~

Measurement of Spectral Anisotropy Using SingleSpacecraft Data

Sean Oughtonl, William H. Matthaeus2, Charles W. Smith21 Department of Mathematics, University College London, London WC1E 6BT, UK

2 Bartol Research Institute, University of Delaware, Newark, DE 19716

Abstract. A technique for analyzing multiple-time, single-point datasets is presented, which yieldsinformation about the (spectral) variation of MIlD fields in directions perpendicular to the mean flow. Theanalysis can be performed for any homogeneous solenoidal fields, including magnetic and incompressiblevelocity fluctuations. A summary of the associated theory is given, along with initial results obtained using

an interval of Voyager data.

THEORYINTRODUCTION

General forms for the correlation functions andspectra associated with homogeneous incompressibleMHD turbulence have been derived elsewhere. See(18) for a summary/review. For example, if (...)is an appropriate averaging procedure, and b(x) =B -(B) is the fluctuating magnetic field, then thecorrelation function for the magnetic fluctuations is~j(r) = (bi(x) bj(x + r»). Taking the Fourier trans-form yields the spectral tensor Sij(k), which consistsof index symmetric (Iij) and antisymmetric (Jij)pieces. The former is determined by three scalarfunctions (18) and contains, among other contribu-tions, the magnetic energy spectrum which has beenstudied extensively in the solar wind context (7, 22).The antisymmetric portion is controlled by a sin-gle scalar function H(k), proportional to the mag-netic helicity spectrum, and has the specific form

(1,15,19)

Almost all in situ observations of solar wind andmagnetospheric parameters are obtained from mea-surements using one spacecraft at a time. Associateddatasets usually consist of measurements of magneticfield B, and plasma velocity V and density n, ob-tained at regular time intervals along the spacecrafttrajectory. These measurements can be used to con-struct correlation functions and spectra, e.g., (4, 11),and the related analysis has added immensely to ourunderstanding of fluctuations at MHD scales.

Nonetheless, because data intervals are approxi-mately equivalent to simultaneous measurements atdiscrete points along a curve in space (see below),there are strong constraints on the extractable in-formation. For example, only limited informationregarding gradients in B oblique to the measure-ment curve can be obtained. Recently we have shownthat suitably defined mean wavenumbers can be cal-culated in directions perpendicular to the reduction(measurement) direction, for certain fields providedthat the associated fluctuations are solenoidal (18).

The approach is an expansion of the technique,introduced in (15), for extracting the magnetic he-licity from one-point, multiple-time datasets. Thus,even though only reduced spectra are available, it isstill possible to obtain information about the scale-lengths characterizing the fluctuations in directionstransverse to the measurement direction. In the fol-lowing section we summarize the associated theoryand apply it to several types of fluctuations believedto be important in the solar wind (16). An intervalof Voyager 2 data is then analyzed in this context.

Jij = it:ijakaH(k). (1)

Thus, if one could simultaneously sample B(x, t)at many locations throughout a given volume, then~j(r) and hence Jij(k) could be estimated. Un-fortunately, this is not achievable using data from asingle spacecraft, which is the usual situation for thesolar wind. Instead, one obtains values of B at asequence of times corresponding to positions alongthe spacecraft trajectory. However, as the mean flowspeed UR is much greater than (a) the spacecraftvelocity, and (b) typical turbulent and wave veloci-ties, one can invoke Taylor's "frozen-flow" hypothesis(8, 21) to relate the time series at a single point to a

CP471 , Solar Wind Nine, edited by S. R. Habbal, R. Esser, J. V. Hollweg, and P. A. Isenberg@) 1999 The American Institute of Physics 1-56396-865-7/99/$15.00

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spatial sampling in the flow direction at a fixed time:b(x = O,tj) == b(-UtjR,O), where R is the unitvector for the heliocentric radius. As a consequenceonly reduced spectra, rather than the more informa-tive and desirable full wavevector spectra, are avail-able (1, 5, 11). For Cartesian coordinate systemswith the x-direction aligned with the radial (reduc-tion) direction, one can show that the antisymmetriccomponent of the reduced spectral tensor is

Table 1. Mean wavenumbers for specific fluctuationsymmetries in the Cartesian coordinate system withx in the radial direction and Bo in the x-y plane.

