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Wave-like phenomena in the interplanetary plasma: Mariner 2
Planet.
Space Sci. 1967, Vol.
15.
pp. 953
to 973.
PergamonPress Ltd. Printed in NorthernIreland
WAVE-LIKE PHENOMENA IN THE INTERPLANETARY PLASMA: MARINER 2 PAUL J. COLEMAN, Jr. Institute of Geophysics and Planetary Physics, University of California, Los Angeles, California 90024, U.S.A. (Received 14 December 1966) Abstract-Some characteristics of the variations of the interplanetary magnetic field and the plasma velocity, observed during the flight of Mariner 2, are considered. First, properties of the power spectra from records of the plasma velocity and the three vector components of the field are described. Next, the properties of the cross spectra between pairs of this set of four variables are described. The variations that would be produced in the plasma velocity and the field components by hydromagnetic waves are determined from a model of a uniform, perfectly conducting, isentropic, ideal gas in a uniform magnetic field. The properties of the variations determined empirically, are compared with those determined from this model. The following conditions, found to be necessary in the model, are found to be satisfied by the observed variations : (1) Significant coherences were usually present between all pairs of measured variables. (2) The phase differences between each pair were independent of frequency. (3) The ratios of the power densities for each pair were independent of frequency. (4) The values of these ratios fall in the ranges established from the model. (5) The values of the phase differences were either 0” or 180”. (6) The phase differences changed by 180’ when the direction of the interplanetary field changed from inward toward the Sun to outward.
The properties of the ionized gases and magnetic fields observed in interplanetary space suggest that the medium would support the propagation of hydromagnetic waves through this region (c.f. Alfvtn, 1950). In a preceding paper (Coleman, 1966a), hereinafter referred to as Paper 1, some properties of the variations in the magnetic field and plasma velocity, observed during the flight of Mariner 2, were described. In this paper, the observed variations will be compared to the variations that would be produced by hydromagnetic waves according to the simplest f&t-order model. The primary purpose of this comparison is to show that most of the properties of the variations in the magnetic field and plasma velocity satisfy certain necessary conditions of hydromagnetic wave behaviour. The results of this comparison were summarized in an earlier letter (Coleman, 1966b). The data to be discussed here were obtained with the magnetometer and theionspectrometer, or plasma probe, on board Mariner 2. The spacecraft was launched on 27 August 1962 and reached its minimum distance from Venus on 14 December 1962. Measurements of three orthogonal vector components of the field and of the velocity of the interplanetary plasma were obtained during the first 4 months of the flight. It will be convenient to employ heliocentric, spherical, polar coordinates with the polar axis coincident with the Sun’s axis of rotation. Thus, the positive r direction is radially outward from the Sun, the I# direction is parallel to the solar equatorial plane and positive in the direction of planetary motion, and the 13direction completes the usual right-handed system. In this sytem, the quantities measured with the magnetometer and plasma probe on board Mariner 2 are the three vector components of the magnetic field, B,, B,, and Bd, and 953 1
PAUL J. COLEMAN, JR.
954
f
I
VECTORS
(pm&in of spacecraft I special
care.
8=90. 8’ 0”
(b)
FIG. la. SPHERICAL POLAR COORDINATE SYSTEM W~II COORDINATES r, 0 AND +.
Orthogonal unit vectors G, 0 and c are shown at the point P, the location of the spacecraft. For simplification, the special case for the point P(r, 0 = 0”, +) is shown along with the magnetic line of force for the ideal spiral field (Parker, 1958) in the solar equatorial plane and passing through P. FIG. lb. ORIENTATION OPTHE X~Z+~~~RELATIVETOTHE~~~ DIRECTIONSATPOINT P FORTHE SPECIAL CASE IN WHICH B0 IS IN THE r$ PLANE. FIG. lc. ORIENTATION OF THE XJWAXES RELATIVE TO THE r@ THE GENERAL CASE.
DIRECTIONS AT A POINT P FOR
the plasma velocity VN V,.. Unit vectors in the r, 13,and 4 directions at the position of the spacecraft, (r, 0, +), are shown in Fig. 1. Estimates of the auto spectra and cross spectra from records of the four measured variables have provided the empirical results which are to be discussed here. In the next section, we will describe a model for hydromagnetic disturbances. We will then compare the observations and the model. THE MODEL We wish to compare the observed variations in the plasma velocity and the magnetic field with the variations that would be produced by hydromagnetic waves in the interplanetary plasma. Following Thompson (1962), with minor changes in notation, we will consider plane-wave perturbations in the plasma displacement, d, moving through a uniform, perfectly conducting, isentropic, ideal gas in a uniform magnetic field B, for the case in which the gas is moving with a bulk velocity V,. For such a system, in the reference frame
WAVE-LIKE PHENOMENA
IN THE INTERPLANETARY
PLASMA:
MARINER
2
955
moving with velocity V,, the linearized equations are
p,@v/at>
apjat+ pov. v =
0
(1)
=
X b> x 5
(2)
aw
-VP -
-I- (b$(v
(YP~~P~~P/~
=
0
E + (v/c) x B, = 0 (I/C)
abjat= -v
x
(3) (4)
E
(5)
where p. is the mean density and p is the density perturbation, V = V, + v is the velocity, B = B, + b is the magnetic field,p, is the mean pressure andp is the pressure perturbation, E is the electric field, y is the ratio of specific heats for the plasma, and c is the velocity of light. It has been assumed that V, and B,, are constant and uniform.
AN ARSI~Y
VECTOR a’
tN THE XYZ SYSTEM
Fm.
2.
ORIENTATIONOF AN ARBXTRARY VECTORa IN TEE xyz SYSTEM.
Initially we will employ rectangular coordinates x, y, z. Since the magnetic field B, is uniform, we assume B, = B,,2 where 2 is a unit vector in the positive x direction. Since the system is symmetrical about the x-axis, it will also be convenient to describe the orientation of an arbitrary vector a in terms of the angles 0, and &. Here 0, = cos-1 (a . $/a) and & = cos-l (aL . g/al) where al is the projection of a on the yz plane. Thus, a, = a,_ cos +a and a, = a,_ sin & and a, = a sin 0,. These quantities are depicted in Fig. 2, From equations (4 and 5), to the first order in b and v, abp
= v X (V x B) ri V x (v x B,)
(6)
Assuming that the displacement d and the velocity v are small so that in this linear approximation ad/at = v, integration of equation (6) yields b = V x (d x B,)
(7)
This expression fixes the relationship between the vector components of b and d transverse to B,. For a plane wave in the displacement, it also fixes the relationship between b and v. Thus, digressing briefly, we may consider the plane wave with propagation velocity U& relative to a system moving with velocity V,, the streaming velocity of the plasma. In this case, d = d,, exp ~~2~~~~)[k. (r - Vat) - Ut]) (8)
PAUL J. COLEMAN,
956
JR.
Note that & is a unit vector in the direction of propagation. moving with the plasma, d = d, exp [(2rri/A)(k . r -
In the frame of reference
Ut)]
@a)
and v =
ad/at = (-27ri/A)Ud
(9)
Also
ad/ax= (2rri/@,d ad,ay= (-k,p)v ad/a2= (--k,pgv
= (-k,p)v (10)
Then, from equation (‘7) and equations (lo),
. T - Ut)] b = (&/U)W,q, + fv3 + 9(-%qJ + .%-k&l = (B,JU)P(k, . vl> + 9(--k,u, cos $,I + 2(-k,u,
b = b, exp [(2+;l)(k
(11) sin &)I
This relationship is based only upon the assumption that the plasma is highly conducting, i.e. it is based upon equation (4 and 5). The additional requirement that equations (l-3) be satisfied, determines the remaining properties of the plane waves which may propagate through such a plasma. Returning now to Thompson’s development, if d = 0 at the time t = 0, integration of equation (1) yields p+p,V,d=O
(121
Integration of equation (3) and substitution for p from equation (12) yields p = - ypoV . d
(13)
Combining equations (2 and 13), we obtain (a2d/at2) - (yp,,/po)v(v.
d) -
(&)(V
X b) X B,-, = o
(14)
Next, we denote the Alfven speed by C, = Bo/(4~po)1/2, the sound speed by C, = (ypo/po)*12 = ($c?‘)~/~, and a unit vector in the direction of B, by &. Equation (14) may then be written
(l~c~2)(a2d~af2) - [(Cs2 + C~2)~C~2]V(V
l
d)
+ V(S . V)(L . d) - (e^. V)2d + (2. V)(V . d)& = 0
(15)
If we consider plane wave solutions of the form given by equation (Sa), the vector components of equation (15) become (U2/Ca2)d,
+ Aijdi = 0
(16)
where the Aij are functions of e”,k, and Cs2/Ca2, and i, j = x, y, z. As suggested by Thompson, the physical significance of this set of equations is more readily seen if we consider the independent variables k . a, e^. a, and L?. fc x d rather than the vector components ds of d. Then equations (16) yield [(Ua -
CA2 -
Cs2)&2]lt
(U2~C~z)~ . d -
. d + (t . k)(e?ad) = 0
(C~2~C~2)(~ . rz>(&. d) = 0
[(U2/CA2) - (& . Ii)->“]@. &x d = 0
(17) (18) (19)
WAVE-LIKE PHENOMENA
IN THE INTERPLANETARY
PLASMA:
MARINER
2
957
From these equations, we see that one root is the velocity of an AlfvCn wave U,a = Cg2 cos2 0, where e”. k = cos Ok. The other two roots are the roots of the quadratic ( 1/Ca4)U4 - [(CA2 + c,a)/c,2]Uz
+ (CS2/CA2)COG8, = 0
(20)
or u* = (1/42){(C,2
+ Cs2) 3 [(CA2 + C,2)2 - 4c,2c,2
co? eJ1/2}1/2
(21)
where the wave with velocity U, is designated the “fast” wave and the wave with the velocity U_ is designated the “slow” wave. From equation (19), we see that the vectors d, &, and B, = B,e” are co-planar. From equation (17), we see that (k,
. dd, = -Ku+z2- G2Y(U,2- CA”- C,s2)lk,,d,, = AJCrrdll
(22)
where the subscripts J_ and 11are used respectively to designate components transverse and parallel to B,. This expression must be satisfied by both fast and slow waves. Note that A, is positive and A_ is negative. Since measurements of v = ad/at and b are available, we must consider the behaviour of v. Since v is 90” out of phase with d, we have v = v, exp (2k/ii)[l2 . r - Ut - J/4]
(23)
with v,, parallel to d,. Since & and B, are constant, the geometrical relationships between d,, &, and B, hold also for v,,, &, and B,. Now from the results obtained thus far, we can obtain v, in terms of vO,ok, &, C,, and C,, i.e. we can determine the direction of v,. Then, from equation (1 l), we can compute b,. For AlfvQ waves, v, = 0 and v, is transverse to both B, and k. Thus, if k is parallel to B,, v, may have any direction transverse to B,. If k, # 0, v, must be transverse to both B, and k. We may arbitrarily select the phase of v so that v, is parallel to B, x fi. Then & = & + 712. Thus, for an AlfvCn wave vz = 0
v, = v cos (& + rr/2)
(24)
v, = v sin (& + 7r/2) For fast and slow waves, v, = v
cos 8,
V, = v sin 8, cos +u v, = v sin 8, sin q%,
(25)
so from equation (22), klv,
(cos & cos 4, + sin & sin d,)* = A,v,k,
(26)
Now B,, k, and v,, are co-planar for fast and slow waves, so that $e = +k or 4k + 7~. Thus, the coefficient of k,v, is & 1. But A+ is positive and A_ is negative, so for fast waves
4, = &, while for slow waves +V = & + z-. Then from equation (25), or (tan 0,), = IA&l cot Ok
(27)
PAUL J. COLEMAN,
958
JR.
