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Fluctuations in the interplanetary plasma
Planet.
Space Sci. 1971, Vol.
19, pp. 421 to 436.
Pergamon
Press.
Printed
in Northern
Ireland
FLUCTUATIONS
IN THE INTERPLANETARY
Department
R. BUCKLEY of Physics, University of Adelaide, S. Australia
PLASMA
(Received 4 January 1971) Abstract-Spacecraft based observations of fluctuations in the interplanetary magnetic field and solar wind speed yield dominant spatial scales of the order 10s km, and negligible structure below about 500 km. Earth based observations of the angular broadening and scintillation of cosmic radio sources have been interpreted in terms of electron density scales of order a few hundred km. It is suggested that for scales below a few hundred km, there exists an enhanced level of small scale density fluctuations not accompanied by comparable magnetic variations. This proposal is shown to be consistent with radio observations, the contribution of the much larger electron density irregularities being quite negligible. A physical mechanism that might account for the small scale fluctuations is described. 1. INTRODUCTION
This paper describes attempts to reconcile measurements of fluctuations of solar wind parameters obtained by two entirely different methods. Spacecraft based observations of variations of the interplanetary magnetic field and the solar wind velocity indicate that almost all the energy is contained in structures whose scale near 1 A.U. is of order lo6 km (Coleman, 1968). Earth based observations of the angular broadening and scintillation of cosmic radio sources when their lines of sight approach the Sun imply, on the other hand, an electron density micro-structure whose scale increases from around 40 km at 0.1 A.U. to around 400 km at 1 A.U. (Hewish and Symonds, 1969). Jokipii and Hollweg (1970) and Hollweg (1970) have recently suggested that the electron density micro-structure observed by radio methods is simply a manifestation of the ‘inner’ or ‘dissipation’ length associated with the large scale phenomena observed by spacecraft. In section 2 of this paper the spacecraft and radio observations are discussed and their results pertinent to the solar wind are summarised. In section 3, the argument proposed by Jokipii and Hollweg is analysed and shown to be inconsistent with the radio observations. A new proposal is put forward in which the electron density fluctuations are uncoupled from those associated with the magnetic field at scales below a few hundred km, and exist at an enhanced level at these scales. For larger scales up to 106 km, the electron density and magnetic fluctuations follow one another. The consequences of this as regards radio observations are analysed and it is shown that the contribution of the larger scales is quite negligible and that angular broadening and scintillation measurements select only the small scale structure. In section 4 a possible physical mechanism that would account for such a small scale electron density structure with no comparable magnetic fluctuation is suggested. This is based on electrostatic waves propagating at right angles to the average magnetic field. 2. IRREGULARITIES
IN THE SOLAR
WIND
We shall discuss in turn direct spacecraft observations of fluctuations in solar wind parameters and the inferences that have been drawn from indirect radio methods. (a) Spacecraft
observations
In the last few years, considerable knowledge has been obtained of the structure of irregularities in the interplanetary magnetic field B (and to a lesser extent in the solar wind velocity VJ from measurements made aboard spacecraft travelling within l-1.5 A.U. from 421 1
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R. BUCKLEY
the Sun, near the ecliptic plane. These measurements have covered the frequency range from 4 x lo-’ Hz (corresponding to the solar rotation period of 27 days) upward, but have mostly been limited to frequencies less than 1O-2 Hz (Ness et al., 1964; Coleman, 1966, 1968; Coleman et al., 1967; Sari and Ness, 1969; Jokipii and Coleman, 1968). From 10m5to low2 Hz the power spectra of magnetic fluctuations approximate to power lawsf+ with y varying in the range l-2 over the various experiments. This variation of y may reflect a secular change of the medium over the solar cycle (Sari and Ness, 1969). At higher frequencies, Siscoe et al. (1968) from magnetic measurements aboard the spacecraft Mariner IV, obtained spectra which followed the law f-3’” from 3 x lo-4 to 5 x 10-l Hz. Holzer et al. (1966) have reported a spectrum from the satellite OGO 1 of the radial component of B (with respect to the Sun) which could be approximated in the range 2 x 10-l to 2 Hz by a power lawf-3s (Coleman, 1968). If the reasonable assumption is made that the wave speeds associated with these low frequency phenomena are of the order of the AlfvCn or proton thermal speeds, which are both roughly 30 km/s, at 1 A.U., then because this figure is much less than the average solar wind speed V, of about 350 km/s, it is clear that the measured time changes in the magnetic field are primarily a consequence of the convection of spatial irregularities past the spacecraft with speed V,. Therefore frequency spectra can be converted to spatial spectra merely by the substitutionf = Vd/27r = V,/2& where q is the spatial wave number, and 1 the spatial scale. With V, = 350 km/s this relation becomes q (m-l) = 2 x 10-5f(Hz).
(1)
Coleman (1968) has grouped the various measurements into three frequency ranges as follows. 1.8 x lo-‘Hz I > 2 x lo9 m. In this extreme low frequency range, the fluctuations parallel to the average (spiral) field. The spectra are sponding variations in v, are interpreted as variations streaming velocity and are presumably associated with that has its origin in the Sun. (b)
in B are observed to be primarily relatively flat. These, and correin the inherent field strength and a filamentary (or sector) structure
2 x 1O-s Hz I > 3 x lo6 m.
In this frequency range, the fluctuations in B are primarily transverse to the average field. The power spectra go roughly as f-1.2. These are interpreted as a manifestation of Alfven waves propagating along the average magnetic field. Below 2 x 10m3Hz, corresponding fluctuations in V, support this view, but measurements of V, are not available above this frequency. 2 x lo-lHzZ>2x104m. In this range, very limited data indicate a spectrum for the radial component which goes as f4".
of B
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Coleman has interpreted the complete spectrum as follows. Instabilities induced by large scale differential streaming in the solar wind generate long wavelength AlfvCn waves in range (a). The energy in these waves ‘cascades’ through range (b) by means of nonlinear interactions. In turbulence terms this is the ‘inertial range’. The energy is eventually dissipated by proton cyclotron damping near 10-r Hz or 500 km scale. This accounts for the rapid drop of the spectrum in range (c) and is analogous to the viscous dissipation range in fluid turbulence. The spectrum in the inertial range may be enhanced toward its upper
POWER DENSIT &I)
10
-9 I
1
WAVENUMBERtm-‘)
IIO5 I
’
I
IcFIG.
“INERTIA;‘RANGE
~~+--II~SSIPATION
1. SKBTCH OF THE SPATtAL POWER SPECTRUM OF MAGNBTIC FLUCTUATIONS OBSERVED BY SPACECRAFT.
end as a result of the firehose instability which occurs because of the existence of unequal proton temperatures parallel and perpendicular to B (Scarf et al., 1967). The complete spectrum is indicated in Fig. 1. In turbulence terminology, the energy containing or ‘outer’ scale is defined by the boundary between ranges (a), (b) and is of order 2 x lo6 km, whereas the dissipation or ‘inner’ scale is the boundary between ranges (b), (c) and is of order 500 km. (b) Earth based radio observations The scattering of radio waves from cosmic sources by irregularities of electron density in the solar wind has two observable consequences. In the first place, sources appear
424
R. BUCKLEY
broadened when their lines of sight approach the Sun. Secondly, diffraction leads to spatial variations of the intensity of the wave. If the irregularities are convected across the line of sight by the solar wind, this results in temporal changes of the observed intensity. This phenomenon is called interplanetary scintillation. Hewish and Symonds (1969) have recently given a useful summary of both areas as they affect the solar wind. Such measurements are not confined to the ecliptic. A brief outline of the relevant theory will introduce several important parameters. A radio wave of wavelength ;Z= 2rr/k is incident normally on a slab of thickness D containing irregularities of refractive index 1 + Ap of average size L. (It is assumed that Ap < 1.) The phase deviation undergone by the wave while traversing one irregularity is kL A,u. The number of irregularities along the line of sight is D/L. Statistically therefore, the RMS phase deviation on emergence from the slab is
For a plasma medium such as the solar wind where the radio frequencyfis than the electron gyro frequency fB= eB/2nm,, p is given by Pa =
1
_f," f"
much larger
’
where f,, = (n,e2/m,q,)*/27r is the plasma frequency. rz, is the electron density. Radio observations of the solar wind use frequencies of order 10’ to lo9 Hz. fBand f,are of order 3 x lo2 Hz and 2 x 104 Hz respectively at 1 A.U. (using B = 10 y = lo-* MKS, n, = 5 x lo6 m-3). If the Parker solar wind model is assumed, both fB and f,will be approximately inversely proportional to solar distance. Use of Equation (3) is therefore amply justified and it follows that Q=!C
3.
2f2J- %
(4)
Substitution into Equation (2) yields after some rearrangement CpO N Q&z
q
(5)
where r, = e2/4nq,m,c2 = 2.8 x lo-l5 m is the classical electron radius. It is clear that if An, follows the same inverse square law with solar distance R, as n, and if the slab thickness D is assumed to be roughly proportional to R,, then (bOis proportional to RF~/~. More detailed analysis, involving integration along the line of sight for a model in which n, and dAn,a are proportional
to R8-2 yields (6)
(Jokipii and Hollweg, 1970). L is measured in Km,&,, is the radio frequency in units of 100 MHz, R, is the distance of closest approach to the Sun of the line of sight, measured in A.U., and E = dhn,“/rrB is the relative RMS electron density fluctuation which by assumption is independent of R,.
