A&A 383, 319-325 (2002)

DOI: 10.1051/0004-6361:20011688

**I. Lerche ^{1,2} - R. Schlickeiser ^{1}**

1 - Institut für Theoretische Physik, Lehrstuhl IV:
Weltraum- und Astrophysik, Ruhr-Universität Bochum,
44780 Bochum, Germany

2 -
*Permanent address*: Department of Geological Sciences, University of South
Carolina, Columbia, SC 29208, USA

Received 16 July 2001 / Accepted 27 November 2001

**Abstract**

In order to describe correctly the heating rate of the interstellar medium by
plasma wave energy losses, we investigate the influence of
non-linear amplitude effects on wave characteristics, using the
Rogister-Mjølhus-Wyller equation, which is the most complete
plasma kinetic description for nonlinear magnetohydrodynamic (MHD) waves
propagating parallel to an ambient magnetic field. It is demonstrated
(1) that any energy change arises only as a consequence of the nonlocal term in
this equation, and (2)
that the energy transfer rate can be evaluated in terms of quadratures
dependent on the initial and/or boundary conditions imposed on exact
solutions to the derivative non-linear Schrödinger equation.
Moreover, the Rogister-Mjølhus-Wyller equation
implies the presence of oscillatory solutions, independent of
the strength of the nonlocal term, indicating that
such waves play a role in the
long-term evolution of the interstellar medium. These waves propagate in
both positive and negative senses
(depending on their wavenumbers), suggesting a bifurcation of energy flux
directions at large and small spatial scales. Such a division is of significance
not only for the long term behavior of the interstellar medium, but also for
particle energization.

**Key words: **ISM: magnetic fields - plasmas - waves -
turbulence - magnetohydrodynamics
(MHD)

Phase | density (cm^{-3}) |
temperature (K) | volume filling factor () |

molecular cloud | 70 | 2 | |

cold neutral medium | 10-100 | 100 | 2 |

HII envelopes | 5-10 | 8000 | 2 |

diffuse intercloud gas | 0.1-0.5 | 8000 | 50 |

coronal | 10^{-3} |
10^{6} |
20-80 |

When investigating the nature of interstellar turbulence, it is necessary to keep in mind that the interstellar medium contains a number of plasmas (commonly referred to as phases) of very diverse characteristics (Spangler 1999). The five major phases are summarized in Table 1. The observed turbulence properties as obtained from radio propagation measurements as dispersion measures, rotation measures and interstellar scintillation are biased towards the high-density ionized interstellar phases with large volume filling factors, i.e. the diffuse intercloud gas and HII envelopes. Especially, dispersion measure and scintillation data are primarily diagnostics of density, and only secondarily of magnetic field. Both diagnostics indicate the existence of interstellar density irregularities with frequency power spectra extending over 11 decades in frequency, much below ( ) the proton gyrofrequency (Armstrong et al. 1995).

The plasma wave models allow consideration
of the damping processes of the turbulence that is important for the
heating of the interstellar medium (Lerche & Schlickeiser 2001).
In this context, in order to
describe correctly wave energy loss in
the interstellar medium, it is necessary to investigate the influence of
non-linear amplitude effects on wave characteristics. Estimates of *small*
amplitude wave energy loss rates are
radically different in spatial and temporal behaviour from estimates pertaining
to *large* amplitude waves (Mjølhus & Wyller 1986,1988;
Spangler 1990,1991; Dawson & Fontan 1990; Flå et al. 1989). Here we investigate this problem on the basis of the
Rogister-Mjølhus-Wyller equation, which, to date, is the most complete
plasma kinetic description for nonlinear magnetohydrodynamic (MHD) waves
propagating parallel to an
ambient magnetic field (Rogister 1971;
Mjølhus & Wyller 1986,1988). With some minor modifications to
the notation presented in Spangler (1991), the wave evolution is described
through

which reduces to the Derivative Nonlinear Schrödinger equation (DNLS) for

where ,

where

In order that this separation of variables holds for arbitrary and

From Eq. (6) it follows that

where constant, and from Eq. (7)

where

where constant.

