Friday, June 27, 2014

A Cavity Resonator Model Applied to Solar Loops and Flare Triggers (1)



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:

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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[2]. 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)[3] 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)[4]. 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:

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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^   +  xi ).  (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 (rs < 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 Öo 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[5])
(More to come)



[1] E.N. Fedorov, V.A. Pilipenko, M.J. Engebretson, and T. J. Rosenber: 2004, ‘Alfven Wave Modulation of the Auroral Acceleration Region’, in Earth Planets Space, 56, p. 649.
[2] E. Schatzman: 1956, Ann. Astrophysics, 19, 45.
[3] Joseph V. Hollweg, Solar Phys., 91, 269, 1984
[4] C. Collado, J. Pozo, J. Mateu and J.M. O’Callaghan: 2007, European Microwave Week.
[5] J. Andries, M. Goosens, J.V. Hollweg, I. Arregui and T. Van Doorselaere,: 2005,  Astronomy & Astrophysics,  430, 1109.

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