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Turbulent Cells in Stars: Fluctuations in Kinetic Energy and

Luminosity

W. David Arnett1,2

wdarnett@gmail.com

Casey Meakin1,3

casey.meakin@gmail.com

ABSTRACT

Three-dimensional (3D) hydrodynamic simulations of shell oxygen burning(Meakin & Arnett 2007b) exhibit bursty, recurrent fluctuations in turbulent kineticenergy. These are shown to be due to a general instability of the convective cell, re-quiring only a localized source of heating or cooling. Such fluctuations are shown to besuppressed in simulations of stellar evolution which use mixing-length theory (MLT).

Quantitatively similar behavior occurs in the model of a convective roll (cell) ofLorenz (1963), which is known to have a strange attractor that gives rise to chaoticfluctuations in time of velocity and, as we show, luminosity. Study of simulationssuggests that the behavior of a Lorenz convective roll may resemble that of a cell inconvective flow. We examine some implications of this simplest approximation, andsuggest paths for improvement.

Using the Lorenz model as representative of a convective cell, a multiple-cell modelof a convective layer gives total luminosity fluctuations which are suggestive of irregularvariables (red giants and supergiants (Schwarzschild 1975)), and of the long secondaryperiod feature in semiregular AGB variables (Stothers 2010; Wood, Olivier & Kawaler2004). This “τ -mechanism” is a new source for stellar variability, which is inher-ently non-linear (unseen in linear stability analysis), and one closely related to in-termittency in turbulence. It was already implicit in the 3D global simulations ofWoodward, Porter, & Jacobs (2003). This fluctuating behavior is seen in extended 2Dsimulations of CNeOSi burning shells (Arnett & Meakin 2011b), and may cause in-stability which leads to eruptions in progenitors of core collapse supernovae prior tocollapse.

Subject headings: stars: evolution hydrodynamics - convection -turbulence -irregular variables-Betelgeuse

1Steward Observatory, University of Arizona, 933N. Cherry Avenue, Tucson AZ 85721

2ICRAnet, Rome, Pescara, Nice

3Theoretical Division, Los Alamos National Lobo-ratory, Los Alamos, NM 875, USA

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1. Introduction

Three-dimensional fluid dynamic simula-tions of turbulent convection in an oxygen-burning shell of a presupernova star showbursty fluctuations which are not seen in one-dimensional stellar evolutionary calculations(which use various versions of mixing-lengththeory, MLT, Bohm-Vitense (1958)). Thispaper explores the underlying physics of thisnew phenomena.

Since the formulation of MLT (Bohm-Vitense1958), there have been a number of significantdevelopments in the theoretical understand-ing of turbulent convective flow.

First, Kolmogorov (1962) and Obukhov(1962) developed the modern version of theturbulent cascade, and published in journalseasily accessible in the west; the original the-ory (Kolmogorov 1941) was not used in MLTalthough it pre-dated it. This explicit expres-sion for dissipation of turbulent velocities1,

ǫturb = u3rms/ℓd, (1)

where urms is the root-mean-square of theturbulent velocity and ℓd is the dissipationlength. It is found both experimentally andnumerically that ℓd ≈ ℓCZ , where ℓCZ isthe depth of the convective zone. Simula-tions for low-Mach number flow show thatthe average of this dissipation over the con-vective zone closely compensates for the cor-responding average of the buoyant power(Arnett, Meakin, & Young 2009). This ad-ditional constraint allows an alternative topresent practice: fixing the free parameter(e.g., the mixing length factor α) directly byterrestrial experiments and numerical simu-lations which deal with the process of turbu-lence itself (Arnett, Meakin, & Young 2010),instead of calibrating it from complex astro-

1This is essentially the “four-fifths law of Kolmogorov”,Frisch (1995), p. 76.

nomical systems (stellar atmospheres) as isnow done.

Second, there has been a considerable de-velopment in understanding the nature ofchaotic behavior in nonlinear systems; seeCvitanovic (1989) for a review and reprintsof original papers, and Frisch (1995); Gleick(1987); Thompson & Steward (1986). Lorenz(1963) presented a simplified solution tothe Rayleigh problem of thermal convection(Chandrasekhar 1961) which captured theseed of chaos in the Lorenz attractor, andcontains a representation of the fluctuatingaspect of turbulence not present in MLT.This advance was allowed by the steady in-crease in computer power and accessibility,which lead to the exploration of solutions forsimple systems of nonlinear differential equa-tions (see Cvitanovic (1989) and referencestherein). It became clear that the Landaupicture (Landau 1944) of the approach toturbulence was incorrect both theoretically(Ruelle & Takens 1971) and experimentally(Libchaber & Mauer 1982). A striking fea-ture of these advances has been the use ofsimple mathematical models, which capturethe essence of chaos in a model with much re-duced dimensionality compared to the physi-cal system of interest.

Third, it has become possible to simulateturbulence on computers. This realizes thevision of John von Neumann (von Neumann1948), in which numerical solutions of theNavier-Stokes equations by computer are usedto inform mathematical analysis of turbu-lence. In this paper we will follow this ideaof von Neumann, in the style which provedsuccessful for chaos studies: building simplemathematical models of a more complex phys-ical system (in this case, the numerical simula-tions of turbulent convection). This approachshould lead to algorithms suitable for imple-mentation into stellar evolution codes, which,unlike MLT, are (1) based upon solutions to

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fluid dynamics equations, (2) non-local, (3)time-dependent, and (4) falsifiable by terres-trial experiment and future improved simula-tions.

Our particular example is a set of simula-tions of oxygen burning in a shell of a starof 23M⊙ (Meakin & Arnett 2007b). This isof astronomical interest in its own right asa model for a supernova progenitor, but alsohappens to represent a relatively simple andcomputationally efficient case, and has gen-eral implications for the convection process inall stars.

Three-dimensional hydrodynamic simula-tions of shell oxygen burning exhibit bursty,recurrent fluctuations in turbulent kinetic en-ergy (Meakin & Arnett (2007b) and below).The reason for this behavior has not been ex-plained theoretically. These simulations showa damping, and eventual cessation, of tur-bulent motion if we artificially turn off thenuclear burning (Arnett, Meakin, & Young2009). Further investigation (Meakin & Arnett2009) shows that nearly identical pulsationsare obtained with a volumetric energy gener-ation rate which is constant in time, so thatthe cause of the pulsation is independent of

any temperature or composition dependence

in the oxygen burning rate. Localized heatingis necessary to drive the convection; even withthis time-independent rate of heating, pulsesin the turbulent kinetic energy still occur.

Such behavior is fundamentally differ-ent from traditional nuclear-energized pulsa-tions dealt with in the literature (e.g., theε-mechanism, Ledoux (1941); Unno, et al.(1989)), and is a consequence of time-dependentturbulent convection (it might be called a ”τ -mechanism”, with τ standing for turbulence).It appears to be relevant to all stellar con-vection. Woodward, Porter, & Jacobs (2003)found, in a very different context, that non-linear interaction of the largest modes excited

pulsations of a red-giant envelope2, which isanother example of the τ -mechanism.

In Section 2 we examine the the physicscontext of the turbulence, including impli-cations of subgrid and turbulent dissipationfor the implicit large eddy simulations (ILES)upon which our analysis is based, and the ef-fect of the convective Mach number on thenature of the flow. In Section 3 we reviewthe 3D numerical results of shell oxygen burn-ing which are relevant to the theory. In Sec-tion 4 we present the results of the classicalLorenz model (Lorenz 1963) for conditionssimilar to those in Section 3. In Section 5we consider implications of turbulent inter-mittency on stellar variability, and provide amodel light curve from this effect alone. Sec-tion 6 summarizes the results. The appendixgives a short derivation of the Lorenz model.

2. Physics Context.

In this section we summarize conceptswhich are needed for the interpretation oflater results.

2.1. Subgrid Damping and Kolmogorov

Approximation of partial differential equa-tions by discrete methods inevitably leads toa loss of information at scales smaller than thegrid size. A single element in space is approxi-mated as a homogeneous entity3; this is equiv-alent to complete mixing at this scale, at eachtime step, of mass, momentum, and energy.The loss of information that occurs with thismixing corresponds to an increase in entropy(Shannon 1948), the mixing of momentum is

2A κ-mechanism, which depends upon variations inopacity, is not required to drive such pulsations.

3Actually most modern simulations (ours included) usehigher order methods which make some further as-sumptions regarding the behavior of variables inside azone. This complicates but does not change the ar-gument; it is still true that information is lost at thezone level.

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equivalent to the action of viscosity, and themixing of internal energy corresponds to thetransport of heat (Landau & Lifshitz (1959),§15 and §49).

