ANGULAR MOMENTUM TRANSPORT VIA INTERNAL GRAVITY WAVES IN EVOLVING STARS

Like any other type of wave, IGW transport energy and AM. The waves extract energy/AM from the region of excitation and deposit it in the region of dissipation. In the case of convectively excited waves propagating in the radiative cores of evolving low-mass stars, the waves extract AM from the convective zone and deposit it in the radiative interior.

Convective motions generate waves with an energy flux of the order of

$$\begin{equation} \dot{E} \sim \mathcal {M} L \end{equation} \tag{ 1 }$$

(Goldreich & Kumar 1990; Kumar et al. 1999), where $\mathcal {M}$ is the convective Mach number near the radiative–convective interface,⁶ and L is the stellar luminosity. For overlying convective zones, the waves carry an energy flux $\dot{E}$ downward. For deep convection zones in low-mass stars, $\mathcal {M} \ll 1$ and the waves have a negligible impact on the net energy transport. The characteristic angular frequency of the waves is the angular convective turnover frequency ω_c near the radiative–convective interface, which we calculate via Equation (5.51) of Hansen & Kawaler (1994):

$$\begin{equation} \omega _c = \frac{v_c}{\lambda }, \end{equation} \tag{ 2 }$$

where λ is the mixing length and v_c is the convective velocity, which for efficient convection is

$$\begin{equation} v_c = \left (\frac{\lambda g}{\rho c_P T} F \right)^{1/3} \simeq \left (\frac{F}{\rho }\right)^{1/3}, \end{equation} \tag{ 3 }$$

and all quantities have their usual meaning. The characteristic AM flux carried by the waves is

$$\begin{equation} \dot{J} \sim \frac{m_c}{\omega _c} \dot{E}, \end{equation} \tag{ 4 }$$

where m_c is a characteristic azimuthal number associated with the waves, which is typically of the order of l ∼ m ∼ several.⁷ For a slowly rotating (Ω ≪ ω_c) star, we may expect prograde (positive m) waves and retrograde (negative m) waves to be excited to equal amplitudes, such that no net AM flux is carried by the waves. As we shall see below, differential rotation naturally produces a wave filter, selectively allowing prograde or retrograde waves to pass through, generating a non-zero net AM flux.

The AM flux of Equation (4) is quite large. The characteristic timescale for waves to change the spin of the radiative region is

$$\begin{eqnarray} &&t_{\rm waves} \sim \frac{I_{\rm rad} \Omega }{\dot{J}}, \end{eqnarray} \tag{ 5 }$$

where I_rad is the moment of inertia of the radiative zone and Ω is the angular rotation frequency. For the Sun, subgiants, and red giants, t_waves ≲ 10⁵ yr. Actual wave spin-up timescales are typically longer (although still much shorter than stellar evolution timescales) because most waves are unable to propagate far into the radiative region. Nonetheless, it is important to realize that IGW are capable of changing the spin of the radiative regions on timescales much shorter than the stellar evolution timescale.

The IGW generated by convection typically have very small frequencies compared to the Brunt–Väisälä frequencies in the radiative region, i.e., ω_c ≪ N (see Figure 1). Consequently, the IGW have very short wavelengths, and their propagation/dissipation is well approximated by WKB scaling relations.⁸ The WKB radial wave number of IGWs is

$$\begin{equation} k_r^2 = \frac{l(l+1) N^2}{r^2 \omega ^2}, \end{equation} \tag{ 6 }$$

and the radial group velocity is

$$\begin{equation} v_{g,r} = \frac{r \omega ^2}{\sqrt{l(l+1)} N}. \end{equation} \tag{ 7 }$$

Lower frequency waves have shorter wavelengths and slower group velocities, making them more prone to damping. The radial wave damping length is (Zahn et al. 1997)

$$\begin{eqnarray} &&L_d = \frac{2 r^3 \omega ^4}{[l(l+1)]^{3/2} N N_T^2 K }, \end{eqnarray} \tag{ 8 }$$

where $N_T^2=N^2-N_\mu ^2$ is the thermal part of the Brunt–Väisälä frequency ($N_\mu ^2$ is the compositional part) and

$$\begin{equation} K = \frac{16 \sigma _B T^3}{3 \rho ^2 c_p \kappa } \end{equation} \tag{ 9 }$$

is the thermal diffusivity.⁹$^,$¹⁰ The opacity κ in the denominator of Equation (9) is the effective opacity due to heat transport by both photons and degenerate electrons.

