Polar Field Puzzle: Solutions from Flux-Transport Dynamo and Surface Transport Models
aa r X i v : . [ a s t r o - ph . S R ] A p r POLAR FIELD PUZZLE: SOLUTIONS FROMFLUX-TRANSPORT DYNAMO AND SURFACE TRANSPORTMODELS
MAUSUMI DIKPATI
High Altitude Observatory, National Center for Atmospheric Research , 3080 CenterGreen, Boulder, Colorado 80301; [email protected] ABSTRACT
Polar fields in solar cycle 23 were about 50% weaker than those in cycle 22.The only theoretical models which have addressed this puzzle are surface trans-port models and flux-transport dynamo models. Comparing polar fields obtainedfrom numerical simulations using surface flux transport models and flux-transportdynamo models, we show that both classes of models can explain the polar fieldfeatures within the scope of the physics included in the respective models. Inboth models, how polar fields change as a result of changes in meridional circula-tion depends on the details of meridional circulation profile used. Using physicalreasoning and schematics as well as numerical solutions from a flux-transportdynamo model, we demonstrate that polar fields are determined mostly by thestrength of surface poloidal source provided by the decay of tilted, bipolar ac-tive regions. Profile of meridional flow with latitude and its changes with timehave much less effect in flux-transport dynamo models than in surface transportmodels.
Subject headings:
Sun: magnetic field, Sun: activity, Sun: dynamo
1. Introduction
Observations from various instruments indicate that the polar fields in cycle 23 wereso weak (see figure 1 of Wang, Robbrecht & Sheeley (2009), see also figure 2 of Arge et al(2002) and figure 2 of Janardhan, Bisoi & Gosain (2010)) that it took a relatively longtime to reverse the magnetic polarity of the Sun’s North and South poles. Even after the The National Center for Atmospheric Research is sponsored by the National Science Foundation. r − θ plane) and they evolve due to advection and diffusion by the actionof both the latitudinal and radial components of the meridional flow as well as by the actionof a depth-dependent turbulent diffusivity. By contrast, surface transport models includea more realistic longitude dependence of the Babcock-Leighton effect in the generation ofpolar fields, and the radial component of the fields generated evolve at the surface by theaction of latitudinal component of the meridional flow and a constant surface diffusivity.The purpose of this paper is to demonstrate that flux-transport dynamo models andsurface transport models, despite some differences in the ingredients, produce remarkablysimilar response in the polar fields’ patterns to the changes in the meridional flow-speed whenthe same latitudinal profile for the poleward surface flow is used. Given that understandingof the polar fields’ behaviour is a key to understanding the recycling of magnetic flux for theoperation of a dynamo, and hence the properties of future solar cycles, we also examine inthis paper whether there are any inconsistencies or contradictions between these two classesof models when applied to the Sun. The polar field puzzle of cycle 23 has become a far moreimportant issue now that the start of cycle 24 has been sluggish.We illustrate in simple numerical terms just how sensitive the amplitude of the polarfield on the Sun and in models is to changes in the strength of the source of its field, namelythe emergence and decay of active regions. We know that the amplitude of cycle 23 was20% weaker than that of cycle 22. If at the end of a given cycle the polar field has oneunit, in the next cycle it takes two units of polar fields coming from the surface poloidalsource to reverse the remnant polar field and have the new polar fields reach minus one unit.But if the surface poloidal source is 20% smaller in the new cycle than the previous one, ascycle 23 was compared to cycle 22, there are only 1.6 units of new, negative, polar fields 3 –available that can be used to reverse the old, positive, polar field and establish the new,negative, polar field. This means the new, negative, polar field will be only 0.6 units, or 40%smaller in amplitude than that of the previous cycle. The percentage decline in polar fieldis, therefore, roughly twice the decline in surface source. The observed decline in polar fieldbetween cycles 22 and 23 was about 50%. Thus at the outset most of this change can beexplained by the change in surface source, whereas other effects are needed to account forthe additional drop of 10%.For both the surface transport and flux-transport dynamo models polar fields are thefollower of a cycle by virtue of their origin, namely the decay of tilted, bipolar active regions,hence in a crude analysis as described above, the 40% reduction in polar fields at the end ofcycle 23 might be true for all such models. However, a potential weakness of the reasoningjust given above is the assumption of a close relation between the strength of a cycle, as mea-sured by the Wolf sunspot number or sunspot area, and the strength of the surface poloidalsource arising from the decay of active regions. That is undoubtedly an oversimplification. Itis well known that there is a strong correlation between Wolf number and sunspot area; thelatter is what has usually been used to drive flux-transport dynamo models and to comparewith model output. As seen for example in figure 1 of Dikpati, de Toma & Gilman (2006)there is a good correlation between sunspot area and average surface magnetic flux, providedboth data sets are averaged over at least six solar rotations. So presuming a good correlationbetween Wolf number and the surface source could be plausible, but recent surface transportmodels (Wang, Robbrecht & Sheeley 2009) do a better job of estimating the surface sourcefrom more detailed distributions of active region sources.