In 3D flow, turbulent energy will cascadefrom large scales to small, at a rate set by thelargest scales (Kolmogorov 1941). At suffi-ciently small scales, microscopic processes ho-mogenize the flow and dissipate the kineticenergy. Thus, there is a deep connectionbetween the turbulent cascade and sub-gridscale mixing.

Sytine, et al. (2000) have demonstratedthat the piece-wise parabolic method (PPM,which we use), based on the Euler equation(which has no explicit viscosity), converges tothe same limit as methods based on Navier-Stokes equation (which do have explicit vis-cosity), as the grid is refined to smaller zonesand smaller effective viscosity (the relevantlimit for astrophysics). The subgrid scale dis-sipation for monotonicity preserving hydro-dynamic algorithms (Boris 2007; Woodward2007), which is implicit in these methods, au-tomatically gives a reasonable treatment ofthe turbulent cascade down to the grid scale.We use this implicit sub-grid dissipation inour large eddy simulation (ILES); this is themost computationally efficient way to dealwith turbulent systems with a large range ofscales. The largest scales, which set the rateof cascade and contain most of the energy areexplicitly calculated, while the sub-grid scalesare dissipated in a way consistent with theKolmogorov cascade.

Woodward et al. (2006) have presented arefinement of the PPM algorithm which hasimproved behavior at the smaller resolvablescales. We have not yet implemented thismodification. The theoretical approach usedhere involves integrated properties of convec-tive cells; we find by direct resolution stud-ies that these properties are well estimatedeven with surprisingly modest resolution, be-

cause they are determined primarily by thelargest scales in the convective region. It ap-pears that the ILES simulations are adequatefor the present analysis.

Arnett, Meakin, & Young (2009) have shownthat the numerical damping at sub-grid scalesin our ILES simulations is quantitatively con-sistent with the introduction analytically ofthe Kolmogorov cascade into the theoreticaldiscussion. The turbulent velocity field wasfound to be dominated by two components:

1. a non-isotropic flow of the largest scalemodes in the convection zone, whichis coupled to the fluctuations. Thishas aspects of a “coherent structure”(Holmes, Lumley, & Berkooz 1996). Thelargest scales are unstable toward break-up, but are least affected by dissipation,and in this sense the most laminar.

2. a more isotropic, homogeneous turbu-lent flow which carries the kinetic en-ergy via the turbulent cascade to scalessmall enough for dissipation to occur(Kolmogorov 1962). Because of thevast size difference, the small scales areweakly coupled to the largest scales,which determine the rate at which en-ergy flows through the cascade.

If we approximate the non-isotropic compo-nent of the flow (the largest scale of con-vection) with that described by the Lorenzmodel, this interpretation captures the oxygen-burning fluctuations.

2.2. Types of Flow

There are two limiting cases for convectiveflow, depending upon the convective Machnumber Mconv (the ratio of the fluid speedto the local sound speed); these are usuallytermed the “incompressible” (Mconv ≪ 1)and “compressible” (Mconv ∼ 1) regimes.

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Stars are stratified in density, so that the no-tion of “incompressiblity” is misleading. Wewill use the term “low-Mach number flow” inplace of “incompressible” when we mean flowsin which acoustic radiation is small, but maybe compressed due to stratification.

For turbulent motion, the pressure pertur-bation P ′ is related to the convective Machnumber by P ′/P ∼ ρu2rms/P ∼ M2

conv.Sound waves outstrip fluid motion, so thatpressure differences quickly become small, ex-cept possibly for a static background strat-ification. Most of the historical researchon convection (e.g., the Benard problem,Chandrasekhar (1961); Landau & Lifshitz(1959)) is done in this limit.

Using the Reynolds decomposition, ϕ =ϕ0 + ϕ′, with horizontal averaging 〈ϕ〉 = ϕ0

and 〈ϕ′〉 = 0, mass conservation for a steadystate flow can be written

〈ρu〉 = 〈ρ0u0〉+ 〈ρ′u′〉 = 0. (2)

The Navier-Stokes equation, using massconservation, is

∂tρu+∇·ρuu = −∇P + ρg + ν∇2u, (3)

where ρuu is the Reynolds stress tensor. Massconservation implies a convenient identity in-volving total co-moving derivatives,

Dtρu = ρDtu. (4)

Taking the dot product of the velocity vec-tor u with the Navier-Stokes equation, givesa kinetic energy equation,

Dtρu · u/2 = −u · ∇P+ρu · g+νu·∇2u. (5)

If on average the system is in a steady state,the time derivative must integrate to zero overthe convective region, and the mass conserva-tion law implies that the total buoyancy powerterm is zero, 〈ρu · g〉 = 0 (assuming constantg on horizontal averaging), and therefore does

not contribute to the production of kinetic en-ergy anywhere in the flow. The only otherterm, which remains to balance the viscousdissipation of the kinetic energy, is the pres-sure term

u · ∇P = u0 · ∇P0 + 〈u′ · ∇P ′〉. (6)

which may be rewritten as

u · ∇P = u0 · ∇P0 + 〈∇ · (P ′u′)〉 − 〈P ′∇ · u′〉.(7)

The divergence term vanishes upon integra-tion over the volume. Using hydrostatic equi-librium for the background state, ∇P0 = ρ0g,and mass conservation, ρ0u0 = −〈ρ′u′〉,

〈−u · ∇P 〉 = 〈ρ′u′ · g〉+ 〈P ′∇ · u′〉 (8)

When the Mach number is small, the secondterm the right hand expression is nearly zerobecause ∇ · u′ ≈ 0 and the turbulent pres-sure is negligible. In this limit the kinetic en-ergy production is best understood as due tothe remaining buoyancy power term 〈ρ′u′ · g〉,which is directly related to the enthalpy flux(Arnett, Meakin, & Young 2009).

When the Mach number is no longer small,the second term on the RHS of Eq. 8 in-creases in importance: both the divergence ofthe fluctuating velocity field and the pressureperturbation begin to play a role. The ve-locity field changes character; it is no longerdominated by rotational flow, but developsan irrotational component (∇ · u′ 6= 0). Theflow becomes diverging (consider the extremelimit of a point explosion which is pure di-vergence). Also, the ram “pressure” (a tensorρuu) is not negligible and must be included inthe momentum equation (“hydrostatic equi-librium”). Sound wave generation increasesrapidly as Mconv → 1 (Landau & Lifshitz(1959), §75). The compressible limit isMconv ≃ 1. Shock formation is the moststartling change in the flow character.

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Which of the Mconv limits is relevant forastrophysics? Both are. Almost all the mat-ter in stellar convection zones, during almostall evolution, is in the limit of low-Machnumber flow, as are our turbulence simula-tions. The exceptions are important: (1)explosions, such as supernovae and novae,(2) vigorous thermonuclear flashes, (3) vig-orous pulsations, especially radial ones, and(4) the sub-photospheric layers of stellar sur-face convection zones, which are strongly non-adiabatic, to name a few.

3. The Oxygen Shell Simulation

Figure 1 illustrates the behavior of two im-portant integral quantities, the total turbu-lent kinetic energy and the total buoyancypower, in the oxygen-burning shell simula-tions for a 23 M⊙ star (Meakin & Arnett2007a). The flow has a low-Mach number(M2

conv ≤ 0.001), although the numericalsimulations use the equations for fully com-pressible fluid flow, and would have correctlytreated high-Mach number flows. Also shownare the same quantities for the Lorenz model,which is discussed in Section 4.

At any instant in time, the total convec-tive kinetic energy is 1

2MCZv2rms, where MCZ

is the mass of the convective zone and vrms isthe rms velocity4, while the kinetic energy inthe isotropic part of the turbulent flow fieldis Eturb =

12MCZu

2t , where ut is the turbulent

velocity, which we define as the isotropic partof the turbulent flow5. The reader is warned

4More precisely, v is the solenoidal velocity in the con-vective region, with overall translational velocity re-moved, so that vrms is its root-mean-square value.

5This follows Arnett, Meakin, & Young (2009), §2.4, inwhich the transverse velocities are used to estimate theisotropic component of the vertical velocity. Thus, weuse u2

t = 3

2(u2

x + u2y) = 3

4v2rms, where ux and uy are

the velocity components perpendicular to the direc-tion of the gravity vector, i.e., the tangential veloci-ties. The vertical velocity also contains a significantcontribution ( 1

4v2rms) from the non-isotropic flow of the

Fig. 1.— Comparison of Convective KineticEnergy Fluctuations in the 3D simulationsand in the Lorenz model, starting from zerovelocity. The Lorenz model is labeled withits Prandtl number σ and its Rayleigh num-ber r. Turbulent velocity squared (solid) andbuoyancy power to the 2/3 power (dotted)are plotted versus time. The axes where cho-sen so that the same number of peaks wouldbe shown, and the average kinetic energy bethe same. The curves are surprisingly simi-lar, even though (1) the Lorenz model usedonly a single mode, (2) σ and r were not ad-justed to give a fit, (3) the Lorenz model im-poses thermal balance while thermal imbal-ance adds to the background slope in the sim-ulation (see Fig. 2), and (4) the simulationshave ∼ 108 more degrees of freedom than theLorenz model.

