A&A 460, 893-899 (2006)
DOI: 10.1051/0004-6361:20065313
M. P. McEwan - G. R. Donnelly - A. J. Díaz - B. Roberts
School of Mathematics and Statistics, University of St Andrews, St Andrews, Fife KY169SS, Scotland
Received 29 March 2006 / Accepted 11 September 2006
Abstract
Aims. With strong evidence of fast and slow magnetoacoustic modes arising in the solar atmosphere there is scope for improved determinations of coronal parameters through coronal seismology. Of particular interest is the ratio
P1/2P2 between the period P1 of the fundamental mode and the period P2 of its first harmonic; in an homogeneous medium this ratio is one, but in a more complex configuration it is shifted to lower values.
Methods. We consider analytically the effects on the different magnetohydrodynamic modes of structuring and stratification, pointing out that transverse or longitudinal structuring or gravitational stratification modifies the ratio
P1/2P2.
Results. The deviations caused by gravity and structure are studied for the fast and slow modes. Structure along the loop is found to be the dominant effect.
Conclusions. The departure of
P1/2P2 from unity can be used as a seismological tool in the corona. We apply our technique to the observations by Verwichte et al. (2004), deducing the density scale height in a coronal loop.
Key words: Sun: oscillations - Sun: magnetic fields - Sun: corona
With the advent of space missions Solar and Heliospheric Observatory (SOHO) and the Transition Region And Coronal Explorer (TRACE) there is convincing evidence of slow and fast magnetoacoustic waves in the corona. There is observational evidence of slow modes occurring as propagating waves (DeForest & Gurman 1998; Robbrecht et al. 2001; Ofman et al. 1999,1997; De Moortel et al. 2002a,b,2000; King et al. 2003; Sakurai et al. 2002; McEwan & De Moortel 2006) and also as standing waves (Ofman & Wang 2002; Wang et al. 2003b,a,2002). Fast kink waves have also been reported as standing modes (Aschwanden et al. 2002,1999; Nakariakov et al. 1999; Wang & Solanki 2004; Verwichte et al. 2004) but may arise also as propagating waves (Verwichte et al. 2005). Williams et al. (2002) found evidence for impulsively generated propagating fast waves. Fast sausage modes have also been identified (Nakariakov et al. 2003; Nakariakov et al. 2005; Melnikov et al. 2005). Extensive reviews of these observations and their theoretical interpretation are provided in Roberts (2004,2000), Wang (2004), Roberts & Nakariakov (2003), Aschwanden (2004), Nakariakov & Verwichte (2005), De Moortel (2006) and Goossens et al. (2006). Waves can be utilised to provide a coronal seismology (Roberts et al. 1984; Roberts 2006,1986,2004,2000; Nakariakov et al. 1999; Nakariakov & Ofman 2001), giving us indirect determinations of various coronal parameters.
The fundamental period P1 of a MHD mode contains information mainly about the average profile of the propagation speed of the mode. Andries et al. (2005a) argued that the frequencies and damping times of a stratified loop are very close to those of an unstratified loop with the same weighted mean density, the weight depending upon the spatial structure of the mode under consideration (see also Díaz et al. 2006).
Observations of standing waves have so far mainly identified the fundamental harmonics of a vibrating loop, with evidence for higher harmonics being rare. However, Verwichte et al. (2004) have identified the fundamental and its first harmonic of the standing transverse kink mode in two cases.
Andries et al. (2005a,b) and Goossens et al. (2006) have pointed out that the identification of harmonics could provide important diagnostic information for the coronal seismology of a loop. In particular, Andries et al. (2005b) studied the ratio P1/P2 of the fundamental oscillation period, P1, and its first harmonic or overtone, P2, of a kink mode oscillation, showing that this ratio falls below 2. For standing waves on an elastic string, P1/P2=2 and so P1/2P2=1. In Andries et al. (2005b) the departure of P1/P2 from 2 is a consequence of the density structure along the loop, and they explore this aspect through numerical modeling of the oscillations. Their work allows a comparison between the model and the observational results of Verwichte et al. (2004), which gave P1/P2=1.81 in one case and P1/P2=1.64 in another case. Andries et al. (2005b) included density structure but ignored the effects of gravity.
We take up the suggestion that P1/2P2 may depart from unity and we argue that such a departure is a natural consequence of the structure and stratification of the medium. We study how the ratio P1/2P2deviates from unity for fast and slow MHD modes in response to such effects as structuring in the longitudinal or transversal directions or gravity. Our main conclusion is that longitudinal structuring is the most important effect and this can be used in coronal seismology to estimate properties such as the density stratification scale.
