ASYMPTOTIC PROPERTIES OF A NONLINEAR aft-DYNAMO D. SOKOLOFF Dept. of Physics, Moscow State University, Moscow, Russia1 A.M. SOWARD Dept, of Mathematics, University of Exeter, Exeter, UK2 S u m m a r y : We discuss various asymptotic regimes of kinematic nonlinear afldynamos in a thin differentially rotating shell as well as conditions for the applicability of the corresponding asymptotic solutions. Key
w o r d s : Mean field dynamo, asymptotic solution, stellar cycles
1. INTRODUCTION The 11-year solar cycle is thought to be connected with dynamo activity in the solar overshoot layer which results in dynamo wave propagation through this layer. The manifestation of this dynamo wave on the solar surface is exhibited by the wellknown butterfly diagrams for sunspots whose latitudinal distribution have the form of waves propagating from the solar poles to the equator. The butterflies do not fill the whole latitudinal extent of the solar overshoot layer, but rather are localised between a critical latitude 6, somewhere between 40° and 50° and the solar equator. Cyclic activity more or less comparable with the solar case is known for many stellar taxonomic classes. This stellar activity is also thought to be connected with a dynamo. All stars (except the Sun) are unresolvable sources. However due to some sophisticated methods of data processing (see e.g. Strassmeier and Linsky, 1996) the latitudinal distribution of starspots is known in some cases. Sometimes starspots occur more or less on the whole stellar surface; sometimes they are confined to a belt near the equator. Stellar and solar dynamos are traditionally studied by numerical methods. The numerical approach has, however, a conceptual shortcoming in the following sense. The morphological distinction between stars with an active belt and the whole surface starspot generation seems to be obvious. On the one hand, however, the ratio of the latitudinal extent of the belt covered by sunspots to the whole latitudinal extent of the overshoot layer is only about 1/3. On the other, it is difficult to insist that the parameters governing dynamo activity are known better than up to a factor of order unity. In particular, the most important generator of mean magnetic field, mean helicity in the overshoot layer, can not be observed directly and is estimated from essentially obscure theories. That is why we should be more interested in qualitative tendencies in stellar dynamos over a wide parameter range rather than in a quantitative fitting of observations by numerical models. 1 2
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From this point of view, the asymptotic approach is powerful. It may be used to obtain analytic solutions of some simplified models of stellar dynamo, which are suitable for this qualitative analysis. Such asymptotic solutions have been obtained in last few years (see Kuzanyan and Sokoloff, 1995; Meunier et al, 1997; Bassom et al, 1997); though the problem of their applicability to stellar dynamos is still a matter of discussion. We try to clarify this problem below. 2. PARKER MIGRATORY DYNAMO The dynamo equations in a thin differentially rotating shell (overshoot layer) with helical convection may be written as
Here B is the azimuthal component of the large-scale magnetic field, A is proportional to the azimuthal component of magnetic potential, D is the dimensionless dynamo number which characterises the intensity of magnetic field generation, p is the inverse dimensionless thickness of the generation layer, 0 is the latitude measured from the equator, a(6,B) is the mean helicity, and G(0) = r - 1 d f l / d r is the radial gradient of angular velocity D. Notice that a and ft are normalized to their largest values, a, and G* respectively. Nonlinear effects can be introduced into Eqs. (2.1, 2.2) via a-quenching as follows: a = a0(0)F(6, B), where a0(0) is the kinematic helicity coefficient. Eqs. (2.1, 2.2) are basically the same as those governing the well-known dynamo in a thin layer (Cartesian) approximation (Parker, 1955). However, the spatial variation in the coefficients correspond to those that follow from the original Mean Field Electrodynamics equations. In particular, the factor cos0 in Eq. (2.1, 2.2) takes into account the shortening of the length of a latitudinal circle B = const from the equator to the pole. This factor plays a key role in the analysis below. Eqs. (2.1, 2.2) do not take into account the curvature of the layer in which magnetic field generation occurs; it is assumed that the layer thickness is much smaller than the stellar radius: the detailed nature of the magnetic field structure across the layer is ignored. Correspondingly, the values of A and B are averaged across the layer and the second radial derivatives are replaced by a simple decay term. The longitudinal gradient of angular velocity is assumed to be much smaller than the radial one. The product aG is supposed to be negative in the generation layer to give the equatorialward dynamo wave. The Parker migratory dynamo can be investigated by a linear kinematic approach when a and G are assumed to be independent of magnetic field. The simplest nonlinear extension of Parker migratory dynamos can be obtained by supposing that helicity is diminished with growing magnetic field B: eg. a = a0(d)F(B),
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where F decays when B grows. This is so-called a-quenching. Of course, many other types of nonlinear influences by the magnetic field on the flow are possible, however a-quenching is thought to be a reasonable parametrisation of nonlinear stellar dynamos. 3. MAIN ASYMPTOTIC REGIMES The basic idea of the asymptotic approach is to explain how the term containing dynamo number D can be compensated in Eqs. (2.1, 2.2), when D is large. The compensation mechanism is different in the kinematic and nonlinear regimes. For nonlinear dynamos, they depend also on the value of fi, i.e. on the role of radial diffusion. That is why stellar dynamos are expected to operate in several different asymptotic regimes. 3 . 1K i n e m a t i c
dynamos
