Thermohaline Mixing: Does It Really Govern the Atmospheric Chemical Composition of Low-Mass Red Giants?
aa r X i v : . [ a s t r o - ph . S R ] N ov THERMOHALINE MIXING: DOES IT REALLY GOVERNTHE ATMOSPHERIC CHEMICAL COMPOSITION OFLOW-MASS RED GIANTS?
Pavel A. Denissenkov , and William J. Merryfield [email protected]@ec.gc.ca ABSTRACT
First results of our 3D numerical simulations of thermohaline convectiondriven by He burning in a low-mass RGB star at the bump luminosity arepresented. They confirm our previous conclusion that this convection has a mix-ing rate which is a factor of 50 lower than the observationally constrained rateof RGB extra-mixing. It is also shown that the large-scale instabilities of salt-fingering mean field (those of the Boussinesq and advection-diffusion equationsaveraged over length and time scales of many salt fingers), which have been ob-served to increase the rate of oceanic thermohaline mixing up to one order ofmagnitude, do not enhance the RGB thermohaline mixing. We speculate on pos-sible alternative solutions of the problem of RGB extra-mixing, among which themost promising one that is related to thermohaline mixing is going to take advan-tage of the shifting of salt-finger spectrum towards larger diameters by toroidalmagnetic field.
Subject headings: stars: abundances — stars: evolution — stars: interiors
1. Introduction
Red giants with M . . M ⊙ are known to experience extra-mixing in their convectivelystable radiative zones that separate the hydrogen burning shell (HBS) from the bottom Department of Physics & Astronomy, University of Victoria, P.O. Box 3055, Victoria, B.C., V8W 3P6,Canada Canadian Centre for Climate Modelling and Analysis, University of Victoria, P.O. Box 3065, Victoria,B.C., V8W 3V6, Canada C/ C ratio and carbon abundance to resume their declines with an increasein luminosity.The most promising physical mechanism of RGB extra-mixing is thermohaline con-vection (Charbonnel & Zahn 2007a). It is driven by a double-diffusive instability that oc-curs when a scalar component that stabilizes a density stratification, e.g. the temperature T , diffuses away faster than a destabilizing component, e.g. the mean molecular weight µ . Eggleton, Dearborn, & Lattanzio (2006) brought our attention to the fact that the re-action He( He, 2p) He produces a local depression of µ , ∆ µ ∼ − − , in the HBS tailabove the bump luminosity. As a result, µ is increasing with the radius, which turns the µ profile into the destabilizing component. Moreover, the radiative thermal diffusivity K ex-ceeds the molecular diffusivity ν mol by many orders of magnitude (the inverse Lewis number τ = ν mol /K ∼ − in the region of µ depression), which should lead to the double-diffusiveinstability. In the ocean, a similar instability develops where warm salty water overlies coldfresh water. The oceanic thermohaline mixing usually takes the form of vertically elongatedfluid parcels of rising fresh and sinking salty water adjacent to each other, called “salt fin-gers” (Stern 1960). It has extensively been studied both experimentally and theoretically(e.g., Kunze 2003).The efficiency of thermohaline mixing in stellar cases can be estimated only theoreti-cally, e.g. via a linear stability analysis of its governing equations (Ulrich 1972) or via directnumerical simulations. Denissenkov (2010, hereafter Paper I) has recently conducted a com-parative study of thermohaline convection in the oceanic and RGB cases using both a linearstability analysis and 2D numerical simulations. Unfortunately, the linear theory yields athermohaline diffusion coefficient that is proportional to the square of the (unknown) aspectratio, a = l/d , of a finger, where l and d are the finger’s length and diameter. Applica-tions of this diffusion coefficient to model the evolutionary decline of carbon abundance inlow-mass red giants demand that a > He-driventhermohaline convection in the vicinity of a µ depression yield a mixing rate equivalent tothat approximated by the linear-theory diffusion coefficient with a <
1, i.e. it is a factor of ∼ / th of its observationally constrained value. The difference in the finger aspect ratiosbetween the oceanic ( a >
1) and RGB ( a <
1) cases is most likely determined by the verylow viscosity ν , or the Prandtl number P r = ν/K ∼ − , in the RGB case. This facilitatesthe development of secondary shear instabilities that strongly limit the growth of salt fingers(Radko 2010, and Paper I). On the other hand, the diffusion coefficient for thermohaline 3 –convection with a > γ -, and intrusive ones, that are known to occur in the ocean. Finally, wespeculate on possible alternative solutions of the problem of RGB extra-mixing.
