A&A 373, 301-306 (2001)
DOI: 10.1051/0004-6361:20010513
L. Nocera^{1} - M. Medrek^{2} - K. Murawski^{2}
1 - Institute of Atomic and Molecular Physics, National
Research Council, Via Moruzzi 1, 56124 Pisa, Italy
2 -
Department of Environmental Physics, Technical University of
Lublin, ul. Nadbystrzycka 40, 20-618 Lublin, Poland
Received 9 November 2000 / Accepted 2 April 2001
Abstract
The influence of a space-dependent random mass density field on the
development of solar p-modes is investigated using analytical and
numerical means. Using a perturbative approach, which is valid for a
weak random field and small amplitude waves, we derive a linear
dispersion relation whose solutions correspond to attenuated
oscillations. The real part of the frequency of these oscillations
exceeds the one of waves propagating in a medium without random
density. We give an interpretation of the "unphysical'' nature of the
frequency shift and of the amplitude attenuation which is similar to
Landau damping.
The analytical findings are compared with the results of the
numerical solution of a model wave equation. We find that, for weak
random fields and for wavelengths which are a few times the
correlation length of the random density fluctuations, numerical
results agree with the analytical theory. Two practical formulas for
deriving the correlation spectrum of the random density field from
observations are also given.
Key words: Sun: oscillations - turbulence - waves
In particular, coupling among different modes due to a random field was considered by e.g. Youakim & Liu (1970) and Razin (1995) who found a strong coupling for small spatial scales of the random field. Additionally, Razin (1995) showed that random inhomogeneities can cause conversion of different elastic modes which propagate in a solid medium. It is interesting that such waves propagate on average faster than in a homogeneous solid medium, a conclusion which is also supported by the results by Roth et al. (1993), Shapiro et al. (1996) and by Müller & Shapiro (2000).
Nonlinear electromagnetic waves which propagate through a plasma containing random fluctuations of mass density were discussed by Liu (1973) who found that the amplitude of a coherent wave was attenuated and that the attenuation was affected by nonlinearity. The Korteweg-de Vries equation with a random forcing term was proposed by Wadati (1990) to show that Gaussian white-noise leads to damping of the soliton solution of that equation.
A method to derive a model random wave equation was presented by Gurevich et al. (1993). Pelinovsky et al. (1998) considered surface water waves which propagate along a channel with a rough bottom. They found that such waves experience attenuation and "deceleration'' due to the random field. Fast magnetosonic waves which are impulsively generated in a plasma with a random mass density were discussed by Murawski et al. (2001) who showed that, due to the random field, the localized pulses experience spatial delay and attenuation. Murawski & Pelinovsky (2000) performed analytical and numerical studies and showed that a space-dependent random flow is able to "speed up'' and enhance sound waves. Murawski & Diethelm (2000) and Lazzaro et al. (2000) indicated that a random field can generate additional modes.
Many of the above mentioned papers are based on asymptotic methods such as the mean field method of Howe (1971b). This approach is valid for weak random fields and it allows a reduction of a set of random differential equations to a dispersion relation. For a thorough discussion of the method, the reader is referred to Frisch (1968) and Konotop & Vásquez (1994).
The motivation to reconsider the classical problem of acoustic oscillations in a random medium (Howe 1971a) stems from the fact that our computer simulations suggest that oscillations with a phase speed larger than the sound speed may be observed even in the linear approximation. This fact, denoted generically as "wave acceleration'', was also reported by other authors (e.g. Collin 1969). It disagrees with Howe's (1971a) asymptotic formulas and it was generally dismissed as "unphysical'', basically on the grounds that it was counter-intuitive.
This issue assumes a particularly important relevance to Solar Physics due to the well known discrepancies between the observed frequencies of p-mode oscillations and the ones predicted by standard models (e.g. Duvall et al. 1998) and due to the turbulent state of the outer layers of the Sun where the p-modes penetrate. Besides the obvious practical implications of an investigation of the shift of p-mode frequencies, a few related questions arise: a) why does the conservative system of equations governing p-modes develop oscillations which appear to be damped in time? b) when and why does the effect of randomness result in an increase or decrease of the frequency? c) are there analogies between these and similar effects in other areas of Plasma Physics? d) what are the limitations of a purely analytical treatment?
To answer these more fundamental questions it is convenient to adopt a simple model for the propagation of p-modes (Swisdak & Zweibel 1999) in which they are basically reduced to one-dimensional acoustic waves. Furthermore, in tackling problem d) above, numerical techniques attain their significance, as already shown by Murawski (2000), Murawski & Pelinovsky (2000) and Murawski et al. (2001).
In Sect. 2 we give the governing equations of our problem and their physical justification. A linear dispersion relation is derived from these equations by means of perturbative methods which is then solved in Sect. 3. Comparison of this solution with results of numerical simulation is presented in Sect. 4. The paper is concluded by a discussion of the main results in Sect. 5.
In the absence of the random density field (
),
sound waves would propagate in the medium whose frequency
and wavevector k satisfy the known dispersion relation:
The effects of the random density field are now analyzed by statistical means. By a random medium we understand a family of media, each with a well-defined value of , and each with a well-defined probability of being realized in an experiment. We denote by the ensemble average (e.g. Ishimaru 1978) over such a family.
Assuming that the density perturbations are small, we expand the
fluid variables around the equilibrium of Eq. (1). Neglecting
nonlinear terms leads to the linear wave equation
Substituting Eq. (7) into Eq. (6), we get
Substituting Eq. (12) into Eqs. (9) and (11), we
arrive at the dispersion relation
Equation (15) shows that, as a consequence of stochastic perturbations, the dispersion relation for the coherent field differs essentially from that for the unperturbed linear problem (see Eq. (5)). The consequences of this difference will be analyzed in Sect. 3.
