Finite Lower Layer: The Chromosphere

The first case for consideration is that of a finite depth of chromosphere, of extenth(see Fig. 2.5). The

coronal extent of the loop is2Lwith two chromospheric footpoints each of extenth. The total loop length

is therefore2(L+h). However, this can be simplified by considering only half the loop, as it is symmetric

about its apex at z = 0(this condition also stabilises the structure, as mentioned before). So the loop

half-length isL+hand the footpoint end lies atz=L+h.

Eq. (2.86) is solved forQ(z)withk2_{(z)given by Eq. (2.87). The functionk(z)has two values,k} cin the

coronal region andkchin the chromospheric layer. The general solution forQ(z)in this two layer model is

Q(z) =    acsin(kcz) +bccos(kcz), 0≤z≤L, achsin(kchz) +bchcos(kchz), L < z≤L+h. (2.88) For typical parameters, coronal temperatureTc = 1MK and a chromospheric temperatureTch = 10000

K, then the sound speeds arecc = 150km s−1andcch = 15km s−1. Consequently,Λc = 50Mm and

Λch = 500km, giving the acoustic cutoff period for the corona to bePc= 2π/Ωc '70minutes, and the

chromospheric cutoff period to bePch = 2π/Ωch '7minutes. Observations by Wang et al. (2006) have

shown that the slow mode in hot coronal loops, excited by an internal nanoflare, oscillates with a period around10−30minutes, so the modes would be evanescent in the chromosphere, though propagating in the

corona.

2.3 Effect of a Lower Layer 41

means that four boundary conditions are required. The line-tying conditions are,

Q(0) = 0, Q(L+h) = 0. (2.89)

These two conditions enforce nodes at the loop apexz= 0and at its footpointz=L+h. This models the

oddstanding modes of oscillation of the loop as a whole. For theevenmodes the first of the conditions in Eq. (2.89) is altered to become

∂Q

∂z(z= 0) = 0. (2.90)

The first two conditions have simple consequences on Eq. (2.88):

Q(z) =    acsin (kcz), corona, achsin kch(z−(L+h)) , chromosphere, (2.91)

for the odd modes, and

Q(z) =    bccos (kcz), corona, achsinkch(z−(L+h)), chromosphere, (2.92) for the even modes.

The third condition requires that the velocity perpendicular to the interface is conserved across the inter- face,

ˆ

n_·[v] = 0. (2.93)

As the model is one dimensional with no flows(v0= 0), this translates to the conservation of velocityvz

acrossz=L; all motion is perpendicular to the interface. Also, asf(z)is continuous (because equilibrium

pressurep0is continuous) then this condition is simplyQcontinuous acrossz =L. Considering the odd

modes then the condition is

acsin(kcz) =achsin(kch(z−(L+h))), z=L.

Thus, for the odd modes

acsin(kcL) +achsin(kchh) = 0. (2.94)

The equivalent condition for the even modes is

bccos(kcL) +achsin(kchh) = 0. (2.95)

differential equation (2.86), rewriting this using Eq. (2.87) gives d2_Q dz2 =− ω2 c2 s(z) Q(z) +Ω 2_(z) c2 s(z) Q(z). (2.96)

Following the analysis of Roberts (1981b) and James (2003) the integral of Eq. (2.96) is considered over the small neighbourhood ofz=L, namely

Z L+ L− d2_Q dz2dz=− Z L+ L− ω2 c2 s(z) Q(z)dz+ Z L+ L− Ω2_(z) c2 s(z) Q(z)dz. (2.97)

James (2003) showed that this integral becomes the fourth boundary condition; the continuity of the total plasma pressure acrossz=L,

_dQ dz − 1 2Λ(z)Q z=L = 0. (2.98)

Consider first theoddmodes; note the definition off(z)andf0

(z)as being f(z) = exp −_2Λ(z_z) , f0(z) =−_2Λ(1_z)f(z). (2.99)

Continuity of the total plasma pressure perturbation across the interface atz=L, given by Eq. (2.98), gives

ac kccos(kcL)− 1 2Λc sin(kcL) −ach ₁ 2Λch sin(kchh) +kchcos(kchh) = 0. (2.100)

Similarly, for the even modes,

bc − 1 2Λc cos(kcL)−kcsin(kcL) +bch − 1 2Λch sin(kchh)−kchcos(kchh) = 0. (2.101)

This is a system of two equations for ach andac; Eqs. (2.94) and (2.100) for the odd modes (and a

similar system for the even modes in Eqs. (2.95) and (2.101)). This can be solved by finding the zero of the determinant of the matrix constructed by this system. This determinant gives the dispersion relation for the odd modes of oscillation in a thin coronal loop structured in sound speed by a step function with two distinct isothermal layers:

1 2 ₁ Λc − 1 Λch

sin(kcL) sin(kchh) − kchsin(kcL) cos(kchh) (2.102) −kccos(kcL) sin(kchh) = 0.

