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Alfven waves in the solar corona and solar wind
Adv. Space Rea. Vol. 14, No. 4, pp. (4)123-(4)133, 1994 Printed in Great Britain. All rights reserved.
0273-1177/9456.00+0.00 Copyright © 1993 COSPAR
ALFVEN WAVES IN THE SOLAR CORONA AND SOLAR WIND M. VeUi Observatoire de Paris-Meudorg 92195 Meudon Cedex, France
ABSTRACT In situ solar wind measurements of MHD turbulence first showed, 20 years ago, that Alfv6n waves propagating away from the sun are a dominant component, at least in high speed streams at solar minimum, with sufficient energy to explain the heating of the distant solar wind. Here we discuss some aspects of the propagation of these waves upward from the s d a r coronal base, where they are presumably generated, with particular emphasis on the effects of the large scale gradients on the transmission, the deveiopment of turbulence and wave dissipation. INTRODUCTION Among the MHD wave-modes which should be generated in abundance through convection and magnetic footpoint shuffling at the photospheric levels, Alfv6n waves have been the prime candidate both for the acceleration of high speed solar wind streams and for wave-based theories of coronal heating. The main reason is that large amplitude "Alfv6nic" fluctuations propagating away from the sun have been shown to be a dominant component of MHD turbulence observed by satellites in situ in the solar w i n d / 1 , 2 , 3 / . The measurements closest to the sun were made by the Helios spacecraft at 0.3 AU; within this distance only very indirect evidence for Alfv6n waves is available: Faraday rotation fluctuations as measured during superior conjunction by the Helios spacecraft are consistent with the presence of large amplitude Alfv6n waves in the region from 2 to 15 R e / 4 / , and radio scintillation observations of the anisotropy of density fluctuations (presumed to be preferentially aligned with magnetic field) also point to the same result
151.
Another reason attention has focused on Alfvln waves is that estimates based on the simplest properties of the other wavemodes (slow and fast magnetoacoustic waves) indicate that the latter should have much difficulty in even reaching the solar corona: (slow) sound waves with periods greater than 5 minutes are evanescent, while waves with shorter periods appear to steepen into shocks within the c h r o m o s p h e r e / 6 / . Fast magnetoacoustic modes which are not strictly propagating along the magnetic field become evanescent in the chromosphere and corona due to the rapid increase of Alfv6n speed with h e i g h t / 7 / . Because of their highly anisotropic dispersion relation which allows t h e m to propagate along the magnetic field independently of the wave-vector, Alfv6n waves do not suffer (at lowest order) from either of the above effects. In reality the situation is much more complex, as the background magnetic field is highly inhomogeneous both in directions parallel to the solar surface as weU as in the radial direction. The overall structuring of the solar atmosphere is therefore instrinsically three dimensional, and the eigenmodes supported may bear only a slight resemblance to the familiar modes of oscillation of a homogeneous m e d i u m / 8 / . Nonetheless to attempt to connect the direct observational evidence in the solar wind with the hypothetical solar source regions in the corona or below it is best to proceed starting from the simplest con-
(4)123
(4)124
M. Vdli
figurations allowing a minimum of realism, introducing first the effects of the gravitational stratification and then the transverse magnetic structure. The basic equations for magnetic field (b) and incompressible velocity (~) fluctuations, in a plasma of density p, may be conveniently expressed in terms of Els/isser variables ~" + = g =]: sign(B0)b'/V/(4~'p), which in a homogeneous medium describe Alfvdn waves propagating in opposite directions along the average magnetic field B0:
0s + + v ~ ' v s + ~ ' : s ~ ' v v ' + ~ 0-7
1 (z. ~ - s + ) # . ¢ o = 0 ,
(1)
where ~7~ is the average Alfv~n velocity. The first two terms in (1) describe wave propagation; the third term describes the reflection of waves by the gradient of the AlfvSn speed along the fluctuations (which vanishes for a vertical field in a planar atmosphere, but is different from zero in the more realistic case of a diverging flux tube); the fourth term describes the WKB amplitude decrease and the isotropic part of the reflection. In equ.(1) gravity and terms involving the gradients of the average density along the fluctuation polarisation are absent, because the average magnetic field and gravity are assumed to be coplanar. Also absent are nonlinear terms, which in the homogeneous limit are proportional to the product of the amplitudes of outward and inward propagating waves. Equ.(1 ) describes the parallel propagation of fluctuations in the plane perpendicular to/~0, or in the case of spherical or cylindrical symmetry, the propagation of toroidal fluctuations in the equatorial plane. In a static medium and in the absence of dissipation, the net wave energy flux Soo is conserved: S + - S-
= Soo,
S + = FV~I~
+12/8,
(2)
where F = pR ~ = pBo/B (R is the radial distance from the base of the atmosphere and is the infinitesimal flux tube expansion factor: o'=0, 2 in a plane and spherical atmosphere respectively). We are interested in the problem of transmission, i.e., in finding how much of the energy in the waves generated below a certain height is transferred above that height as net energy flux. The problem as such is well posed only if purely outward propagating waves escape above the atmospheric layer in question, and the transmission T is therefore defined as T = S o o / S +, (3) where now Soo is the energy flux carried by the outwardly propagating wave at the top of the layer, while S + is the outward propagating energy flux at the atmospheric base. For waves of frequency w and wavevector k = w/Va, equ.(1) becomes, after elimination of the systematic amplitude variation of z + through the renormalization ~.+ = pl/4z+, 1 k I .
~+' ~ ik£'+ - -zk: ~ 2
= 0,
(4)
(length will be normalized to the solar radius R0 i.e., r = R/Ro throughout; a prime denotes differentiation with respect to r, or, without risk of confusion, as the distance along the magnetic field lines). The equations for the velocity and magnetic field fluctuations (also in velocity units) are v" + k ~ v = o,
b" + (k ~ + k"/k - 2k'2/k ~) b = O.