2. 3D Numerical Simulations of Thermohaline Convection
The basic equations and numerical techniques that are employed in our 3D simula-tions of thermohaline convection for the oceanic and RGB cases are described in detail byGargett, Merryfield, & Holloway (2003). They are very similar to those used in Paper I. Themain goal of Gargett et al. was to study differential scalar diffusion in 3D stratified turbu-lence. Therefore, they considered a density stratification stable against the double-diffusiveinstability and used a specified turbulent velocity field as an initial condition. To apply theirmethod and computer code for our purposes, we have made the slower diffusing component,which represents salinity S in the ocean and µ in the current investigation, destabilizing bychanging the sign of the vertical velocity term in their equation (16). We have also reducedthe amplitude of the initial velocity field, so that it serves as a mild perturbation to initi-ate the double-diffusive instability. Our available computational resources have allowed usto run the code within reasonable time intervals with a spatial resolution of 320 . Simpleestimates of the relevant turbulent kinetic energy dissipation rate and the Kolmogorov andBatchelor length scales show that this resolution is close to marginal for our runs, however,not fully resolving the slower-diffusing component has been shown not to have a large effecton the computed fluxes (Traxler et al. 2010). The 3D simulations have been carried out forthe same number of fastest growing fingers in the computational domain and parameter setsfrom Table 1 in Paper I that were used in our 2D computations.The blue and red solid curves in the upper left panel of Fig. 1 show transitions of the ratio D S /k T to its equilibrium values in our 2D and 3D simulations of the oceanic thermohalineconvection. Here, D S and k T are the turbulent salt and microscopic thermal diffusivities.The 2D simulations were performed with the resolution 1024 (Paper I). Our estimated 3Dequilibrium value of D S /k T exceeds its corresponding 2D value merely by a factor of 1.5 4 –(compare the blue and red dashed lines). It is in good agreement with the result obtainedusing a better resolution by Traxler et al. (2010) that we have interpolated for the value ofdensity ratio R ρ = α ∆ T /β ∆ S = 1 . α and β are the coefficients of thermal expansion andhaline contraction) from their Table 1 (star symbol). The upper right panel makes a similarcomparison for the RGB thermohaline convection. In this case, we do not find any differencebetween the 2D and 3D equilibrium values of the ratio D µ /K (the blue and red dashed linescoincide). Therefore, we conclude that our results reported in Paper I seem to be correct. Inparticular, RGB thermohaline convection driven by He burning yields a turbulent mixingrate that is by a factor of 50 lower than the value required by observations.
3. Mean-Field Instabilities
Observations, laboratory experiments, and direct numerical simulations show that small-scale salt-fingering convection in the ocean can lead to instabilities and the formation of dy-namical structures on much larger scales. These include the so-called “collective instability”leading to the generation of internal gravity waves, the intrusive instability developing whenfluid is stratified both vertically and horizontally (in stars, such a situation may result fromthe rotational distortion of level surfaces), and the so-called “ γ -instability” that producesthermohaline staircases (e.g., see Traxler et al. 2010, and references therein). The latterstructures consist of thick well-mixed convective layers separated by thin salt-fingering inter-faces. The staircases are of special interest to us because they have been observed to enhancevertical mixing in the ocean by up to an order of magnitude (Schmitt et al. 2005). Moreover,our preliminary estimates in Paper I indicate that the turbulent flux ratio γ = F T /F µ is adecreasing function of R ρ in the RGB case, which is a sufficient condition for the γ -instability,according to Radko (2003).The large-scale instabilities are studied by averaging the Boussinesq and advection-diffusion equations for T and S ( µ ) over length and time scales of many salt fingers. Asa result, a system of mean-field equations is obtained in which the small-scale turbulentvelocity is incorporated into the mean heat and salinity fluxes F T and F S (Traxler et al.2010). The fluxes are related to one another and to the large-scale temperature gradientvia the