2. Role of meridional circulation in polar fields’ evolution
The structure and strength of the meridional circulation influence the strength of theSun’s magnetic fields to some extent, but meridional circulation cannot be the most impor-tant factor determining the field strength in flux-transport dynamo models. In flux-transportdynamos (and most other dynamos applied to the sun) the spot-producing toroidal fields aregenerated by the Sun’s differential rotation and the poloidal fields are produced by the ac-tion of the so-called α -effect. The α -effect is modeled in different dynamo models in differentways. In Babcock-Leighton type flux-transport dynamo models, such an α -effect arises fromthe decay of tilted, bipolar active regions that emerge to the surface from below. Thus thetwo sources of magnetic fields, the differential rotation and a combination of α -effects arisingfrom helical turbulence as well as from the decay of active regions, are primarily responsiblefor determining the amplitudes of toroidal and poloidal fields, and hence the polar fields. 4 –The primary role of meridional circulation in flux-transport dynamo models is the advec-tive transport of magnetic fields, and hence the structure and strength of the flow are crucialin the dynamo models for determining the timings, namely the duration of a cycle, its rise andfall pattern and the timing of the reversals of the Sun’s polar fields (Dikpati & Charbonneau1999; Dikpati 2004; Dikpati et al 2010). A transport process like meridional circulation ina flux-transport dynamo redistributes the dynamo-generated magnetic flux, a relatively mi-nor effect in creating an increase or decrease of magnetic flux. This is illustrated in Table1 of Dikpati & Charbonneau (1999) in which it is seen that, if the meridional circulationspeed is doubled, the peak polar field changes by only 3%, whereas for the same meridionalcirculation, increasing the surface poloidal source by a factor 2.5 doubles the peak polar field,and decreasing the surface source by 75% decreases the polar field peak by 89%.The question we address is how flux-transport dynamo models and surface transportmodels respond to the changes in the surface poleward flow speed. Detailed calculations showthat a surface transport model produces a weaker polar field (Wang, Lean & Sheeley 2005;Schrijver & Liu 2008), while a flux transport dynamo gives a stronger polar field when thepoleward surface flow-speed is increased (Dikpati & Charbonneau 1999; Dikpati, de Toma & Gilman2008). Are these opposite results in conflict? We address, first with qualitative reasoningand schematics and then in the next section with numerical solution from a flux-transportdynamo model, whether this is a true conflict, and if so, what is the physical origin of thisconflict.Figure 1 illustrates three scenarios: (a) and (b) for surface flux transport models and(c) for a flux transport dynamo. The primary difference between figures 1(a) and (b) is thelatitude where the surface poleward flow is maximum. Figure 1(a) illustrates what happensto surface flux in the declining phase of a cycle when the meridional flow peaks at 6 ◦ asin the model by Wang, Lean & Sheeley (2005); Figure 1(b) illustrates what happens for ameridional flow that peaks at 37 . ◦ (Schrijver, De Rosa & Title 2002). Figure 1(c) showswhat happens to polar fields when a meridional circulation profile in the pole-to-equatormeridional plane peaks at mid-latitudes ∼ ◦ , as has been used in flux-transport dynamomodels by Dikpati & Charbonneau (1999) and Dikpati, de Toma & Gilman (2008). 5 –Fig. 1.— Surface radial field transport mechanisms respectively in a surface flux-transportmodel with a poleward flow peaking at 6 ◦ (frame a) as in the model of Wang et al. (2005),the surface transport model with a poleward flow peaking at 37 . ◦ (frame b) as in the modelof Schrijver et al. (2002), and in a flux-transport dynamo model with a flow peaking atmid-latitudes (frame c) as in the model of Dikpati et al. (2008). In frames (a) and (b), redand blue patches on the surface represent bipolar active regions. Red and blue continuouslines represent radial fields from a bipole, and dashed lines represent the new locations ofradial fields after the meridional flow is increased. For example, by doubling flow speed, alarger increase in