(5)
The regions where eikonal approximations to equ.(5) break down may be found by applying the Liouville Green transformation/9/; this consists in replacing the independent and dependent variables to transform equ.(5) into a Schrodinger equation (or in the time
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Alfv6n Waves in the Solar Corona and Wind
dependent case the Klein'Gordon equation/lO/). This is obtained by substituting r with = f dr/V~(r) and the dependent variables v, b with v / x / ~ , b / v / - ~ . From the resulting equations one may directly read off t h e critical (turning point) frequencies/10/, 2 OJc
~
~
v~,/'"2"4"4- V a V a " / 2 ,
2 b ~Jc ~
~'a/Trl2"4 --
Va Va" /2,
(6)
for the velocity and magnetic field respectively. The critical frequencies are a function of position, but equ.(6) implies that waves will certainly be evanescent when ~ < min(~ ~ , ~b). On this basis it has been suggested that for a given Alfv6n speed profile, transmission of waves with frequencies below the m a x i m u m value (within the atmospheric layer) of the larger of ~ ' , ~ cb is negligible. There are at least two reasons for which detailed computations should be carried out to substantiate such conclusions: first, the independent variable is no longer r, but the travel time at the Alfv~n speed, while the potential is essentially a logarithmic derivative of the Alfv~n speed, which means that exponential behaviour in the latter variable does not translate into spatial evanescence as a function of heliocentric distance (as may be seen from equ.(6) at zero frequency the two independent solutions for the velocity are v =const. and v ..- r). Second, the effective potential tends to zero high in the corona, displaying a fairly sharp peak, and therefore wave tunneling could be important. PROPAGATION THROUGH MODEL SOLAR ATMOSPHERES Let us consider now specific atmospheric models and the consequent profiles for the Alfv6n speed. Ref. / 1 1 / w a s the first to consider the case of a uniform vertical magnetic field in an isothermal plane atmosphere extending to infinity. In this case, the Alfv~n speed increases exponentially with height, Va = Vaoezp(r/2h) where h = R ® / p g R 0 is the (adimensional) scale height, (g is the surface gravity, R the gas constant, the atomic weight is p ~_ 1/2 for a fully ionized plasma and (9 is the temperature) and the travel time for a wave to infinity is finite (twice the time necessary for a wave to travel one scale-height at the base Alfv~n velocity). While the Alfven velocity scale height is constant, the wavelength increases to infinity, and hence the scale height is smaller than the wavelength from a certain point upwards no matter what the frequency: the wave propagation term in equ.(4) may then be neglected, leading to the asymptotic solutions z ± = :l:z ~: ... k±l/2"
(7)
Since k tends to zero with height, requirement of finite energy implies that z + = z there. Hence, the atmosphere forms a closed cavity and stationary solutions must have a vanishing net upward energy flux, S¢¢ = 0. The exact solution may be expressed in terms of Bessel functions, u = uoJo(2hkoezp(-r/2h)),
b = iuoJ1 ( 2 h k o e z p ( - r / 2 h ) ) ,
(8)
where u0, k0 are the velocity fluctuation and wave-vector at the atmospheric base respectively. If one attempts to impose the amplitude of the velocity (or magnetic field fluctuation) at the base, then clearly problems arise if the frequency is such that the value of the argument of J0 (J1) at the base corresponds to a root of J0 = 0 (J1 = 0): there is no stationary solution, and in the time-dependent case forcing with the base velocity causes the energy in the cavity to grow with time as t 2, as typical of a resonantly excited oscillator. A next step towards a more realistic model is to divide the atmosphere into isothermal layers, with differing scale heights, to describe the variations in density between photosphere and corona/12,13,14/. As stressed beforehand in order to define a transmission coefficient it is necessary to assume that the Alfv6n speed is uniform (or at least slowly varying so that the lowest order WKB expansion is valid) above the layer considered. The m o d e l s / 1 2 , 1 3 , 1 4 / w e r e developed for wave propagation in the solar chromosphere,
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M. Velli
transition region, and corona, and the atmospheres were taken to terminate at the height where the Alfv~n speed reaches a value around 2000 km/sec. The transmission coefficient was found to present a series of maxima at certain frequencies (referred to as 'resonances' in the litterature, though they do not coincide precisely with the real resonant frequency discussed above in the context of the infinite isothermal atmosphere) which are related to the typical scales defined in the problem: thickness of the isothermal layers considered, and Alfv~n speed scale heights of the two layers. Outside the resonances the transmission coefficient, for typical solar chromospheric, transition region and coronal parameters was found to be very low. Fig. 1 shows the transmission for a 2-layer model of the solar atmosphere, one for the chromosphere and transition region, the other for the corona. The two staircase curves (a) and (b) are the separate transmission profiles for each layer, while (c) shows the combined transmission with the resonant peaks. The origin of such peaks has since been the subject of some debate: /12,13/attributed them to discontinuities in scale height separating the various atmospheric layers, on the contrary /14/ and /15/ attributed them to the process of continuous reflection. The variation of the transmission with frequency is clearly an integrated effect, which depends on the whole profile of the Alfv~n speed with height. However scale height discontinuities, where the derivatives in velocity and magnetic field fluctuations are also discontinuous, contribute significantly to the appearance of resonances in the transmission, because they define additional characteristic length scales, namely the distance between successive discontinuities. !
i
C'4 0
/
%.,.,.
I
i i
~
/
,:5
I
t~Jl
1000 Fig. 1. Transmission coefficient as a function of frequency for a piecewise isothermal atmosphere made up of two sections: a) the photosphere, and b) the corona. The combined transmission profile c) presents resonant peaks much in the same way as an interference filter. The ratio of coronal to photospheric scale height is 200. Frequency units are arbitrary but w = 1000 typically corresponds to periods of around 5 minutes for the sun. The low-frequency transmission may be calculated from the static solutions of equ.(4), for any finite layer and independently of the detailed profile within the layer (provided it is continuous), to be: T -
4VatV,,,.
(V,~l + Va,,)2'
(9)
(where V~l,r indicate the Alfv~n speed on either side). This result is the same as the transmission across a 6-function potential, because a perturbation of infinite wavelength sees the variation in propagation speed as a discontinuity. Within the framework of linear theory the basic quantity characterizing the energy flux is the transmission coefficient:
Alfv~n Waves in the Solar Corona and Wind
(4)127
clearly, if it were possible to obtain simultaneous observations of velocity and magnetic field at different heights in the solar atmosphere, then one would be able to ascertain the presence of nonlinear interactions and or dissipation from the observed differences with the predictions from the transmission problem, and hence obtain a quantitative estimation of the wave damping in the solar atmosphere. The model atmospheres described above, though giving a more appropriate description of a stellar atmosphere than the plane, infinite, isothermal slab, still lack a number of features typical of the structure of stars with transition regions and coronae: first, the intermittent structure of the flux-tube confined photospheric magnetic field, and its rapid expansion through the chromosphere and transition region; second, the decrease of the magnetic field far from the stellar surface due to the spherical expansion of the atmosphere. Recently / 1 6 / h a v e solved equ.(4) for transverse waves along the field-lines of a strongly diverging fluxtube (appropriately modelling the upper chromosphere and transition region of the sun), in a 2-d (slab) geometry: Alfv~n wave reflection is substantially decreased (Fig. 2), essentially because the expansion of the flux tube reduces the rate of increase of the Alfv~n speed (see a l s o / 1 7 / f o r a piecewise exponential model of an expanding flux tube). Coefficients
Transmission
1
0.1
o.01
0.001
,
,
10
,
,,I
,
lo0
Period
,
,
, , ,
I
!ooo (s)
Fig. 2. Transmission coefficients versus wave period (in seconds) (a) at the center of flux tube where B0 = 1800 G at the base; (b) at the edge of the structure where B0 900 G; (c) for a non expanding flux tube with B0 = 1800 G and (d) with BÜ = 900 G. F r o m / 2 0 / . =
We consider now the transmission through the solar corona starting from above the transition region. This is worthwhile because a significant amount of wave power may be generated directly at the coronal base by reconnection in the magnetic network/18/. In spherical geometry the Alfv~n speed profile for an isothermal static atmosphere may be written v. =
exp((
1>
(1 - ; ),
=
1 < r < oo,
(10)
where C° is the isothermal sound speed and the parameter a, for the sun, typically lies in the range 4 _~ a _~ 15 for coronal temperatures between 8.0 10 5 - 3.0 10s °K. For this famaily of profiles, the Alfv~n speed first increases exponentially, has a maximum in r = a / 4 and then decreases-asymptotically as V, ~ r - 2 . Ref. /19/ investigated the propagation of Alfvdn waves in this configuration numerically as an initial value problem, and showed that, as expected, local reflection becomes strong where the Alfvdn speed gradients are greatest. They did not determine the solutions of the stationary problem however because they stopped their computations before the waves reached the outer edge of their (finite) computational domain. To calculate the transmission coefficient, we integrate equ.(4) backwards numerically, starting at a heliocentric distance large enough b so that an outwardly propagating second order WKB solution is adequate (~o >> ¢0~,wc, the general form in terms of F + may be found/20/). In fig. (3) we plot the transmission
(4)128
M, Velli
coefficient as a function of frequency as the t e m p e r a t u r e of the atmosphere is decreased (a is varied from 4.0 to 14.0 in integer steps) but the ratio ~ of kinetic to magnetic pressure at the coronal base is held constant at a value of 8%. The frequency plotted is normalized to ,,. = cC,(a = 4)/R0, where C0(a = 4) ~_ 215km/sec and the constant c ( ~_ 5.572) has been chosen so that ~ . = 27r hr -1, i.e., ~ / ~ . = 1 corresponds to a wave period of one hour. A strong dependence of transmission on t e m p e r a t u r e for waves of periods greater than one hour is apparent, while waves with periods less than about 15 rains, are completely transmitted by the solar corona.
