Exploring the Physical Basis of Solar Cycle Predictions: Flux Transport Dynamics and Persistence of Memory in Advection versus Diffusion Dominated Solar Convection Zones
aa r X i v : . [ a s t r o - ph ] S e p Exploring the Physical Basis of Solar Cycle Predictions:Flux Transport Dynamics and Persistence of Memory inAdvection versus Diffusion Dominated Solar Convection Zones
Anthony R. Yeates
School of Mathematics and Statistics, University of St Andrews, St Andrews, KY16 9SS,UK [email protected]
Dibyendu Nandy
Department of Physics, Montana State University, Bozeman, MT 59717, USA [email protected] andDuncan H. Mackay
School of Mathematics and Statistics, University of St Andrews, St Andrews, KY16 9SS,UK [email protected]
ABSTRACT
The predictability, or lack thereof, of the solar cycle is governed by numerousseparate physical processes that act in unison in the interior of the Sun. Mag-netic flux transport and the finite time delay it introduces, specifically in theso-called Babcock-Leighton models of the solar cycle with spatially segregatedsource regions for the α and Ω-effects, play a crucial rule in this predictabil-ity. Through dynamo simulations with such a model, we study the physicalbasis of solar cycle predictions by examining two contrasting regimes, one domi-nated by diffusive magnetic flux transport in the solar convection zone, the otherdominated by advective flux transport by meridional circulation. Our analysisshows that diffusion plays an important role in flux transport, even when thesolar cycle period is governed by the meridional flow speed. We further examinethe persistence of memory of past cycles in the advection and diffusion domi-nated regimes through stochastically forced dynamo simulations. We find that 2 –in the advection-dominated regime, this memory persists for up to three cycles,whereas in the diffusion-dominated regime, this memory persists for mainly onecycle. This indicates that solar cycle predictions based on these two differentregimes would have to rely on fundamentally different inputs – which may bethe cause of conflicting predictions. Our simulations also show that the observedsolar cycle amplitude-period relationship arises more naturally in the diffusiondominated regime, thereby supporting those dynamo models in which diffusiveflux transport plays a dominant role in the solar convection zone. Subject headings:
Sun: activity — Sun: interior — Sun: magnetic fields
1. Introduction
Direct observations of sunspot numbers over 400 years, as well as proxy data for muchlonger timescales (Beer 2000), show that both the amplitude and the duration of the so-lar magnetic cycle vary from one cycle to the next. The importance of this phenomenonlies in the contribution of varying levels of solar activity to long-term climate change, andto short-term space weather (Nandy & Martens 2007). While there is now a concensusthat the Sun’s magnetic field is generated by a hydromagnetic dynamo (Ossendrijver 2003;Charbonneau 2005), the origin of fluctuations in the basic cycle is yet to be conclusively deter-mined. Several different mechanisms have been proposed, including nonlinear effects (Tobias1997; Beer et al. 1998; Knobloch et al. 1998; K¨uker et al. 1999; Wilmot-Smith et al. 2005),stochastic forcing (Choudhuri 1992; Hoyng 1993; Ossendrijver et al. 1996; Charbonneau & Dikpati2000; Mininni & G´omez 2002), and time-delay dynamics (Yoshimura 1978; Durney 2000;Charbonneau et al. 2005; Wilmot-Smith et al. 2006). A coupled, equally important, butill-understood issue is how the memory of these fluctuations, whatever may be its origin,carries over from one cycle to another mediated via flux transport processes within the solarconvection zone (SCZ). A unified understanding of all these disparate processes lays thephysical foundation for the predictability (or lack-thereof) of future solar activity. Theseconsiderations motivate the current study.The main flux transport processes in the SCZ involve magnetic buoyancy (timescale onthe order of months), meridional circulation, diffusion and downward flux-pumping (timescalesrelatively larger). Because magnetic buoyancy, i.e., the buoyant rise of magnetic flux tubes,acts on timescales much shorter than the solar cycle timescale, the fluctuations that it pro-duces are also short-lived in comparison. Our focus here is on longer-term fluctuations, onthe order of the solar cycle period, that may lead to predictive capabilities. 3 –Through an analysis of observational data, Hathaway et al. (2003) have shown that thesolar cycle amplitude and duration are correlated with the equatorward drift velocity of thesunspot belts during the cycle. They associate this drift velocity with the deep meridionalcounterflow that must exist to balance the poleward flows that are observed at the surface(Hathaway 1996, 2005; Miesch 2005). The results show a significant negative correlationbetween the drift velocity and the cycle duration, so that the drift is faster in shorter cycles,consistent with the interpretation of meridional circulation as the timekeeper of the solarcycle (Nandy 2004; but see also Sch¨ussler & Schmitt 2004). In addition Hathaway et al.(2003) identified positive correlations between the drift velocity of cycle n and the amplitudesof both cycles n and n + 2. While the two-cycle time lag was a new result, the positivecorrelation between circulation speed and amplitude of the same cycle is supported by severalearlier studies. In their surface flux transport model, Wang et al. (2002) needed a varyingmeridional flow, faster in higher-amplitude cycles, to sustain regular reversals in the Sun’spolar field. They cited observational evidence from polar faculae counts (Sheeley 1991),which peaked early for two of the stronger cycles, coinciding with poleward surges of magneticflux. Furthermore, observations show a statistically-significant negative correlation betweenpeak sunspot number and the duration of cycles 1 to 22 (Figure 1c of Charbonneau & Dikpati2000; see also Solanki et al. 2002). Such a negative correlation between cycle amplitude andduration is also found in the models of Hoyng (1993) and Charbonneau & Dikpati (2000).Taken with the