21D2isotropic 2D slab

k2(kl)

k3(k1)

arbitrarybut equal

0

0

k1 tan t/J

0

,

J[jd(k1) = / dk2dk3 Jij(k1, k2, k3), (2)

so that contributions from all directions perpendicu-lar to the reduction one are integrated out. Substi-tuting from Eq. (1) we have .Jf~d '" Jk3H(k). Thisis a quantity that depends upon the structure of thehelicity spectrum in the z direction, but according toEq. (2) it can be evaluated from information obtainedfrom the frozen-in condition in the x direction! It istempting, qualitatively, to interpret .Jf~d as a meanvalue for k3, weighted by the helicity spectrum. Cau-tion is warranted, however, since the helicity cannotstrictly speaking be a density or weight function as itis not a positive definite quantity. Hence, we define

- (k ) = J k3H(k) dk2dk3 = k:!!~(3)k3 1 J H(k) dk2dk3 1 J2~d(k1)'

ffllf

II~i

11111~f~IWI.nil!!!

!1;; 1 ' i11 \

I

I

slab cases. For most of these symmetries we assumethe existence of a well-defined mean magnetic fieldBo = (B). This preferred direction makes some co-ordinate systems more convenient to work with, andhere we choose x == ft, with Bo in the x-y planeat an angle ~ to R. Thrther details on the full cor-relation and spet1;ral tensors connected with thesesymmetries are given in (I8), for example.

In the absence of any preferred directions one ex-pects on physical grounds that the fluctuations willbe isotropic, for which case it is easily shown thatk2 = k3. Strictly 2D fluctuations, i.e., those withwavevectors and Fourier amplitudes perpendicular toBo, have H(k) = 0, so that the kj also vanish.

The related case of 2 ~ D symmetry (k .Bo = 0but no restriction on amplitude direction) can beshown to have k3 (k1) arbitrary, but k2 (k1) = k1 cot ~independent of H(k). The situation is similar forslab (parallel propagating Alfven wave) fluctuations:k3 = 0 and k2 = k1 tan~. These results are sum-marized in Table 1. It is apparent that there is con-siderable scope for using the k's as diagnostics fordistinguishing between the symmetry states of thefluctuations present in the solar wind.

I

DATA ANALYSIS AND RESULTS

To illustrate the above technique for calculatingmean wavenumbers kj we have analyzed an intervalof Voyager 2 data from days 95-98 of 1978. At thistime the spacecraft was near 2.8 AU, and experienc-ing a reasonably stable mean magnetic field, with1/J ~ 600 and Eo ~ 24km/s in Alfven speed units.The interval is a subset of one used in (11). Thedata points consist of 96 second averages for V, B,and n, with the entire set being rotated to the abovediscussed coordinate system prior to analysis. Cor-relation functions and spectra were computed usingthe procedure described in (11), based on Blackman-Tukey mean lagged product calculations of Rij with20 degrees of freedom. While the stationarity and ho-mogeneity of solar wind fluctuations has not been rig-

and similarly k2 = k1 ~~d / 4~d.As the k are functions of the reduced wavenumber

k1 (component of k in the measurement direction),we obtain helicity-weighted mean wavenumbers ateach reduced wavenumber. One can also integrateover k1 to obtain "bulk" wavenumbers in the y andz directions, e.g., K3 = Jk3(k1) dk1, which provideestimates for the transverse lengthscales characteriz-ing the large-scale (helical) structures present in thefluctuations.

The above quantities are readily computed fromsuitable datasets and can be employed to providepreviously overlooked or neglected information onthe structure and scales of solar wind fluctuations.

In view of H(k) not being positive definite, can-cellation can occur in the averaging. This means thatright-handed and left-handed structures at nearlyequal scales give cancelling contributions, which tendto push the net "mean wavenumber" towards zero.It would perhaps be more useful to have access toJ k3IH(k)1 dk2dk3, etc, but this does not appear tobe achievable using single-spacecraft datasets (see,however, the definitions for K j ) .

Special Cases. When the turbulence has a partic-ular symmetry the kj can often be evaluated further.Here we are interested in the isotropic, 2D, 2!D, and

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FIGURE 1. Plots of the mean wavenumbers and re-lated quantities for the scalar function H(k), which gen-erates the magnetic helicity, for a Voyager 2 interval near2.8 AU. Open circles indicate data points with negativevalues, plus signs those with positive values.

correlation scale Ab === 5.0xIO6km [see (11) for defini-tions]. This suggests that the magnetic fluctuationscontain substantial transverse (helical) structure.

CONCL USIONS

orously established, empirically it appears that whenthe traced correlation function ~Q (r) is of Lanczostype [that is, for some L, RQQ(r > L)jRQQ(O) « 1]then weak homogeneity is an acceptable assumption(11, 12). For the interval considered here RQQ(r)approaches Lanczos type.

