Why does the Sun's torsional oscillation begin before the sunspot cycle?
aa r X i v : . [ a s t r o - ph . S R ] J u l Why does the Sun’s torsional oscillation begin before the sunspot cycle?
Sagar Chakraborty , Arnab Rai Choudhuri , and Piyali Chatterjee Department of Theoretical Sciences, S.N. Bose Centre for Basic Sciences, Kolkata - 700098 and Department of Physics, Indian Institute of Science, Bangalore - 560012 (Dated: November 19, 2018)Although the Sun’s torsional oscillation is believed to be driven by the Lorentz force associatedwith the sunspot cycle, this oscillation begins 2–3 yr before the sunspot cycle. We provide atheoretical explanation of this with the help of a solar dynamo model having a meridional circulationpenetrating slightly below the bottom of the convection zone, because only in such dynamo modelsthe strong toroidal field forms a few years before the sunspot cycle and at a higher latitude.
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There is a small periodic variation in the Sun’s rota-tion with the sunspot cycle, called torsional oscillations.While this was first discovered on the Sun’s surface [1],the nature of torsional oscillations inside the solar con-vection zone was later determined from helioseismology[2-8]. Several authors [9-12] developed theoretical modelsof torsional oscillations by assuming that they are drivenby the Lorentz force of the Sun’s cyclically varying mag-netic field associated with the sunspot cycle. If this istrue, then one would expect the torsional oscillations tofollow the sunspot cycles. The puzzling fact, however, isthat the torsional oscillations of a cycle begin a coupleof years before the sunspots of that cycle appear and ata latitude higher than where the first sunspots are sub-sequently seen. At first sight, this looks like a violationof causality—a classic case of the effect preceding thecause! Our aim is to explain this puzzling observation,for which no previous theoretical model offered any ex-planation. In the models of Covas et al. [10] and Rempel[12], the theoretical butterfly diagrams extend to unreal-istically high latitudes of about 60 ◦ and the low-latitudebranches of torsional oscillations follow the butterfly di-agrams closely, not starting at higher latitudes.Let us summarize some of the other important char-acteristics of torsional oscillations, which a theoreti-cal model should try to explain. (1) Apart from theequatorward-propagating branch which moves with thesunspot belt after the sunspots start appearing, there isalso a poleward-propagating branch at high latitudes. (2)The amplitude of torsional oscillations near the surfaceis of order 5 m s − . (3) The torsional oscillations seemto be present throughout the convection zone, thoughthey appear more intermittent and less coherent as we godeeper down into the convection zone (see Figs. 4, 5 and6 in Howe et al.[7]). (4) In the equatorward-propagatingbranch at low latitudes, the torsional oscillations at thesurface seem to have a phase lag of about 2 yr comparedto the oscillations at the bottom of the convection zone(see Fig. 7 in Howe et al.[7]).The last property of torsional oscillations listed aboveseems to suggest that the bottom of the convection zoneis the source of the oscillations, which then propagateupwards. The property (3) then seems puzzling and con-trary to the common sense. One would expect the os- cillations to be more coherent near the source, becomingmore diffuse as they move upward further away from thesource. The observations indicate the opposite of this.We shall discuss a possible explanation for this observa-tion as well. Spruit [13] proposed thermal effects nearthe surface to be the origin of torsional oscillations—anidea which the property (4) seems to rule out [7].While there may not yet be a complete concensus, themajority of dynamo theorists believe that the sunspot cy-cle is produced by a flux transport dynamo, in which themeridional circulation carries the toroidal field producedfrom differential rotation in the tachocline equatorwardand carries the poloidal field produced by the Babcock–Leighton mechanism at the surface poleward [14-21].Since the differential rotation is stronger at higher lat-itudes in the tachocline than at lower latitudes, the in-clusion of solar-like rotation tends to produce a strongtoroidal field at high latitudes rather than the latitudeswhere sunspots are seen [17-18]. Nandy & Choudhuri [19]proposed a hypothesis to overcome this difficulty. Ac-cording to them, the meridional circulation penetratesslightly below the bottom of the convection zone andthe strong toroidal field produced at the high-latitudetachocline is pushed by this circulation into stable layersbelow the convection zone where magnetic buoyancy issuppressed and sunspots are not formed. Only when thetoroidal field is brought into the convection zone by themeridional circulation rising at lower latitudes, magneticbuoyancy takes over and sunspots finally form. It maybe noted that there is a controversy at the present timewhether the meridional circulation can penetrate belowthe convection zone—arguments having been advancedboth against [22] and for it [23].If the Nandy–Choudhuri hypothesis (hereafter NC hy-pothesis) is correct, then the toroidal field of a particularcycle first forms at a relatively high latitude some timebefore the sunspots of the cycle would start appearing.Assuming that the Lorentz force of the newly formedtoroidal field at the high latitude can initiate the tor-sional oscillations, the NC hypothesis provides a naturalway to explain how the torsional oscillations begin athigh latitudes before the appearance of the sunspots ofthe cycle. Our dynamo model based on the NC hypothe-sis correctly explains the onset of torsional oscillations atthe high latitude before the beginning of the sunspot cy-cle. We, in fact, would like to argue that the early onsetof torsional oscillations provides a compelling evidence insupport of the NC hypothesis.Our theoretical model is based on a mean field ap-proach. However, we know that the magnetic field ishighly intermittent within the convection zone and weneed to take account of this fact when calculating theLorentz force due to the magnetic field. Since the con-vection cells deeper down are expected to have largersizes, Choudhuri [24] suggested that the magnetic fieldwithin the convection zone would look as shown in Fig. 1of that paper. Demanding that the vertical flux tubesgive rise to horizontal flux tubes with magnetic field 10 G (as suggested by flux rise simulations [25-28]) afterstretching in the tachocline, the magnetic field inside thevertical flux tubes at the bottom of the convection zoneis estimated to be of order 500 G [24]. This scenarioprovides a natural explanation for the properties (3) and(4) of torsional oscillations listed above. Presumably thetorsional oscillation gets initiated in the lower footpointsof the vertical flux tubes, where the Lorentz force buildsup due to the production of the azimuthal magnetic field.This perturbation then propagates upward along the ver-tical flux tubes at the Alfven speed. If the magnetic fieldinside the flux tubes is 500 G, then the Alfven speed atthe bottom of the convection zone is of order 315 cms − and the Alfven travel time from the bottom to thetop turns out to be exactly of the same order as thephase lag of torsional oscillations between the bottomof the convection zone and the solar surface. We admitthat the magnetic scenario sketched in Fig. 1 of Choud-huri [24] and adopted here is not yet established throughrigorous dynamical calculations and a proper study ofthe propagation of disturbances through such complexmagnetic structures is unavailable. However, an assump-tion of net upward propagation of magnetic disturbancesin spite of all these complexities is not an unreasonableansatz, which is justified by the success of the theoreti-cal model in matching otherwise unexplained aspects ofobservational data. Since the magnetic field at the bot-tom is highly intermittent and the velocity perturbationsassociated with the torsional oscillations are likely to beconcentrated around the magnetic flux tubes, we expectthe torsional oscillations to be spatially intermittent atthe bottom of the convection zone, as seen in the obser-vational data [7]. Since the magnetic field near the sur-face is less intermittent, the torsional oscillation drivenby the Lorentz stress also appears more coherent there.We thus have the puzzling situation that the torsionaloscillations seem to become more coherent as they movefurther away from the source at the footpoints of fluxtubes at the bottom of the convection zone.To develop the theoretical model of torsional oscilla-tions, we extend our already published solar dynamomodel [20], in which the NC hypothesis has been