transport-rate is obtained respectively on equatorward and poleward sideof the bipoles in frames (a) and (b); thus fields from leader polarity drift closer to fieldsfrom follower polarity in frame (a) and further apart in frame (b). Frame (c) describes avery similar situation as in frame (b), but in terms of poloidal fields in r − θ plane. Solidcontours represent poloidal fields produced from bipolar active regions and dashed contoursrepresent that for an increased flow. Polarity division line of large-scale poloidal fields froma flux-transport dynamo model is shown by a dark line. 6 –We can understand from Figure 1(a) that, when the flow is maximum at low (6 ◦ )latitudes, an increase in poleward flow speed leads to a larger rate of transport on theequatorward side of the leader polarity, and hence a faster poleward transport of the leaderpolarity compared to the follower polarity. Thus the leader polarity gets closer to the followerpolarity to create enhanced annihilation between them. This reduces the fields coming fromthe follower polarity that reach the pole and thus produce weaker polar fields. This iswhat has been obtained by Wang, Lean & Sheeley (2005) and Schrijver & Liu (2008) insimulations using surface transport models. On the other hand, when the flow speed ismaximum at mid-latitudes (Figure 1(b)), an increase in flow speed leads to a larger rate oftransport on the poleward side, causing the follower and the leader polarities to separatemore from each other. Thus there is less annihilation between them. Consequently the polarfield should increase. Schrijver, De Rosa & Title (2002) explained, using a flow peaking at37 . ◦ , that the poleward meridional flow is so effective in maintaining the high latitude fieldthat this flow would have to practically disappear to get a significant decay of the polar flux.We can thus infer that, quite consistently a decrease in flow leads to a decrease in polar flux,and increase in flow to an increase in polar flux when the flow peaks at 37 . ◦ latitude.It follows from the reasoning above that the increase in polar field amplitude with theincrease of meridional flow speed in flux-transport dynamo models Dikpati & Charbonneau(1999) and Dikpati, de Toma & Gilman (2008) and surface transport model of Schrijver, De Rosa & Title(2002), and the decrease in polar field with the increase in meridional flow-speed in the sur-face transport models of Wang, Lean & Sheeley (2005) and Schrijver & Liu (2008), are notin conflict. Rather, it is the choice of meridional flow profile – peaking at low latitudes orat mid-to-high latitudes – that leads to the opposite results which are physically consistent.We recognize that cross-equatorial flux cancellation may also play a role in producing theresults of Schrijver & Liu (2008).There now exist detailed measurements of meridional flow speed by several methods forall of cycle 23, (Ulrich 2010; Basu & Antia 2010; Hathaway & Rightmire 2010) so detailedcomparisons can be made. This has been done most extensively in Ulrich (2010). Allmeasurements show significant variations from year to year within cycle 23, some of whichappear to correlate with variations in rotation, in particular the torsional oscillations, as seenin the helioseismic results of Basu & Antia (2010). In terms of meridional flow averaged overcycle 23 (see figure 1 of Dikpati, Gilman & Ulrich (2010) and figure 10 of Ulrich (2010)) it isclear that surface Doppler (Ulrich 2010) and helioseismic (Basu & Antia 2010) results agreeclosely with each other, while the magnetic feature tracking results (Hathaway & Rightmire2010) are distinctly different. In particular, surface Doppler and helioseismic measures havean average peak at 25 ◦ and the peak shifting with time between 15 and 40 degrees, whilemagnetic feature tracking flow speed peaks closer to 50 degrees and has a distinctly lower 7 –peak value.Early surface transport models (Devore, Boris & Sheeley 1984; Wang, Nash & Sheeley1989) were the first to use a latitudinal meridional flow profile peaking at low latitudes, inparticular, 6 degrees, even before observations could tell us where the flow peaks. Such a low-latitude peak gave the best fit of model-derived surface features with observations of surfacemagnetic fields. In future the much more extensive Doppler based observations of