Cyclic thermal signature in a global MHD simulation of solarconvection

Jean-Francois Cossette1 and Paul Charbonneau1

Physics Department, Universite de Montreal, Montreal, Canada

Piotr K. Smolarkiewicz2

European Centre for Medium-Range Weather Forecasts, Reading, UK

ABSTRACT

Global magnetohydrodynamical simulations of the solar convection zone have recentlyachieved cyclic large-scale axisymmetric magnetic fields undergoing polarity reversals on a decadaltime scale (Beaudoin et al. 2013; Racine et al. 2011; Ghizaru et al. 2010). In this Letter, weshow that these simulations also display a thermal convective luminosity that varies in-phase withthe magnetic cycle, and trace this modulation to deep-seated magnetically-mediated changes inconvective flow patterns. Within the context of the ongoing debate on the physical origin of theobserved 11-year variations in total solar irradiance, such a signature supports the thesis accord-ing to which all, or part of the variations on decadal time scales and longer could be attributedto a global modulation of the Sun’s internal thermal structure by magnetic activity.

Subject headings: Total solar irradiance, solar activity, MHD, solar convection

1. IRRADIANCE VARIATIONS ANDTHE SOLAR DYNAMO

Total solar irradiance (TSI) has been measuredalmost continuously by earth-orbiting satellites formore than three decades and is now known tovary on time scales from minutes to days andmonths as well as on the longer time scale ofthe 11-year solar-cycle; see (Kopp & Lean 2011;Frohlich & Lean 2004). Perhaps the most strikingfeature that emerges from observational measure-ments is the slight increase of the TSI (≈ 0.1%) atsolar activity maxima relative to its value at solarminima. Models that include the contributions ofsunspots, faculae and magnetic network to recon-struct irradiance time series succeed very well atreproducing the observations on the short, intra-cycle time scales (e.g. shorter than a year). Sofar, however, pin-pointing the source of the longerdecadal fluctuations has remained the subject ofcontroversy. In particular, one school of thoughtargues that surface magnetism alone is sufficient toexplain the entire TSI variance (Foukal et al. 2006;

Lean et al. 1998); whereas, the other emphasizesthe effect of a global modulation of thermal struc-ture by magnetic activity (Sofia & Li 2006; Li etal. 2003). Within a broader context, establish-ing the contribution of a global thermal modula-tion to TSI variability relative to that of emerg-ing magnetic flux structures at the solar surface isvaluable for quantifying the coupling between pe-riods of quiet surface magnetism and the Earth’sclimate, such as the postulated relationship be-tween the Maunder Minimum and the Little IceAge (Foukal et al. 2011). It is the potential forthe occurence of such changes which we considerin this Letter.

In principle, magnetically-modulated globalstructural changes could take place throughthe action of the Lorentz force onto the large-scale heat-carrying convective flows, which wouldthen affect temperature and pressure conditionsthroughout the solar interior. Evidence for sucha hypothetical mechanism is partly supported byhelioseismic observations that show a positive cor-

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relation between low-degree p-mode acoustic os-cillations frequencies and the TSI cycle, as wellas recent evidence for a long-term trend in theTSI record that is not seen in indicators of surfacemagnetism; (Frohlich 2013; Bhatnagar et al. 1999;Woodward 1987). Other observations documentpossible variations of the surface temperature andchanges in the solar diameter; (Harder et al. 2009;Thuillier et al. 2005; Gray & Livingston 1997;Kuhn et al. 1988).

Notably, structural models of the Sun haveshown that by incorporating the effect of amagnetically-modulated turbulent mechanisminto the 1D stellar structure equations, it is pos-sible to reproduce the time-dependence of the ob-served p-mode frequency oscillations along withthe expected variations in solar radius, luminos-ity and effective temperature (Sofia & Li 2006; Liet al. 2003). However, these models relied uponan ad-hoc parametrization of the alleged processand therefore solving the full set of equations gov-erning the self-consistent evolution of the solarplasma remains necessary to obtain a completephysical picture of the mechanism at the origin ofthe modulation.

