From an analysis of the incompressible MHD equations, it is concluded that the frequent observation of outward propagating Alfvenic fluctuations in the solar wind can arise from early stages of in situ turbulent evolution, and need not reflect coronal processes

Goldstein, M. L.

Matthaeus, W. H.

Montgomery, D. C.

English

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FNASA
Technical Memorandum 85073
The Turbulent Generation of
Outward Traveling Alfve'onic
Fluctuations in the Solar Wind{	 NASA- jB-85473) THE TURBULENT G ENE,tAT:ICU OF?	 OUTWARD TRAV ELING A.LFVENIC ETUC 3UATIONS INTHE SOLAR WEND (NASA)
	
16	
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HC A02/MF A01
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W. H. Matthaeus
	 oldsteinM. L. G	 G3/92 42135,
f and D. G. Montgomery
JULY 1983
National Aeronautics and
Space Administration
Goddard Space Flight Center
Greenbelt, Maryland 20771
,THE TURBULENT GENERATION OF OUTWARD TRAVELING ALFVfNIC FLUCTUATIONS
IN THE SOLAR WIND
t
William H. MatthwNus
Department of Physics,
The College of William and Mary, Williamsburg, Virgidia 23185
Melvyn L. Goldstein
Code 692, Laboratory for Extraterrestrial Physics
National Aeronautics and Space Administration
Goddard Space Flight Center, Greenbelt, Maryland 20771
i
and
i
David C. Montgomery
Department of Physics,
The College of William and Mary, Williamsburg, Virginia 23185
Abstract
From an analysis of magnetohydrodymic turbulent processes, we conclude that
the frequent observation of outward propagating A fvenic fluctuations in ttie
solar wind can arise from early stages of in situ turbulent evolution, and'
need not reflect corcnal processes as has hitherto been assumed.
PACS numbers .: 94.60.Gd, 52.30.+r
3-2-
Spacecraft observations have indicated that in the interplanetary medium a
high degree of correlation frequently exists between the fluctuating magnetic
field 5 and the fluctuating plasma fluid velocity v. These are "Alfvenie
fluctuations" with rms fluctuation levels comparable to the mean magnetic
field and have nearly aligned v and b. l ' x Finite amplitude Alfvenie fluctua-
tions (with v 2 A, b measured in units of an Alfven speed) are exact
solutions to the equations of ideal incompressible magnetohydrodynamics (MHD)
turbulence' and are candidates for preferred final states of MHD".
Though the evolution of MHD turbulence is complex, it can proceed toward
final states which can be simply described. It has been recently argued' that
MHD turbulence can increase the degree of alignment at large times and that
this process produces interplanetary Alfvenie fluctuations. Evidence in
support of "dynamic alignment" has accumulated in the form of closure
calculations $ and from numerical solutions of the MHD equations."' Under
other conditions, evidence has been presented 111 '" that MHD turbulence will
instead evolve toward a minimum-energy state, with a•decaying ratio of kinetic
to magnetic energy. It is not entirely clear whai, boundaries separate these
two regimes and this remains an unresolved issue in MHD turbulence theory.'
Here we present further evidence for dynamic alignment of v and b in two-
dimensional MHD turbulence from which we see that interplanetary fluctuations
need not represent time asymptotic states. Rather, we argue that inter-
planetary Alfvenie fluctuations are generated in situ in the early stage of
dynamic alignment. Our model not only provides an explJjnation for inertial
range alignment of v and .
 in tiie solar wind, but also accounts for the fact
that the sign of this correlation (relative to the direction of the mean
magnetic field) corresponds to the observed "direction of propagation": away	 I
from the sun s . We find support for the viewpoint that MHD fluctuations in the
-3—
solar wind resembli?'turbulence " 112 arising from local stirring of the medium.
There are at least two important questions involved in the generation of
Alfv6nic states: First, does dynamic alignment occur and why? Second, if a
high degree of alignment initially exists, does it inhibit further development
of turbulent processes, such as spectral transfer? (Perfectly aligned states
exhibit no spectral transfer, which has been used to suggest s ' that inter-
planetary fluctuations are not, properly speaking, evolving turbulence.)
We have investigated these questions numerically using an incompressible
MHD spectral method code " s4. The equations solved are, in a familiar
dimensionless form:
av/at + v • vv = -vp + b • vb + uv=v
ab/at + v • vb
	