Similarly, the dispersion relation for the even modes is

1 2 1 Λc − 1 Λch

cos(kcL) sin(kchh) − kchcos(kcL) cos(kchh) (2.103)

2.3 Effect of a Lower Layer 43

Here, Eqs. (2.102) and (2.103) are solved numerically. On non-dimensionalising the dispersion relations four dimensionless parameters are apparent: ω/Ωcthe dimensionless frequency,L/Λc the dimensionless

loop half-length,h/Λc the dimensionless chromospheric extent of the loop, andΛc/Λch the ratio of the

density scale heights in each region. The fourth parameter is effectively the ratio of the acoustic cutoff frequencies in each region and in these computations it is set equal to10. Any alteration in this parameter

only changes the upper cutoff value, allowing propagation further into the lower region.

Initiallyh/Λcis set close to zero to check that the dispersion relations given by Eqs. (2.102) and (2.103)

match the expected single layer model shown in Fig. 2.2. Fig. 2.6 shows the dimensionless frequencies for the odd and even modes as a function of dimensionless loop half-length whenh/Λc = 0.001Lcompared

to the isothermal case. These curves match exactly; however, they are cutoff by the presence of the lower layer (not present for the dashed single layer curves).

Now consider the effect of other values ofh/Λc. By considering a significant value ofh/Λc, for example

h/Λc= 0.1Lorh/Λc = 0.25L, then the effect of the inclusion of a lower layer can be investigated directly.

Fig. 2.7 shows that the oscillation is evanescent in the lower region, asL/Λc approaches zero, which is

shown by the frequency cutoff atω/Ωc = 10. The dispersion introduced due to gravity is still apparent,

and the frequency oscillation tends to the acoustic cutoff frequency as L → ∞. However, comparing

Fig. 2.7 to Fig. 2.8, the inclusion of a lower layer has reduced the frequency of oscillation significantly. Next, the effect of varyingh/Λcis studied by fixing the value ofL/Λcon the dimensionless frequency.

Fig. 2.9 shows that ash/Λc increases then the shift of the frequency from the isothermal case also

increases. Fig. 2.10 shows the percentage reduction in frequency (∆ω/ω) due to an increasing depth of

footpoint layer h/L when L/Λc = 2.5. It indicates that the shift is small for h/L 1. Also, from

Fig. 2.10, the fundamental odd mode suffers a different shift due to the footpoint structure than the first even mode. It is interesting to note that whenh/Λc= 0then the frequencies of all the modes matches the

frequencies given in Fig. 2.8 forL/Λc= 2.5. A simple structuring in density (by modelling a step function

in the sound speed) causes a shift in the frequency of oscillation in the acoustic (slow) mode.

Another interesting result is apparent in the eigenfunctions ofvz, shown in Fig. 2.11. There is ampli-

fication due to stratification; however, there is no wave propagation in the chromospheric region. Also, as the frequency of oscillation is shifted this suggests that the period is affected by the inclusion of a lower layer. In particular, loops of length similar to the pressure scale height suffer a difference of almost10%

in period in comparison to the single layer case. This model has assumed that the solution is propagating in the lower, chromospheric region. However, typical parameters suggest that the frequency there is below the acoustic cutoff, so it is now convenient to consider the case where evanescent solutions occur in the chromospheric footpoints.

Figure 2.6: Dispersion curves comparing the modes of oscillation of the two layer model (withh= 0.001L)

and the modes of the isothermal model. The modes of the two layer model are given by the solid curves and the modes of the isothermal single layer model are shown by the dashed curves. HereΛ0= Λc,Ω0= Ωc

andΩch= 10 Ωc.

Figure 2.7: Dispersion curves as a function ofL/Λ0 for the two layer model withΛc/Λch = 10(so the

dimensionless cutoff frequencyΩchin the chromosphere is10 Ωc), whenh/Λc = 0.1L. The solid curves

are the modes of oscillation in the two layer model and the dashed curves are the modes of the single layer model. In the diagramΛ0= ΛcandΩ0= Ωc.

2.3 Effect of a Lower Layer 45

Figure 2.8: As in Fig. 2.7 but withh/Λc= 0.25L.

Figure 2.9: Diagram showing the effect of varying the dimensionless chromospheric extent,h/L, for a

Figure 2.10: Diagram highlighting how∆ω/ω= (ωh−ωh=0)/ωh=0varies with varying footpoint extent

h/L. Here the solid line represents the variation in frequency of the first odd mode and the dashed line

represents the first even mode. L/Λc= 2.5in this diagram.

Figure 2.11: Global eigenfunctionsvz =v(z)eiωtfor the two finite isothermal layers, simulating a dense

chromosphere. Here cs = 150km s−1, cch = 15 km s−1 andh = 0.1L. The loop apex is atz = 0

and the loop footpoint is atz/Λ0 = 5.5. The solid vertical line atz/Λ0 = 5indicates the extent of the

2.3 Effect of a Lower Layer 47