c5 tD
c5 c5 C'q
c5 o
10 -3
0.01
0.1
1
10
Fig. 3. Transmiss. coeff, as a funct, of temp. isothermal for atmospheric models of various t e m p e r a t u r e described in the text. a varies in integer steps from a = 4 (curve a) to a = 14 (curve b). Transmission is calculated for a constant value of/3 = 8% at the base of the atmosphere. Frequencies are normalized to w, = cC,(a = 4)/R0 = 2~'hr -1, (c=5.572), so that w = 1 corresponds to a period of one hour for the sun. Stars denote the maximal critical frequency (see text). The m a x i m u m of the critical frequencies equ.(6), defining evanescence for the magnetic and velocity fluctuations (denoted by the stars on the transmission curves), gives a rough estimate of the frequency above which transmission is 100%. For a sufficiently low temperature, the transmission coefficient displays a low-frequency maximum. The dependence of the transmission on the magnetic field base-intensity may be recovered by noticing that the frequency appears in the equations only in the adimensional combination I2 = wRo/V~o. This implies that the curves in fig.(3) should be shifted to the right (left) as the magnetic field base intensity is increased (decreased). Thus, if the magnetic pressure were to decrease tenfold at the base of the solar corona, we would have total transmission of waves with periods below about one hour. The most evident effect of increasing a is the drop in the transmission, in agreement with heuristical arguments based on the critical frequencies (equ.(6)). The low frequency scaling of the transmission may be understood by remarking that the quasi-static solution of equ.(4) (equ.(7)) will be approximately valid out to a distance where k ~_ kt/2k, i.e., ~ _ V~/2. For the given profile (equ. 10), we find that this occurs where (fl is the adimensional frequency defined above) r ~ r~ = fl-1/Sexp(a/6). Beyond this point, we m a y neglect (to lowest order) the reflected wave: imposing that z - vanish in r~ then yields T ..~ f~2/3, as may be seen from equ.(9). The general solution to equ.(4) m a y be written formally in terms of time ordered exponential o p e r a t o r s / 2 1 / . Such an approach allows one to find approximate solutions which, contrary to W K B or multiple scale expansions, exactly conserve the energy flux; they are straightforward to construct when the Alfv~n speed profile may be divided into regions where the propagation t e r m (ik~.±) or reflection t e r m dominates (k'/2k~.~:). In our case
Alfv6n Waves in the Solar Corona and Wind
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a straightforward expansion of the time-orderedpropagators y i e l d s / 2 2 /
T=
2
fl6r
~ 1
• 2" f / 6 r , , 2.1. V~., .]-1 2 [2V---~-~)cosn~,~ i,~tn~)!,,,~V,,o j '
V~o
]cos (2V--Z~)cosh~,,~ (=In.-7--)z ~',~,~ +gsm
(11)
where the functions f and g contain the contributions from the second order terms in the expansion, and depend only weekly on the frequency. As the frequency is increased from zero, the point %, moves inwards, and the value of g~, also increases. When a _> 8, a point is reached where V ~ _~ V~0, and the first hyperbolic cosine in the denominator for T has a m i n i m u m (equal to one) there. This explains the low frequency m a x i m u m in fig. 3. The result foUows also from the static transmission formula equ.(9), which shows that T has a m a x i m u m (=1) when the Alfv4n speed has the same value on the right and left hand side of the layer in question. In equ.(ll) the m a x i m u m is less than one because of the function f. In the limit of large a ~ ~ , all the integrals in the Magnus expansion may be calculated analytically, the sine term in equ.(11) may be neglected, and f -* 1.5525, giving T _~ 0.644, which is dose to the low frequency m a x i m u m value seen in fig. 3. PROPAGATION IN THE SOLAR WIND In the presence of flows the Alfvdn wave equation becomes
b7 +(O+';°)'X~±+~'V(U~:¢')+2
(~--~+)~'(~°~:
U)=O,
(12)
where U is the average wind velocity. We consider here only the simplest, isothermal, spherically symmetric wind model /23/; the roach number M = U/C, and the Alfv4n speed satisfy the well known equations 1
(M - ~-;.)M'
=
2 r
r 2 '
(13)
Va = Vaolr(U/Uo) 112.
As shown in /24/, the asymptotic boundary condition of propagating Alfv4n waves is replaced when the wind is included by the condition of regularity at the Alfv4nic critical point where U = V~ = V~c. The flux conservation equation (2) still holds if we reinterpret S as a wave action per unit frequency: S + - S - = Soo,
S ± = F (U + V~)2
L
[z+l 2,
(14)
and F is unchanged. The transmission coefficient is again defined as T = Soo/So+, where now however the wave action flux is determined by the amplitude of the outwardly propagating wave at the critical point as S ~ = S~ = 4F[~' +[~. The transmission coefficient as a function of frequency is shown in fig. 4 (the wind is taken to be isothermal all the way out to the Alfv6n critical point). Again, the coronal base is fixed at a value of 8%, so the model is still defined completely by the single parameter a (varying from 4 to 14). The shape of the transmission curves is is completely different from that of the static case at low frequencies, where the transmission is enhanced, while at higher frequencies the curves become similar in shape. That the presence of the wind increases the transmission at low frequencies may be understood by examining the static limit of equ.(12), which, rewritten in terms of y ~ = (U :k V~)z ±, becomes ±, 1 It" y ~ _ 1 ,U' VEx ± y= - ~ - ~ ~[-~-+~-~}y, =0,
(15)
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M. Velli
~ •
-
: : : ..... , .
.
.
.
...... ~
~
,
_
I _.,,
(5 (5 0 C'4 0 0
10 -3
0.01
0.I
1
I0
Fig. 4. Transmission coefficient as a function of frequency for the isothermal wind models equ.(13). Normalization of frequency is the same as in figs. 2,3. a varies from a = 4 (curve a) to a = 14 (curve b), in integer steps, while stars denote the cutoff frequency for total asymptotic reflection. with solutions, (as in the static case), given by y+ = -t-y~-. Imposing that y~ vanish at the critical point then gives (/24,25/)
vo +1), ~ ( ' ) ~ +-----~,'u~" u 1 (u31/2 (~Z~
(16)
and the subscript denotes quantities calculated at the Alfv6nic critical point. For the low frequency limit of the transmission we obtain T -
4UoV~0 __~' +2 _
(Vo + voo)~ z +o~
4V,~oV,,¢ (V~o + voo)~"
(17)
Thus we see that it is possible to have perfect transmission at low frequencies, if the Alfv~n speed at the coronal base and the critical point are 'tuned' close to the same value. We remark that the above result is independent of the position of the coronal base, provided the geometry allows for the propagation of a pure Alfv~n-type wave. If we follow the fluctuations down below the transition region, the problem becomes that of correctly estimating the Alfv~n speed within a chromospheric or photospheric flux-tube. Note that there is a complete formal analogy in the expression for the transmission at low frequencies in the static and expanding models. The difference in the behaviour lies in the fact that, while for the static case the position at which the outward propagating wave boundary condition is imposed varies with frequency (leading to a decrease of T with decreasing w), in the expanding case the boundary position is fixed at the Alfv~nic point, and T therefore tends to a constant as w ---, 0. At intermediate frequencies, the transmission still depends very sensitively on the wind temperature . In the solar case this is reminiscent of the strong observed correlation, (at least at solar minimum), between the level of the so called Alfv~nic turbulence and the stream temperature at periods around one h o u r / 2 6 / ; the wind models described above are rather crude, not taking into account important effects such as flux tube expansion, wave pressure, a realistic heat equ., et.c.. However it would be interesting to develop such more realistic models to test whether the observed correlation may be attributed, at least partially, to the transmission properties of the ambient medium. There is a major difference between the propagation properties in the static and expanding case, and in the significance of figs. (3) and (5-6): in the static case, waves
Alfv~n Waves in the Solar Corona and Wind
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of all frequencies are asymptotically outwardly propagating. This may conveniently be expressed by introducing the normalized cross-helicity z +2
_
z -2
tr -- z+ 2 + z -
2 -'* 1, r ~
oo.
