40

Fig. 3.1. Velocity power spectrum of beta Hydri. The solid spectrum displays the

Doppler measurements by Bedding et al. (2001). The dashed spectrum represents a

theoretical expectation, obtained by ¬rst rescaling solar data (Chaplin et al. 1998)

in frequency in the manner of Houdek et al. (1999) and then reducing the ampli-

tudes by a factor of 1.9 (adopted from Gough 2001).

balance between the excitation and damping, and are expected to be rather

low. The turbulent-excitation model predicts not only the right order of

magnitude for the p-mode amplitudes (Gough 1980), but it also explains

the observation that millions of modes are excited simultaneously. Using a

time-dependent generalization (Gough 1977) of a local mixing-length model,

Gough (1980) found most of the radial p modes in the Sun to be stable.

Balmforth (1992a) obtained better agreement between theoretical damping

rates and solar linewidth measurements using the nonlocal generalization of

the mixing-length model by Gough (1976). Moreover, with the help of these

improved damping rate estimates Balmforth (1992b) obtained reasonable

agreement between computed and measured solar velocity amplitudes.

The most convincing evidence to date of solar-type oscillations in other

stars comes from recent observations of β Hydri (Bedding et al. 2001) and

± Cen A (Bouchy & Carrier 2001). In Figure 3.1 the power spectrum of

the Doppler velocity measurements of β Hydri (solid spectrum) is compared

with a theoretical expectation (dashed spectrum). The theoretical spectrum

was obtained by ¬rst scaling solar data (Chaplin et al. 1998) in frequency

by the ratio of the acoustic cut-o¬ frequencies between a model of β Hydri

Acoustic radiation and mode excitation by turbulent convection 41

and the Sun and in amplitude in the manner of Houdek et al. (1999); the

amplitudes were then reduced by a factor of 1.9 to render the total power

in the frequency interval 0.67 mHz < ν < 1.5 mHz the same as that of the

measured β Hydri spectrum.

In this contribution I shall adopt Balmforth™s excitation model to predict

amplitudes of radial p modes for the Sun and β Hydri. Stellar models and

damping rates are computed in the manner of Houdek et al. (1999).

3.2 Linear damping rates, “

Basically, the damping of stellar oscillations arises from two sources: pro-

cesses in¬‚uencing the momentum balance, and processes in¬‚uencing the

thermal energy equation. Each of these contributions can be divided fur-

ther according to their physical origin (see Houdek et al. 1999 and references

therein). Here we limit the discussion to the convection dynamics only.

Vibrational stability is in¬‚uenced crucially by the exchange of energy be-

tween the pulsation and the turbulent velocity ¬eld. The exchange arises ei-

ther via the pulsationally perturbed convective heat ¬‚ux, or directly through

dynamical e¬ects of the ¬‚uctuating Reynolds stress. In fact, it is the mod-

ulation of the turbulent ¬‚uxes by the pulsations that seems to be the pre-

dominant mechanism responsible for the driving and damping of solar-type

acoustic modes. It was ¬rst reported by Gough (1980) that the dynami-

cal e¬ects arising from the turbulent momentum ¬‚ux (also called turbulent

pressure pt ) perturbations contribute signi¬cantly to the mode damping (see

“t in the bottom panel of Figure 3.2). Moreover, he predicted a character-

istic plateau in the damping rates centred near 2.9 mHz (see Figure 3.2).

This plateau was later con¬rmed observationally by Libbrecht (1988) (see

also Christensen-Dalsgaard, Gough & Libbrecht 1989). Detailed analyses

(Balmforth 1992a) reveal how damping is controlled largely by the phase

di¬erence between the momentum perturbation and the density perturba-

tion. Therefore, turbulent pressure ¬‚uctuations must not be neglected in

stability analyses of solar-type p modes.

