A&A 478, 163-174 (2008)
DOI: 10.1051/0004-6361:20077775
K. Belkacem - R. Samadi - M.-J. Goupil - M.-A. Dupret
Observatoire de Paris, LESIA, CNRS UMR 8109, 92190 Meudon, France
Received 2 May 2007 / Accepted 3 October 2007
Abstract
Context. Turbulent motions in stellar convection zones generate acoustic energy, part of which is then supplied to normal modes of the star. Their amplitudes result from a balance between the efficiencies of excitation and damping processes in the convection zones.
Aims. We develop a formalism that provides the excitation rates of non-radial global modes excited by turbulent convection. As a first application, we estimated the impact of non-radial effects on excitation rates and amplitudes of the high-angular-degree modes that are observed on the Sun.
Methods. A model of stochastic excitation by turbulent convection was developed to compute the excitation rates and then successfully applied to solar radial modes. We generalise this approach to the case of non-radial global modes. This enables us to estimate the energy supplied to high-()
acoustic modes. Qualitative arguments, as well as numerical calculations, are used to illustrate the results.
Results. We find that non-radial effects for p modes are non-negligible:
- For high-n modes (i.e. typically n > 3) and for high values of ,
the power supplied to the oscillations depends on the mode inertia.
- For low-n modes, independent of the value of ,
the excitation is dominated by the non-radial components of the Reynolds stress term.
Conclusions. Our numerical investigation of high-
p modes shows that the validity of the present formalism is limited to
due to the spatial separation of scale assumption. Thus, a model for very high-
p-mode excitation rates calls for further theoretical developments; however, the formalism is valid for solar g modes, which will be investigated in a paper in preparation.
Key words: convection - turbulence - Sun: oscillations
Amplitudes of solar-like oscillations result from a balance between stochastic excitation and damping in the outermost layers of the convection zone, which extends nearly to the surface of the star. Accurate measurements of the rate at which acoustic energy is supplied to the solar p modes are available from ground-based observations (GONG, BiSON), as well as from spacecraft (SOHO/GOLF and MDI). From those measurements and a comparison with theoretical models, it has been possible to demonstrate that excitation is due to eddy motions in the uppermost part of the convection zone and by advection of entropy fluctuations.
Stochastic excitation of radial modes by turbulent convection has been investigated by means of several semi-analytical approaches (Balmforth 1992; Samadi & Goupil 2001; Goldreich et al. 1994; Goldreich & Keeley 1977), they differ from each other in the nature of the assumed excitation sources, the assumed simplifications and approximations, and also by the way the turbulent convection is described (see reviews by Stein et al. 2004; Houdek 2006). Two major mechanisms have nevertheless been identified as driving the resonant p modes of the stellar cavity: the first is related to the Reynolds stress tensor and, as such, represents a mechanical source of excitation; the second is caused by the advection of turbulent fluctuations of entropy by turbulent motions (the entropy source term), and as such it represents a thermal source of excitation (Goldreich et al. 1994; Stein & Nordlund 2001). Samadi & Goupil (2001, hereafter Paper I) proposed a generalised formalism, taking the Reynolds and entropy fluctuation source terms into account. In this model, the source terms are written as functions of the turbulent kinetic energy spectrum and the temporal-correlation function. This allowed us to investigate several possible models of turbulence (Samadi et al. 2003a,b). The results were compared with GOLF data for radial modes, and the theoretical values were found to be in good agreement with the observations (Samadi et al. 2003b). Part of the remaining discrepancies has recently been removed by taking into account the asymmetry introduced by turbulent plumes (Belkacem et al. 2006b,a).