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425
When the wavefront emerges from the region containing irregularities, its phase has a random variation perpendicular to the direction of propagation, with an RMS value given by Equations (5) or (6). As a result, the rays are refracted away from the original direction To an observer close enough to the screen the source of waves would of propagation. appear displaced slightly in position. If the irregularities move across the line of sight, the apparent position of the source would fluctuate within a cone of directions centered on the true position. But if (as is usually the case), the time constant of the receiver is too long to resolve such fluctuations of position, the source will simply appear broadened. Similarly if the observer is far enough from the irregularities to receive several refracted rays simultaneously, he will see a broadened source. The amount of such angular broadening is clearly independent of the observers distance from the irregularities. In the latter case however, the intersecting rays can interfere and generate fluctuations of wave intensity, or scintillations. Clearly this phenomenon cannot occur unless the observer is far enough from the irregularities. If the fluctuations in electron density and therefore phase have a single spatial scale L (in practice this has usually meant possessing a Gaussian autocorrelation function emZ*lL*), then the following results apply (Ratcliffe, 1956; Mercier, 1962; Salpeter, 1967). Numerical constants of order unity have been omitted from the following formulae, and it is assumed that L is very much greater than the wavelength. (a) The width 8, of the angular spectrum of scattered waves (and hence the observed angular diameter of the source) is %l - l/kL -$,,/kL
if
do&1
if
#,, > 1.
(7)
(b) Provided that the distance z from the irregularities to the Earth (of order 1 A.U.) is greater than z, where Zf - kLZ if & & 1 - kLa/4,,
if
$0 > 1,
(8)
then the scintillation index (relative RMS intensity fluctuation) for a point source is m N 2/2 &,
(cc llfby
Equation (6))
= 1
if if
$,, < 1 I#J~>~.
(9)
The spatial scale of the intensity pattern at the Earth is L,-L -L/+0
if
&& 1 if
+O>l.
(10)
If the irregularities are convected across the line of sight by the solar wind then the time scale of the pattern at the Earth is T, = L,/V,. (11) On the basis of such formulae it has been concluded (Hewish and Symonds, 1969) from angular broadening measurements over the range O-025 A.U. < R, < O-5A.U. that: (i) Irregularities of electron density with a scale of less than 5000 Km at 0.3 A.U. are a permanent feature of the solar wind. (ii) The irregularities are filamentary and elongated radially away from the Sun. (iii) E (the relative electron density fluctuation) is relatively small.
426
R. BUCKLEY
(iv} The radial variation of 0, indicates that to some extent the irregularities are locally generated and not merely swept out from the Sun by the solar wind. More definite knowledge of the scale of the irregularities cannot be obtained from angular broadening because & is unknown. Scintillation measurements obtained over essentially the whole sky indicate (Hewish and Symonds, 1969; Dennison, 1969) that (a) m cc l/ffrom 80 to 600 MHz. (b) m cc X:1-6 from 0.1 to 15 A.U. (c) On the basis of (a) and equation (9) it is concluded that at 81.5 MHz, & is less than 1 rad. for R, 2 O-3 A.U. (d) From Equation (IO), the irregularity scale increases from 100 to 400 km as R, increases from O-4 to 1 A.U. (e) Below 0.3 A.U., #,, is greater than 1 rad. at 81-5 MHz. Use of Equation (7) combined with angular broadening measurements yields the scale down to 0-I A.U. where it decreases to about 40 km. Scales deduced from scinti~ation and angular broadening measurements overlap approximately near O-3A.U. (f) Scintillation measurements at much higher frequencies are consistent with & remaining less than one radian down to much smaller R,. (Cohen and Gundermann, 1969) and have confirmed the decrease of scale down to 0.1 A.U. (g) Calculations of Fw from Equation (6) confirm that it is propo~onal like n, to l/R,2 and that the ratio E is of order 10 per cent (Cohen et al., 1967). (h) Direct measurements of r,& beyond R, = 0.6 A.U. have been obtained from phase scintillations of Jupiter’s decametric radiation (Slee and Higgins, 1968). These fit reasonably well on to values of &, inferred from intensity scintillations closer to the Sun. At 81-5 MHz the inequality z > z, (for #0 < 1) breaks down if L. 2 800 km. Therefore s~nti~ation meas~emen~ could not resolve scales appreciably greater than this. However the angular broadening measurements are not subject to this restriction. The fact that the two types of measurement give similar scales suggests strongly that there exists in the solar wind a quite definite electron density microstructure whose scale increases from about 40 km at O-1 A.U. to 400 km at 1 A.U., which is rather weak 8 N O-l), which produces phase deviations of order a few radians and less, and (~~~ which is responsible for both scintillation and angular broadening. There is also some indication that these irregularities are filamentary and directed away from the Sun. This is in marked contrast to the in situ measurements of magnetic field fluctuations which as we have seen give at 1 A.U. and beyond an ‘outer’ scale of order 2 x IO6km and an ‘inner’ dissipation scale of about 500 km. 3. RECONCJLWMON
OF THE TWO TYPES OF OBSERVATION
3.1 The proposal of Jokipii and Hollweg In an attempt to reconcile the spacecraft and radio observations, Jokipii and Hollweg (1970) have recently made the following suggestion. They have calculated the effects that would be observed by radio methods if the electron density fluctuations have a power spectrum which is similar in most respects to that of the magnetic fluctuations. In particular, the density spectrum would have an outer scale of order lOa km, and would cut off at an inner scale of a few hundred kilometres. The correlation length that must be used in Equation (6) is then the outer scale: 2 x 108 km. They
FLUCTUATIONS
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have further assumed that E is of order unity. This is based on very large scale observations of the electron density variation which have been reported by Neugebauer and Snyder (1966). Equation (6) then yields values of &, of order 103-lo* rad. Under conditions of very large phase fluctuations, and provided the distance to the observer is not too large, the methods of geometrical optics are applicable These results then follow (Salpeter, 1967). Suppose that the phase fluctuations have transverse autocorrelation function p,+(x) and spatial power spectrum S,(q) (these are Fourier transforms of each other). Define a length scale L4 as follows. L4 = @----I
= [ /_ldq q%+(q)]-li4.
(12)
Provided that the distance z from the irregularities to the observer satisfies the inequality ZQ-
kL,=
(13)
$0
the spatial power spectrum of the intensity fluctuations S,(q) at the observer is given by
S,(q) = and the corresponding
(+cbzq2 12 7
S+(q)
’
(14)
scintillation index is
4oz 2
(kL,2’) m2= Jokipii and Hollweg admit that the geometrical optics approximation is probably not applicable over the entire electron density spectrum, but will break down at wavelengths comparable to the inner scale and below. Nevertheless they claim that this breakdown at very small scales will not significantly affect the argument. Assuming that inequality (13) holds for radio observations of the solar wind, the following conclusions can be drawn from Equation (14). We have seen that S6(q) (for the magnetic fluctuations and by assumption for the electron density and phase fluctuations also) cuts off rapidly beyond the inner scale l/q -=c500 km. The factor q* that multiplies S,(q) in Equation (14) removes power at the large scales and leaves only a residual peak around the inner scale (see Fig. 2). As a result the pattern scale would approximate to 500 km (at 1 A.U.) consistent with the radio observations. Further, because of inequality (13), the scintillation index given by Equation (15) is small despite the large value of +o. It should be pointed out that the results quoted above apply to a one dimensional situation. In the more realistic two dimensional case, the predictions are similar except that the observed frequency power spectrum SI(j) is given by the projection of the two dimensional spectrum SI(qs, qv) onto the direction in which the irregularities are convected by the solar wind,
(Parkin, 1967; Lovelace et al., 1970). Assuming that S,(q,, qJ is not too anisotropic and has a two dimensional form similar to that given by Equation (14), it is clear that this projection
428
R. BUCKLEY
I
1
(106Kmi’
4
(500Km)-‘I
will partially remove the ‘hole’ at low wavenumbers caused by the @ factor, and the result could well be close to the spectra that are frequently observed (see Fig. 2). Early work (Cohen et al., 1967) frequently showed Gaussian spectra but more recent results (Lovelace et at., 1970) indicate power law spectra (p’). The angular broadening results can also be explained this way if it is assumed that geometrical optics is applicable (Holfweg, 1970). fokipii and Hollweg conclude that the radio observations are consistent either with a density structure with dominant scales of a few hundred kilometres and small $s, or with an outer scale of order lo6 km, a cut-off or inner scale of a few hundred kilometres, and very large &,. They favour the latter structure because of its similarity to the i% situ magnetic observations, We now examine their proposal in more detail. In order for inequality (13) to be valid it is necessary that the scale L, be comparable to the large scale structure in the irregularities. Jokipii and Hollweg (1970) do not however attempt to calculate the scale L4. We now attempt to rectify this omission. As well as Z, we shall be interested in two
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other scales L,, L,, defined generally as follows L,’
=
O”d!@%(q) s --m (17)
The last formula applies if the spectrum is not normalised. These integrals can be estimated either graphically from Fig. 1 of Hollweg (1970), or by approximating S+(q) as follows S+(q)=l,
O
= (qL)-312, =o,
l/L
l/l
(18)
where I = 500 km, L = 2 x lo6 km. Then since I < L we get the following results 1 3l4
L2=2
L*
0 -
L
L’L
or
L, -8OOOkm,
Nl-8
I 718
L,
’
L
-
0L
L,w2000km,
L, -1200km.