Using Eq. (18) the dispersion relation in Eq. (15) takes the form
The Cauchy integral is obviously a multivalued function of the
complex frequency .
However, if we assume that the function
f is the restriction to the real axis of an entire function of
,
analytical continuation of the Cauchy integral is possible
into a secondary or "unphysical'' sheet of the complex plane by the
definition (Roos 1969)
Using Eq. (21), we write the principal branch of the dispersion
equation in Eq. (19) as
Thus we introduce the following ordering:
To first order, we set
in the right hand side
of Eq. (22) (already a small quantity). We remind that
,
as defined in Eq. (21), is analytical on the real
axis. Furthermore if, for any real number u, f(u) is Hölder
continuous, we have the Plemelj formula (Roos 1969)
Substituting Eqs. (23) and (25) into
Eq. (22), we have, to
,
Clearly, since, as pointed out in Sect. 2, the correlation spectrum is real, if solves Eq. (19) then and also do. However and lie in the upper half of the unphysical Riemann sheet, whereas lies in the lower half of the physical sheet. None of these roots contribute to the inversion of the causal Fourier transform.
Now we specialize to the case of a Gaussian correlation spectrum of
the random fluctuations:
Substituting Eqs. (32) and (20) into
Eq. (19), we find for the principal branch of the
dispersion relation
Substituting Eq. (32) into Eqs. (30) and
(31) and using the well known relation between the
sine integral of a Gaussian distribution and Dawson's integral
(Gautschi 1972) we find
The real and imaginary parts of
for a weakly
(
)
random medium with Gaussian random fluctuations are
plotted in Fig. 1 as functions of the normalized wavenumber.
Figure 1: The correction to the frequency of sound waves: (solid curve) and (dashed curve). Superimposed are the values of the correction coming from numerical simulations at (). | |
Open with DEXTER |
The random mass density field is represented as follows (Karweit &
Blanc-Benon 1991; Karweit et al. 1995; Chevret et al. 1996):
In our simulations periodic boundary conditions are imposed. At t=0pressure and density are set to their static equilibrium values in Eq. (1); velocity is a Gaussian packet centered at x=Nl_{x}/2and, since the problem being simulated is linear, its amplitude is arbitrarily set to c_{0}.
In this framework Eq. (6) is solved over the interval and for times . The wave distributions v(x,t) output by the simulation are first ensemble-averaged over 120 realization of the random density field ; then the averaged distribution is spectrally analyzed in space and time by means of a Fast Fourier Transform. We made sure that averaging over a larger number of realizations gave no improvement in the results.
In Fig. 1 the frequency of maximum spectral concentration is compared, for each wavenumber, with the real part of the frequency given by the approximate analytical approach of Sect. 3: the span of the error bars equals the variance of the frequency shift over the random density statistical ensemble. The results of this comparison are presented in Sect. 5.
First we attempt a perturbative solution of the linear dispersion
relation in the assumption that the random density fluctuations are
weak. The solution can be written by combining Eqs. (23), (35) and
(36) in the following approximate formula, which
holds for a Gaussian profile of the correlation function of the
random density fluctuations
It is thus important to note how two different and yet both plausible models of the random component of the wave operator lead to conflicting results. In this perspective, direct numerical simulations acquire an important role as indeed shown by Murawski (2000), Murawski et al. (2001) and Murawski & Pelinovsky (2000).
Since the oscillation occurring at the frequency reported in Eq. (38) is not a genuine eigenfunction of the wave operator (see below), its numerical reproduction is not an easy task. Nevertheless we were able to observe frequencies which agree with the theoretical prediction, within statistical error (Fig. 1). We surmise that, amongst the several sources of discrepancy between the numerical results and analytical predictions, a most important one is the breakdown of the statistical properties of the random wave operator which are necessary for the derivation of the random dispersion relation in Eq. (15) (see the review by Mysak 1978).
One further result of our investigation can be summarized as an answer to the question: "How can a solution to the conservative system of Eqs. (9) and (11) be attenuated in time?'' Our interpretation of this apparent paradox is that the root of the dispersion function which lies in the lower half of its unphysical Riemann sheet has no associated eigenfunction; the damped oscillation related to this root is rather similar to the "virtual modes'' found in Landau damping (Sedlácek & Nocera 1992), in inhomogeneous plasmas (Sedlácek 1971, 1994) and in the inhomogeneous vibrating string (Sedlácek et al. 1986): they can be aptly named "random virtual modes''.
In conclusion we reach the following results:
Conclusion ii is of great relevance to Solar global oscillations: it suggests that the random state of the Solar atmosphere may increase the frequency of p-modes. We mention that, in the area of Earth's seismology, positive frequency shifts of elastic waves were found by Roth et al. (1993), Shapiro et al. (1996) and by Müller & Shapiro (2000).
One last important consequence of conclusion iii) is that terms like "wave acceleration'' or "deceleration'' can be misleading: indeed the oscillation associated with the virtual frequency is not an eigenmode of the system of Eqs. (9) and (11). Rather it is a superposition of these eigenmodes. The nature, distribution and possible completeness of these eigenfunctions will be dealt with elsewhere.
Acknowledgements
We thank Prof. E. N. Pelinovsky and Dr. V. M. Nakariakov for their useful comments. We acknowledge financial support by the Polish State Committee for Scientific Research under grant No. PO3D 017 17 and by the Italian National Research Council. The numerical simulations were performed at the Department of Complex Systems, Institute of Physics, UMCS Lublin and at the Poznan Supercomputing Centre.