inverse relation between cycle duration and circulation speed, this is againsuggestive of a positive correlation between circulation speed and cycle amplitude.Meridional circulation plays an important role in a certain class of theoretical solarcycle models often referred to as “flux-transport”, “advection-dominated,” or “circulation-dominated” dynamo models (see, e.g., the review by Nandy 2004). Such models havegained popularity in recent years owing to their success in reproducing various observed fea-tures of the solar cycle (Choudhuri et al. 1995; Durney 1995; Dikpati & Charbonneau 1999;K¨uker et al. 2001; Bonanno et al. 2002; Nandy & Choudhuri 2001, 2002; Chatterjee et al.2004). In these models, a single-cell meridional circulation in each hemisphere (which is ob-served at the solar surface) is invoked to transport poloidal field, first poleward at near-surfacelayers and then down to the tachocline where toroidal field is generated. Subsequently, thereturn flow in the circulation advects this toroidal field belt equatorward through a region atthe base of the SCZ which is characterized by low diffusivity. From this deep toroidal fieldbelt, destabilized flux tubes rise to the surface due to magnetic buoyancy, producing sunspots(Parker 1955). We may point out here that the name “flux-transport dynamo” is somewhatinappropriate to classify a circulation or advection-dominated dynamo (where the diffusiontimescale is much larger than the circulation timescale throughout the dynamo domain).Our results indicate that diffusive flux-transport in the SCZ could play a dominant role in 4 –dynamos even when the cycle period is governed by meridional circulation speed, pointingout that flux-transport is a shared process. So, henceforth, by “flux-transport” dynamo,we imply a dynamo where the transport of magnetic field is shared by magnetic buoyancy,meridional circulation, and diffusion.Flux-transport dynamos offer the possibility of prediction because of their inherentmemory. This arises specifically when the dynamo source regions for poloidal field production(the traditional α -effect) and toroidal field generation (the Ω-effect) are spatially segregated.A brief discussion on important timescales (we identify three significant ones) in the dynamoprocess is merited here. The first is governed by the buoyant rise of toroidal flux tubes fromthe Ω-effect layer to the α -effect layer to generate the poloidal field; since this is a fastprocess on the order of months, no significant memory is introduced here. The secondinvolves the transport of poloidal field back into the Ω-effect layer (either by circulationor diffusion). This could be a slow process where significant memory is introduced whichis dominated by the fastest of the competing processes (advection versus diffusion). Thethird timescale relates to the slow equatorward transport of the toroidal field belt throughthe base of the SCZ, which sets the period of the sunspot cycle. In this class of dynamomodels, with meridional circulation and low diffusivities in the tachocline (at the base ofthe SCZ), the third timescale is almost invariably determined by the circulation speed. It isthe second timescale above, with competing effects of diffusive flux transport and advectiveflux transport, that becomes important in the context of the persistence of memory. In theadvection-dominated, stochastically fluctuating model of Charbonneau & Dikpati (2000),this second timescale (governed by advection of poloidal field due to meridional circulation)was about 17 years, so that the polar field at the end of cycle n correlated strongest withthe toroidal field of cycle n + 2 rather than that of cycle n + 1. The length of memoryof any particular flux-transport dynamo model is unfortunately dependent on the internalmeridional flow profile, and on other chosen properties of the convection zone which are notyet well-determined observationally. A particular problem is the strength of diffusivity inthe convection zone, which strongly affects the mode of operation of the dynamo.Even if one assumes that these flux-transport dynamos capture enough of the realisticphysics of the SCZ to make predictions of future solar activity, these predictions are criticallydependent on the relative role of diffusion and advection in the SCZ. Dikpati & Gilman(2006), in their highly advection-dominated model, show that bands of latitudinal field fromthree previous cycles remain “lined up in the meridional circulation conveyor belt”. Theysuggest that poloidal fields from cycles n − n −
2, and n − n . Based on an assumed proxy for the solar poloidal fields (sunspot area), thisleads them to predict that Cycle 24 will be about 50% stronger than Cycle 23 (Dikpati et al.2006). In stark contrast, Choudhuri et al. (2007), using a flux-transport dynamo model 5 –with diffusion-dominated SCZ, and using as inputs the observed strength of the solar dipolemoment (as a proxy for the poloidal field), predict that Cycle 24 will be about 35% weaker than Cycle 23. Choudhuri et al. (2007) argue that the main contribution to the toroidalfield of cycle n , comes only from the polar field of cycle n − α -effect to self-consistently generate cycle-amplitude variations – as a completelytheoretical construct towards studying cycle-to-cycle variations, in contrast to using diverseobserved proxies for time-varying poloidal fields. Subsequently, we perform a comparativeanalysis of the persistence of memory in this stochastically forced dynamo model in both theadvective and diffusive flux-transport dominated regimes. Therefore, in spirit, this paperdeals with the underlying physics of solar cycle predictability, and is not concerned withmaking a prediction itself. The layout of this paper is as follows. The main features ofthe model are summarised in Section 2, and the results of the parameter-space study arepresented in Section 3. These results are interpreted in Section 4. In Section 5 we analyze thepersistence of memory in the advection versus diffusion dominated regimes. We concludein Section 6 with a discussion on the relevance of this work in the context of developingpredictive capabilities for the solar activity cycle.
