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. The accompanying diagram shows how this might work. (Note l in the equations is the Greek lambda in the diagram).
Here, we assume turbulent regions are terminated on either side by impedances (Z1, Z2) 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, cut-off angular frequencies occur for which:
tan2 kcl = tan2 [(w_c l e^1/2) / c] = 4Z1 Z11/ (Z1 – Z11)^2 = 4Z2 Z12/ (Z2 – Z12)^2
At the base of each footpoint (in the limit of tan theta ~ 0) one finds the cut-off frequency is:
w_c ^2 = 4 Z11 c^2/ Z1 l1^2 e_o = 4 Z12 c2/ Z2 l2^2 e_o
(where e_o is the permittivity of free space, = 8.85 x 10^-12 Farad/m)
This can be simplified to:
w_c = 2 / [(L1 l1) (C2 l2) ]^1/2
and
tan^2 k_c l = tan ^2 [w_c l e^1/2 /c]
where: L1 = V(D) - RI1/ (dI1/dt)
is the loop inductance, dI1/dt the rate of current increase in the loop and R1 the resistance.
and the capacitance:
C2 = e2 [ℓ2 + ℓ2_perp ] = [1 + (i 4pi o2)/ w2 ]( ℓ2 + ℓ2_perp )
where the conductivity (o2) and plasma frequency (w2 ) are assumed to diverge from the values for the other loop BC segment. (In AR2776, the loop designated 'BC' is the one that could actually be observed and measured)
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:
Z(1) = -i(4pi rho(1))^1/2 and Z(2) = -i(4pi rho(2) )^1/2
and:
Y(1) = -i/ (4pi n(1) m _e )^1/2 and Y(2) = -i/ (4pi n(2) m_ e )^1/2
where n(1), n(2) are the respective number (particle) densities, rho(1,2) the energy densities, m_e = 9.1 x 10^-31 kg is the electron mass, and the Y(1,2) are the linear vector admittances in the wave guides. Whence:
Z1’ = [(Z(1)/ Y(1))]^1/2
Z2’ = [(Z(2)/ Y(2))]^1/2
and:
Z(1) = -i{4pi rho(1) (1 +ig1/ w)}^1/2
Z(2) = -i{4pi rho(2) (1 +ig1/ w)}^1/2
where g1 is the “growth factor” such that:
g1 (less than) g/w, where at resonance condition (Z1 = Z2, l1 = l2) , and we take g~ 10^6 s-1. Wave energy increase is determined by (Cromwell, [1]): dW/ dt = gW, or recast ln (W) = gt or W = exp(gt). Where 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 emerges such that dW/dt = 0 (op. cit.) 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 flare event, HXIS measurements disclose Te = 1.04 x 10^ 7 K and T_i = 1.0 x 10 6 K so Te/ Ti > 10 and this meets the condition. The fact that as Crowell et al (op. cit.) 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_perp ; ℓ2 ; ℓ2_perp ; L_k ) are oscillating.
This approximately addressed the model, but to verify if, we would need to observe actual solar loops in the process of oscillation at the scales of resolution for the (ℓ1 ; ℓ1_perp ; ℓ2 etc. components, or about 0."2. So far the only instrument that can deliver this at optical wavelengths is the Hubble space telescope but it's never been aimed at the Sun and likely will never be!
[1] D. Cromwell, P. McQuillan, and J.C. Brown:1988, Solar Phys., 115, 289.
Here, we assume turbulent regions are terminated on either side by impedances (Z1, Z2) 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, cut-off angular frequencies occur for which:
tan2 kcl = tan2 [(w_c l e^1/2) / c] = 4Z1 Z11/ (Z1 – Z11)^2 = 4Z2 Z12/ (Z2 – Z12)^2
At the base of each footpoint (in the limit of tan theta ~ 0) one finds the cut-off frequency is:
w_c ^2 = 4 Z11 c^2/ Z1 l1^2 e_o = 4 Z12 c2/ Z2 l2^2 e_o
(where e_o is the permittivity of free space, = 8.85 x 10^-12 Farad/m)
This can be simplified to:
w_c = 2 / [(L1 l1) (C2 l2) ]^1/2
and
tan^2 k_c l = tan ^2 [w_c l e^1/2 /c]
where: L1 = V(D) - RI1/ (dI1/dt)
is the loop inductance, dI1/dt the rate of current increase in the loop and R1 the resistance.
and the capacitance:
C2 = e2 [ℓ2 + ℓ2_perp ] = [1 + (i 4pi o2)/ w2 ]( ℓ2 + ℓ2_perp )
where the conductivity (o2) and plasma frequency (w2 ) are assumed to diverge from the values for the other loop BC segment. (In AR2776, the loop designated 'BC' is the one that could actually be observed and measured)
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:
Z(1) = -i(4pi rho(1))^1/2 and Z(2) = -i(4pi rho(2) )^1/2
and:
Y(1) = -i/ (4pi n(1) m _e )^1/2 and Y(2) = -i/ (4pi n(2) m_ e )^1/2
where n(1), n(2) are the respective number (particle) densities, rho(1,2) the energy densities, m_e = 9.1 x 10^-31 kg is the electron mass, and the Y(1,2) are the linear vector admittances in the wave guides. Whence:
Z1’ = [(Z(1)/ Y(1))]^1/2
Z2’ = [(Z(2)/ Y(2))]^1/2
and:
Z(1) = -i{4pi rho(1) (1 +ig1/ w)}^1/2
Z(2) = -i{4pi rho(2) (1 +ig1/ w)}^1/2
where g1 is the “growth factor” such that:
g1 (less than) g/w, where at resonance condition (Z1 = Z2, l1 = l2) , and we take g~ 10^6 s-1. Wave energy increase is determined by (Cromwell, [1]): dW/ dt = gW, or recast ln (W) = gt or W = exp(gt). Where 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 emerges such that dW/dt = 0 (op. cit.) 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 flare event, HXIS measurements disclose Te = 1.04 x 10^ 7 K and T_i = 1.0 x 10 6 K so Te/ Ti > 10 and this meets the condition. The fact that as Crowell et al (op. cit.) 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_perp ; ℓ2 ; ℓ2_perp ; L_k ) are oscillating.
This approximately addressed the model, but to verify if, we would need to observe actual solar loops in the process of oscillation at the scales of resolution for the (ℓ1 ; ℓ1_perp ; ℓ2 etc. components, or about 0."2. So far the only instrument that can deliver this at optical wavelengths is the Hubble space telescope but it's never been aimed at the Sun and likely will never be!
[1] D. Cromwell, P. McQuillan, and J.C. Brown:1988, Solar Phys., 115, 289.
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