Parametric instability in the solar wind: numerical study of
the nonlinear evolution
Leonardo Primavera † , Francesco Malara † and Pierluigi Veltri
†
Dipartimento di Fisica, Università della Calabria, 87036 Rende (CS), Italy
†
Istituto Nazionale per la Fisica della Materia, Unità di Cosenza, via P. Bucci, 87036 Rende (CS), Italy
Abstract. A possible mechanism to explain the decrease of the Alfvenic correlation with the distance from the sun, observed
in fast speed streams in the solar wind, is the parametric instability. Starting from a circularly polarized Alfven wave, this
instability naturally produces backward propagating waves and compressive fluctuations, by destroying the initial correlation
of the wave. However, the applicability of this mechanism to the solar wind is debatable, since it works better when the plasma
beta is much lower than 1 and the initial wave is monochromatic. To elucidate better this phenomenon, we numerically
simulated the propagation of a turbulent spectrum of Alfven waves on a uniform background magnetic field by using a
pseudo-spectral, one dimensional, MHD code. We used values of the plasma beta about one and very high values of the
physical diffusivities, by using "artificial" numerical diffusivities to ensure the stability of the numerical scheme. We found
that, even under such conditions, the initial alfvenic correlation of the waves is progressively destroyed. At the saturation of
the instability, the forward and inward propagating waves have comparable energies in the spectra, at the larger scales, while
the forward propagating fluctuations dominate the spectrum at smaller scales. The two spectra tend to approach each to the
other at subsequent times. A tentative comparison of these results with the observations of the alfven waves present in the
solar wind was carried out, with good qualitative agreement.
INTRODUCTION
The turbulence present in the high latitude solar wind
shows very striking Alfvénic characteristics, with high
correlation between velocity and magnetic field fluctuations and relatively low level of density perturbations.
However, several observations ([1, 2, 3, 4]) also showed
that the Alfvénicity of the fluctuations depends on the
distance from the sun. By using the Elsässer notation:
Z
v b ρ , the outward propagating waves can
be identified with Z modes, and the ones travelling
in the opposite direction with Z . Then we define the
pseudo-energies associated with the Elsässer variables:
1
e
Z 2 , and the normalized cross helicity of
2
the fluctuations:
e
e
σc
e
e
where the symbol:
identifies running averages on
a given time scale. Under these assumptions, the radial
evolution of the Alfvénicity in the solar wind can be summarized as follows: a) close to the sun, at a distance of
about 0.3 AU, the hourly average cross helicity σ c has a
value close to 1, meaning that the fluctuations are almost
pure Z (namely outward propagating) fluctuations; b)
with increasing distance from the sun, σ c decreases, because the pseudo-energies e both decrease; c) however,
the decrease rate for e is faster than that for e , up to a
distance of 2 5AU; d) after that distance, the two quantities lower at the same rate. This behaviour is clearly
displayed in fig. 1, where the evolution of the quantity
E e e , i.e. the ratio between the hourly averaged
energies of Z over Z , is shown. Since e decreases
faster than e , according to the point c) above, the ratio increases up to 1 5AU, then it stays approximately
on the same level, since e and e decrease in the same
way.
Moreover, the fluctuations show a strong intermittent
character, as shown from the analysis of the flatness of
the fluctuations carried out by Bruno et al.(2002), in
these proceedings ([5]). They showed that in the fast polar wind, the flatness of both velocity and magnetic field
intensities increases, so they become more and more intermittent with decreasing the length scale, and in general, the intermittency attains higher and higher values as
the distance from the sun increases.
It was conjectured by several authors that in the high
latitude wind, where the medium is more homogeneous,
the radial evolution of the quantities can be influenced by
the parametric instability ([6, 7, 8, 11]). In fact, the parametric decay process of a circularly polarized Alfvénwave (for example a Z mode) leads to the growth of
a forward propagating sound wave, and a backscattered
Alfvén wave with a correlation opposite to that of the
mother wave (a Z mode), thus producing a decrease of
CP679, Solar Wind Ten: Proceedings of the Tenth International Solar Wind Conference,
edited by M. Velli, R. Bruno, and F. Malara
© 2003 American Institute of Physics 0-7354-0148-9/03/$20.00
505
NUMERICAL MODEL
We solved numerically the fully compressible, nonlinear,
MHD equations in a one-dimensional configuration. We
used a pseudospectral numerical code, with periodicity
boundary conditions in the spatial domain x
0 2π .