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that division of the flow into “turbulent” and“large scale” flow is useful but an oversimpli-fication, so that the exact relative values ofthese two kinetic energies depend upon thealgorithm used in their definition, but is oforder unity (Arnett, Meakin, & Young 2009;Meakin & Arnett 2011). Consequently theprecise distinction does not change the quali-tative picture.

The buoyancy power is the rate at whichkinetic energy per unit mass is increased bybuoyant acceleration. If it is integrated overthe space containing the convection zone, wehave qint =

∫

CZ qdr, where q = −gu′ρ′/ρ0 isthe buoyant acceleration times the turbulentvelocity; qint has units of velocity cubed.

In Figure 1 the simulations show a phaselag of about 20 seconds between the peaks

in buoyancy term (q2/3int ) and turbulent kinetic

energy. This is about half the time it takesthe flow to transit a distance ℓCZ , the depthof the convection zone. It also corresponds toan e-folding time for turbulent kinetic energydecay due to Kolmogorov damping, whereεK = u3rms/ℓd = (u2rms/2)(2urms/ℓd). Powerspectra for both variables peak at 89 seconds;an average transit time is 51 s.

Figure 1 shows multimodal behavior inthe 3D simulations; preliminary results froma quantitative analysis (Meakin & Arnett2011) using principal orthogonal decomposi-tion (POD) indicates that a single dominantmode has about 43% of the kinetic energy, thefirst five modes have 75%, and 90% is reachedwith the eleventh mode. There is a strongdominant mode but also significant energy inseveral other modes; the modes interact in anonlinear and dynamic way.

The buoyancy power is a large scale fea-ture and is strongly anisotropic (plumes move

largest eddies. A more careful discussion of the veloc-ities in terms of principal orthogonal decomposition isin preparation.

vertically). The dissipation implied by theturbulence occurs at the Kolmogorov scale(which is tiny); this dissipation is wide-spreadin space (Arnett, Meakin, & Young 2009), in-cluding the entire turbulent region on average,and bounded by the stably stratified layers.Because buoyancy and dissipation occur atvastly different length scales, they are weaklycoupled.

In low Mach number flow, on average overtime, the buoyant driving must balance theturbulent damping for a quasi-steady stateto exist (see Arnett, Meakin, & Young (2009)for a detailed discussion, especially theirEq. 33). The time averaged viscous dissi-pation is

ǫturb = qint/ℓCZ ≈ v3rms/ℓd ≈ 1.54u3t /ℓd, (9)

where ℓd is the dissipation length, ℓCZ isthe depth of the convection zone, and theover-lines indicate an average over time (twoturnover times or four transit times, about200 seconds, in the analysis).

From Equation 9 it is clear that the turbu-lent dissipation is of third order in the tur-bulent velocity, while the buoyancy power,q = −gu′ρ′/ρ0, is second order (because ρ′

is first order). This means that in the turbu-lent regime there is a unique turbulent veloc-ity which satisfies the condition that buoyancypower balance turbulent damping. This hasthe nature of an eigenvalue problem. The as-sumption of a turbulent cascade implies finiteturbulent velocities, with macroscopic struc-ture. Equation 9 applies as an integral overa region much larger than the Kolmogorovscale; at the Kolmogorov scale microscopicdissipation occurs by the Navier-Stokes term(Landau & Lifshitz (1959), §15). If we dotthis Navier-Stokes term with the velocity togenerate a rate of kinetic energy destructionby viscosity (see Equation 5), we have

(u·∂u/∂t)visc = νu·∇2u (10)

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which does scale as the velocity to second or-der. When integrated over the turbulent cas-cade, Equation 10 becomes Equation 9. Theextra factor of velocity comes from the rate atwhich turbulent kinetic energy is delivered tothe Kolmogorov scale. Turbulence is an inher-ently nonlinear process, which sets up its ownscales and structures. Care must be taken inanalyzing it with techniques involving expan-sion in series 6.

The dissipation is seen to be driven by theisotropic part of the flow ut, which accountsfor about three-fourths of the kinetic energyaccording to estimates by Arnett, Meakin, & Young(2009) (which are roughly similar to thosefrom POD analysis by Meakin & Arnett(2011)). Note that v2rms = 4

3u2t which gives

the vertical offset between the solid and dot-ted curves in Figure 1. Consider the averageconditions from 200 to 800 seconds in thesimulations. The average level of buoyancy

(actually q2/3int is plotted) is higher than that

of kinetic energy in isotropic turbulence. If weequate (1) the power generated by buoyancy,to (2) the dissipation due to turbulence, bothaveraged over a few transit times, we find

ℓd/ℓCZ = u3rms/qint ≈ 0.85. (11)

Thus, in the Meakin & Arnett (2007b) simu-lations, the dissipation length is found to be

essentially the depth of the convective zone,consistent with Kolmogorov theory.

These results are far more general than theparticular stellar situation we have discussed.While the neutrino cooling may seem exotic tosome stellar evolutionists, in fact it behavessomewhat like the more familiar cooling byradiative diffusion, and has no strange effecton the turbulence. We note that the originalsimulations (Meakin & Arnett 2007b), which

6See Arnett & Meakin (2011b) for an example of thedanger of using linear stability analysis for turbulencein stars.

included core hydrogen burning cooled by ra-diative diffusion as well as oxygen shell burn-ing cooled by neutrinos, explicitly showed thissimilarity.

In the long term, the thermal state of theconvection zone is supposed to evolve towarda global thermal balance between total heat-ing by nuclear burning and total cooling byneutrino emission (Arnett 1972, 1996). Thisis illustrated in Figure 2, which shows timeaverage luminosities of heating and of cooling,as a function of the entropy of the convectionzone. At lower entropy, as in the 3D simu-lation, the heating is dominant, causing theentropy to increase. This is accomplished pri-marily by expansion, with a small increase intemperature. The decrease in density causesa bigger change in nuclear burning than inneutrino cooling, so that a thermal balancewould be attained (shown as a vertical line)when the heating and cooling curves cross.

The gradual rise in turbulent kinetic en-ergy in the simulation, shown in Figure 1, oc-curs in a shell which is below the entropy ofbalanced heating and cooling, so that heat-ing dominates. The pulses are much fasterthan this secular evolution, which changes ona time scale of t ≥ 103 seconds.

4. The Lorenz Solution

The Lorenz model is a convective roll, orcell, whose dynamics are described by threeamplitude equations. This is a simple ex-ample of an elegant method of reduction ofturbulent flow to a low-order set of dynam-ical equations using amplitudes of a properorthogonal decomposition (POD) of numeri-cal simulations or extensive experimental datasets (Holmes, Lumley, & Berkooz 1996).

The Lorenz model is a better mathemati-cal representation of the dynamics of a con-vective cell than MLT in that the accelera-tion and deceleration over a convective cy-

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cle are integrated to determine the motion,rather than prescribed. Because it uses a sin-gle mode, the Lorenz model does not have sen-sitivity to local variation; it is a global model,unlike MLT, which is local. Lorenz devisedthe model as a point of principle test of me-teorological convection, which like the stel-lar problem, is damped by a turbulent cas-cade. To make use of this extensive literature(Cvitanovic 1989; Frisch 1995; Gleick 1987;Thompson & Steward 1986), we use the orig-inal version of Lorenz (1963), with the samePrandtl and Rayleigh numbers, to explore theimplications on dynamics of the strange at-tractor. The original Lorenz formulation isa 2D, low Mach number, and single modemodel (equivalent to Figure 3). A transi-tion to 3D may be made using the solutionsof Chandrasekhar (1961), but is not neces-sary for present purposes. Real convection(Libchaber & Mauer 1982) is expected to besingle mode only near the onset of convectiveinstability.

