varied(λ)for some fixed transition region extent (shown in Fig. 2.26) then the frequency shift is expected
to be to be similar to, although smaller than the shift produced by a temperature gradient alone (shown in Fig. 2.21). When comparing Fig. 2.26 to Fig. 2.21 it is clear that this is the case. Again, the wave is propagating in the transition region for only a fraction of the total travel time, so the effect is only a fraction of the total effect considered earlier, in the full linear temperature gradient model.
Fig. 2.25 shows that an increase in the extent of the loop embedded in the transition region causes a downward shift in the frequency (with the exception of the fundamental and its first harmonic). However, this shift is negligible for a small transition region extent, e. g. m < 0.2, which is10% of the total loop
length. As the transition region is very thin compared to a typical loop length then in reality0< m <0.1
would be more realistic. This shift is also more significant for higher harmonics. In fact, each harmonic is affected in a unique way: the fundamental suffers a slight frequency increase, and its first harmonic is relatively unaffected; the frequency of all higher harmonics is reduced. For a large extent of loop embedded in the transition region, the frequency shift for higher harmonics is significant, with a reduction of around
30% forn >10harmonics.
Fig. 2.26 indicates that the shift due to the temperature gradient in the transition region is less pronounced than that for a loop that is entirely non-uniform, as in Section 2.5. However, the effect is still present. Again, individual harmonics are affected in individual ways due to the longitudinal structuring.
2.7
Conclusions
The equations describing two dimensional motions in a thin slab of plasma stratified under gravity, with gravity pointing in the direction of thez-axis (similar to Ferraro and Plumpton (1958)1) have been derived.
Using a stretching coordinate the thin slab is modelled as a one-dimensional waveguide governed by the Klein-Gordon equation describing the slow MHD mode of oscillation. A similar equation describes the modes when gravity is antiparallel to thez-axis (Roberts, 2006). The presence of gravity introduces a cutoff
frequency, imposing a natural timescale for the system. In the low-β limit this cutoff frequency becomes
the acoustic cutoff frequency and the governing equation becomes the acoustic Klein-Gordon equation for waves in a stratified atmosphere, first derived by Lamb (1932). The approximation of a low-β plasma in
the corona models the slow MHD mode as field-guided acoustic waves along coronal loops, governed by the acoustic Klein-Gordon equation. In particular, the equation for various sound speed profiles (modelling density structure along the loop) is solved. Initially, it is solved for a loop with constant sound speedcsand
constant cutoff frequencyΩand the effects introduced by gravity are noted.
In the isothermal case stratification by gravity causes a decrease in the oscillation period (increase in frequency). For long loops, L/Λ0 → ∞, then the frequency of oscillation tends to the acoustic cutoff
frequency. Also, gravity introduces amplification by stratification. However, these effects are typically small in the corona asL/Λ0is of order unity. A further consequence is that the frequency shift, introduced
by stratification, is different for each mode - gravity affects the fundamental modes differently to its first harmonic, etc.
Figure 2.25: Diagram showing the effect of increasing the transition region extenth=mLfor a fixed ratio
ofcapex/cbase. Here,(m+c)L/Λc = 2.5andλ= 5. The solid curves represent the odd modes and the
dashed curves are the even modes.
Figure 2.26: Diagram showing the effect of increasing the ratio ofλ= capex/cbasefor a fixed extent of
transition region. Here,c= 0.8andm= 0.2with(m+c)L/Λc = 2.5. The solid curves represent the odd
2.7 Conclusions 67
This study is extended to consider the role of footpoint density structuring on the acoustic oscillations of a coronal loop. A lower layer of dense, cool plasma is introduced, modelling the chromospheric footpoints of the loop, by introducing a step function for the sound speed. The dispersion relations for the even and odd modes are derived and the inclusion of a dense footpoint region causes a decrease in the frequency of oscillation (when compared to the isothermal case). Initially, the system is modelled using real val- ued functions (sinandcos), however, it is found that the system is better modelled using the hyperbolic
functionssinhandcoshas the wave is generally evanescent in the lower region. When considering only
the real valued functions the lower region acts as a ‘brick wall’ to the acoustic oscillations, with no wave energy penetrating the dense footpoint layer. However, it is noted that using the hyperbolic functions, the energy is allowed to penetrate a short distance into the lower layer. An analytical solution to the dispersion relations is derived for the two layer case, when the lower layer is thin. Furthermore, the depth to which the wave penetrates is approximately twice the density scale height in the footpoint region. These findings are supported by considering an infinite depth of footpoint region which shows similar results.
Table 2.1: Table indicating the effect of each loop footpoint structure outlined in this Chapter on the period of oscillation (in seconds). The parameters here assume a coronal loop of length 200Mm (soL = 100
Mm), with a chromospheric footpoint of extent10Mm (where appropriate). The coronal temperature is1
MK and the chromospheric temperature is10000K.