(18)
In the spherically expanding case, figs. (5-6) show the proportion of outward waves that reach the superalfvfinic wind, but the asymptotic behaviour of a depends on the frequency. In isothermal winds the wind speed grows logarithmically with distance from the stellar base. On the other hand, in realistic wind models the speed levels off at a value U = Uoo, so the Alfv~n profile goes asymptotically as Va = V~oo/r. Equ.(12) may now be rewritten as
(U~ -4-V=~/r)z +' - iwRoz ± + ~--~(z+ + z-)(U~ ~: V=~/r)
= 0
(19)
An eikonal expansion which treats the boundary condition at the critical point correctly / 2 7 / t h e n shows that at large distances --1
z+
-((1
- 4,
v2ooR lU )
- 2i, voooRolUoo
z- + O(llr)r.
(20)
For all frequencies greater than ~a0 = U~/2V~ooRo the normalized cross helicity ~ increases with distance beyond the Alfvdn critical point to a frequency-dependent limiting value which tends to one at high frequencies as 1 - (wo/w) 2. At lower frequencies however decreases with distance and tends asymptotically to 0, i.e., we have total reflection at infinity. In the solar case the critical frequency is of the order of a few days, hence it might appear that results within this frequency domain are of only academic interest, were it not for the fact that observations of Alfvdnic turbulence in the solar wind also show a decrease of a with distance from the s u n / 2 8 / . This is difficult to explain within the framework of homogeneous MHD t u r b u l e n c e / 2 , 3 / . The observed decrease in a however occurs at all frequencies, in contradiction with predictions from hnear theory. The observed decrease of the (specific) energy with distance, E = E + + E - ",~ r -1-~, where ~ is a small positive quantity depending on frequency and radius, is also in contradiction with the predictions of the low-frequency hnear theory (equ.(16) showing that E ,,~ const). Refs. /29,30/ have suggested a way in which the combined effects of MHD turbulent evolution and the solar wind expansion might lead to the observed behaviour. Their model is based on an isotropized version of the system equ.(12), written in terms of the energies E ± in the + modes and the residual energy ER:
(v • Vo)E±' ;
(Ca J/2) ±
(¢o + 0/2) = 0
U S a' + E a V . U + ~(E + + E - ) V . U / 2 - ~(E + - E - ) V . 17, = 0,
(21)
where ~ is a parameter which depends on whether we consider the Es as the laterally integrated (i.e., the energies as measured in situ by the satellites) part of the energy tensors, or the modal energy of isotropic spectra (in the former case ~ == 3/14, while in the latter ~ = 1/3). When the Alfvgn speed is neglected everywhere, equs. (21) and (12), with w = 0 are structurally identical, (equs.(12) corresponding to ~ = 1/2), and thus the model can reproduce the decrease in ~, (but not the observed decrease in specific energies) far from the Alfv~nic critical point. Closer, where the Alfv~n speed may not be neglected, equs. (21) and (12) are fundamentally different: by writing (21) in terms of the second order moments, we notice that the Alfvgnic singularity appears both in the equation for E - and E R. This ensures that the Schwartz inequality E R2 < 4 E + E is satisfied. The system equ.(21) on the other hand has lost the singularity in the E R equation (E R is convected by the wind), and the Schwartz inequality is no longer enforced by the equations, as a simple calculation demonstrates. The isotropy condition used to obtain equ.(21), is at the origin of this discrepancy.
JASR 14t4-J
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M. VeUi
DISCUSSION In the previous sections we have discussed the propagation of AlfvSn waves in the radially stratified solar atmosphere. Whether such computations may be used as an accurate guide to understand the problems of atmospheric heating and wind acceleration is a topic for some discussion. Let us compare the variation with distance, for example, of the wave pressure force in the WKB (reflectionless) and non-WKB limit. A number of diagrams concerning the WKB-non WKB comparison for different wind models may be found in /24,27/. Their results for the solar wind show that the wave acceleration is accentuated (weakly) with respect to WKB values in the subsonic (coronal) regions for intermediate frequencies corresponding to the transmission minimum. This moderate enhancement has negligible effects however at greater distances. Do AlfvSn waves may have enough energy to accelerate the solar wind? Observationally, this point is still very uncertain, given the impossibility of differentiating between the different wave-modes at the photosphere. The spectra observed in the solar wind display a fully developed turbulence with a change of slope at periods around one hour (roughly corresponding to the transmission minima in fig.(4)), which might be attributed to the atmospheric filtering, but observational progress on the photospheric base spectrum is necessary before definite answers can be given. The observations we have from radio-scintillation d a t a / 3 1 / , show that in the solar case, the observed fluctuations are of the same order of magnitude as the average wind speed precisely in the domain of interest, so that significant nonlinear modifications are to be expected (in WKB acceleration theories, where the AlfvSnic point plays no role, the presence of waves shifts the effective sonic point closer to the solar surface). Work has only recently begun on the fully turbulent case, and with only partial success, for the case of the solar wind. ACKNOWLEDGEMENTS I would like to thank M.Mignoli for helping in meeting this and other deadlines. REFERENCES 1. Belcher,J.W. and Davis,L, J. Geophys. Res., 76, 3534 (1972). 2. Mangeney, A., Grappin R., and Velli, M.,Advances in Solar System MHD, (E.R. Priest and A.W. Hood, eds., Cambridge, 1991), p. 327. 3. Marsch, E.,Physics of the Inner Heliosphere, (ed. R. Schwenn and E. Marsch, Springer Verlag, 1991), p . . 4. Hollweg, J.V., Bird, M., Volland, H., Edenhofer, P., Sterzried, C., Seidel, B., J. Geophys. Res., 87, 1 (1982). 5. Coles,W.A. and Esser,R., J. Geophys. Res., 97, 19139 (1992). 6. Ulmschneider,P., Schmitz,F., Kalkofen,W., Bohn,H.U., g, 70,487 (1978). 7. Hollweg,J.V., Geophys. Res. Letts., 5, 731 (1978). 8. Einaudi,G., Chiuderi, C. and Califano, F., Adv.SpaceRes., , (1993, in press). 9. Nayfeh, A.,Perturbation Methods, (J. Wiley, 1973), p. 314. 10. R.L. Moore, Z.E. Musielak, S.T Suess and An C.-H., Astrophys. J., 378, 347 (1991). 11. Ferraro,V.C.A. and Plumpton, Astrophys. J., 127,459 (1958). 12. Hollweg, J.V., Cosmic Electrodynamics, 2, 423 (1972). 13. Hollweg, J.V., Solar Phys., 56, 305 (1978). 14. Leroy B. , Astron. Astrophys., 97, 245 (1980). 15. Leer E., Holzer,T.E., and FIa,T., SpaceSci.Rev., 33, 161 (1982). 16. Similon P.L. and S. Zargham in, Mechanisms of chromospheric and coronal heating, ed. P. Ulmschneider E.R. Priest R. Rosner (Springer, 1991), p. 438. 17. HoUweg, J.V., Solar Phys., 70, 25 (1981). 18. Parker, E.N., Astrophys. J., 372, 719 (1991). 19. An,C.-H., Suess,S.T., Moore,R.L. and Musielak,Z.E., Astrophys. J., 350, 309 (1990).