Further details about the numerical technique can be
found elsewhere [7]. The code calculates the time evolution of the density, the velocity, the magnetic field and the
temperature, starting from a given initial condition. The
above quantities are normalized to characteristic values
and the unit time is the Alfvén time. We neglect viscosity and resistivity in the code, although we use hyperviscosity and hyper-resistivity in order to ensure numerical stability.
The initial condition consists of a broad-band, large
amplitude Alfvén wave, polarized in the yz plane, propagating on a uniform background magnetic field b0 . The
total field is chosen to have uniform intensity everywhere:
b ê b cos φ x ê sin φ x ê (1)
Here b
0 5 is the ratio between the amplitude of the
mother wave and the background magnetic field b
1.
The function φ x represents the phase of the pump
FIGURE 1. Radial evolution of the hourly average for the
Alfvénratio e e in the solar wind.
bx0
0 x
y
1
z
1
0
the initial correlation. The parametric instability has been
studied both numerically and analitically in several situations. Recently, Del Zanna et al. ([9, 10]) studied numerically the evolution of a monochromatic Alfvénwave in a
fully three dimensional, compressible situation, subject
to the parametric instability, by studying its fully nonlinear evolution and saturated state. One important result of
them is that the instability is a robust process, namely it
works practically in the same way independently of the
dimensionality of the problem.
However, in order to understand the importance of
such a mechanism for the solar wind, one has to look
at what happens when the initial wave is non monochromatic, but it has an extended spectrum, as it happens in
the solar wind. Malara et al. ([11]) studied the evolution
of the parametric instability when the initial wave has a
broad band spectrum. They found that, when the plasma
β 1, the time evolution of the spectra of Z and Z
has several common characteristics with the evolution
of the spectra with distance, observed in the solar wind
([4, 12, 13]).
In this paper we want to try to understand whether the
parametric instability maight account for the observed
behaviour of the e e ratio with the radial distance and
for the cited high intermittent behaviour found for the
fluctuations in the solar wind. This possibility has been
pointed out in these proceedings by Bavassano ([14]) and
Del Zanna et al. ([15]).
wave. Its form determines the spectrum of the initial
wave. Measures by Helios spacecraft in fast streams
0 3 AU (the closest available disat distances r
tance) show that fluctuations are dominated by outwardpropagating Alfvén waves, with a double-slope spectrum: at frequencies f
f 0 , with f 0 3-5 10 4 Hz,
the spectrum is flatter (slope
1), while at frequencies f
f0 the spectrum is steeper (slope
15 2 ) [12]). In our model the phase φ x has been chosen so that the initial fluctuations have also an approximately double-slope spectrum, though the slopes are
steeper than the measured values. The frequency f 0 corresponds in the model to the wavenumber k 0 60.
The initial wave being Alfvénic, the velocity field is
given by:
cA0
vx0
δb x 0
(2)
b0
where cA0 1 is the Alfvén speed and the δ operator
indicates the fluctuating part of quantity f : δ f f
f ,
f being the spatial average of f . The initial density
ρ x0
1 and temperature T x 0
T0 are uniform.
The value T0 is used to fix the initial plasma β , defined
by β γa T0 cA0 , with γa 5 3 the adiabatic index. In
the simulations we used β 1, which is of the order of
the value measured in the high-latitude solar wind.
The above-described configuration is an exact solution
of the ideal MHD equations, which would propagate
undistorted in the limit of a vanishing dissipation. An
initial noise of amplitude 10 3 is added to the density
to trigger the instability.
506
0.15
Rx
0.1
0.05
0.0
0
20
40
60
80
100
120
140
160
180
t
FIGURE 2.
R∆x as a function of t, for various values of ∆x.
COMPARISON WITH THE
OBSERVATIONS
Our aim is to compare the results of our numerical simulations with the solar wind data. The plot in fig. 1 represents the spatial evolution of the ratio E e e of
the hourly averaged pseudo-energies of the inward and
outward propagating fluctuations. In our simulations, the
evolution with time emulates the change of the quantities
with distance from the sun, whilst the initial broad band
spectrum of the waves mimics the fluctuations present in
the quantities measured in the solar wind during its transit in front of the spacecraft.
In fig. 2 we plotted the ratio R∆x t
e∆x t e∆x t between pseudoenergies, calculated at a given spatial scale
∆x, corresponding to a characteristic wavenumeber k
2π ∆x. We show the behaviour of the ratio R ∆x as a function of t, for different values of ∆x 1 0 0 314 0 1 0 03,
corresponding to the wavenumbers k
6 20 60 100.