Harmonic Uniform Isothermal Two-layer Linear Temperature Transition Region
Period Period Period Model Period Model Period
g6= 0 h= 0.1L λ= 10 λ= 10, h= 0.1L
Fundamental 1334 1314 1468 3430 2533
First harmonic 667 665 742 2140 1313
The work first discussed by James (2003) is extended to include the even modes of oscillation. A stratified coronal loop, with a linear gradient in temperature, decreasing from a maximum at the loop apex to a minimum at the footpoint end is considered. The dispersion relations for both the even and odd modes are derived and solved numerically. As the ratio of apex temperature to footpoint temperature decreases (λ → 1) then this model reduces to the isothermal case, described earlier. As the temperature gradient,
λ, increases the frequency of oscillation decreases (tending to the acoustic cutoff frequency at the loop
apex), with the exception of the fundamental mode which is slightly increased. The density structuring, introduced byλ >1, causes the frequency of each mode to be uniquely shifted – the consequences of this
are discussed in detail in Chapter 4. Also, asλ→ ∞the shift in frequency becomes a constant value.
The final model considers the combined linear temperature gradient model and the isothermal case. A steep linear gradient in temperature over a thin layer of plasma is a good first approximation to the transition region. This is combined with an isothermal coronal part to model a coronal loop embedded in the transition region. The results are similar to the results of each model individually, however, the magnitude is reduced (as each region models a smaller part of the loop here). It is found that increasing the thickness of the transition region causes a decrease in the period, proportional to the thickness of the transition region.
Overall, the inclusion of some kind of footpoint density enhancement in a coronal loop causes a decrease in the frequency of oscillation (see Table 2.7 for details of the effect of structuring on the period of oscilla-
tion). For the most part this modification is small when considering only stratification by gravity. However, a temperature profile or a transition region may affect the frequency of oscillation significantly. This would have severe consequences on any results of coronal seismology. In particular, each harmonic of oscilla- tion may be modified differently to other harmonics. This topic, and its impact for coronal seismology, is returned to in detail in Chapter 4.
Chapter 3
Fast Mode Oscillations in the Corona
3.1
Introduction
In1the discussion of coronal oscillations in Chapter 2 the effect of structure and stratification on the slow
magnetoacoustic mode was considered. To complete the magnetoacoustic picture the fast magnetoacoustic mode is studied. From the polar plot in Fig. 1.6(a) it is clear that the fast mode is roughly isotropic, with the phase speed being highest when the wave is propagating perpendicular to the magnetic field (Cowling, 1976; Priest, 1982). The fast mode propagates with a speedcf, which has a maximum when the angle,θ,
between the wave vector kand the magnetic field isθ = π/2. Generally, for any angleθ,cf lies in the
rangecs< cA≤cffor anyθ. In the lowβplasma of the coronacf→cA.
Using instruments onboard SOHO and TRACE, many observations of the fast modes (both kink and sausage) have been made in recent years (see Section 1.4.1 for an overview). In this Chapter the intention is to develop the theory of the fast MHD mode similar to the development for the slow mode in Chapter 2. Theoretical models describing the magnetoacoustic oscillations in a slab geometry were discussed in Roberts (1981b) and Edwin and Roberts (1982), with an extension to cylindrical magnetic flux tubes in Edwin and Roberts (1983). More recent discussions have included the presence of a twist in the magnetic field (Bennett et al., 1999; Erd´elyi and Carter, 2006), the effect of a steady flow in Erd´elyi and Goossens (1996), the effect of line-tying the oscillations (D´ıaz et al., 2004) and stratification in the vertical direction of the tube (Nakariakov et al., 2000; James, 2003; Mendoza-Briceno et al., 2004; Roberts, 2006). The aim here is to study the effect of gravity on the properties of the oscillation, in particular when gravity is acting acrossthe magnetic field.
Consider the curved geometry of a coronal loop (see Fig. 3.1), with the photosphere-chromosphere mod- elled as a straight plane; the geometry of the loop can be simplified in one of two ways. Either much of the loop is vertical, and therefore aligned either parallel or antiparallel with gravity (for example the geometry considered in Chapter 2), or most of the loop is considered horizontal, thus aligned perpendicular to gravity. In this Chapter the effects introduced by considering gravity acting perpendicularly to a magnetised slab of plasma in the low-β corona (the horizontal loop) are studied. A more realistic model would consider
a gravity force whose direction varies along the slab, and so is neither parallel nor perpendicular to the applied magnetic field. That is a far too difficult problem to tackle analytically, and will not be considered here.
1Aspects of this Chapter have been published in McEwan and D´ıaz (2007, in press): Effect of Gravity on the Fast Modes of a
Horizontal Coronal Slab
2a
h
g
Figure 3.1: Sketch of a coronal loop of heighthand width2a, in the presence of gravityg.
Related studies involving a plasma in a horizontal magnetic field, in the presence of gravity, across an interface have been carried out. In particular, Campbell and Roberts (1989), Evans and Roberts (1990), Miles and Roberts (1992) and Miles et al. (1992) have studied the role of gravity on thep,f andgmodes
used in helioseismology of the solar interior. Jain and Roberts (1994) studied non-parallel propagation of thep-modes in a plasma across a horizontal magnetic interface but stratified vertically by gravity and
a similar model was analysed by Erd´elyi and Taroyan (2001). A recent review of magnetic effects in helioseismology has been given in Erd´elyi (2006). As far as the author is aware there exists no current literature regarding this problem with relation to coronal plasmas. Motivated by this, these studies are extended to a coronal plasma.