These results may be summarized by the following relations: Arft)t!n waves b, = 0 b, = -v(B,lUd
cos e, cos (& + 42)
bz = -v(B&,)
cos 8, sin (+k + 7r/2)
(28)
Fart and slow waves b, = fv(B,/U,J
sin en sin 0,
b, = =Fv(WJ3
cos e, cos & sin 8,
b, = rv(B,Iu3
cos 8, sin & sin e,
(29)
where the upper signs correspond to fast waves, the lower signs correspond to slow waves, and 0,, is given by equation (27). From this model, for each of the three modes, given B,, the direction of propagation defined by 0, and &, the number density of the protons, the temperature, and the magnitude of the velocity perturbation vO,one may calculate b, and the orientation of v,,. During the flight, the average value of B,, was about 5y = 5 . 1O-5 G. The average number density of the protons, n,, near 1 a.u. was about 5 cmm3, and the average proton temperature was 1.7. 106’K. Neugebauer and Snyder, 1962, 1966). Thus, CA ‘Y 5 . 10G cm/set and C, N 4.108 cm/set. From equation (10) and the properties of the different types of waves, we can determine the amplitudes of the components of v and b in terms of the amplitude v,-,. The quantities vo, v,_, b,, and b, are listed in Table 1 for each of the three modes and for 8, = 0, 15, 30,45,60,75, and 90”. The quantities that were measured are B,, BB, B,, and V = IVl. Since the average radial streaming velocity of the plasma was so much greater than the average value of the velocity perturbation as determined from the auto spectra of V, we may assume VN V,. In order to compare the greatly idealized model with the observations, it is necessary to assume that the average interplanetary field, as well as the plasma temperature and density are piecewise uniform. Then we may establish the vector components of v and b in the r&# system, given equations (27 and 28) and the orientation of B, = fB,, + OB,,, + $Boo. Suppose tc = tan-l (-B,,JB,,J and ,!I = tan-l (BW/BL), where BL = (BOF + B,,f)l/“. These angles are shown in Fig. 1. Then from the geometry of the system, for i = r, 0, 4, vi = 2 a,vj
(30)
and bi = i a,b, (31) i where j = x, y, Z, the vj’s are given by equations (24 or 25) and the hi’s are given by equations (28 and 29), and cos /l cos a -sin p cos a sin a cos /l 0 sin B (32) -cos t!lsin a sin p sin a cos a aij
=
The analysis of the Mariner-2 data provided estimates of the auto spectra of the measured variables V,, B,, BO,and B+ and estimates of the cross spectra of all the possible pairs of these variables. From equations (14 and 15) in terms of the scalar amplitude vO,we may determine the power P that would appear in the records of V,, B,, Be, and B+ as a result of these
WAVE-LIKE
PHENOMENA
IN THE IN~RPLANETARY
PLASMA:
MARINER
TABLE 1. RESULTS OF CALCULATIONS OF POWER-DENSITY RATIOS, COHERJZNCES SQUARED,AND PHASEDIFFERENCES FORIDEALIZEDIiyDROMAoNETKCWAvES In these calculations,
B, = 5. 10e5 G, n = 5 protons/cma, T = 106“K. The phase difference, Q, between a pair of variables is given by the sign of R, the square of the coherence, for the pair. Thus, the pair is in phase for R > 0 and 180” out of phase for R < 0. Where two signs are given, the upper sign corresponds to the case of /I = 25” and the lower sign is for j3 = -25”. The quantity P for a particular variable is just the average power that would appear in a record of the variable that was one period of oscillation in length. In these calculations, it was assumed that the variations would be produced by a superposition of randomly polarized waves. For transverse waves the range of 6, indicated by ‘all’ is 0 < @, 5 90”. The results given here are for the case k . B, > 0. For k . B, < 0, all phase differences change by 180”.
Slow mode 1:
-1500.0
30 45 60 75 90
-17.5 -150 -10.4 -59 O-0
Fast mode 0 15
Transverse mode All
0.95 0.97 0.98 0.99 1GO
-
040
0.26 0.30 0.26 0.18 0.10 OGO
103.5 120.0 135.0 150.0 165.0 -
040
040
0.10 0.26 0.41 0.52 0.60 -
0.36 0.45 0.41 0.30 0.16 -
040
-74.8 -60.0
0.25 0.44
0.76
72.5 750 79.6 84.3 90.0
la0 o-97 0.95 0.97 0.98 0.99 1.00
-90-o
0.26 o-30 0.26 0.18 0.10 0.00
- 29.6 -450 -15.0 0.0
0.72 O-60 0.79 0.82
El O-21 0.00
90.0
0.00
I-00
-90.0
0.00
140
90.0 75.0
Z! 60 75 90
l*oo 0.97
Slow mode, @ = 0” 0 15 8:: 30 l-48 45 2.35 60 3.12 75 3.65 90 3.84
040
040
om
-
1.17 1.83 1.48 0.79 0‘22 o+m
1.58 2‘10 2.43 264 2.71
0.00
0.00
1.19 2.06 2.10 1.82 1.54 1.42
1.00 1.95 2.39 2.59 2.68 2.71
-0.52 -0.78 -0.91 -0.97 -0.99 -1.00
180” 180’ 180” 180” 180” 180”
2540 18.30
1490 11.80
-1~00 -0.66
180” 180”
-0.55 -0.82 -0.93 -0.98 -0.99 -1,OO
180” 180” 180” 180° 180” 180”
Slow mode, /I = &25” 0 15 30 45 60 75 90
OGO 0.83 184 264 3.28 3.70 384
Fast mode, /I = 0” 0 1050 15 9.08
0.00
-
2
959
PAUL
960
J. COLEMAN,
JR.
TABLE 1 (cont’d)
30 45 60 15 90
9.74 12.20 15.30 17.90 18.90
12.10 7.71 3.90 1.07 OGO
10.40 10.90 12.00 12.90 13.30
-0.30 -0.11 -0.02 OXKI 0.00
180” 180” 180” -
-l+m
-0.76 -0.48 -0.28 -0.15 -0.06 0.00
180” 180” 180” 180” 180” 180 -
Fast mode, fi = &25” 0 15 30 45 ;!i
10.50 9.08 8.78 9.69 12.00 10.90
1660 13.10 9.84 7.70 498 6.08
90
12.40
4.59
13.40 11.00 9.28 8.75 8.66 8.69 8.71
25.00
15.10
-1.00
180”
16.00
10.80
-1.00
180”
Transverse mode, /3 = 0” All
10.00
Transverse mode, /I = i25” All
7.10
idealized waves. We may also determine the coherences and phase relations among these variables in the idealized case. The power that would appear in the record of a measured variable, due to a single idealized wave, is just the average value of the square of the variable. Thus, P(v,) = (u,2) and P(bJ = (b:), i = r, 8,$. Since the terms auui and ai,& are all related by real multiplicative factors, the cross spectra of the variables will yield real values. Thus, the values of the squares of the coherences, R, and phase differences for each pair of variables may be obtained from the expressions R = (vibj)/(u~)(b~) R = (bibj)/(b~)(b~)
for any i, j for i = j
where k, j = I, 0, t$. Here IRI is equivalent to the coherence squared and the sign of R determines whether the variables are in phase (+) or 180” out of phase (-). During the flight, the preferred value of the angle CIwas about 40”. During periods in which the field was directed predominantly outward from the Sun, the preferred value of B, was about 1.9y. Thus, the preferred value of /? was typically about 22’. However, the measurements of the absolute value of B, are less reliable than those of B,. and B+. As a result, values of P and R were computed for the cases /? = 0 and 25” in order to establish the effects of errors in the estimates of /?. Further, there is evidence, the reliability of which is similarly questionable because of the uncertainty in the absolute values of B,, that the average value of B, remained positive during the periods in which the interplanetary field was directed back toward the Sun. This is the case in which u = 40 + 180”, fi = 25”, which is simply related to the case in which u = 40”, /3 = -25”. As a result, the latter case was also considered. It has been assumed that the waves are circularly polarized. Since B, k, and v are coplanar for fast and slow waves, the angle c$~will be used to indicate the orientation of the plane of polarization of such a wave. There is no reason a priori for the waves to have a preferred plane of polarization in a uniform system. In this case, the values of & would be determined by the properties of the source of the disturbances. In interplanetary space, non-uniformity of B,, V,,, po, andp,, may lead to asymmetries in the system which in turn
WAVE-LIKE
PHENOMENA
IN THE IN~RFLANETARY
PLASMA:
MARINER
2
961
may constrain & to certain allowed values. However, since we were approximating the actual system by a uniform one, we considered first the case in which the disturbances are produced by the superposition of a great many plane waves with random phases and with the orientations of the planes of polarization distributed uniformly. In other words, we considered the case of circularly polarized waves. Thus, in taking the average value of the squares of the variables and of the products of the different variables, we assumed that, when averaged over all the present waves, L&(%(r) sin & cos 8, = 0 {etch
sin #& = W&(0
cos 5&J= 0
i.e. we assumed no coherence on the average between ui and u,, or bi and bj, or vi and bj, i#j,k,j=x,y,z.
Under these assumptions, the values of P(v,), and P(b,), for i = I, 8,+, and the values of R(u+, bi), for i = r, 8, 4, and R(b,, b,), for i # j, and for i, j = Y,8, 4, may be computed in terms of v,, for any field orientation and propagation direction. These values are, of course, independent of the frequency f of the waves. This frequency, measured in the frame of reference moving with the plasma, is f = U/A. If we assume that all the waves originate within a finite region of space centered at the Sun, on the average, disturbances produced by waves propagating outward from the Sun will predominate. However, since a point source in a uniform field will produce a spherically spreading disturbance in the fast mode and disturbances moving aIong the field lines in both directions in the transverse and slow modes, we assume that waves moving back toward the Sun also contribute to the observed variations in v and b. The phase relationships between b and v for a wave moving in the k direction, back toward the Sun, differs by 180” from that for a wave moving outward. Thus, any contribution to b from inwardly travelling waves will reduce the coherences indicated by JR/, between v, and bi, i = r, 6, (6, but will not change the phase relationships so long as the contribution from outwardly travelling waves predominates. Accordingly, the calculated values of R are those for the outwardly travelling waves, or k, = cos ek > 0. The various cases for which P and R have been computed may be summarized as follows : Modes :
Alfvtn, fast, slow
;*
0, 15, 30, 45, 40, 75, 90” &25,0” 40”
tc. Polarization :
Circular
B,:
5.oy
PO: T:
5 mp g cm-3 (m, = 1 proton mass)
105’K
The results are listed in Table 1. For our purposes the most significant parameters listed in the table are the ratios i = r, 8,$, since these quantities are directly measurable from the power P(WW, spectra of V, and B,, i = r, 8, 4. Note that they are also independent of the signs of B. and k,. Another important aspect of these results is the indication that v7 and b, are 180” out of phase regardless of which type of wave is considered, except for fast waves when 8, > 75”. For the transverse and slow waves this result is apparent from an inspection of
PAUL J. COLEMAN,
962
JR.
equations (14 and 15), respectively. Of course, all the listed phase relationships are changed by 180” when B,, is negative. In considering the phase relationships, we have focused our attention on the variables V, and b, because the equations indicate that the phase relationship and the coherence are the least dependent upon the sign and magnitude of j3 in this case. OBSERVATIONS
We will be concerned with data obtained during four lo-day periods and two 6-day periods indicated in Table 2. These periods were selected in order to exclude the major reversals of the polarity of the interplanetary magnetic field, i.e. changes of the algebraic sign of the spiral interplanetary field that occurred twice during each period of solar rotation TABLE 2.