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20. Vel]i, M., R., Grappin, A., Mangeney, Phys. Rev. Letts., 63, 1807 (1989). 21. Kholkunov, V. A., Theor. Math. Phys., 2, 169 (1970). 22. Velli, M., ~, , (1993, in press). 23. Parker, E-N., Cosmical Magnetic Fields, Oxford: Clarendon Press (1991,), p . . 24. M. Heinemann, and S. Olbert, J. Geophys. Res., 85, 1311 (1980). 25. HoUweg, J.V., and Lee, M.A., Geophys. Res. Lefts., 16, 919 (1989). 26. Grappin, R., Mangeney, A., and Marsch, E., J. Geophys. Res., 95, 8197 (1989). 27. Barkhudarov, M.N., Solar Phys., 135, 131 (1991). 28. Roberts, D. A., M. L. Goldstein, L. W. Klein, and W. H. Matthaeus, J. Geophys. Res., 92, 12023 (1987). 29. Zhou Y. and Matthaeus, W.H., Geophys. Res. Lefts., 16, 755 (1989). 30. Zhou Y. and W. H. Matthaeus, J. Geophys. Res., 95A, 10291 (1990).
0273-1177/9456.00+0.00 Copyright © 1993 COSPAR
ALFVEN WAVES IN THE SOLAR CORONA AND SOLAR WIND M. VeUi Observatoire de Paris-Meudorg 92195 Meudon Cedex, France
ABSTRACT In situ solar wind measurements of MHD turbulence first showed, 20 years ago, that Alfv6n waves propagating away from the sun are a dominant component, at least in high speed streams at solar minimum, with sufficient energy to explain the heating of the distant solar wind. Here we discuss some aspects of the propagation of these waves upward from the s d a r coronal base, where they are presumably generated, with particular emphasis on the effects of the large scale gradients on the transmission, the deveiopment of turbulence and wave dissipation. INTRODUCTION Among the MHD wave-modes which should be generated in abundance through convection and magnetic footpoint shuffling at the photospheric levels, Alfv6n waves have been the prime candidate both for the acceleration of high speed solar wind streams and for wave-based theories of coronal heating. The main reason is that large amplitude "Alfv6nic" fluctuations propagating away from the sun have been shown to be a dominant component of MHD turbulence observed by satellites in situ in the solar w i n d / 1 , 2 , 3 / . The measurements closest to the sun were made by the Helios spacecraft at 0.3 AU; within this distance only very indirect evidence for Alfv6n waves is available: Faraday rotation fluctuations as measured during superior conjunction by the Helios spacecraft are consistent with the presence of large amplitude Alfv6n waves in the region from 2 to 15 R e / 4 / , and radio scintillation observations of the anisotropy of density fluctuations (presumed to be preferentially aligned with magnetic field) also point to the same result
151.
Another reason attention has focused on Alfvln waves is that estimates based on the simplest properties of the other wavemodes (slow and fast magnetoacoustic waves) indicate that the latter should have much difficulty in even reaching the solar corona: (slow) sound waves with periods greater than 5 minutes are evanescent, while waves with shorter periods appear to steepen into shocks within the c h r o m o s p h e r e / 6 / . Fast magnetoacoustic modes which are not strictly propagating along the magnetic field become evanescent in the chromosphere and corona due to the rapid increase of Alfv6n speed with h e i g h t / 7 / . Because of their highly anisotropic dispersion relation which allows t h e m to propagate along the magnetic field independently of the wave-vector, Alfv6n waves do not suffer (at lowest order) from either of the above effects. In reality the situation is much more complex, as the background magnetic field is highly inhomogeneous both in directions parallel to the solar surface as weU as in the radial direction. The overall structuring of the solar atmosphere is therefore instrinsically three dimensional, and the eigenmodes supported may bear only a slight resemblance to the familiar modes of oscillation of a homogeneous m e d i u m / 8 / . Nonetheless to attempt to connect the direct observational evidence in the solar wind with the hypothetical solar source regions in the corona or below it is best to proceed starting from the simplest con-
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M. Vdli
figurations allowing a minimum of realism, introducing first the effects of the gravitational stratification and then the transverse magnetic structure. The basic equations for magnetic field (b) and incompressible velocity (~) fluctuations, in a plasma of density p, may be conveniently expressed in terms of Els/isser variables ~" + = g =]: sign(B0)b'/V/(4~'p), which in a homogeneous medium describe Alfvdn waves propagating in opposite directions along the average magnetic field B0:
0s + + v ~ ' v s + ~ ' : s ~ ' v v ' + ~ 0-7
1 (z. ~ - s + ) # . ¢ o = 0 ,
(1)
where ~7~ is the average Alfv~n velocity. The first two terms in (1) describe wave propagation; the third term describes the reflection of waves by the gradient of the AlfvSn speed along the fluctuations (which vanishes for a vertical field in a planar atmosphere, but is different from zero in the more realistic case of a diverging flux tube); the fourth term describes the WKB amplitude decrease and the isotropic part of the reflection. In equ.(1) gravity and terms involving the gradients of the average density along the fluctuation polarisation are absent, because the average magnetic field and gravity are assumed to be coplanar. Also absent are nonlinear terms, which in the homogeneous limit are proportional to the product of the amplitudes of outward and inward propagating waves. Equ.(1 ) describes the parallel propagation of fluctuations in the plane perpendicular to/~0, or in the case of spherical or cylindrical symmetry, the propagation of toroidal fluctuations in the equatorial plane. In a static medium and in the absence of dissipation, the net wave energy flux Soo is conserved: S + - S-
= Soo,
S + = FV~I~
+12/8,
(2)
where F = pR ~ = pBo/B (R is the radial distance from the base of the atmosphere and is the infinitesimal flux tube expansion factor: o'=0, 2 in a plane and spherical atmosphere respectively). We are interested in the problem of transmission, i.e., in finding how much of the energy in the waves generated below a certain height is transferred above that height as net energy flux. The problem as such is well posed only if purely outward propagating waves escape above the atmospheric layer in question, and the transmission T is therefore defined as T = S o o / S +, (3) where now Soo is the energy flux carried by the outwardly propagating wave at the top of the layer, while S + is the outward propagating energy flux at the atmospheric base. For waves of frequency w and wavevector k = w/Va, equ.(1) becomes, after elimination of the systematic amplitude variation of z + through the renormalization ~.+ = pl/4z+, 1 k I .
~+' ~ ik£'+ - -zk: ~ 2
= 0,
(4)
(length will be normalized to the solar radius R0 i.e., r = R/Ro throughout; a prime denotes differentiation with respect to r, or, without risk of confusion, as the distance along the magnetic field lines). The equations for the velocity and magnetic field fluctuations (also in velocity units) are v" + k ~ v = o,
b" + (k ~ + k"/k - 2k'2/k ~) b = O.