The Lorenz equations (see Appendix A)are:

dX/dτ = −σX + σY (12)

dY/dτ = −XZ + rX − Y (13)

dZ/dτ = XY − bZ, (14)

In the classical Lorenz model, X is propor-tional to the speed of convective motion, Y isproportional to the temperature difference be-tween ascending and descending flow, and Zis proportional to the distortion of the verticaltemperature profile from adiabatic (Lorenz1963). Here τ is a time in thermal diffusionunits, σ is the effective Prandtl number, r isthe ratio of the Rayleigh number to its criticalvalue for onset of convection, and a is the ra-tio of the depth to the width of the convectivecell, so that b = 4/(1+a2); see Lorenz (1963).The Prandtl number is the ratio of coefficientsof the viscous dissipation term to the thermal

Fig. 2.— Nuclear heating and neutrino cool-ing in an oxygen burning shell, as a functionof entropy. At lower entropy, as in the sim-ulations, nuclear heating dominates becauseof its density dependence. For the same tem-peratures but higher entropy, the inhibitionof neutrino-antineutrino emission is reduced.The shell entropy slowly increases until cool-ing balances heating, at which point thermalbalance is attained.

g

z

T0T0 − T3 T0 + T3

φ

12ℓ

T0 − T1

T0 − T2

T0 + T1

T0 + T2

u

Fig. 3.— The Lorenz Model of Convection:Convection in a Loop.

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mixing term. Lorenz7 took σ = 10, and chosethe most unstable mode so that a2 = 1/2, andb = 8/3. For this mode, and a Rayleigh num-ber of r = 470/19 = 24.74 times the criticalvalue, steady flow becomes unstable.

Figure 3 illustrates the structure and nota-tion in the Lorenz model. The Prandtl num-ber may be expressed as σ = τrad/τvisc, theratio of the radiative cooling time scale to theviscous time scale. The viscous damping timeis taken to be constant, τvisc = 1/Γ. TheRayleigh number, in units of its value at theonset of laminar convection, is

r = gβTT1/ℓΓKT0, (15)

where K = 1/τrad, βT = −(∂ ln ρ/∂ lnT )P ,and g is the gravitational acceleration. Thelapse rate of the adiabatic background TE isg/CP , so T1 = g(ℓ/2)/CP . In astrophysicalnotation,

∇−∇a = ∇a(T2/T1 − 1), (16)

where T2 is the temperature variation in thevertical direction (Fig. 3). The depth of theroll, in pressure scale heights, is

ℓ/HP =1

∇aln[(T0 + T1)/(T0 − T1)]. (17)

4.1. Energy fluxes

In the meteorological case the fluid motion(wind) is of primary interest, however the en-ergy fluxes8 provide a means to connect withstellar observations.

7A justification is simply that this allows the threeLorenz equations to capture chaotic behavior. See Ap-pendix B for a discussion of the physical implicationsof the Prandtl number for stars.

8In discussion of the Lorenz model, both heat flux andenthalpy flux are mentioned; which is correct? De-pending upon the physics perspective, both may be.In the simplest Lorenz model, the fluid in strictly in-compressible, so that the work term PdV is strictlyzero, and in this case only heat content can carry en-ergy, so heat flux is relevant. However, we may inter-

The vertical enthalpy flux due to radiativediffusion of the background is

FE = −ρνTCP dTE/dz (18)

= ρνTCPT1(2/ℓ), (19)

which is constant in the Lorenz model.The thermal conduction coefficient is νT =(4acT 3/ρκ)/(ρCP ) and K = νT (1/ℓ)

2. Herez = − ℓ

2 cosφ, the radius of the roll being ℓ/2.Because the background temperature is takento be linear in z, the divergence of this flux isidentically zero, and therefore gives no localheating or cooling. The additional verticalenthalpy flux, due to the temperature pertur-bation, is

Fz = −ρνTCP d(T − TE)/dz (20)

= ρνTCP (T2 − T1)(2/ℓ). (21)

If we define a potential temperature9 T4 =T1−T2, the vertical flux is separated into twoparts, so that the adiabatic background valueis denoted by T1 and the changes caused bymotion are contained in T4.

The flux in the horizontal direction is

Fy = −ρνTCPdT/dy (22)

= −ρνTCPT3(2/ℓ), (23)

which averages to zero by symmetry; here y =ℓ2 sinφ. Both Fz and Fy are proportional to apotential temperature which varies in time.

The net vertical enthalpy flux will not beconstant in height z, so that its divergence isnonzero. This implies local heating/cooling,

pret the equations in terms of a stratified system withlow-Mach number flow, having a stratification in den-sity as well as temperature. Then dV is not zero, PdV

work is done, and the relevant flux is the enthalpy flux.9Here potential temperature is used as in fluid dynam-ics and meteorology (Dutton 1986; Tritton 1988); thepotential temperature is measured relative to an adi-abatic background, and may be small even if the con-vective region is strongly stratified (as is often the casein stars).

10

which would have to be compensated for in asteady state, and is a consequence of consid-ering only a single mode. Smaller scale modeswould be needed to deal with the local heat-ing/cooling; see Canuto & Mazzitelli (1991);Canuto et al. (1996) for efforts to include afull spectrum of modes. To get the averageflux through the cell for a single mode, wetake the double projection, of u and of T onthe vertical direction,

Fe(φ) = ρuzCPT

→ ρCPuT3 sin2 φ. (24)

The enthalpy flux is zero at the top and bot-tom of the roll (φ = π and 0), and a maxi-mum at the midpoint (φ = π/2). The averagevalue of sin2 φ over the roll, φ = 0 to 2π, is0.5. This is to be compared with the αE ofArnett, Meakin, & Young (2009), which wasthe range 0.68 to 0.85 for the simulations thenavailable. Because the simulations are multi-mode, this slightly higher value for the corre-lation seems natural.

Integrating over the cell, 0 ≤ φ ≤ 2π, gives

Fe =1

2ρCPuT3, (25)

which is the vertically averaged enthalpy fluxof the cell. This is the “convective flux” instellar models. The up-flow enthalpy fluxequals the down-flow value, and both are pos-itive (hot up-flows and cold down-flows bothtransport energy upward).

The ratio of enthalpy flux to the total ra-diative flux (Fz + FE) is

Fe/Fr = (ℓ/4νT )(uT3/T2), (26)

which is a function of time.

The kinetic flux exactly cancels in this for-mulation, as in MLT, with the up-flow beingthe negative of the down-flow. This is not

true in the simulations. The kinetic flux doesapproach zero for convective regions which

are so thin that they are almost unstrati-fied. However, stratified convection zoneshave an asymmetry in up and down flows, giv-ing a modest net kinetic energy flux (down-ward for pure top driving, up for pure bot-tom driving, and both for more general cases,Meakin & Arnett (2009)).

We may express fluxes in units of the radia-tive flux of the background, FE , and in termsof the variables in the Lorenz equations. Theexcess vertical radiative flux is

Fz/FE = T4/T1

= Z/r, (27)

where T1 = gℓ/2CP , T0 = gHP /(CP − CV )and γ = CP /CV , and

r =γ − 1

2γ

( gQ

HPKΓ

)

. (28)

The horizontal radiative flux is

Fy/FE = T3/T1

= Y/r. (29)

The enthalpy flux of the cell, in the verticaldirection, is

Fe/FE = XY/2r. (30)

In general, without need for any additionalmechanism to cause variability, the net flux

of energy through a turbulent cell varies with

time.

4.2. Steady-state solutions

A steady state solution for the Lorenz equa-tions is

X = Y = Z = 0, (31)

and, for r ≥ 1, a second solution, for a stableconvective roll, appears:

X = Y = ±[b(r − 1)]1/2;Z = r − 1. (32)

11

The second solution is unstable if σ > 2 andr > rc where

rc = σ(σ + b+ 3)/(σ − b− 1), (33)

and this instability gives rise to fluctuationsin velocity (Lorenz 1963). The instability alsogives fluctuations in energy flow through thecell.

The steady state solutions might be ofpractical value for stellar evolutionary codes,to the extent that they provide an estimate ofthe average behavior of the convective cell.However, the fluctuations are large (in nosense are they ”small perturbations”), andmay cause nonlinear complications. For ex-ample, a thermonuclear runaway would besensitive to the largest value of the tem-perature fluctuations because that wouldaffect the burning rate (Arnett & Meakin2011b). The fluctuations drive entrainmentepisodes, and affect the mixing of composition(Meakin & Arnett 2007b). The fluctuationsmay be able to modify the driving of pulsa-tions in stars with vigorous convection zones.In red supergiants, coupling of turbulent fluc-tuations in the surface convection zones toboth radial and non-radial pulsations seemslikely (see below).

4.3. Non-steady Solutions

In order to compare the Lorenz model to3D simulations, it is desirable to have compa-rable starting conditions. Arbitrary choicescan give large initial transients before the at-tractor controls the behavior. The 3D simu-lations start with zero velocity but finite tem-perature deviations from an adiabatic gradi-ent. We take

X = 0, Y = [b(r − 1)]1

2 , Z = r − 1, (34)

as initial conditions, which sets the velocityto zero and uses steady state values for thetemperature fluctuations.