RATIOS OF THE POWER DENSITIES, WHERBNCBS SQUARED, AND PHASE DIF'FERBNCES OBTAINED FROM SIX SETS OF AUTO AND CROSS SPECIRA FOR THE MEASURED VARIABLES B,, Be, B+ AND v,
The power densities, P, the coherences squared, R, and the phase differences, Q, are averages taken over the frequency range l-50 cd. The quantity Rmin is the value of R required for a significant measurement of CD,i.e. for a measurement in which the r.m.s. uncertainty of @ is 45”. A (+) polarity is assigned if the average field was directed outward from the Sun; (-) polarity is assigned if it was inward toward the Sun
WWV’v~
Period
(day4
Polarity
243-252 254-263 270-279 282-291 297-302 313-318 Average Average Average
+ i+ + Both
11.8 7.8 2.3 14.4 6.6 46.1 6.9 22.8 14.5
W’i, BJ
WoYP(K) J”(BM’(I/,)
(lo-28 G2 cm-a se@ 16.5 21.2 5.3 31.7 16.4 32.3 12.7 28.4 20.6
14.5 13.4 4.2 29.3 14.6 32.3 11.1 25.0 18.1
NV,, &I
R min
Meg)
0.223 0.050 0.039 0378 0.487 0.097 0.250 0.175 0.213
0.021 0.023 0.019 0,023 0.029 0.035
181 f 7 -4 * 14 172 f 8 4*4 179 f 4 6zt8 177
(Coleman et al., 1966). Thus, during each of the six periods to be considered, the polarity of the field was nearly always that listed in Table 2. These polarity changes usually occurred at the edges of high-velocity plasma streams. Since shock waves were also observed near the edges of some of the streams that were encountered during the flight (Sonett et al., 1964; Razdan, Colburn and Sonett, 1965; Colburn and Sonett 1966), the selection of these periods of constant polarity eliminated some of the periods during which such major disturbances occurred. As mentioned previously, the variables measured by the magnetometer and plasma probe are B,, Be, B4, and V ‘u V,. In the analysis of the data discussed in Paper 1, amplitude distributions of these variables were examined first. It was found that the distributions were roughly Gaussian. As the next step in the analysis, estimates of power spectra were computed for each of these four variables. Representative composite power spectra for B,, B,, and B4 are shown in Figs. 3-5. The sampling rate of the Mariner-2 magnetometer corresponded to a Nyquist frequency, fN,of 1.35 . 1O-2 c/s or about 1100 cycles per day (cd). The lower limit of the frequency range to be investigated was arbitrarily set at 10e3 fn, or about 1.1 cd, to save computation time. A further saving of computation time was effected by dividing the frequency range 1.1-l 100 cd into two overlapping ranges, 1.1-l 10 cd and 11-1100 cd, and computating spectral estimates separately for each. Spectra estimated for the lower range of frequencies were designated LF, while those estimated for
WAVE-LIKE
PHENOMENA
IN THE INTERPLANETARY
I
PLASMA:
MARINER
2
t
MARINER COMPOSITE
2
POWER
SPECTRUM
4
AVERAGE VALUES FROM LF HR SPECTRA FOR IO-DAY PERIODS BEGINNING ON DAYS
l
% ;
16”
EI 282
AVERAGE VALUES FROM HF LR SPECTRA FOR I-DAY PERIODS BEGINNING ON DAYS
+
i3
270
\
272,274,
\
278,282
iz 283
\ \ \ lo-‘t 16’
I
11111111
I
1
Illlltl
FREQUENCY COMPOS~;~
I
IO-’
IO-’ FIG. 3.
I
(F,. POWER
I
If
IIJ
F,
1.35.I~*cPS) SPECXWJM
FOR
B,.
The spectrum at the lower frequencies is the average of the two low-frequency, high-resolution spectra for the periods from day 270 through day 279 and from day 282 through day 291. The spectrum at the higher frequencies is the average of the five spectra for days 272,274,278, 282 and 283. The resulting composite spectrum is assumed to represent an average property of the field. For comparison, lines showing l/f and l/f’ dependences are also included. For the hip-resolution spectra, Y cz 40 so the 95-per cent confidence band is 0*65-I*‘75 times the computed value. For the low-resolution spectra, v cz: 200 so the band is 0.83-1.23 times the computed value.
963
MARINER COMPOSITE
2 POWER
SPECTRUM
Se
AVERAGE VALUES FROM LF HR SPECTRA FOR IO-DAY PERIODS BEGINNING ON DAYS
270
AVERAGE VALUES FROM HF LR SPECTRA FOR I-DAY PERIODS BEGINNING ON DAYS 272,274,278,282
FREQUENCY
FIG.
4.
COMPOSITE
8 282 \ C/F’
\
& 283
{Fn POWER
= 1.35
l
IO-*
SPECTRUM
CPSf FOR
&L
(See the caption of Fig. 3.)
the higher range were designated HF. For each of the LF spectra of interest here, 100 estimates were computed. For each of the HF spectra only 20 estimates were computed, to save computation time. The lOO-estimate spectra are designated HR, for ‘“higher-resolution”, and the 20-estimate spectra are designated LR, for “lower-resolution”. For the LF-HR spectra, the number of degrees of freedom, Y, was about 40. For the HF-LR spectra, v 1zr200.
WAVE-LIKE PHENOMENA IQ’
I
11
IN THE INTERPLANETARY I
11111,
I I11111 MARINER
COMPOSITE
I
EL 282
AVERAGE VALUES FROM HF LR SPECTRA FOR I-DAY PERIODS BEGINNING ON DAYS 272,274,278,282
a 283
FIG. 5.
COMPOSITE
(See the
( F,=
MARINER
2
965
1 I1111
2 POWER
AVERAGE VALUES FROM LF HR SPECTRA FOR IO-DAY PERIODS BEGINNING ON DAYS 270
FREQUENCY
PLASMA:
SPECTRUM
1.35 . IG’CPS I
POWERSPECTRUM FOR &.
caption of Fig. 3.)
A power spectrum for the measured plasma velocity, V, or V, where it is necessary to distinguish between the theoretical and empirical velocities, is plotted in Fig. 6. This spectrum corresponds to the third period listed in Table 2. The sampling rate of the plasma probe was about one sixth that of the magnetometer. Thus, for the variable V,, only lower-frequency (LF) spectra are available. As the next step, described in Paper 1, the records were examined for correlations between the various pairs of these variables. The procedure employed was the usual one of
966
PAUL J. COLEMAN, JR.
obtaining estimates of the cross spectra of each pair of variables for several sections of the record. It was found that each pair of variables exhibited significant coherence on the average, and that the coherence was particularly strong for the pair I’,, B,. The cross spectra for this pair of variables are shown in Fig. 7 for all six periods listed in Table 2. These cross spectra are low-frequency (LF) spectra, since one of the pair of variables is V,. They are low-resolution spectra (LR) as well, since pilot estimates of the cross spectra indicated very little structure in the phase difference as a function of frequency.
LF- HR FOR
SPECTRUM
IO-DAY
BEGINNING
FIG. 6.
PERIOD ON
DAY
270
POWER SPECTRUMFOR V,.
This spectrum is the low-frequency,high-resolutionspectrum for the periods from day 270 through day 279. For the high-resolution spectra, Y N 40. Thus, the 95-per cent conjidence band for estimates of the power density is 0.65~1.75 times the computed value.
In the plots of phase difference, @, versus frequency for the pair V,, B,, a gradual change of @ with frequency is apparent, even when the coherence is relatively high. This change is an artificial one which arises due to the fact that different timing schemes were employed in recording the measured values of V, and B,. For V,, the time assigned to a particular measurement was the time at the beginning of the 222-set period during which the measurement was obtained. For B,, the time assigned to the corresponding measurement was the midpoint of the same period, or 111 set later. At 1-l cd, this time difference introduces an error of -0~51“ in @(V,, B,). The error increases linearly with frequency so that at 111 cd the error is -51”. Since the phase data in Fig. 7 include these artificial phase differences, lines drawn between the points (f = 0,
480 0.00
I
LEAD
SECMJOARY
PHASE
0.05 1 FREQUENCY
2-
(a)
PRIMARY
(Vp,Brl
(Fn”l.35~IO’CPS)
OVER
MARINE’
J
--I. o.Ia
IO+CPS,
PERlOO BEWWNO
COHERENCE
xx O.O(b-()40
025
0.50
0.75
(b>
SQUARED
(VP, 61)
254n 2820 313.
2
243 a 2700 297.
tF,,=l.35.
El
(D*Y,l962):
FREOUENCY
MARI
PERIOD SESlNNlNG lD*Y,1962>:
1.00
1.25 A.._
FIG. 7. FWASE DIFFERENCES AND COHERENCES FROM CROSS SPECTRA OF THE PAIR OF VARIABLES (v,, &). These spectra are low-frequency, low-resolution spectra. The plots on the left-hand side are those for the three periods during which the polarity of the interplanetary field was predominantly positive. On the right-hand side are those for the three periods during which the polarity was predominantly negative.
2970
t80
PAUL
968
J. COLEMAN,
JR.
These auto and cross spectra were discussed in detail in Paper 1. It was concluded that the significant coherence and the constant phase relationship constituted strong evidence for wave-like motions in the interplanetary plasma. This conclusion led to a somewhat more quantitative comparison of these empirically determined properties with the theoretically determined properties described in the preceding section. The results of this comparison will be described next.
MARINER POWER DENSITY P(Brl/P(Vr)
.-*-.
2 RATIOS
o----o
DAY 243-252
0-o +--m-t
DAY
.-+
DAY 282-291
.-v-.-.
DAY 297-302
.-.
DAY 313-318
DAY 254-263 270-279
/.-.--_.-~-
\.-I
F. MAX. *-•-+-• d_-_-__--o_._
--_._______._------o-------~~._______~_____-
T. MAX. T.MIN. F. M IN. S. MAX.
0
Cl025 FREQUENCY
0.050
F,
iFn=l.35~lb2CPS) (a)
FIG. 8.
RATIOS OF POWER DENSITIES.
The ratios of the power density in the auto spectrum of each vector field component to the power density in the auto spectrum of the radial component of the plasma velocity is plotted versus frequency for six sets of auto spectra. The theoretically allowed ranges for each ratio are also indicated on the right-hand side of the corresponding plot. Separate ranges are shown for each mode of propagation, fast (F), slow (s), and Alfven or transverse (7’).
We will consider in detail the properties of the field and plasma velocity variations in the frequency range from 1 to 50 cd. The lower limit was selected to limit the computer time required for the analysis. As mentioned previously, an upper limit of 100 cd was imposed by the sampling rate of the plasma probe. However, the consistently lower coherences indicated for (V,, B,.) in the frequency range between 50 and 100 cd prompted the selection of 50 cd as the upper limit. The auto spectra of B,, B,, B+, and V, N V,. provided estimates of the power densities for these variables as functions of the frequency. As the first step in the comparison between the measured properties of the variations in B,, B,, B4, and V, and the properties of b,, b,, b,, and v, determined from the model of idealized hydromagnetic waves, various ratios of the spectral densities were computed as functions of the frequency,f, for the six sets of data. In Fig. 8, the ratios P(B,)/P(V,), P(B,)/P(V,), and P(B,&/P(V,.) are plotted versus frequency over the frequency range l-50 cd. In the ideal case, considered in the last section, these ratios are independent of frequency, at least for cases in which the frequencies of the waves, measured in the reference frame
WAVE-LIKE PHENOMENA
IN THE INTERPLANETARY
PLASMA:
--:
I
; %
lG2’
s
-z2
2
969
-
MARINER No
MARINER
POWER DENSITY
2
O___. -0 DAY
RAT I OS
P(B&P(V,)
243-252
o-0 +-----+
DAY 234-263.
+-+
DAY 282-291.
.-m--w.
DAY
297-302
.-.
DAY
313.318.
DAY 270-273
I
1
1
1 F. MAX. T. MAX. T. MIN.
S. MAX. F
I@
I
L
0
0.025 FREQUENCY
0050
FIG.
8b.
1
I
MARINER POWER
2
DENSITY RATIOS P(B+ )/P(Vr 1
7
o----o
DAY 243-252
o-o .a----+
DAY 254-263 DAY
270-279
.-+
DAY
282-291
.----.
DAY
297-302
.-
DAY
3I3-319
_ ____._
f+cy-fq~:>-_
___~--------o-----_o________o
\-p-?~
0
l ________*___-----•-------_*________+_.----
__*__._-----*.-.____T. %. %.
I
I
FREQUENCY
*----____+
I
-_ S. MAX.