2. The Model
We use the solar dynamo code
Surya , which has been studied extensively in differentcontexts (e.g. Nandy & Choudhuri 2002, Chatterjee et al. 2004, Chatterjee & Choudhuri2006), and is made available to the public on request. The major ingredients of the codeinclude an analytic fit to the helioseismically-determined differential rotation profile, a single- 6 –cell meridional circulation in the SCZ, different diffusivities for the toroidal and poloidalfields, a buoyancy algorithm to model radial transport of magnetic flux, and a Babcock-Leighton (BL; Babcock 1961, Leighton 1969) type α -effect localized near the surface layer(signifying the generation of poloidal field due to the evolution of tilted bipolar sunspot pairsunder surface flux transport). The code solves the kinematic mean-field dynamo equationsfor an axisymmetric magnetic field, which may be expressed in spherical coordinates ( r, θ, φ )as B = B ( r, θ ) e φ + B p , (1)where B ( r, θ ) and B p = ∇ × [ A ( r, θ ) e φ ] correspond to the toroidal and poloidal componentsrespectively. The mean-field MHD induction equation (see e.g. Moffatt 1978) then leads tothe following standard equations for the α -Ω dynamo problem: ∂A∂t + 1 s ( v · ∇ ) ( sA ) = η p (cid:18) ∇ − s (cid:19) A + αB, (2) ∂B∂t + 1 r (cid:18) ∂∂r ( rv r B ) + ∂∂θ ( v θ B ) (cid:19) = η t (cid:18) ∇ − s (cid:19) B + s ( B p · ∇ ) Ω + 1 r dη t dr ∂∂r ( rB ) . (3)Here s = r sin θ , and we specify the meridional flow v , the internal angular velocity Ω, thediffusivities η p and η t , and the coefficient α for the BL α -effect which describes the generationof poloidal field at the solar surface from the decay of bipolar sunspots. Note that althoughfor modelling purposes the BL α -effect is mathematically similar to the traditional mean-field α -effect due to small-scale helical turbulence, the former is fundamentally different. The BL α -effect acts on much larger spatial (on the order of active regions or greater) and temporal(surface flux transport) scales, and is quenched at much stronger field strengths (10 G). Theprofiles of Ω and α were described in Chatterjee et al. (2004) and will not be repeated here.We will, however, describe the meridional circulation and diffusivity profiles in more detail. The meridional circulation is defined in terms of a streamfunction ψ ( r, θ ), giving thevelocity by ρ v = ∇ × [ ψ ( r, θ ) e φ ] , (4)where we assume the density stratification ρ = C ( R ⊙ /r − . / . (5) 7 –The streamfunction is given by ψr sin θ = ψ ( r − R p ) sin (cid:18) π ( r − R p ) R ⊙ − R p (cid:19) × (cid:8) − e − β θ ǫ (cid:9) (cid:8) − e β ( θ − π/ (cid:9) e − (( r − r ) / Γ) , (6)where β = 1 . × − m − , β = 1 . × − m − , ǫ = 2 . . × m, and r = ( R ⊙ − R b ) /
4. Here R ⊙ = 6 . × m is the radius of the Sun, R b = 0 . R ⊙ isthe bottom of the simulation domain, and R p = 0 . R ⊙ is the penetration depth of themeridional circulation. We combine the arbitrary constants C and ψ in the parameter v = − ψ / (0 . C ) which gives, approximately, the flow speed near the surface at mid-latitudes. It is this parameter v which we vary to change the circulation speed in thisstudy.The circulation profile is illustrated in Figure 1 for v = 25 m s − . The dots are plottedat yearly intervals for particles moving along the streamlines shown. We use different diffusivities for the toroidal and poloidal fields, defined as follows: η t ( r ) = η RZ + η − η RZ (cid:20) (cid:18) r − r ′ BCZ d t (cid:19)(cid:21) + η − η (cid:20) (cid:18) r − r T CZ d t (cid:19)(cid:21) , (7) η p ( r ) = η RZ + η − η RZ (cid:20) (cid:18) r − r BCZ d t (cid:19)(cid:21) + η − η (cid:20) (cid:18) r − r T CZ d t (cid:19)(cid:21) . (8)Here d t = 0 . R ⊙ , r BCZ = 0 . R ⊙ , r ′ BCZ = 0 . R ⊙ , and r T CZ = 0 . R ⊙ . In the radiativecore we choose a low diffusivity, namely η RZ = 2 . × cm s − , representing moleculardiffusivity only since there is no turbulent convection. We always choose η < η so that thetoroidal field diffusivity η t in the convection zone is lower than the poloidal field diffusivity η p .This is to model the suppression of turbulent diffusivity by strong magnetic fields, as toroidalfield tends to be strong and concentrated in localised flux tubes and therefore subject to lessdiffusion (Choudhuri 2003), whereas poloidal field is weaker and subject to more diffusion.At the surface both diffusivities increase to a high value (of the order of 10 cm s − ), inline with surface flux transport models and observational estimates. Typical profiles areillustrated in Figure 2. 8 – We solve Equations (2) and (3) in a meridional plane 0 . R ⊙ < r < R ⊙ , 0 < θ < π/ r and θ . We use the same boundary conditions as Chatterjee et al.(2004), except that we consider only the Northern hemisphere and set B = 0 and ∂A/∂θ = 0at the equator ( θ = π/ Example solutions for two different runs are shown in time-latitude plots in Figures 3(a)and (b), where the black lines denote contours of toroidal field strength at the base of theconvection zone ( r = 0 . R ⊙ ). This corresponds to the solar butterfly diagram, with thestrongest field located at the active latitudes and migrating equatorward during each cycle.The background shading shows the strength of the radial field at the solar surface ( r = R ⊙ ),which peaks at the pole several years after the toroidal field maxima (of the same sign) at lowlatitudes. The two solutions in Figure 3 characterize the diffusion-dominated SCZ (Figure3a) and advection-dominated SCZ (Figure 3b) regimes of the dynamo. In Figure 3(b) thetoroidal field shows a poleward branch at high latitude which is absent in Figure 3(a), andalso a stronger radial polar field at the surface. The cause of these differences between thetwo regimes will become clear in Section 4.