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Equation (8) demonstrates that low-frequency waves have much smaller damping lengths, and are thus unable to propagate far from the convection zone. Additionally, large values of the N create very short damping lengths, preventing waves from propagating into strongly stratified regions. Characteristic damping lengths are L_d ∼ 10⁻² R_☉ for l = 3, ω = ω_c waves just below the solar convection zone. Most waves therefore damp out long before they reach the stellar core. However, waves of frequency ω ≳ 5ω_c have damping lengths longer than a solar radius and can therefore penetrate all the way to the center of the Sun.

As stars evolve off the main sequence, their cores contract and the value of N near the hydrogen-burning shell increases markedly. The damping lengths of the waves are correspondingly shortened, preventing the waves from propagating to the centers of the stars. IGW are therefore somewhat ineffective at transporting AM into the cores of evolved stars, as we examine in more detail in Section 4.

Differential rotation within a star naturally filters the IGW that propagate within the star. For simplicity, we assume shellular differential rotation at all times.¹¹ The local wave frequency, measured in the co-rotating frame with angular velocity Ω(r), is

$$\begin{equation} \omega (r) = \omega (r_c) - m [ \Omega (r)-\Omega (r_c) ], \end{equation} \tag{ 10 }$$

where ω(r_c) is the wave frequency when launched from the convective zone and Ω(r_c) is the rotation frequency of the convective zone. Consider the case in which the interior layers of the star rotate faster than the surface such that Ω(r) > Ω(r_c). The prograde waves (m > 0) are boosted to lower frequencies by the differential rotation, causing their damping lengths to drastically decrease. If ω → 0, the waves encounter a critical layer and are completely damped out. In contrast, the retrograde waves are boosted to higher frequencies by the differential rotation. Their damping length drastically increases, allowing them to propagate much further within the star than they would have otherwise.

The differential wave damping produces an imbalance in the net AM flux. The AM flux carried inward by a train of waves launched from an overlying convective zone is

$$\begin{equation} \dot{J}(r) = \dot{J}(r_c) e^{-\tau }, \end{equation} \tag{ 11 }$$

where the wave optical depth τ is

$$\begin{eqnarray} \tau &= 2 \int ^{r_c}_r \frac{dr}{L_d} \nonumber \\ &= \int ^{r_c}_r dr \frac{[l(l+1)]^{3/2} N N_T^2 K }{r^3 \omega ^4}. \end{eqnarray} \tag{ 12 }$$

A factor-of-two change in ω changes $\dot{J}(r)$ by a factor of e^15τ, a huge factor for strongly damped waves. Small amounts of differential rotation therefore change the wave frequencies enough to generate a huge difference between AM fluxes carried by prograde and retrograde waves.

Differential rotation thus sets up an efficient wave filter: prograde waves are absorbed before they can propagate far into more rapidly rotating layers of a star. Only retrograde waves pass through, meaning the net AM flux into the rapidly rotating layers is negative. When the retrograde waves dissipate, they deposit their negative AM, spinning down the rapidly rotating layers. The star thus evolves toward a state of rigid rotation.

The wave dynamics presented above do not always proceed so simply. One of the main reasons is the "anti-diffusive" nature of IGW, that can cause IGW to generate shear rather than destroy it. Indeed, there exists no equilibrium rotation rate in the presence of IGW, as waves cause small perturbations in rotation frequency (a small perturbation is defined as ΔΩ ≪ ω/m) to grow in amplitude. In a rigidly rotating star, a small perturbation in spin rate is amplified on a timescale,

$$\begin{equation} t_{\rm grow} = \frac{\pi }{3m^2} \frac{\rho r^4 L_d \omega ^2}{\dot{E}}, \end{equation} \tag{ 13 }$$

for an energy flux $\dot{E}$ carried by waves of frequency ω and azimuthal number m.¹² This is essentially the timescale for waves to change the spin rate of a shell of thickness L_d ≪ r by an amount ω/m.

The shear amplification cannot proceed indefinitely. Once the spin rate has changed by ΔΩ = ω/m, the differential rotation creates a critical layer that absorbs incoming waves. At this point, the shear can no longer be amplified because waves damp out just before reaching the critical layer. The shear thus moves toward the source of the IGW (Goldreich & Nicholson 1989).

IGW AM transport therefore proceeds in two different modes. Small perturbations in spin ($\Delta \Omega \ll {\bar{\omega }}/{\bar{m}}$, where ${\bar{m}}$ and ${\bar{\omega }}$ are the characteristic pattern number and angular frequency of waves which dominate the AM flux) are amplified on the timescale t_grow. Large perturbations in spin ($\Delta \Omega \gtrsim {\bar{\omega }}/{\bar{m}}$) efficiently filter waves in such a manner as to allow them to reduce the differential rotation until $\Delta \Omega \sim {\bar{\omega }}/{\bar{m}}$. Hence, IGW cannot enforce rigid rotation, although they can prevent the build-up of large amounts of differential rotation. The Sun's nearly rigidly rotating radiative zone has ΔΩ ≪ ω_c (Howe 2009), which indicates that some other mechanism (e.g., magnetic torques) prevents the build-up of shear (Denissenkov et al. 2008).