(19) (20)
Far from being dominated by the outer scale 2 x lo6 km, these three values are much closer to the inner scale of 500 km. We may note also that if the pattern scale Lp is defined by L,-2
=
s*
dq@s,(q)
(21)
--co
then from Equation (14), and the definitions of L,, L, we obtain for an arbitrary form of S&l), Lp = L6S/Lp2 -
(22)
500 km with the values in Equation (20).
So the pattern scale is indeed close to the inner scale as discussed by Jokipii and Hollweg. If we assume do = IO4rad., k = 6 m-l (corresponding to 300 MHz), z = 1 A.U. = l-5 x loll m, and L4 = 2 x lo6 m then we get
Thus the inequality (13) is very badly violated. To satisfy it, L, would have to be at least a factor of 10 larger, and it is difficult to see how the estimate in Equation (20) could be in error by such a large factor. The only way to change L4 by such a factor would be to increase the inner scale from 500 to 5000 km which is clearly inconsistent with the observations. What does scintillation theory predict if inequality (13) is badly violated? In two recent papers (Buckley, 1970a,b), the full diffraction theory of a random phase screen (both one and two dimensional) is evaluated asymptotically for large RMS phase deviation do. Three ranges of distance z can be distinguished. (a)
430
R. BUCKLEY
Here geometrical optics is valid as already described.
In this region waves scattered from individual irregularities can intersect. The resultant focussing effects can lead to values of the scintillation index greater than unity. (c)
z > kL,L,I$,.
This is the far (Fraunhofer) diffraction zone. The scintillation index is unity, the intensity power spectrum SI(q) is Gaussian regardless of the form of&(q), and the diffraction pattern scale is L,I1/z& So if the ideas of Jokipii and Hollweg were correct, we would be observing well inside region (c) (z&kL,L, - 15). Observed indices less than unity might be accounted for by ‘diameter broadening’ (Hewish and Symonds, 1969) but one would then expect m to increase away from the Sun, eventually attaining the value unity. The spatial scale would be of order 5 km (with L, - 8000 km, and &, N 10s rad.). In frequency terms this corresponds to a width of Vs/2n (5 m) = 12 Hz. Although Gaussian spectra have been observed (Cohen et al., 1967) their widths are of order 1 Hz at 1 A.U. from the Sun. In addition one would expect angular broadening of order l/kLp rad. which is of order 10 set of arc at 1 A.U. Angular broadening has not been observed further out than O-5 A.U., but if the radial variation (for instance Hollweg, 1970) is extrapolated one would expect values of order 1 set of arc at 1 A.U. It has also been pointed out (A. Hewish, 1970, private communication) that the scintillation index given in Equation (15) is inversely proportional to the square of the radio frequency (#+,cc l/f, k CCf) whereas observations indicate a ll’variation over a wide range of frequency consistent with Equation (9). We conclude therefore that the electron density structure proposed by Jokipii and Hollweg is not consistent with the radio observations. 3.2 A new proposal and its consequences for radio observations
It is fairly clear that the only way out of the contradictions encountered in the last subsection is to relax the assumption of Jokipii and Hollweg, that the power spectra of the magnetic and electron density fluctuations cut off at the same inner scale of a few hundred kilometers. Let us now examine this assumption. It is well known from the ‘frozen field line’ concept that on scales much greater than the proton gyro radius, variations of nP (and therefore n,) and B will tend to follow one another. With a magnetic field of 5-10 y at 1 A.U. and a proton temperature of a few times 106 “K (for instance, Hundhausen et al., 1967) the proton gyro radius is in the range 50100 km. This probably decreases relatively slowly closer to the Sun. It is clear therefore that around the ‘inner’ scale of the magnetic fluctuations (a few hundred km), we are approaching the region where freezing in (and magnetohydrodynamic concepts in general) are breaking down. It seems reasonable to suppose therefore that the magnetic and electron density spectra can ‘uncouple’ for length scales less than a few hundred km. We have seen that radio observations are consistent with a definite but rather weak microstructure in the electron density with a typical scale near 1 A.U. of a few hundred km, and which imposes phase variations of order a few radians or less.
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( , (10'km
PLASMA
431
,.)y (500km+(100km~4
)-' q
RG.
3.
hOPOSED
TWO COMPONENT SPATIAL POWER SPECTRUM OF THE ELECTRON DENSITY FLUCTUATIONS NEAR 1 A.U.
We propose therefore that the complete irregularity structure in the solar wind consists of two roughly independent components, one with a scale measured in millions of km and in which the fluctuations are strong (An, - n,) and follow the magnetic fluctuations, and the other with a scale measured in hundreds of km in which they are weak (An&z, - O-1) and which is not coupled to the magnetic field. A possible physical mechanism that would produce such a structure will be discussed in the next section. Here we shall examine the consequences for radio observations of such a two component spectrum, which is indicated in Fig. 3. The autocorrelation function of the phase fluctuations corresponding to such a spectrum can be written in the form 4(x + x’W(x’) = $02P&) = 9L2PL(X) + 9%%%(x)
(24)
where
402= h2 + 4E2 - &I”. The corresponding
spatial power spectrum is
(25) where S,, Sr are the Fourier transforms of pL, pI and are normalized in such a way that their integrals over all values of q are unity. +z, +r are the separate RMS phase deviations produced by the large and small scale structures respectively. At 1 A.U. we expect #z - 103, $Z < 1 rad. pL(x), pi(x) are functions which are unity at x = 0, and tend to zero for x 2 L, x ;a I respectively, where at 1 A.U., L - 2 x 10 km, I-J 400 km. They can be thought of as autocorrelation functions for the large and small scale structures separately, and they possess only single length scales. We can imagine the small scale, small magnitude phase fluctuations riding on top of the far larger, far stronger variations associated with the large scale structure.
R. BUCKLEY
432
The general formula for the spatial power spectrum S,(q) of the intensity variations at distance z from the phase irregularities is m2&(q) = exp {-2#02(1 - &q/0)X 1’
* dx eiq” [exp %X~PJ~) 2rr s __m
- pd(x + zq/k) - p&x - zq/k))) -
11. (26)
See for instance Buckley (1970a). k = 27r/;1is the radio wavenumber and m2 is the scintillation index, given by the integral over all values of q of the right hand side of Equation (26). For simplicity, we consider a one dimensional situation. Similar conclusions hold for the two dimensional case. The methods developed by Buckley (1970a, b) for analysing Equation (26) do not apply directly to the correlation function given by Equation (24). However, we can proceed as follows. We first estimate the range of values that the parameter zq/k can take. We have z N 1 A.U. N loll m, k - 5 m-l (for 300 HMz). The widths of observed scintillation spectra are typically 1 or 2 Hz. Converting to wavenumber through q = 2rf/ V,, and using V, N 350 Km/s, we get 0 < q sz 2 x 1O-6 m-l for the observed range of values of q. Hence for the values of interest to the radio observations O-=zzG5
x 105m.
lo0 m, and so to a very good approxi-
Now the width of the function pL(x) is of order L mation, pl(zq/k)
= 1- f
(27)
(
2
)
‘, (28)
2p,(x)-pp,(x+$
-p~(x-~&$$‘(x),
where &)(x) = d2pL/dx2 and L, = (--p?)(O))- lj2. Since pL(x) has only a single scale L, it follows that L, is of order L. We assume that $1” is small enough to justify expansion of the various exponentials that occur in Equation (26). This should certainly be valid at 1 A.U. Substitution of Equation (24) into Equation (26) and use of Equation (28) then yields m2Sz(q) = e--(dLIQ’kLa)a [l - 2$:(1 X
- pl(zq/k))]
&/_tdx eiQz [e-(~~z~/s)“p~)(x) (1 + -
or
+T(2pl(x)
I+$))
- p,(~ + !.!)