b.4v + µ7 = b.	 (2)
p is the mechanical plus magnetic pressure and o •v = 0 0 •b. The magr^tie
and mechanical Reynolds numbers are 1/µ and 1/u, respectively. Periodic
boundary conditions in the x-y plane are assumed and a/az = 0. The Fourier
coefficients v(2, -t) and W ,t) are advanced from initial ( t-0) values, for
each wave number k' by integrating the truncated Fourier transformed versions
of (1) and (2). Time is scaled to the Alfvdn transit time of unit distance
corresponding to the initial magnetic field.
The degree -of„alignment is measured by the ratio of two ideal "rugged”
invariants, the cross helicity, He = 04>/2 anO the energy E = <v 2 + b'>/2
(brackets denote a spatial average). Extremal V^^lues of H c!E W /2) indicate
perfect alignment.
Four simulations are discussed hiire,_ 43- ignated A, B, C and D. Runs A, B,
r"
-4-
and C were obtained with a 128028 code with Reynolds numbers of 400. Includ-
ed wavevectors have integer components ranging from 1 to 64 and their negative
counterparts. At ta0, all non -vanishing Fourier amplitudes were confined to a
circular annulus 9 < k s
 < 25 and had a fixed kinetic energy, twice as much	
4
magnetic energy, and phases chosen as follows: The phase of b(;,0) was chosen
randomly and a phase angle 8 was chosen between v ( ;,0) and b(k,0). This
specifies the modal cross helicity, H c (k) = Re v*(k)•b(k). 	 Run A is
initially highly aligned with arccos ( 0.95) = e and He/E = 0.45. Run B is only
moderately aligned with arcc03( . 3) = e and He/E = 0.14. Run C has the phases
of 9 and v both randomly chosen so that H e
 = 0. Initially run D; (64x64 grid;
Reynolds numbers 400) consisted of the homogeneous turbulence used in A, B,
and C superimposed on a large scale shear flow and a correlated large scale b.
At t=0, He/E_ -0.4. Time steps were either 1/256 or 1/512. Other runs, not
described here, displayed similar results.
Direction averaged modal energy spectra for runs A and C are shown in Fig.
	 i
1, both at t_0.19. Initial energy spectra in both runs were identical.
Spectral transfer out of the initially excited annulus is not noticeably
inhibited by the high degree of alignment in run A relative to what it is in
run C. The nearly power-law inertial range in run A occurs at a lower level
than for the He
 N 0 case C, indicating that initial alignment can at most slow	 i
transfer, but cannot prevent turbulence from occurring.
Fig. 2 illustrates the temporal development of dynamic alignment. The
ratio IHa/El for runs A and B increases nearly monotonically, in agreement
with previously reported re3ult3.4'7'6
	
II
A physical explanation for dynamic alignment is suggested by rewriting (1)
and (2) in the Elsas er variables ± o v ± b. We focus on the low-k region,
	