In this paper we take an additional step by extending the Samadi & Goupil (2001) formalism to the case of non-radial global modes. This will enable us to estimate the excitation rates for a wide variety of p and g modes excited in different types of stars. The present model provides the energy supplied to the modes by turbulence in inner, as well as outer, stellar convective regions, provided the turbulent model appropriate for the relevant region is used. Studies of the stochastic excitation of solar radial modes (Samadi et al. 2003a,b) have given us access mainly to the radial properties of turbulence. The present generalised formalism enables us to take the horizontal properties of turbulence into account (through the non-radial components of the Reynolds stress contribution) in the outermost part of the convective zone.
In the Sun, high-angular-degree p modes (as high as one thousand) have been detected (e.g., Korzennik et al. 2004). From an observational point of view, Woodard et al. (2001) found that the energy supplied to the mode increases with , but that above some high-value, which depends on the radial order n(see Woodard et al. 2001, Fig. 2), the energy decreases with increasing . They mention the possibility of an unmodelled mechanism of damping. Hence one of the motivations of this work is to investigate such an issue. As a first step, we develop here a theoretical model of the stochastic excitation taking the -dependence of the source terms into account to seek a physical meaning for such a behaviour of the amplitudes.
Modelling of the mechanisms responsible for excitating non-radial modes is useful not only for high- acoustic modes but also for gravity modes, which are intrinsically non-radial. As for p modes, g modes are stochastically excited by turbulent convection; the main difference is that the dominant restoring force for g modes is buoyancy. We, however, stress that convective penetration is another possible excitation mechanism for g modes (e.g. Dintrans et al. 2005). Such modes are trapped in the radiative interior of the Sun, so their detection promises closer knowledge of the deep solar interior. However, they are evanescent in the convection zone; thus, their amplitudes at the surface are very small and their detection remains controversial. A theoretical prediction of their amplitudes is thus an important issue. It requires an estimation of the excitation rates but also of the damping rates. Unlike p modes, the damping rates cannot be inferred from observations, and this introduces considerable uncertainties; e.g., theoretical estimates of the g-mode amplitudes (Kumar et al. 1996; Gough 1985) differ from each other by orders of magnitudes, as pointed out by Christensen-Dalsgaard (2002). We thus stress that the present work focuses on the excitation rates - damping rates are not investigated. A specific study of gravity modes will be considered in a forthcoming paper.
The paper is organised as follows: Sect. 2 introduces the general formalism, and a detailed derivation of the Reynolds and entropy source terms is provided. In Sect. 3, we demonstrate that the formalism of Samadi & Goupil (2001) is a special case and an asymptotic limit of the present model. In Sect. 4, we use qualitative arguments to determine the different contributions to the excitation rates and identify the dominant terms involving the angular degree ( ). Section 5 presents the numerical results where excitation rates are presented. Section 6 discusses the limitations of the model and some conclusions are formulated in Sect. 7.
Following Paper I, we start from the perturbed
momentum and continuity equation
We use the temporal WKB assumption, i.e. that A(t) is slowly varying with respect to the oscillation period,
(see Paper I for details).
Under this assumption, using Eq. (3) with Eqs. (1) and (2)
(see Paper I) yields:
From Eq. (5), one obtains the mean-squared amplitude
= | |||
(9) |
In the following,
is the large-scale derivative associated
with
,
is the small-scale one associated with
,
and the derivative
operators
and
are associated
with
and ,
respectively.
The mean-squared amplitude can be rewritten in terms
of the new coordinates as
We assume a stationary turbulence, therefore the source
term ( S ) in Eq. (10) is invariant to translation in t_{0}.
Integration over t_{0} in Eq. (10) and using the definition of
(Eq. (6), Eqs. (7) and (8) yields
Equation (12) is first rewritten as
It is then possible to
express the Fourier transform (FT) of the resulting second-order moments
in term of the turbulent kinetic and entropy energy spectrum
(see Paper I for details)
(16) |
The turbulent
Reynolds term Eq. (12) takes the following general expression under the
assumption of isotropic turbulence:
The contribution of the Reynolds stress can thus be written as
(see Appendix A.1)
(24) | |||
(25) | |||
(26) | |||
(27) |
The entropy source term is computed as for the Reynolds contribution
in Sect. 2.1.