- l]+0%9~
(29)
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PLASMA
Now we have seen (Equation (27)) that the largest value of zq/k that need be considered is of order 5 x lo5 m. With #L N lo3 rad. and L, - L - log m we have therefore 0<
4Lzq 2 & 0.3 * (-1kL,
(31)
Over the relevant part of the spectrum therefore we can treat this quantity as a small parameter. The first part of Equation (30) can then be evaluated in terms of S,(q) the ‘large scale’ spectrum (Equation (25)) using the ‘range (a)’ result given in Buckley (1970a, Equations (24), (25)) and the second by introducing the power spectrum S,(q) of the small scale fluctuations as given by Salpeter (1967, Equation (15)). The result is
S&l1 - %Yl -
p,(zq/Wl
1 zq2
+
( )
4h2 sina2 k
S,(q)
i-o
4Lzq2 + K-N kL2
0(&4),
(32)
which we recognise as simply the sum of the geometrical optics result (Equation (14)) and the small phase deviation result that is valid at any distance z (Salpeter, 1967, Equation (15)). Geometrical optics is applicable here since SL(q) is not cut off at the inner scale I. We can write m2&(q) = mL2%Aq) + mz2Mq) m2 = mL2+ mt2
(33)
where
mL2Wq>= mL 2=
( )2 (g2)2(1+ 4Lzq2 k
SLbN
+
0(4,2)) (34)
O(Q))
4
and m,“Sz(q) = 4+,2 sin2
(35) dq sin2 (2-j-J 1 zq2 SZ(q) < 2+,2 where L, = (d4pL(x)/dx4);=$ is now of order L in contrast to the L4 introduced in the last subsection which was estimated to be much less than L. Inserting the values of the various parameters we find that mL- 10-4 (36) ml
434
R. BUCKLEY
postulated two component spectrum is correct, there is no inconsistency between spacecraft and radio observations. Despite the enormous phase deviations generated by the large scales, appreciable intensity fluctuations would result only after the scattered waves have propagated a distance vastly greater than 1 A.U. Similar remarks apply to the angular broadening. It should be remembered however that this derivation relies on the assumption that & small. If this is not so (and we have seen that & derived from scintillation measurements on 81.5 MHz is greater than unity below 0.3 A.U.) the separation of the intensity spectrum into large and small scale components is not clear cut, and there is the possibility that observations close to the Sun are sampling both large and small scales. Recently, Cronyn (1970) has argued that the magnetic and electron density fluctuations have similar small scale structure. Suppose that the three dimensional power spectra of the magnetic or density fluctuations have power law forms pY in the region of interest. Then Cronyn has shown that the frequency spectrum of magnetic fluctuations observed by spacecraft will have the formf-@‘-2), and further that the scintillation spectrum will have the formf(Y-l) (in the weak scattering limit, and ignoring the low frequency dip caused by the ‘sine-squared’ fringing function of Equation (35)). We have seen that in the dissipative range of the magnetic spectrum (Fig. l), very limited observations indicate a spectrum of the form f"'"(Coleman, 1968). The corresponding three dimensional spectrum would therefore be of the form q--5’s,and identifying magnetic and electron density structures we would expect scintillation spectra of the form f-p". Cronyn has noted that the spectra reported by Cohen et al. (1967) which were originally interpreted in terms of Gaussian functions are also consistent at high frequencies with a power law off-” approximately, which is in line with the preceding ideas. However, Lovelace et al. (1970) have observed power law spectra f--n forf > 1 Hz, and in the majority of cases, n lies between 3 and 4. This would seem to imply a form q-4 or q-r’ for the three dimensional density spectra, as opposed to q-6’8 for the magnetic fluctuations. So Lovelace’s results are also consistent with the findings of this paper, namely that the small scale density irregularities are enhanced relative to the magnetic fluctuations. It is possible that the power law spectra are illustrative of a second inertial range associated with a short wavelength regime of the turbulence. 4. A POSSIBLE EXPLANATION FOR THE ELECTRON DENSITY SPECTRUM
We have already noted that for scales much greater than the proton gyro radius which is of order 50-I 00 km at 1 A.U., the magnetic and electron density spectra can be expected to follow one another, and we have also outlined Coleman’s (1968) explanation of the magnetic spectrum, namely that energy generated by differential streaming instabilities cascades down the inertial range from IO6to 500 km where it is destroyed by cyclotron damping. Observations indicate that toward the upper end of this range the magnetic fluctuations are primarily transverse to the average spiral field. Coleman has interpreted them in terms of the left hand circularly polarized proton cyclotron wave travelling along the ambient field. Stix (1962, pp. 194-196) has shown that such waves are strongly damped for frequencies (in the plasma rest frame) near the proton cyclotron frequency and wavenumbers along the main field greater than approximately (fn fD2/ vTc2)1/3 where fB, f,are the proton gyro and plasma frequencies, vT is the proton thermal speed parallel to the main field, and c is the speed of light. At 1 A.U. this is approximately (500 km)-l as required by the inner dissipation scale. Note that this parallel wavelength is rather larger than the proton gyro radius. There is
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probably also some energy in the other mode which propagates along the field: the right hand polarized mode. Generalized firehose instability of this mode due to the anisotropic proton velocity distribution could occur (Scarf et al., 1967) at the upper end of the inertial range. Each of these modes has its electric vector perpendicular to the ambient field, and its propagation vector k along this field. Thus there are no associated density fluctuations. Of course, plasma non-uniformities, and oblique propagation with respect to the field will generate density fluctuations but it is apparent that the close coupling between magnetic and density fluctuations that exists at larger wavelengths need no longer apply. Barnes (1966) has shown that for a plasma with /I 3 0.5 (fi is the ratio of thermal energy density to magnetic energy density and is typically 0.5 in the solar wind at 1 A.U.), most hydromagnetic waves are highly attenuated by combinations of the Landau, and cyclotron damping processes. In fact the only modes not strongly damped are the two transverse waves propagating in a narrow cone centered along the average magnetic field, and the two modes whose wave vectors are perpendicular to the main field. Of these latter, one (the ordinary mode) has its electric vector parallel to the main magnetic field and hence has no associated density perturbations. The other, however (the extraordinary mode) has in general a mixture of transverse and longitudinal electric fields, and therefore has proton and electron density perturbations associated with it. The low frequency behaviour of this latter wave has been analysed by Fredricks (1968a, b) who has called it a ‘Generalised Bernstein mode’. The ratio of the electric fields parallel and perpendicular to the wave vector is of the order of the ratio of the speed of light to the phase speed of the wave (Clemmow and Dougherty, 1969, p. 303). Thus for phase speeds very much less than the speed of light, the perpendicular component is quite negligible and the wave is virtually longitudinal, i.e. electrostatic. The magnetic vector of the wave is in these circumstances negligible. We tentatively suggest therefore that the small scale electron density fluctuations found by radio methods are a manifestation of electrostatic waves propagating across the average magnetic field. They have no magnetic fluctuations associated with them and hence would not be expected to have spectral characteristics similar to the magnetic ones. According to Fredricks (1968a) the spectrum of such waves in the plasma rest frame might be expected to extend roughly from the proton gyro frequency (about 10-r Hz at 1 A.U.) up to the lower hybrid frequency which in the solar wind is equivalent to the geometric mean of the proton and electron gyro frequencies, about 4 Hz at 1 A.U. The wavelength of such waves (perpendicular to the main field) can take on a large range of values, but a typical length would be the proton gyro radius which is 50-100 km at 1 A.U. Unlike waves whose propagation vector has an appreciable component along the magnetic field, these waves are not susceptible to instabilities arising from anisotropy of the velocity distributions. The level of the fluctuations might therefore be expected to be small, which is consistent with the value An&, N 0.1 deduced from radio observations. HOW such waves could be generated is not clear. They might arise from non-linear mechanisms associated with firehose unstable waves propagating along the field. If this is the case, the density perturbations might be expected to have scales around 500 km along the field, and 50-100 km perpendicular to it, and this could account for the filamentary structure mentioned earlier. Verification of these or other ideas must of course await direct spacecraft based measurements of the proton and electron density microstructure in the solar wind.
436
R. BUCKLEY
Ackno~~ledgements--1 am grateful to Dr. P. A. De&son and Dr. B. H. Briggs for useful discussions. supported by a Queen Elizabeth II Fellowship during the period of this work.
I was
REFERENCES BARNES, A. (1966). Physics FIuids 9, 1483. BUCKLEY, R. (197Oa). To be published. BUCKLEY, R. (1970b). To be published. CLEMMOW,P. C. and DOUGHERTY,J. P. (1969). Electrodynamics of Particles and Plasmas. Addison-Wesley. COHEN, M. H., GUNDEMANN,E. J., HARDEBECK,H. E. and SHARP, L. E. (1967). Astrophys. J. 147,449. COHEN, M. H. and GUNDEMANN,E. J. (1969). Astrophys. J. 155,654. COLEMAN,P. J. (1966). J. geophys. Res. 71,5509. COLEMAN,P. J., DAVIS, L., SMITH, E. J. and SONEXT, C. P. (1967). Science 138,1099. COLEMAN,P. J. (1968). Astrophys. J. 153,371. CRONYN, W. M. (1970). Astrophys. J. 161,755. DENNISON, P. A. (1969). Planet. Space Sci. 17, 189. FREDRICKS, R. W. (1968a). J. Plasma Phys. 2,197. FREDRICKS, R. W. (1968b). J. PIasma Phys. 2, 365. HEWISH, A. and SYMONDS,M. D. (1970). Planet. Space Sci. 17, 313. HOLLWEG, J. V. (1970). To be published. HOLZER, R. E., MCLEOD, M. G. and SMITH, E. J. (1966). J.geophys. Res. 71,148l. HUNDHAUSEN,A. J., ASBRIDGE, 3. R., BAME, S. J., GILBERT, H. E, and STRONG, I. B. (1967). J. geophys.