in-corporated. The basic dynamo code SURYA which isextended for the present calculations is available upon request. Apart from the time evolution equations for thetoroidal and poloidal components of the magnetic fieldwhich have to be solved in the dynamo problem, we alsohave to solve an additional simultaneous time evolutionequation of the toroidal velocity component v φ . Thisother equation, which is essentially the φ component ofthe Navier–Stokes equation, is ρ (cid:26) ∂v φ ∂t + v r r ∂∂r ( rv φ ) + v θ r sin θ ∂∂θ (sin θv φ ) (cid:27) =( F L ) φ + 1 r ∂∂r (cid:20) νρr ∂∂r (cid:16) v φ r (cid:17)(cid:21) +1 r sin θ ∂∂θ (cid:20) νρ sin θ ∂∂θ (cid:16) v φ sin θ (cid:17)(cid:21) , (1)where ( F L ) φ is the φ component of the Lorentz force. Weuse the stress-free boundary condition ∂v φ /∂r = 0 at thesolar surface and take v φ = 0 at the bottom, althoughthe bottom boundary condition has no effect when thebottom of the integration region is taken well below thetachocline as we do. The kinematic viscosity ν is pri-marily due to turbulence within the convection zone andis expected to be equal to the magnetic diffusivity. Weuse the exactly same profile of ν as the profile of thediffusivity of poloidal field, which is shown in Fig. 4 ofChatterjee et al. [20]. In other words, we assume themagnetic Prandtl number to be 1. In order to ensure aperiod of 11 yr, we choose some parameters in the dy-namo equations slightly different from what were used byChatterjee et al. [20], as listed in Table 1 of Choudhuriet el. [29]. For the density ρ appearing in (1), we use theanalytical expression used by Choudhuri & Gilman [25],which gives values of density consistent with detailed nu-merical models of the convection zone.If the magnetic field is assumed to have the form B = B ( r, θ, t ) e φ + ∇ × [ A ( r, θ, t ) e φ ] , (2)then the Lorentz force is given by the Jacobian4 π ( F L ) φ = 1 s J (cid:18) sB φ , sAr, θ (cid:19) , (3)where s = r sin θ . We, however, have to take some spe-cial care in averaging this term, since this is the primaryquadratic term in the basic variables ( A, B, v φ ) and hasto be averaged differently from all the other linear terms.The effect of v φ on the magnetic field is also quadratic,and has been added to the similar term giving the ef-fect of differential rotation on magnetic fields in the φ -component of the induction equation. The φ compo-nent of the Lorentz force primarily comes from the ra-dial derivative of the magnetic stress B r B φ / π (the termhaving B θ B φ involves θ derivative and is smaller). Thisstress arises when B r is stretched by differential rotationto produce B φ and should be non-zero only inside the fluxtubes. We assume that B r , B φ are the mean field values,whereas ( B r ) ft , ( B φ ) ft are the values of these quantitiesinside flux tubes. If f is the filling factor, then we have La t i t ude ( deg ) −20020Time (years) La t i t ude ( deg ) FIG. 1: Comparison between observation and theory. Theupper panel superposes the butterfly diagram of sunspots ona time-latitude plot of the observed surface zonal velocity v φ (in m s − ) measured at Mount Wilson Observatory (Courtesy:Roger Ulrich). The comparable theoretical plot is shown inthe lower panel, in which the theoretical butterfly diagramfrom our dynamo model is superposed on the time-latitudeplot of theoretically computed v φ (in m s − ) at the surface. B r = f ( B r ) ft and B φ = f ( B φ ) ft , on assuming the samefilling factor for both components for the sake of simplic-ity. It is easy to see that the mean Lorentz stress is f ( B r ) ft ( B φ ) ft π = B r B φ πf . This suggests that the correct mean field expression for( F L ) φ is given by the expression (3) divided by f .As pointed out by Chatterjee et al. [20], the only non-linearity in the dynamo equations comes from the criticalmagnetic field B c above which the toroidal field withinthe convection zone is supposed to be unstable due tomagnetic buoyancy. Jiang et al. [30] found that we haveto take B c = 108 G (which is the critical value of themean toroidal field and not the toroidal field inside fluxtubes) to ensure that the poloidal field at the surfacehas correct values. Once the amplitude of the magneticfield gets fixed this way, we find that only for a partic-ular value of the filling factor f the amplitude of thetorsional oscillations matches observational values. Ourcalculations give a filling factor f ≈ . f ≈ .
