Table Of ContentDraft version January 30, 2017
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
DRAFT FROM January 30, 2017
SYNERGIES BETWEEN ASTEROSEISMOLOGY AND THREE-DIMENSIONAL SIMULATIONS OF STELLAR
TURBULENCE
W. David Arnett1 and E. Moravveji2
Draft version January 30, 2017
ABSTRACT
Turbulent mixing of chemical elements by convection has fundamental effects on the evolution of
stars. The standard algorithm at present, mixing-length theory (MLT), is intrinsically local, and
must be supplemented by extensions with adjustable parameters. As a step toward reducing this
7
arbitrariness, wecompareasteroseismicallyinferredinternalstructuresoftwoKeplerslowlypulsating
1
B stars (SPB’s; M 3.25M ) to predictions of 321D turbulence theory, based upon well-resolved,
0 (cid:12)
∼
truly turbulent three-dimensional simulations (Arnett, et al. 2015; Cristini, et al. 2016) which include
2
boundary physics absent from MLT. We find promising agreement between the steepness and shapes
n of the theoretically-predicted composition profile outside the convective region in 3D simulations and
a
in asteroseismically constrained composition profiles in the best 1D models of the two SPBs. The
J
structure and motion of the boundary layer, and the generation of waves, are discussed.
6
Subject headings: convection
2
]
R 1. INTRODUCTION (1D) stellar evolution codes.
Despite this progress in computation, simplifications
S Observational and computational advances over the
arenecessary. Attainablenumericalresolutionallowsnu-
. past decade have placed us in a unique era, in which
h mericalReynoldsnumbersRe>104,whicharedefinitely
study of the interiors of stars can be pursued at much
p turbulent, but stars have far higher Reynolds numbers
higher precision than before. The recent observations
- (Arnett&Meakin2016). Toattainhighlyturbulentsim-
o from space of pulsating stars (through the MOST,
ulations, onlyafractionofthestariscomputed(a“box-
r CoRoT,Kepler,K2,andBRITE-constellationmissions),
t in-star” approach) and rotation and magnetic fields are
s andtheplannedfuturemissions(likeTESSandPLATO)
ignored. The simulations extend from the integral scale
a supplyprecisephotometryofstars,withnear-continuous
[ timesamplingfordurationsofweekstoyears. Amongall down into the Richardson-Kolmogorov cascade, and the
subgrid dissipation merges with the Kolmogorov “four-
targets, those more massive than 1.4M have a criti-
1 ∼ (cid:12) fifths” law (Frisch 1995; Arnett, Meakin & Young 2009).
calfeatureincommon: duringtheirmainsequencelives,
v
In addition to the Reynolds number issue, the simula-
3 they harbor a convective core and a radiative envelope.
tions have negligible radiation diffusion (infinite P´eclet
0 The gravity (g) modes propagate in the radiative enve-
number) because of vigorous neutrino cooling, rather
8 lope, are reflected from the convective core, providing
than large but finite P´eclet numbers found during hy-
7 valuable information regarding the physical conditions
drogen and helium burning (Viallet, et al. 2015).
0 near the boundary. Thus, g-mode pulsating stars are
These simulations represent turbulent solutions to the
. excellent tests of the physics of core convection.
1 Navier-Stokes equations, and a step beyond mixing-
In parallel, two- and three-dimensional (2D/3D) im-
0 length theory: they can resolve boundaries.
plicit large eddy simulations (ILES) of turbulent con-
7 It is timely to compare the results of observations, as-
vection at different evolutionary phases (e.g., Baz`an &
1 teroseismic modeling and 2D/3D simulations(e.g. Aerts
Arnett (1994); Nordlund & Stein (1995); Freytag, Lud-
:
v wig, & Steffen (1996); Meakin & Arnett (2006, 2007b); & Rogers 2015). Although these are independent ap-
i Miesch(2005);Nordlund,Stein,&Asplund(2009);Vial- proaches, it is possible to understand many underlying
X
similarities between the state-of-the-art simulations and
let, et al. (2013); Arnett, et al. (2015), and many more)
r modeling. Such synergies shall allow us improve our
haveshedlightonthebehaviorofstellarconvection,and
a
treatment of turbulent convection in 1D models, and ac-
inparticularontheinterfacebetweenconvectiveandra-
count for the convectively-induced mixing in the radia-
diative zones. This allows the development of non-local
tive interior (through overshooting and internal gravity
time-dependent convective theories which are consistent
waves) in a more consistent way.
