A&A 444, 233-244 (2005)
A. Verdini1 - M. Velli1 - S. Oughton2
1 - Dipartimento di Astronomia e Scienza dello Spazio, Firenze, Italy
2 - Department of Mathematics, University of Waikato, Private Bag 3105, Hamilton, New Zeland
Received 24 January 2005 / Accepted 21 June 2005
We investigate the nonlinear evolution of Alfvén waves in a radially stratified isothermal atmosphere with wind, from the atmospheric base out to the Alfvénic point. Nonlinear interactions, triggered by wave reflection due to the atmospheric gradients, are assumed to occur mainly in directions perpendicular to the mean radial magnetic field. The nonlinear coupling between waves propagating in opposite directions is modeled by a phenomenological term, containing an integral turbulent length scale, which acts as a dissipative coefficient for waves of a given frequency. Although the wind acceleration profile is not determined self-consistently one may estimate the dissipation rate inside the layer and follow the evolution of an initial frequency spectrum. Reflection of low frequency waves drives dissipation across the whole spectrum, and steeper gradients, i.e. lower coronal temperatures, enhance the dissipation rate. Moreover, when reasonable wave amplitudes are considered, waves of all frequencies damp at the same rate and the spectrum is not modified substantially during propagation. Therefore the sub-Alfvénic coronal layer acts differently when waves interact nonlinearly, no longer behaving as a frequency dependent filter once reflection-generated nonlinear interactions are included, at least within the classes of models discussed here.
Key words: magnetohydrodynamics (MHD) - turbulence - waves
The main difficulties one has to face for the development of such a scenario is the nature of the nonlinear interactions for the incompressible Alfvén mode. Among the many wave-modes generated by photospheric oscillations, Alfvén waves are the most likely to survive into the corona (Hollweg 1978), although other modes may be generated throughout the atmosphere. Analysis of observed oscillations in coronal structures may eventually lead to their identification (De Moortel et al. 2002a,b), but we concentrate here on Alfvén waves, since they are the dominant modes observed in the solar wind. It is well known that nonlinear terms couple Alfvén waves propagating in opposite directions. Note, however, that in the fast solar wind, outwardly propagating modes seem to dominate (Smith et al. 1995).
The inhomogeneities of the ambient medium suggest a solution for this apparent contradiction between dominantly unidirectional propagation and development of nonlinear interactions (Velli et al. 1989). Variations of the group velocity of the wave (the sum of Alfvén and wind speed gradients) linearly couple the outgoing and ingoing waves producing one from the other and furnishing the trigger for nonlinear interactions to take place. Intensive studies of this mechanism have been carried out in terms of the dynamical time scales which enter the governing equation, while the anisotropic nature of the problem was handled naturally in the context of a Reduced Magneto-Hydrodynamics description (RMHD hereafter) which allows a correct treatment of nonlinear terms (Oughton et al. 2001; Dmitruk & Matthaeus 2003; Dmitruk et al. 2002,2001a; Oughton et al. 2004). This kind of approach has led to the understanding of the ordering of the characteristic times which should effectively favor the development of a turbulent cascade in planes perpendicular to the direction of wave propagation (along the magnetic field) and the efficiency of dissipation. Due to the complexity of the equations describing the propagation of large amplitude Alfvén waves in an inhomogeneous moving medium these analyses were developed in a static stratified atmosphere where the profiles of the ambient magnetic field and the density of the medium were imposed in order to produce the Alfvén velocity gradients which, ultimately, determine the amount of reflection inside the simulation box.
Other authors (Moore et al. 1991; Krogulec & Musielak 1998; Velli 1993; Cranmer & van Ballegooijen 2005; Lou & Rosner 1994; Similon & Zargham 1992; Heinemann & Olbert 1980; Velli et al. 1991; Leroy 1980; Mangeney et al. 1991; Grappin et al. 1991; Krogulec et al. 1994; Leroy 1981, but also Dmitruk et al. 2001b for a phenomenological nonlinear model) have focused their attention on the linear theory of wave propagation.
