MAGNETO-THERMOHALINE MIXING IN RED GIANTS

During their first ascent on the red giant branch (RGB), a majority of low-mass stars (those with M ≲ 2 M_☉) experience extra mixing in their radiative zones separating the H burning shell from the bottom of convective envelope (Sweigart & Mengel 1979; Charbonnel & Do Nascimento 1998; Denissenkov & VandenBerg 2003). Despite 30 years of effort, however, the underlying physical mechanism is still not understood. Observationally, the RGB extra mixing manifests itself through changes of the surface abundances of Li, C, N, and of the isotopic ratio ¹²C/¹³C correlating with an increasing luminosity (Gratton et al. 2000; Smith & Martell 2003). These changes are produced by the joint operation of thermonuclear reactions that take place in the vicinity of the H shell and a nonconvective mixing process that transports reaction products through the radiative zone to the convective envelope. Observations support the idea that this mixing process starts (or, gets much more efficient) when an RGB star reaches a luminosity at which the differential luminosity function for a population of stars having the same age and chemical composition shows a prominent bump (a local pile-up of stars). The luminosity bump results from a temporary slowing down of the star's evolution caused by its structural readjustment. This happens when the H shell crosses and erases a discontinuity in the H-abundance profile left behind by the bottom of convective envelope at the end of the first dredge-up. During the first dredge-up, which occurs on the subgiant branch and the lower RGB, the convective envelope grows in mass, which causes its bottom to penetrate the layers whose chemical composition had been altered yet on the main sequence (MS). This produces changes of the surface abundances of Li, C, N, and of the ¹²C/¹³C ratio similar to but by far less substantial than those incurred from the subsequent operation of the RGB extra mixing.

Until recently, it has been thought that the only reason why the RGB extra mixing does not manifest itself below the bump luminosity is a strong gradient of the mean molecular weight μ caused by the onset of a deep convective envelope (e.g., Charbonnel et al. 1998). Any mixing mechanism has to overcome the stable thermal stratification of the radiative zone, and in the presence of a positive ∇_μ such mixing is correspondingly more difficult. In an RGB star above the bump luminosity, the H shell has already crossed the H-profile discontinuity; therefore, the radiative zone is now chemically uniform everywhere except in a very close neighborhood of the H shell. This circumstance was repeatedly emphasized in the past. In particular, it has been used to model the RGB extra mixing with rotation-driven meridional circulation and turbulent diffusion. It was not until recently that it has become clear that rotational mixing fails to explain the chemical element transport in the radiative zones of upper RGB stars (Chanamé et al. 2005; Palacios et al. 2006). In short, this failure is due to the following main causes: firstly, rotation period measurements for young cluster stars and helioseismic data indicate that MS stars with M ≲ 1 M_☉ lose a great amount of their initial angular momentum via magnetized stellar winds and that they most likely become slow and nearly solid-body rotators before leaving the MS; secondly, the chemical element transport by meridional circulation is strongly hindered by rotation-induced horizontal turbulence in stellar radiative zones (Chaboyer & Zahn 1992); thirdly, the vertical turbulent diffusion powered by differential rotation in the radiative zones of RGB stars operates at a low level too because it also redistributes the angular momentum, thus reducing the degree of differential rotation in a self-regulating way.

A new class of RGB extra mixing models has emerged since Eggleton et al. (2006) noticed that a tiny μ-gradient inversion (∇_μ ≈ −10⁻⁴) becomes visible at the outer tail of the H burning shell precisely at the moment when the H shell erases the H-profile discontinuity. This inversion is produced by the reaction ³He(³He, 2p)⁴He that locally reduces the mean molecular weight by Δμ ≈ μ²ΔX₃/6, where X₃ is the ³He mass fraction. The mechanism can be effective in the low-mass RGB stars because their MS progenitors synthesize large amounts of ³He in their outer radiative cores through nonequilibrium pp burning. Even though this ³He-rich material gets diluted in the convective envelope during the first dredge-up, the radiative zone of a low-mass RGB star above the bump luminosity can still have X₃ increased up to a value of 2 × 10⁻³ (the solar initial ³He abundance is 3 × 10⁻⁵). For this mass fraction, the ³He burning leads to Δμ ≈ −10⁻⁴, assuming that μ ≈ 0.6 and ΔX₃ ≈ −X₃. Below the bump luminosity, the μ-gradient inversion is overridden by the strong positive μ-gradient built up on the MS. It shows up and may come into play only when the ³He burning shell, advancing in mass in front of the major H shell, finds itself in the chemically homogeneous part of the radiative zone. This happens at the bump luminosity. Eggleton et al. (2006) found that a rapid mixing process occurred in their 3D simulations above this point, although the underlying cause was not identified (see Denissenkov & Pinsonneault 2008b).

Inspired by this work, Charbonnel & Zahn (2007a) have proposed that the μ-gradient inversion maintained by the ³He burning drives thermohaline convection in the radiative zones of low-mass RGB stars above the bump luminosity and that this is the long-sought physical mechanism for the RGB extra mixing. Thermohaline convection is a mixing process triggered by a double diffusive instability (e.g., Vauclair 2004). Consider a stratified ideal gas with a stable temperature gradient (∇ ≡ dln T/dln P < ∇_ad, where "ad" stands for adiabatic changes) but with an unstable composition gradient (∇_μ < 0). If we isolate a gas blob and shift it up in the vertical direction then its further motion will depend on how fast the blob exchanges heat and composition with its surrounding medium horizontally. Indeed, the relative difference in density between the surrounding medium and the blob is Δρ/ρ ≈ Δμ/μ −  ΔT/T, assuming that ΔP = 0. For the blob to continue rising, we need Δρ > 0. Our assumptions about the gradients mean that Δμ > 0 and ΔT > 0 in the absence of both heat and molecular diffusion. Because these differences grow when the blob rises, Δρ may stay positive or it may ultimately become negative depending on the ratio r_μ = |∇_μ|/(∇_ad − ∇). In our particular case, r_μ ≪ 1. Therefore, our idealized impermeable and adiabatic blob will rise a short distance and then stop, when the accumulated difference in T compensates that in μ. In reality, the heat exchange, whose rate is specified by the radiative diffusivity K, constantly works to reduce the difference in T. On the other hand, molecular diffusion ν_mol tries to smooth out the difference in μ. The double diffusive instability may therefore develop only if K ≫ ν_mol. In this case, the blob's rising speed can be estimated as v ∼ l/τ_th, where l is the mean path that the blob travels before it gets dissolved, while τ_th ∼ d²/K is the characteristic thermal timescale for a spherical blob of the diameter d. An approximate expression for the thermohaline diffusion coefficient can be obtained as D_thc ∼  lvr_μ ∼  Kr_μ(l/d)². Charbonnel & Zahn (2007a) and Denissenkov & Pinsonneault (2008b) have demonstrated that the observed RGB mixing patterns can be explained by stellar evolutionary models with the ³He-driven thermohaline convection only if l/d ≳ 10–30. A similarly large parameter ratio for thermohaline convection in stellar radiative zones was postulated by Ulrich (1972), as opposed to a ratio l/d ∼ 1 advocated by Kippenhahn et al. (1980). Besides, the double-diffusive instability has been shown to result in formation of elongated (large l to d ratios) structures known as "salt fingers" in laboratory experiments with the saltier and warmer water overlying the fresher and colder water (Stern 1960).