(5)
The regions where eikonal approximations to equ.(5) break down may be found by applying the Liouville Green transformation/9/; this consists in replacing the independent and dependent variables to transform equ.(5) into a Schrodinger equation (or in the time
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Alfv6n Waves in the Solar Corona and Wind
dependent case the Klein'Gordon equation/lO/). This is obtained by substituting r with = f dr/V~(r) and the dependent variables v, b with v / x / ~ , b / v / - ~ . From the resulting equations one may directly read off t h e critical (turning point) frequencies/10/, 2 OJc
~
~
v~,/'"2"4"4- V a V a " / 2 ,
2 b ~Jc ~
~'a/Trl2"4 --
Va Va" /2,
(6)
for the velocity and magnetic field respectively. The critical frequencies are a function of position, but equ.(6) implies that waves will certainly be evanescent when ~ < min(~ ~ , ~b). On this basis it has been suggested that for a given Alfv6n speed profile, transmission of waves with frequencies below the m a x i m u m value (within the atmospheric layer) of the larger of ~ ' , ~ cb is negligible. There are at least two reasons for which detailed computations should be carried out to substantiate such conclusions: first, the independent variable is no longer r, but the travel time at the Alfv~n speed, while the potential is essentially a logarithmic derivative of the Alfv~n speed, which means that exponential behaviour in the latter variable does not translate into spatial evanescence as a function of heliocentric distance (as may be seen from equ.(6) at zero frequency the two independent solutions for the velocity are v =const. and v ..- r). Second, the effective potential tends to zero high in the corona, displaying a fairly sharp peak, and therefore wave tunneling could be important. PROPAGATION THROUGH MODEL SOLAR ATMOSPHERES Let us consider now specific atmospheric models and the consequent profiles for the Alfv6n speed. Ref. / 1 1 / w a s the first to consider the case of a uniform vertical magnetic field in an isothermal plane atmosphere extending to infinity. In this case, the Alfv~n speed increases exponentially with height, Va = Vaoezp(r/2h) where h = R ® / p g R 0 is the (adimensional) scale height, (g is the surface gravity, R the gas constant, the atomic weight is p ~_ 1/2 for a fully ionized plasma and (9 is the temperature) and the travel time for a wave to infinity is finite (twice the time necessary for a wave to travel one scale-height at the base Alfv~n velocity). While the Alfven velocity scale height is constant, the wavelength increases to infinity, and hence the scale height is smaller than the wavelength from a certain point upwards no matter what the frequency: the wave propagation term in equ.(4) may then be neglected, leading to the asymptotic solutions z ± = :l:z ~: ... k±l/2"
(7)
Since k tends to zero with height, requirement of finite energy implies that z + = z there. Hence, the atmosphere forms a closed cavity and stationary solutions must have a vanishing net upward energy flux, S¢¢ = 0. The exact solution may be expressed in terms of Bessel functions, u = uoJo(2hkoezp(-r/2h)),
b = iuoJ1 ( 2 h k o e z p ( - r / 2 h ) ) ,
(8)
where u0, k0 are the velocity fluctuation and wave-vector at the atmospheric base respectively. If one attempts to impose the amplitude of the velocity (or magnetic field fluctuation) at the base, then clearly problems arise if the frequency is such that the value of the argument of J0 (J1) at the base corresponds to a root of J0 = 0 (J1 = 0): there is no stationary solution, and in the time-dependent case forcing with the base velocity causes the energy in the cavity to grow with time as t 2, as typical of a resonantly excited oscillator. A next step towards a more realistic model is to divide the atmosphere into isothermal layers, with differing scale heights, to describe the variations in density between photosphere and corona/12,13,14/. As stressed beforehand in order to define a transmission coefficient it is necessary to assume that the Alfv6n speed is uniform (or at least slowly varying so that the lowest order WKB expansion is valid) above the layer considered. The m o d e l s / 1 2 , 1 3 , 1 4 / w e r e developed for wave propagation in the solar chromosphere,
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M. Velli
transition region, and corona, and the atmospheres were taken to terminate at the height where the Alfv~n speed reaches a value around 2000 km/sec. The transmission coefficient was found to present a series of maxima at certain frequencies (referred to as 'resonances' in the litterature, though they do not coincide precisely with the real resonant frequency discussed above in the context of the infinite isothermal atmosphere) which are related to the typical scales defined in the problem: thickness of the isothermal layers considered, and Alfv~n speed scale heights of the two layers. Outside the resonances the transmission coefficient, for typical solar chromospheric, transition region and coronal parameters was found to be very low. Fig. 1 shows the transmission for a 2-layer model of the solar atmosphere, one for the chromosphere and transition region, the other for the corona. The two staircase curves (a) and (b) are the separate transmission profiles for each layer, while (c) shows the combined transmission with the resonant peaks. The origin of such peaks has since been the subject of some debate: /12,13/attributed them to discontinuities in scale height separating the various atmospheric layers, on the contrary /14/ and /15/ attributed them to the process of continuous reflection. The variation of the transmission with frequency is clearly an integrated effect, which depends on the whole profile of the Alfv~n speed with height. However scale height discontinuities, where the derivatives in velocity and magnetic field fluctuations are also discontinuous, contribute significantly to the appearance of resonances in the transmission, because they define additional characteristic length scales, namely the distance between successive discontinuities. !
i
C'4 0
/
%.,.,.
I
i i
~
/
,:5
I
t~Jl
1000 Fig. 1. Transmission coefficient as a function of frequency for a piecewise isothermal atmosphere made up of two sections: a) the photosphere, and b) the corona. The combined transmission profile c) presents resonant peaks much in the same way as an interference filter. The ratio of coronal to photospheric scale height is 200. Frequency units are arbitrary but w = 1000 typically corresponds to periods of around 5 minutes for the sun. The low-frequency transmission may be calculated from the static solutions of equ.(4), for any finite layer and independently of the detailed profile within the layer (provided it is continuous), to be: T -
4VatV,,,.
(V,~l + Va,,)2'
(9)
(where V~l,r indicate the Alfv~n speed on either side). This result is the same as the transmission across a 6-function potential, because a perturbation of infinite wavelength sees the variation in propagation speed as a discontinuity. Within the framework of linear theory the basic quantity characterizing the energy flux is the transmission coefficient:
Alfv~n Waves in the Solar Corona and Wind
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clearly, if it were possible to obtain simultaneous observations of velocity and magnetic field at different heights in the solar atmosphere, then one would be able to ascertain the presence of nonlinear interactions and or dissipation from the observed differences with the predictions from the transmission problem, and hence obtain a quantitative estimation of the wave damping in the solar atmosphere. The model atmospheres described above, though giving a more appropriate description of a stellar atmosphere than the plane, infinite, isothermal slab, still lack a number of features typical of the structure of stars with transition regions and coronae: first, the intermittent structure of the flux-tube confined photospheric magnetic field, and its rapid expansion through the chromosphere and transition region; second, the decrease of the magnetic field far from the stellar surface due to the spherical expansion of the atmosphere. Recently / 1 6 / h a v e solved equ.(4) for transverse waves along the field-lines of a strongly diverging fluxtube (appropriately modelling the upper chromosphere and transition region of the sun), in a 2-d (slab) geometry: Alfv~n wave reflection is substantially decreased (Fig. 2), essentially because the expansion of the flux tube reduces the rate of increase of the Alfv~n speed (see a l s o / 1 7 / f o r a piecewise exponential model of an expanding flux tube). Coefficients
Transmission
1
0.1
o.01
0.001
,
,
10
,
,,I
,
lo0
Period
,
,
, , ,
I
!ooo (s)
Fig. 2. Transmission coefficients versus wave period (in seconds) (a) at the center of flux tube where B0 = 1800 G at the base; (b) at the edge of the structure where B0 900 G; (c) for a non expanding flux tube with B0 = 1800 G and (d) with BÜ = 900 G. F r o m / 2 0 / . =
We consider now the transmission through the solar corona starting from above the transition region. This is worthwhile because a significant amount of wave power may be generated directly at the coronal base by reconnection in the magnetic network/18/. In spherical geometry the Alfv~n speed profile for an isothermal static atmosphere may be written v. =
exp((
1>
(1 - ; ),
=
1 < r < oo,
(10)
where C° is the isothermal sound speed and the parameter a, for the sun, typically lies in the range 4 _~ a _~ 15 for coronal temperatures between 8.0 10 5 - 3.0 10s °K. For this famaily of profiles, the Alfv~n speed first increases exponentially, has a maximum in r = a / 4 and then decreases-asymptotically as V, ~ r - 2 . Ref. /19/ investigated the propagation of Alfvdn waves in this configuration numerically as an initial value problem, and showed that, as expected, local reflection becomes strong where the Alfvdn speed gradients are greatest. They did not determine the solutions of the stationary problem however because they stopped their computations before the waves reached the outer edge of their (finite) computational domain. To calculate the transmission coefficient, we integrate equ.(4) backwards numerically, starting at a heliocentric distance large enough b so that an outwardly propagating second order WKB solution is adequate (~o >> ¢0~,wc, the general form in terms of F + may be found/20/). In fig. (3) we plot the transmission
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M, Velli
coefficient as a function of frequency as the t e m p e r a t u r e of the atmosphere is decreased (a is varied from 4.0 to 14.0 in integer steps) but the ratio ~ of kinetic to magnetic pressure at the coronal base is held constant at a value of 8%. The frequency plotted is normalized to ,,. = cC,(a = 4)/R0, where C0(a = 4) ~_ 215km/sec and the constant c ( ~_ 5.572) has been chosen so that ~ . = 27r hr -1, i.e., ~ / ~ . = 1 corresponds to a wave period of one hour. A strong dependence of transmission on t e m p e r a t u r e for waves of periods greater than one hour is apparent, while waves with periods less than about 15 rains, are completely transmitted by the solar corona.