Here, three main features have proved to be essential: first, the geometry of the medium, since the profile of Alfvén speed is strongly affected by the structure of the flux tubes in which waves are supposed to propagate; second, the extent of the atmosphere whose global stratification profile determines the transmission, since it is the entire profile of the Alfvén speed, and not simply its local variation, which determines whether or not a wave at a given frequency is capable of escaping at the top of the atmosphere; third, the presence of a wind which separates the atmosphere into two parts. The Alfvén critical point (the distance from the sun at which the wind speed equals the Alfvén speed) represents a natural separation between an internal region where the wind is slow and affects the propagation of the waves only slightly (at least at high frequencies) and an outer region, beyond the critical point, where waves are advected outwards by the wind regardless of their original propagation direction. Even if in the very low corona one might expect the effect of a bulk flow to be negligible, it was shown (Heinemann & Olbert 1980; Velli 1993; Jokipii & Kota 1989) that the wind speed reduces the reflection inside the atmosphere for low frequency waves, carrying them out through the Alfvénic critical point (beyond which they can not propagate backwards), and hence enhancing their transmission.
The aim of this paper is to investigate nonlinear effects on wave propagation once the background medium and the entire (lower) atmosphere are taken into account. Following Dmitruk et al. (2001b), we choose a constant or radially expanding transverse dissipative length scale and a phenomenological nonlinear coupling term. This allows us to give a preliminary estimate of dissipation caused by the turbulent cascade of counter-propagating Alfvén waves. Introducing such a term still allows wave propagation to be handled in a relatively simple way and, despite the roughness of the model, allows one to gain insight into the relative importance of the three features listed above when dissipative nonlinear effects are also present.
The paper is structured as follows. First we review the equations which describe Alfvén wave propagation in an inhomogeneous moving medium, then we briefly review the results derived from linear propagation and finally we introduce the phenomenological terms, which split the nonlinear analysis into two models, one in which interactions take place only among waves of the same frequency, and the other in which interactions between different frequencies are taken into account. For parameter space, particular attention is paid to the solar case. Finally we review these results and discuss the development of a more realistic model.
The isothermal atmosphere is completely defined by setting the values
for temperature, density and magnetic field intensity at the base,
together with the mass and radius of the central object (
The wind speed and Alfvén speed profiles (and their
derivatives) depend on the two parameters
the non dimensional scale height, and ,
plasma parameter at the base,
It is natural
to define the top of the atmosphere at the Alfvénic critical point
(labeled with c), since only an outward flux (
remains at this point, so that
a transmission ()
and a reflection ()
coefficient may be
expressing length in unit
velocities in units of sound speed, and frequencies in unit of
the non-dimensional form for
Eq. (1), linearized and reduced for the spherical symmetry
case, finally becomes:
|Figure 1: Transmission coefficient as function of frequency in logarithmic scale for isothermal layers of different temperatures. The value of ranges from (hot) to (cold) in integer steps. for all the profiles (from Velli 1993).|
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The form of the nonlinear term may be heuristically derived from the following arguments. When Eq. (1) is Fourier decomposed ( ) nonlinear terms couple several wave numbers in the k-space. When a strong magnetic field ( in velocity units) is present, the propagation time of the Alfvén waves is equal or shorter than the characteristic time-scale for nonlinear interaction over most of the Fourier space, the nature of the nonlinear cascade is highly anisotropic, developing preferentially in planes perpendicular to the direction of the mean field (Oughton et al. 1994; Shebalin et al. 1983; Goldreich & Sridhar 1995). It is then useful to decompose local wavenumber in projections along the magnetic field ( ) and in the perpendicular planes ( ) because energy transfer occurs only among the latter, so that Fourier decomposition is exploited only in . When small fluctuations are considered ( ) these arguments lead to the so called RMHD description which can be derived as an expansion of the usual MHD equations in the small parameter with the restriction (see Oughton et al. 2004 and reference therein for more details on RMHD), in which variations along the perpendicular directions are decoupled from those along the magnetic field ( , with ). We can describe the global effect of this perpendicular cascade by means of two quantities at the large scales, namely an integral scale , giving the dimension of the greatest eddies in which energy is injected, and the average velocity difference () among points belonging to the same eddy, which in RMHD turbulence also contains magnetic field fluctuations in velocity units ( ). Identifying these two quantities with the integral turbulent length ( ) and the fluctuations' amplitude of the Elsässer fields we can construct a characteristic timescale which accounts for nonlinear turbulent interactions in Eq. (1) (see Dobrowolny et al. 1980 for a more accurate derivation).