with the 3D simulations. The numerical data provides
We review the recent asteroseismic modeling of two
closure to the Reynolds-averaged Navier-Stokes (RANS)
Kepler slowly pulsating B stars (SPB’s) in 2. In 3 we
equations (Arnett, et al. 2015), converting numerical ex-
§ §
compare the theoretical descriptions in MLT and 321D,
periments into theory. This approach is called 321D,
region by region. In 4 we compare the shapes of the
andaimstoprovidealternatives(ofincreasingsophistica-
§
abundance profiles at the convective boundaries, as in-
tion)totheclassicalMixingLengthTheoryofconvection
ferred from astreroseismology and from the 3D simula-
(MLT, B¨ohm-Vitense 1958), for use in one-dimensional
tionsofO-burningshellina23M model(Arnett,etal.
(cid:12)
1Steward Observatory, University of Arizona, Tucson, AZ 2015)andofC-burningshellina15 M(cid:12)model(Cristini,
85721,[email protected] etal.2015,2016). Ourconclusionsaresummarizedin 5.
2Institute of Astronomy, KU Leuven, Celestijnenlaan 200D, §
3001Leuven,Belgium 2. INPUTMODELS
2
O S W and mass number of each nucleus i. The Ledoux dis-
16 criminant,
0.75
M
14 =∆ = ,
n ad Y
0.70 o L ∇ ∇−∇ −∇
1]− 12 cti reduces to the the Schwarzschild discriminant,
c a
se 10 0.65 fr = ad,
2[cmmix 68 MFCuioxlrleeyd REnadviealotipvee 0.60 enmass wbcoahrtiriccehstpeiomsnpdpeionrsagittduivirmee.egnrsa∇Sidoinealnne∇tsdss,−g∇rre∇aasddpieeacntrtievienalycm,tuaeaanlndam∇nodµleicasudtlihaaer-
D 0.55 g
o weight.
g 4 r
o d The convective velocity is approximately
l 2 0.50 Hy
ExtraDiffusiveMixing u2 g(cid:96)∆ >0, (1)
∼ ∇
0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 where(cid:96)isthefreeparameter,themixinglength,thetem-
perature excess is ∆ = , and g is the
EnclosedMass[M ] ∇ ∇ − ∇a − ∇Y
(cid:12) gravitationalacceleration. Fortheturbulentvelocity,the
Fig. 1.—InternalmixinginKIC7760680. Theblueregionisthe MLTasgivenbyEq.1isanadequateapproximationover
formalconvectivecore(MLT),thegreyregionisthead-hocover-
region OS, and is the commonly used choice. This is a
shoot (braking) layer, and the green shows the radiative interior
which is diffusively mixed at a rate logDmix = 0.75 (cm2sec−1). steady-state approximation which may be inferred from
Theorigin(O),Schwarzschildboundary(S),bottomoffullymixed the turbulent kinetic energy equation (Meakin & Arnett
region (M) and the top of partially mixed (wave) region (W) are 2007b) or alternatively, from the balance of buoyant ac-
annotated. Therightordinateshowsthehydrogenabundancepro- celerationanddecelerationbyturbulentfriction(uu/(cid:96));
file. | |
seeKolmogorov(1941),Kolmogorov(1962),Smith&Ar-
Moravveji,etal.(2015,2016)recentlydidin-depthfor-
nett (2014), and Arnett, et al. (2015).
ward seismic modeling of two SPB stars having the rich-
Using the sound speed s2 = γPV, the Mach number
est seismic spectra known so far. Both KIC10526294
=u/s, andthepressurescaleheightH =PV/g, we
(Papics et al. 2014), and KIC7760680 (Papics et al. M P
may write Eq. 1 as
2015) are of spectral class B8V (3.25M ), and exhibit
(cid:12)
long and uninterrupted series of dipole ((cid:96)=1) g-modes. 2 ((cid:96)/H )∆ .