3. Results
We have carried out a parameter-space study to investigate how the cycle duration andamplitude in our model depend on the speed of meridional circulation and on the diffusivityin the convection zone. In each run of the code the parameters are held constant in time,but they are varied between different runs. Specifically, we vary the parameter v , whichgives the maximum circulation speed, and also η , which affects the diffusive decay andtransport of the poloidal field in the convection zone, but not the toroidal field. In all runswe keep a surface diffusivity of η = 2 . × cm s − , and a toroidal field diffusivity of η = 0 . × cm s − in the convection zone. These choices approximate the fact thatturbulent diffusion is expected to be more efficient in the decay and dispersal of the weakerpoloidal field, but less so for the stronger toroidal field; the latter suppresses the convectivemotions that give rise to turbulent diffusivity in the first place. The α -effect coefficient isnot varied, but is set to α = 30 m s − for each run. This particular value was chosen to 9 –ensure that periodic solutions could be obtained for a wide range of the parameters v and η .Each run was started from an arbitrary initial state, and then evolved until initial transientshad disappeared, leaving a steady periodic dynamo solution. Such a periodic solution wasfound to exist only within a certain range of v for each value of η . The cycle durationand amplitude were then measured from the periodic solutions, in the cases where such asolution was found.We define the cycle duration and amplitude by considering the time evolution of thetoroidal magnetic flux Φ tor in a certain region around the base of the convection zone.Specifically, the toroidal field B ( r, θ ) is integrated at each time step over a region r = 0 . R ⊙ to 0 . R ⊙ , θ = 45 ◦ to 80 ◦ (i.e., over the tachocline and latitudes 45 ◦ to 10 ◦ ). This magneticflux Φ tor should be proportional to the active region magnetic flux at the solar surface,under the assumption that more toroidal flux at the base of the convection zone leads tomore buoyant eruptions. In a steady dynamo solution the flux Φ tor varies in strictly periodicmanner, with its maximum amplitude giving the “cycle amplitude”. We define the “cycleduration” to be the interval between successive peaks of | Φ tor | (half of the full dynamoperiod). This is therefore equivalent to the standard definition of the 11-year solar activitycycle, but of course the simulated periods are different.The resulting cycle duration and cycle amplitude are plotted as functions of the circu-lation speed v in Figures 4 and 5 respectively. In these figures each curve corresponds to adifferent value of the diffusivity η . The range of speeds covered by each curve indicates therange for which the code relaxed to a steady periodic dynamo solution, up to a maximumof v = 38 m s − . In Figure 6 the cycle amplitude is plotted as a function of η , and in thiscase each curve corresponds to a different circulation speed v . Figure 4 shows a clear inverse dependence of the cycle duration on the meridionalcirculation speed v , with faster circulation leading to shorter cycles. A least-squares fit forthe curve with η = 0 . × cm s − gives the dependence of the cycle period on meridionalflow speed T = 217 . v − . , (9)where T is in years and v is in metres per second. This agrees with the T ∼ v − . foundby Dikpati & Charbonneau (1999), and this inverse relation is a well-established result forBabcock-Leighton dynamo models. In these models the circulation, and specifically theequatorward counterflow at the bottom of the convection zone, is the primary determinantof the cycle period (Nandy 2004). 10 –The cycle duration is only weakly dependent on the diffusivity η . A power-law fit for thecurve with η = 2 . × cm s − gives T = 150 . v − . years. The lower power of v hereindicates that for higher-diffusivity solutions the cycle duration is slightly less dependent oncirculation speed, presumably because flux transport by diffusive dispersal starts becomingimportant. Also, it is evident from Figure 4 that, at lower circulation speeds, there is amaximum diffusivity for which a periodic solution can exist. If there is too much diffusionat a low circulation speed, then the poloidal field will decay too much during its transportfrom high to low latitudes, thus generating insufficient toroidal field to sustain a periodicdynamo process. The essential difference between advective and diffusive flux transport isthat the latter also reduces field strength during transport, due to diffusive decay. Now we turn to the dependence of cycle amplitude on the speed of meridional circulation.This is shown in Figure 5, where each curve corresponds to a different diffusivity η accordingto the legend. Rather than being monotonic, the cycle amplitude first increases with v forlow v and then decreases with v for large v , with a turnover at some value of v in between.The location of this turnover shifts to higher speeds as the diffusivity is increased.The dependence of cycle amplitude on diffusivity at any given circulation speed is notentirely clear on Figure 5, but is evident in Figure 6, where cycle amplitude is plotted againstdiffusivity η . In this figure each curve corresponds to a different value of the circulationspeed v . We see a similar behavior in that the cycle amplitude first increases with η forlow η and then decreases with η for high η , with a turnover in between. If the circulationspeed is increased, then the value of η corresponding to this turnover also increases.The behaviour of the cycle amplitude in our model, as illustrated in Figures 5 and6, is more complex than expected a priori . Rather than a simple linear dependence onthe circulation speed v , there is a turnover in cycle amplitude, at a speed which changesdepending on the diffusivity in the convection zone. In the next section we investigate thecause of this behaviour in the model.
4. Advection versus Diffusion Dominated Solar Convection Zones
The turnover of cycle amplitude as depicted in Figure 5 occurs at a higher circulationspeed v as the diffusivity η is increased. The asterisks (joined by a thin line) in Figure7(a) show the location of this turnover as a function of η . We may think of this line in 11 –the ( η , v ) plane as the dividing line between two distinct regimes of the dynamo, which wecall advection-dominated and diffusion-dominated . The advection-dominated regime corre-sponds to high circulation speed and low diffusivity, while the diffusion-dominated regimecorresponds to low circulation speed and high diffusivity. A shift from one of these regimesto another affects flux-transport dynamics in a way that results in contrasting dependence ofcycle amplitude on the governing parameters. Consider how the cycle amplitude varies with v for a fixed value of η , corresponding to a curve on Figure 5. In the diffusion-dominatedregime, a higher circulation speed means less time for diffusive decay of the poloidal fieldduring its transport through the convection zone, leading to more generation of toroidal fieldand hence a higher cycle amplitude. In the advection-dominated regime, a higher circulationspeed