6.2 Excitation of p-modes

Solar oscillations are excited by the convection, both by the entropy fluctuations produced by radiative cooling near the surface and by the Reynolds stresses below the surface where the convective velocities are large (Goldreich et al., 1994Jump To The Next Citation Point). The same processes excite oscillations in convection simulations. In simulations, the spectrum of excited oscillations depends on the size of the computational domain, the wider and deeper it is the richer the spectrum of resonant modes that exist (Figure 33View Image).

Mode excitation may be investigated using the kinetic energy equation for the modes. For radial modes this is the equation for the horizontally averaged variables (indicated by an overbar),

[ ] ¯ρ-D- 1¯u2z = − -∂- ¯uz(P¯g + P¯t − ¯τzz) Dt 2 ∂z ¯ ¯ ∂¯uz- + (Pg + Pt − ¯τzz) ∂z + ¯ρ¯uzg , (44 )
where ¯ Pg is the gas pressure, ¯ Pt is the turbulent pressure, and τ¯zz is the viscous stress tensor, all averaged over horizontal planes. Likewise ¯ρ = ⟨ρ⟩hor is the horizontally averaged mass density, and ¯uz = ⟨ρuz ⟩hor∕¯ρ is the density weighted average vertical velocity (Nordlund and Stein, 2001Jump To The Next Citation Point). Integrate over the spatial domain and time. If there is no net mass displacement the buoyancy work term vanishes. If there is no displacement on the boundaries (e.g., at the bottom), or if the pressure fluctuations vanish on the boundaries (e.g., at the top), then the integral of the divergence term vanishes. The remaining term is the P dV work integral,
∫ ∫ ( ) ( ˙¯) W = dtdz δ ¯Pg + δP¯t ( ∂ξ) , (45 ) Δt z ∂z
where ξ is the displacement and ˙ξ is the velocity. Here δ denotes the “pseudo-Lagrangian” fluctuation, that is the fluctuation in the frame moving with the average radial velocity ¯uz in which the net vertical mass flux vanishes (cf. Nordlund and Stein, 2001). The pressure fluctuations and displacement can be split into modal and convective parts. The gas pressure fluctuations can further be split into an adiabatic part (ad) proportional to the density fluctuation,
∂ ξ δ lnP¯ad = Γ 1δln ¯ρ = − Γ 1- , (46 ) ∂z
and the remaining non-adiabatic part (non-ad). The dominant term is the product of the turbulent pressure plus non-adiabatic gas pressure fluctuations with the divergence of the mode displacement, so the change in the mode amplitude in a time interval Δt is,
∫ ∫ ( ¯ non−ad ¯) ∂ξω- ΔE ω = Δt dt z dz δ Pg + Pt iω ∂z , (47 )
where Eω, the mode energy per unit surface area (at r = R), is
∫ ( ) 1- 2 2 r- 2 2 Eω = 2ω r drρ ξω R = M ωV ω . (48 )
Here M ω is the mode mass and ˙ Vω = ξω(R ) is the mode surface velocity amplitude. The ensemble average < > of the change in mode energy for small changes in amplitude Vω → Vω + ΔV ω over all phases between ΔV ω and Vω is proportional to
⟨| |⟩ ⟨ ⟩ Δ ||Vω2|| = |ΔV ω |2 . (49 )
The result for the energy increase per unit area in time interval Δt is
| | ω2||∫ dz δP∗ξω||2 Δ--⟨E-ω⟩ = ----z-----ω∂z---, (50 ) Δt 8Δ νE ω
where δP ∗ ω is the imaginary part of the sum of the pseudo-Lagrangian fluctuations of the turbulent pressure (Reynolds stress) and non-adiabatic gas pressure (entropy)
non− ad δP = δP⟨t + δ⟩Pg δPt = δ ρu2 z horiz δP ngon−ad = Pg (δlnPg − Γ 1δlnρ ) (51 )
at frequency ω, ξω(z) is the displacement eigenfunction for the mode with frequency ω, and Δ ν = 1∕ ΔT, where T is the time interval over which the expression is evaluated. This factor appears when the discrete Fourier transform is normalized such that the sum of the squares of the absolute values of the discrete amplitudes equals the average of the squares of the fluctuations in time (Parseval’s theorem). If the power spectral density were normalized per unit frequency interval, then the factor would disappear and the total power would be the time integral of the squared amplitude rather than its average. The appearance of E ω in the denominator makes the expression independent of the arbitrary normalization of the mode eigenfunctions ξω. The terms in this expression can then be evaluated using the results of realistic convection simulations (Figure 36View Image) which yields good agreement with observations, or using a model of the convective turbulence properties.
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Figure 36: Comparison of observed and calculated p-mode excitation rates for the entire Sun. Squares are from SOHO GOLF observations for ℓ = 0 − 3 (Roca Cortés et al., 1999) and triangles are the simulation calculations (Equation 45View Equation).