P
In both cases, the relative frequency differences between M ∼ ∇
the observations and models is less than 0.5%. In addi- For quasi-static stellar evolution, the convective Mach
tion to placing tight constraints on the extent of over- numbers are small ( 0.01). Since (cid:96)/HP 1, ∆
M≤ ∼ ∇(cid:28)
shooting beyond the core, and additional diffusive mix- 1 giving ad for a well-mixed convection zone in a
∇ ∼ ∇
ing in the radiative interior, the authors concluded that stellar interior.
the exponentially-decaying diffusive mixing profile for
core overshoot outperforms the step-function prescrip- 3.1.2. 321D
tion. Thus, convectively-induced mixing beyond the for- Fully turbulent 3D simulations of convection may be
mal core boundary seems to have a radial dependence, represented as solutions to a simple differential equation
and decays outwards. foreithertheturbulentkineticenergy,or(asshownhere)
Figure 1 shows the internal mixing profile (colored re- for the turbulent velocity u (Arnett, et al. 2015),
gions) of the best model of KIC7760680, with selected
positions in the enclosed mass coordinate labeled O, S, ∂tu+(u )u= (2)
·∇ B−D
M, and W. We will discuss each region, comparing and
where the buoyancy term3 is gβ ∆ (β =
contrasting the MLT, and a 321D theory based upon (∂lnρ/∂lnT) is the thermodynBami≈c factTor4∇to conTvert
well-resolved numerical simulations (see Arnett, et al. P
temperature excess to density deficit at constant pres-
(2015); Cristini, et al. (2016) and extensive references
sure), and the drag term (chosen to be consistent with
and discussion therein).
the Kolmogorov cascade) is u/τ, with the dissipa-
3. COMPARINGMLTAND321D tion time τ =(cid:96)d/u. Here (cid:96)dDis≈the dissipation length, a
| |
property of the turbulent cascade, and may differ from
There are four distinct regions in Fig. 1, which we
the mixing length used in MLT. We average over angles
elaborate below: (1) a Schwarzschild core, (2) a brak-
andtakethesteadystatelimit,whichgivesfortheradial
ing (overshoot) region, (3) a composition gradient, and
convective speed u ,
(4) a radiative envelope. r
u du /dr =gβ ∆ u u /(cid:96) . (3)
r r T r r d
3.1. Region OS: Schwarzschild core ∇− | |
Away from the convective boundaries the gradient of u
3.1.1. MLT
is small, and this equation resembles Eq. 1 for appro-
In Figure 1, the region extending from the origin at priately chosen (cid:96) . Eq. 3 implies a heating rate due to
d
O to S is well mixed; at S the Schwarzschild criterion thedissipationofflowatsmallscales(Kolmogorov1941,
changes sign, so that within OS buoyancy drives convec-
tion. 3 For strongly stratified media, there is an additional driving
For a composition gradient of zero, = termdueto“pressuredilatation”(Viallet,etal.2013).
µ Y
Σ X /A 0, where X and A are th−e∇mass fr∇actio≡n 4 Sometimesdenotedδ orQ.
i i i i i
→
3
1962),(cid:15) =u2/τ =u3/(cid:96),afeatureignoredinMLT.This occurs,sothatbyasolutionofEq.3,321Dautomatically
K
is the frictional cost to the star for moving enthalpy by determines where the boundary of convection is. This
turbulence. In practice this term is small but not neg- contrasts with MLT for which boundaries are undefined
ligible. The 321D algorithm accounts for this additional and thus a thorny issue.
heatingterminthecomputationof1Dmodels(see 4.1.1
in Arnett, et al. 2015). § 3.2.3. Radiation diffusion effects
Inordertocomparetheasteroseismicmodelsfromhy-
3.2. Region SM: Braking (overshoot) drogenburningtosimulationsfromlaterburningstages,
3.2.1. MLT allowance must be made for the difference in strength
of radiative diffusion in the two cases. The duration of
The Schwarzschild discriminant,
neutrino-cooledstagesismuchshorterthantheradiative
=∆ = leakage time from the core, while core hydrogen burn-
ad
S ∇ ∇−∇ ing takes much longer than its corresponding radiative
changes sign at boundary S, and 0 there, so Eq. 1
Y leakage time. In the terminology of fluid mechanics, the
∇ ≈
impliesanimaginaryconvectivespeed. Todealwiththis
P´ecletnumberissignificantlydifferentinthesetwocases
singular behavior, different physics is traditionally in-
(Viallet, et al. 2015), so their flows may be significantly
troduced. A region SM is defined as the “overshoot”
differenttoo. Thisisespeciallyimportantforthinlayers,
region, in which the luminosity is presumably carried
for which the radiative diffusion time is shorter.