leads to a lower cycle amplitude because there is less time to amplify toroidal field inthe tachocline (through which magnetic fields are swept through at a faster speed). It is thebalance between these conflicting influences that leads to a turnover in cycle amplitude atsome intermediate circulation speed.The bold line in Figure 7(a) shows the transition point between the two regimes thatmay be inferred from a simple balance between circulation and diffusion timescales τ C and τ D . For a given circulation speed v , we define the circulation timescale τ C as the time takenfor meridional circulation to advect poloidal fields from r = 0 . R ⊙ , θ = 45 ◦ to the locationwhere the strongest toroidal field is formed at the tachocline ( θ = 60 ◦ ). The diffusiontimescale is defined as τ D = L /η , where L = 0 . R ⊙ is the radial distance across theconvection zone from the same starting point. The two timescales are compared in Figure7(b), where each horizontal line gives the circulation time τ C for a different speed v , andthe bold curve gives τ D as a function of η . The crossing points of horizontal lines with thiscurve give the transition points between the advection dominated ( τ C < τ D ) and diffusiondominated ( τ C > τ D ) regimes from these simple theoretical considerations – which are ingood agreement with the simulated transition points. We now compare the poloidal and toroidal field evolution in the two regimes. Figure 8shows the poloidal field lines for two runs, at different times through the cycle, starting fromone cycle minimum and finishing at the next cycle minimum, so that the fields reverse in sign.The left-hand column is taken from a run with v = 20 m s − and η = 0 . × cm s − ,which is in the advection-dominated regime. The right-hand column is from a diffusion-dominated run with the same v but with η = 2 . × cm s − . Figure 9 shows the 12 –evolution of the toroidal field for the same two runs.The key difference between the two regimes is the rate at which the poloidal field is ableto diffuse through the convection zone, after it is generated at the surface by the Babcock-Leighton α -effect. This is seen clearly by comparing the poloidal field evolution between 8and 12 years in the two runs (Figures 8c/h and d/i). In the diffusion-dominated run the newclockwise poloidal field diffuses directly down to the tachocline at all latitudes between thesetwo times. However, in the advection-dominated run the new poloidal field does not reachthe tachocline until the end of the cycle (16 years), and reaches the high latitudes before itreaches the tachocline; i.e., in this case, the field evolution follows the meridional circulationconveyor belt. There is still significant anticlockwise poloidal field remaining below thetachocline from the previous cycle, and even a lower band of clockwise field from the cyclebefore that. In the diffusion-dominated case there is only a weak band of anticlockwise fieldremaining from the previous cycle at solar minimum.This suggests that in the advection-dominated regime, poloidal fields from cycles n , n − n − n +1, while in the diffusion-dominatedregime it is produced primarily from cycle n poloidal field, with a small contribution fromcycle n − We have thus far identified the turnover in cycle amplitude to lie at the transition pointbetween advection-dominated and diffusion-dominated regimes of the dynamo. Within the 13 –umbrella of this model, this maximum in the amplitude is understood to be a balancebetween the time available for toroidal field amplification and the time available for poloidalfield decay. Figure 6 shows how the cycle amplitude varies with the diffusivity η for a fixedcirculation speed v . We see that there is a turnover in cycle amplitude for some valueof η , with lower diffusivities corresponding to the advection-dominated regime, and higherdiffusivities to the diffusion-dominated regime. In the diffusion-dominated regime, which isonly reached for lower speeds v in Figure 6, the cycle amplitude decreases with increasingdiffusivity. This is expected due to increased cancellation and decay of the poloidal field.However, in the advection-dominated regime, the cycle amplitude increases as the diffusivity η is increased. This initially seems counter-intuitive, but we show here that it is caused bythe influence of direct diffusive flux transport of poloidal field across the convection zone.This direct diffusion (from the solar surface to the base of the SCZ) was visible inthe poloidal field evolution plots shown in the previous section; we now demonstrate itsquantitative effect as η is varied, by comparing the poloidal field strength | B p | at the baseof the convection zone ( r = 0 . R ⊙ ) with that at the solar surface ( r = R ⊙ ). We takethe ratio | B p (base) | / | B p (top) | , using the peak value of | B p | at each location during thesolar cycle. This ratio is plotted in Figure 10, measured at latitudes 30 ◦ and 60 ◦ , andfor two different circulation speeds. Thin lines correspond to v = 30 m s − , where thedynamo is in the advection-dominated regime for the whole range of η shown. Thick linescorrespond to v = 20 m s − , for which the dynamo changes between the two regimes atabout η = 1 . × cm s − .Consider first the behaviour at 30 ◦ latitude (solid lines in Figure 10). Here the curvesfor both v have positive slope, which implies that a greater proportion of poloidal fieldfrom the surface reaches the bottom of the convection zone as η is increased. Thus directdiffusive flux transport at lower latitudes always acts to increase the amount of poloidal fieldreaching the base of the convection zone. Nearer to the pole, at 60 ◦ latitude (dashed linesin Figure 10), the behaviour is different. Here the ratio decreases as η is increased, bothfor the curve with v = 30 m s − and in the diffusion-dominated regime for v = 20 m s − .In the advection-dominated regime for v = 20 m s − however, the ratio first increases with η . This suggests a more complex relation between the surface and tachocline poloidal fieldsat high latitude, with competing influence from both diffusive and advective flux transport.This is expected because the downflow in the circulation is located at high latitudes.The analysis presented in this section supports the idea that direct diffusive transport ofpoloidal field across the convection zone, especially around mid-latitudes, is responsible forthe trend of increasing cycle amplitude with increasing diffusivity, found in the advection-dominated regime. Although such diffusive transport acts to increase cycle amplitude in 14 –both regimes, diffusion also causes the poloidal field that is being transported by meridionalcirculation to decay, cancelling with field from the previous cycle that is stored below thetachocline. Thus diffusion also has a negative effect on cycle amplitude. It is this negative ef-fect which dominates at higher diffusivities, forcing the dynamo into the diffusion-dominatedregime where cycle amplitude decreases with increasing diffusivity.
5. Persistence of Memory: Cycle-to-Cycle Correlations in Advection versusDiffusion Dominated Regimes in a Stochastically Forced Dynamo