Multiplying the above Eq. (12) by the complex conjugate one obtains the evolution equations for the Elsässer energies at a given frequency . On its RHS one finds , which acts as a sink for the energy of a given mode. To quantify the relative importance of various couplings we first consider only nonlinear interactions between counter-propagating waves with the same frequency. Thereafter we consider the more realistic case of different couplings between a fundamental frequency (a very low frequency, sometimes called the quasi-2D component, labeled with index 0) and other higher frequencies labeled as interacting (index i).
In both cases, once nonlinear terms are introduced we loose the scaling feature of the linearized equation for which given an increment of a factor f in the amplitudes at the critical point we have an equal increment f for the values at the base. Hence, to get realistic values of velocity and magnetic field fluctuations at the base of the atmosphere, we have to tune the amplitude of the outgoing wave imposed at the Alfvénic critical point (since the equations are still integrated backward) for every frequency and coupling considered.
In the following we first present the results concerning the self-interacting case, where only monochromatic waves interact, then we shall consider modification to dissipation induced by different coupling among two or three frequencies. As model parameters we choose corresponding to temperatures for an isothermal layer above the sun's surface ranging from approximately () to (), and we fix the value of the plasma parameter at for every temperature; thus, at the base of the atmosphere the Alfvén speed is always five times the sound speed. Finally we set the value of the phenomenological turbulent length , that is about corresponding to the average size of the supergranules at coronal level which is maintained at a constant value through the entire atmosphere.
|Figure 2: Total wave action density normalized to the base value as a function of radius in and atmospheres, for a high ( , dotted line) and low ( , solid line) frequency wave with an initial wave amplitude. For the cold atmosphere the profile for the wave having the minimum transmission coefficient (dashed line) is also plotted.|
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The reflection rate depends both on temperature and wave frequency and it is higher for cooler atmospheres and lower frequency waves. Moreover, most of the reflection takes place in the lower atmosphere, where the Alfvén speed gradients are stronger. For high enough wave amplitudes, nonlinear terms dominate over linear ones and the profile of is determined uniquely by the local reflection rate, which, for cold atmospheres, is itself dominated by the Alfvén speed gradients.
to increase with ,
to decrease faster with radius in the very low atmosphere.
For a given
this behavior should be more pronounced
for low-frequency waves (those suffering stronger reflection) and
consequently high-frequency dissipation should be less sensitive to
|Figure 3: Dissipation efficiency () as function of frequency for a hot (, continuous lines) and a cold (, dotted lines) corona for a), b) and c).|
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To investigate the amplitude dependence of the solutions
we consider the case of
a frequency small enough (
to remain to the left of
for all temperatures investigated.
In Fig. 4 the values of the outgoing and ingoing wave
amplitudes at the base are plotted as
a function of the outgoing wave amplitude imposed at the critical
|Figure 4: Outgoing and ingoing wave amplitude (solid line) calculated at the base of the atmosphere, as a function of outgoing amplitude at the critical point. The plots are obtained for . Dashed lines show the values obtained in the linear case (the upper and lower line refers to z+ and z- respectively).|
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|Figure 5: Dissipation efficiency as function of initial outgoing wave amplitude for . The different plots refer to atmospheres with .|
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|Figure 6: Flat spectrum. Left panel. Total wave action density, normalized to the base value, for the interacting frequencies in different coupling (labelled with letters, see text), as function of distance from the atmosphere's base expressed in unit of . Temperature is set to and . Right panel. Dissipation efficiency as function of initial outgoing wave amplitude . The different plots refer to atmospheres with ; solid and dotted lines represent respectively and with .|
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Since in a decaying power law spectrum most of the energy is retained in the low frequencies we expect nonlinear interaction to be more important when such frequencies are involved, hence we choose (representative of low frequencies, say ) and varying from about to increasing by a factor 10 at each step. Specifically, we consider the four couplings,
Consider first the flat spectrum case shown in Fig. 6 (left panel). The general dissipation profile as a function of r is similar to that formed in Fig. 2 (right panel). The main result here is that higher frequency waves (b, c ,d) may also dissipate efficiently thanks to their coupling with the reflected mode of the very low frequency component, so that all profiles show a significant decrease with distance. This also has an effect on the decay of the lowest (zero) frequency mode, whose reflected mode is ultimately the trigger for nonlinear evolution. When the amplitudes of the higher frequency modes are large (i.e. of the same order of magnitude of the low frequency mode), they influence the evolution of the zero frequency reflected component, which then also affects the zero-frequency outward component, driving the profiles to convergence as illustrated in Fig. 6 (right panel): at low amplitudes differs from but at high amplitudes , which means a strong coupling.