meridionalflow can be used as input to both surface transport models and flux-transport dynamomodels, capturing the low latitude peak now observed. By contrast, the magnetic feature-tracking speed should be compared with the output of such models in order to estimate thesurface magnetic diffusivity.There is a tracking speed that might also be useful as input to dynamo and surface trans-port models. That is the tracking speed that comes from the Doppler signal of supergranules(ˇSvanda et al 2008). It has been used to measure differential rotation and meridional cir-culation for solar cycle 23. It will be useful to compare the details of flow profiles obtainedby this method with those obtained by more global surface Doppler as well as helioseismicmeasures.
3. Model calculations3.1. Polar field simulations from a self-saturated flux-transport dynamodriven by Babcock-Leighton α -effect only In most flux-transport dynamo models, starting from Dikpati & Charbonneau (1999),the poleward surface flow has generally been taken to be a maximum in midlatitude. Soaccording to the qualitative reasoning given in Figure 1, the fields from the follower polarityseparate out from the leader polarity at a faster rate (see Figure 1(c)) when the flow speed isincreased. Thus there is less annihilation among them; consequently the polar field increases.To test this with flux-transport dynamo simulations, we must take account of the fact thatthere are additional physical processes at work in the model that will change the poloidaland toroidal fields in other parts of the dynamo domain in response to a change in meridionalflow. For example, if the flow toward the poles at the top speeds up, the return flow near thebottom will also speed up. This bottom flow then moves the poloidal field there faster towardthe equator, leaving less time to induce toroidal field at a particular latitude. This reductionin bottom toroidal field in turn leads to a reduced surface poloidal source. Which wins indetermining whether the polar fields increase or decrease? In Figure 2a we show resultsfrom new simulations with our flux-transport dynamo model using ingredients very similar 8 –to those used in Dikpati & Charbonneau (1999) for a pure Babcock-Leighton dynamo (nobottom α -effect) that answers this question. Figure 2a displays measures of the surfacepolar field (blue diamonds) and the tachocline toroidal field (black triangles) for a sequenceof independent simulations of the self-excited dynamo for different peak meridional flowspeeds (all peaking at the same midlatitude). Each simulation is run for about 5 cyclesto reach a nonlinear equilibrium in which each successive cycle has the same period andamplitude.Fig. 2.— (a) Polar fields at ∼ ◦ (left hand scale, blue diamonds and curve) and tachoclinetoroidal fields (right hand scale, black triangles and curve) for self-excited Babcock-Leightonflux transport dynamo solutions with different peak flow speeds (horizontal scale). Flowprofile used is very similar to that taken in Dikpati & Charbonneau (1999), which peaks atmid-latitudes. Green curve is polar fields normalized to the same tachocline toroidal fieldfor all cases, illustrating polar fields obtained for the same tachocline toroidal fields. (b)Polar fields at ∼ ◦ and sub-surface tachocline toroidal fields as a function of quenchingfield strength.Since the dynamo-generated field strengths depend on the choice of quenching fieldstrength also, we plot in Figure 2b the polar and subsurface toroidal field amplitudes asfunction of the quenching field strength in the α -effect. We find a nearly linear relationshipbetween the simulated magnetic field amplitudes and quenching parameter. It is not yetpossible to know from observations what should be the actual quenching field strength,but to obtain an optimum results of polar as well as subsurface toroidal field values from aBabcock-Leighton flux-transport dynamo, it is conceivable that the quenching field strengthsrange within 10-30 kGauss. Thus Figure 2(b) implies that the polar and subsurface toroidalfields also be scaled accordingly when a different quenching field strength is used.We see from Figure 2a that for the example we have chosen, whether the polar fieldis larger or smaller (blue diamonds and curve) for a larger meridional flow depends onthe amplitude of