Several people have derived equations for p-mode excitation using analytic expressions for the convective kinetic energy and entropy spectra, e.g., (Balmforth, 1992Goldreich et al., 1994Samadi and Goupil, 2001). These authors’ formulas and Equation (50View Equation) are similar. In all, the excitation rate is proportional to the turbulent pressure and entropy induced pressure fluctuations times the gradient of the displacement eigenfunction divided by the mode mass. The main difference is that Equation (50View Equation) contains the absolute value squared of the product of the pressure fluctuations with the mode compression, while the others contain the the absolute value squared of the mode compression times the sum of the absolute values squared of the turbulent pressure and entropy fluctuation contributions. This difference results from the assumption (explicit or implicit) that there are no correlations between the turbulence at different scales and that the mode compression, ∂ξ∕∂r, does not change on the length scale of the turbulence and so can be removed from the integral over the local turbulence. Neither of these assumptions is correct and is not necessary. Chaplin et al. (2005) have derived an analytic expression for the mode amplitude where the pressure-mode compression interaction is retained.

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Figure 37: Exponent n(k) and width w(k) of the convective frequency spectrum as a function of horizontal wavenumber for several depths in the convection zone.

Aside from the above simplifying assumptions, there is the question of characterizing the turbulence properties. It is generally assumed that the convective energy spectrum is separable into independent spatial and temporal factors. Convection simulations have revealed that this is not possible. The convection velocity spectrum, Pvel is found to have the form (Georgobiani et al., 2006)

2 2− n(k) Pvel(ν ) = A(ν + w (k) ) (52 )
The width and exponent for the vertical velocity are shown in Figure 37View Image.

Samadi et al. (2003a,b) investigated in detail the effects of different assumptions about the turbulence properties on the p-mode excitation rate. They found that the mode driving is particularly sensitive to the turbulence anisotropy factor and to the dynamic properties of the turbulence as represented by the temporal part of the turbulence spectrum. Using the simulation results, fit by analytic expressions for these factors, increases the excitation rate and the contribution of turbulent pressure relative to that of entropy fluctuations.

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Figure 38: Mode mass increases with decreasing frequency because the modes extend deeper into the Sun. The simulation modes exist only in a shallow box, so the low frequency modes cannot extend deeply.

What physics determines the spectrum of the p-mode driving? Mode driving (Figure 36View Image) decreases at low frequencies for two reasons: first, the mode mass (inertia) increases with decreasing frequency (Figure 38View Image), and second the mode compression decreases with decreasing frequency (Figure 39View Image). The mode driving decreases at high frequencies because convection is a low frequency process (see Figure 33View Image), so the power in the pressure fluctuations decreases at high frequencies (Figure 40View Image).

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Figure 39: Mode compression decreases with decreasing frequency because the eigenfunctions vary more slowly.
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Figure 40: Convective pressure fluctuations decrease at high frequencies because convection is a low frequency process.

In the Sun and cool stars the contributions from the Reynolds stresses and entropy fluctuations are comparable at the peak driving frequencies, while the Reynolds stress excitation dominates at lower frequencies where the excitation occurs over a larger range of depths. In hotter stars and giants the Reynolds stress excitation dominates at all frequencies.

P-mode excitation occurs close to the solar surface where the turbulent velocities and entropy fluctuations are largest (Figure 41View Image). As the frequency increases the driving is more and more confined to a shallow layer just below the surface.

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Figure 41: P-mode excitation as a function of frequency and depth. The image shows the logarithm (base 10) of the absolute value squared of the work integrand, normalized by the factors in front of the integral in expression for the excitation rate, || ∂ξ ||2 ω2|δP∗ω-∂ωr| ∕8Δ νE ω, (in units of erg/cm4/s).

Jacoutot et al. (2008b) and Jacoutot et al. (2008a) have investigated the influence of the presence of magnetic fields on the excitation of p-modes. They find that regions with magnetic field strengths typical of the peripheries of active regions have enhanced excitation of p-modes, particularly at high frequencies. This is consistent with the observations of so-called “acoustic halos” (Braun et al., 1992Brown et al., 1992Hindman and Brown, 1998Jain and Haber, 2002) around active regions.

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