Consider now the power-law spectrum illustrated in Fig. 7. The dissipation profile is the same for all the couplings considered (left panel). Now the energy in the high-frequency waves (mainly propagating outwards) is so small that it has a negligible effect on the evolution of the zero-frequency mode (no back reaction) which, in turn, drives the dissipation of all the coupled modes. On the other hand, in the right panel, one can see that the profiles of and are similar to the flat spectrum case and they begin to converge at almost the same values of .
The two parameters, and , determine how strong the coupling is, independently of the shape of the spectrum. For a given amplitude at the critical point of the zero-frequency wave, temperature controls the amount of reflection produced inside the layer, and hence both the linear coupling among the counter-propagating waves of a given frequency (i.e. differences in the waves' evolution due to frequency differences) and the amplitude of the zero-frequency reflected component (the driver). One finds that increasing the temperature (decreasing ) the coupling becomes weaker for a low and stronger for a high , depending on which of the two above features is dominant. For a given coupling, , at a given temperature, the zero frequency wave amplitude imposed at the critical point determines the importance of nonlinear terms (see Fig. 4), and hence the nonlinear coupling among the waves (the evolution independent of frequencies). Increasing increases the strength of the coupling.
Two factors determine the rate of dissipation for the coupled waves. The first, as in the self-interacting case, comes from inward propagating wave generation and dissipation. Since this mechanism is driven by reflection it is a characteristic feature of low-frequency waves (the fundamental) and it acts approximately as in the previous analysis, hence it depends most on the low-frequency initial wave amplitude. The second comes from the form chosen for the coupling where high-frequency waves can be dissipated too. They are essentially propagating outward and their main effect is to dissipate the reflected low-frequency wave. The ratio formed with two coupled wave amplitudes controls the relevance of this second aspect. For , reflected waves are suppressed as soon as they are generated and the dissipation rates of is increased compared to the self-interacting case but that one of is reduced. When the outward high-frequency wave has little capability in suppressing the reflected waves and both dissipation rates are almost completely determined by the fundamental mode: no matter what the coupling is, dissipation efficiency is practically the same for the coupled waves (strength of the coupling) and also for all the couplings formed with different interacting frequencies (negligible effect of the high frequency waves).
|Figure 8: Dissipation efficiency as function of initial outgoing wave amplitude imposed at the top of the layer in the interacting case. The left panel refers to case a ( and ), and the right panel to case b ( and ). Initial wave amplitude is scaled following a power-law spectrum (see text). The different plots refer to atmospheres with ; solid, dotted and dashed lines represent respectively , and .|
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|Figure 9: Spectra at the top (dotted line) and the bottom (solid line) of the atmosphere for three representative initial values of ( , and marked with crosses, stars and diamonds respectively) and different temperature ( respectively on the left and right panel) in coupling b.|
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Supported by these arguments we shall study the behavior of efficiency and spectra with respect to "initial'' wave amplitude variation of two representative couplings:
For the intermediate frequency coupling (righthand panel),
curves for the coupled waves follow different profiles and
give evidence of how much coupling strength is frequency dependent:
for the "middle'' frequency (dotted line)
dissipation efficiency soon reaches the fundamental mode regime even
in the coolest atmospheres, while the highest frequency mode
(dashed line) follows almost the same profile as in coupling a.
