Preparing the
Hybrid Model:
We begin with a
coronal loop segment of the form depicted in Fig. 1 (Part 1) in which capacitative domains apply, such
that:
C = e [ℓ2||
+ ℓ2^ ] =
[1 + (i 4p s2)/ wo
]( ℓ2||
+ ℓ2^ )
Where wo is
the associated frequency. The region has associated with it an initial
E-field: E(z) = Eo
cos i(wo
t – k|| z)
For the fast-mode kink waves expected to be generated we
have for the wave number vector associated with the resonator of loop BC in AR
2776::
k|| = 2p / ℓ2||
= 6.2 x 10 -7 m-1
Because of the varying E-field, arising from loop
oscillations (for which no tan(noa
- p/4) = mo
), we expect a B-field to form in the region, leading to:
ò B j (r) dl = ¶/ ¶t òc Ez (r)·n dA
Such that:
B j (r)= iwo
r (m
0 e
0)1/2 /2 [Eo cos i(wo t – k|| z)]
With the most critical observation being that we obtain
“corrective” functions in E, based on the zeroth order Bessel function, J o (wo
r (m
0 e
0)1/2)
Assuming a
precise boundary cut-off at the value J o (ar)
= 2.405 where:
ar =(wr Öm 0 Öe 0
) so that the critical radius is:
r = 2.405/ (wo Öm 0
Öe 0)
, the cavity is resonant at:
wo
= 2.405/ (r Öm 0
Öe 0)
Ideally, this should be a harmonic of the kink-mode global
oscillation frequency. This is used here to set up the basic initial tests for
falsification, and will lead to more complete falsifying tests based upon the
twist of the loop and its helicity current density (which ought to be estimated
using proxy indicators)
In the treatment that follows we have in the interior of the
loop (cf. Edwin and Roberts, 1983)
Ñ·v = R(r) exp (wo t + nq + k|| z)
Where R(r) satisfies the Bessel equation:
So: d2R/dr2 + 1/r
(d R / dr ) - (mo2 + n2/r
2) R = 0
Where: mo2 = [(k2
co2 - wo
2)( k2 vA2
- wo 2)
/ (co2 + vA 2)( k2 ck2
- wo 2)
Where ck was previously defined
(Part 1), as was co.
In terms of the solutions as applied to the axis (r= 0)
of typical coronal loops, one has (Edwin
and Roberts, ibid.)
R(r ) = Ao Io(mor) mo2 >
0
Ao
Jo(nor) no2 = -
mo2 > 0
Where Io(mor) and Jo(nor) are Bessel functions
For the conditions in the corona, a fast kink mode will
govern such that:
no tan(noa
- p/4) = mo
with
period t
= 2L/ ck » 9 s
In the case of loop BC in AR2776, we have a = 5.5 x 10 8 cm, so that a/ L »
0.006
This rules out “sausage-mode” waves for the loop BC, since
these propagate only for the condition: ka
> 1.2 and a = L/10 (cf. Edwins and Roberts, 1983)[1].
If the model is correct the coronal cavity for loop BC
initiates frequency “pumping” at:
wo = wcav =
2.405/ (a Öm 0
Öe 0) = 1.3
x 10 2 s-1
This is the theoretical value we expect to obtain. Its relatively low magnitude is surprising at
first glance and in order to see what is happening, one needs to invoke the
conductivity and permittivity tensors,
viz.:
The main interest in terms of the preceding is how the
elements are functions of the various frequencies, e.g. Fitzpatrick, 2004)[2]:
e11
= 1 -
we
2 / w
2 { w/ w -
We }
- wi 2 / w 2 { w/ w + Wi }
e22
= 1 -
we
2 / w
2 { w/ w -
We }
- wi 2 / w 2 { w/ w - Wi }
e22
= 1 -
we
2 / w
2 - wi 2 / w 2
Where e11 and e22 denote
permittivities for right and left circularly polarized waves. Now, let there be
derivative quantities denoted:
S = ½ [e11 + e22]
And: D
= ½
[e11 - e22]
Then for low-frequency wave propagation in
magnetized plasma one finds:
D » 0 S » 1 + wi
2 / Wi 2
and e33 » - we 2 / w 2
With: we
2 / w
2 >> wi 2 / Wi 2 (in low frequency ordering regime)
To find the approximate observed (empirical analog) value for
the loop BC’s coronal cavity resonator angular frequency we use the cavity
resonator prescription of Federov et al, 2004, Sec. 4) such that:
wAR = p VAR/
dr
Where k AR is the cavity-associated wave
–number vector, k AR = wAR / vAR
and VAR is
the Alfven velocity in the cavity, with dr a height-scaled quantity
for the cavity (e.g. when seen in edge-on dimension and with curvature
correction applied). This is taken to be 1.1 x
10 6 m or roughly the minimal value of xi in the range noted earlier.
This yields an
observed value:
w’AR =
1.6 x 10 2 s-1
This difference (w’AR - wAR
) suggests either: 1) the resonant
cavity Alfven speed, VAR is
too high or 2) the quantity d r
is underestimated. Adopting the theoretical value as given (1.3 x
10 2 s-1) one finds the Alfven velocity in the cavity
resonator VAR = 4.5 x 10 7 ms-1 whereas if the observed value is used, one
finds: V’AR = 5.6 x 10 7 ms-1 . The
cavity wavelength parameter is: kAR
= k’AR = 2.8 x 10 -6
m-1 applicable to either VAR
or V’AR . Hence, k AR is a good proxy
wavenumber indicator for the cavity. The Alfven wave conductance for the cavity
SAR = (mo VAR)
-1 = 0.017 W -1
based on the theoretical value of wAR.
(0.014 W
-1 otherwise). The Alfven
wave impedances are, respectively: ZAR = (SAR) -1 = 57 W, and Z’AR = (S’AR) -1 = 70 W. The “characteristic impedance” can also be approximated using:
We are now in position to more realistically assess the quality
factor (Q) for the date using equation 64(a) of Hollweg (1983): Q » k2r/
(2 êk2i
ê)
where here we have k2r » kAR and k2i » kBC . From this we obtain: Q »7.5.
The loop heating rate from the data (EH = 0.60 erg cm-3 s-1) allows us to obtain the amplitude damping rate (w i)
as defined by Hollweg and thence the resonant period: T = 2 p/ (w r). Using the (EH) datum in
conjunction with Hollweg’s equation (31) for the wave energy density (assuming
vA2 = VAR and that the magnetic energy density (B2/8p, in
c.g.s.) is the primary contributor for a
cavity between two nodes cf. Zugzda and Locans, 1982, op.cit.), we get:
T » 69 s
The preceding shows the general approach in obtaining cavity parameters for model testing. With sufficient data resolution and discrimination available for other coronal loops, of the type that was available for loop BC in AR 2776, we should be able to obtain a much better idea of how well this cavity resonator model works and whether it can be suitably generalized.
[4] .Zugzda and Locans, op. cit.
