HALF-WIDTH OF A SOLAR DYNAMO WAVE IN PARKER'S ...

HALF-WIDTH OF A SOLAR DYNAMO WAVE IN PARKER’S MIGRATORY DYNAMO KIRILL M. KUZANYAN and DMITRY SOKOLOFF Department of Physics, Moscow State University, 119899 Moscow, Russia; and Department of Mathematics, University of Exeter, Laver Building, North Park Road, Exeter EX4 4QE, U.K. (Received 19 August 1996; accepted 30 December 1996) Abstract. A kinematic ! -dynamo model of magnetic field generation in a thin convection shell with nonuniform helicity for large dynamo numbers is considered in the framework of Parker’s migratory dynamo. The asymptotic solution obtained of equations governing the magnetic field has the form of an anharmonic travelling dynamo wave. This wave propagates over most latitudes of the solar hemisphere from high latitudes to the equator, and the amplitude of the magnetic field first increases and then decreases with propagation. Over the subpolar latitudes, the dynamo wave reverses; there the dynamo wave propagates polewards and decays with latitude. The half-width of the maximum of the magnetic field localisation and the phase velocity of the dynamo wave are calculated. Butterfly diagrams are plotted and analysed and these show that even a simple model may reveal some properties of the solar magnetic fields.

1. Introduction Solar activity is a quasi-periodic process with period of about 22 years. The magnetic activity of the Sun can be visualised by the well-known Maunder butterfly diagrams which represent the time-spatial distribution of sunspots. The latitudinal width of the area covered by sunspots can be thought of as an observational parameter to be compared with observations. The aim of this paper is to calculate this parameter in the framework of a simple model of Parker’s migratory dynamo. This model is a strong simplification of dynamo processes of the solar interior; in particular it gives only a kinematic description of the solar dynamo wave and does not consider nonlinear processes. Nevertheless, it turns out that this simple model gives a value for the latitudinal width which is comparable with observations. Below we plot a butterfly diagram and calculate the width of Parker’s dynamo wave using the asymptotic solution of Parker’s dynamo equations for the case of large dynamo numbers (Kuzanyan and Sokoloff, 1995). The length of the dynamo wave in the dynamo theory is connected to the width of the area covered by sunspots. However, the concept of the dynamo wavelength is not well defined for the solar dynamo because, although by definition wavelength is the distance between the two closest maxima, one can observe only one half of this wave (one maximum) over one given hemisphere of the Sun (Hale’s polarity law). Nevertheless, a similar idea is used in most relatively simple models (e.g., Parker, 1979), and here we use the following theoretical equivalent. It is known Solar Physics 173: 1–14, 1997. c 1997 Kluwer Academic Publishers. Printed in Belgium.

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KIRILL M. KUZANYAN AND DMITRY SOKOLOFF

that the amplitude of the magnetic field with respect to latitude f ( ) in the vicinity of the dynamo wave maximum has a parabolic form, q and so00we may introduce the half-width of the dynamo wave and define it to be ,2f=f taken at the point of the dynamo wave maximum. Note that this quantity is similar to one used in radio physics, because this is the width of a signal at a level of half maximum amplitude. For Gaussian signals this quantity is larger than the one defined above by factor p 2 log 2 1:67. From the first naive point of view, the half-width of the dynamo wave is of order O(jDj,1=6 ), where D is a dimensionless dynamo number, which characterises the intensity of the generation sources. Indeed, the complex solution for Parker’s migratory dynamo equations in the leading order of the asymptotic expansion for jDj 1 has the form exp fijDj1=3 a( , max )2g (see, e.g., Parker, 1979; Isakov et al., 1981; Kuzanyan and Sokoloff, 1995), where a is a constant of order unity and max is the point of the solution maximum. The half-width of this function is O(jDj,1=6 ). However, we demonstrate below that the scaling for the real part of this solution is surprisingly O (jD j,1=3 ). 2. Asymptotic Solution for a Dynamo Wave Mean field magnetohydrodynamics (Krause and Rädler, 1980) in a thin differentially rotating convection shell for the axisymmetric case at large dynamo numbers gives the following equations governing a dynamo wave (Parker, 1955; see also Stix, 1989):

