A&A 405, 779-786 (2003)
DOI: 10.1051/0004-6361:20030664
M. Roth^{} - M. Stix
Kiepenheuer-Institut für Sonnenphysik, Schöneckstr. 6, 79104 Freiburg, Germany
Received 18 October 2002 / Accepted 5 May 2003
Abstract
We investigate the effects of a large-scale time-dependent
flow in the solar convection zone on the solar p-mode
oscillations. The theory of time-dependent perturbations
is applied, and we concentrate on flow fields that can
be described by a single harmonic in space and time.
An iterative method of obtaining approximate analytical
solutions to the equations of the coupled oscillator is
outlined. Example calculations are presented for the
special case of two coupling partners. Special attention
is paid to the resonance that occurs when the time
dependence of the flow meets the beat frequency of two
p modes.
We conclude that time-dependent flow fields in the solar
convection zone may diminish the height of the peaks in
the oscillation power spectrum, and may contribute to their
asymmetry, broadening, and effective shift.
Key words: Sun: convection - Sun: oscillations
The interaction of the large-scale velocity field on the Sun with the global solar oscillations is best known from the rotationally induced splitting of p-mode frequencies into multiplets. Typically this splitting occurs for a toroidal velocity field such as rotation, and results from self-coupling of individual eigenoscillations. In contrast to this, a poloidal velocity field may couple two different oscillations, especially when their frequencies lie close together, which is called quasi-degeneracy. Lavely & Ritzwoller (1992) have outlined the application of perturbation theory to quasi-degenerate oscillations, and Roth & Stix (1999) have calculated the effect of a steady-state poloidal velocity upon the oscillation frequencies. This theory has been used to derive upper limits for large-scale poloidal velocity fields in the solar convection zone (Roth et al. 2002).
On the other hand, the velocity field in the solar convection zone is
not steady. Especially the small-scale flow, e.g. the granulation, is
strongly variable, on time scales of minutes. Larger-scale flow
components have longer lifetimes, e.g. up to several days for the
supergranulation, and weeks to months for velocity fields that have a
scale comparable to the depth of the convection zone. Thus, since the
coupling velocity varies with time, its effects on the solar
oscillations must be time-dependent, too. We have pointed out earlier
(Roth & Stix 2001)
that a time-dependent perturbation may lead to a transfer of oscillation
energy between the modes that participate in the coupling. In the
present paper we shall further study this subject. But we shall restrict
the case to large-scale motions that vary slowly in comparison with the
oscillations themselves. Moreover, in spite of the time-dependence of
the velocity field ,
we use the condition
The coupling of solar p-mode oscillations has a well-known analogy
in the coupling of pendulums. Let two un-coupled pendulums with masses m_{1} and m_{2} oscillate with frequencies
and ,
respectively. Coupling with a spring constant k yields an
equation
In the subsequent two sections we shall outline the theory for time-dependent perturbations and describe an iterative method of solution. In Sect. 4, then, we apply the general scheme to the special case of two coupling solar p modes, and discuss the possible effects on the appearance of such modes in the power spectrum.
The starting point is the unperturbed equation of oscillation,
We expand the general solution
in terms of
the set
,
The system (11) of ordinary differential equations is of second order with non-constant coefficients. It describes the evolution of the system of oscillations under the influence of the perturbation , and constitutes an initial-value problem by the specification of the coefficients at the instant t=0, i.e., c_{m}(0)=c_{m}^{(0)}. There are various processes on the Sun that could be accounted for the perturbation . Here we shall concentrate on a time-dependent flow field in the convection zone.
The initial-value problem can be solved numerically for any configuration of the flow field. On the other hand, we may gather more insight into the system with the aid of approximate analytical solutions. A general analytical solution does not exist for an arbitrary time-dependence of H_{1} (Walter 1993).
According to our choice (6) above we have
A closed analytical solution of the system (14) is not possible;
but approximative methods can be used. One possibility to obtain an
approximate solution is the ansatz
Comparison of the coefficients that correspond to the zero-order in
yields the differential equation
In the first order of the expansion (15) in powers of
we find the equation
The second order yields the differential equation
For higher orders we can continue in this manner.
If the coupling is weak, i.e., if
A close view to the solutions (20) and (23) makes clear that in the diverse orders we obtain large contributions for eigenstates that are quasi-degenerate. In comparison to these, the contributions that result from oscillations with a large distance in frequency can be neglected.
The iterative method yields the next higher order from the preceding one by multiplication with and subsequent integration. Because of this, ever higher harmonics of the fundamental frequencies and appear, as well as combinations of these. Therefore the amplitude of the oscillation varies additionally with these harmonic "overtones''.
