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ISSN 0016 7932, Geomagnetism and Aeronomy, 2015, Vol. 55, No. 8, pp. 10761080. © Pleiades Publishing, Ltd., 2015.

Features of the Spatial Distribution of Spots in the Solar Cycle and a Model of Dynamo in a Thin Layer
V. G. Ivanov and E. V. Miletsky
Central (Pulkovo) Astronomical Observatory, Russian Academy of Sciences, Pulkovskoe sh. 65, St. Petersburg, 196140 Russia e mail: vg.ivanov@gao.spb.ru
Received February 8, 2015; in final form, February 17, 2015

Abstract--In our earlier works (Ivanov and Miletsky, 2012, 2014), we demonstrated that (1) the evolution of average sunspot latitudes can be described by a universal latitude curve, the shape of which does not depend on the cycle amplitude and (2) at the cycle decline phase, these latitudes correlate well with the current level of solar activity. In this work, we demonstrate that these features of the latitude evolution of the cycle, as well as the empirical Waldmeier rule, can be described by a simple model of a convective dynamo in a thin spherical layer with the addition of some nonlinearities. DOI: 10.1134/S0016793215080113

1. INTRODUCTION It is well known that the latitude evolution of solar spots in the 11 year solar cycle is described by the SpЖrer law: the first spots appear at high latitudes, and then the activity center drifts to the equator. In this process, it was found (see, e.g., Eigenson et al., 1948; Vitinskii et al., 1986; Hathaway, 2011) that the behav ior of the average sunspot latitude (ASL) varies from cycle to cycle much more weakly than the spot ampli tude characteristics. In our earlier works (Ivanov and Miletsky, 2012; Ivanov and Miletsky, 2014) based on data of the extended Greenwich catalogue for 18742013, we demonstrated that ASL behavior can be described suf ficiently accurately by a universal latitude curve, the shape of which very slightly depends on the cycle power: (t) = a exp(b(t T0)), where a = 26.6° and b =0.126 year1 are parameters common for all cycles and T0 is the latitude phase reference point for the given cycle (for brevity, we below call this regularity R1). The variable , which indicates the characteristic ASL, can be used here as a characteristic of the cycle phase; it is an alternative to the commonly used time passing from the moment of the cycle minimum. In the same works, it was noted that ASLs have one more feature (we call it R2): in the second half of the cycle decline phase, the ASL still does not depend on the cycle amplitude but correlates well with the current level of solar activity. The second feature is illustrated in Fig. 1, in which panels (a) and (b) depict the index of the sunspot group number G smoothed over 13 solar rotations with a sinusoidal filter from the 12th to 23th solar cycles as a function of the cycle phases and , respectively; panel (c) depicts root mean square deviations of G

from the mean for all cycles ( and ) as functions of these variables; and panel (d) depicts the ratio of these deviations /. It is seen from a comparison of pan els (a) and (b) that the scatter of trajectories in the sec ond half of the cycle decline phase is less in the second case. In other words, a definite cycle phase in differ ent cycles can be associated with values of the index G from a rather wide range, and the characteristic lati tude at the cycle decline phase is in correspondence with a more narrow interval of this index. The same is testified by panels (c) and (d) showing that, at this phase, < . The obtained regularities certainly must impose constraints on the possible mechanisms of solar peri odicity. In this work, we study the question of whether the observed features of ASL evolution can be repro duced by the model of the solar convective dynamo. At the initial stage of the study, we are interested mainly in the qualitative aspect of the problem; for this rea son, we chose for the study the simplest model of the dynamo in a thin spherical layer. When required, the model is complemented with nonlinearities. 2. LINEAR MODEL OF THE DYNAMO IN A THIN SPHERICAL LAYER The dependences of the toroidal field strength B and vector potential of the poloidal field A on the radius r and colatitude for the case of a thin layer can be written in the form F(r, , t) = f(, t)exp(kr)/kr. The dimensionless equations of the dynamo for A(, t) and B(, t) in this model have the form (see, e.g., (Schmitt and SchЭssler, 1989; Solanki et al., 2008))

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FEATURES OF THE SPATIAL DISTRIBUTION OF SPOTS G() 9 8 7 6 5 4 3 2 1 0 0 G() 9 8 7 6 5 4 3 2 1 0 30° (a) 0 2.0 1.5 ,

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(c) 2 = 1/b log (/a), years 4 6 8 10 12

1.0 0.5

2

4

6 (b)

8

10 12 , years

0

2

4

6 , years (d)

8

10

12

2.0

1.5 1.0 0.5

/

25°

20°

15°

10°

5°

0

2

4

6

8

10 12 , years

Fig. 1. Relation between the average sunspot latitude and current level of the solar activity in the second half of the cycle phase (see explanations in the text).