r.^^
neglecting dissipative terms and combine egs. (1) and (2) into az
±
/at =
-3-
i+•yzt - vp. p is determined by using v •z± 0 to yield a Poisson equation.
COnsider now the case 12+ 1 >> 11-1. Note that z+ is nearly indepeofdent of
time. mince pp is of order z-•vz+ , the fractional rate Jaz+/ atl /110+ 1 goes to
zero as z- -o 0. However, the fractional rate 13z 	 z-) remains of 0(1) as
z + 0. The spectrum of z+ is effectively frozen, but that of z continues
co evolve and to squire higher wave number components. In fact, at every
time step the z- spectrum will spread to all additive combinations of wave
numbers that comprised those present in 10+ (0) and z- at the previous time
step. The transfer of the z- spectrum to higher wave numbers will continue at
a fractional rate that remains finite even as z- becomes small. , Eventually,
z will be dissipated as it reaches the higher k values, (.,chancing the
inequality 11+ 1 » Iz I. We believe this mechanism, by wh,ch the majority
species cannibalizes the minority one by sending it to high wave numbers to be
dissipated, is the essential mechanism involved in dynamic alignment.
The simulations reflect this qualitative behavior: Fig. 3 shows the
ti
normali,ze,d direction-averaged cross helicity spectrum 2H c (k)/E(k) vs. k at an
early stage of runs A, B and D. The transfer of excitation to k's higher than
those initially present consists almost entirely of cross helicity contribu-
tions of opposite sign tr) those in the energy-containing low-k part of the
spectrum. For runs A and B, z-
 is transferred to toward the dissipation
range, as is z+ for run D.
This effect apparently dominates; for times of the order of an Alfven
transit time or an 't eddy turnover time." Subsequently, the simulations
indicate that the normalized cross helicity spectra undergo a further
systematic evolution as illustrated for run A in Fig,, 4. The inertial range
gradually loses its one-signed character as the majority species slowly moves
out of the low k's. For highly correlated runs (such as A), minority species
}
domination of the highest computed Ws persists for very long times (beyond
11f=20, not shown).
In runs with less aligned initial correlation such as B, behavior similar
to that of in Fig. 4 is observed, but with less pront)unced minority sP^cies
domination of high k's ( not shown). The high k part of the spectrum can
become more or less equally populated with z+ and z- . Because z+ and i- have
equal dissipation rates, dyrlamic alignment continues, leaving an excess of
majority species in the energy containing scales. In no case have we seen
broad -band domination of the inertial range by the majority species before
very long times D 20). However, the early time appearance of an inertial
range populated by the minority species and the very slow spread of the
majority species from energy containing to hiar Jr k are features common to all
our simulations.
These results suggest a mod-1 for the turbulent production of interplaneta-
ry Alfvenic fluctuations. Consider a volume of solar wind plasma containing a
mean magnetic field lying entirely in a single magnetic sector. is
 The large
scale velocity field is radially outward, so that the sign of the radial
component of the mean magnetic field determines the sign of the large scale
contribution to He = Ha ( large). Ha (large) is generally nonzero in the
vicinity of high speed streams (where Alfvenic fluctuations are usually
observed) even in the reference frame moving with the mean solar wind
velocity. Now assume that inertial range fluctuations are due, at least in
part, to nonlinear couplings which drain energy in the large scale fields and
pump the small scale fluctuations. If the energy transfer produces dynamic
alignment consistent with the results described above, then at "early times"
the inertial range of interplanetary fluctuations would contain a sign c:.
cross helicity opposite to that present in the large scale energy containing
c
r
f
y
1
4-7-
structurea; i.e., the cross helicity of the inertial range fluctuations, say
Hc (inertial), would have a sign corresponding to outward propagation.
This model is consistent with spectral analysis of highly Alfvenic periods
which often shows that long wavelength fluctuations have Hc (k) opposite in
sign to the inertial range fluctuations." These "inward ,propagating" long
wavelength waves would be the transition in the spectrum between the large
scale energy containing structures which are driving the turbulence and the
inertial range fluctuations which are being driven.
Durir,ig the later stage, as the turbulence gets "older", the inertial range
may become equally populated with both x+ and z r;s envisioned i,n Ref. 5. One
might anticipate that interplanetary plasma which has spent a longer time
being "stirred" is less 'likely to be Alfvenic. One reported observation; at 5
AU behind a high speed stream is possible evidence of this.. In that spectrum,
the cross helicity in the inertial range oscillated in sign and was generally
not close to its maximal values (Ref. 11, Fig. 11), indicating weak inward and
outward propagating Alfvenic fluctuations.
Another possibility, $ is that dynamic alignment enhances a seed correla-
tion of v and t so that by 1 AU highly Alfv gnic fluctuations would be seen.
In contrast to our model, the majority z-field would represent the Alfvenic
fluctuations. One difficulty with this model is the limited time available
for turbulent evolution as the solar wind flows between the sun and 1 AU. A
substantial fractional increase of the 'v—'b correlation requires tens of Alfven
transit times (cf. Fig. 2). For nominal interplanetary conditions at 1 AU,
the energy containing scale is approximately 10 12 cm ' 'L6 and the Alfven
speed is approximately one tenth the flow speed of about 4 x 10 7 cm/s. Thus
only a few Alfven transit times elapse for each AU of mean plasma motion and
it is unlikely that there is time for a substantial degree of dynamic
I
A
i
1 g!
-8-
alignment to occur by 1 AU. Furthermore, an additional mechanism would be
required to account for the outward traveling sense of the v-b correlation.
Our model naturally generates outward propagating fluctuations. The early
stages of MHD turbulence produce an inertial range with fluctuating velocity-
magnetic field correlation opposite in sign to the correlation of the large
scale velocity and magnetic fields which di--ive the turbulence. This can
account for the widespread occurrence of Alfvenic fluctuations in the solar
wind. The analytical model presented is essentially unchanged by the addition
,a a mean magnetic field and is applicable to any dimensionality. Simulations
in two dimensions have shown the mechanism to act starting from a variety of
initial conditions.
We acknowledge early and helpful discussions with U. Frisch and the
hospitality of'the NASA/Goddard Space Fright Center Laboratory for Extra-
terrestrial Physics to WHM. This research was supported in part by NASA Grant
NSG-7416 and DOE Contract DE-A505-76ET53045 at William and Maryand and by a
NASA Solar Terrestrial Theory grant to the Goddard Space FlightCenter.
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1f
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Figure Captions
Figure 1. Modal total energy spectra at t s 0.19 for runs C (uncorrelated
initial fields) and A (highly correlated initial fialdt-4. The inital
spectra were identical except for-the values of H c (k). Spectral transfer
occurs in both cases.
Figure 2. The ratio He/E vs. time for runs A and B, showing the near monotone
increase in the degree of alignment of v and 5 due to turbulence.
Figure 3. Normalized cross helicity spectrum for runs A and B at t _ 0.19 and
D at t a 0.4. Data in Figs. 3 and 4 were smoothed by a 9 point moving
average before plotting. In all three cases, the transfer of cross
helicity to high k is almost exclusively opposite in sign to that of the
low k t_0 state.
Figure 4. (a) Run A at t = 0.98 when transfer of the initially dominant
z+
-mode into the inertial range begins resulting in a H. ( k) mixed in sign.
(b) Run A in a later stage (t m T.$) where the z+ ' field dominates the
cross helicity spectrum except at the highest k1s.
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The turbulent generation of outward traveling Alfvenic fluctuations in the solar wind

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