Figure 1 shows the behavior of the Lorenzmodel of convection (panel labeled LORENZ),for a similar number of pulses as shown forthe 3D simulation, and the same variables:buoyancy power XY/(b(r − 1)) to the two-thirds power, and kinetic energy per unit massX2/(b(r − 1)). The factors b(r − 1) are cho-sen so that the steady state values are nor-malized to unity, but are qualitatively cor-rect for comparison with the 3D simulations.The velocities in Fig. 1 are not precisely thesame, the 3D simulations giving turbulent ve-locity while the Lorenz speeds are more ap-propriate to “coherent structures”. They arerelated (see § 3.1), as the large scale ve-locities become unstable and turbulent, andtheir kinetic energies are the same order ofmagnitude (Arnett, Meakin, & Young 2009;Meakin & Arnett 2011).

The time is measured in the dimension-less units of Lorenz; a turnover time is twotransit times, and roughly the time betweenpeaks. We see that the peaks in buoyancypower slightly precede those in kinetic energy,as in the 3D simulations but less dramatically.This difference is related to the fact that theLorenz model has viscous dissipation actingdirectly on the large scale velocities, while the3D simulations have dissipation at the Kol-mogorov scale, which is separated from thelarge scale by the turbulent cascade, involv-ing many, many reductions in length scales(see Section B.3). Additional modes wouldfill in the “valleys” in the Lorenz model (see§5). Over the time shown, the average kineticenergy is 0.968 of the formal steady state solu-tion, which is shown in Figure 1 as the dashedhorizontal line in the Lorenz panel.

The degrees of freedom in the three-dimensional simulations, and in the Lorenzmodel, are dramatically different. The floating-point operation count differs by a large factor:∼ 108 (several times 108 zones times 7 effec-tive scalar variables for simulations, versus

12

three amplitudes for the Lorenz model). Withsuch an extreme compaction, it is striking thatthey give a similar picture for fluctuations inconvection.

4.4. Long Term Behavior

Unlike the full 3D simulations, it is triv-ial to extend the Lorenz model to later times.Figure 4 shows an extension in time by a fac-tor of 10. There is a relatively steady growthin amplitude up to time ∼7, at which pointchaotic behavior begins to appear. The fluc-

tuations in kinetic energy (and q2

3

int) are large,sometimes exceeding the steady state value(∼1) by a factor of 4. The convective lumi-nosity from a single Lorenz model shows thesame qualitative behavior.

This drastic behavior raises two interestingpossibilities:

1. The numerical simulations will followthe solutions of the Lorenz equation,and exhibit vigorous and chaotic fluc-tuations at later times, or,

2. The multimode behavior of the simula-tions will allow even stronger dissipa-tion, preventing extreme behavior.

Either way, the result is important for the evo-lution of supernova progenitors, especially re-garding the effects of fluctuations on mixing(yields) and outbursts. Simulations of mul-tiple burning shells (C, Ne, O and Si) in 2Dhave been continued further than the 3D sim-ulations for the oxygen burning shell. Theyappear to follow the first option, so that theprediction from 2D simulations is that corecollapse progenitors will have violent erup-tions prior to core collapse (Arnett & Meakin2011b). Full 3D simulations need to be ex-tended to later times at which the Lorenzmodel predicts chaotic behavior.

4.4.1. Duration

The oxygen shell might not last for the∼70 transit times shown in Figure 4; a lin-ear estimate suggests that it consumes its fuelin about 100 traversal times (the Damkohlernumber isDa ≤ 0.01, Arnett, Meakin, & Young(2009)), so the background evolution shouldnot be neglected for such time intervals.Oxygen burning is likely to be a more dy-namic event than previously supposed (e.g.,by Woosley, Heger, & Weaver (2002)). Aver-aging over multiple cells may moderate thenet fluctuation (see below).

For convection zones in other stars, thenumber of traversal times available may bemuch larger. The sun has a deep surfaceconvection zone (20 pressure scale heights); aplume would, if unimpeded, fall through theconvection zone in about 2 hours; it wouldtake about 7 centuries to process all the massof the convection zone through the surface lay-ers. These times bracket the effective mixingtime, so that of order 107 to 1013 mixing timeswould occur over the age of the present-daysun.

4.4.2. Chaotic Behavior

Figures 5 and 6 show the familiar long termbehavior of the Lorenz model (Gleick 1987).Figure 5 shows the trajectory in X and Yspace for the first 70 dimensionless time in-tervals. X is the dimensionless speed andY the dimensionless temperature variation inthe horizontal direction. Both X and Y switchsigns, but there is an overall correlation be-cause there is a net enthalpy flux, which isproportional to XY. The correlation is in thesense that hot regions rise and cool regionssink. The radiative flux associated with thehorizontal temperature variation is propor-tional to Y, and while on average over longtimes is zero, it achieves this in segments oftime in which first one then the other direc-

13

Fig. 4.— Later Behavior of the Lorenz Modelof Convection, for σ = 10, r = 28, andb = 8/3. The format is the same as Figure 1,but the time interval is longer by a factor of10. Chaotic behavior is beginning to appear,with fluctuations in kinetic energy and buoy-ant driving sometimes exceeding the steadystate value by a factor of 4.

Fig. 5.— Long Term Behavior of the classicLorenz Model for X (speed) and Y (horizontaltemperature fluctuation), for σ = 10, r = 28,and b = 8/3. The trajectory is shown for 70Lorenz time units. Both X and Y switch signs,but there is an overall correlation (hot zonesrise while cool zones sink). This correlationgives rise to positive values for buoyancy fluxand enthalpy flux.

tion (sign) are favored. This is the strangeattractor at work.

Figure 6 shows the same time interval, butplots Y and Z (the dimensionless temperaturein the vertical direction, parallel to the grav-ity vector). Notice that Z has positive val-ues about which it orbits. The stratificationbreaks the symmetry in vertical versus hor-izontal directions. The radiative flux in thevertical direction has a steady part and a fluc-tuating part. The latter is proportional to Z,and is positive (outward energy flow).

In MLT, the Schwarzschild discriminant(S ≡ ∇−∇a) is the temperature excess aboveadiabatic; this excess is assumed to be pro-portional to the enthalpy flux. This is incon-sistent with the Lorenz model, because thevelocity X can have both signs while Z doesnot. The error arises because in MLT thespeed of convection is taken to be intrinsicallypositive (to avoid this problem), and may betraced back to a lack of conservation of mass(”blobs dissolve” into the environment ratherthan flow back to complete a cycle).

4.4.3. MLT

What happens if we reduce the three equa-tions of Lorenz to two, forcing one variableto be at its steady state value? This is simi-lar to the MLT approach (assuming the mix-ing length parameter is chosen so that thekinetic energy scale is physically correct; seeArnett, Meakin, & Young (2010)). Enforcingthe steady state value for the vertical tem-perature excess, Z = r − 1, but allowingX and Y to vary, corresponds to a modelwith a single temperature variable (like MLT).Such integrations are shown in Fig. 7. Thereare no pulses in kinetic energy or buoyancy;the curves quickly approach a constant value.Convection proceeds by steady motion in aroll; a finite XY is required to give torque tomake the roll. This ”two equation model” nolonger has a strange attractor; the pulses have

14

Fig. 6.— Long Term Behavior of the clas-sic Lorenz Model, for σ = 10, r = 28, andb = 8/3. Same as Figure 5, but with Y andZ (vertical temperature fluctuation) shown.Z is a positive for these parameters. Y hasboth positive and negative steady state val-ues, about which it orbits.

Fig. 7.— Initial Behavior of the two EquationModel of Convection, for σ = 10, r = 28, andb = 8/3. The critical value for instability ofthe convective roll is rc = 24.74. The formatis the same as Figure 1. The turbulent fluc-tuations are eliminated, as they are in MLT.

been eliminated. This explains why stellarevolutionary calculations which use MLT donot show these fluctuations10.

5. Cells and Shells

5.1. Multiple Modes in Cascade

Figure 1 shows that the primary differ-ence in the Lorenz model and the 3D sim-ulations is that the Lorenz model has onlya single mode, while the simulations are ob-viously multimodal. This difference may besuperficial. The Lorenz model in this appli-cation (as in the original meteorological one)has additional modes implied by the turbu-lent cascade which mediates the damping (i.e.,they are implicitly in Γ; see Appendices §Aand §B). A simple Richardson cascade wasdiscussed in §B.3, in which f , the fractionalchange in length scale for each step in the cas-cade, is assumed to be a constant (Frisch 1995;Davidson 2004). This is not very plausible forthe largest scale modes because they are themost sensitive to boundary conditions (theymust fit into the convective region), but is sim-ple and instructive. The fractional time spentin the cascade for each mode may be shown tobe the fractional kinetic energy in that mode.Using §B.3, this gives f (2/3)(n−1)(1− f2/3) forn = 1, 2, · · ·, or roughly 0.37, 0.23, and 0.14for the first three. There is a dominant modeaccompanied by several weaker but significantones.