It is expected that the memory of a flux-transport dynamo is much longer in theadvection-dominated regime than in the diffusion-dominated regime, and solar cycle pre-dictions have been based on this expectation (Dikpati & Gilman 2006; Jiang et al. 2007).However, a detailed comparative analysis of persistence of memory in these different regimesunder the umbrella of the same model had not been previously performed. Our analysisin the previous section has brought us closer to understanding the flux transport dynamics(in these two regimes) that is the physical basis for any memory mechanism. In this sec-tion, we consider how the persistence of this memory differs between the two regimes, bylooking at the correlation between peak polar and toroidal fields of subsequent cycles. Sincethe simulations considered earlier relaxed to a regular periodic cycle, we cannot use theseto study correlations between different cycles. Therefore, we now introduce self-consistentfluctuations in the cycle properties by means of a stochastically varying α -effect, and explorethe resulting correlations between different cycles. We introduce fluctuations in the model by varying the coefficient α of the α -effect (seeChatterjee et al. 2004 for the full expression). We set α = α base + α fluc σ ( t ; τ cor ) , (10)where α base = 30 m s − is the mean value, α fluc = 30 m s − gives the maximum amplitudeof the fluctuations (corresponding to the 200% level), and σ is a uniform random deviateselected from the interval [ − , τ cor . Althoughfor our purposes this is essentially a device for changing the cycle properties from one cycleto the next, there is a strong physical basis for stochastic variations in α which have beeninvoked in several previous studies (Choudhuri 1992; Hoyng 1993; Ossendrijver et al. 1996;Charbonneau & Dikpati 2000; Mininni & G´omez 2002). Our model uses a Babcock-Leighton 15 – α -effect where poloidal field is generated at the surface from the decay of tilted active regions(Babcock 1961; Leighton 1969). Thus stochastic variations in the α coefficient are natural,because it arises from the cumulative effect of a finite number of discrete flux emergenceevents (active region eruptions with varying degrees of tilt).To compare the two regimes we consider two runs, both with η = 1 . × cm s − .The circulation speed v is kept constant throughout each run, and only the α effect is varied.Run 1 has v = 15 m s − , so is diffusion-dominated, while run 2 has v = 26 m s − and isadvection-dominated. The coherence time τ cor is set to 2 . . α -effect is a result of surface flux transportprocesses (diffusion, meridional circulation and differential rotation) which can take up to ayear to generate a net radial (component of the poloidal) field from multiple flux emergenceevents (Mackay et al. 2004). In this section we compare the peak surface radial flux Φ r for cycle n with the peaktoroidal flux Φ tor for cycles n , n + 1, n + 2, and n + 3. The toroidal flux is defined as beforeby integrating B ( r, θ ) over the region r = 0 . R ⊙ to 0 . R ⊙ , θ = 45 ◦ to 80 ◦ . The radialflux Φ r is found by integrating B r ( R ⊙ , θ ) over the solar surface between θ = 1 ◦ to 20 ◦ , (i.e.,latitudes 70 ◦ to 89 ◦ ). Note that the peak toroidal flux precedes the peak surface radial fluxfor the same cycle, which has the same sign. The poloidal field then produces the toroidalfield for cycle n + 1 with the opposite sign. We measure the correlation of the surface radialflux for cycle n with the toroidal flux of different cycles, comparing the absolute value ofeach total signed flux.Both runs were computed for a total of 275 cycles with fluctuating α , so as to producemeaningful statistics for each of the dynamo regimes. The results are illustrated in Figures11 and 12 as scatter-plots of Φ tor for different cycles against Φ r ( n ). The (non-parametric)Spearman’s rank correlation coefficient r s is given above each plot, along with its significancelevel. The correlation coefficients are summarised in Table 1, where the Pearson’s linearcorrelation coefficient r p is also given for comparison. Although the latter is less reliable, asit assumes a linear relation, it agrees well with r s in each case.The results show a clear difference between the two regimes. The advection-dominated 16 –regime shows significant correlations at all 4 time delays, apparently suggesting that thememory of past poloidal field survives for at least 3 cycles; however, more on this later. Thediffusion-dominated regime has a strong correlation only between Φ r ( n ) and Φ tor ( n + 1),suggesting that the dominant memory relates to just a one cycle time-lag, although veryweak correlations are also found with Φ tor ( n ) and Φ tor ( n + 3).In both regimes the strongest relation is the positive correlation between Φ r ( n ) andΦ tor ( n + 1). This is to be expected as this is the more deterministic phase of the cycle– where toroidal fields (of cycle n + 1, say) are inducted from the older cycle n poloidalfield via the relatively steady differential rotation. Note however that the two fluxes do nothave to be directly coupled, in that the two fluxes may be positively correlated becausethey are both created from the mid-latitude poloidal field of cycle n (generated by the α -effect). The polar flux Φ r ( n ) arises through poleward meridional transport of the cycle n poloidal field, while the toroidal flux Φ tor ( n + 1) is generated from cycle n poloidal fieldthat is diffusively transported across the convection zone. This is particularly the case inthe diffusion-dominated regime. Nonetheless, even this indirect scenario suggests that thestrongest correlation should in fact be between the cycle n poloidal field and cycle n + 1toroidal field in this class of α -Ω dynamo models.The other phase of the cycle, in which the poloidal field is generated by the α -effect, isinherently more random due to the fluctuating α -effect in these runs. Nevertheless, there isa strong positive correlation between Φ tor ( n ) and Φ r ( n ) in the advection-dominated regime,while this correlation is largely absent in the diffusion-dominated regime. This we attributeto the relatively stronger role of advective flux transport in the advection-dominated regime– which implies that a larger fraction of the original toroidal flux that has buoyantly eruptedis transported to the polar regions by the circulation. In effect therefore, the advection-dominated regime allows correlations to propagate in both phases of the cycle , whereas thediffusion-dominated case allows correlations to propagate only in the poloidal-to-toroidalphase. The other correlation is broken in the diffusion-dominated regime because the advec-tion is short-circuited by direct diffusion, which transports flux downwards and equatorward– where it is cancelled by oppositely signed flux from the other hemisphere. This explainshow the correlations can survive for multiple cycles in the advection-dominated regime, butnot in the diffusion-dominated regime.
6. Conclusion