However, there appears to exist observational and theoretical arguments challenging this model. First of all, a large number of old metal-poor MS stars with M ≲ 0.9 M_☉, both in globular clusters and in the halo field, which had accreted He- and C-rich high-μ material from their evolved cluster or binary companions do not seem to have been thoroughly mixed by thermohaline convection (Newsham & Terndrup 2007; Denissenkov & Pinsonneault 2008a; Aoki et al. 2008), as it would be expected even if the less efficient prescription by Kippenhahn et al. (1980) were used for D_thc. Second, thermohaline convection is expected to be suppressed by the rotation-induced horizontal turbulence that works together with the molecular diffusion to reduce the μ contrast between the rising gas blob and its surroundings (Denissenkov & Pinsonneault 2008b). Third, strong differential rotation is predicted to hinder thermohaline convection as well, because the "salt fingers" may be tilted by the rotational shear so rapidly that they will get damped before they produce significant mixing (Canuto 1999, and references therein). We anticipate that a similar effect is also produced by the Coriolis force in a uniformly rotating radiative zone. Contrary to these expectations, a much larger fraction of Li-rich objects has been found among rapidly rotating (vsin  i ⩾ 8 km s⁻¹) K giants than among their more common slowly rotating (vsin  i ≲ 1 km s⁻¹) counterparts (Drake et al. 2002). The Li-rich K giants are low-mass stars located above the bump luminosity (Charbonnel & Balachandran 2000) in which large amounts of Li are thought to be synthesized via the ⁷Be-transport mechanism (Cameron & Fowler 1971). To be efficient, this mechanism needs a 10 to 100 times faster mixing than that required to reproduce the abundance patterns in the majority of upper RGB stars (Denissenkov & Herwig 2004). It is not clear how thermohaline convection can explain the phenomenon of Li-rich K giants given that its efficiency should be lower in the more rapidly rotating stars. These arguments have motivated our search for an alternative RGB mixing mechanism.

In this paper, we use a simple model of toroidal magnetic field generation in a differentially rotating radiative zone of a bump luminosity RGB star to obtain order-of-magnitude estimates demonstrating that the buoyant rise of magnetic flux rings being formed close to the local minimum in μ may be a good alternative to the ³He-driven thermohaline convection. A similar model has recently been proposed by Busso et al. (2007) (hereafter, referred to as BWNC). However, they assumed that a rising ring always stays in thermal equilibrium with its surrounding medium, and we argue that this leads to a substantial overestimate of the ring's radial velocity. We account for the impact of μ gradients and discuss the origin of the magnetic rings. We show that, as a mechanism for the RGB extra mixing, magnetic buoyancy has some advantages over thermohaline convection and, therefore, it is worth further investigating by means of multidimensional MHD simulations.

Our background RGB stellar model, in the radiative zone of which the formation and buoyant rise of magnetic flux rings are studied, represents a typical metal-poor upper RGB star. It has the initial mass M = 0.8 M_☉, helium and heavy-element mass fractions Y = 0.24 and Z = 0.0005, and the luminosity log  L/L_☉ = 2.085 corresponding to the age of 13.65 Gyrs. The RGB extra mixing is free to work in this star because the H burning shell has already erased the H-profile discontinuity in it. The model has been computed using the stellar evolution code described by Denissenkov et al. (2006). Its chemical element mass fraction, μ, and ∇_μ profiles immediately above the H shell are plotted in Figure 1.

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For extra mixing to dredge up material deficient in C but not enriched in Na, as required by the observed abundance patterns in the metal-poor field RGB stars (Gratton et al. 2000), it has to reach a depth between 0.045 R_☉ and 0.055 R_☉ (Figure 1(a)). We will assume the mixing depth r_mix = 0.05 R_☉ (shown by the vertical solid line in Figure 1). In previous works (e.g., Denissenkov & Weiss 1996; Denissenkov & VandenBerg 2003), the mixing depth was specified using either the relative mass coordinate δM_mix = (M_mix − M_c)/(M_bce − M_c) or the logarithmic temperature difference Δlog T = log T(r_c) − log T(r_mix), where the subscripts "c" and "bce" refer to the He core boundary and to the bottom of convective envelope, respectively. However, we have noticed that, when plotted as functions of r, the abundance profiles remain nearly stationary in spite of the slow mass inflow from the radiative zone that feeds the H burning shell. Therefore, we have decided to simply use the radius for the RGB mixing depth specification. The value of r_mix = 0.05 R_☉ corresponds to δM_mix = 0.135 and Δlog T = 0.245.

For subsequent estimates of various quantities characterizing the efficiencies of magnetic flux ring formation and buoyant rise in the radiative zone of our RGB model, we need to know some of its structure parameters at r = r_mix. These are summarized in Table 1 along with the parameter values at r = r_bce. We have also listed the values used by BWNC. Note that they have considered a half-solar metallicity bump luminosity model with the initial mass of 1.5 M_☉.

Table 1. RGB Model Structure Parameters

		Our Model		BWNC Model
Parameter	Units	rmix = 0.0500 R☉	rbce = 0.996 R☉	rmix = 0.0495 R☉	rbce = 0.912 R☉
Mr	M☉	0.309	0.331	0.252	0.256
HP	r	0.257	0.449	0.371	0.622
μ	AMU	0.593406	0.593474
T	K	2.16 × 107	2.02 × 106	2.51 × 107	2.26 × 106
ρ	g cm−3	14.8	1.03 × 10−2	5.19	2.55 × 10−3
P	dyn cm−2	4.49 × 1016	2.93 × 1012	1.87 × 1016	8.49 × 1011
K	cm2 s−1	9.35 × 107	8.15 × 1010
N2T	s−2	5.40 × 10−4	1.13 × 10−9
N2μ	s−2	1.48 × 10−7	0.00
νmol	cm2 s−1	1.80 × 102	6.76 × 102
νrad	cm2 s−1	1.74 × 102	1.43 × 104
η	cm2 s−1	19.3	6.90 × 102
κ	cm2 g−1	0.382	0.728
$|\dot{r}|$	cm s−1	3.91 × 10−5	8.50 × 10−5
Ω	rad s−1	10−3	10−6

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In our magnetic buoyancy model, it is assumed that a toroidal magnetic field in the radiative zone of an RGB star is generated by its differential rotation that stretches a preexisting poloidal field around the rotation axis. To elaborate on the model, we therefore need an estimate of the degree of differential rotation in the radiative zone. For this, we use the rotation profile M2 presented by Palacios et al. (2006) in their Figure 2 (the dotted curve in upper panel C). It shows the angular velocity Ω as a function of δM = (M_r − M_c)/(M_bce − M_c) in the radiative zone of a bump luminosity model with initial values of M, Y, and Z almost identical to ours. The model's rotational evolution was computed using the stellar evolution code STAREVOL designed for 1D simulations of the angular momentum and chemical element transport in stellar radiative zones (Siess et al. 2000; Palacios et al. 2003). The code takes proper account of the rotation-induced meridional circulation, horizontal and vertical turbulent diffusion, as well as atomic diffusion and mass loss. The initial zero-age MS model for the M2 run has been assumed to be a slow solid-body rotator with the surface rotational velocity of 5 km s⁻¹. This choice is a reasonable replacement for detailed computations of the pre-MS and early MS evolution during which a low-mass star experiences a strong magnetic breaking of its much faster initial surface rotation and a core-envelope rotational coupling leading to its nearly uniform internal rotation on a much shorter timescale than its MS lifetime.