One way to proceed would be to introduceadditional modes into the Lorenz model (cho-sen with guidance by the 3D simulations),and to generate an expanded set of amplitudeequations which generalize the three of Lorenz(Eq. 12, 13, and 14). This would make the sys-tem multimodal, and allow for modal interac-

10 In analogy, the two-body (Kepler) problem in celes-tial mechanics is well behaved, while the unrestrictedthree-body problem is far more complex (Poincare1893).

15

Up-draft

Toroidal Flow

Fig. 8.— The Lorenz Model extended: con-vection in a sphere composed of a cell oftoroidal shape. With no rotation to break thesymmetry, the direction of the upflow vectoris not constrained, and will be chaotically cho-sen.

Up-draft

Down-draft

Up-draft

Fig. 9.— The Lorenz Model extended: Con-vection in a shell composed of cells. Noticethe alternation of the sign of rotation. Thismay be thought of as a cross sectional view ofinfinitely long cylindrical rolls, or of a set oftoroidal cells, with pairwise alternating vor-ticity (see Chandrasekhar (1961), §16). Eachcell may exhibit independent fluctuations intime and space.

tions. This approach is a simplified version ofan elegant proposal by Lumley and collabora-tors (Holmes, Lumley, & Berkooz 1996): em-pirical eigenfunctions are constructed by PODfrom simulations, introduced into the differ-ential equations to derive a set of nonlinearODE’s for the amplitudes of each mode (aGalerkin projection), and this set of ODE’s issolved to generate the evolution of the aver-age properties of the turbulence. This is beingexplored (Meakin & Arnett 2011) as a way tocapture more fully the dynamics and multi-modal behavior seen in the 3D simulations.

5.2. Multiple Cells

Suppose we envisage the convective re-gion to be populated by cells, each of whichis a separate Lorenz model representing thelargest mode, a convective roll, in that cell.A special case is the convective core: the ge-ometry is not that of a layer, but a sphere11,as is indicated in Figure 8. The flow has atoroidal structure. In this case the gravita-tional acceleration goes to zero at the origin(the center of mass of the star), but the veloc-ity need not be zero there. If there is no netrotation, then the direction of the up-flow isundetermined, and will be chosen chaoticallyby the turbulent flow.

For a convective shell, we imagine that thelayer is filled by cells, as shown schematicallyin Figure 9. For the oxygen burning shell,the inner and outer radii are about 4 × 108

and 8 × 108 cm, respectively. The area ofthe spherical shell, evaluated at the midpointin radius, is 4π(6 × 108)2, and the cell istaken to be roughly square, so that its areais ℓ2 = (4 × 108)2. The ratio of shell areato cell area is about 9π, so we assume thereare roughly 30 cells spread over the spheri-

11Woodward, Porter, & Jacobs (2003) found such a ”gi-ant dipole” behavior in their 3D simulations of almostfully convective spheres.

16

cal shell. In general, the number of cells ina shell will depend upon the geometry of theconvection zone.

If the cells are not synchronous12, but actindependently, the effect of the pulses willbe smoothed when summed over the wholeshell. However, the cells may interact con-structively; the solution to this more complexproblem remains open13. At issue are boththe interactions between cells in a single con-vective region, and interactions between mul-tiple convective regions associated with differ-ent burning shells.

The individual cells exhibit fluctuationsnot only in time, but also in space. Each cellrepresents a mode which is unstable, and de-stroys itself and reforms, usually somewhereelse. Because the medium is fluid, the pat-tern of cells is much more dynamic and lessregular than that of a crystalline solid, so theFig. 9 should be interpreted as representing asnapshot of a system which fluctuates in bothspace and time. Hinode/SOT observations ofthe solar surface (Berger, et al. 2010) revealsuch a highly dynamic, complex structure14.

5.3. Irregular Variables

Figure 10 explores an idea of MartinSchwarzschild (Schwarzschild 1975), who es-timated the number of convective cells in thesun and in red giants and supergiants. He ar-

12See Alligood, Sauer & Yorke (1996) for a discussion ofsynchronization of chaotic orbits.

13 Figure 23 in Meakin & Arnett (2007b) suggests thecomplexity of the cell interaction within a single con-vective region. The original simulations were on awedge, of 27o in theta and in phi. Simulations withlarger aspect ratio (larger angle wedges) do show amoderation of the total fluctuation, in qualitativeagreement with the discussion above. Figure 2 inMeakin & Arnett (2006) indicates the complexity ofinteraction between multiple burning regions even in2D.

14We thank Dr. A. Title for providing a copy of hisspectacular movies of data and simulations.

gued that only a modest number of cells (∼12)would exist at any given time in a red giant orsupergiant. To illustrate the point, we use theLorenz model to approximate the behavior ofa convective cell. We have estimated the con-vective flux for 12 cells at random phase, byadding 12 time sequences from an computa-tion like Figure 4 (but extended to t=800),starting at 12 randomly chosen times in thisinterval. This time sampling is intended to ap-proximate a spatial ensemble average. Theirflux, summed and normalized, is shown as asolid line, for a dimensionless time from 60 to70, which corresponds to roughly 20 pulses.The signal is noisy and looks “chaotic”.

These fluctuating cells make up a convec-tive region, and will couple to the normalmodes of the star to cause both radial andnon-radial pulsations. The amplitude of thesepulsations will depend upon the overlap inte-grals between the normal modes and the cellmotion, and the stellar damping. This sug-gests that the noisy behavior will be combinedwith the relatively cleaner periodicity of thenormal modes, giving a power spectrum witha base like that shown in Figure 11, but withsuperimposed spikes corresponding to the ex-cited normal modes. While turbulent convec-tion alone is sufficient to cause luminosity fluc-tuations, it occurs in regions of high opacityand partial ionization, which also drive pulsa-tion, so that composite behavior and multipleperiods may be expected.

Joel Stebbins (pioneer of photoelectric as-tronomy) monitored the brightness of Betel-geuse (α Orionis) from 1917 to 1931, andconcluded that ”there is no law or order inthe rapid changes of Betelgeuse” (Goldberg1984), which seems apt for Fig. 10 (the Lorenzstrange attractor) as well. More modernobservations (Kiss, Szabo, & Bedding 2006)show a strong broad-band noise componentin the photometric variability. The irregularfluctuations of the light curve are aperiodic,

17

Fig. 10.— Fluctuations of Luminosity in Con-vective Layer of 12 Cells of Random Phase,for σ = 10, r = 28, and b = 8/3. Thedimensionless flux (luminosity) is shown fora convective layer with 12 visible Lorenzcells. The luminosity variations are largeand seemingly chaotic, suggestive of irreg-ular variables and Betelgeuse in particular(Kiss, Szabo, & Bedding 2006).

1 10 100

0.1

1

10

1x102

1x103

1x104

Frequency

SpectralPower

Fig. 11.— Power spectrum of the luminos-ity fluctuations implied by turbulence alone.There is no sharp peak, but a broad distri-bution of power, as would be expected froma chaotic source. Resonant interaction withnormal modes of the star could add peaks tothe spectrum, which would provide an obser-vational constraint on interior structure, anal-ogous to astro-seismology of solar-like stars.

and resemble a series of outbursts. Direct 3Dsimulations of Betelgeuse (Chiavassa, et al.2010) show the same complex behavior (andhave the advantage that they predict the de-tailed spectral behavior as well). This shouldbe no surprise; the 3D equations have embed-ded in them a strange attractor.

Increasing the number of cells reduces thelevel of fluctuation about the mean. Averag-ing over the 2 million granules of the sun givesa very stable luminosity, which would plot asa straight line in Fig. 10 (even 2000 randomcells do this). However, the size of the cells atthe bottom of the solar convection zone will belarger (hence fewer cells), and if chaotic mightgive a long term modulation to the solar lumi-nosity. Full star simulations of the whole so-lar convective zone, with sufficient numericalresolution to give well developed turbulence,should shed light on this issue.

Application to turbulent stellar atmo-spheres requires surmounting two difficulties:(1) the flow is no longer low-Mach number(see § 2.2), and (2) the ionization zone causesdramatic changes in opacity (assumed con-stant in the Lorenz model). Fortunately 3Datmospheres exist, and analysis such as wehave done on stellar interior convection simu-lations is feasible.