From the requirement that
be real for all ,
it follows that
*F*_{0}, *c*_{2}, *c*_{1} and *h*_{0}^{2} all be real. Further *c*_{1} must be greater than
or equal to zero to avoid
turning negative, which would
drive
complex, and thus violate the ansatz. From Eqs. (8)
and (11) it then follows that

must be real and positive, and that

The imaginary part of

With
,
real, Eq. (3) can be written in
the general form

Now note from Eq. (11) that

and that from Eq. (8)

so that

For *c*_{1}=0, however, one has ,
and then one is interested in
solutions to

for all solutions of Eq. (1) which are spatially bounded with as for any finite . Thus, if there is to be any energy change then it arises

If *N*_{2} is small, then the global energy loss rate
is
obtained by taking the solutions of Eq. (1) in the *absence* of
the term containing *N*_{2}, i.e. solutions of the standard DNLS, and then using
them directly in the right hand side of Eq. (18). According to
property (16) only solutions with *c*_{1}=0 are of interest
here. We denote solutions to Eq. (17) obtained with *N*_{2}=0 by *D*(*s*).
Then with *N*_{2}=0,

where

and the energy loss rate then is approximately

Equation (20) can we rewritten as

The general solution to Eq. (22) is not difficult to obtain but is somewhat involved. First re-write variables with

where

Collecting real and imaginary parts separately we obtain

and

Note that both Eqs. (25) and (26) involve only and but not explicitly. Then set when Eq. (26) takes the form (using the abbreviation etc.)

which has an immediate first integral

or

where is the constant of integration.

Using the solution (29) in Eq. (25) yields

Equation (30) is of the generic form

where

The general method of solving Eq. (31) is to write

when

Hence Eq. (31) yields

so that

and so

From Eq. (29) one has

(38) |

yielding

(39) |

Hence

The general problem has, therefore, been reduced to simple quadratures whose evaluation depends explicitly on the conditions chosen to determine the constants of integration. Hence, all spatially bounded structural forms with have been obtained.

To obtain wave solutions to Eq. (17) is relatively easy. Set

Then, remembering , inserting Eq. (41) into Eq. (17) gives

for

The waves are forward propagating if

(44) |

and reverse propagating otherwise. Thus long wavelengths (

and the group velocity is

(46) |

For positive sense waves one has and is oppositely directed to for

This difference in propagation character at short and
long wavelengths, dependent
on the amplitude of the wave (through
), would
suggest that there should be at least one choice of *k* that could lead to
particle acceleration. This part of the problem is now considered.

where we have restored the dimensionality. Here is the relative phase on

For a particle of rest mass

with

in terms of the momentum and with the connection . Moreover one has

Multiplying Eq. (49) with , Eq. (50) with , adding the results and using Eq. (51) yields the relation

Also, multiplying Eq. (49) with , Eq. (50) with , where

and adding the results yields

where we use Eq. (51). Equation (56) has the complete integral

which can be written as

Consider particles starting at

This value will be independent of time if

Then, Eqs. (49)-(50) simply solve as

and Eq. (58) yields for the relation of final (f) and initial (i) values

which is equivalent to

Hence: if the initial particle velocity

The condition for acceleration
can be written
with Eq. (45) as

it follows that acceleration can occur at least when

Considerations of particle motion other than that given by Eq. (60) may also provide selective velocity ranges for particle acceleration. However, it is sufficient to show here that at least some particles are accelerated by the wave solutions of the nonlinear Landau damping equation.

(1) First, some of the basic properties of spatially bounded soliton-like solutions were provided indicating how such structures can develop.

(2) Second, particular structured forms, spatially bounded, were reduced to simple quadratures, thereby providing general analytic solutions in quadrature form. The patterns of behaviour for such soliton-like solutions were shown to be dependent on the initial and/or boundary conditions one chooses to impose, leading to a rich diversity of possible behaviors.