A convenient starting point for our analysis of fast waves is the set of linearised ideal MHD
equations for a straight uniform magnetic field
,
constant plasma pressure and an equilibrium density profile
that is structured along the z-axis (Roberts 1991; Díaz 2004; Díaz et al. 2002; see Appendix A for a derivation),
Equations (1)-(3) are valid in the internal and external region separately. We use boundary conditions of continuous total pressure and velocity across the interface to model the radial dependence. In Roberts (1991) neither a uniform magnetic field nor uniform pressure was assumed but the density
was taken to be independent of z.
Equations (1)-(3) may be used to obtain the modes of
oscillation of an unbounded homogeneous cylindrical magnetic flux tube of
radius a with constant densities
and
inside and outside the tube. For waves of frequency
and longitudinal wavenumber k, the modes of oscillation of
a magnetic flux tube embedded in a magnetised plasma have been discussed by
Edwin & Roberts (1983), who obtained a dispersion diagram for the modes. Figure 1 displays such a diagram, obtained here for a flux tube
with internal Alfvén speed
embedded in an environment with Alfvén
speed
;
the sound speed
inside the tube is
and the sound speed
in the environment is
.
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Figure 1:
The dispersion diagram for magnetoacoustic waves in a magnetic flux
tube of radius a. The diagram gives the phase speed
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The ratio P1/2P2 departs from unity even in the case of a straight homogeneous loop. To see this, consider the fast kink mode which in
Fig. 1 has a phase speed
which in
the limit of a thin tube (
)
has speed ck, where
Consider then the ratio
P1/2P2 of the fundamental fast kink oscillation of
period P1 to its first harmonic of period P2. Since
,
we have an associated period
.
Consider a line-tied coronal loop of length 2L.
Line-tying determines the values of the longitudinal wavenumber k
that allow the oscillation to fit within the loop, so that (see Roberts et al. 1984)
,
for integer n.
Then we obtain a period P=Pn given by
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Figure 2:
P1/2P2 for the kink mode in a uniform coronal loop in a uniform environment. The dotted curve is for the case
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We can understand more fully the departure of
P1/2P2 from unity
in the kink mode if we focus on a thin tube ()
with zero plasma
.
With
(sound speeds are set to zero),
the kink mode in a thin tube has a phase speed
given by (Edwin & Roberts 1983)
We now consider the role of structuring along the magnetic field. This is the
effect discussed by Andries et al. (2005b) for a different equilibrium profile.
Consider again a zero-
plasma, taking an exponential density profile
for coronal density scale height
.
The density increases from a value
at the loop apex
to a value
at the loop base
,
which are related to the density scale height
as
In the zero-
limit, Eqs. (1) and (2) can be combined
to obtain a single partial differential equation for the perturbed total
pressure (Donnelly et al. 2006; Díaz et al. 2002),
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Figure 3:
P1/2P2 as a consequence of combined longitudinal and
transverse structuring. The density is exponentially structured along the loop.
The solid line has a base density that is 8 times the density at the apex
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Figure 4:
P1/2P2 as a function of the inverse scale height
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The general solution is a result of a combination of two effects, radial and
longitudinal structuring. Two typical curves for various ratios
and
are shown in Fig. 3. We highlight the
fact that due to the presence of the exponential density profile the ratio
P1/2P2 is now no longer equal to unity for any value of a/L.
In fact, longitudinal structuring shifts the ratio even for
(Fig. 4). However, in addition the effect of the structuring across the loop shifts it further (though in a similar way to that shown in
Fig. 2 for an unstructured loop) as a/L is increased and the
dispersive nature of the mode is included. Notice in Fig. 3 that the shift due to longitudinal structuring is larger than that due to radial structuring, especially since for solar coronal loops
.
The previous case of an unstructured loop follows from Fig. 4 by taking the limit
,
so
.