( v B ) B ( A sin ) = D ( B ) + R Rv , t v ( A sin ) A = + S ( , t ) , D ( A ) + BR R v sin t where the diffusion operator D= 1 sin 1 ( kR ) 2 , sin sin 2 m R vR 'R , Rm = m , , R =

Reynolds numbers R =

R is the radius of the spherical shell, is the convective viscosity, ' = /r is the radial gradient of the angu lar velocity, = cos and v sin 2 are normalized pro files of the alpha effect and meridional velocity, respectively, m and vm are the maximum values of these quantities, and S(, t) is the source of the poloi dal field potential.
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In this model, it is natural to associate the spot for mation level observed at a given latitude with the tor oidal field strength b of a given sign; to associate the activity level with the same strength averaged over the hemisphere b; and to associate the ASL with the lat itude = 90° weighted with a weight |b|. To obtain cycles with different amplitudes, we complement the model with "secular" periodic variations in a certain parameter with a period Tmod = 3.0 (in conventional units of time), which, as is seen below, exceeds the main period of the dynamo model T0 by more than an order of magnitude. For this parameter, we use the source S (below, this type of the model is denoted as VS), the intensity of the effect m (V), or the meridional circulation vm (Vm). It should be noted that such an artificial introduction of the "secular periodicity," is certainly meant to extend the phase space of the stud ied models, not to explain actually observed long peri odic solar activity variations. For all of the models, we set Rk = 3. We specify quasi dipole initial conditions B(, 0) = 0 and A(, 0) = sin2 and study the evolution of the fields on the time interval of 0 t 6. In doing this, beginning from cri
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1078 90° 45° 0° 45° 90° 6 4 2 0 b

IVANOV, MILETSKY

60° 30° 0°

0

1

2

3 t

4

5

6

Fig. 2. Latitudetime diagram (upper panel), as well as average strength b (middle panel) and average latitude (lower panel) as functions of time for the linear model.

ticial Reynolds numbers R = R 37, we then grad ually increase R until the dynamo reaches a steady regime. We begin the investigation with the linear model of the VS type, in which the critical values R = R = 37.43 are associated with a cycle period of T0 = 0.1086. For this model, Fig. 2 depicts the latitudetime dia gram, as well as the dependences of b and on time; Fig. 3, the dependence of b on cycle phases (a) and (b) , as well as (c) of spreads of b for given values of and and (d) ratios of these spreads (these panels are similar in their meaning to the corresponding panels of Fig. 1), as well as (e) dependences of on the cycle phase . Comparing Figs. 1 and 3, we see that the ASL curve (R1) for the linear model is universal, as should be expected, and R2 regularity does not take place. 3. INTRODUCTION OF NONLINEARITIES INTO THE DYNAMO MODEL It is evident that the model must be complemented by nonlinearities to reproduce R2 regularity. One well
Latitudinal regularities and correlation for the Waldmeier rule in different types of models VS rw B A R2 Rl,R2 +0.94 +0.66 V r
w

Vm r
w

0.78 R2 +0.77 Rl 0.83 R2 +0.69 0.84 R2 +0.64

A + buoyancy Rl, R2 0.61

known way to do this is to assume that the effect is suppressed by the toroidal magnetic field (see, e.g., (Charbonneau, 2010)). We take this effect into account by writing m = m0 exp(B/Bc), where Bc is the critical value of the strength (the B model). In addition, we study a model with a nonlinearity of another type, in which the effect is suppressed in a similar way by the poloidal field: m = m0 exp(A/Ac) (the A model). For these types of models, we every where set Bc = 0.4 and Ac = 0.4. The first two rows of the table show the results of studying these two nonlinearity variants for the three aforementioned types of "secular variations" (VS, V, and Vm). The valid latitudinal regularities are shown in corresponding cells of the table. It is seen that both of the regularities above (R1 and R2) take place only for the A + VS model. Let us complicate the model requirements by recalling one more effect that must manifest itself in the 11 year cycle model corresponding to observa tions. This is the Waldmeier rule (Waldmeier, 1935), according to which the coefficient rW of the correla tion between the length of the cycle growth phase and cycle amplitude for observed solar cycles is negative and exceeds 0.6 in absolute value (its exact value depends on the epoch under study and the method of averaging the solar activity index). It is seen from the table, which presents this coefficient, that it has the correct sign for V models but none of the three effects (R1, R2, and rW < 0) match in any of the six models. Let us introduce in the model one more modifica tion by assuming (see, e.g., Solanki et al., 2008) that field tubes of the toroidal field begin to rise to the sur face efficiently only if the strength of the field B exceeds the critical value Bb. This situation can be described by assuming that the toroidal fields observed
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GEOMAGNETISM AND AERONOMY

FEATURES OF THE SPATIAL DISTRIBUTION OF SPOTS G (a) , 1.5 4 1.0 2 0.5 0 0 G 6 0.05 (b) 0 35 / 1.0 4 0.8 0.6 2 0.4 0.2 0 60 50 40 30 20 10 0 24 22 20 18 16 14 12 0 0.1 0 0.1

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(c)

6

60 20 15 40

(e)

