A Study of Shock-Associated Magnetohydrodynamic

Waves in the Solar Wind

Final Report on NASA Grant NAGW-1594

Steven R. Spangler

Department of Physics and Astronomy

The University of Iowa

Iowa City, Iowa 52242-1479

June 1992

(NASA-CR-190567) A STUDY OF N92-29631

SHOCK-ASSOCIATED MAGNETOHYDRODYNAMIC WAVES

IN THE SOLAR WIND Final Report (Iowa Univ.)

12

 P Unclas

G3/92 0109248
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I. Introduction

The above-named grant was used to further investigate the properties of plasma waves near

the Earth's bow shock and to apply such knowledge to plasma turbulence in the interstellar

medium. The principal reason for choosing this region of space is that collisionless shock

waves create unstable ion and electron particle distributions. These unstable distributions

create large-amplitude hydromagnetic waves. By making observations in this part of space,

we are therefore close to the source of these waves. An associated argument in favor of this

part of space is the extensive set of data from the ISEE spacecraft, which yields plasma

measurements such as vector magnetic field measurements, plasma densities, and particle

distributions. The availability of these data makes for a particularly fruitful union of theory

and observations.

Three major topics were addressed-by. the proposal, one theoretical and two observational. These topics were: (a) an attempt to understand the evolution of the largeamplitude magnetohydrodynamic (MHD) waves in the foreshock, using a nonlinear wave

equation^called the Derivative Nonlinear Schrodinger equation (henceforth DNLS) as a

model, (,b) using the extensive set of ISEE data to test for the presence of various nonlinear wave processes which might be present, and (c) a study of plasma turbulence in the

interstellar medium which might be physically similar to that in the solar wind. For these

investigations we used radioastronomical techniques. Good progress was made in each of

these areas and a separate discussion of each is given .below.

II. The Theory of Nonlinear Alfven Waves with

Application to the Earth's Foreshock Waves

I feel that very good progress was made during the grant period in the understanding of

the nonlinear evolution of waves in the Earth's foreshock. To effectively communicate the

significance of these results, a little review of the problem is in order. Magnetohydrodynamic waves are generated at shallow distances in the Earth's foreshock by unstable ion

distributions. They are rapidly amplified to large amplitudes so that nonlinear processes

should be important. At shallow depths in the foreshock, these waves remain relatively

monochromatic and circularly polarized. A spacecraft further back in the foreshock, at a

point where the waves have had a chance to evolve somewhat, sees different characteristics.

Waves deep in the foreshock show a number of differences with the monochromatic,

circularly polarized wave at shallow foreshock depths. The polarization is no longer circular,

the wave components show evidence of steepening, and there is the formation of highfrequency wave packets called shocklets in the time series.

In my opinion, the main goal of the theory of nonlinear magnetohydrodynamic waves

should be to understand these characteristics, and by understand I mean the nomination

of physical processes which are responsible for them, and not just their reproduction in

computer simulations. The development of such an understanding has been the primary

goal of my research program since 1985. The mathematical tool for this research program

has been the Derivative Nonlinear Schrodinger Equation, a nonlinear wave equation for the

magnetic field in a magnetohydrodynamic wave. This equation is
In equation (1) the complex magnetic wave field is given by <f> = bv + ibz, x is the direction

of (one-dimensional) wave propagation and a and /j are the coefficients of nonlinearity and

dispersion. If a two-fluid theory is used to derive (1), a = l/[8jr/>oVyi(l — /?)]. The coefficient

p, is positive for RCP waves and negative for LCP waves. The magnitude of \i = V^/flj.

The effect of wave growth and damping can be approximated in an ad hoc manner by

appending a linear growth and damping term to the right-hand side of equation (1), as was

done by Spangler (1986).