Res. 72,87. JOKIPII, J. R. and COLEMAN,P. J. (1968). J. geophys. Res. 73,5495. JOKIPII, J. R. and HOLLU~EG,J. V. (1970). Astrophys. J. 160, 745. LO~ELACE, R. V. E., SALPETER,E. E., SHARP, L. E. and HARRIS, D. E. (1970). Astrophys. J. 159,1047. MERCIER, R. P. (1962). Proc. Camb. Phil. Sot. 58,382. NESS, N. F., SCEARCE, C. S. and SEEK, J. B. (1964). J. geophys. Res. 69,3351. NEUGEBAUER, M. and SNMER, C. W. (1966). The Solar Wind (Eds. R. J. Mackin and M. Neugebauer), p. 3. Pergamon Press. PARKIN, I. A. (1967). Ph.D. Thesis, University of Adelaide. RATCLIFFE, J. A. (1956). Rep. Prog. Phys. 19,188. SALPE~R, E. E. (1967). Astrophys. J. 147,433. SARI, J. W. and NESS, N. F. (1969). Solar Phys. 8, 155. SCARF, F. L., WOLFE, J. H. and SILVA, R. W. (1967). J. geophys. Res. 72,993. SISCOE, G. L., DAMS, L., COLEMAN,P. J., SMITH, E. J. and JONES, D. E. (1968). J. geophys. Res. 73.61. SLEE, 0. B. and HIGGINS, C. S. (1968). Aust. J. Phys. 21, 341. Snx, T. H. (1962). The Theory of Plasma Waves. McGraw-Hill.
Space Sci. 1971, Vol.
19, pp. 421 to 436.
Pergamon
Press.
Printed
in Northern
Ireland
FLUCTUATIONS
IN THE INTERPLANETARY
Department
R. BUCKLEY of Physics, University of Adelaide, S. Australia
PLASMA
(Received 4 January 1971) Abstract-Spacecraft based observations of fluctuations in the interplanetary magnetic field and solar wind speed yield dominant spatial scales of the order 10s km, and negligible structure below about 500 km. Earth based observations of the angular broadening and scintillation of cosmic radio sources have been interpreted in terms of electron density scales of order a few hundred km. It is suggested that for scales below a few hundred km, there exists an enhanced level of small scale density fluctuations not accompanied by comparable magnetic variations. This proposal is shown to be consistent with radio observations, the contribution of the much larger electron density irregularities being quite negligible. A physical mechanism that might account for the small scale fluctuations is described. 1. INTRODUCTION
This paper describes attempts to reconcile measurements of fluctuations of solar wind parameters obtained by two entirely different methods. Spacecraft based observations of variations of the interplanetary magnetic field and the solar wind velocity indicate that almost all the energy is contained in structures whose scale near 1 A.U. is of order lo6 km (Coleman, 1968). Earth based observations of the angular broadening and scintillation of cosmic radio sources when their lines of sight approach the Sun imply, on the other hand, an electron density micro-structure whose scale increases from around 40 km at 0.1 A.U. to around 400 km at 1 A.U. (Hewish and Symonds, 1969). Jokipii and Hollweg (1970) and Hollweg (1970) have recently suggested that the electron density micro-structure observed by radio methods is simply a manifestation of the ‘inner’ or ‘dissipation’ length associated with the large scale phenomena observed by spacecraft. In section 2 of this paper the spacecraft and radio observations are discussed and their results pertinent to the solar wind are summarised. In section 3, the argument proposed by Jokipii and Hollweg is analysed and shown to be inconsistent with the radio observations. A new proposal is put forward in which the electron density fluctuations are uncoupled from those associated with the magnetic field at scales below a few hundred km, and exist at an enhanced level at these scales. For larger scales up to 106 km, the electron density and magnetic fluctuations follow one another. The consequences of this as regards radio observations are analysed and it is shown that the contribution of the larger scales is quite negligible and that angular broadening and scintillation measurements select only the small scale structure. In section 4 a possible physical mechanism that would account for such a small scale electron density structure with no comparable magnetic fluctuation is suggested. This is based on electrostatic waves propagating at right angles to the average magnetic field. 2. IRREGULARITIES
IN THE SOLAR
WIND
We shall discuss in turn direct spacecraft observations of fluctuations in solar wind parameters and the inferences that have been drawn from indirect radio methods. (a) Spacecraft
observations
In the last few years, considerable knowledge has been obtained of the structure of irregularities in the interplanetary magnetic field B (and to a lesser extent in the solar wind velocity VJ from measurements made aboard spacecraft travelling within l-1.5 A.U. from 421 1
422
R. BUCKLEY
the Sun, near the ecliptic plane. These measurements have covered the frequency range from 4 x lo-’ Hz (corresponding to the solar rotation period of 27 days) upward, but have mostly been limited to frequencies less than 1O-2 Hz (Ness et al., 1964; Coleman, 1966, 1968; Coleman et al., 1967; Sari and Ness, 1969; Jokipii and Coleman, 1968). From 10m5to low2 Hz the power spectra of magnetic fluctuations approximate to power lawsf+ with y varying in the range l-2 over the various experiments. This variation of y may reflect a secular change of the medium over the solar cycle (Sari and Ness, 1969). At higher frequencies, Siscoe et al. (1968) from magnetic measurements aboard the spacecraft Mariner IV, obtained spectra which followed the law f-3’” from 3 x lo-4 to 5 x 10-l Hz. Holzer et al. (1966) have reported a spectrum from the satellite OGO 1 of the radial component of B (with respect to the Sun) which could be approximated in the range 2 x 10-l to 2 Hz by a power lawf-3s (Coleman, 1968). If the reasonable assumption is made that the wave speeds associated with these low frequency phenomena are of the order of the AlfvCn or proton thermal speeds, which are both roughly 30 km/s, at 1 A.U., then because this figure is much less than the average solar wind speed V, of about 350 km/s, it is clear that the measured time changes in the magnetic field are primarily a consequence of the convection of spatial irregularities past the spacecraft with speed V,. Therefore frequency spectra can be converted to spatial spectra merely by the substitutionf = Vd/27r = V,/2& where q is the spatial wave number, and 1 the spatial scale. With V, = 350 km/s this relation becomes q (m-l) = 2 x 10-5f(Hz).
(1)
Coleman (1968) has grouped the various measurements into three frequency ranges as follows. 1.8 x lo-‘Hz
in B are observed to be primarily relatively flat. These, and correin the inherent field strength and a filamentary (or sector) structure
2 x 1O-s Hz
In this frequency range, the fluctuations in B are primarily transverse to the average field. The power spectra go roughly as f-1.2. These are interpreted as a manifestation of Alfven waves propagating along the average magnetic field. Below 2 x 10m3Hz, corresponding fluctuations in V, support this view, but measurements of V, are not available above this frequency. 2 x lo-lHz
of B
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Coleman has interpreted the complete spectrum as follows. Instabilities induced by large scale differential streaming in the solar wind generate long wavelength AlfvCn waves in range (a). The energy in these waves ‘cascades’ through range (b) by means of nonlinear interactions. In turbulence terms this is the ‘inertial range’. The energy is eventually dissipated by proton cyclotron damping near 10-r Hz or 500 km scale. This accounts for the rapid drop of the spectrum in range (c) and is analogous to the viscous dissipation range in fluid turbulence. The spectrum in the inertial range may be enhanced toward its upper
POWER DENSIT &I)
10
-9 I
1
WAVENUMBERtm-‘)
IIO5 I
’
I
IcFIG.
“INERTIA;‘RANGE
~~+--II~SSIPATION
1. SKBTCH OF THE SPATtAL POWER SPECTRUM OF MAGNBTIC FLUCTUATIONS OBSERVED BY SPACECRAFT.
end as a result of the firehose instability which occurs because of the existence of unequal proton temperatures parallel and perpendicular to B (Scarf et al., 1967). The complete spectrum is indicated in Fig. 1. In turbulence terminology, the energy containing or ‘outer’ scale is defined by the boundary between ranges (a), (b) and is of order 2 x lo6 km, whereas the dissipation or ‘inner’ scale is the boundary between ranges (b), (c) and is of order 500 km. (b) Earth based radio observations The scattering of radio waves from cosmic sources by irregularities of electron density in the solar wind has two observable consequences. In the first place, sources appear
424
R. BUCKLEY
broadened when their lines of sight approach the Sun. Secondly, diffraction leads to spatial variations of the intensity of the wave. If the irregularities are convected across the line of sight by the solar wind, this results in temporal changes of the observed intensity. This phenomenon is called interplanetary scintillation. Hewish and Symonds (1969) have recently given a useful summary of both areas as they affect the solar wind. Such measurements are not confined to the ecliptic. A brief outline of the relevant theory will introduce several important parameters. A radio wave of wavelength ;Z= 2rr/k is incident normally on a slab of thickness D containing irregularities of refractive index 1 + Ap of average size L. (It is assumed that Ap < 1.) The phase deviation undergone by the wave while traversing one irregularity is kL A,u. The number of irregularities along the line of sight is D/L. Statistically therefore, the RMS phase deviation on emergence from the slab is
For a plasma medium such as the solar wind where the radio frequencyfis than the electron gyro frequency fB= eB/2nm,, p is given by Pa =
1
_f," f"
much larger
’
where f,, = (n,e2/m,q,)*/27r is the plasma frequency. rz, is the electron density. Radio observations of the solar wind use frequencies of order 10’ to lo9 Hz. fBand f,are of order 3 x lo2 Hz and 2 x 104 Hz respectively at 1 A.U. (using B = 10 y = lo-* MKS, n, = 5 x lo6 m-3). If the Parker solar wind model is assumed, both fB and f,will be approximately inversely proportional to solar distance. Use of Equation (3) is therefore amply justified and it follows that Q=!C
3.