@A = ()B + @ 2A ; @t @2 @B = ,DG() cos @A + @ 2B : (1) @t @ @2 Here B is the azimuthal component of the mean magnetic field, A is proportional to the azimuthal component of the magnetic potential, D is the dimensionless

dynamo number which characterises the intensity of the sources of the magnetic field generation and is the latitude in the shell that is measured from the solar equator. Furthermore, ( ) is the mean helicity and G( ) the radial gradient of the angular rotation, and these quantities are normalised with respect to their maximum values, say, and G , respectively. The asymptotic solution of Equations (1) for jD j 1 has been obtained by Kuzanyan and Sokoloff (1995) using the WKB method, which shows that at leading order with respect to small parameter jD j,1=3

0 @

A

jDj,2=3 B

1 A = exp (ijDj1=3 S + jDj2=3 ,0t + )(f0 + ) ;

(2)

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where S and vector f0 are complex functions of latitude . Below we use the form of function S (see Figure 1) and eigenvalue ,0 obtained by Kuzanyan and Sokoloff (1995). For the sake of definiteness, we use the simplest form of function G( ): G = const = 1. We consider hereafter the case when D < 0, which yields a basically equatorward dynamo wave. The asymptotic solution under consideration should enable the dynamo wave to decay at locations remote from the maximum of the sources of generation. As shown by Kuzanyan and Sokoloff (1995) the function ^ = () cos appears in an asymptotic analysis of Equations (1) and it corresponds to the source of generation of magnetic field. So the solution should vanish remote from the domain where ^ is maximum. In other words, we consider a dynamo wave decaying to ‘remote boundaries’ at the pole and the equator. Note, however, that for some helicity profiles the role of ‘remote’ boundaries can be more sophisticated (see, e.g., Worledge et al., 1996). Let us describe briefly those properties of the solution which will prove to be important. The growth rate Re ,0 and the frequency of oscillations Im ,0 are principally determined by the location of the maximum of the generation sources 0; however the maximum of the solution is situated at point 1 < 0 where Im S () is a minimum and for which

p ^ (1 ) 9 3 ^1 = ^ = p qp 0:81 ; 16 2 3,1

where ^ = ^ (0 ) is the maximum of function ^ () (the sources of generation). For ( ) = sin we have 0 = =4 = 45 and 1 27 (see Figure 1(a)). The dynamo wave propagates mostly equatorwards. However, at point 2 > 0 , for which

p ^ (2 ) 9 6 ^2 = ^ = 64 0:35 ; the quantity Re S 0 ( ) changes its sign, and the dynamo wave reverses. For ( ) = sin we have 2 80 (see Figure (1)b). This poleward dynamo wave has an

amplitude much less than the equatorward one, and decays while propagating polewards. This reversal is actually observable over the Sun (e.g., Makarov and Sivaraman, 1983); see comparison with observations in Kuzanyan and Sokoloff (1995). 3. Dynamo Number To compare this asymptotic solution with observations, an estimate of the solar dynamo number is needed. The following definition for the dynamo number is used in Equations (1):

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Figure 1. Asymptotic solution of Equations (1) with helicity () = sin . The following dependencies upon latitude are shown: (A) amplitude of exponential term of solution of Equations (1) for D = 103 ; (B) real part of the wave number of the dynamo wave Re S 0 . The point of reversal of the dynamo wave propagation is circled (A). To the left from this point the dynamo wave propagates equatorwards, and to the right – polewards. The direction of the dynamo wave propagation is shown by the arrows (A). As a normalization condition Im S (0 ) = 0 is accepted.