Equation (1), with or without terms to approximate wave growth and damping, has

been used in previous papers to model the foreshock waves (Spangler, Sheerin, and Payne,

Phys. Fluids 28, 104, 1985; Spangler, Astrophys. J. 299, 122, 1985; Spangler, Phys. Fluids

29, 2535, 1986). At the time of the last of these papers, I felt that we had developed a

good understanding of how shocklets and wave steepening might arise. My interpretation

was that the modulational instability of circularly polarized, parallel-propagating waves was

responsible. In a modulational instability, an initial amplitude modulation of a wave train

will tend to grow with time. The results of the above papers, obtained using numerical

solutions to the DNLS, showed that for waves with initial spatial modulation, as expected

for real waves in nature, the modulational instability caused wave steepening and the formation of contracted wave pulses. Mathematically, these pulses correspond to solitons, and

I identified them with the shocklets.

Within a short period of time, however, it became clear that there was a major theory

derivation problem with this interpretation. The modulational instability occurs only when

the product of nonlinear and dispersive coefficients a/i < 0. In the case of right-hand

polarized waves (p, > 0), which the foreshock waves are observed to be, this would indicate

that wave steepening and shocklet formation occurs only when a < 0. In the context

of a fluid described above, this would require ft > 1. While this condition is somewhat

restrictive, it applies at least some of the time in the solar wind.

A real problem emerged from work proposed in my 1988 proposal and carried out

under this grant. This paper (Spangler, Phys. Fluids B 2, 407, 1990) showed that a kinetic

theory derivation of the DNLS resulted in a nonlinearity coefficient a which varied rather

little with the plasma /?. The sign reversal necessary for the modulational instability to

"go" occurred only for extreme ratios of the electron-to-ion temperature. Taken at face

value, this would indicate that the evolution of upstream waves cannot be interpreted in

terms of modulationally unstable wave packets and casts doubt on the utility of the DNLS

as a model for such phenomena.

There are two possible interpretations of these developments. The first is that the

DNLS does not have an adequate physics content to describe these types of waves. The

second possibility is that there may be additional physical phenomena, absent from the

analyses discussed above but able to be incorporated in a modified version of the DNLS,

which could account for the observations. My viewpoint during the period of this grant was

the latter.

During the grant period, I therefore considered the types of physical processes which

are important in the foreshock region, but not contained in the DNLS as written in equation (1). I came up with three such processes, and a discussion of their import for solar wind

magnetohydrodynamic waves was discussed in Spangler (1992; Proceedings of the Seventh

Solar Wind Conference, Goslar, Germany, in press).
1. Wave obliquity. The foreshock waves are observed to be propagating at slight angles

with respect to the static magnetic field. These angles are quite small, being of order

5° to 20° generally. Such small propagation angles motivated the approximation of

parallel wave propagation which was used in my earlier work. However, a result of

work carried out during the grant period showed that such small propagation angles

can be quite important. This is demonstrated in a paper produced during the period

of NASA support (Spangler and Plapp, Phys. Fluids B, in press, Sept. 1992). We

pointed out that for obliquely propagating waves, the wave equations for bv and

bz contain Korteweg-deVries type nonlinearities which vanish in the limit of purely

parallel-propagating waves. Since the Korteweg-deVries equation is well known to

produce steepening of wave trains, we felt that this result gave considerable insight

into the nature of nonlinear wave evolution in the foreshock.

2. Wave growth and damping. Equation (1) conserves wave energy; wave growth and

damping processes are not included. The real waves in space are being amplified due

to resonant ion instabilities, and damped due to various linear and nonlinear mechanisms. As discussed in Spangler (1992), there are qualitative arguments for believing

that wave growth and damping might facilitate the processes of wave steepening and

shocklet formation, even when modulationally stable solutions to (1) produced no

such effects.