Above, we ignored viscous effects that are important in cases where IGW are able to produce large amounts of shear. Viscosity coupled with IGW-induced shear can produce shear-layer oscillations (SLOs) near the base of the convection zone (see Kumar et al. 1999; Kim & MacGregor 2001; Talon et al. 2002; Talon & Charbonnel 2005) that may have been detected in the Sun (Howe et al. 2000).¹³ The timescale of the SLO is approximately equal to t_grow evaluated for ω = ω_c, and is on the order of years in the Sun.

For the purposes of the secular evolution of global scale differential rotation, many of the anti-diffusive effects of IGW, such as the SLO, can be ignored. The SLO has a short timescale and likely does not qualitatively affect evolution on longer timescales. Instead, secular evolution arises from wave filtering due to the steady-state (or averaged) differential rotation. This filtering allows IGW to reduce the differential rotation until its amplitude is of the order of $\Delta \Omega \sim \bar{\omega }/\bar{m}$.

An additional complication is that we must include the effects of a broad spectrum of waves (consisting of large ranges in l, m, and ω), whose shape is not well constrained (see Section 4.1) and which has a stochastic nature. The stochastic nature of the wave excitation is likely to average out into a smooth wave spectrum over comparatively long ($t \gg \omega _c^{-1}$) spin-down timescales.¹⁴ Although the general tendency for waves to reduce large amplitude background differential rotation is not strongly dependent on the wave spectrum, the details of the process can be.

Finally, we have ignored the influence of the Coriolis force on the IGWs even though the local spin frequency Ω(r) can be comparable to or greater than the local wave frequency ω(r). We expect the effect of rotation on propagating IGWs to be well captured by the traditional approximation (Bildsten et al. 1996; Lee & Saio 1997), which has been explored in previous works (Pantillon et al. 2007; Mathis et al. 2008, 2013; Mathis 2009). The main effect of the Coriolis force is to increase the effective value of l for the IGW, decreasing their damping length. This will introduce some quantitative corrections to our findings, although uncertainties in the wave spectrum are likely to be more important. It is also possible that rotation will change the nature of convection and the spectrum of IGW that it generates (Mathis et al. 2014); however, since even the non-rotating spectrum is poorly understood, we ignore this issue in this work.

Our goal is to obtain general results that are robust against the details of the effects above. We proceed with a simplified analysis that produces order-of-magnitude estimates for secular wave spin-down timescales, and defer a more precise description of AM redistribution via IGW to future work.

There is broad agreement that convective motions most efficiently generate IGW when the length scales and timescales of the convection and the IGW are comparable (Lighthill 1978). The dominant source of IGW are large-scale convective rolls with size H and with coherence times $\omega _c^{-1}$. This generates waves with frequencies ω ∼ ω_c, horizontal mode number l ∼ r_c/H, and radial wavenumber (N/ω_c)H⁻¹ ≫ H⁻¹, where N is a typical Brunt–Väisälä frequency in the radiative zone. The luminosity of these low-frequency waves is

$$\begin{equation} \dot{E} = \eta \mathcal {M} L. \end{equation} \tag{ 17 }$$

Here, η is an efficiency factor of the order of unity. This is the prediction of, e.g., Press (1981), Garcia Lopez & Spruit (1991), Goldreich & Kumar (1990), Kumar et al. (1999), Lecoanet & Quataert (2013), and is consistent with recent numerical simulations (Alvan et al. 2014).

Although the peak of the excitation spectrum is well understood, the rest of the spectrum is poorly constrained. The primary difficulty is that high-frequency waves are excited by small length scale convective motions, which are difficult to resolve in simulations or experiments. For instance, the power spectra of convective motions presented in the simulations of Belkacem et al. (2009) and Alvan et al. (2014) using the ASH code (Clune et al. 1999; Brun et al. 2004), look very different from the power spectra measured in far more turbulent experiments (e.g., Niemela et al. 2000). Even theoretically, there is no consensus on whether the small-scale motions follow a Kolmogorov (Kolmogorov 1941) energy cascade, or a Bolgiano–Obukhov (Bolgiano 1959; Obukhov 1959) entropy cascade (e.g., Lohse & Xia 2010). Note, however, that the most turbulent simulations and experiments suggest that small-scale fluctuations follow a Kolmogorov cascade (e.g., Boffetta et al. 2012; Lohse & Xia 2010, and references within).