,

jDj = R04 G2 ;

(3)

where R0 7 1010 cm is the solar radius and the turbulent magnetic diffusivity. For a crude estimation we use G =R0 , where 2:7 10,6 s,1 is the angular velocity of the Sun. So that certainly jD j = R03 = 2 . Other definitions of D are possible, e.g., Kleeorin, Ruzmaikin, and Sokolov (1983) use the quantity R0 h2 instead of R03 , where h 0:3R is the thickness of the convection shell, so the corresponding value of D is thus an order of magnitude lower than the one used here. Other authors (e.g., Belvedère, Proctor, and Landzafame, 1991) use d3 instead of R03 , where d is the thickness of the layer at which the dynamo wave is generated, and this is even thinner than h. Mixing-length theory for turbulent diffusivity (see, e.g., Spruit, 1974; Zeldovich, Ruzmaikin, and Sokoloff, 1983) gives lv0 =3, where l is the size, and v0 the small-scale velocity of turbulent vortices. Using the scale of supergranulation (7 108 cm) as l and the corresponding velocity (105 cm s,1 ) as v0 (see, e.g., Howard et al., 1991; also Kleeorin, Rogachevsky, and Ruzmaikin, 1989), we obtain

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2 1012 cm2 s,1. This estimate is confirmed by modern advances in the theory

(e.g., Parker, 1993; Pipin, 1995). Mixing length theory gives l2 =h0 (Krause, 1967; see also, e.g., Zeldovich, Ruzmaikin, and Sokoloff, 1983), where h0 is the vertical characteristic scale of velocity and density distribution. Assuming the standard hydrodynamic model of the Sun (see, Köhler, 1973; also Stix, 1981; Gilman and Miller, 1981; DeLuca and Gilman, 1991) one may suppose h0 to be less than the smallest size of a solar granule 7 107 cm (Roudier and Miller, 1986). This gives the very high estimate that is of order 2 104 cm s,1. Schmitt, Rosner, and Bohn (1984) suggest even the upper estimate of h0 of order 109 cm, but here we use a higher value of h0 . Taking an upper estimate of h0 to be of the order of the convection shell thickness h0 h R0 =3 2 1010 cm results in 70 cm s,1 and 2 2 jDj 9 Rh 0 Rv0 2 102 : 0

0

One can obtain larger values of jD j by taking lower values of the vertical stratification scale, by considering magnetic field generation in a thin overshoot layer (see, e.g., Schmitt, Rosner, and Bohn, 1984, or Rüdiger and Brandenburg, 1995), or by proposing more developed models of the solar convection (see, e.g., Howard et al., 1991). One can derive another estimate of jD j by noting that for large jD j the period of the solar cycle To according to Parker’s migratory dynamo approach is (in dimensional units) 2 To = Im2, jDj,2=3 R 0 :

(4)

0

For ( ) = sin Kuzanyan and Sokoloff (1995) obtained Im ,0 0:26, and since To 22 yr, the estimate for given above leads to jDj 800 : (5) Because Equations (1) do not take into account various kinds of dissipation except a simple diffusion in the latitudinal direction, e.g., the dissipation due to magnetic field diffusion in the radial direction, they underestimate the generation threshold value of jD j. Consequently, the above-mentioned values are supercritical for Parker’s migratory dynamo, but they can be sub-critical for more detailed models. These models of the magnetic field generation, which do not contradict the models of the solar interior, use larger values of jD j, which can force the cycle period to be shorter than that observed. This is the well-known problem of the duration of the cycle in solar dynamo models. Taking into account the order-of-magnitude nature of these estimates, we now take jD j = 103 and artificially identify the dynamo wave period with the observed

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duration of the cycle. In view of estimate (5), this approximation can be considered to be reasonable. Furthermore, we suppose that magnetic field growth is somehow suppressed by a back-reaction of magnetic field on the motion and that we have Re ,0 = 0. 4. Butterfly Diagram for the Asymptotic Solution Let us now calculate the location and the half-width of the zone of increased value of the toroidal field (the zone of the dynamo wave maximum) at various times. We introduce the following function that is proportional to the magnitude of the toroidal magnetic field in the leading order of asymptotic expansion:

