A description is provided of how fast magnetoacoustic waves are ducted along regions of low Alfven velocity (high density) in the corona, exhibiting a distinctive wave signature which may be used as a diagnostic probe of in situ coronal conditions (magnetic field strength, density inhomogeneity, etc.). Some observational knowledge of the start time of the impulsive wave source, possibly a flare, the start and end times of the generated wave event, and the frequency of the pulsations in that event permits a seismological deduction of the physical properties of the coronal medium in which the wave propagated. With good observations the theory offers a new means of probing the coronal atmosphere

Roberts, B.

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GUIDED MHD WAVES AS A CORONAL DIAGNOSTIC TOOL
B. Roberts +
Department of Physic8 and Aetronomy
The University of Iowa
Iowa City, IA 52242
INTRODUCTION
The coronal atmosphere is observed to be highly inhomogeneous with marked
density and temperature variations; coronal loops abound. Magnetic forces
dominate the atmosphere providing thermal insulation across field lines and
almost rigid wave guides for the propagation of MHD waves. In a homogeneous
low beta plasma, the slow magnetoacoustic wave gives one-dimensional
propagation of sound whereas the fast magnetoacoustic wave gives isotropic
propagation at the Alfven speed. (See Weitzner (1983) and Roberts (1984,
1985) for recent discussions of the properties of MHD waves.) If, however,
the low beta plasma is inhomogeneous in density, as with the corona, then fast
magnetoacoustic waves are guided by regions of low Alfven speed (Habbal, Leer,
Leer, and Holzer 1979; Edwin and Roberts 1982, 1983; Roberts, Edwin, and Benz
1983, 1984; see also Newcomb 1957). For a general overview see Edwin and
Roberts (1986a). Regions of low Alfven speed occur in both coronal loops and
in open field regions (coronal holes). We note, too, that current sheets
(regions of the plasma where field reversal occurs) provide wave guides for
fast magnetoacoustic waves (Edwin, Roberts, and Hughes 1986).
An important thing about such wave guides is that they preferentially select
certain ranges of frequency and wavenumber for guided propagation. If a fast
mode is generated impulsively, such as by a flare, it is guided along a
region of low Alfven speed and will exhibit frequencies of the order of the
Alfven speed divided by the width of the inhomogeneity. For typical coronal
conditions, this will give rise to frequencies of about 1Hz or higher.
The occurrence of preferred frequencies in an impulsively generated fast
magnetoacoustic wave raises the interesting possibility that such distinctive
signatures could be used as a seismological probe of the coronal atmosphere,
allowing us to determine magnetic field strengths and/or spatial extents of
density inhomogeneities. We discuss this possibility here (see also Edwin and
Roberts 1986b).
_On leave from Dept. of Applied Mathematics, The University, St.
Andrews, Fife, Scotland.
_F.DtNG pAGE BLAN_ NOT FILMED
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https://ntrs.nasa.gov/search.jsp?R=19870011488 2020-03-20T12:03:59+00:00Z
IMPULSIVELY GENERATED FAST WAVES
Consider a field-aligned cylindrical tube of dense gas embedded in a uniform
magnetic field. In a low beta atmosphere, impulsively generated fast waves
exhibit tlmescales (periodicities) of the general order (Roberts et al.
1984)
2.6a (I Pe) I/2
- -- , (I)
1: VA Po
where Po and vA are the gas density and Alfven speed within the cylinder of
radius a, and Pe is the gas density in the environment (where the Alfven
speed is VAe , with PoVA 2 = PeVAe2). Ducting occurs provided VAe > VA, which
corresponds to Po > Pc. In a very dense region (Po >> Pc), equation (I)
reduces to
2.6a
- . (2)
VA
For example, with a = 500 km and vA = 103km s-I we obtain, from (2),
= 1.3s.
In fact, the timescale _ is the maximum timescale that an impulsive event
exhibits. Suppose that the source is located on the axis of the cylinder at
z = 0, the impulse occurring at time t = 0. Then, at an observation point z
= h (>> a) far down the axis of the cylinder, pressure and magnetic field
variations, due to the passage of the impulsively generated fast magneto-
acoustic wave (taken to be symmetric, sausage mode, oscillations within the
cylindrical inhomogeneity), will exhibit three distinct phases (Roberts et
al. 1984), analogous to sound waves in ocean layers (Pekerls, 1948). The
first phase--called the periodic phase--begins at time t = h/VAe; in this
phase the wave's amplitude is low and its period is given by (i). The second
phase commences at a time t = h/v A when both the amplitude and the frequency
of the wave undergo a substantial increase. This is the quasi-periodic
phase. It lasts, until the time t = h/cgmin, where Cgmin is the minlmum, value
of the mode s group velocity, when the amplitude of the wave beglns to
decline as the wave passes by the observation point z = h; this is the decay
(or Airy) phase. A sketch of these three phases is given in Roberts et al.
(1983, 1984).
DIAGNOSTICS
The distinctive theoretical profile of impulsively generated oscillations
within a density enhancement (e.g., coronal loop) suggests that such oscilla-
tions might be used as a diagnostic probe of in sltu coronal conditions. The