A major difference between cool and hot atmospheres is the
behavior of the middle frequency which, for low enough initial
amplitude shows a more efficient dissipation than the fundamental
mode. This can be attributed to the wind effect which manifests for
cooler atmospheres and weak nonlinear self-interaction
or coupling (low amplitudes). It separates very low frequency
behavior (wave mainly transmitted, poor dissipation) and intermediate
frequency behavior (wave mainly reflected and strong dissipation).
|Figure 10: Ratio between the spectra at the top and the bottom of the atmosphere for (respectively left and right panel) in the case of coupling b. The plots are labeled with the initial amplitudes imposed at the top of the layer. Also shown is the ratio for the linear case (dotted line).|
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It is then interesting to track the modifications of the imposed (at )
back to the base of the atmosphere, for varying "initial'' wave amplitudes.
Only coupling b will be considered, for which one expects greater
Figure 9 plots
imposed at the
top of the atmosphere (dotted lines) and the spectra obtained by
integration to the base of the atmosphere (solid lines). Results are shown
for two different temperatures (
Three representative top-boundary wave amplitudes of the fundamental frequency
are considered whose values are
Since we impose a power-law spectrum at the top of the layer
all dotted lines have slope -5/3 and can be
used as reference to see the modification induced by wave
It is striking how much the spectra remain unchanged for practically
all the temperature and all the initial amplitude considered and even if
couplings with highest frequency waves are formed (not shown here).
With a more accurate inspection one actually finds that the spectra
change slightly (see for example solid the line marked with diamonds in the
In Fig. 10 we plot such ratios normalized to the value of the fundamental mode, i.e. and (the fundamental being normalized to its value) for the cases shown in Fig. 9. Since the spectrum at the top is fixed by the initial condition, a normalized lower (greater) than one means the spectrum is steepening (flattering).
For low initial amplitudes in hot atmospheres the spectrum becomes flatter at high frequencies, since the coupling is not so strong and frequency dependence displays its influence. As we increase the strength of the coupling, by increasing , differences are smoothed and the spectrum first adjusts itself to the slope imposed at the top of the atmosphere and then finally steepens. The same behavior is observed for the higher frequency coupling case a (not shown), so one can attribute such a general feature to the high temperature atmospheres.
For low temperatures, the picture is more complicated. Frequency differences are more important and the strength of the coupling is reduced, the intermediate frequency waves contribute to dissipation with their self-interacting part so that the spectrum steepens at intermediate frequencies and flattens at high frequency. To quantify this tendency, we have calculated the exponent of the resulting power-law scaling at the base for the low and high frequency branches. In the worst of the cases ( for in Fig. 10) the slopes are 1.61 (spectrum steepening) and 1.76 (spectrum flattening) to compare with 1.667, but generally the slope is 1.65 (steepening) or 1.67 (flattering). One can then conclude that even if the spectra evolves during propagation the differences between the slopes at the top and bottom are always very small or even negligible.
Note how for lower
and low amplitude the spectrum is unchanged
by propagation, even though consistent differences in the dissipation
efficiencies of the coupled modes are found.
The origin of such discrepancies is to be found in the
which involves total wave action density at
the boundaries of the layer, including a contribution from
inward and outward fluxes at the boundaries which are not simply wave
amplitudes (which indeed determine the spectrum).
One can in fact rewrite as follows:
|Figure 11: Same as in Fig. 8 for coupling b and power-law initial spectrum except that integrations are performed using a spherically expanding turbulent length with L0 fixed by the average dimension of supergranules at coronal level. Results are shown for atmospheres.|
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For low , dissipation efficiency is considerably altered (reduced) with respect to the non-expanding case (compare Figs. 11 and 8). Since nonlinear term influences are now reduced, we need greater initial amplitudes to efficiently couple the frequencies considered; as a consequence the profiles of are shifted to the right with respect to the non-expanding model (by a factor of 3 or 4). For coupling a, the plots of show the same kind of variations as discussed above and the profile of the interacting frequencies ( and ) again coincides for every amplitude and temperature. Their values are identical to the high frequency one in coupling b (dashed lines in Fig. 11).