Because direct numerical simulations of wave excitation by convection (Rogers & Glatzmaier 2005; Brun et al. 2011; Rogers et al. 2013; Alvan et al. 2014) may not have fully resolved the turbulent motions generating high-frequency waves, we instead turn to theoretical predictions based on the assumption of a Kolmogorov power spectrum of convective motions. Lecoanet & Quataert (2013) predict a wave power spectrum

$$\begin{equation} \frac{d \dot{E}(\omega,l)}{d\omega \ d l} \sim \frac{\dot{E}}{l \omega _c} \left(\frac{\omega }{\omega _c}\right)^{-a} \left(l \frac{H}{r_c}\right)^b \left(l \frac{d}{r_c}\right)^c, \end{equation} \tag{ 18 }$$

where d is the width of the transition regime between the radiative and convective regions, and $\dot{E}$ is the total wave energy flux given by Equation (17). Here, $k_{\perp } = \sqrt{l(l+1)}/r$ is assumed to be less than H⁻¹(ω/ω_c)^3/2, and a, b, c are power-law coefficients. Depending on the details of the transition region, Lecoanet & Quataert (2013) find a between 7.5 and 8.5, b = 4, and c between 0 and 1. Goldreich & Kumar (1990) predict a = 7.5 and b = 3.

There is a sharp decline in wave luminosity with frequency because only small eddies have high frequencies, and there is very little power in the small eddies. Furthermore, the waves most efficiently excited by these small eddies have small horizontal wave lengths, and thus damp very quickly. The least damped waves have small l, and are excited due to the (low probability) coherent superposition of many small eddies (Garcia Lopez & Spruit 1991). These waves have

$$\begin{equation} \frac{d \dot{E}(\omega)}{d\omega } \sim \frac{\dot{E}}{\omega _c} \left(\frac{\omega }{\omega _c}\right)^{-a}. \end{equation} \tag{ 19 }$$

We allow a to be a free parameter, and expect most probable values to lie in the range 3.5 ≲ a ≲ 7.5.

Up to this point, we have not considered the effects of stratification. Kumar et al. (1999) suggest that the stratification of a convection zone above a radiative zone can enhance excitation of high-frequency waves. Because the scale height decreases with increasing radius, they argue that the energy-bearing convective motions will shift to smaller length scales and higher frequencies with increasing radius. Under these assumptions, high-frequency, low l waves can be excited by the coherent superposition of many small, energy-bearing eddies. This allows for much more efficient excitation of high-frequency waves. Their analytic calculations predict a wave spectrum with a ∼ 3.3, whereas semi-analytic work has suggested a ∼ 4.5 (see also Talon et al. 2002; Denissenkov et al. 2008). Stratification will not enhance the excitation of high-frequency waves for convection zones below radiative zones.

However, recent high-resolution simulations of strongly stratified convection do not show this shift of the energy-bearing motions to smaller scales—rather, they find that the kinetic energy is peaked at large scales throughout the convection zone (Hotta et al. 2014). If convection in stars is dominated by motions much larger than the local scale height, then it is unlikely that stratification will amplify the excitation of high-frequency waves. In this work, we adopt a = 4.5 as a fiducial value, but we caution that both steeper (larger a) and shallower (smaller a) frequency spectra are certainly possible.

Requiring $\int ^\infty _{\omega _c} \dot{E}_\omega d \omega = \eta \mathcal {M} L$ yields the wave energy flux per unit frequency

$$\begin{equation} \dot{E}_\omega \sim \frac{a-1}{\omega _c} \left (\frac{\omega }{\omega _c}\right)^{-a} \eta \mathcal {M} L. \end{equation} \tag{ 20 }$$

The total AM flux of waves with azimuthal number m and frequency near ω is

$$\begin{equation} \dot{J}(r_c,\omega) \sim \frac{m}{\omega _c} \left (\frac{\omega }{\omega _c}\right)^{-a} \eta \mathcal {M} L. \end{equation} \tag{ 21 }$$

Equations (19)–(21) only apply for IGW with ω ≳ ω_c, and they are valid at the radiative–convective interface (r = r_c). Further into the radiative zone where r < r_c, the wave spectrum will shift toward higher frequencies since lower frequency waves damp on short length scales.