the flow. This effect was also seen in Baumann et al (2004) who, using a 9 –surface transport model, reported non-monotonic behavior of the polar fields as a function offlow-speed. This confirms that surface transport models and flux-transport dynamo models,despite operating with slightly different ingredients, produce similar basic surface radial fieldfeatures. For flux-transport dynamos, the amplitude of polar fields is affected greatly by thedecrease in induced toroidal field at the bottom (black triangles and curve), because thisreduces the surface poloidal source from which the polar fields are produced. If we normalizeout this effect by adjusting the bottom toroidal fields to the same value for all meridionalflow speeds, so the surface poloidal source is independent of meridional flow speed, then thepolar field would be given by the green curve. Thus for the same surface source, the polarfield does increase markedly with an increase in meridional flow speed for the choice of flowprofile that peaks at mid-latitudes, consistent with the schematic description provided inFigure 1.One important issue about a flux-transport dynamo driven by a Babcock-Leighton α -effect is that it cannot really reproduce a best match of polar field amplitudes with obser-vations. This fact has been noted by many Babcock-Leighton dynamo modelers since early1990’s (Durney 1995, 1996, 1997; Dikpati & Charbonneau 1999). In some attempts forreproducing a correct polar field value using such models, the dynamo has actually diedafter a few cycles, because a recycled polar flux of only ten Gauss is not enough to maintainthe dynamo, for example see the discussion of Table IV of Dikpati et al (2002). That con-sideration led to the development of a calibrated flux-transport dynamo driven by a smallamount of the tachocline α -effect in addition to a Babcock-Leighton α -effect, results fromwhich we discuss next. We carry out further numerical simulations making a sudden change in meridional flowin the model and finding the change in the polar field in the first one or two cycles that follow.This is done for the case of a self-excited flux-transport dynamo that achieves saturation onlyby ’alpha-quenching-like’ amplitude-limiting nonlinearity, run with a steady meridional flowand with the fixed dynamo source terms. In this case, for a given set of input parameters,the model produces all cycles of the same amplitude.Starting from such a saturated dynamo solution, we do numerical experiments for peakmeridional circulation amplitudes ranging from 6 −
32 ms − for two distinct circulation pro-files, one peaking at 25 ◦ as used in Dikpati et al (2010a), and the other peaking at 50 ◦ .Following expression (1) of Dikpati et al. (2010a), we prescribe a meridional flow profile as: 10 – ψr sin θ = ψ ( θ − θ ) sin (cid:20) π ( r − R b )( R − R b ) (cid:21) (cid:0) − e − β rθ ǫ (cid:1) (cid:0) − e β r ( θ − π/ (cid:1) e − (( r − r ) / Γ) , (1)in which, R b = 0 . R , β = 0 . / (1 . × ) cm − , β = 0 . / (1 . × ) cm − , ǫ =2 . r = ( R − R b ) /
5, Γ = 3 × . × cm and θ = 0. This choice of theset of parameter values produce a flow pattern that peaks at 24 ◦ as shown in the bluecurve in Figure 3(a). The dimensionless length in our calculation is 1 . × cm andthe dimensionless time is 1 . × s, which respectively come from taking the dynamowavenumber, k = 9 . × − cm − , as one dimensionless length, and the dynamo frequency, ν = 9 . × − s − , as one dimensionless time. In other words, the dynamo wavelength(2 π × . × cm) is 2 π and the mean dynamo cycle period (22 years) is 2 π in our di-mensionless units. Thus, in nondimensional units, the above parameters are: R b = 4 . β = 0 . β = 0 . ǫ = 2 . r = ( R − R b ) /