entirely by radiative diffusion ( = ) and a new
rad For oxygen burning and carbon burning, the flow fol-
algorithm is defined, replacing t∇he var∇iable u by a new
lows a nearly adiabatic trajectory, having an entropy
variable, the effective diffusion rate D ; see Moravveji,
ov deficit in the braking region (Fig. 4 in Arnett, et al.
et al. (2016) for a short and clear discussion. Inside SM,
(2015)). If radiative diffusion is not negligible, heat
0. The coordinate M designates the
rad ad will flow into this region of entropy deficit, reducing the
∇ −∇ ≤S ≤
layer at which this effective “convective diffusion” is no
strength of the buoyancy braking and thus widening the
longer able to destroy the composition gradient, so over
overshoot layer. This causes the actual gradient to devi-
region SM we still have 0.
Y ate from the adiabatic gradient ( ) and approach the
∇ ≈ ad
∇
radiative one ( ). The temperature gradient in the
rad
3.2.2. 321D ∇
overshooting region is expected to lie between the adia-
Because the Ledoux discriminant, batic and radiative ones rad < < ad (Zahn 1991;
∇ ∇ ∇
Zhang 2016).
=∆ =
ad Y Calibration of overshooting algorithms using observa-
L ∇ ∇−∇ −∇
changessignatS, <0, andtheregionSMissubjected tions of hydrogen-burning stars may be inadequate for
to buoyant decelerBation. Mixing continues over SM so later burning stages because of the differences in P´eclet
number (Viallet, et al. 2015). This could be a problem
that all of the region OSM is mixed, even though is
negative over SM. Thus SM is the overshoot regionL, in for helium burning (Mosser et al. 2014; Schindler, et al.
2015;Constantino,etal.2015;Ghasemi,etal.2016),and
which the flow turns back to complete its overturn. The
vector velocity u has different signs in upflow and down- will certainly get worse for the neutrino-cooled stages.
flow; it becomes horizontal at coordinate M, not zero;
3.3. Region MW: Composition Gradient
(see 3.8 in Arnett, et al. 2015). The coordinate M is a
shear§layer. Moravveji,etal.(2015,2016)founditdesirabletoadd
Here 0,sowehave <0,andg(cid:96)∆ <0. an extra diffusive mixing (their D ) beyond the well-
Y ad ext
∇ → ∇−∇ ∇
Eq. 1 is not possible. Near the boundaries the velocity mixed region OSM, because the agreement between ob-
gradient terms dominate over the Kolmogorov term in served and modeled frequencies improve – in χ2 sense –
Eq. 2, so by a factor 11. This is an important clue regarding the
structure of this region.
urdur/dr =d(u2r/2)/dr ∼gβT(cid:96)∆∇. (4) Thephysicalbasisforextramixingseemstobeatleast
two-fold. Because coordinate M is a shear layer, Kelvin-
For negative buoyancy (∆ <0), the buoyant decelera-
∇ Helmholtz instabilities will mix matter above and below
tion acts to decrease the radial component of the turbu-
coordinate M. In Eq. 3 we saw that the radial velocity
lentkineticenergy. Arnett,etal.(2015)showthatthisis
could be decelerated at a boundary. As the flow turns, a
essentiallythesameasdefiningtheboundarybythegra-
dient Richardson criterion (Ri=N2/(∂u/∂r)2 >1/4). horizontalvelocityuh mustdevelop(Arnett,etal.2015).
This variable does not appear in MLT (the “blob disap-
The Schwarzschild criterion is only a linear instability
pears back into the environment”). A finite u implies
condition, derived by assuming infinitesimal perturba- h
a shear instability may occur. A necessary condition for
tions. The Richardson criterion is a nonlinear condition,
instability (due to Rayleigh and to Fjørtoft, see 8.2 in
whichindicateswhethertheturbulentkineticenergycan
§
Drazin (2002)) is that
overcome the potential energy implied by stable strati-
fication. For a well-mixed region L → S. Use of L d2uh/dr2(uh u0)<0, (5)
in a stellar code can give fictitious boundaries due to −
small abundance gradients, which are blown away in a somewhere as we move in radius through the flow at the
3D fluid dynamic simulation; the Richardson criterion boundary. Here u is the horizontal velocity and u is
h 0
insures against these. that velocity at the radius at which d2u /dr2 = 0 (the
h
The boundary of a convective region has a negligible point of inflection). While a stably-stratified composi-
radial velocity of turbulence; Eq. 4 indicates where this tion gradient may tend to inhibit the instability, Eq. 5
4
illustrates a basic feature: the velocity field drives the
instability.