The Klein-Gordon equation arises in a variety of wave studies (Roberts 2004): it describes the slow mode in a loop and its reduction to a one dimensional sound wave in the low limit of a rigid magnetic field (Roberts 2006). It also describes both sausage and kink modes in a thin photospheric flux tube in which gravitational stratification is allowed for (Rae & Roberts 1982; Spruit & Roberts 1983). The Klein-Gordon equation may be written in the form
The simplest case to discuss is that of a medium for which the
propagation speed c and the cutoff frequency
are constants. This case, for example, arises for an acoustic wave propagating vertically in an isothermal atmosphere. Then Eq. (17) has solution
Similarly, we may consider the odd modes which leave the loop apex undisturbed, so Q=0 at z=0 and z=L. Then
Thus the ratio of the fundamental and first harmonic frequencies,
,
leads to
P1/P2, with
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Figure 5:
P1/2P2 for a slow (or acoustic) mode in an isothermal coronal
loop as a function of loop half-length L in units of the pressure scale height
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Figure 5 displays the ratio
P1/2P2 as a function of loop
half-length L (measured in units of the density scale height
), as determined by Eq. (24). Stratification
of density causes
P1/2P2 to fall off from unity, with the effect being
most marked in very long loops (
). In general, coronal loops
have
,
and the departure of
P1/2P2 from unity is
only slight. In fact the magnitude of the shift of
P1/2P2 due to stratification by gravity for the slow mode for a loop of typical half length (
)
is comparable to the magnitude of the shift brought about by radial magnetic structuring for the fast mode. The slow mode is much less dispersive than the fast mode so the correction due to radial structuring is even smaller. For example, a loop with internal density
,
half-length L=5
104 km and radius a=5000 km, so a/L=1/20,
produces a kink mode ratio of
P1/2P2=0.995 (see Fig. 2). This
may be compared with an acoustic wave in an isothermal atmosphere with sound speed
km s-1 for which the acoustic cutoff frequency is
and
105 km, resulting in
P1/2P2=0.988. This is a shift from unity of 0.012 or
.
Typically, slow or acoustic modes produces a small harmonic shift due to gravity in all but extremely long loops.
Consider Eq. (15) for the case when c and
vary with z. To be specific, we discuss the case of an acoustic wave propagating vertically in an atmosphere with a linear temperature profile for which the propagation speed c is the sound speed
:
In a similar way, we can obtain the dispersion relation for the even modes which satisfy Q=0 at z=L and have
at z=0:
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Figure 6:
The ratio
P1/2P2 for a sound wave in a non-isothermal loop of length 2L. The sound speed squared varies linearly with distance, falling from a value
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Equations (33) and (35) determine the dimensionless
frequency
for various values of
and
.
The actual frequency
is determined once the base sound speed
and the temperature gradient are specified. It is interesting to note
that the structure of Eqs. (33) and (35) remains
even in the absence of gravity (g=0), though the order
of the Bessel functions
then reduces to unity. Thus a shift in the ratio of
P1/2P2 occurs as a consequence of non-isothermality, even if gravity is ignored.
Equations (33) and (35) are solved numerically for
various values of
,
the ratio of the sound speed
at the loop apex to the sound speed
at its base. Equation (35) provides the period P1 and its first harmonic
gives P2 and is determined by Eq. (33). The ratio
P1/2P2 is displayed in Fig. 6. When
is close to unity, the loop is almost isothermal and
P1/2P2 is close to unity (though decreasing with increasing loop length). But for a more strongly structured sound speed, the shift from unity in
P1/2P2 is stronger. For example, for a base sound speed of
km s-1 and an apex sound speed of
km s-1, so
,
Fig. 6 shows that
in short loops (
)
and falling to approximately 0.58 in extremely long loops (
). For loops with a larger temperature gradient (
), the immediate deviation of
P1/2P2 from unity becomes more significant for short loops; however, for long loops the behaviour of
P1/2P2 is similar to the isothermal case.
We have seen in the above that structuring along the loop introduces a shift in
P1/2P2, even in the absence of gravity (which reduces the cutoff
frequency
to zero). Accordingly, consider Eq. (15) in
the absence of a cutoff frequency,
:
We may determine P1/2P2 using dispersion relation Eq. (39) and a similar relation for the odd modes. The results are displayed in Fig. 7. Notice that for small W/L (e.g. W/L=0.1), for which the exponential variation covers most of the loop, we obtain results similar to Fig. 4, as the density profile for each case is similar. However, a direct comparison with Fig. 4 is not possible as it refers to the kink mode whereas here we consider the slow mode. On the other hand, for a thin chromospheric layer (for which W is comparable to L), P1/2P2 returns to unity unless the base density is very high.
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Figure 7:
P1/2P2 as a function of the inverse scale height
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Finally, comparing these results with those of the Klein-Gordon equation we can
see that the longitudinal structure alone reproduces the profiles obtained with
the inclusion of gravity. For example, comparing Fig. 5 with the
plot for a fully stratified loop in Fig. 7 (solid line) we see
that the shape is similar, since the density ratio is related to the inverse scale
length
by Eq. (38) with W=0.
We have shown that some properties of the equilibrium can be obtained by studying the shift of P1/2P2 from unity. Currently, observations have indicated this effect (without interpretation) purely for the fast modes of coronal loops. We have shown in Sect. 2 that the main cause for the shift in these modes lies in the structure along the magnetic field. For an exponentially stratified loop the ratio P1/2P2 depends on the density scale height.