3. Beam modification of the wave dispersion relation. The prediction of modulationally stable wave packet evolution in (1) is based on the dispersion relation for

MHD waves in a fluid or Maxwellian plasma. In this case, for right-hand polarized

waves (d^u/dk2) > 0, where u is the wave frequency and k is the wavenumber. This

is illustrated in Figure 1 as "normal dispersion, stable." The dispersion coefficient /i

is nothing other than the sign of d2u>/dk2. For a > 0 modulational instability and

thus wave steepening cannot occur.

However, large-amplitude MHD waves near shocks in the solar system do not occur

in an unperturbed fluid or Maxwellian plasma. They are found in the presence of the dense

unstable beams that generated them. These beams of unstable ions typically have a number

density of one to several percent of the background density. As has been pointed out by

Peter Gary and others (e.g., Gary et a/., Phys. Fluids 27, 1852, 1984) such dense beams

can produce a major modification to the wave dispersion relation. This is illustrated in

Figure 1 by the line indicated as "beam-modified dispersion," and is a stylized version of

the results presented in Gary et al. (1984). As may be seen, the beam-modified dispersion

relation includes a range of wavenumber in which (d^u/dk2) < 0. While a rigorous study

of modulational instability including the effects of beam modification has not been carried

out, the heuristic ideas mentioned above suggest that modulational instability and wave

steepening might be expected.

The first two of these possibilities were quantitatively and rigorously examined during

the course of the grant period. The computer code for solving the DNLS subject to arbitrary initial conditions used in the investigations of Spangler, Sheerin, and Payne (1985),

Spangler (1985), and Spangler (1986) was modified to simultaneously investigate the effects

of oblique propagation and growth and damping. In Figure 2 we present the results of a
(Jj

NORMAL

DISPERSION,

STABLE

BEAM-MODIFIED

DISPERSION

Figure 1. Stylized representation of the effect of an unstable ion beam on the dispersion

relation u(k) for magnetohydrodynamic waves. The heavy solid line gives the dispersion relation for right-hand polarized waves in a fluid plasma. The dashed line

gives the imaginary part of the dispersion relation (growth rate) given an unstable ion

distribution. The light solid line shows the corresponding real part of the dispersion

relation in this case. It shows substantial differences with the fluid dispersion relation,

including an apparent reversal in the sign of d2u/dk2 near the wavenumber where

is a maximum.

sample calculation. The upper row shows the initial conditions in wave intensity &J + 6|,

y component of the wave field, and spatial power spectrum. The wave packet then evolved

subject to oblique propagation at an angle (typical for the real waves) of 11.5°, and growth

at the primary wavelength of 7max = O.OOSftf

 l

. The state of the wave train at a time of

lOOfi,- is shown in the bottom row of Figure 2.

The remarkable feature of this plot is that it shows essentially all of the major observed

features of upstream waves, such as steepening of the wave packet (most clearly visible in

the plot of wave intensity), shocklet formation (plot of 6y), polarization evolution (clearly

apparent in a comparison of the plots of by and bz), and spectral transfer of wave energy to

higher wavenumbers (plot of power spectrum).

The conclusion to be drawn from this and other calculations is that the DNLS, despite

its somewhat limited physics content, seems capable of describing the main observed characteristics of upstream waves. Use of the DNLS also allows identification of the physical

processes responsible for this evolution, which are (a) the DNLS nonlinearity, which causes

a spectral transfer of wave energy, (b) obliquity of propagation, which causes polarization
0.12 0.62 1.12

MBMENUMBER

Figure 2. Evolution of a spatially modulated, obliquely propagating wave packet. Upper

row shows initial condition, bottom row shows wave field at t = 100/ftj. The columns

show, from left to right, respectively, wave intensity, one component of the wave field,

and the spatial power spectrum. The units of wave intensity and field are BQ and BQ,

respectively, while the units of the power spectrum are arbitrary.

evolution and, through the Korteweg-deVries nonlinearities, a tendency towards wave steepening, and (c) wave growth, which via linear processes keeps the wave amplitude large so

that effective spectral transfer can occur, thus offsetting the demodulation and reduction

in wave amplitude which occurs in the absence of growth and damping.