Another important assumption made by Palacios et al. (2006) is uniform specific angular momentum distribution in the RGB convective envelope. This is required to explain the origin of rapidly rotating red horizontal branch stars (Sills & Pinsonneault 2000). This empirical demand for a strong differential rotation in the convective envelopes of low-mass RGB stars is supported by recent 3D hydrodynamic simulations of the interaction of turbulent convection and rotation performed by Palacios & Brun (2006).

The M2 bump luminosity model has Ω(r_bce) ≈ 10⁻⁶ rad s⁻¹, and Ω(r_mix) ≈ 10⁻³ rad s⁻¹. We will use these values in our following discussion. So, in the absence of transport processes other than those considered by Palacios et al. (2006), the angular velocity in the radiative zone of our RGB stellar model could vary with the radius approximately as Ω(r) = Ω(r_mix)(r_mix/r)². This steep rotation profile results from the conservation of angular momentum in the mass inflow and modest angular momentum redistribution by meridional circulation and vertical turbulent diffusion.

Denissenkov & Pinsonneault (2008b) have derived a general criterion for convective instability in the presence of a negative ∇_μ and strong horizontal turbulence, the latter being characterized by a diffusion coefficient D_h ≫ ν_mol. They have allowed for a possibility that a convective element of the diameter d can travel a distance l>d. One of the consequences of this criterion is that thermohaline convection may be suppressed by the horizontal turbulent diffusion unless

$$\begin{eqnarray} &&\frac{|\nabla _\mu |}{\nabla _{\rm rad}-\nabla _{\rm ad}} > \frac{1}{3}\frac{D_{\rm h}}{K+D_{\rm h}}, \end{eqnarray} \tag{ 1 }$$

where ∇_rad is the radiative temperature gradient (compare this inequality with condition 5 from Vauclair 2004). Note that a similar result can be obtained from the dispersion relation (10) derived and analyzed by Ulrich (1972). One should only take into account that, in the presence of the strong horizontal turbulence, Ulrich's molecular diffusion coefficient D has to be replaced with D_h. Then, for the thermally limited modes, one readily finds

$$\begin{eqnarray} &&D_{\rm thc} \approx D_{\rm Ulrich}\times \left(1-\frac{D_{\rm h}}{K}\frac{\nabla _{\rm rad}-\nabla _{\rm ad}}{|\nabla _\mu |}\right), \end{eqnarray} \tag{ 2 }$$

where D_Ulrich is Ulrich's original thermohaline diffusion coefficient. From the last expression it follows that thermohaline convection may operate (D_thc>0) only when |∇_μ|/(∇_ad − ∇_rad)>D_h/K, which is close to the requirement (1), provided that D_h < K. The difference in the right-hand-side ratios between the two conditions comes about from the fact that, unlike Denissenkov & Pinsonneault (2008b), Ulrich neglected the contribution of D to the heat diffusion. That was warranted for the nonrotating case considered by him, in which D_h = 0, and therefore D = ν_mol ≪ K.

Palacios et al. (2006) have shown that D_h always stays comparable to K at r ≈ r_mix in their M2 RGB model. Moreover, the ratio D_h/K turns out to be larger than ∼10⁻³ at all radii between r_mix and r_bce, both in the bump luminosity model and in the more evolved model shown in their Figure 5 (panels C and D). Given that the ³He burning can make the ratio |∇_μ|/(∇_rad − ∇_ad) ∼ 10⁻³ at most, while the RGB extra mixing will necessarily reduce it below this limit (by smoothing out the μ-gradient), the linear analysis predicts that thermohaline convection may be suppressed in these models, especially in the vicinity of r_mix. It would be inconsistent to ignore this prediction because similar eroding effects of the horizontal turbulence on the model rotational and mixing properties have already been included in the simulations performed by Palacios et al. (2006). They are responsible for the significant reduction of the efficiency of mixing by meridional circulation (Chaboyer & Zahn 1992) and for the erasing of the latitudinal differential rotation, the latter effect allowing to consider Ω as a function of r alone (Zahn 1992). Besides, Zahn's concept of the rotation-induced anisotropic turbulence in stellar radiative zones, with horizontal components of the turbulent viscosity strongly dominating over those in the vertical direction, was repeatedly used in stellar evolution computations to facilitate the penetration of μ-gradient barriers by the vertical turbulent diffusion in both MS stars and upper RGB stars (Talon & Zahn 1997; Talon et al. 1997; Maeder 2003; Palacios et al. 2003, 2006).

Following Mestel & Weiss (1987), we assume that, like in the solar-type MS stars, the differential rotation in the radiative zone of the RGB star creates a toroidal magnetic field B_φ by shearing a preexisting constant poloidal field B_p = {B_r,  B_θ,  0} (we use the spherical polar coordinates). As a result of its buoyancy-related undular instability (e.g., Acheson 1978; Spruit & van Ballegooijen 1982), the toroidal field is prompted to form magnetic flux rings that will rise toward the bottom of convective envelope, thus producing chemical mixing and also participating in angular momentum redistribution.

At the stellar equator (θ = 90°), the momentum and induction equations (Charbonneau & MacGregor 1993) can be reduced to

$$\begin{eqnarray} \frac{\partial \Omega }{\partial t} & = & \omega _{{\rm A},r}^2\,\frac{\partial b}{\partial r}, \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} \frac{\partial b}{\partial t} & = & r^2\,\frac{\partial \Omega }{\partial r}, \end{eqnarray} \tag{ 4 }$$

where $\omega _{{\rm A},r} = B_r/(\sqrt{4\pi \rho }\,r)$ is the local Alfvén frequency associated with the radial field component B_r, and b = rB_φ/B_r. In Equations (3) and (4), we have omitted the viscosity and magnetic diffusivity, for these will be shown to work on much longer timescales than the formation and buoyant rise of magnetic rings. For the sake of simplicity, we have additionally assumed that the poloidal field's configuration is such that B_θ = 0 at the equator. Neglecting changes with the radius of the coefficients in Equations (3) and (4), we find that the variations of Ω and b can locally be described by the same Alfvén wave equation

$$$ \frac{\partial ^2 f}{\partial r^2} - \frac{1}{\big(r^2\omega _{{\rm A},r}^2\big)}\,\frac{\partial ^2 f}{\partial t^2} = 0. $$$

So, at a given radius, the initial (continuing while t ≪ ω⁻¹_A,r) decrease of Ω and increase of |b| can be approximated as Ω(t, r) ≈ Ω_max(r)cos(ω_A,r t), and b(t, r) ≈ b_max(r)sin(ω_A,r t). From Equations (3) and (4), we can also estimate the ratio of the wave amplitudes Ω_max/b_max ≈ ω_A,r/r, hence (B_φ)_max ≈ B_r(Ω_max/ω_A,r). This ensures that the sum of the rotational kinetic and toroidal field potential energy (per unit gram) is conserved,

$$\begin{eqnarray} &&\frac{1}{2}\left(r\Omega \right)^2 + \frac{B_\varphi ^2}{8\pi \rho } = \frac{1}{2}\left(r\Omega _{\rm max}\right)^2. \end{eqnarray} \tag{ 5 }$$