Ifthishorizontalflowisstable, acompositiongradient
can be maintained. Turbulent fluctuations may cause
the stability criterion to be violated, and thus erode the
layeruntilitisagainstable. Ifthereisentrainmentatthe
boundary,asthe3Dsimulationsshow(Meakin&Arnett
2007b), then the boundary moves away from the center
of the convection zone. The result is that the composi-
tion gradient will be left near the margin of instability.
Thisgivesthesigmoidshape, foundin3Dsimulationsof
oxygenburning(Arnett,etal.2015)andcarbonburning
(Cristini, et al. 2016).
At mass coordinate M the average turbulent velocity
in the radial direction is zero. Because the flow is tur-
bulent, it is zero only on average, so u = 0, but has a
finitermsvalue, u2 =0. Overthereg(cid:104)io(cid:105)nsSMandMW,
∆ < 0, so the(cid:104)Bru(cid:105)n(cid:54) t-V¨ais¨al¨a frequency is real. Thus,
∇ Fig. 2.— Fitting Eq.6 to the abundance profiles (X) of
this region can support waves, which the velocity fluc-
KIC10526294, and KIC7760680. The fit parameters are given in
tuations necessarily excite. The turbulent fluctuations Table1. Thecompositiongradientsinthe3Dsimulations(Arnett,
couplebesttog-modesatlowMachnumbers(Meakin& et al. 2015; Cristini, et al. 2015) quickly approach a self-similar
Arnett 2007a). The mixing induced here is much slower shapeofthesameform;asshowninCristini,etal.(2016).
thanintheconvectionregion5,andallowsacomposition
gradient to be sustained over MW. Because the mixing Radiative transfer isnegligible for carbon burning and
isslow, thecompositionstructuremaybeacombination oxygen burning, so the Pecl´et number is essentially infi-
of previous history and slow modification. nite (Viallet, et al. 2015), while radiative transfer does
affect hydrogen burning. In the braking region, radial
3.4. Region W: Radiative envelope deceleration(turningoftheflow)isduetonegativebuoy-
Totheextentthatnonlinearinteractionsofwavesgen- ancy. At the top of a convection zone, rising matter be-
erate entropy, the circulation theorem implies that slow comes cooler (has lower entropy) than its surroundings.
currentswillbeinduced,leadingtomore“extramixing” Radiative transfer tends to heat this cooler matter, re-
inradiativeregionssuchasthoseabovecoordinateW.In ducingthenegativebuoyancy(radialbraking)sothatthe
1D stellar evolution, this type of mixing is treated diffu- matter must continue further before it is turned. This
sively(D ),andasteroseismologycantightlyconstraint resultsinawiderbrakinglayer(smallerη);seediscussion
ext
it. in (Arnett, et al. 2015). Boundary layers are narrower
for neutrino-cooled stages, so that calibration of bound-
4. COMPARINGABUNDANCEPROFILES ary widths from photon-cooled stages are systematically
Figure 2 shows the inferred profile of hydrogen in the in error. Simulations in 3D of H burning are difficult
two Kepler SPB’s. This may be well-fitted by a logis- without artificial scaling of the heating rate (Meakin &
tic function strikingly similar to the composition profile Arnett 2007b). Because of negligible radiative diffusion
shapes resulting from 3D simulations. We choose to fit in carbon and oxygen burning, those corresponding val-
the abundance profiles X (in mass fraction) to ues for η in Table1 should tend to be higher than would
be expected in hydrogen burning, as they do.
φ θ
X(z)=θ+ − , (6) Simulationsofoxygenburninggivebehaviorsimilarto
1+exp[ η(z 1)] carbon burning, but are more complicated to interpret
− −
becauseofasignificantinitialreadjustmentofconvective
where z = r/r is the normalized radius and r is
mid mid shellsize,andanepisodeofingestionof20Ne;seeArnett,
the radius at the mid-point of the abundance gradient.
et al. (2015) and references therein to earlier works. De-
The steepness of the composition profile is encoded in
spite this, the large value of η reasonably represents the
η. At the “‘core”, η(z 1) 1, and X(z) θ. At
r =r ,X =(X− +−X (cid:29))/2,soz =1and→X = time- and angle-averaged O-profiles.