In order to demonstrate the coupling of solar oscillations by a
variable flow field, we now consider a system that consists only
of two coupling modes.
Figure 1: Comparison of the numerical solution (solid line) of Eqs. (25) and (26) of the two-coupler system with the approximate solutions according to expansions (27), in zero ( ), first ( ), and second (--) order. The two modes have frequencies Hz and Hz. The initial conditions are c_{1}(0)=1, c_{2}(0)=0.7, and . The perturbation has the magnitude and a frequency (81 days). | |
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In the case of two coupling modes we must solve the equations
The iterative method described above is based on the ansatz
For a special example we compare the approximate solutions with a numerical solution (Fig. 1). We take two oscillations with frequencies Hz and Hz, respectively, that are coupled by a time-dependent perturbation. This perturbation can be regarded as the real part of (6), with and (81 days); is defined by an integral of form (12). Initially the two oscillations have prescribed amplitudes, given by c_{1}(0)=1, , and c_{2}(0)=0.7, . For the perturbation we use the initial condition H_{1}(r,0)=0.
If there was no coupling, the coefficients c_{1} and c_{2} would keep their initial values. In fact this state is represented by the zero-order approximate solution. The approximate solutions of higher orders, as well as the numerical solution, show the variation of c_{1} and c_{2} around those initial values. Figure 1 demonstrates that the numerical solution is rather well approximated in the second order already.
At first sight it would appear that the first and second-order solutions, as well as the numerical solution, fail to satisfy the initial conditions . Actually they do so by means of contributions that vary on the fast time scale of the two p modes, and that have amplitudes of magnitude or relative to the amplitudes of the leading terms. The scales of Fig. 1 are such that these contributions are suppressed.
The example shows that time-dependent coupling of solar oscillations
can cause large variations of the amplitude of the diverse modes.
Moreover, the approximate solution makes clear that these variations
are superpositions of oscillations with certain frequencies, namely
,
,
,
etc., - all the combinations of
and
that appear in the diverse approximations.
Figure 2: Squared magnitude of the coefficient c_{1}(t) for the coupling of two oscillations with frequencies Hz and Hz, respectively, and a perturbation in resonance, with , and . A numerical solution of (25) and (26), with initial conditions c_{1}(0)=1, c_{2}(0)=0, and , (i=1,2). | |
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In order to obtain the general solution we have to multiply the coefficients c_{i} by . We see that, beginning with the first order, the eigenvector gets an admixture of its coupling partner, with frequency , and vice versa. The magnitude of this admixture is time-dependent, and can be considered as a contribution of a virtual state with frequency . The strength of the admixture depends on the frequency difference, the initial excitation of the oscillations and on the strength of the coupling, given by the matrix element H_{12}. The properties of the matrix elements have been discussed by Lavely & Ritzwoller (1992).
For certain frequencies
there are singular terms in the
approximate solutions (20) and (23), or (29)
and (30). In the first and second order of the two-coupler
system this is the case for
The zero-order solution is again given by (28). In the first
order we find
In the second order we have
In contrast to the non-resonant case discussed earlier, we have included into the approximate solutions (32)-(34) all the terms required to satisfy the initial conditions in the first and second order, namely for i=1,2 and j=1,2. Those terms can cause growing terms in subsequent higher orders. Thus, only the four integration constants that arise in the zero-order solution are left open.
In Fig. 2 we show the behavior of the coefficient c_{1}(t)for an example resonance case, as obtained from the numerical solution of (25) and (26). It clearly shows the linear growth with time of the oscillation amplitude, which was found in the approximate solutions. The other coefficient, c_{2}(t), also grows beyond limits. The simultaneous growth in amplitude of both oscillations is possible because the perturbation is not Hermitian, and hence the oscillation energy is not conserved.
Figure 3: Power spectrum (upper solid curve) of a solar p-mode with frequency Hz, calculated for a perturbation by a time-dependent flow, with contributions up to order . The frequency of the coupling partner (not shown) is Hz; the flow parameters are m/s, s=t=5 and s^{-1}. (1): Unperturbed p-mode. (2): p-mode with side lobes at . (3): Additional side lobe at , where . (4): Additional side lobes at . (5): Additional side lobes at . (6): Additional side lobes at (very small, at bottom). In addition to the sum, each panel shows the actually added power (lower dash-dotted black curve), and the power added in the preceding panels ( dash-dotted grey). | |
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Of course, the growth will be limited at some stage. We suppose that this limitation is not achieved by one of the two classical mechanisms that counteract resonance, namely damping and non-linearity. Damping is very weak in the solar interior, and only in the atmosphere it might have a small effect; it is well-known that adiabatic theory reproduces the solar eigenfrequencies with a precision of 10^{-3}and better. The non-linear terms of the hydrodynamic equations could become crucial in principle. But the observed amplitudes of the solar oscillations are so small that the frequency shift that goes along with the non-linear saturation (Landau & Lifschitz 1963) does not take place. Instead, we think that the growth shown in Fig. 2 proceeds only for a limited amount of time because the coupling velocity changes. The assumption (6) of a purely periodic coupling is too crude. The initial growth time of our resonances is of order , as can be seen from the approximate solutions (32)-(34) as well as from Fig. 2; during this time the coupling velocity in the solar convection zone might change, so that the mutual growth and decay of eigenoscillations is a transitory phenomenon. Indeed, Roth (2001) found some evidence for the exchange of oscillatory power among the Sun's oscillations.