The strong dependence of wave optical depth on frequency (Equation (12)) implies the frequency of waves that dominate the energy/AM flux at a given radius is sharply peaked at a characteristic value, ω_*(r). At the radiative–convective interface, ω_*(r_c) ≃ ω_c. Below the interface, ω_* is set by waves whose optical depth is of the order of unity at that location. Lower frequency waves have been attenuated and higher frequency waves carry less AM via Equation (21). To solve for ω_*(r), we find the peak in the value of

$$\begin{equation} \dot{J}(r,\omega) = \dot{J}(r_c,\omega) e^{-\tau (r,\omega)}. \end{equation} \tag{ 22 }$$

Taking the derivative of Equation (22) with respect to ω and setting equal to zero yields

$$\begin{equation} \tau _{*} = \frac{a}{4}. \end{equation} \tag{ 23 }$$

Then, using Equation (12) we find

$$\begin{equation} \omega (r) = \left [\frac{4}{a} \int ^{r_c}_r dr \frac{[l(l+1)^{3/2} N_T^2 N K }{r^3} \right ]^{1/4}. \end{equation} \tag{ 24 }$$

The right-hand side of Equation (24) depends primarily on the stellar structure with only a very weak dependence on the wave spectrum. Equation (24) applies for frequencies above the convective turnover frequency ω_c, and an expression valid at all frequencies is

$$\begin{equation} \omega _*(r) = {\rm max} \left [ \omega _c \, \ \left (\frac{4}{a} \int ^{r_c}_r dr \frac{[l(l+1)^{3/2} N_T^2 N K }{r^3} \right)^{1/4} \right ]. \end{equation} \tag{ 25 }$$

The AM flux carried by waves with frequency ω ≈ ω_* is, using Equation (21),

$$\begin{equation} \dot{J}_*(r) \sim \frac{ m \eta \mathcal {M} L}{\omega _c} \left (\frac{\omega _*(r)}{\omega _c}\right)^{-a} e^{-a/4}. \end{equation} \tag{ 26 }$$

Because the AM flux is dominated by waves with frequency ω_*(r), Equation (26) gives an approximate value for the total AM flux carried by IGW at radius r. Since low l waves have the longest damping lengths, Equation (26) should be evaluated using l ∼ |m| ∼ 1. The associated spin-up timescale for regions below a radius r is

$$\begin{equation} T_{*}(r) = \frac{\Omega _s I(r)}{\dot{J}_*(r)}. \end{equation} \tag{ 27 }$$

The timescale T_*(r) indicates the timescale on which waves could change the spin rate of the region below radius r, if the surface of this region contains differential rotation that creates a wave filter. It is different than the spin evolution timescale t_w(r) that indicates the wave spin-down timescale of a spherical shell at radius r. Unfortunately, t_w(r) is strongly dependent on the stellar rotation profile, which is generally unknown. We find that T_*(r) is a better diagnostic because it provides an estimate for the timescale on which rotation rates at radii below r could change, in the presence of a wave filter at radius r. For our purposes, the most important quantities are the maximum value of T_*(r) and its value at the top of the g-mode cavity (for subgiants and red giants these values are approximately the same). Since the AM flux of Equation (26) is somewhat dependent on the slope of the frequency spectrum, we cannot expect Equation (27) to yield exact results. Nonetheless, it provides a simple method of estimating IGW spin evolution timescales.

We would like to compare the wave synchronization timescale to the spin-up timescale due to stellar evolution. This timescale is found from the spin evolution in the absence of AM transport:

$$\begin{equation} \frac{d}{dt} (I \Omega _s ) = \Omega _s \dot{I} + \frac{I \Omega _s}{t_s} = 0, \end{equation} \tag{ 28 }$$

which entails a spin-up timescale of

$$\begin{equation} \tau _{\rm spin}(r) = - \frac{r}{2 \dot{r}}, \end{equation} \tag{ 29 }$$

where $\dot{r}$ is the rate of change of the radius of the mass contained in a spherical shell at r.

Figure 3 shows the values of ω_*(r), J_*(r), T_*(r), and τ_spin(r) for a few different stellar models, using a = 4.5. We have used l = m = 1 and η = 0.1 as an estimate for the wave energy carried by these limited values of l and m. For solar-like stars, the value of ω_*(r) reaches a maximum of ∼5ω_c. Although this reduces the value of J_*(r) to ∼10⁻³ its value at the convection zone, the remaining AM is still capable of changing the spin frequency on timescales of ∼10⁸ yr. Even for steep frequency spectra a ∼ 7.5, the wave timescales are less than the age of the Sun. We therefore agree with previous results (Talon & Charbonnel 2005; Charbonnel & Talon 2005) that have found that IGW affect the solar angular velocity profile on short timescales.