2f2J- %
(4)
Substitution into Equation (2) yields after some rearrangement CpO N Q&z
q
(5)
where r, = e2/4nq,m,c2 = 2.8 x lo-l5 m is the classical electron radius. It is clear that if An, follows the same inverse square law with solar distance R, as n, and if the slab thickness D is assumed to be roughly proportional to R,, then (bOis proportional to RF~/~. More detailed analysis, involving integration along the line of sight for a model in which n, and dAn,a are proportional
to R8-2 yields (6)
(Jokipii and Hollweg, 1970). L is measured in Km,&,, is the radio frequency in units of 100 MHz, R, is the distance of closest approach to the Sun of the line of sight, measured in A.U., and E = dhn,“/rrB is the relative RMS electron density fluctuation which by assumption is independent of R,.
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When the wavefront emerges from the region containing irregularities, its phase has a random variation perpendicular to the direction of propagation, with an RMS value given by Equations (5) or (6). As a result, the rays are refracted away from the original direction To an observer close enough to the screen the source of waves would of propagation. appear displaced slightly in position. If the irregularities move across the line of sight, the apparent position of the source would fluctuate within a cone of directions centered on the true position. But if (as is usually the case), the time constant of the receiver is too long to resolve such fluctuations of position, the source will simply appear broadened. Similarly if the observer is far enough from the irregularities to receive several refracted rays simultaneously, he will see a broadened source. The amount of such angular broadening is clearly independent of the observers distance from the irregularities. In the latter case however, the intersecting rays can interfere and generate fluctuations of wave intensity, or scintillations. Clearly this phenomenon cannot occur unless the observer is far enough from the irregularities. If the fluctuations in electron density and therefore phase have a single spatial scale L (in practice this has usually meant possessing a Gaussian autocorrelation function emZ*lL*), then the following results apply (Ratcliffe, 1956; Mercier, 1962; Salpeter, 1967). Numerical constants of order unity have been omitted from the following formulae, and it is assumed that L is very much greater than the wavelength. (a) The width 8, of the angular spectrum of scattered waves (and hence the observed angular diameter of the source) is %l - l/kL -$,,/kL
if
do&1
if
#,, > 1.
(7)
(b) Provided that the distance z from the irregularities to the Earth (of order 1 A.U.) is greater than z, where Zf - kLZ if & & 1 - kLa/4,,
if
$0 > 1,
(8)
then the scintillation index (relative RMS intensity fluctuation) for a point source is m N 2/2 &,
(cc llfby
Equation (6))
= 1
if if
$,, < 1 I#J~>~.
(9)
The spatial scale of the intensity pattern at the Earth is L,-L -L/+0
if
&& 1 if
+O>l.
(10)
If the irregularities are convected across the line of sight by the solar wind then the time scale of the pattern at the Earth is T, = L,/V,. (11) On the basis of such formulae it has been concluded (Hewish and Symonds, 1969) from angular broadening measurements over the range O-025 A.U. < R, < O-5A.U. that: (i) Irregularities of electron density with a scale of less than 5000 Km at 0.3 A.U. are a permanent feature of the solar wind. (ii) The irregularities are filamentary and elongated radially away from the Sun. (iii) E (the relative electron density fluctuation) is relatively small.
426
R. BUCKLEY
(iv} The radial variation of 0, indicates that to some extent the irregularities are locally generated and not merely swept out from the Sun by the solar wind. More definite knowledge of the scale of the irregularities cannot be obtained from angular broadening because & is unknown. Scintillation measurements obtained over essentially the whole sky indicate (Hewish and Symonds, 1969; Dennison, 1969) that (a) m cc l/ffrom 80 to 600 MHz. (b) m cc X:1-6 from 0.1 to 15 A.U. (c) On the basis of (a) and equation (9) it is concluded that at 81.5 MHz, & is less than 1 rad. for R, 2 O-3 A.U. (d) From Equation (IO), the irregularity scale increases from 100 to 400 km as R, increases from O-4 to 1 A.U. (e) Below 0.3 A.U., #,, is greater than 1 rad. at 81-5 MHz. Use of Equation (7) combined with angular broadening measurements yields the scale down to 0-I A.U. where it decreases to about 40 km. Scales deduced from scinti~ation and angular broadening measurements overlap approximately near O-3A.U. (f) Scintillation measurements at much higher frequencies are consistent with & remaining less than one radian down to much smaller R,. (Cohen and Gundermann, 1969) and have confirmed the decrease of scale down to 0.1 A.U. (g) Calculations of Fw from Equation (6) confirm that it is propo~onal like n, to l/R,2 and that the ratio E is of order 10 per cent (Cohen et al., 1967). (h) Direct measurements of r,& beyond R, = 0.6 A.U. have been obtained from phase scintillations of Jupiter’s decametric radiation (Slee and Higgins, 1968). These fit reasonably well on to values of &, inferred from intensity scintillations closer to the Sun. At 81-5 MHz the inequality z > z, (for #0 < 1) breaks down if L. 2 800 km. Therefore s~nti~ation meas~emen~ could not resolve scales appreciably greater than this. However the angular broadening measurements are not subject to this restriction. The fact that the two types of measurement give similar scales suggests strongly that there exists in the solar wind a quite definite electron density microstructure whose scale increases from about 40 km at O-1 A.U. to 400 km at 1 A.U., which is rather weak 8 N O-l), which produces phase deviations of order a few radians and less, and (~~~ which is responsible for both scintillation and angular broadening. There is also some indication that these irregularities are filamentary and directed away from the Sun. This is in marked contrast to the in situ measurements of magnetic field fluctuations which as we have seen give at 1 A.U. and beyond an ‘outer’ scale of order 2 x IO6km and an ‘inner’ dissipation scale of about 500 km. 3. RECONCJLWMON
OF THE TWO TYPES OF OBSERVATION
3.1 The proposal of Jokipii and Hollweg In an attempt to reconcile the spacecraft and radio observations, Jokipii and Hollweg (1970) have recently made the following suggestion. They have calculated the effects that would be observed by radio methods if the electron density fluctuations have a power spectrum which is similar in most respects to that of the magnetic fluctuations. In particular, the density spectrum would have an outer scale of order lOa km, and would cut off at an inner scale of a few hundred kilometres. The correlation length that must be used in Equation (6) is then the outer scale: 2 x 108 km. They
FLUCTUATIONS
IN THE
INTERPLANETARY
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427
have further assumed that E is of order unity. This is based on very large scale observations of the electron density variation which have been reported by Neugebauer and Snyder (1966). Equation (6) then yields values of &, of order 103-lo* rad. Under conditions of very large phase fluctuations, and provided the distance to the observer is not too large, the methods of geometrical optics are applicable These results then follow (Salpeter, 1967). Suppose that the phase fluctuations have transverse autocorrelation function p,+(x) and spatial power spectrum S,(q) (these are Fourier transforms of each other). Define a length scale L4 as follows. L4 = @----I
= [ /_ldq q%+(q)]-li4.
