The Waldmeier effect and the flux transport solar dynamo

0 Department of Physics, Indian Institute of Science , Bangalore 560012 , India A B S T R A C T We confirm that the evidence for the Waldmeier effect WE1 (the anticorrelation between rise times of sunspot cycles and their strengths) and the related effect WE2 (the correlation between rise rates of cycles and their strengths) is found in different kinds of sunspot data. We explore whether these effects can be explained theoretically on the basis of the flux transport dynamo models of sunspot cycles. Two sources of irregularities of sunspot cycles are included in our model: fluctuations in the poloidal field generation process and fluctuations in the meridional circulation. We find WE2 to be a robust result which is produced in different kinds of theoretical models for different sources of irregularities. The Waldmeier effect WE1, on the other hand, arises from fluctuations in the meridional circulation and is found only in the theoretical models with reasonably high turbulent diffusivity which ensures that the diffusion time is not more than a few years. 1 I N T R O D U C T I O N Waldmeier (1935) noted an anticorrelation between the rise times of sunspot cycles and their strengths. In other words, a cycle with a longer rise time is expected to have a weaker peak at the maximum. This is known as the Waldmeier effect. We shall refer to this as WE1. There is another related effect. The rise rates of cycles show a correlation with their strengths: a faster rising cycle is likely to be stronger. We shall call it WE2. Occasionally one uses the term Waldmeier effect to also mean this second effect WE2, causing some amount of confusion in the literature. For example, sometimes one talks of using the Waldmeier effect to predict the strength of a sunspot cycle after it has just begun. In this case, clearly WE2 which involves rise rates is meant rather than WE1 which involves rise times. Shortly after a sunspot cycle has begun, it becomes possible to estimate its rise rate, but it is not possible to know the rise time until the cycle has reached its maximum. The aim of this paper is to explore whether the Waldmeier effect can be explained with the flux transport dynamo model, which presently appears to be the most promising model for explaining the solar cycle. The flux transport dynamo model involves several parameters, some of which are rather poorly known at the present time. One important question is whether the Waldmeier effect is reproduced theoretically only for certain combinations of parameters. If that is the case, then it should be possible to put some constraints on the parameters of the flux transport dynamo by demanding that the theoretical model accounts for the Waldmeier effect. We also E-mail: (BBK); . ernet.in (ARC) present a short discussion of the observational data, in view of a recent controversy whether the Waldmeier effect really exists. Hathaway, Wilson & Reichmann (2002) found evidence for WE1 in both the Z urich sunspot numbers and the group sunspot numbers, but Dikpati, Gilman & de Toma (2008) claim that this effect does not exist in sunspot area data. We argue that the rise time has to be properly defined to obtain the Waldmeier effect. In our opinion, Dikpati et al. (2008) failed to discover WE1 in the sunspot area data because they had not taken proper rise times. With a proper definition of the rise time, we show that WE1 is present in various kinds of sunspot data. The other effect WE2 seems more robust. Cameron & Schussler (2008) found evidence for WE2 in various kinds of sunspot data, which we also confirm. Thus, in our view, both WE1 and WE2 are real effects which a satisfactory theoretical model of the sunspot cycle should explain. Let us mention some of the salient features of the flux transport dynamo model, which has been developed by many authors during the last few years (Choudhuri, Schussler & Dikpati 1995; Durney 1995; Dikpati & Charbonneau 1999; Kuker, R udiger & Schultz 2001; Nandy & Choudhuri 2002; Choudhuri 2003; Chatterjee, Nandy & Choudhuri 2004; Choudhuri, Chatterjee & Nandy 2004; Munoz-Jaramillo, Nandy & Martens 2009). The toroidal magnetic field is produced in the tachocline by the action of differential rotation on the poloidal field and eventually rises to the solar surface due to magnetic buoyancy to produce sunspots. The decay of tilted bipolar sunspots gives rise to a poloidal field near the surface by the mechanism first elucidated by Babcock (1961) and Leighton (1969). The meridional circulation, which is observed to be poleward in the upper half of the solar convection zone (SCZ) and must have a hitherto unobserved equatorward component at the bottom of SCZ to conserve mass, advects the toroidal field equatorward at the bottom of the SCZ and advects the poloidal field poleward at the surface. This provides the theoretical explanation of the equatorward drift of sunspot belts as well as the poleward migration of the weak diffuse magnetic field on the solar surface. Lastly, we need a mechanism to transport the poloidal field from the surface where it is generated by the BabcockLeighton mechanism to the bottom of SCZ where differential rotation can act on it. This transport of the poloidal field can be achieved by two means: through advection by the meridional circulation or through diffusion. Yeates, Nandy & Mackay (2008) have divided flux transport dynamo models into two classes: advection dominated and diffusion dominated, depending on the transport mechanism of the poloidal field from the surface to the bottom of SCZ. Jiang, Chatterjee & Choudhuri (2007) and Yeates et al. (2008) were the first to point out many qualitative differences between these two kinds of models. Many authors (Chatterjee et al. 2004; Chatterjee & Choudhuri 2006; Jiang et al. 2007; Choudhuri & Karak 2009; Goel & Choudhuri 2009; Hotta & Yokoyama 2010a,b) have given several independent arguments that the solar dynamo is likely to be diffusion dominated. We shall show in this paper that only diffusion-dominated dynamos and not advection-dominated dynamos can account for the Waldmeier effect WE1, further strengthening the case that the solar dynamo is diffusion dominated. The readers should be cautioned that in the early years of flux transport dynamo research sometimes the term advection dominated was used rather loosely and may not always conform with our present usage. In this paper, we shall use the terms advection dominated and diffusion dominated following the careful classification introduced by Yeates, Nandy & Mackay (2008; see their fig. 7a). It should also be noted that at the bottom of SCZ the advection of the toroidal field by the equatorward meridional circulation has to be the dominant process over diffusion, as emphasized by Choudhuri et al. (1995). If this were not the case, then the dynamo wave would propagate poleward, following the dynamo sign rule (Parker 1955; Yoshimura 1975; Choudhuri et al. 1995; see also Choudhuri 1998, (...truncated)