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Next, we examine the results for a 1.5 M_☉ terminal-age main sequence (TAMS) star. A star of this mass develops a surface convection zone as it begins to evolve off the main sequence toward cooler surface temperatures. When this convection zone first forms, it is relatively shallow, although it still extends several scale heights and carries nearly all the stellar flux.¹⁵ Because the bottom of the convection zone exists at low densities where the scale height is small, the convective turnover frequency at its base is quite large. The waves generated by the convection therefore have high frequencies and are easily capable of traversing the entire radiative zone (in this case they reflect at the core convection zone and form standing oscillation modes). Moreover, because the convective Mach numbers are larger near the surface convection zone than in the convective core, we expect the surface-generated IGW to dominate the AM flux.¹⁶ These waves are capable of redistributing AM on short timescales (T_* ∼ 10⁶ yr), allowing more massive stars M ≳ 1.4 M_☉ to undergo rapid spin evolution as they initially evolve off the main sequence. This situation was also noted in Talon & Charbonnel (2008).

The results are much different for subgiants and red giants. In these stars, the large values of N near the hydrogen-burning shell result in large values of ω_*. Only high-frequency waves (relative to the convective turnover frequency) are capable of penetrating into the g-mode cavities probed by asteroseismic measurements. These waves only carry small amounts of AM, assuming the frequency spectrum is reasonably steep (a ≳ 3). Consequently, the IGW which are able to propagate into the core cannot change its spin on short timescales, and T_* ≫ τ_spin near the cores of these stars. Therefore, IGW on their own are likely not capable of efficiently spinning down the cores of ascending RGB stars. This result is reassuring, as the observed rapid core rotation in subgiants and red giants indicates IGW have not been able to spin down the cores of these stars. However, we note that in the upper radiative zone (r/R ≳ 10⁻¹ for the subgiant model and r/R ≳ 2 × 10⁻² for the red giant model) the wave spin-down timescales are short (T_* ≪ τ_spin), implying that IGW can still affect the spin of these regions of the star.

The results presented above indicate that IGW can help reduce differential rotation for stars leaving the main sequence, but they cannot keep the inner core (regions at and below the hydrogen-burning shell) synchronized as the star evolves up the subgiant/RGB. We would like to know the moment in the evolution at which the waves can no longer penetrate into the core, allowing it to decouple from the surface convection zone.

To determine the epoch of decoupling, we find the stellar evolutionary state at which IGW spin evolution timescales become longer than stellar evolution timescales. We generate stellar models with MESA (Paxton et al. 2011, 2013) and evolve them from the zero-age main sequence, calculating profiles of T_*(r) and t_s(r) at each step. We then find the first stellar model that contains a location below the surface convection zone where T_*(r) > τ_spin(r), and we define this to be the moment of decoupling.

Figure 4 shows evolutionary tracks for stars of different mass and indicates the moment of decoupling for each model. We calculate T_* using waves of l = |m| = 1, α = 4.5, and η = 0.1. For stars in the mass range M_☉ < M < 1.5 M_☉ that comprise most of the observed subgiant/ascending RGB Kepler sample (Schlaufman & Winn 2013), the moment of decoupling occurs at effective temperatures T_eff ≈ 5500 K. At decoupling, the radii of the stars are typically ∼1.75 their main-sequence radius for M_☉ ≲ M ≲ 1.5 M_☉. The large-frequency separation of stars, Δν, is approximately half its main-sequence value at the time of decoupling (see bottom panel of Figure 4).

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The actual stellar parameters at decoupling will depend on stellar metallicity, spin frequency, wave spectrum, etc., but should typically occur in the 5200 K ≲ T_eff ≲ 6200 K temperature range. In particular, low-metallicity stars (such as KIC 7341231, analyzed in Deheuvels et al. 2012) have larger T_eff for the same mass, and will have correspondingly warmer temperatures at decoupling. For steep wave spectra (a ∼ 7.5) the decoupling occurs earlier in the stellar evolution, very soon after core hydrogen exhaustion, and at larger T_eff. Shallow wave spectra (a ≲ 3) do not decouple until later in the stellar evolution, further up the RGB, at evolutionary stages beyond those of the subgiants observed by Deheuvels et al. (2012, 2014).

The decoupling of the core occurs for three reasons. First, the stellar evolution timescales decrease from ∼10⁹ yr to ∼10⁷ yr as a star evolves from the main sequence to the RGB, meaning waves have to act on shorter timescales to keep up with the spin up of the contracting core. Second, as stars evolve across the subgiant region, their surface convective zone deepens, penetrating further into the star where the convective turnover frequencies are smaller. The frequencies of the convectively excited waves correspondingly decrease, meaning they cannot propagate as far into the core. Third, as the core contracts, its peak Brunt–Väisälä frequency N increases by over an order of magnitude. The inner core therefore becomes optically thick to the waves, prohibiting efficient core–envelope coupling.