(12)
Provided that the distance z from the irregularities to the observer satisfies the inequality ZQ-
kL,=
(13)
$0
the spatial power spectrum of the intensity fluctuations S,(q) at the observer is given by
S,(q) = and the corresponding
(+cbzq2 12 7
S+(q)
’
(14)
scintillation index is
4oz 2
(kL,2’) m2= Jokipii and Hollweg admit that the geometrical optics approximation is probably not applicable over the entire electron density spectrum, but will break down at wavelengths comparable to the inner scale and below. Nevertheless they claim that this breakdown at very small scales will not significantly affect the argument. Assuming that inequality (13) holds for radio observations of the solar wind, the following conclusions can be drawn from Equation (14). We have seen that S6(q) (for the magnetic fluctuations and by assumption for the electron density and phase fluctuations also) cuts off rapidly beyond the inner scale l/q -=c500 km. The factor q* that multiplies S,(q) in Equation (14) removes power at the large scales and leaves only a residual peak around the inner scale (see Fig. 2). As a result the pattern scale would approximate to 500 km (at 1 A.U.) consistent with the radio observations. Further, because of inequality (13), the scintillation index given by Equation (15) is small despite the large value of +o. It should be pointed out that the results quoted above apply to a one dimensional situation. In the more realistic two dimensional case, the predictions are similar except that the observed frequency power spectrum SI(j) is given by the projection of the two dimensional spectrum SI(qs, qv) onto the direction in which the irregularities are convected by the solar wind,
(Parkin, 1967; Lovelace et al., 1970). Assuming that S,(q,, qJ is not too anisotropic and has a two dimensional form similar to that given by Equation (14), it is clear that this projection
428
R. BUCKLEY
I
1
(106Kmi’
4
(500Km)-‘I
will partially remove the ‘hole’ at low wavenumbers caused by the @ factor, and the result could well be close to the spectra that are frequently observed (see Fig. 2). Early work (Cohen et al., 1967) frequently showed Gaussian spectra but more recent results (Lovelace et at., 1970) indicate power law spectra (p’). The angular broadening results can also be explained this way if it is assumed that geometrical optics is applicable (Holfweg, 1970). fokipii and Hollweg conclude that the radio observations are consistent either with a density structure with dominant scales of a few hundred kilometres and small $s, or with an outer scale of order lo6 km, a cut-off or inner scale of a few hundred kilometres, and very large &,. They favour the latter structure because of its similarity to the i% situ magnetic observations, We now examine their proposal in more detail. In order for inequality (13) to be valid it is necessary that the scale L, be comparable to the large scale structure in the irregularities. Jokipii and Hollweg (1970) do not however attempt to calculate the scale L4. We now attempt to rectify this omission. As well as Z, we shall be interested in two
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429
other scales L,, L,, defined generally as follows L,’
=
O”d!@%(q) s --m (17)
The last formula applies if the spectrum is not normalised. These integrals can be estimated either graphically from Fig. 1 of Hollweg (1970), or by approximating S+(q) as follows S+(q)=l,
O
= (qL)-312, =o,
l/L
l/l
(18)
where I = 500 km, L = 2 x lo6 km. Then since I < L we get the following results 1 3l4
L2=2
L*
0 -
L
L’L
or
L, -8OOOkm,
Nl-8
I 718
L,
’
L
-
0L
L,w2000km,
L, -1200km.
(19) (20)
Far from being dominated by the outer scale 2 x lo6 km, these three values are much closer to the inner scale of 500 km. We may note also that if the pattern scale Lp is defined by L,-2
=
s*
dq@s,(q)
(21)
--co
then from Equation (14), and the definitions of L,, L, we obtain for an arbitrary form of S&l), Lp = L6S/Lp2 -
(22)
500 km with the values in Equation (20).
So the pattern scale is indeed close to the inner scale as discussed by Jokipii and Hollweg. If we assume do = IO4rad., k = 6 m-l (corresponding to 300 MHz), z = 1 A.U. = l-5 x loll m, and L4 = 2 x lo6 m then we get
Thus the inequality (13) is very badly violated. To satisfy it, L, would have to be at least a factor of 10 larger, and it is difficult to see how the estimate in Equation (20) could be in error by such a large factor. The only way to change L4 by such a factor would be to increase the inner scale from 500 to 5000 km which is clearly inconsistent with the observations. What does scintillation theory predict if inequality (13) is badly violated? In two recent papers (Buckley, 1970a,b), the full diffraction theory of a random phase screen (both one and two dimensional) is evaluated asymptotically for large RMS phase deviation do. Three ranges of distance z can be distinguished. (a)
430
R. BUCKLEY
Here geometrical optics is valid as already described.
In this region waves scattered from individual irregularities can intersect. The resultant focussing effects can lead to values of the scintillation index greater than unity. (c)
z > kL,L,I$,.
This is the far (Fraunhofer) diffraction zone. The scintillation index is unity, the intensity power spectrum SI(q) is Gaussian regardless of the form of&(q), and the diffraction pattern scale is L,I1/z& So if the ideas of Jokipii and Hollweg were correct, we would be observing well inside region (c) (z&kL,L, - 15). Observed indices less than unity might be accounted for by ‘diameter broadening’ (Hewish and Symonds, 1969) but one would then expect m to increase away from the Sun, eventually attaining the value unity. The spatial scale would be of order 5 km (with L, - 8000 km, and &, N 10s rad.). In frequency terms this corresponds to a width of Vs/2n (5 m) = 12 Hz. Although Gaussian spectra have been observed (Cohen et al., 1967) their widths are of order 1 Hz at 1 A.U. from the Sun. In addition one would expect angular broadening of order l/kLp rad. which is of order 10 set of arc at 1 A.U. Angular broadening has not been observed further out than O-5 A.U., but if the radial variation (for instance Hollweg, 1970) is extrapolated one would expect values of order 1 set of arc at 1 A.U. It has also been pointed out (A. Hewish, 1970, private communication) that the scintillation index given in Equation (15) is inversely proportional to the square of the radio frequency (#+,cc l/f, k CCf) whereas observations indicate a ll’variation over a wide range of frequency consistent with Equation (9). We conclude therefore that the electron density structure proposed by Jokipii and Hollweg is not consistent with the radio observations. 3.2 A new proposal and its consequences for radio observations
It is fairly clear that the only way out of the contradictions encountered in the last subsection is to relax the assumption of Jokipii and Hollweg, that the power spectra of the magnetic and electron density fluctuations cut off at the same inner scale of a few hundred kilometers. Let us now examine this assumption. It is well known from the ‘frozen field line’ concept that on scales much greater than the proton gyro radius, variations of nP (and therefore n,) and B will tend to follow one another. With a magnetic field of 5-10 y at 1 A.U. and a proton temperature of a few times 106 “K (for instance, Hundhausen et al., 1967) the proton gyro radius is in the range 50100 km. This probably decreases relatively slowly closer to the Sun. It is clear therefore that around the ‘inner’ scale of the magnetic fluctuations (a few hundred km), we are approaching the region where freezing in (and magnetohydrodynamic concepts in general) are breaking down. It seems reasonable to suppose therefore that the magnetic and electron density spectra can ‘uncouple’ for length scales less than a few hundred km. We have seen that radio observations are consistent with a definite but rather weak microstructure in the electron density with a typical scale near 1 A.U. of a few hundred km, and which imposes phase variations of order a few radians or less.
FLUCTUATIONS
IN THE INTERPLANETARY
( , (10'km
PLASMA
431
,.)y (500km+(100km~4
)-' q
RG.
3.
hOPOSED
TWO COMPONENT SPATIAL POWER SPECTRUM OF THE ELECTRON DENSITY FLUCTUATIONS NEAR 1 A.U.
We propose therefore that the complete irregularity structure in the solar wind consists of two roughly independent components, one with a scale measured in millions of km and in which the fluctuations are strong (An, - n,) and follow the magnetic fluctuations, and the other with a scale measured in hundreds of km in which they are weak (An&z, - O-1) and which is not coupled to the magnetic field. A possible physical mechanism that would produce such a structure will be discussed in the next section. Here we shall examine the consequences for radio observations of such a two component spectrum, which is indicated in Fig. 3. The autocorrelation function of the phase fluctuations corresponding to such a spectrum can be written in the form 4(x + x’W(x’) = $02P&) = 9L2PL(X) + 9%%%(x)
(24)
where
402= h2 + 4E2 - &I”. The corresponding
spatial power spectrum is
(25) where S,, Sr are the Fourier transforms of pL, pI and are normalized in such a way that their integrals over all values of q are unity. +z, +r are the separate RMS phase deviations produced by the large and small scale structures respectively. At 1 A.U. we expect #z - 103, $Z < 1 rad. pL(x), pi(x) are functions which are unity at x = 0, and tend to zero for x 2 L, x ;a I respectively, where at 1 A.U., L - 2 x 10 km, I-J 400 km. They can be thought of as autocorrelation functions for the large and small scale structures separately, and they possess only single length scales. We can imagine the small scale, small magnitude phase fluctuations riding on top of the far larger, far stronger variations associated with the large scale structure.
R. BUCKLEY
432
The general formula for the spatial power spectrum S,(q) of the intensity variations at distance z from the phase irregularities is m2&(q) = exp {-2#02(1 - &q/0)X 1’
* dx eiq” [exp %X~PJ~) 2rr s __m
- pd(x + zq/k) - p&x - zq/k))) -
11. (26)
See for instance Buckley (1970a). k = 27r/;1is the radio wavenumber and m2 is the scintillation index, given by the integral over all values of q of the right hand side of Equation (26). For simplicity, we consider a one dimensional situation. Similar conclusions hold for the two dimensional case. The methods developed by Buckley (1970a, b) for analysing Equation (26) do not apply directly to the correlation function given by Equation (24). However, we can proceed as follows. We first estimate the range of values that the parameter zq/k can take. We have z N 1 A.U. N loll m, k - 5 m-l (for 300 HMz). The widths of observed scintillation spectra are typically 1 or 2 Hz. Converting to wavenumber through q = 2rf/ V,, and using V, N 350 Km/s, we get 0 < q sz 2 x 1O-6 m-l for the observed range of values of q. Hence for the values of interest to the radio observations O-=zzG5
x 105m.
lo0 m, and so to a very good approxi-
Now the width of the function pL(x) is of order L mation, pl(zq/k)
= 1- f
(27)
(
2
)
‘, (28)
2p,(x)-pp,(x+$
-p~(x-~&$$‘(x),
where &)(x) = d2pL/dx2 and L, = (--p?)(O))- lj2. Since pL(x) has only a single scale L, it follows that L, is of order L. We assume that $1” is small enough to justify expansion of the various exponentials that occur in Equation (26). This should certainly be valid at 1 A.U. Substitution of Equation (24) into Equation (26) and use of Equation (28) then yields m2Sz(q) = e--(dLIQ’kLa)a [l - 2$:(1 X
- pl(zq/k))]
&/_tdx eiQz [e-(~~z~/s)“p~)(x) (1 + -
or
+T(2pl(x)
I+$))
- p,(~ + !.!)