Substituting the values of r = r_mix and ρ = ρ(r_mix) from our RGB model (Table 1) into the above relations, we find that

$$\begin{eqnarray} \omega _{{\rm A},r}^{-1} & = & 1.50\times 10^3\,B_r^{-1}\ \ {\rm yr\;rad}^{-1}, \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} (B_\varphi)_{\rm max} & = & 4.75\times 10^5\,(\Omega _{\rm max})_{-5}\,B_r, \end{eqnarray} \tag{ 7 }$$

where (Ω_max)₋₅ ≡ Ω_max/(10⁻⁵ rad s⁻¹). Equation (7) gives an order-of-magnitude estimate of the toroidal field maximum strength that may be generated by the differential rotation at r = r_mix, while Equation (6) estimates its growth time. For example, we may expect (from Equation (4)) that, after a time Δt ≪ ω⁻¹_A,r, a toroidal field B_φ ≈ Ω_maxB_r q Δt ≈ (B_φ)_max ω_A,r q Δt = 3.16 × 10²(Ω_max)₋₅B_r q Δt will be created, where q = (∂ln Ω/∂ln r) < 0 is the initial rotational shear, and Δt is expressed in years. Assuming that Ω_max = Ω(r_mix) = 10⁻³ rad s⁻¹ (Table 1) at the moment when the differential rotation begins to stretch the poloidal field, we obtain the estimate

$$\begin{eqnarray} &&B_\varphi (\Delta t,\,r_{\rm mix}) \approx 3.16\times 10^4 B_r\,q\,\Delta t, \end{eqnarray} \tag{ 8 }$$

where Δt ≪ 1.50 × 10³ B⁻¹_r yrs. The relative decrease of Ω for the same period of time is |ΔΩ/Ω_max| ≈ 4.42 × 10⁻⁷ B²_r q²(Δt)² ≪ 1. In the general case of B_θ ≠ 0 and Ω = Ω(r, θ) we would have B_φ ≈ rsin θ(∇Ω, B_p) Δt for Δt ≪ ω⁻¹_A,r (Spruit 1999), i.e., both poloidal field components would be involved into the winding up of toroidal field.

If the uniform rotation of the solar radiative core and the core-envelope rotational coupling in the low-mass MS stars are both produced by the back reaction of the azimuthal component of the Lorentz force emerging in the process of generation of toroidal magnetic fields by the shearing of preexisting poloidal fields, as proposed by Charbonneau & MacGregor (1993), then the MS progenitors of low-mass RGB stars are required to possess poloidal magnetic fields with strengths of the order of 0.01 G to 10 G in their radiative interiors. The low magnetic diffusivity η ∼ 10² sm² s⁻¹ in the low-mass MS stars excludes the ohmic dissipation of these fields. Therefore, the post-MS contraction of the H-exhausted core from its MS radius of ∼ 0.2 R_☉ down to its RGB radius r_c ≈ 0.02 R_☉ and magnetic flux conservation might lead to the amplification of B_r up to the values 17 G to 1 kG. From the same considerations, it also follows that B_r in the radiative zone of the RGB star may decrease with the radius as B_r ≈ B_r(r_mix)(r_mix/r)². It is true that the mass inflow in the radiative zone may sweep the frozen-in poloidal field toward the H burning shell on a long timescale of the order of $\Delta r/|\dot{r}| = (r_{\rm bce}-r_{\rm mix})/|\dot{r}|\sim 3\times 10^7$ yrs (Table 1). However, the poloidal field may be replenished by a dynamo operating at the bottom of convective envelope. It may then be entrained and redistributed all over the radiative zone by the mass inflow, which could also produce the dependence B_r ∝ r⁻². These estimates will be used in our further analysis.

6.1. General Results

Because we assume a differential rotation in the radiative zone, the inertial frame of reference has been chosen. The magnetic flux rings are assumed to be axisymmetric with respect to the rotation axis. In the spherical polar coordinates, their radial and latitudinal accelerations are

$$\begin{eqnarray} &&\frac{du_r}{dt} = \frac{u_{\theta }^2}{r}+\left(\frac{u_{\varphi }^2}{r}-\Omega ^2r\sin ^2\theta \right)+ \frac{\rho _{\rm e}-\rho }{\rho _{\rm e}+\rho }\left(\frac{GM_r}{r^2} \right.\nonumber \\ &&\left.-\Omega ^2r\sin ^2\theta \right) -\frac{B_{\varphi }^2}{4\pi r(\rho _{\rm e}+\rho)} - \frac{C_{\rm D}}{\pi a}\frac{\rho _{\rm e}}{\rho _{\rm e}+\rho }u_r\sqrt{u_r^2+u_{\theta }^2}, \end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray} \fl \frac{du_\theta }{dt} = &-&\frac{u_ru_\theta }{r}+\left(\frac{u_{\varphi }^2}{r}\cot \theta - \Omega ^2r\sin \theta \cos \theta \right) \nonumber \\ \fl\hphantom{\frac{du_\theta }{dt} =} &-& \frac{\rho _{\rm e}-\rho }{\rho _{\rm e}+\rho }\,\,\Omega ^2r\sin \theta \cos \theta - \frac{B_{\varphi }^2}{4\pi r(\rho _{\rm e}+\rho)}\cot \theta\nonumber\\ \fl\hphantom{\frac{du_\theta }{dt} =} &-& \frac{C_{\rm D}}{\pi a}\frac{\rho _{\rm e}}{\rho _{\rm e}+\rho }u_\theta \sqrt{u_r^2+u_{\theta }^2}. \end{eqnarray} \tag{ 10 }$$

These equations (Choudhuri & Gilman 1987; MacGregor & Cassinelli 2003; MacDonald & Mullan 2004) take into account the centrifugal reduction of the local gravitational acceleration, the buoyant force, the magnetic tension force, and the aerodynamic drag force (we employ the drag coefficient C_D = 1). They are supplemented with the equation

$$\begin{eqnarray} &&\frac{d}{dt}(u_\varphi r\sin \theta) = 0 \end{eqnarray} \tag{ 11 }$$

describing the conservation of the azimuthal component of the specific angular momentum. The subscript "e" means that the respective quantity is referred to the external medium surrounding the rings, while all other quantities, except Ω, refer to ring properties.

At its starting position (r₀,  θ₀), a ring is specified by its initial cross-section radius a₀, the strength of the frozen-in toroidal magnetic field B_φ,0, and internal thermodynamic properties P₀, T₀, ρ₀, and μ₀. The dynamic equilibrium between the ring and its surrounding medium requires that

$$\begin{eqnarray} &&\fl P_{\rm e} = P + \frac{B_{\varphi }^2}{8\pi },\ \ {\rm or}\ \ \frac{\Delta P}{P_{\rm e}} \equiv \frac{(P_{\rm e}-P)}{P_{\rm e}} = \frac{1}{(1+\beta)}\approx \frac{1}{\beta }, \end{eqnarray} \tag{ 12 }$$

where β ≡ P/(B²_φ/8π) ≫ 1 is the ratio of the thermodynamic to magnetic pressure in the ring. The conservation of mass and magnetic flux of the ring determine how its radius and toroidal field evolve during its motion

$$\begin{eqnarray} \left(\frac{a}{a_0}\right)^2 & = & \frac{\rho _0r_0\sin \theta _0}{\rho r\sin \theta } = \frac{B_{\varphi,0}}{B_\varphi }, \end{eqnarray} \tag{ 13 }$$

where r₀ ⩽ r ⩽ r_bce. For the equation of state, we use the ideal gas law. Finally, the entropy change in the ring is described by the simple equation

$$\begin{eqnarray} &&\frac{dS}{dt} = \frac{32(\gamma -1)\sigma T_{\rm e}^4}{3\kappa _{\rm e}\rho _{\rm e}a^2P}\,\delta T = 2\gamma K\,\frac{\delta T}{a^2} \end{eqnarray} \tag{ 14 }$$

derived by MacGregor & Cassinelli (2003). Here, S = ln  [(P/P₀)(ρ₀/ρ)^γ], δT = (T_e − T)/T_e, and γ = 5/3.