F (; t) = Re fexp[ijDj1=3 S () + ijDj2=3 Im ,0t]g ;

(6)

here t is the dimensionless time measured in units of the diffusion time R02 = . This function is plotted in Figure 2. The picture seems to be comparable with Maunder butterfly diagrams (see, e.g., Wilson et al., 1988). The extreme condition for function F gives Im S 0 cos + Re S 0 sin = 0 ;

(7)

where = jD j1=3 Re S [e (t)] + jD j2=3 Im ,0 t. Then at leading order with respect to the small parameter jD j,1=3 we obtain the latitude of the extreme points of the dynamo wave as a function of time e (t) via the following equation:

jDj1=3 Re S [e(t)] + jDj2=3 Im ,0 t = n; n = 0; 1; 2; : : : : (8) Maxima correspond to even values of n and minima correspond to odd n. Following a definite extremum, say, a maximum (n = 0), we study the propagation and the phase velocity of the dynamo wave. Suppose that e = 2 at t = 0,

i.e., the dynamo wave commences at the point of reversal. In other words, we use Re S (2 ) = 0 as a normalisation condition for function Re S ; the temporal evolution of e (t) is shown in Figure 3. Numerical integration of the asymptotic solution obtained by Kuzanyan and Sokoloff (1995) yields the estimate Re S (0) ,0:348. Note that the point = 0 (the equator) is approximately attained by the dynamo wave under assumption (4). Indeed, at instant t = To =2 we have

jDj1=3 Re S [e(To =2)] = , (in dimensionless units). This corresponds to some latitude f = e (To =2) 6:3 which is quite close to the equator. Then the maximum of the dynamo wave finishes its migration over the solar hemisphere and a new half-cycle commences.

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Figure 2. Values of function F (; t) measured in relative units over the solar cycle. The picture is qualitatively the same as the well-known Maunder butterfly diagrams.

Thus, assumption (4) yields at one and the same time only one equatorward wave of the solar sunspot migration. Hale’s polarity law is valid for our model. For larger values of the dynamo number we may expect latitude f to be far from the equator, so we may suppose a violation of this law. Let us approximate function F in the vicinity of its extremum = e in a parabolic form and introduce the half-width of the dynamo wave (t) as follows:

v u 2F [e(t); t] u (t) = [e(t)] = u : t, @ 2 @2 F [e(t); t]

Substituting condition (7) into this formula results in

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Figure 3. The latitude of the maximum of the equatorward dynamo wave e (t) over the solar cycle (n = 0). The bold dashed line indicates the poleward dynamo wave. The equatorward dynamo wave is marked ‘A’, the poleward ‘B’. Thin dashed line indicates points of termination of the dynamo wave, f = e (To =2) 6:3 (D = 103 ).

,

p ,1=3 [e (t)] = jS2[jD(j t)]j : e

(9)

The half-width of the dynamo wave maximum is shown as a function of e in Figure 4. It is noted that the width of a butterfly diagram over latitude is of order 15 to 30 , and this seems to be in accord with available observations. In fact, one can calculate the width of a butterfly diagram at any latitude, while it is observable via sunspot data only for latitudes lower than approximately 50 . Note that Equation (8) for the poleward dynamo wave ( 2 ) is also applicable. This formula enables one to calculate the time of the dynamo wave propagation polewards over the high latitudes, which appears to be about 4 months. This time is

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Figure 4. The half-width of the maximum of the dynamo wave (e ) versus the latitude of the maximum.