Then Eq. (13) becomes
(29) |
The final expression for the contribution of entropy fluctuations
reduces to (see Appendix A.2)
We show in this section that we recover the results of Paper I providing that:
(36) |
= | (37) | ||
= | 0 . | (38) |
With Eqs. (41), (39) and (40) simplify as
We derive asymptotic expressions for the excitation source terms (Eqs. (22) and (31)) in order to identify the major nonradial contributors to the excitation rates in the solar case.
Let us consider the equation of continuity and the transverse
component of the
equation of motion for the oscillations. Let us neglect the Lagrangian
pressure
variation and Eulerian gravitational potential variation at
r=R (the surface). The ratio of the horizontal to the vertical
displacement at the surface boundary is then approximately given by (Unno et al. 1989, p. 105)
(46) |
In what follows, we introduce the complex number f,
which is the degree of non-adiabaticity, defined by the relation
(48) |
Let now compare the derivatives. Under the same assumptions above,
neglecting the
term in
in the radial
component of the
equation of motion (standard mechanical boundary condition),
one gets, near the surface,
Finally, we can group the different terms of Eqs. (23) and (31)
into four sets
Reynolds stress contribution:
We start by isolating non-radial effects in the range . Note that the limit is justified in Sect. 6.1 by the limit of validity for the present formalism. We investigate two cases, and respectively. The condition for which is satisfied for around the f mode for and in the gap between the g_{1} and f mode, for .
Using the set of inequalities Eqs. (55) to (58), for a typical
frequency of 3 mHz (i.e.
), R(r)(Eq. 23) becomes for high-n modes (
):
For low-n modes (
,
i.e. for instance
)
some additional dependency must be retained (see Eq. (57)). One gets
Entropy contribution:
Numerical investigation shows that the non-radial component of the entropy source term does not affect the excitation rates significantly except for , which is out of the validity domain of the present formalism (see Sect. 6.1). The non-radial effects appear through the mode compressibility, (Eq. (33)). From Eq. (57) one can show that non-radial contributions play a non-negligeable role for low-n modes. However, such low-frequency modes are not enough localised in the superadiabatic zone, where the entropy source term is maximum, to be efficiently excited by this contribution.
In the following, we compute the excitation rates of p modes for a solar model. The rate (P) at which energy is injected into a mode per unit time is calculated according to the set of Eqs. (11)-(13). The calculation thus requires knowledge of four different types of quantities:
(62) |
Finally, for the quantities in point 3, the total kinetic energy contained in the turbulent kinetic spectrum (E(k)) is obtained following Samadi et al. (2006).
Figure 1: Top: the rate (P) at which energy is supplied to each mode for is divided by the excitation rate ( ) obtained for the mode. Computation of the theoretical excitation rates is performed as explained in Sect. 5.1. Bottom: ratio where I is the mode inertia. | |
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The rate (P) at which energy is supplied to the modes is plotted in Fig. 1, normalized
to the radial excitation rate (
).
It is seen that the higher the ,
the more energy is supplied to the mode.
This is explained by additional contributions (compared to the radial case) due to
mode inertia, the spherical symmetry (departure from the plane-parallel assumption),
and the contribution of horizontal excitation.
Note that, as discussed in Sect. 3 (see Eq. (41)), the departure from the plane-parallel
approximation is negligible for p modes. Then,
to discuss the other two contributions,
one can rewrite Eq. (4) as
The second term of the product Eq. (63) depends on the non-radial effects through the excitation source terms (Eqs. (31) and (22)). To investigate this quantity independent of the mode mass (defined as ), we plot the ratio in Fig. 1. One can then discuss two types of modes, namely low-n () and high-n (>3) modes (see Fig. 1).
Another quantity of interest is the theoretical surface velocity,
which can be compared to observational data.