5, Γ = 3 and θ = 0. Thus in nondi-mensional units, by changing β from 0.1 to 0.8 and β from 0.3 to 0.1, a flow patternpeaking at 50 ◦ can be obtained. This means the dimensional values for β and β will be β = 0 . / (1 . × ) cm − and β = 0 . / (1 . × ) cm − in order to create a polewardsurface flow peaking at 50 ◦ , as shown in the red curve in Figure 3(a).Here ρ ( r ) = ρ b [( R/r ) − . m , with m = 1 . ρ b taken as 1 gm/cc for practicalpurposes. Then using the constraint of mass-conservation, the velocity components can beobtained as, v r = ρr sin θ ∂∂θ ( ψr sin θ ) and v θ = − ρr sin θ ∂∂r ( ψr sin θ ) . All other ingredients ( α -effects, differential rotation, and quenching) remain the sameas in Dikpati et al (2010). In both cases the peak velocity before the abrupt change is 18m/s, which is set by adjusting the values of ψ . In both cases, θ = 0 ensures that there isa single meridional flow cell that extends all the way to the poles from the equator. In allcases, the change in meridional flow is timed to occur at the epoch of polar reversal. We getsimilar results if the change in flow speed occurs at other phases of a cycle.The polar field amplitude produced by the model for all these cases is plotted in Figure3(b). This amplitude is the maximum, area weighted average of the radial field from 55 ◦ latitude to the pole for the first polar field peak to occur after the change in meridional flowspeed (the same average as used in Dikpati et al (2010)). We see in Figure 3(b) that theamplitude of the change in polar fields with a sudden change in meridional flow is rather The parameter values for β , β and Γ were given in dimensionless units in the GRL paper by Dikpati etal. (2010a), whereas other parameters were given in dimensional units; we thank Dr. Luis Eduardo AntunesVieira for helping us catching that.
11 –Fig. 3.— (a) Typical profiles of poleward surface flow in meridional circulation used in setof self-excited flux transport dynamo simulations; blue curve from Dikpati et al (2010a)with peak near 25 ◦ ; red curve a profile with peak near 50 ◦ . (b) Polar field amplitudes inthe polar cap (latitudes 55 ◦ to 90 ◦ ) from model simulations, for cycle immediately followingsudden change in meridional flow speed from 18 m/s to value shown on horizontal axis. NineGauss has been added to the red curve so it can be compared to the blue curve on the samescale. The much lower actual polar fields in the red curve are probably caused by the longertransport time from the source in active latitudes to the poles, since the meridional flowpeaks at a much higher latitude in this case.small – no more than 10% for the full range of circulation amplitude change for either profile,perhaps within the measurement error for average polar fields. Thus the sudden change inmeridional flow has little effect on the polar field peak in the cycle immediately following thechange. Thus in a flux-transport dynamo model, a change in peak polar fields of as muchas several tens of percent between one cycle and the next can not come from a change inmeridional flow speed. It must come from a significant change in the amplitude of the sourceof polar fields, namely the eruption and decay of active region magnetic flux. 12 –Fig. 4.— Periods of first five cycles following a sudden change in meridional flow speed in aflux-transport dynamo model. The period sequence from each simulation is marked with theamount of the change. All simulations start from a previous simulation with peak meridionalflow speed of 18 ms − , which gives a period of about 9 years, shown on the left hand scalefor cycle number zero. 13 –How the polar field maximum of the next cycle changes with the change in meridionalflow is quite different for the two profiles. For the low latitude peak in flow, both faster andslower meridional flow leads to a decrease in polar field strength, while for the high latitudeflow peak, the faster(slower) the new flow, the larger (smaller) subsequent polar field peak.The changes in polar field that occur for an increase in meridional flow speed for the twomeridional flow profiles are completely consistent with the qualitative arguments made usingFigure 1 of §
2. So is the decline in polar field when the meridional circulation with highlatitude peak is reduced. Only the decline in polar fields when the flow pattern peaking atlow latitudes is reduced requires a different explanation – perhaps the dynamo cycle-periodgets so long that there is more time for the surface flux moving to the poles to be diffuseddown and not reach the pole.These simulations of changes in polar fields due to drastic changes in meridional circula-tion raise lead to an additional question – how quickly does the dynamo period adjust to thechanged meridional flow. Figure 4 plots the period of the first five cycles computed followingthe abrupt change in meridional flow speed without altering the form of the streamlines. Notsurprisingly, the