Significant uncertainties remain in our understanding of the physics of the solar dynamomechanism, implying that prediction of future solar activity based on physical models is a 17 –challenging task. Here we have demonstrated how a flux-transport dynamo model behavesdifferently in advection and diffusion dominated regimes. Such differences, amongst others,have previously led to conflicting predictions of the amplitude of Cycle 24. Dikpati et al.(2006) use an advection-dominated model to predict a much stronger cycle than Cycle 23,whereas Choudhuri et al. (2007) use a diffusion-dominated model to predict a much weakercycle. The latter prediction is somewhat similar in spirit to the precursor methods (Schatten2005; Svalgaard et al. 2005), which use the polar field at cycle minimum to predict theamplitude of the following cycle. Owing to the lack of observations of conditions inside theconvection zone, opinions differ as to whether the real solar dynamo is weakly or stronglydiffusive (e.g. Dikpati & Gilman 2006; Jiang et al. 2007).We find that for low circulation speeds v (in the diffusion-dominated regime), the cycleamplitude is an increasing function of v , as in the observations of Hathaway et al. (2003).However, the amplitude curve has a turnover point and is a decreasing function of v athigher v (in the advection-dominated regime), opposite to the observed correlation. Whenthe diffusivity in the convection zone is increased, the location of this turnover moves toa higher v . Our extensive analysis shows that this turnover corresponds to the transitionbetween the diffusion and advection dominated regimes. In the diffusion-dominated regime,faster circulation means less time for decay of the poloidal field, leading to a higher cycleamplitude, whereas in the advection-dominated regime diffusive decay is less important anda faster circulation means less time to induct toroidal field, thus generating a lower cycleamplitude. If the observed statistics of the past 12 cycles as reported by Hathaway et al.(2003) reflect a true underlying trend, then our results imply that the solar dynamo is infact working in a regime which is dominated by diffusive flux transport in the main bodyof the SCZ (although the cycle period is still governed by the slow meridional circulationcounterflow at the base of the SCZ). This conclusion supports the analysis of Jiang et al.(2007).Through a correlation analysis in a stochastically forced version of our model, we havealso explored the persistence of memory in the solar cycle for both the diffusion-dominatedand advection-dominated regimes. It is this memory mechanism which is understood to leadto predictive capabilities in α -Ω dynamo models with spatially segregated source regionsfor the α and Ω effects. This understanding is based on the finite time delay required forflux transport to communicate between these different source regions. We find that thepolar field of cycle n correlates strongest with the amplitude (toroidal flux) of cycle n + 1 inboth the regimes. In the diffusion-dominated regime this is the only significant correlation,indicative of a one-cycle memory only. However, in the advection-dominated case, there arealso significant correlations with the amplitude of cycles n , n + 2, and n + 3. In contrastto the correlations that we infer, Charbonneau & Dikpati (2000) found that the strongest 18 –correlation in their advection dominated model was with a two-cycle time lag. Since suchcorrelations lead to predictive capabilities, and obviously seem to be model and parameter-dependent as suggested by our results, such a correlation analysis should be the first steptowards any prediction, the latter being based on the former. In hindsight, however, bothDikpati et al. (2006) – who use an advection-dominated model and inputs from multipleprevious cycles, and Choudhuri et al. (2007) – who use a diffusion-dominated model andinput from only the past cycle to predict the next cycle, appear to be have made the correctchoices within their modelling assumptions.Note that the memory mechanism in our advection-dominated case appears to have adifferent cause than that implied by Dikpati & Gilman (2006), who invoke the survival ofmultiple old-cycle polar fields feeding into a new cycle toroidal field. All of the survivingcorrelations in our advection-dominated regime (Figure 12) are positive; they do not alternatein sign. This alternation in sign would be expected if bands of multiple older cycle poloidalfield survive in the tachocline – odd and even cycle poloidal fields would obviously contributeoppositely because of their alternating signs. In that case we would expect the absolute valueof Φ r ( n ) to correlate positively with Φ tor ( n +1) and Φ tor ( n +3) and so on, but negatively withΦ tor ( n + 2) and Φ tor ( n + 4) and so on, as evident in the results of Charbonneau & Dikpati(2000, Figure 9; after accounting for the fact that they use signed magnetic fields). Rather,in the advection-dominated regime of our model, the correlations appear to persist simplybecause fluctuations in field strength are passed on in both the poloidal-to-toroidal andtoroidal-to-poloidal phases of the cycle, as evidenced by the correlation between amplitudeand polar flux of cycle n . In a recent analysis, Cameron & Sch¨ussler (2007) find that thepredictive skill of a surface flux transport model – similar in spirit to the advection-dominateddynamo model of Dikpati et al. (2006) – is contained in the input information of sunspotareas in the declining phase of the cycle. They argue that memory of multiple past cycles, inthe form of surviving bands of poloidal field (its surface manifestations in their case), neednot be the only reason behind the predictive capability of the advection-dominated dynamomodel of Dikpati et al. (2006). Our analysis of the advection-dominated regime supportsthis suggestion of Cameron & Sch¨ussler (2007).Coming back to the diffusion-dominated regime, our comparative analysis indicatesthat in this case, the memory of past cycles is governed by downward diffusion of poloidalfield into the tachocline – which primarily results in a one-cycle memory. The fact thatdiffusion is an efficient means for transporting flux is often ignored, especially in this era ofadvection-dominated models; however, we find that diffusive flux transport is quite efficient.The identification of this one cycle memory in our stochastically forced model contradictsDikpati & Gilman (2006) – who claim that prediction is not possible in this regime. As longas the source regions are spatially segregated, and one of the source effects is observable and 19 –the other deterministic, flux transport α -Ω dynamos will inherently have predictive skills nomatter what physical process (i.e., circulation, or diffusion, or downward flux-pumping) isinvoked to couple the two regions. We may also point out that in the context of cycle-to-cycle correlations, downward flux pumping (Tobias et al. 2001) would have the same effectas diffusion in that it also acts to short-circuit the meridional circulation conveyor belt. Soalthough downward flux pumping differs from diffusive transport because in the latter casethe fields may reduce in strength due to decay, the overall persistence of memory is expectedto be similar if diffusive flux transport was replaced or complemented by downward fluxpumping.In summary, our analysis has served both to explore the diffusion dominated and ad-vection dominated regimes within