Fortunately, global magnetohydrodynamical(MHD) simulations of the solar convection zone(SCZ) have recently been able to produce regu-lar, solar-like magnetic cycles undergoing hemi-spheric polarity reversals on a decadal timescale (Beaudoin et al. 2013; Racine et al. 2011;Kapyla & al. 2010; Ghizaru et al. 2010), and there-fore offer the opportunity for the search and inves-tigation of potential global structural changes. Inthis paper we report on a global MHD simulationof the SCZ that shows a modulation of the con-vective heat flux by a solar-like cyclic large-scalemagnetic field. In section 2 we briefly documentthe physical setup of our experiment and describethe main results. In section 3, we further explorethe simulation data in search of plausible physi-cal mechanisms that could explain the observedthermodynamic signal and we conclude the pa-per in section 4 by pointing out a few possibleobservational signatures.

2. RESULTS: MODULATION OF THECONVECTIVE HEAT FLUX

We solve the anelastic form of the idealMHD equations for momentum, potential tem-perature fluctuations (tantamount to fluctua-tions of specific entropy) and magnetic induc-tion inside a thick spherical shell of electrically-conducting fluid extending from r = 0.602R⊙ tor = 0.96R⊙ in radius, rotating at the solar rateΩ⊙ = 2.69×10−6rad s−1 and spanning 3.4 densityscale-heights. Rather than forcing convection byapplying large-amplitude solar heating and cool-ing fluxes at bottom and top boundaries, as it iscustomary in simulations of stellar convection, aNewtonian-cooling term in the potential temper-ature equation damps naturally-occuring depar-tures from an ambient state representative of thecurrent-day Sun on a user-defined time scale τ .The ambient state is subadiabatically-stratified inthe lower portion of the shell corresponding tothe radiative interior (0.602R⊙ ≤ r ≤ 0.71R⊙)and is weakly superadiabatic in the range as-sociated with the bulk of the convective layer(0.71R⊙ ≤ r ≤ 0.96R⊙). This approach has leadto convective dynamo solutions exhibiting a num-ber of solar-like properties, the most prominent ofwhich is a large-scale axisymmetric magnetic fieldcomponent undergoing polarity reversals aboutthe equatorial plane on a 40-year time scale; seethe following publications for an in-depth explo-ration of various solar-like features found to thisday: (Beaudoin et al. 2013; Racine et al. 2011;Ghizaru et al. 2010). Solutions are generatedwith the MHD-extended version of the EULAGmodel (Smolarkiewicz & Charbonneau 2013), anall-scale high-performance hydrodynamical codeused primarily in atmospheric and climate re-search (Prusa et al. 2008). We hereby focus ona recent simulation covering 32 magnetic polar-ity reversals (similar in design and the resultingflow regime to the one presented in Ghizaru etal. 2010) in which global structural changes havebeen observed.

Convection and Newtonian cooling are thedominant modes of thermal energy transport inthe unstable layer and nearly cancel each other outto yield a quasi-stationnary turbulent state, withthe contribution from radiation being negligible.The means by which we quantify the efficiency of

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convective energy transfer is the convective heatflux, or enthalpy flux, which in the anelastic ap-proximation is given by F ≡ cpρourT , where cpis the specific heat at constant pressure, ρo is thedensity associated with the reference state of theapproximation, ur is the vertical component ofthe velocity and T is the temperature perturba-tion with respect to the mean horizontal state.The convective luminosity is the integrated flux Fover a spherical surface Lcv ≡

∫F dσ.

The top and bottom panels of figure 1 show, re-spectively, a snapshot of F in a Mollweide projec-tion (longitude vs latidude) at r = 0.87R⊙ and acorrelation plot of F versus ur at the same height,with numbers indicating the fraction of points con-tained in each quadrant. The two dashed lines inthe top plot delimit, respectively, the inside andthe outside regions (hereafter, the mid/high lati-tudes) of the equatorial band comprised betweenthe latitude circles θ = ±30o. Notably, the flux atthe mid/high latitudes is spatially very intermit-tent, despite the low grid resolution of the experi-ment (Nϕ×Nθ×Nr = 128×64×47), whereNϕ, Nθ