flux ratio γ = F T /F S and the Nusselt number N u , which are assumed to dependonly on the local value of the density ratio R ρ ; specification of these dependences closes themean-field equations. Large-scale changes in R ρ thus result in modulations of F T and F S viathe dependences of γ and N u on R ρ that can either amplify or damp initial perturbations.A linear stability analysis of the mean-field equations yields a third-order dispersionrelationship for the growth rates of all the three large-scale instabilities mentioned above 5 –(equation 2.13 of Traxler et al. 2010). Coefficients of the cubic equation depend on thePrandtl and inverse Lewis numbers as well as on N u , γ and their derivatives with respectto R ρ . The coefficients are also functions of the horizontal and vertical wavenumbers, l and k . To calculate them, we first need to estimate the dependences of N u and γ on R ρ .Traxler et al. (2010) estimated those dependences via a body of 3D numerical simulations ofthe oceanic salt-fingering convection. We have used data from their Table 2 to reproduce, fora comparison, their “flower plot” depicting the growth rates of fastest growing instabilitiesat R ρ = 1 . γ -instability modes are the ones dominating inthe region beneath the leaves. To obtain estimates of γ ( R ρ ) and N u ( R ρ ) for the RGBcase (Fig. 2), we have performed additional 2D numerical simulations of the He-driventhermohaline convection for the same parameter set that was used in Paper I but for anumber of different values of R ρ . The resulting flower plot is shown in the lower right panelof Fig. 1. We see that in the RGB case only the salt-fingering mode is unstable in spite ofthe flux ratio γ being a decreasing function of R ρ , as in the oceanic case. This differenceis explained by the fact that the Nusselt number is extremely small, N u ∼ − , in theRGB case. Because of this, it is necessary to include horizontal diffusive fluxes in the mean-field stability analysis — something which was not done by Radko (2003) but was treatedby Traxler et al. (2010). In the RGB case, the inclusion of reasonable lateral gradientsproportional to their corresponding vertical gradients multiplied by the ratio of centrifugalto gravitational acceleration only produces an asymmetry of the bulb with respect to thevertical axis.
4. Possible Alternative Solutions
Our 2D and 3D numerical simulations of thermohaline convection driven by He burninghave demonstrated that it is unlikely to be the sole mechanism of RGB extra-mixing. Itscorresponding mixing rate in the vicinity of a µ depression, D µ ≈ × − K (the upper rightpanel of Fig. 1), is too low compared to the observationally constrained value of ( D µ ) obs ≈ . K (Paper I). Therefore, for its efficiency to be consistent with observations, the RGBthermohaline mixing has to be either assisted or modified by other processes. In Paper I,we have shown that a turbulent viscosity ν t exceeding the microscopic one by a factor of10 could enhance the rate of He-driven thermohaline mixing up to the observed value,provided that it was not itself associated with strong turbulent mixing. The enhancementof thermohaline mixing in this case is caused by the fact that the higher viscosity suppresses 6 –the development of secondary shear instabilities that limit the growth of salt fingers. Onthe other hand, if the increase of viscosity is accompanied by a similar increase of the rateof turbulent mixing, the latter will reduce salt-finger buoyancy and vertical transports bysmoothing out the chemical composition contrast between rising and sinking fluid parcels.The additional source of turbulence in the radiative zones of RGB stars could arise fromthe differential rotation that is produced by the mass inflow that feeds the HBS (Paper I). Wehave employed the COMSOL Multiphysics software package to solve the angular momentumtransport equation for the radiative zone of the low-metallicity bump-luminosity RGB modelfrom Paper I in the presence of mass inflow and turbulent shear mixing. Their respectiverate ˙ r and diffusion coefficient ν v are given by equation (23) from Paper I and equation(3) with the uncertainty factor f v from the paper of Denissenkov, Chaboyer, & Li (2006).The angular velocity at the BCE was assumed to have the same value, Ω BCE = 6 . × − rad s − , as in the M2 bump-luminosity RGB model of Palacios et al. (2006). Simple estimatesshow that for f v & r and ν v leads to a stationary solution with ν v ≈ × − K in thevicinity of a µ depression that very weakly depends