(3) Third, a class of wave-like solutions were derived that do not
damp with time, independent of the value for the so-called "non-local" term in
the equation that controls energy loss.
These wave-like solutions were shown to be capable of accelerating charged
particles. It may be that more general conditions for particle acceleration can
be found than those developed here. However, the fact that some particles, at
the least, can reach higher energies using the waves, suggests that a more
general investigation of the spatially bounded soliton-like behaviours should
also provide acceleration capabilities.
The complexity and diversity of possible structural soliton patterns has so far
precluded a detailed investigation of the particle acceleration capabilities of
*all* classes of solutions to the non-linear Landau damping wave kinetic
equation. But that one class of solutions *does* provide for such
acceleration augers well for any further investigation
involving more general classes of solutions.

The main point is that the influence of non-linear effects on wave
characteristics for energy gain and/or loss in the interstellar medium
is not to be lightly dismissed. In particular, for heating of the interstellar
medium, the investigation in Sect. 2.2 shows that the energy transfer rate (as
approximated by Eq. (21)) can be evaluated in terms of quadratures
dependent on the initial and/or boundary conditions imposed. This use of exact
solutions to the DNLS equation is a major improvement over the ad hoc
suggestions (Spangler 1991) for the structured solutions. The ad hoc
behaviors do not come even close to satisfying the basic equations under
consideration by direct insertion into the equations. For the classes of
solutions described by decaying soliton behaviors, the influence of such heating
(cooling) events can seriously affect the understanding of the balance of wave
heating and radiative cooling in the interstellar medium. The basic solutions
exhibited here would, seemingly, be a significant step forward in our long-term
attempts to understand this problem.

Equally, the presence of oscillatory solutions, independent of *N*_{2} for any and
all values of *N*_{2}, would seem to indicate that such waves play a role in the
long-term evolution of the interstellar medium. If the soliton-like structures
decay (for )
then one is eventually left with a body of non-linear
oscillating waves. These waves propagate in both positive and negative senses
(depending on their wavenumbers), suggesting a bifurcation of energy flux
directions at large and small spatial scales. Such a division is of significance
not only for the long term behavior of the interstellar medium, but also for
particle energization.
This paper has shown how all of these problems can be addressed using the
one-dimensional non-local DNLS equation. It is clear that the richness of
patterns can only increase with multi-dimensional equations of similar sort.

The *total* effect of all such possibilities on the long term evolution of
the interstellar medium is seen but murkily at present, but also represents an
exciting challenge.
The soliton behaviors, the forward and reverse propagating
oscillatory waves, and the particle acceleration by the non-linear waves, all
provide processes that are fundamentally beholden to the non-linear terms,
without which all such effects would vanish. And these non-linear processes form
the start of a significant combined integration of information in our attempts
to understand the intertwined evolution of waves and charged particles in the
interstellar medium.

We are grateful to Drs. M. Goossens and Y. Voitenko for pointing out to us the underlying assumptions in the derivation of the Rogister-Mjølhus-Wyller equation. We gratefully acknowledge support by the Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 191.

The Rogister-Mjølhus-Wyller Eq. (1)
is a "Kinetically Modified Derivative
Nonlinear Schrödinger equation" (KMDNLS) where the modification arises
from the nonlocal term scaling with *N*_{2}. It reduces to the
Derivative Nonlinear Schrödinger equation (DNLS) when
*N*_{2}=0.

The derivation of the DNLS equation has been reviewed by
Dawson & Fontan (1990) and Mjølhus & Wyller (1986).
It has been originally
obtained from the set of
nondissipative MHD equations, where all variables are supposed to depend only on
the *z*-coordinate (Sakai & Sonnerup 1983; Spangler & Sheerin
1982) by means of a reductive perturbation method (Taniui & Wei
1968). Due to this geometric restriction, the model obtained is
unidirectional and *B*_{z}=*B*_{0} always, so that longitudinal perturbations of
the magnetic field cannot be described.
The complex variable
represents the real valued transverse field
components *B*_{x,y} according to

(66) |

and the profile and evolution coordinates and are stretched according to

(67) |

The small parameter characterizes the simultaneous weak nonlinearity and weak dispersion (i.e. long wave length) requiring that the characteristic scale of turbulence

(68) |

which is equivalent to the low frequency limit . The smallness of also imples that the magnetic field perturbation is larger than the density fluctuation. As long as we apply the DNLS equation to parallel propagating Alfven waves at frequencies much less than the proton gyrofrequency, both limiting restrictions are appropriate.