0.10

30

25 (d)

20

Fig. 3. Relation between the average sunspot latitude and current level of the solar activity (see explanations in the text).

in the photosphere are related to fields generated by the dynamo mechanism in terms of a quasi step func tion Bobs = B exp(B/Bb)8), where we set Bb = 0.1. An analysis of A models with this additional effect shows that the VS + A type + buoyancy model satisfies all necessary requirements, including the Waldmeier rule (with a correlation coefficient rW = 0.61). Figure 4, which is similar to Fig. 2, depicts the model character istics of interest. 4. DISCUSSION AND CONCLUSIONS Thus, we have demonstrated that the simplest models of the dynamo in a thin spherical layer can describe observed regularities of the latitudinaltem poral evolution of the activity cycle. It is easiest to reproduce the R2 feature (the relation between the activity level and average latitude at the cycle decline phase); it is absent only in models with secular varia tion in the alpha effect (V). To obtain the R1 feature (universality of the latitu dinal drift law), which naturally arises in linear mod els, one should restrict oneself to models with suppres sion of the alpha effect by the poloidal field. It should be noted that, if the suppression of the alpha effect by
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the toroidal field naturally arises in the theory, the assumption about the participation of the weaker poloidal field in this process can look rather artificial. However, our goal in this work was only to show that this regularity can be easily reproduced even in sim plest "toy models." Part of the poloidal field in this case is reduced to decreasing the quantity R and, therefore, the rate at which toroidal fields are trans formed into poloidal. Upon a more detailed investiga tion with the use of realistic models, this part can be played by some other factor in phase with the poloidal field. Note also that a more complex model must describe the real periodicity feature, according to which > at the cycle growth phase (see Fig. 1d); this is not reproduced by the simple model that we have described. Thus, the simple model of the dynamo in a thin layer with secular variation in the source and suppres sion of the alpha effect by the poloidal field (VS + A) can qualitatively reproduce both of the observed latitu dinal regularities. If it is complemented by critical buoyancy, it is also able to reproduce the Waldmeier rule. One can suppose that the development of the aforementioned dynamo model with the transition to realistic two dimensional models will make it possible
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IVANOV, MILETSKY , 0.10 0.3 0.2 0.1 0 0 G 0.4 0.3 0.2 0.1 0.2 0 60 50 40 30 20 10 0 35 30 25 20 15 0.05 (b) 0.08 0.06 0.04 0.02 0.10 0 35 / 1.0 0.8 0.6 0.4 0 0.1 0 0.1

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60 20 15 40

(e)

25 (d)

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Fig. 4. The same as in Fig. 3 for the VS + A type + buoyancy nonlinear model.

to find a way also to provide a quantitative description of these effects. ACKNOWLEDGMENTS This work was supported in part by the Russian Foundation for Basic Research, project no. 13 02 00277, and Presidium of the Russian Academy of Sci ences, programs 9 and 41. REFERENCES
Charbonneau, P., Dynamo models of the solar cycle, Living Rev. Sol. Phys., 2010, vol. 7, p. 3. Eigenson, M.S., Gnevyshev, M.N., Ol', A.I., and Rubashov, B.M., Solnechnay aktivnost' i ee zemnye proyavleniya (Solar Activity and Its Terrestrial Manifes tations), MoscowLeningrad, OGIZ, 1948. Hathaway, D.H., A standard law for the equatorward drift of the sunspot zones, Sol. Phys., 2011, vol. 273, p. 221. Ivanov, V.G. and Miletskii, E.V., Reference moments of the 11 year solar activity cycles and the universality of the law of latitude drift in solar spots, Trudy Vserossiiskoi konferentsii "Solnechnaya i solnechno zemnaya fizika

2012" (Proceedings of the All Russian Conference "Solar and SolarTerrestrial Physics 2012"), 2012, pp. 5154. Ivanov, V.G. and Miletsky, E.V., SpЖrer's law and relation ship between the latitude and amplitude parameters of solar activity, Geomagn. Aeron. (Engl. Trasnl.), 2014, vol. 54, no. 7, pp. 907915. Schmitt, D. and Schussler, M., Non linear dynamos. I. One dimensional model of a thin layer dynamo, Astron. Astrophys., 1989, vol. 223, p. 343. Solanki, S.K., Wenzler, T., and Schmitt, D., Moments of the latitudinal dependence of the sunspot cycle: A new diagnostic of dynamo models, Astron. Astrophys., 2008, vol. 483, p. 623. Vitinskii, Yu.I., Kopetskii, M., and Kuklin, G.V., Statistika pyatnoobrazovatel'noi deyatel'nosti Solntsa (Statistics of Solar Spot Generation Activity), Moscow: Nauka, 1986. Waldmeier, M., Neue Eigenschaften der Sonnenflecken kurve (New qualities of the sunspot curves), Astron. Mitt., 1935, vol. 14, pp. 105130.

Translated by A. Nikol'skii

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2015