Figure 1 ( a) displays the spectrum for the magneticenergy and kl times the helicity spectrum obtainedfrom the Voyager 2 interval. The mean radial solarwind speed for the interval, 442kmjs, has been usedto convert from frequency to reduced wavenumber.The behavior shown is fairly typical of near eclip-tic fluctuations, see, for example, (6). Figure l(b)shows the ratio I = k3jk2, with the isotropic valueof unity overplotted as a dashed line. Clearly, thisresult is not compatible with isotropy of the fluctu-ations. Since the interval is characterized by a sig-nificant Bo this lack of isotropy is consistent withexpectations based on the dynamical emergence ofanisotropy in the presence of mean magnetic fields(10, 17,20). Nonetheless ~ 38% of the fluctuationshave ~ < III < 2, so that much of the departurefrom isotropy is not extreme. The plot also gives anindication of the split between 2~D and slab fluctu-ations. Pure slab fluctuations are characterized byI = 0, whereas the 2~D symmetry has I arbitrary(Table 1). Only about 1% of the If I values fall below0.01, which suggests that slab fluctuations are not adominant component in this interval [d. Fig. 2(b)].

Turning to the plots of k2 and k3 (Fig. 2), we ob-serve general increasing trends as kl increases, withthe scatter about best-fit straight lines being morepronounced for kl ~ 10-6 kIn-I, which is of order thecorrelation scale for the magnetic field (see below).The straight lines overplotted on Figure 2(a) are thescalings applicable for fluctuations which are purely2! D (dashed) and slab (solid) for the same mean fielddirection as the actual data interval. The agreementat low kl between the data and the 2~D prediction isstrikingly good, however the interpretation of thesescales (larger than the correlation scale) is not un-ambiguous, e.g., (13). At smaller scales a single puresymmetry state seems an unlikely option. As 2! Dmodes have arbitrary values for k3, it is harder todraw conclusions from Fig. 2(b), although, clearlyslab modes are not a dominant component.

The bulk wavenumbers are calculated to be K 2 ==-3.2 X 10-9 km-2, and K3 == -4.2 X 10-9km-2.Taking the square roots and converting to the associ-ated lengthscales gives, respectively, 1.1 x 105 km and1.0 x 105 kill. These values are substantially smallerthan the lengthscales associated with the kl and kimoments of H;:d, both"", 107 kill, and the magnetic

The technique presented above complements ex-isting methods (2, 9) which help discriminate be-tween various types of fluctuation and may providefurther support for the perspective that MHD-scalesolar wind fluctuations consist of (at least) two-components: quasi-2D turbulence and slab Alfvenwaves (2, 14,22), or perhaps some other mixture ofpure symmetries (3).

We have focussed here on mean wavenumbers as-sociated with the magnetic helicity spectrum. Thetheory holds, however, for the index antisymmetricpart of any homogeneous auto or cross-correlation

509

for valuable assistance, and T. Horbury, for help-ful and provocative criticisms. Research supportedin part by grants from the NSF (ATM-9713595),NASA (NAG5-3026 and NAG5-6570), JPL (959167),PPARC, and the Nuffield Foundation.

REFERENCES

10-210-3 (8)

:::-" 1 0-4'E _5~ 10

L.1.:'" 10-6

10-7 010-8 ,.. -0 ~

10- 10.4

k1 [km-1)

100~(b) '" ~ o~.~..~

10-2~ 0 0*+ I

":-E=.'"

LX

0 ..0

~~?O~-6::-;o"';.;:-6 :;:5 ~ ..~J' -' ~ + ~ .+ ..

10.8 ooocxxtP .0

10- 10. 10- 10. 10-4

k1 [km-1]

FIGURE 2. Helicity-weighted mean wavenumbers forthe 2.8AU Voyager interval. See Figure 1 (and text).

tensor, when the underlying fields are solenoidal (18).For example, mean wavenumbers can be defined forincompressible velocity fluctuations via the velocityhelicity spectrum, and for the induced electric fieldE = -(v x b) using the antisymmetric piece (Jij) ofthe "minus" correlation tensor Rij(r) = (vibj-bivj),where primes indicate evaluation at x+r rather thanx. As shown in (18), Jij is directly related to theelectric potential for E. On the assumption that so-lar wind velocity fluctuations are approximately in-compressible, one could then obtain kj for these andsimilar fields and such an investigation is in progress.

Finally, we note that there are many questions stillbe answered regarding the mean wavenumbers. Forexample, their variation with heliocentric distance,stream speed, and solar cycle. Data from the Ulyssesmission may be particularly useful in this regard asit contains numerous (long) intervals where Bo is rel-atively stable.

We are grateful to N.F. Ness for providing accessto the Voyager 2 magnetometer data, R.J. Leamon

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