If we again construct the ratio , then when we consider high temperatures we obtain the same features as in the non-expanding case, but in the "worst cases'' (low temperature and low amplitudes) differences in the slopes are more evident suggesting some spectral in such conditions.
As might be expected, lower temperature atmospheres, with higher gradients, and lower frequency waves allow a stronger dissipation of outwardly propagating waves. On the other hand our results seem to imply that a well developed turbulent spectrum does not change appreciably during propagation.
The dissipation rate has been studied, varying the temperature of the layer and the frequency and amplitude (imposed at the critical point) of the waves. We find that for a given amplitude and frequency, the dissipation rate is stronger in the lower part of the atmosphere and depends strongly on the temperature, which ultimately determines the amount of reflection via the density gradients. As the wave amplitude is increased, the dissipation rate is enhanced, reaching a saturation value which depends on temperature and frequency. Saturation is an effect of nonlinear-dissipative interactions which limit the inward propagating (reflected) wave amplitude once the outgoing amplitude is increased beyond a given value (which also depends on temperature and frequency). Below a critical frequency (whose temperature dependence is negligible), dissipation assumes a constant rate all the way down to "zero-frequency'' fluctuations, while above the critical frequency the dissipation rate tends to zero as frequency is increased, since the amount of reflection decreases with frequency (WKB behavior). The local interaction analysis shows that even if the amplitudes of the inward propagating waves at the base are negligible, continuous reflection due to wind and Alfvén speed gradients can produce significant dissipation (see comments in Fig. 5 at the end of Sect. 3.1); however, only low frequency waves are efficiently dissipated for reasonable wave amplitudes.
We have also investigated a more realistic calculation, considering non-local interactions, where an outwardly propagating Alfvén wave is allowed to interact nonlinearly with the total rms value of inward fluctuations summed over all frequencies (and vice versa). Our approach differs from what has been done in a recent paper of Cranmer & van Ballegooijen (2005) in two related aspects concerning the nonlinear-dissipative interactions. We consider the full expression of the wave action density, hence including the downward propagating flux, or in other words, the nonlinear equations of the Elsässer fields are integrated simultaneously to include the effect of the dissipation in a consistent way. Nonlinear interactions act on both types of (counterpropagating) waves, thereby producing the attenuation of the reflected component which affects the driving of the global dissipation. In our analysis, the description of the total rms value of the fluctuation is approximated by two or three representative waves with different frequencies spread across a spectrum, the "zero frequency'' wave for the low frequencies, and the higher frequencies selected from spectra with different slopes. The strength of the coupling, i.e., the total energy in the outward modes at the lower frequencies (for which reflection is efficient) is crucial for the way in which energy is dissipated along the spectrum. If the amplitude of the low frequency wave is high enough (say at the base for a atmosphere) dissipation of all the outward modes is driven by the low-frequency (quasi-2D) reflected waves: the coupling may be considered strong. A second important aspect is the slope of the spectrum or equivalently the relevant energy residing in the higher frequency waves. Its effect is to enable dissipation of the low-frequency reflected component, since there is little reflected energy at high frequencies, and ultimately to reduce the dissipation efficiency of all the waves coupled. In summary, for a given total outward energy, dissipation is more efficient if the spectra have higher energies at lower frequencies, i.e. steeper spectral slopes.
Setting the lowest frequency to higher values, say , produces some differences in the results, but the global analysis remains unchanged. From Fig. 3 one can guess how the strength of the coupling is affected and hence how the spectra change. In fact the higher the frequency is, the lower the amount of reflected waves; hence the lowest frequency wave is less efficient in driving the dissipation of all the waves coupled. Imposing the same amplitude at the critical point the strength of the coupling is lower, linear effects (i.e. differences in the wave propagation due to different reflection rates) become more important and the spectra show a somewhat higher modification. However, increasing the amplitude of the lowest frequency wave restores the importance of nonlinear terms (the coupling) which overcome the linear effects. These considerations remain valid if one chooses the lowest frequency in the low-frequency plateaus of the curves in Fig. 3. Note that the rightward extension of the plateau increases with increasing wave amplitude.