and Nr stand for, respectively, the number of gridpoints in the longitudinal, latitudinal and radialdirections. In particular, the strongest heat fluxes(i.e. F ≥ 1.5×107 W ·m−2) appear in the form oflocalized concentrations or ‘hot spots’; whereas,cooling regions (i.e. F < 0) correspond to theweaker fluxes that are interspersed among the lat-ter. By contrast, the equatorial band is dominatedby North-south positive and negative flux lanes,which are the thermal signature of the convectivemodes commonly known as ‘banana cells’ and arecharacteristic of flows in a rotating spherical shell(see Miesch and Toomre 2009 for an in-depth dis-cussion of global convective dynamics). The bot-tom plot shows that approximately 70% of upflowsand downflows contribute to positive fluxes, whilethe remaining 30% is responsible for cooling byfluid entrainment; a landmark of flows operatingin a highly turbulent regime. Here the asymme-try between distributions associated with ur > 0and ur < 0 can be traced back to the pattern ofbroad upflows and narrow downflow lanes typicalof thermal convection in a density-stratified envi-ronnement; see, e.g., fig. 1 of (Ghizaru et al. 2010)and fig. 2 of (Miesch & Toomre 2009).

Figure 2 displays time-latitude plots of thezonally-averaged toroidal component of the mag-

Fig. 1.— A simulation snapshot: Mollweide pro-jection of the convective heat flux F (θ, ϕ) at r =0.87R⊙ (top) and its correlation with the verticalflow velocity at the same height (bottom). Eachdiamond corresponds to the value of F at a gridpoint on the spherical shell, and the numbers givethe fraction of grid point values appearing in eachquadrant.

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netic field (top panel) and zonally-averaged devia-tion about the temporal mean of F at r = 0.87R⊙(bottom) for a segment of the simulation span-ning slightly more than 150 years. The ax-isymmetric toroidal field is clearly antisymmet-ric about the equator and is split into a large-scale component at mid/high latitudes revers-ing polarities on a period varying between 30and 40 years and another flux concentration lo-cated in the equatorial band evincing a shorter,3 to 5 year oscillatory signal. Interestingly, he-lioseismic frequency analysis and indicators ofgeomagnetic activity suggest the presence of ashort-term magnetic cycle operating inside theSun, in addition to the well-known 11-year signal(Fletcher & al. 2010; Mursula et al. 2003). Thebottom plot shows the global thermal modula-tion. In particular, at mid/high latitudes, theconvective heat flux is temporally well correlatedwith the absolute value of the zonally-averagedtoroidal magnetic field. Likewise, the signatureof the short-term magnetic cycle can be seen inthe flux comprised within the equatorial band, al-though it appears to be in anti-phase with the fluxvariation at the mid/high latitudes. Here, how-ever, we postpone a thorough discussion of theshort-term magnetic signal to a future publicationand focus on changes related to the long-termcycle.

The top panel of fig. 3 displays histories of boththe unstable layer’s normalized total magnetic en-ergy E⋆

m ≡ Em/max(Em) (black curve) and tem-poral perturbation of the normalized convectiveluminosity ∆L⋆

cv ≡ (Lcv−Lcv)/Lcv at r = 0.87R⊙(red curve) and its 5-year running mean (yellowcurve), where the maximum of magnetic energyis taken over the full time-sequence of the simu-lation and Lcv is the time-averaged convective lu-minosity. Thus, the combination of the mid/highlatitudes in-phase flux modulation and the equa-torial band’s signal gives a total convective fluxvariation that has the same phase as Em. Its am-plitude is of the order of 10% of Lcv, which itselfonly reaches 3% of the true solar luminosity L⊙at this given height, thereby yielding a modula-tion of amplitude 0.003L⊙. Although the ampli-tude of the modulation is comparable to that ofthe actual decadal TSI fluctuations, it must bekept in mind that this specific simulation is quitesubluminous, with Lcv peaking at≈ 0.1L⊙ at mid-

depth of the domain. We shall comment furtheron this issue in section 4. Nevertheless, the signof the correlation between E⋆

m and ∆L⋆cv is posi-

tive (r = +0.63) and therefore consistent with anincrease of TSI at solar maxima, as shown by thebottom panel, in which ∆L⋆

cv is plotted as a func-tion of E⋆

m. Notably, the scattering of points alongthe vertical axis can be attributed to the influenceof the short-term magnetic cycle on the total con-vective luminosity. The fact that convective heattransport correlates positively with magnetic ac-tivity forcibly points to an intricate interplay be-tween the flow and magnetic field topologies, sincethe presence of a magnetic field generally tends toinhibit, rather than enhance, the flow of an elec-trically conducting fluid, at least in the parameterregime relevant to solar interior conditions.