We compute the mean-squared surface velocity for each mode
according to the relation (Baudin et al. 2005):
Figure 2 displays the surface velocities for , and 50. Note that the surface velocities are normalized to the maximum velocity of the modes ( 8.5 cm s^{-1}using MLT). This choice is motivated by the dependence of the absolute values of velocities on the convective model that is used, and it is certainly imperfect. However, its influence disappears when considering differential effects. As an indication, 3 error bars estimated from GOLF for the modes are plotted (see Baudin et al. 2005, for details). The differences between the radial and non-radial computations are indeed larger than the uncertainties for . For a more significant comparison, error bars for non-radial modes should be used, but they are difficult to determine with confidence (work in progress). For larger than 50, we do not give surface velocities; as derived, those here depend on the assumption of approximately constant damping rate that is not confirmed for .
When available, observational data should allow us to investigate the two regimes that have been emphasised in Sect. 5.2, namely the high- and low-n modes.
Figure 2: Surface rms velocities of modes calculated using Eq. (65) and normalized to the maximum velocity of the radial modes (see text). Note that the damping rates are taken from GOLF (Baudin et al. 2005) and are chosen to be the same for all angular degrees ( ). Three error bars derived from GOLF are plotted on the curve. | |
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The main assumption in this general formalism appears in Eq. (11),
where it has been assumed that the spatial variation of the eigenfunctions is large compared
to the typical length scale of turbulence, leading to what we call the separation of scales.
In order to test this assumption, one must compare the oscillation
wavelength to the turbulent one or, equivalently, the wavenumbers. To this end,
we use the dispersion relation (see Unno et al. 1989)
For the turbulent wavenumber, we choose to use, as a lower limit, the convective wavenumber
,
where
is the typical convective length scale.
Thus, the assumption of separation of scales is fulfilled, provided
(67) |
Figure 3: Top: ratio of the horizontal oscillation wavenumber to the convective wavenumber ( ), versus the normalized radius (r/R). is computed using the mixing length theory such that ( is the mixing length) and k_{r} is computed using the dispersion relation Eq. (66). Note that the ratio is computed for a frequency of around mHz, depending on the angular degree ( ). Bottom: the same as in the top but for the ratio . | |
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Concerning the radial component of the oscillation wavenumber, the limiting value of seems to be the same (i.e. ). Thus, we conclude that, for modes of angular degree lower than 500 one can use the separation of scales assumption. For , the characteristic length of the mode becomes shorter than the characteristic length of the energy bearing eddies. Those modes will then be excited by turbulent eddies with a length-scale smaller than , i.e. lying in the turbulent cascade. These eddies inject less energy into the mode than the energy bearing eddies do, since they have less kinetic energy. We can then expect that - at fixed frequency - they received less energy from the turbulent eddies than the low-degree modes. A theoretical development is currently underway to properly treat the case of very high modes.
A second approximation in the present formalism is the use of a closure model. The uppermost part of the convection zone is a turbulent convective system composed of two flows (upward and downward), and the probability distribution function of the fluctuations of the vertical velocity and temperature does not obey a Gaussian law (Lesieur 1997). Thus, the use of the quasi-normal approximation (QNA, Millionshchikov 1941), which is exact for a normal distribution, is no longer rigorously correct. A more realistic closure model has been developed in Belkacem et al. (2006a) and can be easily adapted for high- modes. This alternative approach takes the existence of two flows (the up- and downdrafts) within the convection zone into account. However, the QNA is nevertheless often used for the sake of simplicity as is the case here. Note that, when using the closure model with plumes, it is no longer consistent to assume that the third-order velocity moments strictly vanish; however, as shown by Belkacem et al. (2006b,a), their contribution is negligible in the sense that their effect is weaker than the accuracy of the presently available observational data.
