It has been known for decades that solar coronal loops undergo oscillations. The problem has been in finding a consistent way to quantify and represent them. Compelling evidence for loop fluctuations attributed to a driven processes was first reported by Duijveman et al (1983). They noted that observations showed variation in brightness before onset time for a particular solar flare (the 1B/M4 event of Nov. 1980). Thus, preliminary MHD oscillations likely would have gone on for hours, with the oscillation amplitude increasing to the point a flare became inevitable (see their Figs. 5a – 5d) As the authors pointed out, this would also disclose a particular loop in the region (BC) not to be in hydrodynamic equilibrium. This means variable fluid (plasma) motions would be present well before the impulsive phase.

W =
exp(g t )

where g is the linear growth rate for
ion-acoustic waves and W is the wave energy. We postulate here that by time t =
t

_{A}1,2 where:
t

_{A}1 = 2 ∫_{x1’}^{x1}^{ }dℓ1_{|| }/ c_{A}ℓ1_{|| }and t_{A}2 = 2 ∫_{x2’}^{x2}^{ }dℓ2_{|| }/ c_{A}ℓ2_{|| }
and the modified Alfven velocity is: c

_{A}= v_{A }/ [1 + (v^{2}_{A }/ c^{2})]^{1/2}
so the
critical amplitude has been reached.
Duijveman et al (1983) record that the injected beam flux F

_{o}for footpoint B is about “twice as large as for footpoint C” (MacKinnon et al, 1985, note 1.7 x 10^{35}s^{-1}vs. 1.1 x 10^{35}s^{-1}, which explains the divergence observed for the respective loop half angles, and also the offset of footpoint B relative to the chromosphere (and footpoint C) as portrayed in Fig. 3 of Duijveman et al (op.cit) and also reproduced below. This is, observationally-speaking, where the beam instability and tandem*Pearlstein loss-cone instability*is incepted.The offset is depicted below by reference to the right hand side and the shortening of ℓ2

_{|| }in the transmission line ansatz . This is with the curvature of the loop factored in.

Note that use has been made of the

*Euler coordinate*(ℓ) and its relation to the column depth x (from Duijveman et al, 1983, Fig. 9). The Euler coordinate for loop BC is defined between the limits of

*the effective beam*– from column depth x1 = 1.9 x 10

^{17}E1

^{2}» 4.9 x 10

^{19}cm

^{-2 }at beam injection (E1 = 16 kev) to x

_{tr}» 1.6 x 10

^{20}cm

^{-2 }for

*the transition zone*. The quantity x

_{tr}is the approximate penetration depth of the beam electrons, which after conversion from Fig. 9 in Duijveman et al (1983), yields an Euler coordinate (along the loop BC) » 10

^{3}km » 10

^{6}m

How much it has been shortened can be estimated from the Fermi (I) acceleration-energy equation:

DE =
1/ 2m[p

_{^}^{2 }+ (L1_{o}/ L1)^{2}p_{êê}^{2 }]
We use
the form:

(L

_{o}/ L) = {DE - p_{^}^{2 }/ 2m/ p_{êê}^{2 }/ 2m}^{1/2}
where
L

_{o}is the original distance between magnetic mirrors . L is the instantaneous distance at the time of beam inception, and DE is the beam energy (20 keV) and the perpendicular-velocity and parallel velocity kinetic energies (p_{^}^{2 }/ 2m; p_{êê}^{2 }/ 2m ) are apportioned to the last velocity ratio readings displayed and computed from the data of Duijveman et al (1983) and Stahl (1984). Inserting this value, given that we know: v_{^}= ℓ2 v_{||}_{ }^{ }, we have:
L » (L

_{o })/ 2.3 » (4.6 x 10^{9}cm) / 2.3 » 2.0 x 10^{9}cm.
Note
that this is still the change in the total mirror distance: M= ℓ2

_{|| }+ ℓ2_{^}+ l2, which therefore ought to have diminished by nearly 50% or one fourth the total loop dimension.( 9.3 x 10^{9}cm).
Now,
from the data for column depth from Duijveman et al (1982, op. cit.) and its
relation to the Euler coordinate for the loop (Fig. 9) one can infer the
effective length of the fracture region, over [(x2’ – x2) - ℓ2

_{^}].(Fig. 4). This assumes the magnetic fracture occurs at a column depth of x » 2.7 x 10^{19}cm^{-2}. From this, the estimated length of loop for (x2’-x2) - ℓ2_{^}is: 4 0 x 10^{9}cm.
The
dimension ℓ2

_{|| }= ℓ2, meanwhile, can be estimated using*the beam-angle condition*(q = 0) at the footpoint, for which:
ℓ2
= dv m

_{e}v_{c}/ e E
for
which dv = [8.4 x
10

^{9}cm/s - 6 x 10^{9}cm/s] and v_{c}(cutoff velocity)= 7.4 x 10^{9}cm/s, and E = 1 V/cm. These parameters yield:
ℓ2

_{|| }= 1.0 x 10^{9}cm
From
the condition for the

*transverse potential drop*:
f

_{^}= ∫_{a}^{b }E_{^}(dx_{^}) » E_{^}[(ℓ2_{^})_{b}- (ℓ2_{^})_{a }] we know as per the previous result.
[(ℓ2

_{^})_{b}- (ℓ2_{^})_{a }] » 1.6 x 10^{9}cm
assuming
as before that: E

_{^}» 1V/ cm and [(ℓ2_{^})_{b}- (ℓ2_{^})_{a }] » l or the wavelength at cutoff, we find:
k

_{c}= 3.9 x 10^{–9}m^{-1}
for
which, using a phase velocity v

_{ph}» 3.5 x 10^{5}ms^{-1}:
w

_{c}» 1.3 x 10^{-3}s^{-1}
To
provisionally check these results, consider

*Landau resonance*for which s = 0 for:_{|| }v

_{|| }= 0

so: w
- sW
»
k

_{|| }v_{||}
or k

_{|| }= w / v_{|| }(for s = 0)
at
resonance condition (impedance Z1 = Z2, l1 = l2) , we see that for the waveguide treatment:

w

_{c}= k_{|| }v_{||}
Now,
the kink mode of oscillation for a higher density coronal loop has a typical
period (Edwin and Roberts, 1983):

t =
2L/ c

_{k}_{}
where
c

_{k}denotes the hybrid Alfven speed for the loop, as a function of ion and electron density in at before rise time (to flaring) and the respective Alfven speeds. with the exception of this “c_{k}mode” fast body waves have a low-k cutoff allowing only l < D(L) where D(L) is the diameter of the coronal loop[1]. For loop BC, we have D(L) = 1.1 x 10^{9}cm, so no wavelengths can be shorter unless of the “c_{k}mode” variety. This means that the waveguide values l1, l2 must be spurious because these are not related to any kink-modes.
Using
the period for kink mode oscillation in the preceding formula, we obtain:

t
= 52 s

for a
total loop length L = 9.3 x 10

^{9}cmThus have we computed a period of oscillation for one particular solar loop subject to very specific parameters. The problem for the solar physics researcher is to

*generalize treatments*for a whole range of loops. All I have shown here is one method for treating one loop of historical importance in a specific context.

[1] There
are also

*slow modes*which exist with characteristic velocity c_{o}. This is a factor 10-11 times slower than c_{k }. See Figs. 5-8 in: P.M. Edwin and B. Roberts: 1988,*Solar Phys*., 88, p. 179.
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