The status of publication of these results is as follows. A preliminary report of all

this material was presented as an invited talk at the Solar Wind 7 Conference and will

appear in the published conference proceedings. As mentioned above, a paper discussing

the properties of obliquely propagating nonlinear Alfven waves is in press in the Physics

of Fluids B. Finally, in the course of the next few months, I plan to write a paper for the

Journal of Geophysical Research, which will be an expanded version of the Solar Wind 7

Conference proceedings paper.

III. Observational Studies of Large-Amplitude

MHD Waves in the Earth's Foreshock

One of the major areas of research supported by grant NAGW-1594 was an extensive

investigation of wave observations made by the ISEE spacecraft when it was in the Earth's

ion foreshock. This project has been intended as a major expansion of the study of Spangler

et a/., J. Geophys. Res. 93, 845, 1988. Like Spangler et al. (1988), a major emphasis is to
deduce the presence of various linear and nonlinear wave mechanisms through the presence

of density compressions, but the investigation is far more general and has discovered a

number of properties of these waves which were hitherto unnoticed. This work comprised

the Ph.D. thesis research of Dr. James Leckband, who successfully defended his thesis in

April 1992.

The data consist of 20 "episodes" chosen as times when the ISEE spacecraft was in

the Earth's foreshock, when large-amplitude wave activity was visible on survey plots, and

when all wave and plasma characteristics appeared to be time-stationary for periods of at

least 30 minutes. Wave data consisted of magnetometer data with temporal resolution of

4 seconds, plasma density data with a resolution of 4 (high data rate mode) or 12 (low rate

mode) seconds, plus information on average electron and ion temperature for each period.

An example of a data interval is shown in Figure 3. The top three panels show the three

components of the wave magnetic field, and the bottom panel shows the plasma density.

The x component (top panel) is the variable magnetic field in the direction of the static

magnetic field, while the y and z components comprise the transverse wave field. In the case

of a parallel-propagating plane wave, all fluctuations would be in the y and z components.

From this figure one can readily see that the properties of the magnetic fluctuations

in the y and z components are quite different from those in the x components. From the

bottom panel, one can see also that these waves are quite compressive, with a typical density

modulation index = an/n of order 15%. One of the principal goals of this research program

has been to understand the mechanisms responsible for these density fluctuations.

The work carried out in this program has consisted of computing and studying autospectra, cross spectra, and cross correlations of the data series shown in Figure 3, e.g.,

cross correlation and cross spectra of by with n. This analysis has yielded a host of interesting results which will be published in the papers resulting from Dr. Leckband's Ph.D.

thesis. A brief synopsis of some of these results is given below.

• Upstream solar wind magnetic field and density data from IMP 8 were used to estimate

the "pristine" solar wind fluctuations in these quantities. These calculations allowed

us to calculate, for each of the 20 episodes, the frequency and thus length scale at

which foreshock turbulence began to dominate that of the undisturbed solar wind.

This typically occurred at or slightly below a frequency of 0.01 Hz, corresponding to

a wavelength of about 40 thousand kilometers.

• Power spectra of the transverse magnetic field components typically show a welldefined spectral maximum corresponding to the dominant frequency of the waves. In

a new result, we found that the fall off of spectral power with frequency above this peak

is almost always power law with a spectral index of about 2.5. This is considerably

steeper than the value of 5/3 corresponding to the one-dimensional reduction of a

Kolmogorov spectrum.

• A careful study was made of the dependence on frequency of the angle of propagation

with respect to the static magnetic field, &kB- This investigation also featured a

rigorous consideration of the errors involved in these propagation angles. It was found

that these angles typically ranged from about 5° to 15°. This information is directly

relevant to the theory presented in Section n.
seii

I «•

I i £ '<f

1°

1 -: I I

§

I

IJ

•5*2

01 9
• The principal finding of this study has been the existence of a very strong correlation between 6n, the fluctuation in density, and bx. This correlation was strong and

present in each of the episodes, with correlation coefficients ranging from about 40 to

70 percent.