The findings above appear to be consistent with asteroseismic measurements. The subgiant stars studied by Deheuvels et al. (2014) have temperatures of T_eff ∼ 5000 K and radii in the range 2–3 R_☉ range, and therefore have likely evolved past the moment of decoupling. Indeed, Deheuvels et al. (2014) find that these stars appear to have cores which are spinning up with time. Our findings are also consistent with those of Tayar & Pinsonneault (2013) who find that the internal rotation rate of a low-mass subgiant (KIC 7341231; Deheuvels et al. 2012) requires decoupling to occur at stellar radii of R ∼ 1.5–1.9 R_☉.

Finally, we can estimate the extent of the decoupled core by searching for the first radial location at which T_*(r) > τ_spin(r) as one travels from the convective envelope inward (marked by a square in Figure 3). Regions below this radial location are decoupled from convectively excited IGW. Figure 5 shows the extent of the decoupled region in terms of radius R_dc and mass M_dc as a function of the stellar radius as stars evolve up the RGB. Typically only the inner part of the radiative core is decoupled, with M_dc ≈ 0.2 M_☉ as the star evolves up the lower RGB. We therefore expect any differential rotation to be restricted to mass coordinates M(r) < M_dc ≈ 0.2 M_☉. The decoupled region includes the helium core, the hydrogen-burning shell, and a small fraction of the radiative envelope. Steeper wave spectra have larger decoupled regions (comprised by the bulk of the radiative region) while shallower wave spectra have smaller decoupled regions (but still including the helium core and hydrogen-burning shell).

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In the Sun, low l IGW with frequencies larger than the convective turnover frequency ω ≳ 5ω_c can traverse the entire radiative zone, and can reduce differential rotation on short timescales (T_* ∼ 10⁷–10⁸ yr). This result is in accordance with a series of studies (e.g., Talon & Charbonnel 2005; Charbonnel & Talon 2005) examining IGW AM transport in solar-like stars, whose more detailed calculations/simulations found that waves reduce differential rotation within the Sun on similar timescales. This result is not strongly dependent on the IGW wave spectrum generated by convection, nor does it require the existence of a SLO. The solar IGW AM transport timescale of T_* ∼ 10⁷–10⁸ yr also appears to be consistent with observations of the spin-down of young cluster stars (Stauffer & Hartmann 1987; Keppens et al. 1995; Bouvier et al. 1997; Krishnamurthi et al. 1997; Barnes 2003).

However, IGW by themselves are unlikely to produce the observed rigid rotation of the solar interior. In the Sun, the IGW that reach r = 0 have angular frequencies ω_* ≈ 4 μHz, whereas the rotation rate of the radiative zone is Ω_☉ ≃ 2.6 μHz. In the absence of other AM transport mechanisms, we may therefore expect to observe differential rotation of the order of ΔΩ ∼ ω_*/m ∼ 4 μHz, in contrast to the nearly rigid rotation which is observed (ΔΩ ≪ Ω_☉). We conclude that IGW are capable of performing the bulk of the AM transport required to keep the radiative interior of the Sun synchronous with the convective envelope (in agreement with Talon & Charbonnel 2005 and Charbonnel & Talon 2005), but that another source of torque is required to enforce rigid rotation (in agreement with Gough & Mcintyre 1998 and Denissenkov et al. 2008).

As stars evolve across the subgiant branch and up the RGB, their cores become opaque to incoming IGW waves and decouple from the surface convection zone. After decoupling, the cores are able to spin up as they ascend the subgiant branch, as observed by Deheuvels et al. (2014). However, asteroseismic measurements of the core rotation rate of stars ascending the RGB (Mosser et al. 2012) indicate that the cores of these stars slowly spin down as they evolve. IGW on their own are likely incapable of producing this spin down.

However, IGW are capable of changing the stellar spin down to radii of r ∼ 10⁻¹ R_☉ (in comparison, the base of the convective zone resides at radii r ∼ 0.75 R_☉, while the hydrogen-burning shell is located at r ∼ 3 × 10⁻² R_☉). This implies that convectively excited IGW can remove most of the AM from the contracting radiative zone and are able to couple the slowly rotating convective zone with the bulk of the moment of inertia of the radiative zone. We predict that only the inner core (i.e., the inner ∼0.2 M_☉ comprising the helium core, hydrogen-burning shell, and a small fraction of the radiative outer core) of RGB stars rotate rapidly, whereas layers exterior to this can be spun down by IGW. Moreover, our results imply that other AM transport mechanisms (e.g., magnetic torques) need only remove the relatively small amount of AM contained in the inner ∼0.2 M_☉ of the core in order to allow it to spin down on the RGB.