The initial conditions for the ring are specified by r₀ = r_mix =  0.05 R_☉, 0 <  θ₀ ⩽  90°, u_r =  0, u_θ =  0, u_φ = r₀sin θ₀ Ω(r₀,  θ₀), β = β₀, a = a₀, S = 0, δP ≡ (P_e − P)/P_e = (1 + β₀)⁻¹, δρ ≡ (ρ_e − ρ)/ρ_e = (1 + β₀)⁻¹, and δT = 0. The ring's radius a will be measured in units of the local pressure scale height H_P = P_e/gρ_e, where g = GM_r/r² is the gravitational acceleration. The implicit assumption of uniform pressure inside the ring requires that a ≪ H_P (the thin ring approximation). For brevity, we will denote a ≡ (a/H_P) and, given the previous remark, will only consider cases with a ≪ 1.

The above differential and algebraic equations have been solved numerically. The motion of the ring in the equatorial plane (θ₀ = 90°) is shown in Figure 2 for our assumed rotation law Ω(r, θ) = Ω(r_mix)(r_mix/r)², in which case θ(t) = θ₀. Solid curves correspond to the initial value of a₀ = 10⁻⁴, while dashed curves have a₀ = 10⁻³. The powers of ten near the curves give their specified values of β₀, the respective toroidal field strengths (in MG) being displayed in parentheses. Our computations have shown that for β₀ ≳ 10⁸ (in other words, for (B_φ)₀ ≲ 0.1 MG) and a₀ ≳ 2 × 10⁻⁴ the dependence of the total buoyant rising time on β₀ and a₀ can be approximated as

$$\begin{eqnarray} &&t_{\rm b} \approx 1.4\times 10^2\,\left(\frac{\beta _0}{10^8}\right)\, \left(\frac{a_0}{10^{-4}}\right)^2\ \ {\rm yr}. \end{eqnarray} \tag{ 15 }$$

This is transformed into the average buoyant velocity

$$\begin{eqnarray} &&\langle v_{\rm b} \rangle = \frac{\Delta r}{t_{\rm b}} = \frac{(r_{\rm bce} - r_{\rm mix})}{t_{\rm b}} \approx 15\,\left(\frac{10^8}{\beta _0}\right)\,\left(\frac{10^{-4}}{a_0}\right)^2 \;{\rm cm\;s}^{-1},\nonumber\\ \end{eqnarray} \tag{ 16 }$$

which is nearly five orders of magnitude smaller than the appropriately scaled average velocity used by BWNC (for the scaling, we have used the data from the fifth column of Table 1 and the column RGB-1 of Table 2). We have tested that this big difference is entirely caused by the assumption of thermal equilibrium between the ring and its surroundings made in the cited paper.

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Table 2. Magnetic Flux Ring Parameters

		Our Model (Δμ = 5 × 10−5)		BWNC Model (Δμ = 0)
Parameter	Units	Br = 7.3 G	Br = 73 G	RGB-1	RGB-2
a0	cm	1.3 × 107	9.7 × 106	6.5 × 106	1.5 × 107
a0	HP	1.5 × 10−2	1.1 × 10−2	5.1 × 10−3	1.2 × 10−2
tb	yr	2.9 × 102	1.6 × 102	0.19	1.1
〈vb〉	cm s−1	7.3	13	9.8 × 103	1.8 × 103
(Bφ)0	kG	254	254	380	48
β0		1.8 × 107	1.8 × 107	3.3 × 106	2.0 × 108
(Bφ)bce	kG	3.5	3.5	3.5	0.44
N		1.3 × 102	1.3 × 102	1.0	1.0
$\dot{N}=N/t_{\rm b}$	yr−1	0.45	0.83	5.3	0.91

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The dashed and dot-dashed curves in Figure 3 show the dependences of t_b on a₀ for β₀ = 10⁸ and β₀ = 10⁷ obtained by solving Equations (9)–(11), while the dotted lines represent their approximations by (15). It is seen that the exact solutions strongly deviate from the approximate ones at a₀ ≲ 2 × 10⁻⁴. This limit corresponds to a regime in which the aerodynamic drag force comes into play. Indeed, the last terms on the right-hand sides of Equations (9) and (10), which describe the drag force, are negative and inversely proportional to a. They lead to a slowing down of the ring's buoyant rise at small a, which is not reflected in Equation (15).

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6.2. Rings with μ Reduced by ³He Burning

6.3. Comparison with Results Obtained by BWNC

Given that one rising magnetic flux ring carries the mass m_b = 2π²r₀ a²₀ ρ₀sin θ₀ = 2π²ra²ρsin θ, N such rings present in the radiative zone at the same time will provide chemical mixing with the mass rate

$$\begin{eqnarray} &&\dot{M}_{\rm b} = \frac{Nm_{\rm b}}{t_{\rm b}}, \end{eqnarray} \tag{ 17 }$$

where t_b is their buoyant rising time. The quantity $\dot{M}_{\rm b}$ has to match the observationally constrained rate of the RGB extra mixing $\dot{M}_{\rm mix}\approx 4\times 10^{-8}\;M_\odot$ yr⁻¹ (BWNC). The right-hand side of Equation (17) is a function of the ring parameters N, a₀, and (B_φ)₀. For a ring with a reduced μ, the third parameter should be replaced with Δμ, unless δμ ≪ 1/β₀ (Section 6.2). In the case of Δμ = 0, considered by BWNC, the equating of $\dot{M}_{\rm b}$ to $\dot{M}_{\rm mix}$ gives a relationship between N, a₀, and (B_φ)₀. To estimate a reasonable value of a₀ at r = r_mix, they have referred to the characteristic dimension a_☉ ∼ 1000–2000 km of magnetic flux tubes that are believed to exist deep in the solar convective zone. Besides the RGB model, they have also considered a representative model for low-mass asymptotic giant branch (AGB) stars, whose structure above the H burning shell resembles that of upper RGB stars. There is indirect evidence, such as distinctive ¹⁸O/¹⁶O, ¹⁷O/¹⁶O, ¹²C/¹³C, and N/C abundance ratios in the meteorite grains of AGB circumstellar origin (Wasserburg et al. 2006) and in the atmospheres of carbon-enhanced metal-poor stars (Ryan et al. 2005; Sivarani et al. 2006; Masseron et al. 2006; Denissenkov & Pinsonneault 2008a; Lebzelter et al. 2008), indicating that extra mixing may also operate in the radiative zones of these stars (Nollett et al. 2003). In their RGB-1 and RGB-2 magnetic buoyancy models, BWNC have employed the values of a₀ obtained assuming that a(r_bce) = a_☉ and using Equation (13) with the stellar structure parameters from their AGB and RGB stellar models, respectively. These values are listed in our Table 2 along with the corresponding estimates for our RGB model.