• From study of power spectra and the angle of wave propagation 0kB, we believe that

there are two classes of waves in the foreshock MHD turbulence. Typically, the "wave

band" magnetic field fluctuations from about 0.014 Hz to 0.030 Hz are propagating

in a more or less parallel manner. Typical propagation angles are 5°-20°. However,

at low frequencies between about 3 and 10 mHz, where the foreshock turbulence still

dominates over the pristine solar wind, the propagation angles are consistent with

nearly perpendicular propagation. We thus identify this second class of wave as low

frequency (at frequencies perhaps one-third or one-fourth of the wave band frequency)

and of perpendicular propagation.

• Dr. Leckband interprets these low-frequency waves as the result of a decay or filamentation instability in which parallel-propagating MHD waves decay, producing

perpendicular-propagating daughter magnetosonic waves. This theory has recently

been presented in a paper by Vifias and Goldstein (J. Plasma Phys. 46, 107, 1991).

Dr. Leckband finds agreement between the frequencies predicted by the VinasGoldstein theory and the observed second class of foreshock waves.

• Using a novel method of data display for MHD waves, i.e., a dynamic spectrum,

method, we have found cases in which there is slight evidence for the presence of the

three-wave decay instability in which an initial parallel-propagating MHD wave decays

into a forward-propagating ion-acoustic wave and a backward-propagating MHD wave.

We believe these results are of considerable value in obtaining a better understanding

of the foreshock waves. These results have been presented at a number of AGU meetings,

and as a poster at the Solar Wind 7 meeting. In the coming months we will be writing a

major paper for the Journal of Geophysical Research using the results of this thesis.

IV. A Radioastronomical Study of MHD Turbulence

in the Interstellar Medium

One of the benefits of research in space plasma physics is that it can help us understand

plasma processes in more remote astrophysical plasmas, in which important phenomena

such as the acceleration of cosmic rays and formation of stars maybe occurring.

This viewpoint, adopted in my research program in which theoretical and observational studies of turbulence in the solar wind, is used to provide insight into the turbulent

plasma in the interstellar medium.

The observational technique used is radio wave scintillation, in which we look for

phase and amplitude fluctuations in radio waves which have been induced by propagation

through a medium with a stochastically varying plasma density.

Considerable progress in this area was made during the period of support by grant

NAGW-1594. Some of the highlights are as follows.
10

We continued our development of a "scattering map" of the galaxy, obtaining a measurement of a quantity related to the path integral of the electron density variance.

These measurements showed that the intensity of turbulence is far greater in the inner

galaxy than at the location of the sun, indicated that there are "clumps" of intense

turbulence which are of size a few parsecs or less, and indicated that galactic regions of

intense turbulence might be associated with regions of star formation (Fey, Spangler,

and Mutel, Astrophys. J. 337, 730, 1989; Fey, Spangler, and Cordes, Astrophys. J.

372, 132, 1991).

We pointed out that previously published observations indicated that the inner scale

of the turbulence in the interstellar medium is of order 50—200km. We noted that this

is of the order of the ion-inertial length in the warm ionized phase of the interstellar

medium. This result strengthens the analogy of the interstellar medium with the solar

wind, where the inner scale is also of the order of the ion-inertial length. This result

also provides information on the phase (density and temperature) of the interstellar

medium in which the plasma turbulence resides. This result was published in Spangler

and Gwinn (Astrophys. J. 353, L29, 1990).

I continued my investigation, in a collaboration with colleagues at the Italian Institute for Radio Astronomy, of the phenomenon called low-frequency variability of

extragalactic radio sources. A new investigation was carried out utilizing data on intensity scintillations and measurements of the brightness distribution of those sources.