/
0.050
0.025
0
F,
(Fn=l.35~IO-2CPSl
Fn
(F.=l.35.162CPS)
FIG. 8c.
moving with the streaming velocity of the plasma, are much smaller than the proton gyro frequency which is about 0.08 c/s in a field of strength 5y. In the empirical cases, the powerdensity ratios are evidently independent of frequency, since the ratios plotted in Fig. 8 show no apparent frequency dependence. In the ideal case, the values of these power density ratios for a particular plane wave depend upon the geometry of the system and upon the field magnitude, B,, the temperature, of lB,,l. The T, and density, pO, of the gas. For Alfven waves, the values are independent 2
970
PAUL J. COLEMAN, JR.
values computed for the ideal case, as described in the preceding section, are also indicated on the right-hand side of Fig. 8. In the empirical cases, the values of the power-density ratios were generally within or close to the ranges determined for the ideal case. The variations among the empirical cases are probably due primarily to variations in lB,,l, p,,, and T. Comparison of the measured values and the theoretical values of the power-density ratios also indicates that waves in the transverse and/or fast modes were present, since the measured ratios are considerably greater than the ratios expected for slow waves, except for 19,near 90”, but are generally within the ranges of the ratios expected for the other two modes. This result does not imply that slow waves were absent but rather that AlfvCn and/or fast waves were present and, therefore, contributed most to the variations in the field. Evidence that the observed field variations could not be produced by purely transverse waves was presented in Paper 1, in which it was shown that the variations in B,,the magnitude of the vector component of the field in the direction of the ideal spiral field, were of roughly the same magnitude as the variations in the other vector components. However, to this point, the most significant results of this comparison are simply that the measured values of the power-density ratios are evidently independent of frequency and that these values are in the ranges calculated for the idealized waves using only the measured values of the mean proton density, the mean field strength, and the mean proton temperature. Having considered the properties of the auto spectra of the measured variables B,,B,, B4,and I’,., let us turn to the cross spectra of various pairs of these variables. The cross spectra of a pair of variables provides estimates of IRl, the magnitude of the square of the coherence between the two time series corresponding to the two variables, as a function of frequency, and estimates of @, the difference in the phase of the predominant coherent components of the two variables, as a function of frequency. In effecting the comparison between the empirical cases and the ideal case, the quantities (RI and a’, calculated for pairs of the measured variables were compared, respectively, with IRI and the algebraic sign of R calculated for the corresponding pair of variables for the ideal case, as described in the preceding section. Thus, for the ideal case, if R > 0, 0 = 0". If R < 0,(3= 180”. Also, for the ideal case, R is independent of frequency. Cross spectra were computed for all pairs of the measured variables for all six periods of interest. The values of JR(V,,, BJ, the square of the coherence for the pair of variables V, and B,., was greater by factors in the range from 5 to 8 than the values of IRIfor any of the other pairs, although on the average, all pairs exhibited significant coherences. Accordingly, only values of (R( V,., &)Iand @(V,.,B,), taken from the cross spectra shown in Fig. 7, are reproduced in Table 2. Values for all pairs are given in Paper 1. In most cases, the quantity JR(Vr, &)I exhibited considerable frequency dependence over the range l-100 cd, as shown in Fig. 7. However, in the frequency range l-50 cd, the average value is sufficient for our purposes. Furthermore, the quantity @(V,, B,.) is roughly independent of frequency. Thus, only average values of IRIand 0, taken over the frequency range l-50 cd, are listed in Table 2. From Table 2, it is apparent that the coherence of V,.and B,was relatively high and that these two variables were nearly 180” out of phase when the average field was directed from the Sun (Spolarity) and nearly in phase when the average field was directed toward the Sun (-polarity). The method by which this polarity was reckoned is described in Paper 1. The apparent differences in 0 of -10” to -15” from 0” to 180” is due primarily to the phase error discussed earlier.
WAVE-LIKE
PHENOMENA
IN THE INTERPLANETARY
PLASMA:
MARINER
2
971
Phase differences of 180” and 0” for positive and negative polarities, respectively, are consistent with the results of the computations in the preceding section for the ideal case in which the waves propagate primarily outward, i.e. propagate with k, > 0. As mentioned previously, we might expect the coherences to be well below the values for the ideal waves because of the effects of waves propagating back toward the Sun, i.e., propagating with k, < 0. SUMMARY
AND DISCUSSION
In the foregoing, various properties of simultaneous variations observed in the interplanetary magnetic field and in the velocity of the solar-wind plasma were compared to the properties of hydromagnetic waves as determined from an idealized model. The qualitative study of the observed variations yielded the following results over the frequency range, l-50 cd: (1) Significant coherences were usually exhibited between all pairs of the set of variables B,, &, B+, and V,. (2) The coherences generally varied with frequency. For the cases in which the values were consistently significant, the coherences decreased with increasing frequency. (3) The phase differences of the predominant coherent components in each pair of variables were roughly independent of frequency. (4) The ratios of the power densities for each pair of variables were similarly independent of frequency. Comparison with the properties of the idealized waves showed that the first, third, and fourth properties listed above would be expected for any complex group of waves as long as the amplitudes of the waves in one mode relative to the amplitudes of the waves in each of the other two modes are roughly the same at any of the frequencies studied. The second property listed above would be expected if; for example, the waves that were encountered were to include two such groups of waves, each with a different power spectrum and a different phase relationship. The quantitative study of the observed variations yielded the following results : (5) Most of the measured values of the power-density ratios, P(B,)/P(V,), i = r, 8, 4, were in the ranges of the values that were calculated for the ideal waves. The measured values were generally close to the maxima of these calculated ranges. (6) The measured values of the power-density ratios, given in Table 2, show that P(B,)/P(V,) > P(B&‘(V,J > P(B,)/P(V,). The results calculated for the ideal waves, listed in Table 1, show that this relationship would be expected for all Alfvtn waves and for fast waves with ek < 45”. (7) The phase differences of VP and B,, as measured by their cross spectra, were either 0” or 180”. The value of 0” was observed when the spiral interplanetary field was directed toward the Sun. The value of 180” was observed when the field was directed away from the Sun. (8) The measured values of the coherences, listed in Table 2, were found to be considerably smaller than the values calculated for the ideal waves and listed in Table 1. The fifth and sixth properties listed above would be expected for AlfvCn and/or fast waves. The seventh property would be expected if waves propagating outwardly along the field predominated. The two groups, mentioned in connection with the frequency dependence of the coherence, may be inward and outward-bound waves with somewhat different
972
PAUL J. COLEMAN,
JR.
power spectra. In this case, the properties of this frequency dependence would indicate that the outward-bound waves were somewhat less predominant at the higher frequencies. The presence of two such groups, i.e. inward and outward-bound waves, would also account for the eighth property listed above. We remarked above that fast waves and/or Alfven waves would produce the measured power-density ratios. In Paper 1, it was shown that AlfvCn waves alone could not produce the observed variations because the variations along B, were of about the same magnitudes as those in the transverse directions. Thus, it would appear that the field variations were produced primarily either by fast waves and AlfvCn waves or by fast waves alone. These results do not indicate whether Alfvtn waves were definitely present. Further, they do not indicate whether slow waves were present, but undetected due to the presence of waves in one or both of the other modes. Since the fast and slow modes are coupled in the ideal case, and since the AlfvCn mode is not likely to be strictly decoupled in the interplanetary medium, it is probable that all three types of waves were encountered. It should be mentioned also that the magnetic field measurements suggests that qualitatively similar variations occur in the range of frequencies up to at least 500 cd. These field measurements provided information in the frequency range O-1000 cd. The data indicate that the properties of the field variations in the range 50-500 cd are the same as those in the region l-50 cd, except that the absolute power densities decrease with increasing frequency, reaching the noise level of the magnetometer at about 500 cd. The auto spectra of the field variables were described in Paper 1. At frequencies near 500 cd, the spectra, in general, showed a frequency dependence somewhat weaker than f-2.Of course, a cutoff would be expected near the proton cyclotron frequency of 0.08 c/s. For AlfvCn waves, this cutoff could appear in our computed spectra at frequencies up to about 1 c/s due to Doppler shifts (compare equations 8 and Sa). There is some evidence from OGO-1 (Holzer, McLeod and Smith, 1966) that a steepening of the spectra of the field variables to a dependence stronger than f-” occurs near 0.2 c/s. There are several likely sources for waves in the interplanetary medium. It has been suggested that the Sun’s chromosphere and corona result from the convective motion in the granulation zone of the photosphere (Leighton, 1963). Acoustic waves generated by this motion are believed to travel upward or outward, through the atmosphere, transporting sufficient energy to provide heating of the upper layers. There is evidence that the periods of these waves are about 6 min, which is roughly the mean lifetime of the granulation pattern. The preferred spacing of the cells in the pattern is about 2000 km. There is another pattern or convective flow in the Sun which corresponds to the more The coarser supergranulation is physically similar to recently detected supergranulation. the finer granulation, but behaves independently. The mean lifetime of the supergranulation pattern is evidently about 10 hr. The preferred spacing of these larger cells is about 35,000 km. Waves produced by these two patterns of convective motion may produce disturbances in the interplanetary medium. In this case, we might expect variations which would be modulated with periods of about 10 hr and less, with the shorter periods due to harmonics, and variations with periods of 6 min and less. The Sun’s rotation would, of course, complicate the frequency spectrum that would be observed at a fixed point. Thus, from the waves corresponding to the granulation pattern, one would expect components with periods of 17 min and shorter and from the waves corresponding to the supergranulation pattern, one would expect components with periods of 4 hr and shorter.
WAVE-LIKE
PHENOMENA
IN THE INTERPLANETARY
PLASMA:
MARINER
2
973
In a discussion of interplanetary dynamical processes, Parker (1963) described several other likely sources for disturbances in the interplanetary medium. Interactions between streams of solar plasma travelling at different velocities will occur since the coronal temperature varies with time and position. Such interactions may produce disturbances travelling in both directions from the boundaries between the streams. Further, several types of instabilities are predicted for the case in which the solar wind is composed of many streams in each of which the plasma is in a different state. In addition, the solar-wind model of Parker (1958) for the coronal expansion requires that the outwardly flowing plasma expand anisotropically. In the absence of collisions in the plasma, this is expected to generate plasma instabilities that could result in wave-like phenomena. Acknowledgements-1 wish to express my appreciation to M. M. Neugebauer and C. W. Snyder for permission to use data from the Mariner-2 plasma probe; to L. Davis, Jr., C. P. Sonett, and E. J. Smith with whom I collaborated on the magnetometer experiment, and to E. A. Kraut who has been kind enough to discuss with me many aspects of this analysis. The work was supported by the National Aeronautics and Space Administration under research grant NsG 249. REFERENCES ALFVBN,H. 1950. Cosmical Electrodynamics. Clarendon Press, Oxford. COLBURN,D. S.and SONET~,C. P. 1966. Space Sci. Rev. 5,439-506. COLEMAN,P. J.,Jr.1966a. J.geophys. Res. 71, 5509. COLEMAN, P. J.,Jr.1966b.Phys. Rev. Lett. 17,207-211. COLEMAN,~. J.,J~.,DAvIs,L.,J~., SMITH,E.J.and Jo~m,D.E.1966. J.geophys. Res. 71,2831-2839. HOLZER,R. E., MCLEOD, M. G. and SMITH,E. J. 1966. J.geophys. Res. 71,1481-1486. LEIGHTON,R. B. 1963. A. Rev. Astr. & Astrophys. 1, 19-40. NEIJGEBAUER, M. M. and SNYDER,~.W. 1962.Science 13&l-2. NEUGEBAUER,M. M. and SNYDER,~.W. 1966.Mariner-2 observations ofthe solar wind,Part1. Average properties, Preprint, Jet Propulsion Laboratory, Pasadena, California, April. PARKER,E. N. 1958.Astrophys. J. 128,664-676. PARKER,E. N. 1963. Interplanetary Dynamical Processes, pp. 151-160. Interscience, New York. RAZDAN,H.,COLBI_JRN,D. S.and SONET~,C. P.1965. Planet. Space Sci. 13,1111-1123. SONETT, C. P., COLBFJRN,D.S.,DAVIS,L.,J~.,SMITH,E. J.and COLEMAN, P.J.,Jr.1964. Phys. Rev.Lett. 13,153-156. THOMPON,W. B. 1962. An Introduction to Plasma Physics, Chapter 5. Pergamon Press, New York. Pe3IoMe--PaCCMaTpElBaIoTCs HeKOTOpbE XapaKTt?pHCTElKIl B3pIlaI@t MfSKIIJIaHeTHOFO MWHHTHOrO IlOJlHI4 CKOPOCTIl lIJl33MbI,H36JIKg3eMbIe BO BpeMH noneTa Mapxaepa 2.