BWNC have also assumed that there is only one magnetic flux ring floating in the radiative zone at any time. Substituting the number N = 1 together with the values of a₀ from Table 2 and $\dot{M}_{\rm mix}=4\times 10^{-8}\;M_\odot$ yr⁻¹ into Equation (17) constrains the required values for t_b and 〈v_b〉 (Table 2). Under the assumptions, made by BWNC, that the ring always stays in the thermal equilibrium with its surrounding medium (δT = 0) and that the only force impeding its motion is the aerodynamic drag force, it is easy to show that $t_{\rm b}\propto \sqrt{\beta _0/a_0}\propto (B_\varphi)_0/\sqrt{a_0}$. It is from the last relation that BWNC have determined the toroidal field strengths needed to drive the RGB extra mixing by magnetic buoyancy (Table 2, (B_φ)₀ is transformed to (B_φ)_bce using Equation (13)). They pointed out that these results are consistent with existing observations of magnetic fields in red giants (e.g., Blackman et al. 2001).

However, as we have noted, BWNC underestimated the ring's rising time by assuming that the heat exchange between the surrounding medium and the ring occurs instantaneously, which maintains ΔT = 0 all the time. Our more conservative assumption explicitly takes into consideration the radiative heat exchange, which leads to a much longer ring's rising time, as approximated by Equation (15). Employing the values of a₀, t_b, and β₀ from columns RGB-1 and RGB-2 of Table 2, we find that the rising times have been underestimated by the factors 6 × 10⁴ and 4 × 10⁶ for these buoyancy models. To keep the same estimates for the toroidal field strength, the ring number N has to be increased by the corresponding factors. However, the volume occupied by ∼10⁵–10⁶ rings is comparable to the total volume of the radiative zone, in which case the stellar structure would be greatly disturbed, especially at the bottom of convective envelope where the rings encounter turbulent convection. Here, the frozen-in toroidal field is either quickly dissipated via the strong turbulent diffusion, thus depositing its energy at the radiative/convective interface, or it inhibits convective motions if its potential energy exceeds the turbulent kinetic energy (Moss 2003). In either case, the stellar structure would be strongly modified. In the next section, we will show that a reasonably small number of N can be obtained only for rings with a reduced μ.

6.4. There Still May Be a Solution

By assigning N = 1, BWNC have implicitly assumed that the average time t_form needed to form a magnetic flux ring is equal to its buoyant rising time t_b. Indeed, instead of (17) the buoyancy mixing mass rate ought to be calculated as $\dot{M}_{\rm b} = m_{\rm b}/t_{\rm form} = Nm_{\rm b}/t_{\rm b}$, where N = t_b/t_form. We assume that, like in the case of the solar tachocline (e.g., Schmitt & Rosner 1983), the appropriate MHD mechanism responsible for the formation of magnetic flux rings in the vicinity of the H burning shell is the undular buoyancy instability. The criterion for its development has been extensively discussed in the literature, e.g., by Acheson (1978), Spruit & van Ballegooijen (1982), Spruit (1999), and Fan (2001). It has been shown that a diffusive toroidal magnetic field gets broken into distinct arching flux tubes when

$$\begin{eqnarray} &&B_\varphi > (B_\varphi)_{\rm crit} \approx \sqrt{4\pi \rho \,r^2 N^2\,\frac{H_P}{r}\,\frac{\eta }{K}}, \end{eqnarray} \tag{ 18 }$$

provided that (∂ln B_φ/∂ln r) = O(1). A profile of (B_φ)_crit in the radiative zone of our RGB model is plotted with the dashed curve in Figure 4(a). Its corresponding profile of β_crit is shown with the dot-dashed curve. At r = r_mix (log  r_mix/R_☉ ≈ −1.30), the critical toroidal field is 254 kG and β_crit = 1.8 × 10⁷ (Table 2).

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In the presence of rapid rotation, such that $\Omega \gg \omega _{{\rm A},\varphi } \equiv B_\varphi /(\sqrt{4\pi \rho }\,r)$, the Coriolis force reduces the instability's growth rate ω_A,φ by the factor ω_A,φ/Ω (Pitts & Tayler 1985; Spruit 1999). In our RGB model, Ω(r_mix) = 10⁻³ rad s⁻¹ strongly exceeds ω_A,φ(r_mix) = 2.11 × 10⁻¹¹ B_φ rad s⁻¹ for all reasonable strengths of B_φ ≪ 47 MG. Therefore, we use the estimate of t_inst ≈ Ω/ω²_A,φ for the instability growth time. The winding up of a toroidal field by differential rotation continues until B_φ ≈ ΩB_rΔt q reaches the critical value (18). This takes (Δt)_crit ≈ 3.17 × 10⁻³(B_φ)_crit/(B_r Ω₋₅ q) years. After that, it will take another t_inst ≈ 1.93 × 10¹⁰ [ ρ(r/R_☉)² Ω₋₅]/(B_φ)²_crit years for the buoyancy instability to occur and assemble the rings. So, the total ring-formation time is t_form = (Δt)_crit + t_inst. For our simple estimates to be true, t_form should be shorter than $\omega _{{\rm A},r}^{-1} = 7.82\times 10^3\,\sqrt{\rho }\,(r/R_\odot)/B_r$ yrs, which allows us to consider that the rotational shear is nearly constant and that, for our assumed rotation law in the radiative zone Ω₋₅ = 10²(r_mix/r)², q = O(1). For this case, and assuming that B_r = 7.32(r_mix/r)² G, the four characteristic timescales are plotted in Figure 4(b). In particular, we have t_form = 2.20 yrs at r = r_mix. Note that (Δt)_crit ∝ (B_rΩ)⁻¹, t_inst ∝ Ω, and ω⁻¹_A,r ∝ B⁻¹_r. We have used the parameter B_r(r_mix) = 7.32 G for which the timescales (Δt)_crit and t_inst coincide. Its value scales as Ω⁻²(r_mix). At a fixed value of Ω(r_mix), B_r(r_mix) determines which of the two timescales, (Δt)_crit or t_inst, makes a predominant contribution to t_form.

It is very likely that the buoyancy instability forms not just one but a number n>1 of magnetic flux rings during the time t_form, therefore N = n(t_b/t_form). The equating of $\dot{M}_{\rm b}=n m_{\rm b}/t_{\rm form}$ to $\dot{M}_{\rm mix} = 4\times 10^{-8}\;M_\odot$ yr⁻¹ constrains the ring's minimum radius as a function of t_form, n, and the initial colatitude

$$\begin{eqnarray} \left(\frac{a_0}{H_P}\right)^2 &=& \frac{\dot{M}_{\rm mix}\,t_{\rm form}}{n\,2\pi ^2r_0^3\rho _0\sin \theta _0}\left(\frac{H_P}{r_0}\right)^{-2}\nonumber\\ &=& 9.78\times 10^{-5}\frac{t_{\rm form}}{n\,\sin \theta _0}, \end{eqnarray} \tag{ 19 }$$

where r₀ = r_mix, and t_form is expressed in years.

For the values of t_form = 2.20 yrs, n = 1, and θ₀ = 90°, we calculate a₀ = 1.47 × 10⁻² (in units of H_P, as usual), and the total number of rings in the radiative zone N = 0.454 t_b. As mentioned before, the ring with the radius a₀ = 10⁻² and μ reduced by Δμ = 5 × 10⁻⁵ has the buoyant rising time t_b ≈ 132 yrs. This means that the ring with a₀ = 1.47 × 10⁻² and the same mean molecular weight would cross the radiative zone in t_b ≈ (1.47)² × 132 = 285 yrs (Equation (15)). Hence, we have N = 129. Note that this number does not change if the values of the parameters t_form and n are varied, provided that our buoyancy mixing mass rate is still constrained to match $\dot{M}_{\rm mix}$. Indeed, in this case both the ratios t_form/n (Equation (19)) and t_b (Equation (15)) are proportional to a²₀, which leads to its cancellation in the relationship N = n(t_b/t_form).