Rogister (1971) was the first in deriving the DNLS from the full three
dimensional quasi-neutral Vlasov equations together with the pre-Maxwell's
equations (without the displacement current). A perturbation expansion
equivalent to the reductive perturbation method led to an equation containing
all the terms of the DNLS, but also with the additional non-local term
representing resonant particles. Mjølhus & Wyller (1988) started
from the guiding center plasma model to infer the KMDNLS Eq. (1). At least two assumptions are hidden in the
derivation of the KNDLS equation
that can make the applicability of the KNDLS equation to nonlinear MHD waves
in the interstellar medium less than optimal:

(1) The assumption that the velocity distribution of resonant particles
is constant in time leads to the dissipative coefficient *N*_{2} in Eq.
(1) being independent of time. If the velocity
distribution of the resonant particles, which causes the wave-particle
interaction, evolves much faster than the wave amplitudes then the
quasilinear diffusion smoothes out the velocity
distribution, thus quenching the Landau resonance and the coefficient *N*_{2}.
Then one again would have the DNLS Eq. (1) with *N*_{2}=0.

(2) The assumption of time reversibility of Landau damping, which
gives rise to the Hilbert transform operator in the nonlocal term of Eq.
(1), might be violated. Such an effect
can happen because of a change of the
dissipative term due to scattering of particles at small angles.
A non-local term would persist but would be more complexly described than by a
Hilbert transform.

Both assumptions become especially crucial for the asymptotic states of the turbulence at long times.

Unfortunately, compared to the interplanetary medium, where in-situ measurements
on spacecrafts of the turbulent properties of the interplanetary medium
are available, the observational data for interstellar medium turbulence is
scarce, and, as mentioned already, is mainly based on line-of-sight electron
density fluctuation measurements through a multi-phase interstellar medium
from scintillation, dispersion measure and Faraday rotation measure
observations. Therefore no direct
observational clue on the validity of the underlying assumptions (1) and (2)
in the interstellar medium can be provided. It is, however, important to keep
these hidden assumptions in mind when applying the solution properties of the
KNDLS equation presented in this work to the interstellar medium.

We finally mention that Dawson & Fontan (1990) have investigated the
soliton-behaviour of Eq. (1) with *N*_{2}=0, whereas Flå et al. (1989) found numerically that with the soliton-like modes damp with time.

- Armstrong, J. W., Rickett, B. J., & Spangler, S. R. 1995, ApJ, 443, 209 In the text NASA ADS
- Dawson, S. P., & Fontan, C. F. 1990, ApJ, 348, 761 In the text NASA ADS
- Flå, A. L., Mjølhus, E., & Wyller, J. 1989, Phys. Scr., 40, 219 In the text NASA ADS
- Lerche, I., & Schlickeiser, R. 2001, A&A, 366, 1008 In the text NASA ADS
- Mjølhus, E., & Wyller, J. 1986, Phys. Scr., 33, 442 In the text NASA ADS
- Mjølhus, E., & Wyller, J. 1988, J. Plasma Phys., 40, 299 In the text NASA ADS
- Rogister, A. 1971, Phys. Fluids, 14, 2733 In the text
- Sakai, J., & Sommerup, B. U. Ö. 1983, JGR, 88, 9069 In the text NASA ADS
- Spangler, S. R. 1990, Phys. Fluids, B2, 408 In the text
- Spangler, S. R. 1991, ApJ, 376, 540 In the text NASA ADS
- Spangler, S. R. 1999, in Plasma Turbulence and Energetic Particles in Astrophysics, ed. M. Ostrowski, & R. Schlickeiser, Krakow, ISBN 83-908592-0-3, 1 In the text
- Spangler, S. R., & Sheerin, J. P. 1982, J. Plasma Phys., 27, 193 In the text NASA ADS
- Taniuti, T., & Wei, C. 1968, J. Phys. Soc. Jpn., 24, 941 In the text

Copyright ESO 2002