c5 tD
c5 c5 C'q
c5 o
10 -3
0.01
0.1
1
10
Fig. 3. Transmiss. coeff, as a funct, of temp. isothermal for atmospheric models of various t e m p e r a t u r e described in the text. a varies in integer steps from a = 4 (curve a) to a = 14 (curve b). Transmission is calculated for a constant value of/3 = 8% at the base of the atmosphere. Frequencies are normalized to w, = cC,(a = 4)/R0 = 2~'hr -1, (c=5.572), so that w = 1 corresponds to a period of one hour for the sun. Stars denote the maximal critical frequency (see text). The m a x i m u m of the critical frequencies equ.(6), defining evanescence for the magnetic and velocity fluctuations (denoted by the stars on the transmission curves), gives a rough estimate of the frequency above which transmission is 100%. For a sufficiently low temperature, the transmission coefficient displays a low-frequency maximum. The dependence of the transmission on the magnetic field base-intensity may be recovered by noticing that the frequency appears in the equations only in the adimensional combination I2 = wRo/V~o. This implies that the curves in fig.(3) should be shifted to the right (left) as the magnetic field base intensity is increased (decreased). Thus, if the magnetic pressure were to decrease tenfold at the base of the solar corona, we would have total transmission of waves with periods below about one hour. The most evident effect of increasing a is the drop in the transmission, in agreement with heuristical arguments based on the critical frequencies (equ.(6)). The low frequency scaling of the transmission may be understood by remarking that the quasi-static solution of equ.(4) (equ.(7)) will be approximately valid out to a distance where k ~_ kt/2k, i.e., ~ _ V~/2. For the given profile (equ. 10), we find that this occurs where (fl is the adimensional frequency defined above) r ~ r~ = fl-1/Sexp(a/6). Beyond this point, we m a y neglect (to lowest order) the reflected wave: imposing that z - vanish in r~ then yields T ..~ f~2/3, as may be seen from equ.(9). The general solution to equ.(4) m a y be written formally in terms of time ordered exponential o p e r a t o r s / 2 1 / . Such an approach allows one to find approximate solutions which, contrary to W K B or multiple scale expansions, exactly conserve the energy flux; they are straightforward to construct when the Alfv~n speed profile may be divided into regions where the propagation t e r m (ik~.±) or reflection t e r m dominates (k'/2k~.~:). In our case
Alfv6n Waves in the Solar Corona and Wind
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a straightforward expansion of the time-orderedpropagators y i e l d s / 2 2 /
T=
2
fl6r
~ 1
• 2" f / 6 r , , 2.1. V~., .]-1 2 [2V---~-~)cosn~,~ i,~tn~)!,,,~V,,o j '
V~o
]cos (2V--Z~)cosh~,,~ (=In.-7--)z ~',~,~ +gsm
(11)
where the functions f and g contain the contributions from the second order terms in the expansion, and depend only weekly on the frequency. As the frequency is increased from zero, the point %, moves inwards, and the value of g~, also increases. When a _> 8, a point is reached where V ~ _~ V~0, and the first hyperbolic cosine in the denominator for T has a m i n i m u m (equal to one) there. This explains the low frequency m a x i m u m in fig. 3. The result foUows also from the static transmission formula equ.(9), which shows that T has a m a x i m u m (=1) when the Alfv4n speed has the same value on the right and left hand side of the layer in question. In equ.(ll) the m a x i m u m is less than one because of the function f. In the limit of large a ~ ~ , all the integrals in the Magnus expansion may be calculated analytically, the sine term in equ.(11) may be neglected, and f -* 1.5525, giving T _~ 0.644, which is dose to the low frequency m a x i m u m value seen in fig. 3. PROPAGATION IN THE SOLAR WIND In the presence of flows the Alfvdn wave equation becomes
b7 +(O+';°)'X~±+~'V(U~:¢')+2
(~--~+)~'(~°~:
U)=O,
(12)
where U is the average wind velocity. We consider here only the simplest, isothermal, spherically symmetric wind model /23/; the roach number M = U/C, and the Alfv4n speed satisfy the well known equations 1
(M - ~-;.)M'
=
2 r
r 2 '
(13)
Va = Vaolr(U/Uo) 112.
As shown in /24/, the asymptotic boundary condition of propagating Alfv4n waves is replaced when the wind is included by the condition of regularity at the Alfv4nic critical point where U = V~ = V~c. The flux conservation equation (2) still holds if we reinterpret S as a wave action per unit frequency: S + - S - = Soo,
S ± = F (U + V~)2
L
[z+l 2,
(14)
and F is unchanged. The transmission coefficient is again defined as T = Soo/So+, where now however the wave action flux is determined by the amplitude of the outwardly propagating wave at the critical point as S ~ = S~ = 4F[~' +[~. The transmission coefficient as a function of frequency is shown in fig. 4 (the wind is taken to be isothermal all the way out to the Alfv6n critical point). Again, the coronal base is fixed at a value of 8%, so the model is still defined completely by the single parameter a (varying from 4 to 14). The shape of the transmission curves is is completely different from that of the static case at low frequencies, where the transmission is enhanced, while at higher frequencies the curves become similar in shape. That the presence of the wind increases the transmission at low frequencies may be understood by examining the static limit of equ.(12), which, rewritten in terms of y ~ = (U :k V~)z ±, becomes ±, 1 It" y ~ _ 1 ,U' VEx ± y= - ~ - ~ ~[-~-+~-~}y, =0,
(15)
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M. Velli
~ •
-
: : : ..... , .
.
.
.
...... ~
~
,
_
I _.,,
(5 (5 0 C'4 0 0
10 -3
0.01
0.I
1
I0
Fig. 4. Transmission coefficient as a function of frequency for the isothermal wind models equ.(13). Normalization of frequency is the same as in figs. 2,3. a varies from a = 4 (curve a) to a = 14 (curve b), in integer steps, while stars denote the cutoff frequency for total asymptotic reflection. with solutions, (as in the static case), given by y+ = -t-y~-. Imposing that y~ vanish at the critical point then gives (/24,25/)
vo +1), ~ ( ' ) ~ +-----~,'u~" u 1 (u31/2 (~Z~
(16)
and the subscript denotes quantities calculated at the Alfv6nic critical point. For the low frequency limit of the transmission we obtain T -
4UoV~0 __~' +2 _
(Vo + voo)~ z +o~
4V,~oV,,¢ (V~o + voo)~"
(17)
Thus we see that it is possible to have perfect transmission at low frequencies, if the Alfv~n speed at the coronal base and the critical point are 'tuned' close to the same value. We remark that the above result is independent of the position of the coronal base, provided the geometry allows for the propagation of a pure Alfv~n-type wave. If we follow the fluctuations down below the transition region, the problem becomes that of correctly estimating the Alfv~n speed within a chromospheric or photospheric flux-tube. Note that there is a complete formal analogy in the expression for the transmission at low frequencies in the static and expanding models. The difference in the behaviour lies in the fact that, while for the static case the position at which the outward propagating wave boundary condition is imposed varies with frequency (leading to a decrease of T with decreasing w), in the expanding case the boundary position is fixed at the Alfv~nic point, and T therefore tends to a constant as w ---, 0. At intermediate frequencies, the transmission still depends very sensitively on the wind temperature . In the solar case this is reminiscent of the strong observed correlation, (at least at solar minimum), between the level of the so called Alfv~nic turbulence and the stream temperature at periods around one h o u r / 2 6 / ; the wind models described above are rather crude, not taking into account important effects such as flux tube expansion, wave pressure, a realistic heat equ., et.c.. However it would be interesting to develop such more realistic models to test whether the observed correlation may be attributed, at least partially, to the transmission properties of the ambient medium. There is a major difference between the propagation properties in the static and expanding case, and in the significance of figs. (3) and (5-6): in the static case, waves
Alfv~n Waves in the Solar Corona and Wind
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of all frequencies are asymptotically outwardly propagating. This may conveniently be expressed by introducing the normalized cross-helicity z +2
_
z -2
tr -- z+ 2 + z -
2 -'* 1, r ~
oo.