We have shown that the reduced mean molecular weight in the magnetic flux rings formed in the region of ³He burning accelerates their buoyant rise quickly enough for their total number needed to maintain the RGB extra mixing to be reasonably small (N ≈ 10² rings would only occupy ∼10⁻³–10⁻⁴ part of the radiative zone). However, the estimated value of N ≈ 1.3 × 10² is in fact a lower limit obtained under the most favorable assumptions. In particular, we have silently assumed that all rings have the same formation time and cross-section radius related by Equation (19), and that no rings are formed above the radius r_mix. These assumptions are obviously not true. Relaxing either of them will certainly increase the total ring number in the radiative zone. To figure out what effect may be produced by relaxing the first assumption, we would have to determine a relationship between t_form and a₀ for a spectrum of rings created by the buoyancy instability, e.g., like it has been done for the solar tachocline by Schmitt & Rosner (1983). This problem is out of scope of the present preliminary study.

We find it more important to address here the second issue. Indeed, given that the critical strength of toroidal magnetic field for triggering the buoyancy instability decreases rapidly with the radius (the dashed curve in Figure 4(a)), while our estimated ring-formation time stays much shorter than the toroidal field growth time ω⁻¹_A,r (Figure 4(b)), the formation of magnetic flux rings at r₀>r_mix appears to be unavoidable. These rings would contribute to the total ring number present in the radiative zone at the same time but they would not participate in the chemical element transport because their constituent material has not been nuclearly processed. In fact, the buoyancy of these "parasitic" rings should be reduced compared to the buoyancy of the rings originating at r ≈ r_mix because their mean molecular weight does not differ from that of the surrounding medium through which they rise. On the other hand, they are formed in a region where the thermal diffusivity K is higher than at r = r_mix and, therefore, their heat exchange with the surroundings goes faster, which should accelerate their buoyant rise. Our computations show that the rings formed in the region of constant μ (at r ≳ r_mix + 0.05 R_☉) rise 10³ to 10⁴ times slower than the rings formed in the μ-depression domain. Given that their formation takes between 10 and 100 yrs (the solid curve in Figure 4(b)), a number of the "parasitic" rings present in the radiative zone at the same time may be as large as 10⁵. Unfortunately, our model is too simple to be able to predict if and how the "parasitic" rings could impede the large-scale magnetic buoyancy mixing, but we do realize that they may create a real problem for the mechanism proposed by BWNC that has been revised in our paper. Indeed, although BWNC did not discuss a formation of magnetic rings, it is difficult to understand why the rings cannot be formed at r>r_mix as easily as they are created at r = r_mix.

Our approximate analysis of the magnetic flux ring formation and buoyant rise in the differentially rotating radiative zone of the bump luminosity RGB star gives some support to a combined "magneto-thermohaline" mode of the RGB extra mixing, as opposed to the pure thermohaline and pure magnetic buoyancy modes proposed by Charbonnel & Zahn (2007a) and BWNC, respectively. A key component to the operation of our mixing mechanism, which is certainly present in all upper RGB and low-mass AGB stars, is the mass inflow in the radiative zone. It makes two important things. First, the conservation of the specific angular momentum in the mass inflow results in a steep Ω-profile with a rotational shear q ≈ O(1) that does not appear to be strongly reduced by the meridional circulation and turbulent diffusion (Denissenkov & Tout 2000; Palacios et al. 2006). Second, the mass inflow entrains a diffusive magnetic field that is probably generated via an α–Ω or α² dynamo just beneath the bottom of convective envelope (Nordhaus et al. 2007; Nordhaus & Blackman 2007; Mestel 2001). A very high ratio of the ohmic dissipation time to the average inflow time $((\Delta r)^2/\langle \eta \rangle)/(\Delta r/\langle |\dot{r}|\rangle) \approx 1.2\times 10^4$ guarantees that this field will be redistributed over the whole radiative zone. Because the volume occupied by this field is squeezed when the flow approaches the H burning shell, the field may become stronger at a smaller radius. In general, this random field has an unstable configuration that will decay on a short Alfvén timescale (Spruit 1999). However, in their 3D MHD simulations Braithwaite & Nordlund (2006) have shown that such unstable random field may evolve into a stable "twisted torus" configuration with toroidal and poloidal field components of comparable strength. The poloidal field lines get wrapped around an axisymmetric torus, thus forming an approximate dipole. These simulations have provided the first plausible explanation of the origin and stability of dipole magnetic fields in presently nonconvective stars. It is assumed that the seed random magnetic field in those stars had been a dynamo left over from the period of their protostellar convective contraction.

Our RGB extra mixing model has much in common with the solar magnetic spindown model that postulates the presence of a weak poloidal field in the solar radiative core (Mestel & Weiss 1987; Charbonneau & MacGregor 1993). Note that even the values of r_bce = 0.996 R_☉ and Ω(r_bce) = 10⁻⁶ rad s⁻¹ used in our RGB model are close to the corresponding solar values. The important difference is the assumption of strong differential rotation in the radiative zone of our model, while the present-day Sun is known to be a nearly solid-body rotator, at least above r = 0.2 R_☉ (Couvidat et al. 2003). However, the young Sun did possess a strong differential rotation in the core which had resulted from its spinning up during the pre-MS contraction and angular momentum loss from the surface via a magnetized stellar wind. The magnetic spindown model assumes that the differential rotation in the young Sun was broken by the back reaction of the Lorentz force that emerged when the differential rotation was winding up a toroidal field from the preexisting poloidal field (Charbonneau & MacGregor 1993), or by the magneto-rotational instability (Menou & Le Mer 2006).

Whereas an important role in damping large-scale toroidal field oscillations in the young Sun is thought to be played by the phase mixing (Charbonneau & MacGregor 1993), we do not think that this is also true for our magnetic buoyancy model. Indeed, the phase mixing timescale at r = r_mix in our RGB stellar model is

$$\begin{eqnarray} &&t_{\rm p} = \left(\frac{3\pi ^3r^2}{\eta \omega _{{\rm A},r}^2q_{\rm A}^2}\right)^{1/3}\approx 2.9\times 10^5\ \ {\rm yrs}, \end{eqnarray} \tag{ 20 }$$

where we have used an estimate of t_p obtained by Spruit (1999), assuming that q_A = 1. This is comparable with the time needed for the whole radiative zone to be thoroughly mixed, $t_{\rm mix} = (M_{\rm bce} - M_{\rm mix})/\dot{M}_{\rm mix}\approx 5.5\times 10^5$ yrs. Hence, long before the toroidal field oscillations on neighboring magnetic surfaces get out of phase, the magnetic buoyancy instability will come into play, the magnetic flux rings will be formed and reach the convective envelope.

Compared with the pure thermohaline mixing, our model has the following advantages. First, the horizontal turbulent diffusion is unlikely to hinder the buoyant rise of magnetic flux rings with a reduced μ because the frozen-in toroidal field will not allow turbulence to penetrate plasma in the rings and decrease the μ contrast between the ring and surrounding material. On the contrary, there is nothing to prevent the horizontal turbulent diffusion from eroding the μ contrast in thermohaline convective elements (Section 4).