The results of this investigation are the most compelling information to date that

low-frequency variability is indeed an interstellar scintillation phenomenon, and furthermore indicated that such measurements must provide a means of determining

the velocity of the density irregularities responsible for interstellar scintillations. The

results of this investigation have been written up, submitted to Astron. Astrophys.,

favorably reviewed, revised and resubmitted. We expect publication in late 1992 or

early 1993.

An investigation was made of the heating of the interstellar medium by dissipation of

irregularities (plasma waves) responsible for interstellar scintillations. In contrast to

some previous investigations, I found that there need not be a thermodynamic embarrassment in which wave heating exceeds the cooling capability of the medium, provided

that the irregularities responsible for scintillations are in envelopes of H II regions,

which have a substantial cooling capability. Nonetheless, wave heating may be a significant energy input in parts of the interstellar medium. A distinctive feature of this

study was the consideration of a number of nonlinear wave dissipation mechanisms,

such as the parametric decay instability and wave steepening which have been widely

discussed in the context of solar wind turbulence, but never previously in association

with interstellar turbulence. This work is presented in Spangler (Astrophys. J. 376,

540, 1991).
11

V. Publications Resulting from NASA-Supported Research

During the Period of Grant NAGW-1594

During the grant period of grant NAGW-1594, a number of papers were produced in the

fields of the theory of nonlinear Alfven waves, observed characteristics of waves near the

Earth's foreshock, and radioastronomical studies of plasma turbulence in the interstellar

medium. These papers are indicated below.

1. A. L. FEY, S. R. SPANGLER, and R. L. MUTEL

VLBI Angular Broadening Measurements in the Cygnus Region

Astrophys. J. 337, 730-738, 1989

2. S. R. SPANGLER

Kinetic Effects on Alfven Wave Nonlinearity I: Ponderomotive

Density Fluctuations

Phys. Fluids B 1, 1738-1746, 1989

3. S. R. SPANGLER

Kinetic Effects on Alfven Wave Nonlinearity II: The Modified

Nonlinear Wave Equation

Phys. Fluids B 2, 407-418, 1990

4. S. R. SPANGLER and C. R. GWINN

Evidence for an Inner Scale to the Density Turbulence in the

Interstellar Medium

Astrophys. J. Lett. 353, L29, 1990

5. T. J. LAZIO, S. R. SPANGLER, and J. M. CORDES

Faraday Rotation Measure Variations in the Cygnus Region and the

Spectrum of Interstellar Plasma Turbulence

Astrophys. J. 363, 515, 1990

6. S. R. SPANGLER and R. J. REYNOLDS

A Comparison of H Alpha Intensity and Radio Wave Scattering

on Eight Low-Latitude Lines of Sight

Astrophys. J. 361, 116, 1990

7. A. L. FEY, S. R. SPANGLER, and J. M. CORDES

VLA and VLBI Angular Broadening Measurements: The Distribution

of Interstellar Scattering at Low Galactic Latitudes

Astrophys. J. 372, 132, 1991

8. S. R. SPANGLER

The Dissipation of MHD Turbulence Responsible for Interstellar

Scintillation and the Heating of the Interstellar Medium

Astrophys. J. 376, 540, 1991
12

9. J. M. CORDES, J. M. WEISBERG, D. A. FRAIL, S. R. SPANGLER,

and M. RYAN

The Galactic Distribution of Free Electrons

Mature 354, 121, 1991

10. S. R. SPANGLER and B. B. PLAPP

Characteristics of Obliquely Propagating Nonlinear Alfven Waves

Phys. Fluids B [in press, September 1992]

11. S. R. SPANGLER, W. A. EASTMAN, L. GREGORINI, F. MANTOVANI,

and L. PADRIELLI

Refractive Interstellar Scintillations and Low Frequency Variability:

A Detailed Analysis Using Measured Source Structures

Astron. Astrophys. [submitted]