mid mid core surf mid In Table 1 we compare upper boundaries because core
0.5(θ+φ). Atthestellar“surface”(z 1)andforlarge
η, X = φ. Convective boundaries(cid:29)in stars are often H burning has only an upper boundary. The large val-
surf ues of η 1 imply that the composition boundaries
narrow with respect to radius. The mathematical form are narrow(cid:29)relative to the radius, in all cases. The η
inEq.6stretchestheabundancegradientoverthewhole
for KIC10526294 is roughly twice that of KIC7760680.
z axis. Thus, it is straightforward to fit the sigmoid
The former star is very young (X = 0.63) with a
shapetoanyabundanceprofilefromstellarmodelsatany core
steeper composition jump; the latter is more evolved
evolutionary phase from 1D or 2D/3D (after temporal
(X = 0.50) with broader µ-gradient zone. Further
andangularaveraging). ThenormalizationusedinEq.6 core
investigation of precisely which physical processes deter-
tends to separate shape from absolute scale, so that it
mine the parameter η is underway (see 3.3).
may be used for different compositions (burning stages). §
In Fig. 2, the maximum and minimum mass fractions
5Mostoftheturbulentenergyisatlargewavelengths(g-modes), of fuel are simply a function of the chosen consumption
so the induced waves are less able to cause mixing than shorter by burning, fixing φ and θ. The position of the bound-
wavelengthsmight. ary is normalized relative to the center of the gradient
5
velocitiesinthebrakingregions,whicharerelatedtothe
TABLE 1
Comparison of composition gradients (θ,φ and η in Eq.6) development of singularities in boundary layers (Landau
toHprofileoftwoKeplerSPBs,andtoCandOprofiles &Lifshitz(1959) 40,Gough(1977)). The321Dcanpro-
from 3D simulations (resolutions ∼10243). videacontinuous§descriptionoftheconvectiveboundary
– from fully convective to radiative – avoiding the awk-
ward patching characteristic of MLT.
Source Burning θ φ η
The 321D procedure promises a dynamical treatment
Asteroseismology:
of the overshooting and wave generation in stellar mod-
KIC10526294 coreH 0.63 0.71 112
KIC7760680 coreH 0.50 0.71 57 els. Possibleeffectsonconvectiveflowfromradiativedif-
fusion should be further explored for regions with mod-
3Dsimulations: erate P´eclet numbers (Viallet, et al. 2013), between ex-
upperboundary shellC 0.727 0.828 184 isting deep interior and atmospheric simulations. Better
upperboundary shellO 0.42 0.77 103 boundaries can provide a possible solution to the behav-
ior of convective helium burning cores, which the MLT
fails to represent well.
The sigmoid fits to the hydrogen profiles of the two
r (which avoids the important issue of entrainment).
mid
Kepler targets come from 1D (MESA) including ad-hoc
What is important here is the narrowness of the com-
coreovershootingandextradiffusivemixingtomatchthe
position gradient (the large value of η) and the shape
observedg-modefrequencies. Incontrastthesimulations
of the curve joining the high and low fuel abundances,
ofCandOburningshellshavenofreeparameterstoad-
both of which are predicted by the 3D simulations, and
just. Despite dealing with such different burning stages,
independently inferred from asteroseismology, giving an
the similarity shown in Table 1 and Fig. 2 is striking.
encouraging agreement.
5. SUMMARY Special thanks are due to Simon Campbell and to An-
Asteroseismically-inferred composition gradients are drea Cristini for access to their simulation data prior
strikinglysimilartothosefoundin3Dsimulations; MLT to publication, and to Raphael Hirschi, Casey Meakin,
cannot predict boundary structure. We compare, zone Cyril Georgy, Maxime Viallet, John Lattanzio and Miro
by zone, the predictions of MLT and 321D, from the Moc´ak for helpful discussions. We thank an anonymous
convective core to the radiative mantle. Advantages of referee who helped to improve the manuscript. This
321D are its time-dependence, non-locality, incorpora- work was supported in part by the Theoretical Pro-
tion of the Kolmogorov cascade and turbulent heating, gram in Steward Observatory, and by the People Pro-
and resolution of convective boundaries. The properties gramme (Marie Curie Actions) of the European Union’s
of the fully convective regions are similar in 321D and Seventh Framework Programme FP7/2007-2013/ under
MLT. Use of the 321D can avoid imaginary convective REA grant agreement N◦623303 (ASAMBA).
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