For stars other than the Sun, with different excitation mechanisms, the non-linear saturation might be crucial (Christy 1964). In addition, the superposition of several resonances may lead to chaotic behavior, and to irregular stellar variability (Perdang 1985). For the Sun we do not expect such effects.
We shall now discuss the possible consequences of a time-dependent large-scale flow to the interpretation of an observed power spectrum of solar oscillations.
The coupling of the oscillations depends on the variation of the flow field, which in the present paper is represented by the frequency and the coupling matrix H_{nm}. As a result of the coupling the oscillations are mixtures of eigenstates of the unperturbed system. Thus, it is generally not possible to measure pure states. However, as long as the coupling is weak, a mixed state still consists predominantly of one of the original eigenstates, with a small admixture of other states.
In addition to the coupling, we must realize that in general the oscillations are damped, and hence the peaks in the power spectrum have a natural line width. This is true for pure as well as for mixed states. Usually the mode frequencies are determined from the observed data by fitting Lorentzians to the peaks in the power spectrum.
Our results suggest that the power of a state with frequency leaks into the admixed states, but also that some of the power in the admixed states originates from leakage of other states (those that couple). That means that side lobes emerge with heights that depend on the strength of the coupling and on the initial conditions. We conclude from the iterative method that these side lobes are located at , , , , , etc., where is the frequency of a coupling partner.
For a mode with frequency Hz and l=22, n=14, m=16, Fig. 3 displays the power distribution according to the expansion (27). The coupling partner has a frequency Hz, and l'=27, n'=13, m'=21. For this calculation we assumed that, due to the natural damping, the single peaks can be represented by Lorentzians. Side lobes to the primary peak appear at frequencies where the diverse denominators of expressions (20) and (23) become small (notice that this example is not a resonant case). The 6 panels illustrate how the diverse side lobes contribute. We have truncated the expansion after the second order. The error of the expansion is therefore of the order . The flow chosen for this example has a parabolic depth dependence of u_{s}^{t}(r) according to Eq. (6), and m s^{-1}; the harmonic degree and azimuthal order are s=t=5, the time dependence is . In this case the first side lobes are already about 100 times smaller than the main peak.
Due to the asymmetry of the admixture emerging from the coupling
partners the center of gravity of the peak is shifted. Hence there is
an effective frequency shift relative to the frequency of a mode of
the equilibrium model. This means that the considerations concerning
the frequency shifts that we had made for a steady-state flow, must
be generalized. Time-dependent flow fields lead to diminished,
asymmetric, and broadened and effectively shifted peaks in the power
spectrum. The frequency shift is equivalent to an average effect
of the flow. As in the case of a steady flow we find that the
shift of the frequency of one mode is accompanied by a frequency
shift of its coupling partner with the same magnitude, but with
opposite sign. This means that, of two coupling modes with
frequencies
and ,
one mode has side lobes at
,
the other at
(cf. Fig. 4).
Figure 4: Power spectra of two modes, at Hz (black) and Hz (grey), under time-dependent coupling by a flow field with s^{-1}, m/s, and initial conditions A_{1}^{(0)}=A_{2}^{(0)}=1 and B_{1}^{(0)}=B_{2}^{(0)}=0. The dash-dotted curves are the Lorentzian profiles of the two coupling modes, the vertical dotted lines mark the centers of gravity of the two resulting spectra. | |
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As for the interpretation of an observed power spectrum, we may
conclude that a peak consists of a prominent main peak and several
side lobes. As an example, Fig. 5 compares the result
shown in Fig. 3 (lower right panel) with the power spectrum
of the same mode (l = 22, n = 14, m = 16), as derived from MDI
data.
Figure 5: Comparison of the perturbation calculation, truncated after the second order, for the p mode shown in Fig. 3 ( upper curve), and a power spectrum for the same p mode as derived from MDI data ( lower curve). The dotted line marks the center of gravity of the upper spectrum. | |
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