dotted line). Symbols are the approximate solution, given by Equation (33View Equation), for &tidle;μp = 0.8. (b) Results for &tidle;μ = 0.8 p and different thread widths: a = 100 km (solid line), a = 50 km (dotted line) and a = 200 km (dashed line). The shaded zone corresponds to the range of typically observed wavelengths of prominence oscillations (from Soler et al., 2009d).
View Image Figure 54:
Wave damping by resonant absorption and ion-neutral effects in an infinitely long cylindrical prominence thread. Ratio of the damping time to the period of the kink mode as a function of kza for a thread with an inhomogeneous transitional layer. (a) Results for μ&tidle;p = 0.8 and different transitional layer widths: l∕a = 0 (dotted line), l∕a = 0.1 (dashed line), l∕a = 0.2 (solid line), and l∕a = 0.4 (dash-dotted line). Symbols are the solution in the thin boundary approximation (Equation [33View Equation]) for l∕a = 0.2. (b) Results for l∕a = 0.2 and different ionization degrees: &tidle;μp = 0.5 (dotted line), &tidle;μp = 0.6 (dashed line), &tidle;μp = 0.8 (solid line), and &tidle;μp = 0.95 (dash-dotted line). In both panels a = 100 km (from Soler et al., 2009d).
View Image Figure 55:
Ratio of the damping time to the period of the kink mode as a function of k a z in an infinitely long thread with a = 100 km and l∕a = 0.2. The different line styles represent the results for a partially ionized thread with &tidle;μp = 0.8 and considering all the terms in the induction equation (solid line), for a partially ionized thread with &tidle;μp = 0.8 and neglecting Hall’s term (symbols) and for a fully ionized thread (dotted line) (from Soler et al., 2009d).
View Image Figure 56:
Spatial damping of kink waves due to ion-neutral effects in an infinitely long prominence thread. Results for the kink mode spatial damping in the case l∕a = 0: (a) λ∕a, (b) LD∕a, and (c) L ∕λ D versus P ∕τ Ap for &tidle;μ p = 0.5, 0.6, 0.8, and 0.95. Symbols in panels (a), (b), and (c) correspond to the analytical solution given by Equations (12), (13), and (14) in Soler et al. (2011) in the thin tube approximation, while the horizontal dotted line in panel (c) corresponds to the limit of LD ∕λ for high frequencies. The shaded area denotes the range of observed periods of thread oscillations.
View Image Figure 57:
Results for the kink mode spatial damping in an infinitely long prominence thread, in the case l∕a ⁄= 0: (a) LD ∕a and (b) LD ∕λ versus P ∕τAp for l∕a = 0.05, 0.1, 0.2, and 0.4, with &tidle;μp = 0.8. Symbols in panel (b) correspond to the analytical solution in the thin tube approximation, while the vertical dotted line is the approximate transitional period for l∕a = 0.1. The shaded area denotes the range of observed periods of thread oscillations.
View Image Figure 58:
Results for the kink mode spatial damping in an infinitely long prominence thread, in the case l∕a ⁄= 0: (a) LD ∕a and (b) LD ∕λ versus P ∕τAp for &tidle;μp = 0.5, 0.6, 0.8, and 0.95, with l∕a = 0.2. Symbols in panel (b) correspond to the analytical solution in the thin tube approximation. The shaded area denotes the range of observed periods of thread oscillations.
View Image Figure 59:
Model used by Soler et al. (2010a) to represent a finite length thread. A partially filled magnetic flux tube, with length L and radius a, is considered. The tube ends are fixed by two rigid walls representing the solar photosphere. The tube is composed of a dense region of length Lp surrounded by two much less dense zones corresponding to the evacuated parts of the tube. In the prominence region a transversely inhomogeneous layer of length l is considered. The plasma in the prominence region is assumed to be partially ionized with an arbitrary ionization degree &tidle;μ p. Both the evacuated part and the corona are taken to be fully ionized.