02 by Choud-huri [23]. Theoretical values of velocity in all our figuresare computed by using f = 0 . v r = 300 cm s − in (2) toaccount for the upward transport by Alfven waves whensolving our basic equation (1) for v φ . Note that this addi-tional v r does not represent any actual mass motion anddoes not have to satisfy the continuity equation whichthe meridional circulation satisfies. Because of our lack La t i t ude ( deg ) La t i t ude ( deg ) La t i t ude ( deg ) abc 1994 1996 1998 2000 2002 2004−50050 −10010 FIG. 2: Theoretical torsional oscillations ( v φ in m s − ) intime-latitude plots at different depths of the convection zone:(a) 0.95 R ⊙ , (b) 0.9 R ⊙ , (c) 0.8 R ⊙ . of knowledge about this upward transport, we assume theupward velocity to be independent of depth and allow itto transport the magnetic stresses from the bottom to thesurface where they freely move out due to the stress-freeboundary condition, mimicking what we believe must behappening in the real Sun.Figure 1 presents a comparison of theory with observa-tions by putting the butterfly diagram of sunspots on thetime-latitude plot of torsional oscillations at the surface.The theoretical plot correctly reproduces the initiation ofthe low-latitude branch of torsional oscillations about 2yr before the sunspot cycle, starting at a latitude higherthan typical sunspot latitudes. Apart from the NC hy-pothesis, the assumption of the upward advection of theperturbations at Alfven speed is crucial. On switching offthe Alfven wave, even though the torsional oscillationsbegin at a high latitude at the bottom of the convectionzone before the starting of the sunspot cycle, the distur-bance has to reach the surface through diffusion and wedo not see the correct initiation of torsional oscillations atthe surface. We also note that the phase of the torsionaloscillations (i.e. regions of positive and negative v φ inthe time-latitude plot) with respect to the sunspot cycleis reproduced quite well. On decreasing (increasing) theAlfven speed, the phase of the torsional oscillations withrespect to the butterfly diagram gets shifted towards theright (left). While our main aim was to explain the prop-erties of the low-latitude branch of torsional oscillations,our theoretical model has reproduced the high-latitudebranch as well, without our having to do anything spe-cial for it. The physics behind this branch will becomeclear when we discuss Figure 3 later. Figure 2 showingtorsional oscillations at different depths has to be com-pared with the observational Figs. 4–6 of Howe et al. [7].A careful look shows a small phase delay in the upper lay- −2 −1.5 −1 −0.5 0 1990 1995 2000 20050.60.70.80.9 Time (years) D ep t h −8 −6 −4 −2 FIG. 3: Theoretical torsional oscillations ( v φ in m s − ) asfunctions of depth and time at latitudes 20 ◦ (left) and 70 ◦ (right). The contours indicate the Lorentz force ( F L ) φ , thesolid and dashed lines denoting positive and negative values. ers compared to the lower layers. The observational plotsbecome more intermittent at the greater depths due tothe more intermittent nature of the magnetic field there.This is not reproduced in the theoretical model based onmean field equations.Figure 3 shows how torsional oscillations evolve indepth and time at 2 different latitudes. The plot for lati-tude 20 ◦ compares very well with Fig. 4(D) of Vorontsovet al. [5] or Fig. 7 of Howe et al. [7]. It is clear in theplot of 20 ◦ that the Lorentz force is concentrated in thetachocline at 0.7 R ⊙ , where the low-latitude torsional os-cillations are launched to propagate upward. The physicsof the high-latitude branch is, however, very different,with the Lorentz force contours for latitude 70 ◦ indi-cating a downward propagation and not a particularlystrong concentration at the tachocline. As the poloidal field sinks with the downward meridional circulation atthe high latitudes, the latitudinal shear d Ω /dθ in the con-vection zone acts on it to create the toroidal component[21] and thereby the Lorentz stress. With the downwardadvection of the poloidal field, the region of Lorentz stressmoves downward. In the case of the low-latitude branch,the plot for latitude 20 ◦ shows that the amplitude of thetorsional oscillations becomes larger near the surface dueto the perturbations propagating into regions of lowerdensity, which is consistent with observational data [5].If the upward Alfven propagation is switched off, thenthe disturbances from the bottom of the convection zonereach the top by diffusion (the diffusion time being about5 yr in our model), but the amplitude of torsional oscilla-tions in the upper layers of the convection zone generallyfalls to very low values.Compared to the earlier theoretical models of torsionaloscillations, the two novel aspects of our model are (1)the NC hypothesis, which allows the formation of strongtoroidal field in the high-latitude tachocline before thebeginning of the sunspot cycle; and (2) the assump-tion that the perturbations propagate upward along fluxtubes at the Alfven speed. 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