lnclusion of resistivity
6.2.2 WKB approximation
Tbe inner solution obtained by boundary layer tbeory bas only a local validity.
Far from tbe Alfven resonance we can use tbe WKB metbod (13](14] to obtain an approximate expression for tbe solution of (6.10).
Consider tbe original equation (6.10). We seek a solution in tbe form of a WKB sene:
(6.29) ó =ft and S(z) is given by
!
zooS(z) = l)nr/!n(z')dz'
0
(6.30) In our case, we truncate tbe series given by (6.30) after n =1. Tbis means tbat S(z) now becomes
(6.31) lnserting (6.31) into (6.10) and cernparing powers of f give tbe following two equations for r/Jo and r/!1
(6.32)
dr/!o 1
dz +2r/!o(<Pl-L)=0 (6.33)
From (6.33) and (6.32) it tben fellows tbat
(6.34) Note tbat </J1 is real. Tbe bounded WKB solutions can now be written as
Vy "'So exp(fz rPl(z')dz') x exp(/z -(1 + i)f-t
L~
(k2a2- w2)tdz') z >Za (6.35)Vy "'sl exp(/z rPl(z')dz') x exp(/z (i- 1)f-t
L~
l(k2a2 - w2)ltdz') z <Za (6.36) wbere Za stands for tbe location of tbe Alfven resonance. Tbe WKB solutions (6.35) and (6.36) must be matebed to (6.24) and (6.25). In tbe neigbbourbood of tbe Alfven resonance tbe WKB approximation takes tbe formS -..L -
1. V2 (
1 ') -1.
i!vy"' of 12r 4 exp-- +z f 2 r 2
3
r>O
(6.37)S
_.!_I 1_1. V2(.
1)_1.1 I;!
Vy "' l f 12 T 4 exp - Z - f 2 T 2
3
r<O
(6.38)In order to match (6.37) and (6.38) with (6.24) and (6.25) we require that
Sa
=
f.h exp(2 1
4 1ri)E1 and (6.39)
From (6.39) it then follows that
!Sol= IS1I·
This principle result indicates that there is total transmission. Note that the detailled behavior of the solution in the neighbourhood of the Alfven resonance can not be obtained from the WKB solution. However after it has passed through the resonance ( which is now a turning point) and reaches a region where the WKB solution is again applicable, the solution in this region is predicted by the matching procedure. In chapter 9 we will see that in a lossless case, we have total reflection instead of total transmission.6.3 Magnetoacoustic-gravity waves
In this section we derive a differential equation for Vz. We saw that in the ideal case, Vz is described by a second order singular differential equation. By including resistivity we obtain a fourth order regular differential equation. This implies that a new mode is created. The calculation of this new mode is the subject of this section.
After trivia! manipulations we can derive the following differential equation for
Vz:
(6.40)
f. is defined by (6.9). Note that due to the inclusion of resistivity the equation ( 6.40) is regular. In addition to magnetoacoustic-gravity waves,(6.40) contains also a new mode which is a result of resistivity. By setting this latter equal to zero, we get our familiar secoud order singular differential equation back. The equation (6.40) is not so easy to handle, because its coefficients depend on z .
To get some insight in the behaviour of the solution of (6.40) we investigate it for z - t oo and z - t -00. In the limit z - t oo, the Alfven speed becomes very large. In this limit the differential equation (6.40) becomes
(6.41)
38
equation (6.41) provides no propagating solutions. A fact which is not Hence, in this limit we obtain a fourth order differential equation with con-stant coefficients. Thus, there exist solutions of the form
A ~z
This equation takes the form of equation(5.41) when we subtitute À=~ L
(6.46)
(6.47)
(6.45)
According to chapter 5, equation (5.41) has only complex solutions. But this contradiets our initia! assumption that À is real. The condusion is then: equa-tion (6.44) has no real valued roots. In the same way it can he proved that these roots arealso not imaginary. For the differential equation (6.42) this means that it has evanescent wave solutions.
6.3.1 Inner and outer solution
The highest derivative in equation (6.40) is multiplied by t which we will assume small. The smallness of t allows us to consider the equation (6.40) as a singular perturbation problem. Therefore we can use boundary layer theory to approxi-mate its solution. It's convenient to perfom the following transformations
and
( = 1- (oexp(--), z L
Vz = <I>exp(-kz) With these transformations (6.40) becomes
~<I> ~<I> ~<I>
it(1- ()4 d(4 - it(1- ()3(9
+
4kL) d(3+
(1- ()(O(t)+ ()
d(2(6.48)
(6.49)
d<I>
+[O(t)
+
1-(a+ f3+
1)(] d( - (a/3+
O(t))<I> = 0 (6.50)where a and f3 are given by
(a, /3)
= ~ +
kL±
ilL, (6.51) l is determined by(6.52) where Wb is the Brunt Vaisla frequency and Wa is the cut-off frequency. Equation (6.50) is almost the same as Kamp (1989) (19](20] found when he described mag-netoacoustic gravity waves in an atmosphere with a nearly horizontal field. Yet, a fundamental difference has to he noted. In our case, the highest derivative is multiplied by the complex number i. This is typical for every loss mecha-nism. Note that by letting t -t 0, we find the hypergeometrie equation which , as we saw in chapter 5, describes the magnetoacoustic gravity waves in the ideal (singular) case. The parameter t is assumed to he small. The dissipation
40
process is then assumed to he important only in a narrow layer around the cusp negligible anymore near the resonance. In order to determine the region where these rapid variations appear, we perform (as we did in the previous chapter) a stretching of the ( coordinate according to
(=!...
é'' (6.55)
Substitution of this stretching in (6.50) gives fort.-+ 0 a resulting equation that depends upon the value of v. By applying the method of dominant balance it's found that the only acceptable limit fort. -+ 0 is the so-called distinguished limit for v =1/3, confirming the result found in the previous chapter. Subtitution of
The solution given by (6.58) is regular at the cuspresonance. The real part of the integral is an even function of r and is given by
1
00 1 1 3R = -e-38 (cos(sr)- 1)ds
0 s (6.59)
Equation (6.58) gives the the quantity <I> through the resonance layer; therefore, it only remains to show that this inner solution matches the outer solution outer and the inner solution behave in the same way. This observation leads to the following matching procedure [13) :
lim <l>outer = lim <l>inner
In the inner region, we can't speak about the magnetoacoustic gravity wave and the new mode. In that region we can't dinstinguish between the two waves.
Only ifwe are far enough from the cusp resonance, we see from (6.61) and (6.60) that the inner solution is composed of a magnetoacousticgravity wave and a new mode. it analitically to the outer region. Consicier therefore the original differential equation (6.40). We seek a solution in the form of a WKB serie:
Aftersome trivial algebraic manipulations we find from (6.70) 5 . A. dr/;o
3 . A. 2 1
2 . A. A. 2 1
z'l'o dz - z'l'o
L +
z'l'l '1'0+
L3 = 0 (6.71)Again, we see that the WKB approximation doesn't hold near the cusp reso-nance, because equation (6.71) yields a contradiction whenever r/Jo equals 0. For