new periods are about what we would expect for an advection-dominatedflux-transport dynamo in which the dynamo cycle-period is inversely proportional to themeridional flow-speed. What is perhaps surprising about the results seen in Figure 4 is howquickly the model adjusts to the new period established by the changed meridional flow.Except for the extreme case when the flow peak is reduced from 18 ms − to 6 ms − , we seethat the adjustment occurs almost entirely within the first cycle. A forthcoming paper onsequential data assimilation for solar dynamo models is addressing this issue in more detail;preliminary results indicate that the ’response time’ of a flux-transport dynamo to a changein meridional flow is as short as about 8 months.The Figure 4 shows the settlement of the dynamo cycle-period for a calibrated dynamo asdiscussed in § α -effect, the cycle-period changes in a similar way. However,for the same meridional flow-speed the cycles are little faster in that case.In diffusion-dominated dynamos the cycles are faster than that in advection-dominateddynamos, due to enhanced diffusive transport added to the advective transport of magneticflux. An extensive analysis by Hotta & Yokoyama (2010) shows how the dynamo cycle-period would change when the magnetic diffusivity in the bulk of the convection zone isincreased. From the above study we anticipate that diffusion-dominated dynamos wouldrespond to a sudden change in meridional flow-speed in an analogous way to that seen inFigure 3(b). However, it would be worthwhile in the future to do an investigation of theresponse of advection-dominated dynamos of Mu˜noz-Jaramillo et al (2010) that used a more 14 –sophisticated buoyancy mechanism than was used by Dikpati & Charbonneau (1999).
4. Discussion
While both surface flux-transport models and flux-transport dynamo models include theadvective-diffusive transport of magnetic flux, and both can consistently explain polar fieldpatterns, there are inherent differences – surface transport models simulate the evolution ofradial fields on the photospheric latitude-longitude surface, whereas flux transport dynamomodels solve the axisymmetric dynamo equations for the toroidal field, B T , and the vectorpotential, A ( ∇ × A represents poloidal fields) in the meridian plane in the convection zone.There are additional physical effects operating in the evolution of magnetic fields in a flux-transport dynamo model, due to the presence of the radial flow in the circulation pattern,radial diffusion, and depth-dependent diffusivity. Similarly additional physical effects arecaptured in surface transport models, namely their more realistic treatment of longitudedependence and hence estimation of the Babcock-Leighton surface source term for poloidaland therefore polar fields.When poleward surface flow is increased in a flux-transport dynamo, if there is nochange in the profile of meridional flow, the upward flow near the equator and the downwardflow near the high-latitudes will also increase, causing the poloidal fields produced from theleader polarity to move up to a slightly higher diffusivity region compared to where theywould have been if the flow would not have increased. The poloidal fields from the followerspots, on the other hand, do the opposite — they sink down towards the lower diffusivityregion (see Figure 1c). As a consequence, the equatorward side of the polarity division line ofthose poloidal fields undergoes faster diffusive decay, while the poleward side undergoes lessdecay and therefore remains more frozen. Since the dynamo equations solve for the vectorpotential, A , the changes in A described above get reflected in the radial component of thepoloidal field, given by r sin θ ∂∂θ ( A r sin θ ). Thus the presence of depth-dependence in thediffusivity profile and the radial component in the meridional flow profile both contributeto the increase in the poloidal fields on the poleward side of the bipoles. This effect is notpresent in surface transport models.Nevertheless, we have shown through physical description and numerical calculationsthat the surface flux-transport models and flux-transport dynamo models, despite somedifferences in their physics, produce similar results when run with a very similar latitudinalprofile of surface poleward meridional flow, although surface-transport models can betterreproduce polar field patterns in latitude and time compared to any dynamo model. 