the framework of a BL type dynamo, and to demonstratehow the memory of the dynamo may be different in these two regimes. Based on our analysiswe assert that diffusive flux transport in the SCZ plays an important role in flux transportdynamics, even if the dynamo cycle period is governed by the meridional flow speed. Infact, the observed solar cycle amplitude-period dependence may arise more naturally in thediffusion-dominated regime, as discussed earlier. Taken together therefore, we may concludethat diffusive flux transport is a significant physical process in the dynamo mechanism andthis process leads primarily to a one-cycle memory which may form the physical basis forsolar cycle predictions, if other physical mechanisms involved in the complete dynamo chainof events are well understood. Separate, detailed examinations of these other related physicalmechanisms will be performed in the future.This research was funded by NASA Living With a Star grant NNG05GE47G. We alsoacknowledge support from the UK STFC and the Solar Physics NSF-REU program at Mon-tana State University. 20 – REFERENCES
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23 – v θ ( m s − ) Fig. 1.— Streamlines of the meridional circulation profile in our model ( left ), and latitudinalvelocity profile across a radial cut at θ = 45 ◦ ( right ). Dots on the streamlines show yearlypositions for particles with v = 25 m s − , starting from the squares at θ = 45 ◦ and movinganti-clockwise. 24 – D i ff u s i v it y ( c m s − ) η p η t Fig. 2.— Example diffusion profiles as a function of r for the toroidal ( η t ) and poloidal ( η p )fields. The dotted line shows the location of the tachocline. Here η = 2 . × cm s − , η = 0 . × cm s − , and η = 1 . × cm s − . 25 – L a tit ud e ( d e g ) (a) 0 20 40 60Time (years)020406080 L a tit ud e ( d e g ) (b) Fig. 3.— Two example solutions: (a) with v = 20 m s − and η = 2 . × cm s − (characterizing a diffusive flux-transport dominated SCZ); and (b) with the same v but η = 0 . × cm s − (characterizing an advective flux-transport dominated SCZ). In eachcase black lines are contours of toroidal field B at the base of the convection zone (solid linesfor positive values, dashed for negative). The grayscale in the background shows surfaceradial field strength B r ( r = R ⊙ ), with white for positive and black for negative. The samecontour levels are used in both plots. 26 –
10 15 20 25 30 35Peak Flow Speed v0 (ms -1 )1015202530 C y c l e D u r a ti on ( y ea r s ) η cm s -1 ) : 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Fig. 4.— Dependence of cycle duration on the meridional circulation speed v . Each linestyle corresponds to a different value of η (the poloidal diffusivity in the convection zone)as given in the legend. 27 –
10 15 20 25 30 35Peak Flow Speed v0 (ms -1 )0.60.70.80.91.0 C y c l e A m p lit ud e ( M x ) η cm s -1 ) :0.500.751.001.251.501.752.00 Fig. 5.— Dependence of cycle amplitude on the meridional circulation speed v . Each linestyle corresponds to a different value of η (the poloidal diffusivity in the convection zone)as given in the legend. 28 – η cm s -1 )0.60.70.80.91.0 C y c l e A m p lit ud e ( M x ) v0 (ms -1 ) :1520253035 Fig. 6.— Dependence of cycle amplitude on the poloidal diffusivity η in the convectionzone. Each line style corresponds to a different value of the meridional circulation speed v ,as given in the legend. 29 – η cm s -1 )1520253035 P ea k F l o w S p ee d v0 ( m s - ) ADVECTIONDOMINATEDDIFFUSIONDOMINATED (a) η cm s -1 )510152025 T i m e s ca l e ( y ea r s ) τ D v0 (ms -1 ) :1520253035 (b) Fig. 7.— Transition between the advection-dominated and diffusion-dominated regimes. In(a) asterisks indicate the flow speeds v corresponding to turnover of cycle amplitude for fixedvalues of η (inferred from the simulations shown in Figures 5 and 6). The bold line showsthe transition point that may be inferred from simple theoretical comparison of circulationand diffusion timescales. Panel (b) shows the diffusion timescale τ D as a function of η (boldline), and circulation timescales τ C for selected speeds v (horizontal lines), as defined in thetext. 30 – (a) (f)0 years(b) (g)4 years(c) (h)8 years(d) (i)12 years(e) (j)16 years Fig. 8.— Comparison of poloidal fields in advection-dominated (left column/panels a to e)and diffusion-dominated (right column/panels f to j) regimes. Each row corresponds to atime through the solar cycle, running from one cycle minimum to the next. Solid lines showclockwise field lines and dashed lines show anti-clockwise field lines. Also indicated is thebase of the SCZ at 0 . R ⊙ . 31 – (a) (f)0 years(b) (g)4 years(c) (h)8 years(d) (i)12 years(e) (j)16 years Fig. 9.— Comparison of toroidal field in advection-dominated (left column/panels a toe) and diffusion-dominated (right column/panels f to j) regimes. Each row correspondsto a different time through the solar cycle, running from one cycle minimum to the next.Grayscale contours show toroidal field strength, with black corresponding to the strongestnegative field and white to the strongest positive toroidal field. Also indicated is the base ofthe SCZ at 0 . R ⊙ . 32 – η cm s -1 )012345 B p ( b a s e ) / B p ( t op ) v0 = 30 ms -1 ° lat.v0 = 30 ms -1 ° lat. v0 = 20 ms -1 ° lat.v0 = 20 ms -1 ° lat. Fig. 10.— Ratio of poloidal field | B p | at base of convection zone ( r = 0 . R ⊙ ) to that atthe surface ( r = R ⊙ ), measured as a function of diffusivity at latitudes 30 ◦ (solid lines) and60 ◦ (dashed lines). Thick lines correspond to runs with v = 20 m s − and thin lines to runswith v = 30 m s − . 33 – rs = 0.185 99.8% Φ r(n)0.550.600.650.700.750.800.85 Φ t o r( n ) rs = 0.737 100.0% Φ r(n)0.550.600.650.700.750.800.85 Φ t o r( n + ) rs = -0.040 49.1% Φ r(n)0.550.600.650.700.750.800.85 Φ t o r( n + ) rs = 0.195 99.9% Φ r(n)0.550.600.650.700.750.800.85 Φ t o r( n + ) (a) (b)(c) (d) Fig. 11.— Cycle-to-cycle correlations in the diffusion-dominated regime (run 1), betweenradial flux Φ r ( n ) and (a) toroidal flux Φ tor ( n ), (b) Φ tor ( n + 1), (c) Φ tor ( n + 2), and (d)Φ tor ( n + 3). The Spearman’s rank correlation coefficient is given along with its significancelevel for 275 cycles. All magnetic fluxes are in units of 10 Mx. 34 – rs = 0.653 100.0% Φ r(n)0.20.40.60.8 Φ t o r( n ) rs = 0.805 100.0% Φ r(n)0.20.40.60.8 Φ t o r( n + ) rs = 0.356 100.0% Φ r(n)0.20.40.60.8 Φ t o r( n + ) rs = 0.237 100.0% Φ r(n)0.20.40.60.8 Φ t o r( n + ) (a) (b)(c) (d) Fig. 12.— Cycle-to-cycle correlations in the advection-dominated regime (run 2), betweenradial flux Φ r ( n ) and (a) toroidal flux Φ tor ( n ), (b) Φ tor ( n + 1), (c) Φ tor ( n + 2), and (d)Φ tor ( n + 3). The Spearman’s rank correlation coefficient is given along with its significancelevel for 275 cycles. All magnetic fluxes are in units of 10 Mx. 35 –Table 1. Cycle-to-cycle correlationsΦ r ( n ) for run 1 Φ r ( n ) for run 2(diffusion-dominated) (advection-dominated) r s r p r s r p Φ tor ( n ) 0.185 99 . . Φ tor ( n + 1) 0.737 100 . . Φ tor ( n + 2) -0.040 49 . . Φ tor ( n + 3) 0.195 99 . . Φ tor ( n + 4) 0.036 44 . . Φ tor ( n + 5) 0.107 92 . . Note. — Correlation coefficients and significance levels for peak surfaceradial flux Φ r versus peak toroidal flux Φ tortor