Modeling the Sun's Large-Scale Magnetic Field during the Maunder Minimum

We use a flux transport model to simulate the evolution of the Sun’s magnetic dipole moment, polar fields, and open flux under Maunder minimum conditions. Even when the rate of active region emergence is taken to be a factor of � 30 smaller than in recent solar cycles, regular polarity oscillations of the axial dipole and polar fields can be maintained if the speed of the poleward surface flow is reduced from � 20 to � 10 m s � 1 and the source flux emerges at very low latitudes (� 10 � ). The axial dipole is then found to have an amplitude of the order of 0.5 G, as compared with � 4 G during solar cycle 21. The strength of the radial interplanetary field component at Earth is estimated to be in the range � 0.3–0.7 nT, about a factor of 7 lower than contemporary values. We discuss the implications of these weak fields for our understanding of geomagnetic activity and cosmic-ray modulation during the Maunder minimum. Subject headings: interplanetary medium — solar-terrestrial relations — Sun: activity — Sun: corona — Sun: magnetic fields — Sun: photosphere In a previous paper (Wang, Lean, & Sheeley 2002a), we simulated the evolution of the Sun’s large-scale magnetic field during solar cycles 13–22 (1888–1997). The model included the effect of emerging flux and its subsequent transport over the solar surface by differential rotation, supergranular convection, and a poleward bulk flow. When the flux eruption rates were scaled according to the observed sunspot numbers and the meridional flow speed was fixed, we found that the polar fields built up during the more active cycles were too strong to be reversed during the less active ones. Regular polarity reversals were obtained only if the flow rate was assumed to be positively correlated with the cycle amplitude. The inferred poleward velocities varied from � 15 m s � 1 for cycles 14, 16, and 20 to � 27.5 m s � 1 for cycle 19; the corresponding maximum yearly sunspot numbers for these cycles range from � 60 to � 200. During the period of the Maunder minimum (1645– 1715), sunspot numbers peaked at values even lower than those recorded during present-day activity minima (see Eddy 1976, 1983; Ribes & Nesme-Ribes 1993; Hoyt & Schatten 1996; Beer, Tobias, & Weiss 1998; Cliver, Boriakoff, & Bounar 1998; Letfus 2000). We now ask whether, by suitably adjusting the meridional flow speed, we can maintain the polarity reversals even with such a dramatic reduction in the source flux. If so, how strong are the derived polar and interplanetary fields, and how do they vary with time? How much open and closed magnetic flux remains during intervals when no activity is present at all? In a related investigation, Mackay (2003) has presented simulations of idealized ‘‘ grand minima ’’ using a flux transport model. Mackay’s approach differs substantially from ours, however, in that he fixes the flow amplitude and does not require that the polar fields reverse their sign during every cycle. The organization of this paper is as follows. We begin with a brief discussion of the sunspot data used to constrain our simulations of the Maunder minimum (x 2). After outlining our basic model and procedure (x 3), we show how the evolution of the Sun’s axial dipole and polar fields depends on the meridional flow velocity and on the source latitudes (x 4) and then illustrate how the large-scale magnetic field might have evolved during the Maunder minimum (x 5). Our results and their implications are discussed in x 6.