CIIepBaAaeTCR OHEiCaHkIe CBOtiCTBCIIeKTpOBMOrqHOCTElCOrJIaCHOMepaM CKOpOCTkl 3aTeMonncbIBaIoTcHcBoticTBanonepeuIIJIa3MbItrTpeXCOCTaBJIRIO~llXBeKTOpaIIOJI% HbIX CIIeKTpOBMem,Q' IIapaMElTaKOI'OKOMIIJIeKTa geTbIpeXIIepeMeHHbIX.rIpoIzaBOAMbIerHHpOMaI'HHTHblMH BOJIHaMIl IIepeMeHbI B CKOpOCTH IIJIa3MbI H COCTaBJIHIOIIGIX Ha 0cKoBaHHIiMoaenH onaoponrroro,BnonHe npoBonslqer0 nonHI, 0npefienInoTcfx M3eHTpOIIHOrOUAeaJIbHOrOra3a B OAHOpOAHOM MarHUTHOM none. OnpenenseMbIe 3MIIHpHqeCKH CBOZtCTBaBapHaqllt CpaBHHBaIOTCR CO CBOtiCTBaMH,KOTOpbIe 6bun4 YTBepmAeHbI Ha OCHOBaHElll 3TOtiMOAeJIH. CJIeAyIOIIVie YCJIOBIIH, KOTOpbIe 6buni npE13HaHbI He06XOAIiMbIMH B MORenIT,IIOBIlAElMOMY yF[OBJIeTBOpHIOTCfI Ha6JIIOHaeMbIMIi BapHaqHHMH: (1) Memny BCeMn IIapaMHK3MepHeMbIXnepeMeHHhlx06bIqHOCyrrleCTBOBaJIa3HaqIF TeJIbHaRKOI'epeHTHOCTb. (2) Pa3HOCTb +a3 Memay Ka?KAOtIIapOZt 6bIaHe3aBnCnMaOT =IaCTOTbI. (3) ~Oa$+iq~eHTbI yAeJIbHO&MOlqHOCTIl AJIFI KaxcgOt IlapbI He 6b1nK3aBHCIIMbI OT 9aCTOTbI. K YCTaHOBneHHbIM COrJIaCHO (4) 3HaseHHH 3TEIX K03+&iqHeHTOB llp%IypOYeHbl Monena pa3pfqaM. (6)3HaseHnH paaHocTI% aa3 COOTBeTCTBOBaJIH nn60 O",nn60 180". (6) Pa3HOCTb @a3 H3MeHFIJIaCb IIOCpepCTBOM 180",Korna KanpaBneHHe MemnnaHeTHOI-0IIOJIR MeHRJIOCbOT BHYTpeHHerO,B CTOpOHY COJIHqa,K HapymHOMy.
Space Sci. 1967, Vol.
15.
pp. 953
to 973.
PergamonPress Ltd. Printed in NorthernIreland
WAVE-LIKE PHENOMENA IN THE INTERPLANETARY PLASMA: MARINER 2 PAUL J. COLEMAN, Jr. Institute of Geophysics and Planetary Physics, University of California, Los Angeles, California 90024, U.S.A. (Received 14 December 1966) Abstract-Some characteristics of the variations of the interplanetary magnetic field and the plasma velocity, observed during the flight of Mariner 2, are considered. First, properties of the power spectra from records of the plasma velocity and the three vector components of the field are described. Next, the properties of the cross spectra between pairs of this set of four variables are described. The variations that would be produced in the plasma velocity and the field components by hydromagnetic waves are determined from a model of a uniform, perfectly conducting, isentropic, ideal gas in a uniform magnetic field. The properties of the variations determined empirically, are compared with those determined from this model. The following conditions, found to be necessary in the model, are found to be satisfied by the observed variations : (1) Significant coherences were usually present between all pairs of measured variables. (2) The phase differences between each pair were independent of frequency. (3) The ratios of the power densities for each pair were independent of frequency. (4) The values of these ratios fall in the ranges established from the model. (5) The values of the phase differences were either 0” or 180”. (6) The phase differences changed by 180’ when the direction of the interplanetary field changed from inward toward the Sun to outward.
The properties of the ionized gases and magnetic fields observed in interplanetary space suggest that the medium would support the propagation of hydromagnetic waves through this region (c.f. Alfvtn, 1950). In a preceding paper (Coleman, 1966a), hereinafter referred to as Paper 1, some properties of the variations in the magnetic field and plasma velocity, observed during the flight of Mariner 2, were described. In this paper, the observed variations will be compared to the variations that would be produced by hydromagnetic waves according to the simplest f&t-order model. The primary purpose of this comparison is to show that most of the properties of the variations in the magnetic field and plasma velocity satisfy certain necessary conditions of hydromagnetic wave behaviour. The results of this comparison were summarized in an earlier letter (Coleman, 1966b). The data to be discussed here were obtained with the magnetometer and theionspectrometer, or plasma probe, on board Mariner 2. The spacecraft was launched on 27 August 1962 and reached its minimum distance from Venus on 14 December 1962. Measurements of three orthogonal vector components of the field and of the velocity of the interplanetary plasma were obtained during the first 4 months of the flight. It will be convenient to employ heliocentric, spherical, polar coordinates with the polar axis coincident with the Sun’s axis of rotation. Thus, the positive r direction is radially outward from the Sun, the I# direction is parallel to the solar equatorial plane and positive in the direction of planetary motion, and the 13direction completes the usual right-handed system. In this sytem, the quantities measured with the magnetometer and plasma probe on board Mariner 2 are the three vector components of the magnetic field, B,, B,, and Bd, and 953 1
PAUL J. COLEMAN, JR.
954
f
I
VECTORS
(pm&in of spacecraft I special
care.
8=90. 8’ 0”
(b)
FIG. la. SPHERICAL POLAR COORDINATE SYSTEM W~II COORDINATES r, 0 AND +.
Orthogonal unit vectors G, 0 and c are shown at the point P, the location of the spacecraft. For simplification, the special case for the point P(r, 0 = 0”, +) is shown along with the magnetic line of force for the ideal spiral field (Parker, 1958) in the solar equatorial plane and passing through P. FIG. lb. ORIENTATION OPTHE X~Z+~~~RELATIVETOTHE~~~ DIRECTIONSATPOINT P FORTHE SPECIAL CASE IN WHICH B0 IS IN THE r$ PLANE. FIG. lc. ORIENTATION OF THE XJWAXES RELATIVE TO THE r@ THE GENERAL CASE.
DIRECTIONS AT A POINT P FOR
the plasma velocity VN V,.. Unit vectors in the r, 13,and 4 directions at the position of the spacecraft, (r, 0, +), are shown in Fig. 1. Estimates of the auto spectra and cross spectra from records of the four measured variables have provided the empirical results which are to be discussed here. In the next section, we will describe a model for hydromagnetic disturbances. We will then compare the observations and the model. THE MODEL We wish to compare the observed variations in the plasma velocity and the magnetic field with the variations that would be produced by hydromagnetic waves in the interplanetary plasma. Following Thompson (1962), with minor changes in notation, we will consider plane-wave perturbations in the plasma displacement, d, moving through a uniform, perfectly conducting, isentropic, ideal gas in a uniform magnetic field B, for the case in which the gas is moving with a bulk velocity V,. For such a system, in the reference frame
WAVE-LIKE PHENOMENA
IN THE INTERPLANETARY
PLASMA:
MARINER
2
955
moving with velocity V,, the linearized equations are
p,@v/at>
apjat+ pov. v =
0
(1)
=
X b> x 5
(2)
aw
-VP -
-I- (b$(v
(YP~~P~~P/~
=
0
E + (v/c) x B, = 0 (I/C)
abjat= -v
x
(3) (4)
E
(5)
where p. is the mean density and p is the density perturbation, V = V, + v is the velocity, B = B, + b is the magnetic field,p, is the mean pressure andp is the pressure perturbation, E is the electric field, y is the ratio of specific heats for the plasma, and c is the velocity of light. It has been assumed that V, and B,, are constant and uniform.
AN ARSI~Y
VECTOR a’
tN THE XYZ SYSTEM
Fm.
2.
ORIENTATIONOF AN ARBXTRARY VECTORa IN TEE xyz SYSTEM.
Initially we will employ rectangular coordinates x, y, z. Since the magnetic field B, is uniform, we assume B, = B,,2 where 2 is a unit vector in the positive x direction. Since the system is symmetrical about the x-axis, it will also be convenient to describe the orientation of an arbitrary vector a in terms of the angles 0, and &. Here 0, = cos-1 (a . $/a) and & = cos-l (aL . g/al) where al is the projection of a on the yz plane. Thus, a, = a,_ cos +a and a, = a,_ sin & and a, = a sin 0,. These quantities are depicted in Fig. 2, From equations (4 and 5), to the first order in b and v, abp
= v X (V x B) ri V x (v x B,)
(6)
Assuming that the displacement d and the velocity v are small so that in this linear approximation ad/at = v, integration of equation (6) yields b = V x (d x B,)
(7)
This expression fixes the relationship between the vector components of b and d transverse to B,. For a plane wave in the displacement, it also fixes the relationship between b and v. Thus, digressing briefly, we may consider the plane wave with propagation velocity U& relative to a system moving with velocity V,, the streaming velocity of the plasma. In this case, d = d,, exp ~~2~~~~)[k. (r - Vat) - Ut]) (8)
PAUL J. COLEMAN,
956
JR.
Note that & is a unit vector in the direction of propagation. moving with the plasma, d = d, exp [(2rri/A)(k . r -
In the frame of reference
Ut)]
@a)
and v =
ad/at = (-27ri/A)Ud
(9)
Also
ad/ax= (2rri/@,d ad,ay= (-k,p)v ad/a2= (--k,pgv
= (-k,p)v (10)
Then, from equation (‘7) and equations (lo),
. T - Ut)] b = (&/U)W,q, + fv3 + 9(-%qJ + .%-k&l = (B,JU)P(k, . vl> + 9(--k,u, cos $,I + 2(-k,u,
b = b, exp [(2+;l)(k
(11) sin &)I
This relationship is based only upon the assumption that the plasma is highly conducting, i.e. it is based upon equation (4 and 5). The additional requirement that equations (l-3) be satisfied, determines the remaining properties of the plane waves which may propagate through such a plasma. Returning now to Thompson’s development, if d = 0 at the time t = 0, integration of equation (1) yields p+p,V,d=O
(121
Integration of equation (3) and substitution for p from equation (12) yields p = - ypoV . d
(13)
Combining equations (2 and 13), we obtain (a2d/at2) - (yp,,/po)v(v.
d) -
(&)(V
X b) X B,-, = o
(14)
Next, we denote the Alfven speed by C, = Bo/(4~po)1/2, the sound speed by C, = (ypo/po)*12 = ($c?‘)~/~, and a unit vector in the direction of B, by &. Equation (14) may then be written
(l~c~2)(a2d~af2) - [(Cs2 + C~2)~C~2]V(V
l
d)
+ V(S . V)(L . d) - (e^. V)2d + (2. V)(V . d)& = 0
(15)
If we consider plane wave solutions of the form given by equation (Sa), the vector components of equation (15) become (U2/Ca2)d,
+ Aijdi = 0
(16)
where the Aij are functions of e”,k, and Cs2/Ca2, and i, j = x, y, z. As suggested by Thompson, the physical significance of this set of equations is more readily seen if we consider the independent variables k . a, e^. a, and L?. fc x d rather than the vector components ds of d. Then equations (16) yield [(Ua -
CA2 -
Cs2)&2]lt
(U2~C~z)~ . d -
. d + (t . k)(e?ad) = 0
(C~2~C~2)(~ . rz>(&. d) = 0
[(U2/CA2) - (& . Ii)->“]@. &x d = 0
(17) (18) (19)
WAVE-LIKE PHENOMENA
IN THE INTERPLANETARY
PLASMA:
MARINER
2
957
From these equations, we see that one root is the velocity of an AlfvCn wave U,a = Cg2 cos2 0, where e”. k = cos Ok. The other two roots are the roots of the quadratic ( 1/Ca4)U4 - [(CA2 + c,a)/c,2]Uz
+ (CS2/CA2)COG8, = 0
(20)
or u* = (1/42){(C,2
+ Cs2) 3 [(CA2 + C,2)2 - 4c,2c,2
co? eJ1/2}1/2
(21)
where the wave with velocity U, is designated the “fast” wave and the wave with the velocity U_ is designated the “slow” wave. From equation (19), we see that the vectors d, &, and B, = B,e” are co-planar. From equation (17), we see that (k,
. dd, = -Ku+z2- G2Y(U,2- CA”- C,s2)lk,,d,, = AJCrrdll
(22)
where the subscripts J_ and 11are used respectively to designate components transverse and parallel to B,. This expression must be satisfied by both fast and slow waves. Note that A, is positive and A_ is negative. Since measurements of v = ad/at and b are available, we must consider the behaviour of v. Since v is 90” out of phase with d, we have v = v, exp (2k/ii)[l2 . r - Ut - J/4]
(23)
with v,, parallel to d,. Since & and B, are constant, the geometrical relationships between d,, &, and B, hold also for v,,, &, and B,. Now from the results obtained thus far, we can obtain v, in terms of vO,ok, &, C,, and C,, i.e. we can determine the direction of v,. Then, from equation (1 l), we can compute b,. For AlfvQ waves, v, = 0 and v, is transverse to both B, and k. Thus, if k is parallel to B,, v, may have any direction transverse to B,. If k, # 0, v, must be transverse to both B, and k. We may arbitrarily select the phase of v so that v, is parallel to B, x fi. Then & = & + 712. Thus, for an AlfvCn wave vz = 0
v, = v cos (& + rr/2)
(24)
v, = v sin (& + 7r/2) For fast and slow waves, v, = v
cos 8,
V, = v sin 8, cos +u v, = v sin 8, sin q%,
(25)
so from equation (22), klv,
(cos & cos 4, + sin & sin d,)* = A,v,k,
(26)
Now B,, k, and v,, are co-planar for fast and slow waves, so that $e = +k or 4k + 7~. Thus, the coefficient of k,v, is & 1. But A+ is positive and A_ is negative, so for fast waves
4, = &, while for slow waves +V = & + z-. Then from equation (25), or (tan 0,), = IA&l cot Ok
(27)
PAUL J. COLEMAN,
958
JR.