While this paper was in preparation,Stothers (2010) re-examined the idea that gi-ant convective cell turnover is the explana-tion of the long secondary period observed insemiregular red variable stars (Stothers & Leung1971), including Betelgeuse and Antares.Stothers used MLT to derive a velocity scalefor the overturn, also relying on general fea-tures15 of simulations of Chan & Sofia (1996);see §3 in Stothers (2010). This theory ap-pears to work directly as a complement tothe discussion above. The use of a Lorenz

15Arnett, Meakin, & Young (2009) discuss and contrastthese and other simulations.

18

model for the giant cells already implies realdynamics. The strange attractor necessarilyprovides variability in luminosity, with its ownquasi-period (Wood, Olivier & Kawaler 2004)and velocity scale (see §4 in Stothers (2010)).Several giant cells16 are at work, each with aquasi-period of order of the transit time, andtherefore similar to the estimate of Stothers,and the observations. The larger convectivevelocities needed are simply a necessary con-sequence of the dynamics implied by the con-vective luminosity (Arnett, Meakin, & Young2010).

The introduction of turbulence as an active

agent in the discussion of stellar variability(e.g., Stothers (2010); Wood, Olivier & Kawaler(2004)) seems timely. An interesting improve-ment would be to use POD empirical eigenval-ues from simulations (e.g., Chiavassa, et al.(2010), § 5.1 above), and develop a low or-der dynamical model to explore long durationbehavior.

6. Summary

We have identified a major new feature ofstellar physics: chaotic behavior due to tur-bulent fluctuations in stellar convection, andcorresponding luminosity fluctuations. Whilethe simulations upon which the analysis wasbased were fully compressible, the theory usesthe approximation of sub-sonic flow. Both nu-merically and analytically, a strange attrac-tor like that of Lorenz (1963) seems to appearnaturally in stellar convection.

As a first approximation to more rigorousanalysis, we have applied the Lorenz modelto kinetic energy fluctuations in the oxygenburning shell, to the turbulent energy cascade,and to fluctuations in luminosity in irregular

16This idea can be tested observationally and numeri-cally; the structure of the convection region will con-strain the number of cells, which may be comparedwith the amplitude of luminosity fluctuations.

variables.

Figure 1 shows a comparison of the tur-bulent kinetic energy fluctuations in 3D sim-ulations of turbulent flow and in the Lorenzmodel. No parameters were adjusted to give afit. Additional modes, appropriate for turbu-lent flow, would improve the comparison fur-ther.

This suggests a new, inherently nonlin-

ear mechanism for variability in stars, theτ -mechanism, which is caused by luminos-ity fluctuations directly associated with tur-bulent convective cells. Because the mech-anism is nonlinear, it is not captured bylinear stability analysis, which is a main-stay of variable star theory. Such lumi-nosity fluctuations may have been observedalready in the broad-band noise seen inBetelgeuse (alpha Orionis (Goldberg 1984;Kiss, Szabo, & Bedding 2006), and in thelong secondary periods in pulsating AGB stars(Wood, Olivier & Kawaler 2004; Stothers 2010),and are expected to be observable in principlein all stars with extensive surface convectionzones, including those with “solar-like” vari-ability. This mechanism is probably the causeof the strongly driven pulsations found byWoodward, Porter, & Jacobs (2003) in their“red giant” model; the development of thoselarge pulsations was a clue which may now bemore fully understood.

Such fluctuations provide a source of per-turbations for instabilities, and may inducemixing not presently accounted for in stellarevolutionary calculations. The fluctuations inconvective velocity are comparable to averagevalues. If these fluctuations couple to nuclearburning, as for example in cases of degenerateignition, shell flashes, or later stages of oxygenand silicon burning17, outbursts may develop.

17For new developments since this paper was submitted,see Arnett & Meakin (2011b), where it is suggestedthat core collapse progenitors are dynamically active

19

This work was supported in part by NSFGrant 0708871 and NASAGrant NNX08AH19Gat the University of Arizona, and by theCLEAR sub-contract from University of Michi-gan. We wish to thank Fr. J. Funes (SpecaloVaticano), Prof. R. Ruffini (ICRAnet), andProf. J. Lattanzio (Monash), P. Wood (Aus-tralian National University), and the AspenCenter for Physics for their hospitality, ProfF. Timmes and S. Starrfield for discussions,R. Stothers for helpful email, and the secondanonymous referee for constructive comments.

prior to collapse.

20

A. APPENDIX: Physical Basis of the Lorenz Model

Mass conservation. Conservation of mass is enforced by use of a stream function (Landau & Lifshitz(1959), §9); the simplest solution is a two dimensional, cylindrical ”roll” (Chandrasekhar (1961),p. 44). The flow is assumed to be subsonic.

Momentum conservation. The Navier-Stokes equation may be written as

Du/Dt = ν∇2u− g∆ρ/ρ, (A1)

in the low Mach number limit; the last term is the buoyant acceleration. The flow executes a circleof radius ℓ/2; see Figure 3. The density fluctuation is related to the temperature fluctuation by∆ρ/ρ0 = βT (T − T0)/T0, where βT = −(∂ ln ρ/∂ lnT )P .

We separate the variable u by considering the flow to be a mass flux which is constant aroundthe ring at any given moment, represented by an average speed u which is a function of time only18.The hydrostatic background in the convective region has an entropy S which is constant. Then,

1

ρ

dP

dr=

dW

dr= −g, (A2)

where W = E + PV is the enthalpy per unit mass, and W = CPT , and CP = γγ−1RY . Now

∫

gdr = g∆r if g is constant. Lorenz assumed a linear temperature decrease with height, whichcorresponds to constant gravitational acceleration g. Using a height z = (ℓ/2) cos φ,

W (φ) = W (0) + g(ℓ/2) cos φ, (A3)

or

T (φ) = T0 +g(ℓ/2)

CPcosφ. (A4)

We denote T1 = g(ℓ/2)/CP , soTE = T0 + T1 cosφ, (A5)

which corresponds to the environmental temperature. For T1 > g/CP the system is convectivelyunstable, while for T1 < T0, the background temperature can never be negative. We represent thetemperature by

T = T0 + T2 cosφ+ T3 sinφ. (A6)

Viscous damping. The viscous damping term may be approximated by ν∇2u → −Γu =−ν(2/ℓ)2u. Here the constant Γ is the inverse of the viscous dissipation time scale τvis.

The buoyant acceleration in the vertical direction is

Bz = −g·(δρ/ρ) → gβT (2T3/T0) sin2 φ. (A7)

Only a temperature (buoyancy) difference in the horizontal direction (φ = π/2) gives a net torqueto turn the convective roll.

18 Variations in density due to change in hydrostatic stratification could be included as well as the implied changes invelocity and cross-sectional area; see Tritton (1988), p. 188-196).

21

With damping due to a linear viscosity term,

du/dt = −Γu+gβTT0

(T − TE) sinφ. (A8)

Integrating Eq. A8 over a complete cycle in 0 ≤ φ ≤ 2π, gives19

du/dt = −Γu+[gβT2T0

]

T3 (A9)

The temperature terms in cosφ integrate to zero because of the sinφ factor in the buoyancy term.Thus we have just two terms in Eq. A9, a sink from the viscous damping and a source frombuoyancy.

In the turbulent case, we might identify this rate of dissipation as an integral over the turbulentregion, so it is transformed into a global quantity, εK/ρ = u3/ℓ. This is the deceleration timesthe velocity, giving a deceleration of −u|u|/ℓ, so that ν∇2u → −u|u|/ℓ. The absolute value |u|is used here because the characteristic time is τvis = ℓ/|u|, and the deceleration is thus −u/τvis.The length scale is the depth of the convective shell (i.e., ℓ). The convective speed u is constant inspace, so we may use a nonlinear damping term, as implied by Kolmogorov (1941),

du/dt = −u|u|/ℓ+[gβT2T0

]

T3. (A10)

Energy conservation. At constant pressure, the first law of thermodynamics simplifies to

dW/dt = CPdT/dt = ǫ− 1

ρ∇ · F+

εKρ, (A11)

where ǫ is the net heating-cooling rate from nuclear burning and neutrino emission, CP the specificheat at constant pressure, T the temperature, ρ the mass (nucleon) density, F the flux of radiativeenergy, and εK the volumetric heating by turbulence. Ignoring ǫ and εK ,

∂T/∂t+ u · ∇T = −(1

ρ∇ · F)/CP . (A12)

The divergence of flux of radiative luminosity is

1

ρ∇ · F = −1

ρ∇·[−(c/3ρκ)∇aT 4], (A13)

so that

∂T/∂t+ u · ∇T = νT∇2T, (A14)

where νT = (4acT 3/3ρκ)/(ρCP ) is the thermal conductivity for radiative diffusion, for constant κand CP .