View Image Figure 60:
Results for the thread model in Figure 59 without transverse transitional layer and for the prominence thread located at the central part of the magnetic tube. (a) Period, P, of the fundamental kink mode in units of the internal Alfvén travel time, τ Ap, as a function of L ∕L p. The horizontal dotted line corresponds to the period of the kink mode in a homogeneous prominence cylinder. The symbols are the analytic solution (Equation (24) in Soler et al., 2010a). (b) Damping time, τd, in units of the internal Alfvén travel time, τAp, as a function of Lp∕L. The different lines denote μ&tidle;p = 0.5 (dotted), 0.6 (dashed), 0.8 (solid), and 0.95 (dash-dotted). The symbols are the analytic approximation for &tidle;μp = 0.8 (Equation (27) in Soler et al., 2010a). (c) τd∕P versus Lp ∕L. The line styles have the same meaning as in panel (b) and the symbols are the approximation given by Equation (34View Equation) (from Soler et al., 2010a).
View Image Figure 61:
Results for a thread configuration with a transverse transitional layer and for the prominence thread located at the central part of the magnetic tube. (a) τd in units of the internal Alfvén travel time, τ Ap, and (b) τ ∕P d as a function of L ∕L p. The different lines in both panels denote l∕a = 0.05 (dotted), 0.1 (dashed), 0.2 (solid), and 0.4 (dash-dotted). The symbols in panels (a) and (b) correspond to the analytic approximations with l∕a = 0.2, Equations (34) and (32) in Soler et al. (2010a).
View Image Figure 62:
Attenuation of prominence oscillations by wave leakage. (a) Quality factor (Q ≡ πτ ∕P 0 d) of stable IP (solid curves) and NP (dashed curves) prominence oscillations as a function of the coronal Alfvén speed. (b) Quality factor of the horizontally (squares) and vertically (diamonds) polarized stable oscillations versus the coronal density (from Schutgens, 1997b and Schutgens and Tóth, 1999).
View Image Figure 63:
(a) Ratio 2 2 ck∕v Ap (solid line) as a function of the density contrast, ζ. The dotted line corresponds to the value of the ratio 2 2 ck∕vAp for ζ → ∞. (b) Magnetic field strength as a function of the internal density, ρp, corresponding to four selected threads (from Lin et al., 2009).
View Image Figure 64:
Left: Analytic inversion of physical parameters in the (ζ, l∕a, vAp) space for a filament thread oscillation with P = 3 min, τ d = 9 min, and a wavelength λ = 3000 km (see, e.g., Lin et al., 2007). Right: Magnetic field strength as a function of the density contrast and transverse inhomogeneity length-scale, derived from the analytic inversion for a coronal density ρc = 2.5 × 10–13 kg m–3.
View Image Figure 65:
Determination of (a) prominence Alfvén speed and (b) magnetic field strength from the computation of periods and damping times for standing kink oscillations in two-dimensional prominence thread models and observations of period and damping times in transverse thread oscillations. The observed period and damping time are 20 and 60 min, respectively, and L = 105 km (from Soler et al., 2010a).
View Image Figure 66:
Plot of the solution lines satisfying P1∕2P2 = constant in the parameter space. The upper line corresponds to P1∕2P2 = 1.25 and the lower one to P1∕2P2 = 3, with each line showing an increment in P1∕2P2 of 0.25 from the previous one (from Díaz et al., 2010).
View Image Figure 67:
Sketch of the magnetic and plasma configuration used to represent a flowing thread (shaded volume) in a thin magnetic tube. The two parallel planes at both ends of the cylinder represent the photosphere (from Terradas et al., 2008).
View Image Figure 68:
Dependence of the Alfvén velocity in the thread as a function of the coronal Alfvén velocity for the six threads observed by Okamoto et al. (2007). In each panel, from bottom to top, the curves correspond to a length of magnetic field lines of 100,000 km, 150,000 km, 200,000 km, and 250,000 km, respectively. Asterisks, diamonds, triangles, and squares correspond to density ratios of the thread to the coronal gas ζ ≃ 5, 50, 100, and 200 (from Terradas et al., 2008).