3. THE HUNT FOR THE PHYSICALMECHANISM

As a first step towards the identification of thephysical mechanism causing the observed modula-tion, we shall examine how strongly the convectiveflux at each grid point is modulated by the mag-netic field. The top panel of figure 4 shows cu-mulative convective heat flux PDFs correspond-ing to, respectively, every minima (black contin-uous curve) and maxima (red dashed curve) ofthe simulation for the spherical surface located atr = 0.87R⊙; cf. fig. 1. Observe that at peak timeof the magnetic cycle, the amount of flux valuescorresponding to the hot spots increases with re-spect to its value at minimum time; whereas, theopposite phenomenon characterizes the negativeflux values. The bulk of the positive flux distri-bution (i.e. 0 ≤ F < 1.5 × 107 W · m−2), on theother hand, shows no obvious change. To sub-stantiate this, we have computed the correlationcoefficient of the time sequence of each of the con-vective heat flux PDF’s histogram bins with thetime series of Em for grid points on the full spher-ical shell (black continuous curve), the equatorialband (black dotted curve) and the mid/high lat-itudes (black dashed curve) and plotted the re-sult in the bottom panel of fig. 4. Each bin con-tributes to some fraction of the total convectiveluminosity variation (cf. fig. 3), and red curvesdisplay their correlation with Em. The correla-tion analysis confirms what could already be in-ferred from the cumulative PDFs, namely, that

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there is a significant modulation of hot spot andnegative flux populations by the magnetic cycle.Notably, the strong anti-correlation (r ≈ −0.6) ofthe mid/high latitudes negative fluxes with Em isthe sign of a pronounced diminution of fluid en-trainment at peak cycle time, which could be dueto the suppression of turbulence by the intensi-fication of the magnetic field. The bulk of thepositive flux distribution does show a weak posi-tive correlation with Em, even though this changecannot be easily detected in the minima-maximaPDFs. Interestingly, this fluctuation translatesinto a decrease of the luminosity at cycle maxi-mum, as shown by the anti-correlation with Em

near F = 0, and therefore implies a correspondingdecrease of the filling factor of the associated fluxregions. The means and variances of the luminos-ity variations implied by hot spots, elements fromthe bulk of the positive flux distribution and neg-ative flux elements are, respectively, as expressedin terms of fractions of Lcv: (0.320; 0.039), (0.781;0.024) and (−0.102; 0.007). The PDF of verticalvelocities (not shown), on the other hand, is invari-ant with respect to magnetic energy and thereforeimplies that the action of the Lorentz force on theflow is not simply a straightforward suppressionof buoyancy-driven flows by the magnetic field,which again suggests that the interplay betweenflow and field topologies is magnetohydrodynami-cally complex.

4. REMARKS

We have presented a basic analysis of a globalMHD simulation of solar convection evincing amodulation of thermal structure by regular, solar-like, large-scale cyclic magnetic fields undergoingpolarity reversals. Most importantly, the convec-tive luminosity correlates positively with the mag-netic cycle, which is consistent with the enhancedvalue of the TSI observed at solar maxima. Thetime analysis of the distribution of convective heatfluxes on a spherical surface allows to classify eachflow feature according to the size of its contri-bution to the variance of the luminosity modu-lation; the most important effect being associatedwith the intense and localized positive flux ele-ments known as hot spots, followed by a luminos-ity deficit coming from the bulk of the positiveflux distribution, and a positive contribution dueto a diminution of cooling via fluid entrainment.