Second, thermohaline convection may fail to explain the operation of enhanced extra mixing in rapidly rotating Li-rich K-giants, which is needed to activate the Cameron–Fowler mechanism, because one would expect that the growth of "salt fingers" is impeded by rotation (Canuto 1999). Oppositely, it would be natural to suppose that magnetic flux rings are formed more efficiently in the more rapidly rotating stars. This hypothesis is supported by the fact that the fastest rotators among young cluster solar-type stars appear to have the shortest timescale of rotational coupling between the core and envelope (e.g., Irwin et al. 2007).

Third, whereas the efficiency of thermohaline convection in upper RGB stars is dependent exclusively on the abundance of ³He left in the radiative zone, magnetic buoyancy can, in principle, be driven by differential rotation alone, provided that it succeeds in winding up a sufficiently strong toroidal field. By the end of the RGB evolution of a low-mass star, its envelope ³He abundance gets depleted by a factor of ten or more (Charbonnel & Zahn 2007a). So, what will then drive extra mixing in this star on the AGB?  The higher rate of heat exchange between "salt fingers" and their surrounding medium increased in proportion to the luminosity will not help because the evolutionary timescale decreases inversely proportional to the luminosity which cancels the former effect. Thus, there is a need for an additional driving parameter that would not let extra mixing die out. Such parameter might be the rotational velocity. Indeed, it is known that a surprisingly fast rotation has somehow survived in red horizontal branch stars in spite of the RGB mass loss (Peterson 1983; Sills & Pinsonneault 2000; Behr et al. 2003). It could be used on the subsequent AGB evolutionary phase to drive our magneto-thermohaline mixing. The much lower (compared to the RGB phase) ³He abundance left in these stars could be compensated by a stronger toroidal magnetic field or a larger number of magnetic flux rings generated in the presence of a higher mass inflow rate ($\dot{M}_{\rm mix}\approx 10^{-6}\;M_\odot$ yr⁻¹, according to BWNC).

Fourth, thermohaline convection cannot penetrate below the radius r_min ≈ 0.063 R_☉ at which μ has a minimum (the right vertical dotted line in Figure 1). However, mixing down to this depth would be too shallow to reduce the surface carbon abundance (Figure 1(a)), as required by observations (Gratton et al. 2000; Smith & Martell 2003). An overshooting on a length scale of order H_P could solve the problem (the left vertical dotted line in Figure 1 is placed at a distance H_P below r_min) but then thermohaline "fingers" would have to penetrate a region of higher μ where they experience a strong breaking. It should also be noted that the penetration of a region with the positive ∇_μ below r_min would reduce the average mixing rate D_thc ∝ |∇_μ| by decreasing the slope of the negative ∇_μ in the mixed radiative zone. In the magnetic buoyancy model, there is at least a potential possibility to dredge up the nuclearly processed material from below r_min. This can be done by rings with a frozen-in magnetic field of a few MG if such are formed closer to the major H burning shell. Of course, all the above assumptions need to be verified by more rigorous models.

Fifth, it is interesting to note that the operation of our RGB mixing mechanism seems to produce environment in which the functioning of the ³He-driven thermohaline convection is impossible. Indeed, Charbonnel & Zahn (2007b) have estimated that a toroidal magnetic field of B_φ ≈ 100 kG would entirely suppress the thermohaline mixing at r ≈ r_mix, while a field of 10 kG would be sufficient to inhibit the thermohaline instability in the upper one-third of the narrow μ-depression domain. These fields are weaker than those needed for our mechanism to work.

It is also important to note that the flux rings considered in our paper are a convenient proxy for the more realistic Ω-shaped loops that are the likely products of the magnetic buoyancy instability that operates in a layer of strong toroidal field (e.g., Caligari et al. 1995). Downflows/upflows that take place in the loop legs during formation and rise may affect the efficiency of transport of processed material relative to the behavior obtained in the case of a flux ring. It is not clear how the strong rotational shear that is present within the radiative zone would affect this process. In principle, the only place where significant shear is required is within the layers where μ is depressed and flux tube formation takes place. If the overlying portion of the radiative zone were in a state of near-uniform rotation, the loop legs might form a kind of conduit connecting the region containing processed material with the bottom of the convection zone. In any case, it is difficult to give an airtight argument for what the rotational state of the radiative zone should be, either differential rotation of the kind assumed in our paper or near-uniform rotation enforced by the fields that permeate the region.

Another aspect of our computations that should be noted is that the results of Sections 6.1 and 6.2 were obtained for rings initially in thermal equilibrium with their surroundings, a state of maximal initial buoyancy. If, alternatively, the rings were in an initial mechanical equilibrium state (zero net force), the rise would be slower by virtue of the heating required to overcome the neutral buoyancy at t = 0. This would also increase the total ring number.

In this work, we have presented a simple model of the formation and buoyant rise of magnetic flux rings in the radiative zone of the bump luminosity RGB star. Our model is based on ideas and equations published by Spruit & van Ballegooijen (1982), Schmitt & Rosner (1983), Mestel & Weiss (1987), Choudhuri & Gilman (1987), Charbonneau & MacGregor (1993), Spruit (1999), Eggleton et al. (2006), BWNC, and Denissenkov & Pinsonneault (2008b). It qualitatively describes a possible mechanism for the RGB extra mixing, which we call the magneto-thermohaline mixing, as an alternative to the pure ³He-driven thermohaline convection that has recently been proposed by Charbonnel & Zahn (2007a). For our mechanism to work, the radiative zone has to possess a strong differential rotation and a poloidal magnetic field B_p ≳ 1–10 G. We assume that the differential rotation stretches the poloidal field around the rotation axis, thus creating a strong toroidal magnetic field B_φ ≈ 0.1–1 MG. When the latter exceeds a critical value, the buoyancy-related undular instability comes into play to form magnetic flux rings. These rings turn out to be buoyant, therefore they rise toward the bottom of convective envelope.

We have shown that, when the radiative heat exchange between the ring and its surrounding medium is taken into account, the ring's buoyant rising time increases by about five orders of magnitude compared to the case considered by BWNC, when the ring and its surrounding medium are assumed to be in thermal equilibrium all the time. However, given that our model still neglects possible internal heating of the ring's material by residual nuclear reactions and anisotropic thermal exchanges in the presence of strong oriented magnetic field, while it uses the aerodynamic drag coefficient that is 20 times as large as the one employed by BWNC, it is fair to say that BWNC might have fixed a safe upper limit while our paper fixes a conservative lower limit for the magnetic ring's rising velocity.

We have found that the number of rings needed to be present in the radiative zone at the same time to produce the observationally constrained rate of the RGB extra mixing is unrealistically large unless these rings originate from the region of the μ inversion maintained by the ³He burning. Such rings have a deficit of the mean molecular weight compared to the bulk of the radiative zone through which they move. Their buoyancy is mainly caused by the difference in μ rather than by a deficit in density due to the excess magnetic pressure. The frozen-in toroidal magnetic field is still needed for the rings to remain cohesive while rising. That is why we have coined the term "magneto-thermohaline" mixing. Our model has some advantages over the pure thermohaline mixing model, the most important of which being the robustness of the magnetic rings against the eroding effect produced by the horizontal turbulent diffusion. Leaving aside the problem of the "parasitic" rings that are formed at r ≳ r_mix + 0.05 R_☉, our model looks promising. However, because it is based on a number of assumptions whose legality is impossible to confirm in the framework of our 1D computations we call for its future verification by 3D MHD simulations.

P.A.D. and M.P. acknowledge support from the NASA grant NNG05 GG20G. The National Center for Atmospheric Research is sponsored by the National Science Foundation.