We have also considered a spherically expanding length scale . In this case, the qualitative features discussed above remain essentially the same (for high enough initial amplitudes). The main effect of the expansion is to reduce the phenomenological gradient 1/L(r) entering the nonlinear term of Eq. (12) as we move further out in the atmosphere so that dissipation in the higher part of the layer is greatly reduced. As a result the amount of energy dissipated is decreased and one can actually attribute differences in the dissipation rate of waves at different frequencies almost entirely to differences in their propagation through the low atmosphere where gradients are greater and hence where most of reflection takes place.
Of course, to better understand the spectral evolution in a stratified atmosphere one should also include the chromosphere, the photosphere and the transition region. Here different physical conditions are encountered and it is not clear how the evolution of the waves in these regions affects the development of "Alfvénic turbulence''.
Modeling the deeper stratified layers as a set of isothermal layer with different temperatures (with a discontinuity across the transition region) produces changes of the parameter which has been held fixed in our analysis. In our case, how changes in affect the result may be discussed qualitatively. First note that for a given temperature, the wind solutions are selected (see Eq. (2)) and hence the wind profile and its gradients remain the same. As is increased the Alfvén speed decreases (still maintaining its characteristic profile, i.e. a maximum at ), its gradients decrease too and the Alfvén critical point moves to lower radii. In the linear case the net effect is an enhanced transmission at low and intermediate frequencies. When nonlinear interactions are taken into account the amplitudes must be scaled to get the same quantitative results, i.e. the same amplitude may be regarded as low in a high plasma or high in a low plasma as far as the strength of the coupling is concerned because, for a given temperature, the strength depends entirely on the amount of reflection. When multiple isothermal layers with different temperatures and different thicknesses are considered, other characteristic length scales and gradients are introduced in the equation describing the wave propagation. The transmission properties of the entire atmosphere are altered, as are the properties of wave dissipation and the spectral slope (we leave this question to a following paper).
Our results concern the evolution of a spectrum formed with only three
frequencies which might not capture the whole shape modification
in a complex atmosphere.
In our model atmosphere, even if highly stratified, the density gradients,
change gradually without discontinuities.
However, the Alfvén waves observed in the solar wind do not necessarily
originate at a photospheric level
and can be generated directly in the corona.
Moreover coronal structures can produce localized gradients so that
the propagation of the waves can be dramatically altered with respect to the
simple case studied here.
One can then say that many processes, different from a
are able to transform a
into a simple spectrum as observed in situ in the solar wind.
We acknowledge financial assistance from the European RT network TOSTISP contract No. HPRN-CT-2001-00310 and from the Italian Ministry of Research contract COFIN No. 2002025872, and the New Zealand Marsden Fund (02-UOW-050 MIS). We would like to thank the IPAM program "Great Challenge Problems in Computational Astrophysics'' at UCLA where this work was completed. A. Verdini would like to thank S. Landi for carefully reading the manuscript and for useful suggestions.
Table A.1: Values of the outgoing wave amplitude (km s-1) imposed at the top of the atmosphere which, once integrated backward, give rms velocity field fluctuations values at the base between and . L(r) signifies that an expanding turbulent lengthscale has been used. Except where indicated a flat spectrum is imposed at the top boundary.In Table A.1 we display the amplitudes at the top of the layer for various frequencies and coupling models which yield the "realistic'' amplitudes at the coronal base. Since the equations are integrated backward, the energy we impose at the critical point decreases as the dissipation increases. In the first box, base values are given for the local wave interaction model; in the second box base values are given for the global interaction model with a flat spectrum, and finally in the third box we display the results obtained when a power-law initial spectrum is imposed at the top of the atmosphere. Presence of an expanding turbulent length is specified directly in the table (labeled with an L(r)).
Note how the differences in the amount of dissipated energy among different couplings (second box) disappear as the power-law scaling is introduced in the initial (top) condition (compare the values relative to cases I, II, III and IV with those ones in the last box). Generally we can say that for lower temperature and lower frequencies (of the coupled waves) the dissipation is higher. Comparing again the results obtained when flat or power-law spectra are imposed as the initial ( ) condition, we can conclude that in the former dissipation efficiency is stronger (Fig. 7 right panel) even if the total amount of dissipation is actually reduced (because the flat spectrum contains a greater amount of energy to start with).