Solar p-mode frequencies vary with the solar cycle, the frequencies being

largest at sunspot maximum. Theoretical studies have shown that variations

in the thermal structure of the super¬cial layers of the Sun cause positive p-

mode frequency variations to be associated with a reduction in the e¬ective

temperature, and hence irradiance (Gough & Thompson 1988; Gough 1990;

Goldreich et al. 1991; Balmforth, Gough & Merry¬eld 1996). Additional

information of variations of p-mode properties over the solar cycle can be

gained by studying theoretical p-mode damping rates. Today fairly accu-

Houdek

42

Fig. 3.2. Top: Linear damping rates for a solar model, computed (curve) in the

manner of Houdek et al. (1999), are compared with linewidth measurements (sym-

bols) by BiSON (Chaplin et al. 1998) and from the LOI instrument (Appour-

chaux et al. 1998). Bottom: Contributions to the computed damping rates arising

from the gas (“g ) and turbulent (“t ) pressure perturbations.

rate linewidth measurements of the spectral peaks in the acoustical power

spectrum are provided by ground-based and space-born instruments, cover-

ing a substantial time period. Over the 11-year period of the solar cycle,

the radiative interior of the sun will barely be a¬ected by thermal distur-

bances, because it relaxes di¬usively on a Kelvin-Helmholtz time of about

Acoustic radiation and mode excitation by turbulent convection 43

Fig. 3.3. Solar-cycle changes in linewidths from activity minimum to maximum.

The plotted data (with error bars) were obtained from BiSON (Chaplin et al. 2002)

measurements. The solid curve shows the modelled variations which result from

decreasing the eddy shape parameter ¦ by 6 per cent, while keeping the entropy

in the deep layers of the convection zone constant during the cycle (Houdek et al.

2001).

3 — 107 years. As discussed by Gough (1981) the convective envelope adjusts

itself internally to any perturbation on a timescale of a month (which is much

less than 11 years), whereas its thermal cooling timescale is about 2 — 105

years (which is much greater than 11 years). Consequently, on a timescale

of 11 years, the heat ¬‚ux remains divergence-free, and the entropy is essen-

tially invariant. In modelling the solar-cycle variation, one has to keep the

entropy in the deeper convectively unstable layers constant in models for

which the structure is varied to simulate the changes. In Figure 3.3 simula-

tion results of damping rate variations are compared with measurements of

linewidth variations over the cycle 23. The solar-cycle changes are modelled

by varying the horizontal extend of the convective eddies, represented by

the eddy shape parameter ¦ (Gough 1977), which is of order unity. The

parameter ¦ had to be reduced in order to obtain a fair agreement with the

BiSON measurements, which suggests that the horizontal granule size is de-

creasing with activity. This is indeed in agreement with the observations by

Muller (1988). Moreover, the decrease of 6 per cent in ¦ is in fair agreement

with the measured value of 5 “ 10 per cent (Roudier & Reardon 1998).

Houdek

44

3.3 Stochastic excitation

Huge progress on stellar convection has been made recently with the help

of hydrodynamical simulations of the outer parts of the convection zone in

solar-type stars (e.g., Stein & Nordlund 2001). However, such simulations

are very time consuming and consequently simple convection models are still

needed in evolutionary computations. Essentially in all such simple models

the anelastic (or Boussinesq) approximation (Gough 1969) to the ¬‚uid equa-

tions is assumed. In this approximation the time derivative of the density

¬‚uctuation in the continuity equation is neglected, which is equivalent to

¬ltering out high-frequency phenomena such as sound waves. Consequently

a separate model is needed to describe approximately the acoustical noise

generated by the turbulent motion of the convective eddies. Such a model

was proposed by Lighthill (1952): in this model the density ¬‚uctuations are

the same between a real ¬‚uid with highly nonlinear motion and a ¬ctitious

acoustic medium with linear motion upon which an external stress system is

acting. Balmforth (1992b) reviewed the theory of acoustical excitation in a

pulsating atmosphere, and, following Goldreich & Keeley (1977), he derived

the following expression for the rate of energy injected into a mode with fre-

quency ω by quadrupole emission through the ¬‚uctuating Reynolds stress:

M

2

π 1/2 ‚ξ

ρ 3 w4 „ S(m, ω) dm ,

PR = (3.1)

8I ‚r

0