on both f v and Ω BCE , although the finalrotational shear is a decreasing function of f v . This value exceeds the microscopic viscosityby a factor of 10 , which is not enough to enhance D µ to its observed value (Paper I). Itis also difficult to understand why the rotation-induced small-scale turbulence should notproduce a mixing rate of the same magnitude as ν v , in which case D µ will not be enhancedat all (Paper I). Therefore, we do not think that the proposed viscosity enhancement is apossible solution of the problem.Our angular momentum transport computations confirm the conclusion that radiativezones of low-mass RGB stars above the bump luminosity should possess strong differen-tial rotation. This may lead to the generation of high-amplitude toroidal magnetic fields,especially close to the HBS, provided that a weak poloidal field is present in the radia-tive zone. Under these circumstances, a possible alternative solution of the problem couldbe magneto-thermohaline mixing (Denissenkov, Pinsonneault, & MacGregor 2009). We in-tend to investigate this possibility using 2D numerical simulations in one of our forthcomingpapers.Toroidal magnetic fields in radiative zones may also filter out salt fingers with smalldiameters, so that the maximum growth rate is shifted towards thicker fingers. Since thefinger growth rate decreases with an increase in diameter as σ ∝ d − (e.g., Paper I), thesquare of the vertical velocity shear between the axes of neighbouring rising and sinking 7 –fingers, ∼ ( σl/d ) , is proportional to a d − . This parameter is likely to control the de-velopment of secondary shear instabilities (Kunze 1987); consequently, thicker fingers willprobably reach higher aspect ratios before they are destroyed. Following Charbonnel & Zahn(2007b), we have complemented the dispersion relationship (13) from Paper I (note that theterm νk T k S k was erroneously omitted) with terms that take into account the Lorentz forceassociated with a specified toroidal magnetic field B . Its solutions are shown in Fig. 3 forthe field strengths B = 0, 10, and 100 G (the solid blue curves from left to right). In thecase of B = 100 G, the fastest growing fingers transverse to the field have a diameter that isnearly 40 times as large as that of fingers in the absence of a magnetic field. This diameteris also almost four times larger than the diameter of fastest growing fingers in the case of aviscosity that is amplified by the factor of 10 (the red solid curve). However the influenceof such a field on the planform of the fastest growing fingers still needs to be studied. In ournext study, we will perform 2D numerical simulations of the RGB thermohaline convectionin the presence of a toroidal magnetic field to find out if the magnetic shift of the salt-fingergrowth-rate spectrum towards larger diameters can really enhance its associated mixing rateup to the empirically constrained value.PAD is grateful to Don VandenBerg who has supported this work through his DiscoveryGrant from Natural Sciences and Engineering Research Council of Canada. REFERENCES
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This preprint was prepared with the AAS L A TEX macros v5.2.
Time (s) l og ( D S / k T ) Ocean (3D vs. 2D) 0 2 4 6 8 10 12 14x 10 −3.5−3.25−3−2.75−2.5−2.25−2 Time (s) l og ( D µ / K ) RGB (3D vs. 2D) k − l og ( l ) −0.05 0 0.05−101234567 −6−5−4−3−2−10 k − l og ( l ) −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−101234567 −9−8−7−6−5−4−3 Fig. 1.— Upper panels compare the ratios of thermohaline diffusion coefficient to thermaldiffusivity obtained in our 3D (red curves) and 2D (blue curves) numerical simulations forthe oceanic (left) and RGB (right) cases. Black star symbol shows a value interpolatedfor the density ratio R ρ = 1 . − ) of the fastest growing large-scale instabilities of salt-fingering mean-field as functions of the horizontal l and vertical k wavenumbers, both measured in cm − , for the oceanic (left) and RGB (right) cases. 10 – R ρ γ − R ρ N u ( − ) Fig. 2.— The inverse flux ratio, γ − = F µ /F T , and Nusselt number as functions of densityratio estimated from our 2D numerical simulations of the RGB thermohaline convection. 11 – log d (cm) l og σ ( s − ) Fig. 3.— The RGB salt-finger growth-rate spectra computed for the same parameter setas in Table 1 from Paper I under the assumptions that the toroidal magnetic field has thestrengths B = 0, 10, and 100 G (the blue curves from left to right), and for B = 0 G butwith the viscosity amplified by the factor of 104