Solutions to the Dispersion Relations

4.3.1 Asymptotic Solution

It is possible to derive an asymptotic solution to equation (4.17), valid for Iczq-^ 0, in exactly the same way as in the previous chapter. This w ill be useful so that a comparison with the equivalent solutions of Chapter 3 w ill allow a determination o f what the additional effect of the extra layer in the model atmosphere might be. This additional effect is likely to be the same for both uniform field and uniform Alfvén speed variants and so it is expedient just to look at the one case. Here, the uniform field model w ill be examined asymptotically.

F irstly the p-modes w ill be examined. It is assumed that as K —>0, 0 2 —>0*2^ as before, with 0*2;6Q, ±1. Then it is easy to show that

E-> E* = exp

(4.26)

(4.27) (p-> 1 1 +P _{V a},2 ' 7c y (4.28) 165

Yc(Yc+2p)/2

p(i - (4.29)

as K -^0. Thus the right hand side of the dispersion relation (4,17) has a well behaved lim it as K -^0, namely

(m + l)E ^(Y , + 2P ) Q ^ (l-n t) E Î- 1

(4.30)

+ 2P

e

J (l-£2^) + 2Y,nJ

In order to expand the left hand side of (4.17) it is convenient to assume that

(4.31)

so that expansions for a and Q2 are given by equations (3.47) and (3.48) for n > l. For n=l the expansions (3.51)-(3.53) must be used. Then

0,

s..

^(m +1)0^, s<m + 1, s=m + 1, s>m+ 1, (4.32) where a _ r^2 ( -1)" ‘ ( m + l) r ( l+ m + n)9n * * 2

(m+l) [r(m +l)] +(-l)"‘ V(l+m + n)e„

In just the same way as before an examination of the consequences o f each o f these three cases shows that only if s=m+l are any sensible results obtained. W ith s=m +l, equating the lim its for both sides of the dispersion relation gives an equation for 8„ which can be rearranged to yield

8n = (“ l)n-i E^(Y,+ 2p)(l -n ^ ) (m+ l)[r(m + _{r ( l+ m + n)}1) ]' (4.33)

Then the asymptotic solution for Q2 jg

2

r(l+m+n)y,((l-n^) -

2

£îJ) E^

m+l _{m - m}+1 n (2K)m+l

(n-l)!r(m+2)r(m+l)(y,+2P)(l-ay

(4.34)

valid for n = l, 2, 3,... This gives the asymptotic solution for the p-modes with the added effect o f the variable canopy height. The expression for given in equation (3.55), is reproduced here for convenience:

7 m+l

2r(l+m+n)y,((l-£2^)-2f2j)

m- m+l 'p J (2K)"’^1

n:

(n-l)!r(m+2)r(m+l)(y^+2P)(l-îi)

Comparison o f (4.34) with the equivalent result (3.55) for the model without the field free isothermal middle layer shows that the difference is simply the factor E*'^. Rewriting E* in dimensional form gives

E* = eZc/2H„

_(4.35)

being the density scale height in the middle region o f the atmosphere. Thus it is clear that the frequency shift due to the magnetism in the atmosphere decreases exponentially with the height of the canopy. For parameters typical of the temperature minimum, H^=100 km and so the frequency shift w ill decrease by a factor of ten with a change o f canopy height of approximately 250 km, everything else being kept equal.

The asymptotic solution for the f-mode can also be calculated for this model. W ith the assumption that

a ^ ~ l+ r(2 K )“, (4.36)

the expansions for the left hand side of the dispersion relation give the same result as before, namely that the asymptotic expansion for K —>0 is (see equation (3.70) in Chapter 3)

0(K ^ + o ( K ') - 5 H j^ + o(K ""''). (4.37)

Now, however, the right hand side of the dispersion relation has the following expansion:

-2 (m + l)E jr

2y, (2K)' + o(K'). (4.38)

Comparing (4.76) and (4.38) again implies that s=m +l, and then r can be obtained. The result is that for the f-mode

Again, the effect o f the middle region is to simply introduce the extra E*-2 factor as for the p-modes.

4.3.2 Numerical Solutions

The dispersion relations (4.17) and (4.25) can be solved numerically in the same way as for the the simpler models of the previous chapter. This time though the main concern is how the frequencies vary as a function of z^. For the basic atmosphere the parameters given in Table 3.2, representing a low lying canopy, have been used with suitable modifications for

the generalisation of the model. These are summarised in Table 4.1. Tp 4460 K T ^ 4170 K Tc 10,000 K pp 512.5 kg m*^ s'2 Yp» Yc’ Ym m 3/2 R 6425.97 n f s'2 K 'l g 274.0 m s 2 Rsun 6.96x10® m

Table 4.1. Parameters for the atmospheric model used in the numerical solution o f the dispersion relations (4.17) and (4.25).

Two sets o f data have been illustrated here. Figures 4.2-4.4 show the variation o f mode frequency as is increased from 0 to 600 km, keeping the magnetic field strength at the base of the chromosphere constant at 20 G. The modes shown are /=300 for the f-mode and the first two p-modes. Higher n modes follow the same pattern. Throughout the numerical results the behaviour of the mode concerned is given for a magnetic field which is uniform in the chromosphere (solid line), one which has a uniform Alfvén speed in the chromosphere (dotted line) and one in which there is no magnetic field (dashed line). For Bq constant, the maximum permissible field strength at the base o f the chromosphere, given by equation (4.5), is a decreasing function of the canopy height. The curves for the uniform

731.

■V

( p H z )

1731.

f-mode

0

1 0 0

200

300

400

500

600

z (km )_C

Figure 4.2. The variation of the f-mode frequency as a function o f the canopy height for

1=300, As Zjj is varied the magnetic field at the base of the chromosphere is kept constant at 20 G. The solid line shows the case of a uniform field, the dotted line gives the variation for a chromosphere in which the Alfvén speed is constant and the dashed line is for a field free atmosphere. The curves stop where the maximum permissible magnetic field strength at the base o f the chromosphere, as given by equation (4.5), falls below 20 G.

2644

•V

(pHz)

2642

B„=20 G

n=1 p-mode

0

1 0 0

200

300

400

500

600

2

(km)

C

Figure 4.3. The variation o f the frequency o f the first p-mode as a function o f the canopy height Zg, for /=300. As is varied the magnetic field at the base o f the chromosphere is kept constant at 20 G. The solid line shows the case o f a uniform field, the dotted line gives the variation for a chromosphere in which the Alfvén speed is constant and the dashed line is for a field free atmosphere. The latter two are almost indistinguishable at this scale. The uniform field case stops where the maximum permissible field strength is less than 20 G but the uniform Alfvén speed case is affected by magnetoacoustic cut-off at a lower value of z^.

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In document The effects of magnetic fields on oscillations in the solar atmosphere (Page 177-184)