In the kinematic case, Eqs. (2.1, 2.2) subject to suitable boundary conditions can be considered as an eigenvalue problem and terms with ju can be incorporated into the eigenvalue. The only way to compensate the large dynamo number term is to suppose that the latitudinal scale of dynamo wave decreases with D as |D| -1/3 , The dynamo wave frequency (the cycle length) depends on the spatial structure of aG near the latitude 9* at which this product is maximal. The dynamo wave propagates from the pole to the equator. Simultaneously its amplitude grows with maximum growth rate at the point 0* but continues to grow at lower latitudes. The actual maximum of dynamo wave amplitude occurs at some lower latitude Q1 < 6* (Kuzanyan, Sokoloff, 199$). Of course, direct applications of this kinematic solution are very restricted. As a thought experiment, let us consider a star with a very small seed magnetic field, In its early linear stage of development, we expect it to exhibit a very thin belt at low latitudes. Indeed, such a kinematic solution may well be applicable to the first solar cycle following the Maunder minimum. Due to sunspot monitoring performed during the first normal cycle after the Maunder minimum by La Hire at Observatoire de Paris, we know, that corresponding butterfly diagram is extremely narrow and concentrated very near to the equator. 3.2W e a k g e n e r a t i o n a s y m p t o t i c If the dynamo number D exceeds the generation threshold Dcr only slightly, we can suppose that the role of nonlinearity is to suppress the growth of magnetic field. Once achieved, we expect the spatial magnetic field distribution as well as its oscillation frequency to be the same as that for the corresponding kinematic solution but with quenched a. Such an asymptotic solution was been obtained for Eqs. (2.1, 2.2) by Brauer (1979) but the appropriate weakly nonlinear extension of Kuzanyan and Sokoloff 's (1995) model is given by Bassom et al (1997). In both Studia geoph. et geod. 42 (1998)
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cases, the amplitude of this weakly nonlinear solution is proportional to \/D — Dcr. Presumably, weak solar cycles could be identified with this solution. The generation threshold is relatively large for the solar dynamo. In other words, the radial diffusion described by the /z-terms in Eqs. (2.1, 2.2) plays an important role. Due to this fact, there is a separation of latitudinal length scales, and so we can use the kinematic solution of Parker migratory dynamo to obtain the spatial magnetic field structure. By this we mean that the requirement for the validity of this application is that the latitudinal extent A0 of the dynamo wave at a given instant is much smaller then 90°, i.e. the latitudinal extent of a given hemisphere. In practice, the relevant ratio is only about 1/3. However, practical experience with WKB-asymptotics in quantum mechanics shows that such a ratio may be small enough to reproduce the qualitative features of the actual solution. Kuzanyan and Sokoloff (1997) demonstrated that this extrapolation of asymptotic results to solar parameter values gives butterfly diagrams comparable with those from observations. 3.3 S t r o n g
generation
asymptotic
If the dynamo number D greatly exceeds Dcr so that the generation terms are still comparable with radial diffusion, yet another asymptotic regime arises (Meunier et al, 1997). Then magnetic field steady state configuration has the form of a wave, localized inside a latitudinal belt 01 < 0 < 02. Presumably, this regime can be connected with strong solar cycles. The corresponding wavelength is governed by the same scaling as in the previous asymptotic regime and the applicability condition for the regime under discussion is that A0 - 90°; evidently this solution has marginal applicability to the Sun. Meunier et al. (1997) give simple equations for #1 and #2. For definiteness, we now consider the simplest form of helicity consistent with its symmetry properties, specifically a0(0) = sin0 with quenching F(B) = 1/(1 + B2). This model is characterised by 01 < 45° < (02; the belt, however, is not symmetric in respect to 0 = 0* = 45°, specifically 01 < 90° - 02. The origin of this asymmetry can be explained as follows: The dynamo wave propagating from the pole to the equator needs stronger generation to switch on from a negligible to equipartition values than is required to survive once equipartition has been achieved - i.e. it requires to be kick-started! The values of 01 and 02 decrease and grow respectively with increasing dynamo number. 3.4E x t r e m e l y
strong generation
If the dynamo number is even larger, the boundaries 01 and 02 of the belt in the solution described above in §3.3 move close to the equator and pole respectively. Then the conditions for validity of that asymptotic solution are no longer met. In this new asymptotic regime, the corresponding asymptotic solution of Eqs. (2.1, 2.2) is still unknown. Nevertheless, recently the situation has been clarified numerically by Proctor et al. (1998) (private communication). They point out that, once the Parker dynamo wave fills the entire region from the equator to the pole, a secondary bifurcation occurs manifest as a second wave with a new frequency riding on the 312
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primary wave and developing in much the same way. With yet further increase of the generation source complicated nonlinear dynamics ensues leading to chaos. Acknowledgements: We are grateful for M.R.E. Proctor for important discussions. A financial support from RFBR project 97-05-64797 and organizing committee is acknowledged (DS). Manuscript received: 31st January, 1998;
Revisions accepted: llth May 1998
References Bassom, A.P., Kuzanyan, K.M. and Soward, A.M., 1997: A nonlinear dynamo wave riding on a spatially varying background. Proc. R. Soc. Lond. A, (submitted) Brauer, H., 1979: The nonlinear dynamo problem: Small oscillatory solutions in a strongly simplified model. Astron. Nachr., 300, 43-49 Kuzanyan, K.M. and Sokoloff, D.D., 1995: A dynamo wave in an inhomogeneous medium. Geophys. Astrophys. Fluid Dynam., 81, 113-129 Kuzanyan, K.M. and Sokoloff, D.D., 1997: Half-width of a solar dynamo wave in Parker's migratory dynamo. Solar Phys., 173, 1-14 Meunier, N., Proctor, M.R.E., Sokoloff, D.D., Soward, A.M. and Tobias, S.M., 1997: Asymptotic properties of a nonlinear ow-dynamo wave: period, amplitude and latitude dependence. Geophys. Astrophys. Fluid Dynam., 86, 249-285 Parker, E.N., 1955: Hydromagnetic dynamo models. Astrophys. J., 122, 293-314 Strassmeier, K.G. and Linsky, J.L. (eds.), 1996: Stellar Surface Structure. IAU, pp. 289
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