The latter may be attributed, for instance, to thesuppression of turbulence by the enhanced mag-netic field; whereas, the other two could be inter-preted as manifestations of the flow’s response tothe modulation of turbulent entrainment by thecycle (recall the strong anti-correlation r ≈ −0.6)while simultaneously satisfying the constraint ofmass conservation and energy transport require-ment imposed by the thermal forcing of the sys-tem. If such a speculative mechanism turned outto be real, it could well explain the weaker correla-tion of positive flux values with magnetic energy.

We have carried out our analysis of the fluxvariation at the given height 0.87R⊙, and find thatthe amplitude of the luminosity fluctuations im-plied by the modulation at this specific level isof the order of 0.3% of the true solar luminosity.However, the mean convective luminosity at thisspherical shell is only of the order of 0.03L⊙ andthe fact that our domain stops below the photo-sphere limits the type of comparisons that can bemade with respect to the real Sun. For instance,the amplitude of the TSI variation implied by theglobal modulation may vary depending upon thesubphotospheric layers’ ability to store/release theheat flowing up from below as a result of the deep-seated flux signal (Foukal et al. 2006). Moreover,the use of an impenetrable velocity boundary con-dition at the top of our model, a common fea-ture of global MHD simulations, suppresses con-vective motions in the outer layers and thereforeprevents one from making accurate predictions asto potential impacts of the flux modulation on theupper SCZ’s dynamics. Nevertheless, our obser-vation of a positive correlation between convec-tive energy transport efficiency and magnetic en-ergy suggests that magnetically-modulated globalstructural changes could contribute to an enhance-ment of the TSI at peak cycle time. A quantitativeestimate of the corresponding amplitude contri-bution will require further modeling of the upperSCZ’s physical connection to the photosphere aswell as carrying out a similar analysis at higherluminosity.

We are currently engaged in a detailed anal-ysis of the modulation and its underlying phys-ical mechanism, briefly discussed in this Letter.The overarching aim is to assess the implicationsthat these results might have for the existenceof activity cycle-driven structural changes inside

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the Sun, and predict potential observational signa-tures. In this respect, the results presented here al-ready indicate that the response of the convectiveflux to the magnetic activity cycle shows a signifi-cant latitudinal dependency, which could translatein cyclic variations of the surface pole-to-equatortemperature contrast (viz. Rast et al. 2008; Kuhnet al. 1988), and/or asphericity of the solar pho-tosphere (Thuillier et al. 2005).

Comments from an anonymous referee helpedto improve the presentation. The numerical sim-ulations reported in this paper were carried outprimarily on the computing facilities of CalculQuebec, a member of the Compute Canada con-sortium. This work is supported by Canada’sFoundation for Innovation, Natural Sciences andEngineering Research Chair program and Discov-ery Grant program (PC), and the NSERC Gradu-ate Fellowship Program (JFC). PKS is supportedby the European Research Council under the Eu-ropean Union’s Seventh Framework Programme(FP7/2012/ERC Grant agreement No. 320375).

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Fig. 2.— Temporal evolution: Zonally-averagedtoroidal component of the magnetic field (top) andzonally-averaged deviation of the convective heatflux F (bottom) at r = 0.87R⊙. At mid/high lat-itudes, the absolute value of the zonally-averagedtoroidal field has a period and phase comparableto that of the flux variation.

Fig. 3.— Top: Histories of normalized total mag-netic energy (black curve) and temporal pertur-bation of the normalized convective luminosity(red curve) with its 5-year running mean (yellowcurve). Bottom: Correlation between the tem-poral perturbation of convective luminosity andmagnetic energy.

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Fig. 4.— Top: Cumulative convective heat fluxPDFs corresponding to, respectively, every min-ima (black continuous curve) and maxima (reddashed curve) of the simulation at grid points lo-cated at r = 0.87R⊙. Here, maxima and minimaare defined as extremal points in time of magneticenergy. Bottom: The black curves show the cor-relation coefficient of each bin of the convectiveheat flux’s PDF with the time series of total mag-netic energy for regions consisting of the full spher-ical shell (solid curve), the equatorial band (dottedcurve) and the mid/high latitudes (dashed curve).Red curves display the correlation coefficient ofeach bin’s associated luminosity input variationwith total magnetic energy.

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