periodic phase is of lower amplitude than the quasi-perlodlc phase. Thus, it
is possible that the periodic phase of the motion be obscured by noise and so
the fact that it is a precursor to the quasi-periodic phase may pass
unnoticed. Considering the quasi-periodic phase, we note that its duration,
_dur, is given by
326
_dur = h I 1 I) = h rVAe (_ I/2]rain vA _ _n ) "c VAe c
g g
(3)
Thus the duration depends both on the distance h between the source and the
observation level and also on the magnitude of Cgmln, which in turn depends
upon the magnitude of the density enhancement. In fact, Zdur is related to
the frequency _min within the quasi-periodlc phase (strictly, the frequency
at the end of the quasl-periodlc phase): for fixed h, a and VAe , the higher
the frequency _min the shorter the duration time _dur (Roberts et al. 1984).
There is some observational support for this relation in Tapplng's (1978)
record of meter wavelength pulsating bursts, but further studies are needed.
The greater the density enhancement (i.e., the larger the value of Po/Pe),
the lower the frequency _mln and the lower the group velocity minimum, Cgmin,
thus suggesting that very dense inhomogeneitles have long quasl-perlodic
phases. By contrast, much smaller density enhancements (Po _ Pe) have short
quasl-perlodic phases but extended decay phases, giving an event with the
appearance of a decaying oscillation of extended duration. This may be an
explanation of the gradually decaying wave train observed by McLean and
Sheridan (1973).
A numerical illustration of equation (3) may be helpful. With Po = I0 Pe,
inspection of Figure 4 in Roberts et al. (1984) gives _min a _ VAe for which
(from their Figure 7) _dBr _ h/VAe. Thus, with VAe = 2 x 103 km s-I (so
vA = 630 km s-i), a = I0a km and h = 5 x 104 km, the quasi-periodic phase
has a period of about 3s and a duration of 25s (or some 8 periods). Other
choices of Po/Pe, h and a will clearly give somewhat different numbers.
In summary, then, we have described how fast magnetoacoustic waves are
ducted along regions of low Alfven velocity (high density) in the corona,
exhibiting a distinctive wave signature which may be used as a diagnostic
probe of in sltu coronal conditions (magnetic field strength, density
inhomogeneity, etc.) Some observational knowledge of the start time of the
impulsive wave source, possibly a flare, the start and end times of the
generated wave event, and the frequency of the pulsations in that event
permit a seismological deduction of the physical properties of the coronal
medium in which the wave propagated. With good observations the theory
offers a new means of probing the coronal atmosphere.
AKNOWLEDGEMENTS
B.R. is greatful for support from the NASA Solar Maximum Mission Guest
Investigator Program, and to Dr. Art Poland for his kind invitation to
attend the CPP Workshop. B.R. was also supported by NASA through grant
NGL-16-O01-043. It is a pleasure to acknowledge the kind hospitality of
Professor Donald Gurnett of the University of Iowa, and the efficient and
helpful assistance from Kathy Kurth.
327
REFERENCES
Edwin, P. M. and B. Roberts, 1982, "Wave Propagation in a Magnetically
Structured Atmosphere; III: The Slab in a Magnetic Environment,"
Solar Phys., 76, 239.
Edwin, P. M. and B. Roberts, 1983, "Wave Propagation in a Magnetic Cylinder,"
Solar Phys., 88, 179.
Edwin, P. M. and B. Roberts, 1986a, "Guided Waves in Solar Magnetic
Structures," in Proc. Second Intern. Conf. on 'Surface Waves in Plasmas
and Solids," Lake Ohrid, Yugoslavia (Sept. 1985).
Edwin, P. M. and B. Roberts, 1986b, "Impulsively Generated Fast Coronal
Pulsations," in Proc. SMM Workshop 'Rapid Oscillations in the Corona' (ed.
B. R. Dennis), NASA, in press.
Edwin, P. M., B. Roberts and W. J. Hughes, 1986, "Dispersive Ducting of MHD
Waves in the Plasma Sheet: A Source of Pi2 Wave Bursts,"
Geophys. Res. Lett., 13, 373.
Habbal, S. R., E. Leer and T. E. Holzer, 1979, "Heating of Coronal Loops by
Fast Mode MHD Waves," Solar Phys., 64, 287.
McLean, D. J. and K. V. Sheridan, 1973, "A Damped Train of Regular Meter-Wave
Pulses from the Sun," Solar Phys., 32, 485.
Newcomb, W. A., 1957, "The Hydromagnetic Wave Guide," in 'Magnet.hydro-
dynamics' (ed. R. Landshoff), Stanford University Press, p. 109.
Pekeris, C. L., 1948, "Theory of Propagation of Explosive Sound in Shallow
Water," in 'Propagation of Sound in the Ocean,' Geol. S.c. Amer. Memoir,
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Roberts, B., 1984, "Waves in Inhomogeneous Media, in 'The Hydromagnetics of
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Roberts, B., 1985, "Magnet.hydrodynamic Waves," in 'Solar System Magnetic
Fields' (ed. E. R. Priest), Reidel, p. 37.
Roberts, B., P. M. Edwin and A. O. Benz, 1983, "Fast Pulsations in the Solar
Corona, Nature, 305, 688.
Roberts, B., P. M. Edwin and A. O. Benz, 1984, "On Coronal Oscillations,"
Ap. J., 279, 857.
Tapping, K. F., 1978, "Meter Wavelength Pulsating Bursts During the May 21,
1972, Solar Noise Storm, Solar Phys., 59, 145.
Weitzner, H., 1983, "Linear Wave Propagation in Ideal Magnet.hydrodynamics,"
in 'Basic Plasma Physics,' (eds. A. A. Galeev and R. N. Sudan),
North-Holland, Vol. i, p. 201.
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Guided MHD waves as a coronal diagnostic tool

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