Examining The Transmission Line Approach To Solar Flare Triggering

 

There are numerous ways by which we can approach the transfer of wave energy in solar coronal loops, and one of the ways proposed (originally in my Ph.D. thesis) was via adopting the transmission line model. Such transmission lines are most often employed in the transfer of electric energy from a power grid to specific locations, but with some adaptations it's feasible to arrange the model for solar considerations.

Here, we assume the turbulent regions are terminated on either side by impedances that cause partial reflection and standing Alfven waves each with characteristic wavelengths, l1 and l2.  For simplicity of treatment, we let ℓ1 = l1  and ℓ2 = l2.  So that, Z1 = Z1’ and Z2 = Z2’.  Then the input impedance referenced to the base of the loop for each case is:

Z(1)_in = 

 j/2(Z1 – Z11) tan kl1   + [Z1 Z11 – ¼ (Z1 – Z11)² tan² kl1   ]^1/2

 and

Z(2)_in =

  j/2(Z2 – Z12) tan kl2   + [Z2 Z12 – ¼ (Z2 – Z12)² tan² kl2   ]^1/2

On each side,  we define cut-off angular frequencies occur for which:

tan² k_c l   =  tan² [(vc l Öe) / c]  =  

  4Z1 Z11/ (Z1 – Z11)²   =    4Z2 Z12/ (Z2 – Z12)²

At the base of each footpoint (in the limit of tan q » 0) one finds the cut-off frequency is:

w_c ² = 4 Z11 c²/ Z1 l1²  e_o  =   4 Z12 c²/ Z2 l2²  e_o 

 This can be simplified to:    w_c  = 2 / Ö(L1 l1) Ö(C2 l2)  and

tan² k_c l =  tan ² [w_c lÖe /c]

where as before: L1 =     V(D)  - RI₁/ (dI₁/dt)

and: 

C2 = e2 [ℓ2_||    +  ℓ2_^ ]   =  [1  +  (i 4p s2)/ w2 ]( ℓ2_||    +  ℓ2_^ )

where the conductivity (s2) and plasma frequency (w2 )  are assumed to diverge from the values for the other loop BC segment.

It bears looking more closely here at the associated wave impedances, Z1’ and Z2’ in the context of the theory of long lines. In particular: for the specific wave turbulent regions (wave guides) let (cf. Zugzda and Locans, 1982)[1]:

Z(1) =  -iÖ(4pr(1))   and  Z(2) =  -iÖ(4pr(2))  

and:

Y(1) = -i/ Ö(4pn(1) m _e )  and   Y(2) = -i/ Ö(4pn(2) m _e ) 

where n(1), n(2) are the respective particle number densities and the Y(1,2) are the linear vector admittances in the wave guides.  Whence:

 

Z1’ =  Ö(Z(1)/ Y(1)),      Z2’ = Ö(Z(2)/ Y(2))

and:

Z(1) =  -iÖ{4pr(1) (1 +ig₁/ w)},     Z(2) =  -iÖ{4pr(2) (1 +ig₁/ w)}  

where g₁ is the “growth factor” such that:

g₁  <    g/ w

where, at resonance condition (Z1 = Z2, l1 = l2) , we take: g »  10 ⁶ s^-1  and the aggregated (lumped) frequency of all modes in the loop is estimated to be w »  0. 043 s^-1

The resulting rate of wave energy increase, e.g. determined by (Cromwell, 1988)[2]:

dW/ dt =  gW  

 

here g is the linear growth rate for ion-acoustic waves and W is the wave energy. The key point here is that when (T_e/ T_i) > 4.8   a mean oscillatory condition (consistent with marginal stability )  emerges such that dW/dt = 0 (Cromwell et al, 1988)[3]. In other words, the wave energy will oscillate between maximal and minimal amplitudes and with it the dimensions of the wave region.

For the 1B/M4 event, HXIS measurements disclose T_e =   1.04 x 10 ⁷ K and T_i =   2.0 x 10 ⁶ K so T_e/ T_i > 5 and this meets the flare triggering condition.  The fact that as Crowell et al (1988) note that the beam stopping length varies considerably in simulations can be explained by the fact that scale lengths in  the turbulent regions as well as  (ℓ1_||  ;  ℓ1_^  ;  ℓ2_||  ;  ℓ2_^   ; L_k  )  are oscillating.

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[1] Y.D. Zugzda and V. Locans: 1982, Solar Phys., 76, p. 77.

[2] D. Cromwell, P. McQuillen and J.C. Brown, Solar Phys., 115, 289, 1988.

[3] D. Cromwell, P. McQuillan, and J.C. Brown:1988,  op. cit..