The background temperature TE is chosen such that ∇2TE = 0. The radiative diffusion term is

νT∇2T ≈ K(TE − T ), (A15)

19The two terms constant in space (φ) generate a factor of 2π while the integral over sin2 φ gives a factor of π.

22

where K(TE −T ) is the extra radiative diffusion due to the deviation of temperature from TE, andK = νT (2/ℓ)

2 is the inverse of the radiation diffusion time scale τrad.

The classical Lorenz equations. Following Lorenz (1963), we correct for an aspect ratio a by afactor b = 4/(1 + a2) which deals with the excess in vertical over horizontal heat conduction (bKversus K). Substituting for T and TE in Equation A14, we have

dT2

dtcosφ+

dT3

dtsinφ

−2uT2

ℓsinφ+

2uT3

ℓcosφ

= bK(T1 − T2) cosφ−KT3 sinφ. (A16)

Coefficients of orthogonal functions must separately sum to zero, so

dT3/dt− 2uT2/ℓ = −KT3 (A17)

dT2/dt+ 2uT3/ℓ = bK(T1 − T2). (A18)

We introduce a potential temperature

T4 = T1 − T2, (A19)

and eliminate T2. If T1 = gℓ/2CP is independent of time,

dT3/dt = −KT3 + 2uT1/ℓ− 2uT4/ℓ (A20)

dT4/dt = −bKT4 + 2uT3/ℓ. (A21)

We define dimensionless variables

τ = t K

X = u (2/ℓK)

Y = T3 (gβT /ℓΓKT0)

Z = T4 (gβT /ℓΓKT0), (A22)

and use Equations A9, A20, and A21 to get the Lorenz equations in their usual form: Equations 12,13, and 14.

dX/dτ = −σX + σY (A23)

dY/dτ = −XZ + rX − Y (A24)

dZ/dτ = XY − bZ, (A25)

where τ is a time in thermal diffusion units, σ = Γ/K is the effective Prandtl number, and r =(gβTT1/ℓΓKT0) = (gβTT1/ℓΓ

2T0)σ is the ratio of the Rayleigh number to its critical value for onsetof convection. The Prandtl number is the ratio of coefficients of the viscous dissipation term to thethermal mixing term, σ = ν/νT = Γ/K = τrad/τvis, where τvis = 1/Γ is the viscous damping time.

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B. APPENDIX: What Should The Prandtl Number Be?

The original Lorenz parameters appear to be a fair choice to represent the flow in 3D simulations(see Figure 1). This is consistent with an effective Prandtl number for the numerical simulation ofσt ≈ 10; however, the severe reduction in degrees of freedom from the simulations to the Lorenzmodel warns against taking the numerical value too literally. We note for comparison that waterhas σ ≈ 6 and air has σ ≈ 0.7. Apparently Lorenz felt he was lucky (Lorenz (1993), p. 137);he took a suggested value (σ = 10) which gave chaotic behavior instead of a lower value, actuallyappropriate for air, for which his equations give stable rolls. As a mathematical example thisquantitative difference in the Prandtl number is not significant, but for physical applications, it is.

B.1. The Microscopic Value

Using the ratio of microscopic thermal diffusion to viscous time scale, Hansen & Kawaler (1994)(page 178 and 185) suggest that σ ≈ 10−8. Lorenz (1963) finds the critical value for r for instabilityof steady convection to be rc = σ(σ + b+ 3)/(σ − b− 1). For σ < b+ 1 = 2.666, steady convectionis always stable, so that the Hansen & Kawaler (1994) value would never give turbulence. Ifσ > b+1, steady convection is unstable for sufficiently large Rayleigh numbers. This precise valuefor instability is a characteristic of the canonical Lorenz model, and is affected by the particularchoice of dissipation function (see Appendix A).

B.2. The Simulation Value

In the numerical simulations, the effective Prandtl number is dominated by the turbulent cascade,which gives mixing of both momentum and heat at a rate determined by the largest eddy size. Thus,in numerical simulations, the turbulent flow defines its own effective value of this parameter σ → σt,and whatever value σt has, it is clearly above the threshold for instability for the system we havenumerically simulated. Numerical simulations and experiment suggest, for developed turbulence athigh Reynolds numbers, σt ≈ 0.7, a value typical of many common gases.

We note that in the “Reynolds analogy” (Monin & Yaglom (1971), p. 341), Osborne Reynoldsargued that the mechanisms for transport of heat and momentum were essentially the same in aturbulent medium, so that the effective turbulent Prandtl number should be of order unity (σt ≈ 1).

B.3. A Cascade Estimate

Suppose we think of the Prandtl number as the relative strength of the process which makes thevelocity field isotropic to that which converts the kinetic energy into heat. These occur at differentends of the turbulent cascade. To better understand what this might mean, consider σ =(timeto change direction)/(time to heat). We approximate the time to change direction by the time tohalve the kinetic energy, (ℓ/2vrms). At any length scale λ the turbulent cascade has a transfer ratefor kinetic energy of v3λ/λ = v3/ℓ. This implies a time spent at each length scale of τλ = λ/vλ. Ifeach level in the cascade is smaller on average in length scale by a factor λ(n + 1)/λ(n) = f , thetotal time for the cascade is (ℓ/v)(1+ f2/3 + · · ·) = (ℓ/v)/(1− f2/3), where the geometric series hasbeen used for the summation (see also Frisch (1995), §7.8). This gives an effective Prandtl numberσt ≈ 2/(1 − f2/3). Some representative values: for f = 1/e, this gives σ ≈ 4.1, for f = 1/2 thisgives σ ≈ 5, and f = 1/

√2 gives σ ≈ 10. This argument may tend to overestimate the Prandtl

number, so that σt would be smaller than the historical value chosen by Lorenz (σ = 10).

24

B.4. Renormalization Group

The separation in size of the large scale eddies with those at the dissipation scale suggests thatthis coupling might be approximated in some ingenious way. The microscopic viscosity might beconsidered a “bare” value, to be renormalized to a “dressed” value, in analogy to the field-theoretictreatment of interacting particles. Kadanoff (1966) proposed the idea of reducing the size of asystem a step at a time by grouping neighboring entities (in this case molecules) and treatingeach group as a single interaction. Wilson (1970) has successfully implemented the general ideaof coarse-graining, or “weeding out the small scales”. These general ideas may be applied to theturbulent cascade. Yakhot & Orszag (1986) use renormalization-group (RG) analysis of turbulencewith some success, and in particular, estimate the effective Prandtl number for turbulent flow tobe σt = 0.7179; see also Kraichnan (1987).

However there is some debate: in their review, Smith & Woodruff (1998) warn “. . . the RGmethod . . . leads to suggestive results when applied to turbulence. . . However, its application toturbulence cannot yet be called a major success, owing to the uncontrolled approximations currentlyrequired to implement it.” A similar sentiment is found in McComb (2004), p. 290.

B.5. A Perspective

We have captured the behavior of the pulses in the simulations, by a Lorenz model using theparameters that Lorenz used. In this sense, these values are relevant to our problem, although it isunclear from first principles what the precise value of the effective turbulent Prandtl number shouldbe. The form for dissipation that Lorenz used is not the same as implied by Kolmogorov, so that thethreshold for instability changes (Arnett & Meakin 2011c). We may interpret the Prandtl numberσ and the Rayleigh number r in terms of a renormalization in which the existence of turbulenceimplies effective Prandtl and Rayleigh numbers for the convective cell. The actual values of themicroscopic viscosity and thermal conductivity have little feedback on the behavior of the largesteddies; see Section 2.1. Magnetic fields in real stars may affect the effective Prandtl number further,which in the classical Lorenz model affects in turn the development of instability in the convectiveroll.

For problems which are insensitive to the details of the small scale flow, the values of themicroscopic Prandtl do not matter; the turbulent system bootstraps to an effective dissipationfor which the rules of Kolmogorov hold. This approximation applies to hydrostatic (and mildlydynamic) stellar evolution, in which the burning times are long compared to sound travel times.For these problems, ILES is appropriate. There are notable exceptions, such as a flame frontin a medium of unmixed fuel and ash (SNIa progenitor models), for which the small scales areimportant, and direct numerical simulations (DNS) of the small scales is necessary.

Finally, we recall how the Lorenz model was constructed: a low order mode was chosen, aRayleigh number was chosen just above the onset of instability for that mode, and a Prandtl numberwas chosen which gave interesting behavior. While different parameter choices are mathematicallyinteresting (Sparrow 1982), their physical relevance must be re-evaluated. The Lorenz equationsare a spartan subset of the fluid equations which contain the germ of chaos; it is probably betterto use them as a guide rather than a gospel.

25

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This 2-column preprint was prepared with the AASLATEX macros v5.2.

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