(18)
In the spherically expanding case, figs. (5-6) show the proportion of outward waves that reach the superalfvfinic wind, but the asymptotic behaviour of a depends on the frequency. In isothermal winds the wind speed grows logarithmically with distance from the stellar base. On the other hand, in realistic wind models the speed levels off at a value U = Uoo, so the Alfv~n profile goes asymptotically as Va = V~oo/r. Equ.(12) may now be rewritten as
(U~ -4-V=~/r)z +' - iwRoz ± + ~--~(z+ + z-)(U~ ~: V=~/r)
= 0
(19)
An eikonal expansion which treats the boundary condition at the critical point correctly / 2 7 / t h e n shows that at large distances --1
z+
-((1
- 4,
v2ooR lU )
- 2i, voooRolUoo
z- + O(llr)r.
(20)
For all frequencies greater than ~a0 = U~/2V~ooRo the normalized cross helicity ~ increases with distance beyond the Alfvdn critical point to a frequency-dependent limiting value which tends to one at high frequencies as 1 - (wo/w) 2. At lower frequencies however decreases with distance and tends asymptotically to 0, i.e., we have total reflection at infinity. In the solar case the critical frequency is of the order of a few days, hence it might appear that results within this frequency domain are of only academic interest, were it not for the fact that observations of Alfvdnic turbulence in the solar wind also show a decrease of a with distance from the s u n / 2 8 / . This is difficult to explain within the framework of homogeneous MHD t u r b u l e n c e / 2 , 3 / . The observed decrease in a however occurs at all frequencies, in contradiction with predictions from hnear theory. The observed decrease of the (specific) energy with distance, E = E + + E - ",~ r -1-~, where ~ is a small positive quantity depending on frequency and radius, is also in contradiction with the predictions of the low-frequency hnear theory (equ.(16) showing that E ,,~ const). Refs. /29,30/ have suggested a way in which the combined effects of MHD turbulent evolution and the solar wind expansion might lead to the observed behaviour. Their model is based on an isotropized version of the system equ.(12), written in terms of the energies E ± in the + modes and the residual energy ER:
(v • Vo)E±' ;
(Ca J/2) ±
(¢o + 0/2) = 0
U S a' + E a V . U + ~(E + + E - ) V . U / 2 - ~(E + - E - ) V . 17, = 0,
(21)
where ~ is a parameter which depends on whether we consider the Es as the laterally integrated (i.e., the energies as measured in situ by the satellites) part of the energy tensors, or the modal energy of isotropic spectra (in the former case ~ == 3/14, while in the latter ~ = 1/3). When the Alfvgn speed is neglected everywhere, equs. (21) and (12), with w = 0 are structurally identical, (equs.(12) corresponding to ~ = 1/2), and thus the model can reproduce the decrease in ~, (but not the observed decrease in specific energies) far from the Alfv~nic critical point. Closer, where the Alfv~n speed may not be neglected, equs. (21) and (12) are fundamentally different: by writing (21) in terms of the second order moments, we notice that the Alfvgnic singularity appears both in the equation for E - and E R. This ensures that the Schwartz inequality E R2 < 4 E + E is satisfied. The system equ.(21) on the other hand has lost the singularity in the E R equation (E R is convected by the wind), and the Schwartz inequality is no longer enforced by the equations, as a simple calculation demonstrates. The isotropy condition used to obtain equ.(21), is at the origin of this discrepancy.
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M. VeUi
DISCUSSION In the previous sections we have discussed the propagation of AlfvSn waves in the radially stratified solar atmosphere. Whether such computations may be used as an accurate guide to understand the problems of atmospheric heating and wind acceleration is a topic for some discussion. Let us compare the variation with distance, for example, of the wave pressure force in the WKB (reflectionless) and non-WKB limit. A number of diagrams concerning the WKB-non WKB comparison for different wind models may be found in /24,27/. Their results for the solar wind show that the wave acceleration is accentuated (weakly) with respect to WKB values in the subsonic (coronal) regions for intermediate frequencies corresponding to the transmission minimum. This moderate enhancement has negligible effects however at greater distances. Do AlfvSn waves may have enough energy to accelerate the solar wind? Observationally, this point is still very uncertain, given the impossibility of differentiating between the different wave-modes at the photosphere. The spectra observed in the solar wind display a fully developed turbulence with a change of slope at periods around one hour (roughly corresponding to the transmission minima in fig.(4)), which might be attributed to the atmospheric filtering, but observational progress on the photospheric base spectrum is necessary before definite answers can be given. The observations we have from radio-scintillation d a t a / 3 1 / , show that in the solar case, the observed fluctuations are of the same order of magnitude as the average wind speed precisely in the domain of interest, so that significant nonlinear modifications are to be expected (in WKB acceleration theories, where the AlfvSnic point plays no role, the presence of waves shifts the effective sonic point closer to the solar surface). Work has only recently begun on the fully turbulent case, and with only partial success, for the case of the solar wind. ACKNOWLEDGEMENTS I would like to thank M.Mignoli for helping in meeting this and other deadlines. REFERENCES 1. Belcher,J.W. and Davis,L, J. Geophys. Res., 76, 3534 (1972). 2. Mangeney, A., Grappin R., and Velli, M.,Advances in Solar System MHD, (E.R. Priest and A.W. Hood, eds., Cambridge, 1991), p. 327. 3. Marsch, E.,Physics of the Inner Heliosphere, (ed. R. Schwenn and E. Marsch, Springer Verlag, 1991), p . . 4. Hollweg, J.V., Bird, M., Volland, H., Edenhofer, P., Sterzried, C., Seidel, B., J. Geophys. Res., 87, 1 (1982). 5. Coles,W.A. and Esser,R., J. Geophys. Res., 97, 19139 (1992). 6. Ulmschneider,P., Schmitz,F., Kalkofen,W., Bohn,H.U., g, 70,487 (1978). 7. Hollweg,J.V., Geophys. Res. Letts., 5, 731 (1978). 8. Einaudi,G., Chiuderi, C. and Califano, F., Adv.SpaceRes., , (1993, in press). 9. Nayfeh, A.,Perturbation Methods, (J. Wiley, 1973), p. 314. 10. R.L. Moore, Z.E. Musielak, S.T Suess and An C.-H., Astrophys. J., 378, 347 (1991). 11. Ferraro,V.C.A. and Plumpton, Astrophys. J., 127,459 (1958). 12. Hollweg, J.V., Cosmic Electrodynamics, 2, 423 (1972). 13. Hollweg, J.V., Solar Phys., 56, 305 (1978). 14. Leroy B. , Astron. Astrophys., 97, 245 (1980). 15. Leer E., Holzer,T.E., and FIa,T., SpaceSci.Rev., 33, 161 (1982). 16. Similon P.L. and S. Zargham in, Mechanisms of chromospheric and coronal heating, ed. P. Ulmschneider E.R. Priest R. Rosner (Springer, 1991), p. 438. 17. HoUweg, J.V., Solar Phys., 70, 25 (1981). 18. Parker, E.N., Astrophys. J., 372, 719 (1991). 19. An,C.-H., Suess,S.T., Moore,R.L. and Musielak,Z.E., Astrophys. J., 350, 309 (1990).
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