Monday, June 30, 2014

A Cavity Resonator Model Applied to Solar Loops and Flare Triggers (2)

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:

wAR    =  1.6 x 10 2 s-1

This difference  (wAR    -  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  = (SAR) -1  = 70 W.  The “characteristic  impedance” can also be approximated using:

 Zch »  ZAR ( kBC/ k AR)

This is modified from the auroral cavity version given by Federov et al (2004, op. cit.). In the above, kBC denotes a wavelength number vector applicable to the loop, and we use the observational (cavity) values for  ZAR , since k BC is an observed value.  For the date on which the 1B/M4 event occurred (11- 5- 80) we have k BC = 1.9 x 10 -7 m-1  so that Zch » 4.6 W.

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.

 The reader should be aware of why the period found here  (»  69 s) differs from that estimated in the June 22 post (»  52 s). In particular, note the latter assumed global kink mode oscillations for a total loop length L = 9.3 x 10 9 cm

[1] P.M. Edwin and B. Roberts, Solar Phys., 88, 179, 1983.
[2] Richard Fitzpatrick: 2004, Introduction to Plasma Physics, Lulu Books, p. 109.
[3] Hollweg, op. cit
[4] .Zugzda and Locans, op. cit.

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