R∆x t is initially very small, the initial velocity and magnetic field being highly correlated. With increasing time,
the instability tends to reduce the Alfvénic correlation
and R∆x t increases at all scales ∆x. This growth stops at
time tsat 90-100, corresponding to the instability saturation. For subsequent times, the ratio R ∆x t remains almost constant or it oscillates around a final average value
that increases as the averaging length scale decreases.
In order to try a comparison between these results and
measures in the high-latitude solar wind, we consider the
ratio e e , which has been calculated by [8], shown in
fig. 1, for fluctuations averaged on a time basis of 1 hour,
obtained from Ulysses measures. In order to establish
a corrispondence between frequencies measured in the
solar wind and wavevectors in the numerical model,
we assume that the frequency f 0 5 10 4 Hz
1
h 1 , where the slope of the e spectrum changes, at
r 0 29 AU, corresponds to the wavenumber k 0 60;
the behavior of R ∆x t for k 60 is among those plotted
in Figure 2. However, the saturation value of the ratio
ff xx ll ff xx 4
Ff l
2
(3)
1. the flatness for both the velocity and magnetic field
has a value of 3 at large scales l (that indicates a
Gaussian distribution of the strongest gradients of
the fluctuations), while it increases for decreasing
the length scale l, pointing out that intermittency
dominates the turbulence at small scales;
2. the flatness for the magnetic field intensity is higher
than the corresponding values at the same scale l for
the velocity field, demonstrating that the magnetic
field is in general “more intermittent” than velocity;
3. the flatness increases when the distance from the
sun increases, meaning that the fluctuations become
more and more intermittent during the travel in the
heliosphere.
In fig.s 3 and 4 we show the plots of the flatness, defined as in (3), for several length scales l, ranging from
the highest (the one of the numerical box), down to the
smallest length scales available in the simulation, at several times. We recall that time evolution in the simulation
is analogous to evolution with distance in the observational data. A look at the result shows a striking analogy
between the results of the simulations and the observation by Bruno et al. (2002) [5]. In particular, we point out
that: a) the flatness increases with decreasing the length
scale (see point 1 above), b) the flatness increases with
increasing time, likewise it does with distance from the
sun in the data. However, we point out that the final values of the flatness are much higher in our simulation than
in the data. Moreover, the magnetic field in our simula-
2
They showed that:
e e shown in fig. 1 is slightly higher (e e
0 5)
with respect to the one obtained in our simulations for the
corresponding length scale (e e
0 1 for ∆x 0 1).
Nevertheless, we stress that, in spite of the crudeness of
the extimations made for evaluating the length scale in
the initial spectrum of the waves, the magnitude order
of the values are in fairly good agreement. We conclude
that, even from a quantitative point of view, the behavior
of e e found in the solar wind is partially reproduced
by the numerical model.
Finally we remark that the evolution of the instability produces shock waves that, in turn, give rise to an
intermittent behaviour for the intensity and direction of
the fluctuating fields, due to the presence of strong gradients in those quantities. Bruno et al. (2002), analyzed
the behaviour of the flatness of the velocity and magnetic
field fluctuations observed in the solar wind at several
distances from the sun. The flatness at a length scale l,
for a generic quantity f (the intensity of the velocity or
magnetic field fluctuations), is defined as:
x=1.0
x=0.314
x=0.1
x=0.03
507
REFERENCES
tions does not appear to be more intermittent than the
velocity field, as found in the data (see point 2 above).
We conclude that the agreement of the values found in
the simulations with the observational data is only qualitative, then our simplified model does not account for
all the characteristic features of the polar wind. Some ingredient is missing, such as the expansion of the wind
and the three-dimensionality of the problem. Further investigation in these directions are necessary to draw a
conclusion.
1.
2.
3.
4.
ACKNOWLEDGMENTS
5.
The authors are grateful to B. Bavassano, P. Pietropaolo,
R. Bruno and R. Grappin for clarifying details about
the comparison with the solar wind data and interesting
discussions.
6.
7.
8.
9.
10.
11.
FIGURE 3. Flatness (as defined in (3)) of the magnetic field
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12.
13.
14.
15.
FIGURE 4. Flatness (as defined in (3)) of the velocity field
intensity fluctuations at several length scales l from the simulations.
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