As an illustration, we consider the observational data in Verwichte et al. (2004) (Table II). They reported two periods in two of their time series (labelled "C'' and "D'') which were
interpreted as the fundamental and first harmonics of the loop. The values of
P1/2P2 obtained from the wavelet analysis are (Verwichte et al. 2004)
Notice that this procedure gives us a value of the ratio
which is independent of other considerations. This is an important advantage over other quantities deduced from coronal seismology, such as the determination of the magnetic field strength (Nakariakov & Ofman 2001), for which the values of other unknowns (e.g. the equilibrium coronal density) need to be assumed.
In this work we have explored the various effects which cause the ratio P1/2P2 to depart from unity, its value in a homogeneous medium. Magnetic structuring, due principally to density contrast between the interior and exterior of a loop, causes fast magnetoacoustic waves to be dispersive, and this manifests itself in the ratio P1/2P2. Longitudinal structuring or stratification has a more significant effect than radial structuring, producing a larger departure from unity in P1/2P2. Longitudinal structure has also been considered by Andries et al. (2005b). We have illustrated the effect for a simple flux tube with a discrete density profile, but we can anticipate similar results for any radial structure (e.g. the Epstein profile). Of course, other effects such as magnetic flux tube expansion or non-adiabatic damping may also produce a shift in P1/2P2 from unity; however, such effects are left for a future study.
Slow magnetoacoustic waves are only very weakly dispersive, so shifts in P1/2P2 due to radial structuring are small. However, longitudinal structuring or stratification has a more important role here too, reducing P1/2P2 below unity (becoming 0.5 in the limit of an infinitely long loop). The presence of a gravitational force (as opposed to longitudinal structuring by whatever effect) complicates the behaviour of P1/2P2, but the effects are generally small in the corona (because of the high pressure scale height).
The results presented here can be used to extract information about the equilibrium state of a coronal loop. Previous work (e.g. Nakariakov & Ofman 2001) have studied the relevance for coronal seismology of the fundamental period, which allows us to deduce global properties of the loop, such as the mean density or the magnetic field strength. However, observational measurements of P1/2P2 gives information about smaller scales, and we have used this to estimate the structure's length scale for the fast mode (or the ratio between the footpoint and apex density). In principle, if all the harmonics could be observed, we could invert the problem and obtain a density profile (as it is currently done in helioseismology, where thousands of modes are reported). But with two coronal modes only currently observed we are not able to obtain such detailed information. Our method can also be applied to slow modes, but there are currently no observations of P1/2P2 for slow modes. On the other hand, it is interesting to note that more than one mode has been detected in prominences (Régnier et al. 2001; Pouget et al. 2006). Currently, only information relating to the fundamental harmonics of each prominence oscillation family is used for seismology, but similar techniques could be applied in the future for extracting information from the first (and higher) harmonics.
In conclusion, we have demonstrated how the individual contributions cause a deviation of P1/2P2 from unity, an effect highlighted in Andries et al. (2005b). Lateral structure, longitudinal structure and density stratification all play a part in forming P1/2P2, but we conclude that longitudinal structure is the key ingredient for magnetoacoustic modes.
Acknowledgements
M.P.M. and G.D. acknowledge financial support from the Particle Physics and Astronomy Research Council. A.J.D. acknowledges support from PPARC on the St Andrews Solar Theory Rolling Grant. The authors would also like to thank Erwin Verwichte, Jesse Andries and the anonymous referee for their useful comments in improving this paper.
We derive Eqs. (1)-(3), following Díaz (2004).
The starting point is the set of linearised ideal MHD equations for a uniform
magnetic field
,
in absence of gravity and
with constant plasma pressure
and a density profile that is stratified along the z-axis,
:
The induction Eq. (A.2), may be expanded to yield
In the following development, the symbol
denotes the components of the perturbed quantities and gradients perpendicular to
.
Using Eq. (A.5), the
perpendicular component of Eq. (A.3) can be rewritten as
Before dealing with the parallel component another expression for the
perturbed total pressure is required. From Eq. (A.4) we have
Finally, the component of Eq. (A.3) along the field gives
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(A.11) |
Equations (A.6), (A.9) and (A.10) are Eqs. (1)-(3). They are formally the same as in Roberts (1991), although in that paper the details of the derivation are slightly different: they
were deduced with the assumption of Cartesian coordinates and with B0(x) and
instead of
.
Since
in the solar corona, we can
restrict ourselves to studying the oscillatory
modes in the low-beta limit. This assumption implies
,
and
.
Now, selecting the velocity components as our dependent variables leads to a pair of coupled partial differential equations, although by choosing the total
pressure perturbation,
,
as our dependent variable a single partial
differential equation is obtained.
First of all, from Eq. (A.10) in the low-beta limit we have
vz=0, pointing out that the slow mode is removed in this limit.
Then, we take the gradient in the perpendicular
plane of Eq. (A.6) and use
and
,
giving
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(A.15) |