however much shorter than that observed in practice which typically is of the order of a few years (e.g., Makarov and Sivaraman, 1983). A possible explanation of this discrepancy is that details of the generation of the dynamo wave in the vicinity of the solar pole ( =2, cos 1) is beyond the framework of our approach, i.e., Parker’s migratory dynamo (see, e.g., Sokoloff, Fiot, and Nesme-Ribes, 1995, for derivation of equations governing Parker’s migratory dynamo). The phase velocity of the dynamo wave can be calculated to be

Dj1=3 Im ,0 ; vph (t) = ddte (t) = , jRe S 0[ (t)] e

(10)

and this is shown in Figures 5 and 6. Finally the area covered by the maximum amplitude of a dynamo wave can be estimated according to

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Figure 5. Phase velocity of the dynamo wave vph measured in degrees per year over the solar cycle versus latitude. The equatorward dynamo wave is marked ‘A’, the poleward ‘B’. The thin dashed line indicates an asymptote for reversal point = 2 .

(t) = 2R02 [e (t)] cos e(t) :

(11)

This dependence is given by Figure 7. This function is proportional to the sunspot number for times later than about 2–3 years.

5. Discussion We have considered the ! -dynamo problem, which includes inhomogeneity in a localization of the sources of the magnetic field generation. We have used a kinematic formulation with a simple dependence of ( ), no meridional flow, no radial dependence (Parker’s migratory dynamo). The ! -effect has been taken to be

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Figure 6. Phase velocity of the dynamo wave vph versus time. The notations used are the same as in Figure 5.

constant (G = const: = 1). Nevertheless, we have obtained a qualitative picture of the dynamo wave propagation over the solar convection zone in one hemisphere. The solution constructed is quite similar to Maunder butterfly diagrams and exhibits both the growing equatorward migration of the magnetic activity and the poleward decaying branch. We choose the simplest profiles of generation sources ( ) and G( ). Evidently we could modify these profiles in order to fit observational data concerning the half-width of the dynamo wave. In principle, this idea could be directly developed to a mathematical position of an inverse problem of reconstruction of product G from sunspot observational data. However, such a problem requires much deeper analysis of the observational data and, in particular, finding out an observational equivalent of the half-width of the solar dynamo wave. We believe that such a task

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Figure 7. Area of the maximum of the equatorward dynamo wave (t) in percent of the total surface of one solar hemisphere 0 = 2R02 versus the solar cycle. This value is supposed to be proportional to the area mostly covered by sunspots for times later than about 2–3 years.

requires the contribution of observers and so is beyond the scope of the present paper. Let us also discuss the result obtained in the context of the general properties of dynamo waves. The order jD j,1=3 in formula (9) with respect to the dynamo number arises because we only study the real part of the exponential term in

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formula (6). A simple demonstration of this type of behaviour arises from the simple function f (x) = exp [(ax2 + ibx)=], in which a and b are real constants and a small parameter.pHere the half-width of its envelope, i.e., the absolute value jf (x)j has the order O( ), while the half-width of its real part Re f (x) is only of order O (). Let us now consider a gedanken experiment with a dynamo wave propagating through an infinite medium with ( ) decaying to infinity. Our results indicate that the finite amplitude part of the solution consists of a modulated wave localised to a packet. The wavelength is of order jD j,1=3 , while the packet length is of order jDj,1=6 (see Figure 1(a)), and so there are of order of jDj1=6 waves in the packet. Note, that numerical simulations for nonlinear dynamo waves in a long tube by Proctor, Tobias, and Knobloch (1997) give comparable results.

Acknowledgements The authors are grateful to the Royal Society for supporting their visits to the Mathematics Department of Exeter University (D. S., 2 June, 1996 to 31 August, 1996 and K. K., 2 October, 1996 for one year). The authors are grateful to A. P. Bossom, V. N. Obridko, M. R. E. Proctor, E. Schatzman, A. M. Shukurov, A. M. Soward, and S. Tobias for valuable discussions. The work is supported in part by the Russian Foundation for Fundamental Research, grants No. 96–02–16252a and No. 95–02 –03724 and the Foundation ‘Russian University’.

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