- l]+0%9~
(29)
FLUCTUATIONS
IN THE
INTERPLANETARY
433
PLASMA
Now we have seen (Equation (27)) that the largest value of zq/k that need be considered is of order 5 x lo5 m. With #L N lo3 rad. and L, - L - log m we have therefore 0<
4Lzq 2 & 0.3 * (-1kL,
(31)
Over the relevant part of the spectrum therefore we can treat this quantity as a small parameter. The first part of Equation (30) can then be evaluated in terms of S,(q) the ‘large scale’ spectrum (Equation (25)) using the ‘range (a)’ result given in Buckley (1970a, Equations (24), (25)) and the second by introducing the power spectrum S,(q) of the small scale fluctuations as given by Salpeter (1967, Equation (15)). The result is
S&l1 - %Yl -
p,(zq/Wl
1 zq2
+
( )
4h2 sina2 k
S,(q)
i-o
4Lzq2 + K-N kL2
0(&4),
(32)
which we recognise as simply the sum of the geometrical optics result (Equation (14)) and the small phase deviation result that is valid at any distance z (Salpeter, 1967, Equation (15)). Geometrical optics is applicable here since SL(q) is not cut off at the inner scale I. We can write m2&(q) = mL2%Aq) + mz2Mq) m2 = mL2+ mt2
(33)
where
mL2Wq>= mL 2=
( )2 (g2)2(1+ 4Lzq2 k
SLbN
+
0(4,2)) (34)
O(Q))
4
and m,“Sz(q) = 4+,2 sin2
(35) dq sin2 (2-j-J 1 zq2 SZ(q) < 2+,2 where L, = (d4pL(x)/dx4);=$ is now of order L in contrast to the L4 introduced in the last subsection which was estimated to be much less than L. Inserting the values of the various parameters we find that mL- 10-4 (36) ml
434
R. BUCKLEY
postulated two component spectrum is correct, there is no inconsistency between spacecraft and radio observations. Despite the enormous phase deviations generated by the large scales, appreciable intensity fluctuations would result only after the scattered waves have propagated a distance vastly greater than 1 A.U. Similar remarks apply to the angular broadening. It should be remembered however that this derivation relies on the assumption that & small. If this is not so (and we have seen that & derived from scintillation measurements on 81.5 MHz is greater than unity below 0.3 A.U.) the separation of the intensity spectrum into large and small scale components is not clear cut, and there is the possibility that observations close to the Sun are sampling both large and small scales. Recently, Cronyn (1970) has argued that the magnetic and electron density fluctuations have similar small scale structure. Suppose that the three dimensional power spectra of the magnetic or density fluctuations have power law forms pY in the region of interest. Then Cronyn has shown that the frequency spectrum of magnetic fluctuations observed by spacecraft will have the formf-@‘-2), and further that the scintillation spectrum will have the formf(Y-l) (in the weak scattering limit, and ignoring the low frequency dip caused by the ‘sine-squared’ fringing function of Equation (35)). We have seen that in the dissipative range of the magnetic spectrum (Fig. l), very limited observations indicate a spectrum of the form f"'"(Coleman, 1968). The corresponding three dimensional spectrum would therefore be of the form q--5’s,and identifying magnetic and electron density structures we would expect scintillation spectra of the form f-p". Cronyn has noted that the spectra reported by Cohen et al. (1967) which were originally interpreted in terms of Gaussian functions are also consistent at high frequencies with a power law off-” approximately, which is in line with the preceding ideas. However, Lovelace et al. (1970) have observed power law spectra f--n forf > 1 Hz, and in the majority of cases, n lies between 3 and 4. This would seem to imply a form q-4 or q-r’ for the three dimensional density spectra, as opposed to q-6’8 for the magnetic fluctuations. So Lovelace’s results are also consistent with the findings of this paper, namely that the small scale density irregularities are enhanced relative to the magnetic fluctuations. It is possible that the power law spectra are illustrative of a second inertial range associated with a short wavelength regime of the turbulence. 4. A POSSIBLE EXPLANATION FOR THE ELECTRON DENSITY SPECTRUM
We have already noted that for scales much greater than the proton gyro radius which is of order 50-I 00 km at 1 A.U., the magnetic and electron density spectra can be expected to follow one another, and we have also outlined Coleman’s (1968) explanation of the magnetic spectrum, namely that energy generated by differential streaming instabilities cascades down the inertial range from IO6to 500 km where it is destroyed by cyclotron damping. Observations indicate that toward the upper end of this range the magnetic fluctuations are primarily transverse to the average spiral field. Coleman has interpreted them in terms of the left hand circularly polarized proton cyclotron wave travelling along the ambient field. Stix (1962, pp. 194-196) has shown that such waves are strongly damped for frequencies (in the plasma rest frame) near the proton cyclotron frequency and wavenumbers along the main field greater than approximately (fn fD2/ vTc2)1/3 where fB, f,are the proton gyro and plasma frequencies, vT is the proton thermal speed parallel to the main field, and c is the speed of light. At 1 A.U. this is approximately (500 km)-l as required by the inner dissipation scale. Note that this parallel wavelength is rather larger than the proton gyro radius. There is
FLUCTUATIONS
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PLASMA
435
probably also some energy in the other mode which propagates along the field: the right hand polarized mode. Generalized firehose instability of this mode due to the anisotropic proton velocity distribution could occur (Scarf et al., 1967) at the upper end of the inertial range. Each of these modes has its electric vector perpendicular to the ambient field, and its propagation vector k along this field. Thus there are no associated density fluctuations. Of course, plasma non-uniformities, and oblique propagation with respect to the field will generate density fluctuations but it is apparent that the close coupling between magnetic and density fluctuations that exists at larger wavelengths need no longer apply. Barnes (1966) has shown that for a plasma with /I 3 0.5 (fi is the ratio of thermal energy density to magnetic energy density and is typically 0.5 in the solar wind at 1 A.U.), most hydromagnetic waves are highly attenuated by combinations of the Landau, and cyclotron damping processes. In fact the only modes not strongly damped are the two transverse waves propagating in a narrow cone centered along the average magnetic field, and the two modes whose wave vectors are perpendicular to the main field. Of these latter, one (the ordinary mode) has its electric vector parallel to the main magnetic field and hence has no associated density perturbations. The other, however (the extraordinary mode) has in general a mixture of transverse and longitudinal electric fields, and therefore has proton and electron density perturbations associated with it. The low frequency behaviour of this latter wave has been analysed by Fredricks (1968a, b) who has called it a ‘Generalised Bernstein mode’. The ratio of the electric fields parallel and perpendicular to the wave vector is of the order of the ratio of the speed of light to the phase speed of the wave (Clemmow and Dougherty, 1969, p. 303). Thus for phase speeds very much less than the speed of light, the perpendicular component is quite negligible and the wave is virtually longitudinal, i.e. electrostatic. The magnetic vector of the wave is in these circumstances negligible. We tentatively suggest therefore that the small scale electron density fluctuations found by radio methods are a manifestation of electrostatic waves propagating across the average magnetic field. They have no magnetic fluctuations associated with them and hence would not be expected to have spectral characteristics similar to the magnetic ones. According to Fredricks (1968a) the spectrum of such waves in the plasma rest frame might be expected to extend roughly from the proton gyro frequency (about 10-r Hz at 1 A.U.) up to the lower hybrid frequency which in the solar wind is equivalent to the geometric mean of the proton and electron gyro frequencies, about 4 Hz at 1 A.U. The wavelength of such waves (perpendicular to the main field) can take on a large range of values, but a typical length would be the proton gyro radius which is 50-100 km at 1 A.U. Unlike waves whose propagation vector has an appreciable component along the magnetic field, these waves are not susceptible to instabilities arising from anisotropy of the velocity distributions. The level of the fluctuations might therefore be expected to be small, which is consistent with the value An&, N 0.1 deduced from radio observations. HOW such waves could be generated is not clear. They might arise from non-linear mechanisms associated with firehose unstable waves propagating along the field. If this is the case, the density perturbations might be expected to have scales around 500 km along the field, and 50-100 km perpendicular to it, and this could account for the filamentary structure mentioned earlier. Verification of these or other ideas must of course await direct spacecraft based measurements of the proton and electron density microstructure in the solar wind.
436
R. BUCKLEY
Ackno~~ledgements--1 am grateful to Dr. P. A. De&son and Dr. B. H. Briggs for useful discussions. supported by a Queen Elizabeth II Fellowship during the period of this work.
I was
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