Additionally, while on the RGB, the material accreting onto the helium core may have been previously spun down by IGW, meaning the core will not rapidly spin up as it accretes. This possibility is somewhat dependent on the wave spectrum and so we do not investigate it in detail. However, the impeded spin-up of the core would once again allow other AM transport mechanisms to spin down the inner core with only a relatively small amount of AM transport. Last, the IGW could enforce large angular velocity gradients between the inner core (which is unaffected by IGW) and the outer core (which is spun down by the IGW). The IGW-induced shear could then enhance the potency of other AM transport mechanisms. Successful descriptions of AM transport may therefore require the simultaneous interplay between IGW and other sources of torque.

Our results, combined with asteroseismic measurements, may place some constraints on the viability of some surprising results from simulations of IGW wave generation/propagation (see, e.g., Rogers et al. 2008, 2013; Alvan et al. 2014). These authors suggest that a radial damping length increased by ∼(N/ω) ≫ 1 better describes IGW attenuation, speculating that nonlinear wave–wave interactions may be at play. However, if we use this modified damping length in Equation (12), we find that low-frequency IGW can penetrate all the way into the cores of subgiants/red giants. These IGW could spin down the cores on short timescales, in contrast with their observed rapid rotation. Thus, this weaker IGW damping appears to be inconsistent with observations.

We may also be able to constrain the frequency spectrum of the convectively excited IGW. If the wave spectrum is somewhat flat (a ≲ 3.5) as suggested by Rogers et al. (2013), then the AM flux carried by high-frequency waves (ω ≫ ω_c) is much greater. This would allow high-frequency IGW to change the spin of the cores of subgiants/red giants on short timescales. In the absence of additional AM transport mechanisms, the IGW would generate differential rotation of the order of ΔΩ ∼ ω_*, causing the cores to spin faster than observed, at P ∼ 2π/ω_* ∼ 2 days. This scenario seems unlikely, as it would require the presence of an additional AM transport mechanism which would mostly erase the IGW-induced shear, yet allow the smaller degree of observed differential rotation to persist. We find it more plausible that the wave spectrum is steep enough (a ≳ 3.5) to prevent IGW from significantly altering the spin profile of the g-mode cavity in red giants.

Recent studies (Winn et al. 2010; Dawson 2014) of the tidal evolution of hot Jupiters around main-sequence stars have suggested that some observed features of the hot Jupiter distribution can be explained by weak AM transport within the stellar interiors. In particular, these studies have suggested that tides only couple a small piece of the stellar moment of inertia (e.g., a solar-like convection zone) to the planetary orbit. Our results suggest such a decoupling to be extremely unlikely, as IGW can reduce differential rotation on timescales shorter than the ages of the hot Jupiter systems.

Asteroseismic analyses of clump stars (Mosser et al. 2012; see also Tayar & Pinsonneault 2013) burning helium in their core reveal slower core rotation rates, with rotation periods of P ∼ 100 days. In clump stars, IGW are excited at the top of the helium-burning core and the base of the convective envelope, and both sources of IGW must be included in AM transport calculations. Preliminary results reveal that IGW may be sufficient to couple the core and envelope of clump stars. However, these results are somewhat dependent on the wave spectrum, so we defer a more detailed investigation to future publications.

In massive stars nearing core collapse, the stellar structure becomes complex, with onion-like shells of convective/radiative zones. Although stellar evolutionary timescales become extremely short, the vigorous convection generated by nuclear burning in the cores of these stars generates large fluxes of IGW (Quataert & Shiode 2012; Shiode & Quataert 2014). It is therefore likely that IGW play an important role in AM transport for these stars, and we hope to explore this issue in a future paper.

The main uncertainty involved in IGW AM transport is the spectrum of convectively driven waves. Theoretical predictions suffer from our poor understanding of stellar convection, in particular, the inadequacy of mixing length theory. Moreover, they can be sensitive to frequently discarded factors of the order of unity (e.g., a change in the value of ω_c by a factor of π in Equation (26) will alter wave timescales by orders of magnitude). In turn, results from simulations are difficult to interpret and a detailed physical understanding/justification of their outcomes is often lacking. We hope that future simulations of convectively driven IGW can either confirm or deny current expectations and lead to a genuine understanding of convectively driven IGW dynamics. (1)