These results may be summarized by the following relations: Arft)t!n waves b, = 0 b, = -v(B,lUd
cos e, cos (& + 42)
bz = -v(B&,)
cos 8, sin (+k + 7r/2)
(28)
Fart and slow waves b, = fv(B,/U,J
sin en sin 0,
b, = =Fv(WJ3
cos e, cos & sin 8,
b, = rv(B,Iu3
cos 8, sin & sin e,
(29)
where the upper signs correspond to fast waves, the lower signs correspond to slow waves, and 0,, is given by equation (27). From this model, for each of the three modes, given B,, the direction of propagation defined by 0, and &, the number density of the protons, the temperature, and the magnitude of the velocity perturbation vO,one may calculate b, and the orientation of v,,. During the flight, the average value of B,, was about 5y = 5 . 1O-5 G. The average number density of the protons, n,, near 1 a.u. was about 5 cmm3, and the average proton temperature was 1.7. 106’K. Neugebauer and Snyder, 1962, 1966). Thus, CA ‘Y 5 . 10G cm/set and C, N 4.108 cm/set. From equation (10) and the properties of the different types of waves, we can determine the amplitudes of the components of v and b in terms of the amplitude v,-,. The quantities vo, v,_, b,, and b, are listed in Table 1 for each of the three modes and for 8, = 0, 15, 30,45,60,75, and 90”. The quantities that were measured are B,, BB, B,, and V = IVl. Since the average radial streaming velocity of the plasma was so much greater than the average value of the velocity perturbation as determined from the auto spectra of V, we may assume VN V,. In order to compare the greatly idealized model with the observations, it is necessary to assume that the average interplanetary field, as well as the plasma temperature and density are piecewise uniform. Then we may establish the vector components of v and b in the r&# system, given equations (27 and 28) and the orientation of B, = fB,, + OB,,, + $Boo. Suppose tc = tan-l (-B,,JB,,J and ,!I = tan-l (BW/BL), where BL = (BOF + B,,f)l/“. These angles are shown in Fig. 1. Then from the geometry of the system, for i = r, 0, 4, vi = 2 a,vj
(30)
and bi = i a,b, (31) i where j = x, y, Z, the vj’s are given by equations (24 or 25) and the hi’s are given by equations (28 and 29), and cos /l cos a -sin p cos a sin a cos /l 0 sin B (32) -cos t!lsin a sin p sin a cos a aij
=
The analysis of the Mariner-2 data provided estimates of the auto spectra of the measured variables V,, B,, BO,and B+ and estimates of the cross spectra of all the possible pairs of these variables. From equations (14 and 15) in terms of the scalar amplitude vO,we may determine the power P that would appear in the records of V,, B,, Be, and B+ as a result of these
WAVE-LIKE
PHENOMENA
IN THE IN~RPLANETARY
PLASMA:
MARINER
TABLE 1. RESULTS OF CALCULATIONS OF POWER-DENSITY RATIOS, COHERJZNCES SQUARED,AND PHASEDIFFERENCES FORIDEALIZEDIiyDROMAoNETKCWAvES In these calculations,
B, = 5. 10e5 G, n = 5 protons/cma, T = 106“K. The phase difference, Q, between a pair of variables is given by the sign of R, the square of the coherence, for the pair. Thus, the pair is in phase for R > 0 and 180” out of phase for R < 0. Where two signs are given, the upper sign corresponds to the case of /I = 25” and the lower sign is for j3 = -25”. The quantity P for a particular variable is just the average power that would appear in a record of the variable that was one period of oscillation in length. In these calculations, it was assumed that the variations would be produced by a superposition of randomly polarized waves. For transverse waves the range of 6, indicated by ‘all’ is 0 < @, 5 90”. The results given here are for the case k . B, > 0. For k . B, < 0, all phase differences change by 180”.
Slow mode 1:
-1500.0
30 45 60 75 90
-17.5 -150 -10.4 -59 O-0
Fast mode 0 15
Transverse mode All
0.95 0.97 0.98 0.99 1GO
-
040
0.26 0.30 0.26 0.18 0.10 OGO
103.5 120.0 135.0 150.0 165.0 -
040
040
0.10 0.26 0.41 0.52 0.60 -
0.36 0.45 0.41 0.30 0.16 -
040
-74.8 -60.0
0.25 0.44
0.76
72.5 750 79.6 84.3 90.0
la0 o-97 0.95 0.97 0.98 0.99 1.00
-90-o
0.26 o-30 0.26 0.18 0.10 0.00
- 29.6 -450 -15.0 0.0
0.72 O-60 0.79 0.82
El O-21 0.00
90.0
0.00
I-00
-90.0
0.00
140
90.0 75.0
Z! 60 75 90
l*oo 0.97
Slow mode, @ = 0” 0 15 8:: 30 l-48 45 2.35 60 3.12 75 3.65 90 3.84
040
040
om
-
1.17 1.83 1.48 0.79 0‘22 o+m
1.58 2‘10 2.43 264 2.71
0.00
0.00
1.19 2.06 2.10 1.82 1.54 1.42
1.00 1.95 2.39 2.59 2.68 2.71
-0.52 -0.78 -0.91 -0.97 -0.99 -1.00
180” 180’ 180” 180” 180” 180”
2540 18.30
1490 11.80
-1~00 -0.66
180” 180”
-0.55 -0.82 -0.93 -0.98 -0.99 -1,OO
180” 180” 180” 180° 180” 180”
Slow mode, /I = &25” 0 15 30 45 60 75 90
OGO 0.83 184 264 3.28 3.70 384
Fast mode, /I = 0” 0 1050 15 9.08
0.00
-
2
959
PAUL
960
J. COLEMAN,
JR.
TABLE 1 (cont’d)
30 45 60 15 90
9.74 12.20 15.30 17.90 18.90
12.10 7.71 3.90 1.07 OGO
10.40 10.90 12.00 12.90 13.30
-0.30 -0.11 -0.02 OXKI 0.00
180” 180” 180” -
-l+m
-0.76 -0.48 -0.28 -0.15 -0.06 0.00
180” 180” 180” 180” 180” 180 -
Fast mode, fi = &25” 0 15 30 45 ;!i
10.50 9.08 8.78 9.69 12.00 10.90
1660 13.10 9.84 7.70 498 6.08
90
12.40
4.59
13.40 11.00 9.28 8.75 8.66 8.69 8.71
25.00
15.10
-1.00
180”
16.00
10.80
-1.00
180”
Transverse mode, /3 = 0” All
10.00
Transverse mode, /I = i25” All
7.10
idealized waves. We may also determine the coherences and phase relations among these variables in the idealized case. The power that would appear in the record of a measured variable, due to a single idealized wave, is just the average value of the square of the variable. Thus, P(v,) = (u,2) and P(bJ = (b:), i = r, 8,$. Since the terms auui and ai,& are all related by real multiplicative factors, the cross spectra of the variables will yield real values. Thus, the values of the squares of the coherences, R, and phase differences for each pair of variables may be obtained from the expressions R = (vibj)/(u~)(b~) R = (bibj)/(b~)(b~)
for any i, j for i = j
where k, j = I, 0, t$. Here IRI is equivalent to the coherence squared and the sign of R determines whether the variables are in phase (+) or 180” out of phase (-). During the flight, the preferred value of the angle CIwas about 40”. During periods in which the field was directed predominantly outward from the Sun, the preferred value of B, was about 1.9y. Thus, the preferred value of /? was typically about 22’. However, the measurements of the absolute value of B, are less reliable than those of B,. and B+. As a result, values of P and R were computed for the cases /? = 0 and 25” in order to establish the effects of errors in the estimates of /?. Further, there is evidence, the reliability of which is similarly questionable because of the uncertainty in the absolute values of B,, that the average value of B, remained positive during the periods in which the interplanetary field was directed back toward the Sun. This is the case in which u = 40 + 180”, fi = 25”, which is simply related to the case in which u = 40”, /3 = -25”. As a result, the latter case was also considered. It has been assumed that the waves are circularly polarized. Since B, k, and v are coplanar for fast and slow waves, the angle c$~will be used to indicate the orientation of the plane of polarization of such a wave. There is no reason a priori for the waves to have a preferred plane of polarization in a uniform system. In this case, the values of & would be determined by the properties of the source of the disturbances. In interplanetary space, non-uniformity of B,, V,,, po, andp,, may lead to asymmetries in the system which in turn
WAVE-LIKE
PHENOMENA
IN THE IN~RFLANETARY
PLASMA:
MARINER
2
961
may constrain & to certain allowed values. However, since we were approximating the actual system by a uniform one, we considered first the case in which the disturbances are produced by the superposition of a great many plane waves with random phases and with the orientations of the planes of polarization distributed uniformly. In other words, we considered the case of circularly polarized waves. Thus, in taking the average value of the squares of the variables and of the products of the different variables, we assumed that, when averaged over all the present waves, L&(%(r) sin & cos 8, = 0 {etch
sin #& = W&(0
cos 5&J= 0
i.e. we assumed no coherence on the average between ui and u,, or bi and bj, or vi and bj, i#j,k,j=x,y,z.
Under these assumptions, the values of P(v,), and P(b,), for i = I, 8,+, and the values of R(u+, bi), for i = r, 8, 4, and R(b,, b,), for i # j, and for i, j = Y,8, 4, may be computed in terms of v,, for any field orientation and propagation direction. These values are, of course, independent of the frequency f of the waves. This frequency, measured in the frame of reference moving with the plasma, is f = U/A. If we assume that all the waves originate within a finite region of space centered at the Sun, on the average, disturbances produced by waves propagating outward from the Sun will predominate. However, since a point source in a uniform field will produce a spherically spreading disturbance in the fast mode and disturbances moving aIong the field lines in both directions in the transverse and slow modes, we assume that waves moving back toward the Sun also contribute to the observed variations in v and b. The phase relationships between b and v for a wave moving in the k direction, back toward the Sun, differs by 180” from that for a wave moving outward. Thus, any contribution to b from inwardly travelling waves will reduce the coherences indicated by JR/, between v, and bi, i = r, 6, (6, but will not change the phase relationships so long as the contribution from outwardly travelling waves predominates. Accordingly, the calculated values of R are those for the outwardly travelling waves, or k, = cos ek > 0. The various cases for which P and R have been computed may be summarized as follows : Modes :
Alfvtn, fast, slow
;*
0, 15, 30, 45, 40, 75, 90” &25,0” 40”
tc. Polarization :
Circular
B,:
5.oy
PO: T:
5 mp g cm-3 (m, = 1 proton mass)
105’K
The results are listed in Table 1. For our purposes the most significant parameters listed in the table are the ratios i = r, 8,$, since these quantities are directly measurable from the power P(WW, spectra of V, and B,, i = r, 8, 4. Note that they are also independent of the signs of B. and k,. Another important aspect of these results is the indication that v7 and b, are 180” out of phase regardless of which type of wave is considered, except for fast waves when 8, > 75”. For the transverse and slow waves this result is apparent from an inspection of
PAUL J. COLEMAN,
962
JR.
equations (14 and 15), respectively. Of course, all the listed phase relationships are changed by 180” when B,, is negative. In considering the phase relationships, we have focused our attention on the variables V, and b, because the equations indicate that the phase relationship and the coherence are the least dependent upon the sign and magnitude of j3 in this case. OBSERVATIONS
We will be concerned with data obtained during four lo-day periods and two 6-day periods indicated in Table 2. These periods were selected in order to exclude the major reversals of the polarity of the interplanetary magnetic field, i.e. changes of the algebraic sign of the spiral interplanetary field that occurred twice during each period of solar rotation TABLE 2.
