Magnetohydrodynamic Instabilities of Double Magnetic Bands in a Shallow-water Tachocline Model. I. Cross-equatorial Interactions of Bands

Along with a butterfly diagram of sunspots, combined observational studies of ephemeral active regions, X-ray and EUV bright points, plage, filaments, faculae, and prominences demonstrate a pattern, which is known as the Extended Solar Cycle. This pattern indicates that the wings of the sunspot butterfly could be extended to much higher latitudes (up to similar to 60 degrees), to an earlier time than the start of a sunspot cycle, hence yielding a strong overlap between cycles. Thus, during the ongoing cycle's activity near 30 degrees latitude in each hemisphere, the next cycle kicks off at around 60 degrees. By representing these epochs of overlaps by oppositely directed double magnetic bands in each hemisphere, we compute the unstable eigenmodes for MHD Rossby waves at the base of the convection zone and study how the properties of these energetically active Rossby waves change as these band pairs migrate equatorward. We find that in each hemisphere the low-latitude band interacts with the high-latitude band and drives the MHD instability as the solar activity progresses from 35 degrees-15 degrees latitude, which is essentially the rising phase. When the activity proceeds further equatorward from 15 degrees, the interaction between low- and high-latitude bands weakens, and the cross-equatorial interaction between two low-latitude bands in each hemisphere starts. The eigenmodes in the latitude-longitude plane also reflect such changes in their pattern as the bend of the active cycle moves below 15 degrees latitude.