We have shown that the excitation rates for high- and n modes are sensitive to the variation in the mode inertia (I). The value of I depends on the structure of the stellar model and the properties of the eigenfunctions in these external regions. Samadi et al. (2006) have shown that different local formulations of convection can change the mode inertia by a small amount. This sensitivity then affects the computed excitation rates (P). However, the changes induced in P are found to be smaller than the accuracy to which the mode excitation rates are derived from the current observations (see Baudin et al. 2005; Belkacem et al. 2006b). Furthermore, concerning the way the modes are obtained, we have computed non-adiabatic eigenfunctions using the time-dependent formalism of Gabriel for convection (see Grigahcène et al. 2005). The mode inertia obtained with these non-adiabatic eigenfunctions exhibits a dependency different from those obtained using adiabatic eigenfunctions (the approximation adopted in the present paper). On the other hand, the mode inertia using non-adiabatic eigenfunctions (see Houdek et al. 1999, for details) obtained according to Gough's time-dependent formalism of convection (Gough 1977) shows smaller differences with the adiabatic mode inertia. Accordingly, the way the interaction of oscillation and time-dependent convection is modelled affects the eigenfunctions differently. As explained in Sect. 5.3, the formalism developed in this paper can be an efficient tool for deriving constraints on the mode inertia to distinguish between the different treatments of convection. Further work is thus needed on that issue.
We extended the Samadi & Goupil (2001) formalism in order to predict the amount of energy that is supplied to non-radial modes. In this paper, we focused on high- acoustic modes with a particular emphasis on the solar case. The validity of the present formalism is limited to values of the angular degree lower than , due to the separation of scale assumption that is discussed above in Sect. 6.1. We have demonstrated that non-radial effects are due to two contributions, namely the effect of inertia that prevails for high-order modes (n > 3) and non-radial contributions in the Reynolds source term in (see Eq. (22)) that dominate the radial one for low-order modes (n < 3).
Contrary to Belkacem et al. (2006b) who used 3D simulations to build an equilibrium model, we restricted ourselves to the use of a simple classical 1D MLT equilibrium model. Indeed, we were interested in deriving qualitative conclusions on nonradial contributions. Forthcoming quantitative studies will have to use more realistic equilibrium models, particularly for the convection description, such as models including turbulent pressure (e.g. Balmforth 1992) or patched models (e.g. Rosenthal et al. 1999).
From a theoretical point of view, several improvements and extensions of the present formalism remain to be carried out. For instance, one must relax the assumption of the separation of scales if one wants to model very high- modes. Such an investigation (which is currently underway) should enable us to draw conclusions about the observational evidence that, beyond some value of the energy supplied to the modes decreases with frequency (see Woodard et al. 2001, Fig. 2). Another hypothesis is the isotropic turbulence that has been assumed in the present work as a first approximation. Such an assumption needs to be given up to get a better description of the nonradial excitation of modes by turbulent convection, which requires further theoretical developments.
The present work focuses on p modes, but the formalism is valid for both p and g modes. We will address the analysis of gravity modes in a forthcoming paper.
The eigenfunctions ( ) are developed in spherical coordinates
and expanded in spherical harmonics.
Hence, the fluid displacement eigenfunction for
a mode with given
is written as
(A.2) |
The Reynolds stress contribution can be written as (see Sect. 2.1)
We now consider the covariant
and the
contravariant
natural base coordinates
where the eigenfunction can be expanded:
g_{rr} | = | ||
g_{ij} | = | (A.10) |
(A.14) |
(A.16) |
(A.17) |
Using the expression Eq. (A.6) for T^{ijlm}, we write
(A.21) |
(A.23) |
= | |||
(A.25) |
= | (A.26) | ||
= | (A.27) | ||
= | (A.28) |
(A.29) |
(A.30) |
(A.31) |
(A.32) | |||
(A.33) | |||
(A.34) |
(A.36) |
(A.38) |
We start from Eq. (28), and to proceed further in the derivation of the entropy
fluctuation source term, one has to compute
(A.43) |
(A.46) | |||
(A.47) | |||
(A.48) |