RATIOS OF THE POWER DENSITIES, WHERBNCBS SQUARED, AND PHASE DIF'FERBNCES OBTAINED FROM SIX SETS OF AUTO AND CROSS SPECIRA FOR THE MEASURED VARIABLES B,, Be, B+ AND v,
The power densities, P, the coherences squared, R, and the phase differences, Q, are averages taken over the frequency range l-50 cd. The quantity Rmin is the value of R required for a significant measurement of CD,i.e. for a measurement in which the r.m.s. uncertainty of @ is 45”. A (+) polarity is assigned if the average field was directed outward from the Sun; (-) polarity is assigned if it was inward toward the Sun
WWV’v~
Period
(day4
Polarity
243-252 254-263 270-279 282-291 297-302 313-318 Average Average Average
+ i+ + Both
11.8 7.8 2.3 14.4 6.6 46.1 6.9 22.8 14.5
W’i, BJ
WoYP(K) J”(BM’(I/,)
(lo-28 G2 cm-a se@ 16.5 21.2 5.3 31.7 16.4 32.3 12.7 28.4 20.6
14.5 13.4 4.2 29.3 14.6 32.3 11.1 25.0 18.1
NV,, &I
R min
Meg)
0.223 0.050 0.039 0378 0.487 0.097 0.250 0.175 0.213
0.021 0.023 0.019 0,023 0.029 0.035
181 f 7 -4 * 14 172 f 8 4*4 179 f 4 6zt8 177
(Coleman et al., 1966). Thus, during each of the six periods to be considered, the polarity of the field was nearly always that listed in Table 2. These polarity changes usually occurred at the edges of high-velocity plasma streams. Since shock waves were also observed near the edges of some of the streams that were encountered during the flight (Sonett et al., 1964; Razdan, Colburn and Sonett, 1965; Colburn and Sonett 1966), the selection of these periods of constant polarity eliminated some of the periods during which such major disturbances occurred. As mentioned previously, the variables measured by the magnetometer and plasma probe are B,, Be, B4, and V ‘u V,. In the analysis of the data discussed in Paper 1, amplitude distributions of these variables were examined first. It was found that the distributions were roughly Gaussian. As the next step in the analysis, estimates of power spectra were computed for each of these four variables. Representative composite power spectra for B,, B,, and B4 are shown in Figs. 3-5. The sampling rate of the Mariner-2 magnetometer corresponded to a Nyquist frequency, fN,of 1.35 . 1O-2 c/s or about 1100 cycles per day (cd). The lower limit of the frequency range to be investigated was arbitrarily set at 10e3 fn, or about 1.1 cd, to save computation time. A further saving of computation time was effected by dividing the frequency range 1.1-l 100 cd into two overlapping ranges, 1.1-l 10 cd and 11-1100 cd, and computating spectral estimates separately for each. Spectra estimated for the lower range of frequencies were designated LF, while those estimated for
WAVE-LIKE
PHENOMENA
IN THE INTERPLANETARY
I
PLASMA:
MARINER
2
t
MARINER COMPOSITE
2
POWER
SPECTRUM
4
AVERAGE VALUES FROM LF HR SPECTRA FOR IO-DAY PERIODS BEGINNING ON DAYS
l
% ;
16”
EI 282
AVERAGE VALUES FROM HF LR SPECTRA FOR I-DAY PERIODS BEGINNING ON DAYS
+
i3
270
\
272,274,
\
278,282
iz 283
\ \ \ lo-‘t 16’
I
11111111
I
1
Illlltl
FREQUENCY COMPOS~;~
I
IO-’
IO-’ FIG. 3.
I
(F,. POWER
I
If
IIJ
F,
1.35.I~*cPS) SPECXWJM
FOR
B,.
The spectrum at the lower frequencies is the average of the two low-frequency, high-resolution spectra for the periods from day 270 through day 279 and from day 282 through day 291. The spectrum at the higher frequencies is the average of the five spectra for days 272,274,278, 282 and 283. The resulting composite spectrum is assumed to represent an average property of the field. For comparison, lines showing l/f and l/f’ dependences are also included. For the hip-resolution spectra, Y cz 40 so the 95-per cent confidence band is 0*65-I*‘75 times the computed value. For the low-resolution spectra, v cz: 200 so the band is 0.83-1.23 times the computed value.
963
MARINER COMPOSITE
2 POWER
SPECTRUM
Se
AVERAGE VALUES FROM LF HR SPECTRA FOR IO-DAY PERIODS BEGINNING ON DAYS
270
AVERAGE VALUES FROM HF LR SPECTRA FOR I-DAY PERIODS BEGINNING ON DAYS 272,274,278,282
FREQUENCY
FIG.
4.
COMPOSITE
8 282 \ C/F’
\
& 283
{Fn POWER
= 1.35
l
IO-*
SPECTRUM
CPSf FOR
&L
(See the caption of Fig. 3.)
the higher range were designated HF. For each of the LF spectra of interest here, 100 estimates were computed. For each of the HF spectra only 20 estimates were computed, to save computation time. The lOO-estimate spectra are designated HR, for ‘“higher-resolution”, and the 20-estimate spectra are designated LR, for “lower-resolution”. For the LF-HR spectra, the number of degrees of freedom, Y, was about 40. For the HF-LR spectra, v 1zr200.
WAVE-LIKE PHENOMENA IQ’
I
11
IN THE INTERPLANETARY I
11111,
I I11111 MARINER
COMPOSITE
I
EL 282
AVERAGE VALUES FROM HF LR SPECTRA FOR I-DAY PERIODS BEGINNING ON DAYS 272,274,278,282
a 283
FIG. 5.
COMPOSITE
(See the
( F,=
MARINER
2
965
1 I1111
2 POWER
AVERAGE VALUES FROM LF HR SPECTRA FOR IO-DAY PERIODS BEGINNING ON DAYS 270
FREQUENCY
PLASMA:
SPECTRUM
1.35 . IG’CPS I
POWERSPECTRUM FOR &.
caption of Fig. 3.)
A power spectrum for the measured plasma velocity, V, or V, where it is necessary to distinguish between the theoretical and empirical velocities, is plotted in Fig. 6. This spectrum corresponds to the third period listed in Table 2. The sampling rate of the plasma probe was about one sixth that of the magnetometer. Thus, for the variable V,, only lower-frequency (LF) spectra are available. As the next step, described in Paper 1, the records were examined for correlations between the various pairs of these variables. The procedure employed was the usual one of
966
PAUL J. COLEMAN, JR.
obtaining estimates of the cross spectra of each pair of variables for several sections of the record. It was found that each pair of variables exhibited significant coherence on the average, and that the coherence was particularly strong for the pair I’,, B,. The cross spectra for this pair of variables are shown in Fig. 7 for all six periods listed in Table 2. These cross spectra are low-frequency (LF) spectra, since one of the pair of variables is V,. They are low-resolution spectra (LR) as well, since pilot estimates of the cross spectra indicated very little structure in the phase difference as a function of frequency.
LF- HR FOR
SPECTRUM
IO-DAY
BEGINNING
FIG. 6.
PERIOD ON
DAY
270
POWER SPECTRUMFOR V,.
This spectrum is the low-frequency,high-resolutionspectrum for the periods from day 270 through day 279. For the high-resolution spectra, Y N 40. Thus, the 95-per cent conjidence band for estimates of the power density is 0.65~1.75 times the computed value.
In the plots of phase difference, @, versus frequency for the pair V,, B,, a gradual change of @ with frequency is apparent, even when the coherence is relatively high. This change is an artificial one which arises due to the fact that different timing schemes were employed in recording the measured values of V, and B,. For V,, the time assigned to a particular measurement was the time at the beginning of the 222-set period during which the measurement was obtained. For B,, the time assigned to the corresponding measurement was the midpoint of the same period, or 111 set later. At 1-l cd, this time difference introduces an error of -0~51“ in @(V,, B,). The error increases linearly with frequency so that at 111 cd the error is -51”. Since the phase data in Fig. 7 include these artificial phase differences, lines drawn between the points (f = 0,
480 0.00
I
LEAD
SECMJOARY
PHASE
0.05 1 FREQUENCY
2-
(a)
PRIMARY
(Vp,Brl
(Fn”l.35~IO’CPS)
OVER
MARINE’
J
--I. o.Ia
IO+CPS,
PERlOO BEWWNO
COHERENCE
xx O.O(b-()40
025
0.50
0.75
(b>
SQUARED
(VP, 61)
254n 2820 313.
2
243 a 2700 297.
tF,,=l.35.
El
(D*Y,l962):
FREOUENCY
MARI
PERIOD SESlNNlNG lD*Y,1962>:
1.00
1.25 A.._
FIG. 7. FWASE DIFFERENCES AND COHERENCES FROM CROSS SPECTRA OF THE PAIR OF VARIABLES (v,, &). These spectra are low-frequency, low-resolution spectra. The plots on the left-hand side are those for the three periods during which the polarity of the interplanetary field was predominantly positive. On the right-hand side are those for the three periods during which the polarity was predominantly negative.
2970
t80
PAUL
968
J. COLEMAN,
JR.
These auto and cross spectra were discussed in detail in Paper 1. It was concluded that the significant coherence and the constant phase relationship constituted strong evidence for wave-like motions in the interplanetary plasma. This conclusion led to a somewhat more quantitative comparison of these empirically determined properties with the theoretically determined properties described in the preceding section. The results of this comparison will be described next.
MARINER POWER DENSITY P(Brl/P(Vr)
.-*-.
2 RATIOS
o----o
DAY 243-252
0-o +--m-t
DAY
.-+
DAY 282-291
.-v-.-.
DAY 297-302
.-.
DAY 313-318
DAY 254-263 270-279
/.-.--_.-~-
\.-I
F. MAX. *-•-+-• d_-_-__--o_._
--_._______._------o-------~~._______~_____-
T. MAX. T.MIN. F. M IN. S. MAX.
0
Cl025 FREQUENCY
0.050
F,
iFn=l.35~lb2CPS) (a)
FIG. 8.
RATIOS OF POWER DENSITIES.
The ratios of the power density in the auto spectrum of each vector field component to the power density in the auto spectrum of the radial component of the plasma velocity is plotted versus frequency for six sets of auto spectra. The theoretically allowed ranges for each ratio are also indicated on the right-hand side of the corresponding plot. Separate ranges are shown for each mode of propagation, fast (F), slow (s), and Alfven or transverse (7’).
We will consider in detail the properties of the field and plasma velocity variations in the frequency range from 1 to 50 cd. The lower limit was selected to limit the computer time required for the analysis. As mentioned previously, an upper limit of 100 cd was imposed by the sampling rate of the plasma probe. However, the consistently lower coherences indicated for (V,, B,.) in the frequency range between 50 and 100 cd prompted the selection of 50 cd as the upper limit. The auto spectra of B,, B,, B+, and V, N V,. provided estimates of the power densities for these variables as functions of the frequency. As the first step in the comparison between the measured properties of the variations in B,, B,, B4, and V, and the properties of b,, b,, b,, and v, determined from the model of idealized hydromagnetic waves, various ratios of the spectral densities were computed as functions of the frequency,f, for the six sets of data. In Fig. 8, the ratios P(B,)/P(V,), P(B,)/P(V,), and P(B,&/P(V,.) are plotted versus frequency over the frequency range l-50 cd. In the ideal case, considered in the last section, these ratios are independent of frequency, at least for cases in which the frequencies of the waves, measured in the reference frame
WAVE-LIKE PHENOMENA
IN THE INTERPLANETARY
PLASMA:
--:
I
; %
lG2’
s
-z2
2
969
-
MARINER No
MARINER
POWER DENSITY
2
O___. -0 DAY
RAT I OS
P(B&P(V,)
243-252
o-0 +-----+
DAY 234-263.
+-+
DAY 282-291.
.-m--w.
DAY
297-302
.-.
DAY
313.318.
DAY 270-273
I
1
1
1 F. MAX. T. MAX. T. MIN.
S. MAX. F
I@
I
L
0
0.025 FREQUENCY
0050
FIG.
8b.
1
I
MARINER POWER
2
DENSITY RATIOS P(B+ )/P(Vr 1
7
o----o
DAY 243-252
o-o .a----+
DAY 254-263 DAY
270-279
.-+
DAY
282-291
.----.
DAY
297-302
.-
DAY
3I3-319
_ ____._
f+cy-fq~:>-_
___~--------o-----_o________o
\-p-?~
0
l ________*___-----•-------_*________+_.----
__*__._-----*.-.____T. %. %.
I
I
FREQUENCY
*----____+
I
-_ S. MAX.
