Phase speed of magnetized Rossby waves that cause solar seasons

Motivated by recent analysis of solar observations that show evidence of propagating Rossby waves in coronal holes and bright points, we compute the longitudinal phase velocities of unstable MHD Rossby waves found in an MHD shallow-water model of the solar tachocline (both overshoot and radiative parts). We demonstrate that phase propagation is a typical characteristic of tachocline nonlinear oscillations that are created by unstable MHD Rossby waves, responsible for producing solar seasons. For toroidal field bands placed at latitudes between 5° and 75°, we find that phase velocities occur in a range similar to the observations, with more retrograde speeds (relative to the solar core rotation rate) for bands placed at higher latitudes, just as coronal holes have at high latitudes compared to low ones. The phase speeds of these waves are relatively insensitive to the toroidal field peak amplitude. Rossby waves for single bands at 25° are slightly prograde. However, at latitudes lower than 25° they are very retrograde, but much less so if a second band is included at a much higher latitude. This double-band configuration is suggested by evidence of an extended solar cycle, containing a high-latitude band in its early stages that does not yet produce spots, while the spot-producing low-latitude band is active. