where r is distance to the centre, m is the mass interior to r, and M is the

total mass; also ,w,„ are respectively the length, velocity and correlation

time scales of the most energetic eddies, determined by the mixing-length

model, I is the mode inertia and ρ is the density. The function S(m, ω)

accounts for the turbulent spectrum, which approximately describes con-

tributions from eddies with di¬erent sizes to the noise generation rate PR ,

and which we implemented as did Balmforth (1992b). The displacement

eigenfunction of a p mode is described by ξ.

A similar expression is obtained for the emission of acoustical radiation

by low-order multipole sources through the ¬‚uctuating entropy. The ratio of

the noise generation rate between the ¬‚uctuating entropy, Ps , and Reynolds

stress is (Goldreich, Murray & Kumar 1994):

1/2

Ps 4

≈ γ1 , (3.2)

PR ±¦

where ± is the mixing-length parameter (which is the ratio between the mix-

ing length and the local pressure scale height) and γ1 is the ¬rst adiabatic

Acoustic radiation and mode excitation by turbulent convection 45

Fig. 3.4. Noise generation rate as a function of frequency for a solar model. Left:

Results, obtained with equation (3.1), are compared with observations by BiSON

(Chaplin et al. 1998) and from the BBSO (Libbrecht 1988). The contribution

from the ¬‚uctuating entropy Ps is about one order of magnitude larger than the

contribution from the ¬‚uctuating Reynolds stress PR . Right: Results are obtained

from hydrodynamical simulations (adopted from Stein & Nordlund 2001). The

contribution from the ¬‚uctuating Reynolds stress (Pturb ) is on average about four

times larger than the contribution from the ¬‚uctuating entropy (Pgas ).

exponent. Assuming typical values for ±, γ1 and ¦ for the solar case, the

value of this ratio is ∼ 3. Consequently the noise generation rate due to the

¬‚uctuating entropy is about one order of magnitude larger than the contri-

bution from the ¬‚uctuating Reynolds stress (see left panel of Figure 3.4).

In contrast, the hydrodynamical simulations by Stein & Nordlund (2001),

depicted in the right panel of Figure 3.4, found the Reynolds-stress contri-

bution to be the larger. From this comparison it is obvious that there is

still controversy as to whether the ¬‚uctuating entropy or Reynolds stress is

the dominating source of excitation and consequently further studies seem

warranted.

With the estimates of “ and PR , the mean-square oscillation amplitudes,

Vs , are obtained from the expression Vs2 = PR /(“ I) (e.g. Houdek et al.

1999). Figure 3.5 shows the mean-square velocity amplitudes for a model

of the Sun and β Hydri. For the Sun results are plotted for computations

in which both observed (solid curve) and theoretical (dashed curve) damp-

ing rates were assumed. Both results are calibrated to the BiSON (Chap-

lin et al. 1998) observations (symbols) by scaling the maximum values of the

computed amplitudes to the measurements. For β Hydri the mean-square

velocity amplitudes are scaled by the factor 1.57, obtained from the scaled

solar model using the theoretical damping rates: the estimated peak value

Houdek

46

Fig. 3.5. Mean-square velocity amplitudes for models of the Sun and β Hydri,

assuming equation (3.1) for the total noise generation rate.

of 65 cm s’1 for the velocity amplitude is in reasonable agreement with the

measured value of 60 cm s’1 by Bedding et al. (2001).

3.4 Acoustic radiation in the equilibrium model

In the previous section acoustic radiation was discussed in a pulsating atmo-

sphere. This very mechanism is also working in the static model: through

the generation of sound waves, kinetic energy from the turbulent motion

will be converted into acoustic radiation and thus reduce the e¬cacy with

which the motion might otherwise have released potential energy originating

from the buoyancy forces. This will result in a di¬erent strati¬cation of the

convectively unstable layers.

The implementation of the acoustic ¬‚ux in the dynamical equations de-

scribing the convective motion of the turbulent eddies is accomplished in a

straightforward way by adopting the phenomenological picture of an over-

turning eddy. In this picture the ¬‚uid element maintains balance between

buoyancy forces and turbulent drag by continuous exchange of momentum

with other elements and its surroundings (e.g., Unno 1967). The equation

of motion for a turbulent element can then be written as

2w2 w2 µ

δ

= g T ’ Λ Mt , (3.3)

T

where T and T denote the mean temperature of the background ¬‚uid and

the convective temperature ¬‚uctuations, respectively, g is the acceleration

due to gravity, δ = ’(‚ ln ρ/‚ ln T )pg , and the constants Λ and µ are the

emissivity coe¬cient and Mach-number dependence, respectively. The tur-

bulent Mach number is de¬ned as Mt = w/c (c being the adiabatic sound

speed). The last term on the right-hand side of equation (3.3) is derived

Acoustic radiation and mode excitation by turbulent convection 47

Fig. 3.6. Maximum value of the turbulent Mach number as a function of model mass

along the ZAMS. Results are displayed for model computations in which acoustic

radiation was either omitted (Λ = 0) or included assuming di¬erent values for the

acoustic emissivity Λ and Mach-number dependence µ.

from the Lighthill-Proudman formula (Proudman 1952) and represents a

drag due to the acoustic radiative losses, additional to the turbulent drag

expressed by the left-hand side, which together are balanced by the potential

energy coming from the buoyancy forces (¬rst term on the right-hand side).

We consider two simple models for the emission of acoustic waves by ho-

mogeneous, isotropic turbulence: in the ¬rst model the acoustic emission

is dominated by the energy-bearing eddies and is thus scaled by a Mach-

number dependence of µ = 5 (Lighthill 1952). In the second model the

acoustic radiation is predominantly emitted by inertial-range eddies, as sug-

gested by Goldreich & Kumar (1990), who derived a Mach-number depen-

dence of µ = 15/2. For the emissivity coe¬cient we adopt for the model

with µ = 5 the value Λ = 100, as suggested by Stein (1968) for a solar model,

and for the model with µ = 15/2, a value of Λ = 1000, which provides a

value for the acoustic ¬‚ux similar to the model with µ = 5.

The turbulent Mach numbers for models along the ZAMS with masses

of 1.0 “ 1.9 M are depicted in Figure 3.6. The results are displayed for

computations in which the acoustic ¬‚ux was either omitted (Λ = 0; con-

tinuous curve) or included assuming di¬erent values for the emissivity co-

e¬cient Λ and the Mach-number dependence µ (see eq. 3.3). The e¬ect of

acoustic radiation is essentially negligible for models with masses less than

Houdek

48

Fig. 3.7. Noise generation rate as a function of p mode frequency for a 1.6 M star

of age 1.49 Gy. Results are displayed for various values of Λ and µ.

about 1.5 M , and becomes largest for models with masses M in the range

1.5 M < M < 1.8 M . The e¬ect of acoustic radiation in the equilibrium

∼ ∼

model on the rate at which energy is injected into the individual p modes,

PR , is displayed in Figure 3.7 for a 1.6 M model of age 1.49 Gy. Interest-

ingly, at frequencies near where PR is largest, it becomes larger for models

in which acoustic radiation was included in the computations; however, the

di¬erences are still small. This comes about because PR depends also on

the shape of the modal eigenfunctions (particularly at the top of the con-

vection zone), which are slightly modi¬ed in a compensating way between

models obtained with and without the inclusion of acoustic radiation in the

equilibrium structure. Computed damping/growth rates for a model of

the Delta Scuti star BW Cnc are portrayed in Figure 3.8. These results

can be compared with observations from campaigns of the STEPHI network

in the Praesepe cluster (Michel et al. 1999). Michel et al. suggest for the

observed Delta Scuti star BW Cnc a mass of 1.6 “ 1.65 M and an age of

700 “ 800 My. The top panel of Figure 3.8 displays the results for a model

in which acoustic radiation was omitted in the computations (Λ = 0) and