15 –As a consequence of the absence of depth-dependent diffusivity, the radial componentof the meridional flow and the latitudinal component of the poloidal fields, surface-transportmodels produce a more visible difference in the polar field amplitudes than the polar fields ob-tained from a flux-transport dynamo model when a high-latitude reverse flow-cell is present.Jiang et al (2009) have shown that the surface radial fields get significantly reduced atthe pole if the poleward surface flow reverses beyond 70 ◦ . However, flux-transport dynamomodels change the latitude location of the maximum surface radial fields from pole to theboundary of the two cells when such models are run with two flow cells (Bonanno et al2005; Jouve & Brun 2007; Dikpati et al 2010). The observed polar fields from Wilcox So-lar Observatory are instrumentally averaged over the latitude range from 55 ◦ to the pole.Thus the model-derived radial fields integrated over the latitudes from 55 ◦ up to the poleand weighted by the surface area, (i.e. < B r > = R π R π/ . B r R sin θdθdφ R π R π/ . R sin θdθdφ ) are not actuallymuch different in the two cases whether a high-latitude reverse-cell is present or absent (seethe discussion in (Dikpati et al 2010)).In this analysis, we confined ourselves to consideration of advection-dominated flux-transport dynamos and surface transport models. How polar fields respond to changesin meridional flow in the cases of diffusion-dominated dynamos (Yeates, Nandy & Mackay2008) and flux-transport dynamos with more complexities included, such as turbulent pump-ing (Guerrero & de Gouveia Dal Pino 2008), diffusivity quenching (Guerrero, Dikpati & de Gouveia Dal Pino2009) and buoyancy-induced delay in surface poloidal field generation (Jouve, Proctor & Lesur2010), have not yet been explored.An alternative explanation for the weak polar fields and long minimum of cycle 23 hasbeen given by Nandy, Mu˜noz-Jaramillo & Martens (2011). Using an advection-dominatedflux-transport dynamo (Mu˜noz-Jaramillo, Nandy & Martens 2009) that operates with atwo-step diffusivity profile, Nandy, Mu˜noz-Jaramillo & Martens (2011) do a large numberof simulations in which the meridional flow amplitude is changed randomly, once per cycle,at cycle maximum. The peak amplitude falls in the range of 15 −
30 ms − . They find thatcycles in which the meridional flow speed is larger in the ascending phase than in the declin-ing phase tend to be followed by longer, deeper minima, and the polar field strength of suchcycles tend to be weaker. Both these correlations are modest, with correlation coefficient r ∼ .
5, which corresponds to a variance of one-quarter only. This means that the other75% of the variance in polar fields’ weakening must be due to other effects. In addition,no observational evidence was given in Nandy, Mu˜noz-Jaramillo & Martens (2011) that themeridional circulation in cycle 23 actually was higher in the ascending phase than in thedescending phase, and, indeed, the best measures of surface plasma flow for cycle 23 thatexists, namely that of Ulrich (2010) and Basu & Antia (2010), does not support the as- 16 –sumption of speed-up in meridional flow-speed in the ascending phase of cycle 23 (see figure6 of Ulrich (2010) and figure 3 of Basu & Antia (2010)). Thus it follows that, at least forcycle 23, it is not possible to explain the polar field drop or the long minimum using thecorrelation found by Nandy, Mu˜noz-Jaramillo & Martens (2011).Currently all benchmarked flux-transport dynamos operate in the 2D axisymmetricregime (see Jouve et al (2008)). It is also necessary to investigate the role of longitude-dependence from the tilted, bipolar active regions in the generation of evolution of the Sun’spolar fields using a 3D version of flux-transport dynamos.
5. Acknowledgements:
We thank Peter Gilman for reviewing the manuscript. We extend our thanks to ananonymous referee for helpful comments which helped improve the manuscript. This work ispartially supported by NASA’s Living With a Star program through the grant NNX08AQ34G.The National Center for Atmospheric Research is sponsored by the National Science Foun-dation.
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