The problem of identifying a unique trigger for solar flares has been pursued for over 4 decades, but with little to show for it. In this post I examine a possible approach that might be productive, especially after higher resolution images become available from a planned solar telescope.
1. Background
to Cavity Resonator Approaches:
On the Sun itself, the 5-minute oscillations, more
figuratively described from time to time as a “ringing of the photosphere” were
first tied to waves trapped in a resonant cavity by Schatzman, 1956
.
The gist of the model is that the upper and lower cavity boundaries reflect
waves into the cavity and thereby engender a standing wave, which may either be
acoustic or gravity.
Properties of generic coronal cavity resonators were
elucidated by Hollweg (1984)
whereby a coronal loop is treated
as three relatively disjoint regions,
separated by discontinuities. The standing waves produced are invoked to
account for energy dissipation and heating in the corona. Some aspects of
Hollweg’s model (e.g. reflection properties of Alfven waves, quality factor Q,
relation to wave number vectors) are also employed in my own resonator model
for solar flare inception.
Where I diverge from both the (e.g. Federov et al) auroral
cavity resonator and the coronal one proposed by Hollweg is that in this flare
trigger model I adopt a dual resonator
for a given compact flare loop. The basic sketch is shown above. Thus, to trigger a specific (e.g. compact)
flare the conditions must be such that the Q-(quality) value in each resonator
reinforce the other. In my generic model, I include a small coronal arch cavity
resonator with some resemblance to Hollweg’s and a large scale loop resonator
which depends on the oscillations arising from magnetic (Alfven) waves in
combination with the loop’s twist (and associated kink instability)
The difference is that I go into much more detail to
incorporate ex-post facto data into my model to show how the magnitudes of the
changing physical quantities vary not only in the corona in its pre-flare
state, but during the flare as well. As an application ansatz,
dual
resonators in the electrical engineering setting often use closed-loop
resonators in order to shift down the original resonator and arrive at a very
small structure (e.g. Collado et al, 2007)
.
In the flare trigger model context this would be the kernel or coronal loop
apex resonator.
In the engineering
context, mirrors are sometimes employed to linear cavities (analogous to the
extended coronal loop with its primary cavity at the apex) to obtain
simultaneous dual wavelength oscillations. It is precisely within the scope of
these dual oscillations that the flare trigger can be conceived – e.g. for
specific cases when a dual resonance is achieved and with it the maximum
instability.
2. Motivation:
Having established that a hybrid flare model is the most
plausible one to approach the 1B/M4 flare of November 5, 1980, I now single out
the key feature for the flare trigger. Before proceeding, let us inspect the
gestalt for what this article is all about, as depicted in the schematic below:
The diagram depicts the generic inputs and processes entering the hybrid
flare model (in the main rectangle), which includes components for R
(reliability statistics), H (helicity considerations) and Poisson statistics.
Thus, the hybrid model seeks to reconcile all of these, as well as recognizing
the inputs from two paradigms: the
E-J
and the
B-v.
For example, the
B-v paradigm and its assumptions figure more prominently in the
reliability analyses as well as the magnetic helicity. (E.g. see:
http://brane-space.blogspot.com/2010/10/look-at-magnetic-helicity.html ) The
E-J paradigm
factors more into the basis for Poisson variations based on the manner in which
the current densities (J) arise and how the E-field is generated.
The putative flare trigger attempts to make use of all of
these.
What is desired is a model that approximately replicates the
event sequence for the region AR2776 such that the flares occurring conform to the average
Poisson
activity:
l (av) =
2.6 x 10
-5 s
-1 and the length, resonance variations described
above imply a twisted cavity (dual) resonator over the region defined by (
l1 +
ℓ1
^
+
ℓ2
^ +
x
i
).
(See e.g. my post of June 22nd: http://brane-space.blogspot.com/2014/06/quantifying-solar-loop-oscillations.html
Where: 0 < xi <
1.1 x 106 m
The basic geometry is shown in Fig. 1 (top) in the region of the apex,
and primary coronal cavity. It is assumed that with compressional Alfven waves
the loop aperture can vary, from a1 to the outer radius taken as r1. This
variation could well account for the uncertainty in source-kernel dimensions:
The electric field E(z) shown in Fig. 1 is described according to:
E(z) = Eo cos w(t – z/ vp)
Where Eo denotes the uniform (non-varying field magnitude)
and vp = c sin(J)
is the phase velocity with J the pitch angle of the
twist component for relative helicity (H(R) [T]).
The model works via the basic loop changing its effective
resonator length (for which there is an associated resonator angular frequency
wo)
and twist (
F(r)).
Radial surfaces (r
s < r)
form in the loop apex (small resonant cavity)
for E-field resonating corrections modeled after the
J o (ar) Bessel
function. The J o (ar)
induced field in turn generates azimuthal corrections in the axial
B-field that alters the twist of the gross loop.
Thus, the
twist dependence is (see also
http://brane-space.blogspot.com/2013/04/looking-at-bessel-functions-applications.html):
F(r) =
L J1(ar)/ r J o
(ar)
= L B j (r) / r Bz (r) = L E z
(r) /
r E j (r)
Such that: E z
(r) ® B j (r) ® E1 z (r) ® B1
j (r) ® E2 z (r) ® B2
j (r)
E j (r) ® B z (r) ® E1
j (r) ® B1 z (r) ® E2
j (r) ® B2 z (r) . . . . En j (r) ® Bn z (r)
From the
secondary, tertiary etc. fields inner nested radii aij are generated
which conform to ratios related to the Bessel functions. The key point of the
mutually generated fields is that they operate according to a positive feedback
which ultimately incepts a resonance condition and explosive release of energy.
The radii in turn can be used to obtain wave modes associated with a given
oscillation period for the resonator. The relative E-field strengths
successively generated for the ideal cavity coronal resonator are defined by
the Bessel series:
Jm
(x) = (1/ 2m m!) xm [1 -
x 2/ 22 1! (m + 1)
+ x4/ 242!
(m + 1) (m + 2) - ….
.(-1)j
x2j / 2 2j j! (m + 1) (m + 2)……(m + j) + …]
Which may be
simplified for the m = 0 case to:
J0 (x) = 1 -
(x/2)2 + 1/(2!)2 (x/4)4 – 1/ (3!)2
(x/2)6 + .
Where x for the
E-field is defined x = Ö mo Öℰo w r
= 2.405
For which a key
cut-off radius is defined at the surface rs = r . Other surfaces (si) may be defined
for zeros of J0 (x). Meanwhile, coronal loop oscillation
periods and emergence have been well explicated by a number of authors (e.g. Edwin
and Roberts, 1983, op. cit., Andries
et al, 2005)
(More to come)