/
0.050
0.025
0
F,
(Fn=l.35~IO-2CPSl
Fn
(F.=l.35.162CPS)
FIG. 8c.
moving with the streaming velocity of the plasma, are much smaller than the proton gyro frequency which is about 0.08 c/s in a field of strength 5y. In the empirical cases, the powerdensity ratios are evidently independent of frequency, since the ratios plotted in Fig. 8 show no apparent frequency dependence. In the ideal case, the values of these power density ratios for a particular plane wave depend upon the geometry of the system and upon the field magnitude, B,, the temperature, of lB,,l. The T, and density, pO, of the gas. For Alfven waves, the values are independent 2
970
PAUL J. COLEMAN, JR.
values computed for the ideal case, as described in the preceding section, are also indicated on the right-hand side of Fig. 8. In the empirical cases, the values of the power-density ratios were generally within or close to the ranges determined for the ideal case. The variations among the empirical cases are probably due primarily to variations in lB,,l, p,,, and T. Comparison of the measured values and the theoretical values of the power-density ratios also indicates that waves in the transverse and/or fast modes were present, since the measured ratios are considerably greater than the ratios expected for slow waves, except for 19,near 90”, but are generally within the ranges of the ratios expected for the other two modes. This result does not imply that slow waves were absent but rather that AlfvCn and/or fast waves were present and, therefore, contributed most to the variations in the field. Evidence that the observed field variations could not be produced by purely transverse waves was presented in Paper 1, in which it was shown that the variations in B,,the magnitude of the vector component of the field in the direction of the ideal spiral field, were of roughly the same magnitude as the variations in the other vector components. However, to this point, the most significant results of this comparison are simply that the measured values of the power-density ratios are evidently independent of frequency and that these values are in the ranges calculated for the idealized waves using only the measured values of the mean proton density, the mean field strength, and the mean proton temperature. Having considered the properties of the auto spectra of the measured variables B,,B,, B4,and I’,., let us turn to the cross spectra of various pairs of these variables. The cross spectra of a pair of variables provides estimates of IRl, the magnitude of the square of the coherence between the two time series corresponding to the two variables, as a function of frequency, and estimates of @, the difference in the phase of the predominant coherent components of the two variables, as a function of frequency. In effecting the comparison between the empirical cases and the ideal case, the quantities (RI and a’, calculated for pairs of the measured variables were compared, respectively, with IRI and the algebraic sign of R calculated for the corresponding pair of variables for the ideal case, as described in the preceding section. Thus, for the ideal case, if R > 0, 0 = 0". If R < 0,(3= 180”. Also, for the ideal case, R is independent of frequency. Cross spectra were computed for all pairs of the measured variables for all six periods of interest. The values of JR(V,,, BJ, the square of the coherence for the pair of variables V, and B,., was greater by factors in the range from 5 to 8 than the values of IRIfor any of the other pairs, although on the average, all pairs exhibited significant coherences. Accordingly, only values of (R( V,., &)Iand @(V,.,B,), taken from the cross spectra shown in Fig. 7, are reproduced in Table 2. Values for all pairs are given in Paper 1. In most cases, the quantity JR(Vr, &)I exhibited considerable frequency dependence over the range l-100 cd, as shown in Fig. 7. However, in the frequency range l-50 cd, the average value is sufficient for our purposes. Furthermore, the quantity @(V,, B,.) is roughly independent of frequency. Thus, only average values of IRIand 0, taken over the frequency range l-50 cd, are listed in Table 2. From Table 2, it is apparent that the coherence of V,.and B,was relatively high and that these two variables were nearly 180” out of phase when the average field was directed from the Sun (Spolarity) and nearly in phase when the average field was directed toward the Sun (-polarity). The method by which this polarity was reckoned is described in Paper 1. The apparent differences in 0 of -10” to -15” from 0” to 180” is due primarily to the phase error discussed earlier.
WAVE-LIKE
PHENOMENA
IN THE INTERPLANETARY
PLASMA:
MARINER
2
971
Phase differences of 180” and 0” for positive and negative polarities, respectively, are consistent with the results of the computations in the preceding section for the ideal case in which the waves propagate primarily outward, i.e. propagate with k, > 0. As mentioned previously, we might expect the coherences to be well below the values for the ideal waves because of the effects of waves propagating back toward the Sun, i.e., propagating with k, < 0. SUMMARY
AND DISCUSSION
In the foregoing, various properties of simultaneous variations observed in the interplanetary magnetic field and in the velocity of the solar-wind plasma were compared to the properties of hydromagnetic waves as determined from an idealized model. The qualitative study of the observed variations yielded the following results over the frequency range, l-50 cd: (1) Significant coherences were usually exhibited between all pairs of the set of variables B,, &, B+, and V,. (2) The coherences generally varied with frequency. For the cases in which the values were consistently significant, the coherences decreased with increasing frequency. (3) The phase differences of the predominant coherent components in each pair of variables were roughly independent of frequency. (4) The ratios of the power densities for each pair of variables were similarly independent of frequency. Comparison with the properties of the idealized waves showed that the first, third, and fourth properties listed above would be expected for any complex group of waves as long as the amplitudes of the waves in one mode relative to the amplitudes of the waves in each of the other two modes are roughly the same at any of the frequencies studied. The second property listed above would be expected if; for example, the waves that were encountered were to include two such groups of waves, each with a different power spectrum and a different phase relationship. The quantitative study of the observed variations yielded the following results : (5) Most of the measured values of the power-density ratios, P(B,)/P(V,), i = r, 8, 4, were in the ranges of the values that were calculated for the ideal waves. The measured values were generally close to the maxima of these calculated ranges. (6) The measured values of the power-density ratios, given in Table 2, show that P(B,)/P(V,) > P(B&‘(V,J > P(B,)/P(V,). The results calculated for the ideal waves, listed in Table 1, show that this relationship would be expected for all Alfvtn waves and for fast waves with ek < 45”. (7) The phase differences of VP and B,, as measured by their cross spectra, were either 0” or 180”. The value of 0” was observed when the spiral interplanetary field was directed toward the Sun. The value of 180” was observed when the field was directed away from the Sun. (8) The measured values of the coherences, listed in Table 2, were found to be considerably smaller than the values calculated for the ideal waves and listed in Table 1. The fifth and sixth properties listed above would be expected for AlfvCn and/or fast waves. The seventh property would be expected if waves propagating outwardly along the field predominated. The two groups, mentioned in connection with the frequency dependence of the coherence, may be inward and outward-bound waves with somewhat different
972
PAUL J. COLEMAN,
JR.
power spectra. In this case, the properties of this frequency dependence would indicate that the outward-bound waves were somewhat less predominant at the higher frequencies. The presence of two such groups, i.e. inward and outward-bound waves, would also account for the eighth property listed above. We remarked above that fast waves and/or Alfven waves would produce the measured power-density ratios. In Paper 1, it was shown that AlfvCn waves alone could not produce the observed variations because the variations along B, were of about the same magnitudes as those in the transverse directions. Thus, it would appear that the field variations were produced primarily either by fast waves and AlfvCn waves or by fast waves alone. These results do not indicate whether Alfvtn waves were definitely present. Further, they do not indicate whether slow waves were present, but undetected due to the presence of waves in one or both of the other modes. Since the fast and slow modes are coupled in the ideal case, and since the AlfvCn mode is not likely to be strictly decoupled in the interplanetary medium, it is probable that all three types of waves were encountered. It should be mentioned also that the magnetic field measurements suggests that qualitatively similar variations occur in the range of frequencies up to at least 500 cd. These field measurements provided information in the frequency range O-1000 cd. The data indicate that the properties of the field variations in the range 50-500 cd are the same as those in the region l-50 cd, except that the absolute power densities decrease with increasing frequency, reaching the noise level of the magnetometer at about 500 cd. The auto spectra of the field variables were described in Paper 1. At frequencies near 500 cd, the spectra, in general, showed a frequency dependence somewhat weaker than f-2.Of course, a cutoff would be expected near the proton cyclotron frequency of 0.08 c/s. For AlfvCn waves, this cutoff could appear in our computed spectra at frequencies up to about 1 c/s due to Doppler shifts (compare equations 8 and Sa). There is some evidence from OGO-1 (Holzer, McLeod and Smith, 1966) that a steepening of the spectra of the field variables to a dependence stronger than f-” occurs near 0.2 c/s. There are several likely sources for waves in the interplanetary medium. It has been suggested that the Sun’s chromosphere and corona result from the convective motion in the granulation zone of the photosphere (Leighton, 1963). Acoustic waves generated by this motion are believed to travel upward or outward, through the atmosphere, transporting sufficient energy to provide heating of the upper layers. There is evidence that the periods of these waves are about 6 min, which is roughly the mean lifetime of the granulation pattern. The preferred spacing of the cells in the pattern is about 2000 km. There is another pattern or convective flow in the Sun which corresponds to the more The coarser supergranulation is physically similar to recently detected supergranulation. the finer granulation, but behaves independently. The mean lifetime of the supergranulation pattern is evidently about 10 hr. The preferred spacing of these larger cells is about 35,000 km. Waves produced by these two patterns of convective motion may produce disturbances in the interplanetary medium. In this case, we might expect variations which would be modulated with periods of about 10 hr and less, with the shorter periods due to harmonics, and variations with periods of 6 min and less. The Sun’s rotation would, of course, complicate the frequency spectrum that would be observed at a fixed point. Thus, from the waves corresponding to the granulation pattern, one would expect components with periods of 17 min and shorter and from the waves corresponding to the supergranulation pattern, one would expect components with periods of 4 hr and shorter.
WAVE-LIKE
PHENOMENA
IN THE INTERPLANETARY
PLASMA:
MARINER
2
973
In a discussion of interplanetary dynamical processes, Parker (1963) described several other likely sources for disturbances in the interplanetary medium. Interactions between streams of solar plasma travelling at different velocities will occur since the coronal temperature varies with time and position. Such interactions may produce disturbances travelling in both directions from the boundaries between the streams. Further, several types of instabilities are predicted for the case in which the solar wind is composed of many streams in each of which the plasma is in a different state. In addition, the solar-wind model of Parker (1958) for the coronal expansion requires that the outwardly flowing plasma expand anisotropically. In the absence of collisions in the plasma, this is expected to generate plasma instabilities that could result in wave-like phenomena. Acknowledgements-1 wish to express my appreciation to M. M. Neugebauer and C. W. Snyder for permission to use data from the Mariner-2 plasma probe; to L. Davis, Jr., C. P. Sonett, and E. J. Smith with whom I collaborated on the magnetometer experiment, and to E. A. Kraut who has been kind enough to discuss with me many aspects of this analysis. The work was supported by the National Aeronautics and Space Administration under research grant NsG 249. REFERENCES ALFVBN,H. 1950. Cosmical Electrodynamics. Clarendon Press, Oxford. COLBURN,D. S.and SONET~,C. P. 1966. Space Sci. Rev. 5,439-506. COLEMAN,P. J.,Jr.1966a. J.geophys. Res. 71, 5509. COLEMAN, P. J.,Jr.1966b.Phys. Rev. Lett. 17,207-211. COLEMAN,~. J.,J~.,DAvIs,L.,J~., SMITH,E.J.and Jo~m,D.E.1966. J.geophys. Res. 71,2831-2839. HOLZER,R. E., MCLEOD, M. G. and SMITH,E. J. 1966. J.geophys. Res. 71,1481-1486. LEIGHTON,R. B. 1963. A. Rev. Astr. & Astrophys. 1, 19-40. NEIJGEBAUER, M. M. and SNYDER,~.W. 1962.Science 13&l-2. NEUGEBAUER,M. M. and SNYDER,~.W. 1966.Mariner-2 observations ofthe solar wind,Part1. Average properties, Preprint, Jet Propulsion Laboratory, Pasadena, California, April. PARKER,E. N. 1958.Astrophys. J. 128,664-676. PARKER,E. N. 1963. Interplanetary Dynamical Processes, pp. 151-160. Interscience, New York. RAZDAN,H.,COLBI_JRN,D. S.and SONET~,C. P.1965. Planet. Space Sci. 13,1111-1123. SONETT, C. P., COLBFJRN,D.S.,DAVIS,L.,J~.,SMITH,E. J.and COLEMAN, P.J.,Jr.1964. Phys. Rev.Lett. 13,153-156. THOMPON,W. B. 1962. An Introduction to Plasma Physics, Chapter 5. Pergamon Press, New York. Pe3IoMe--PaCCMaTpElBaIoTCs HeKOTOpbE XapaKTt?pHCTElKIl B3pIlaI@t MfSKIIJIaHeTHOFO MWHHTHOrO IlOJlHI4 CKOPOCTIl lIJl33MbI,H36JIKg3eMbIe BO BpeMH noneTa Mapxaepa 2.
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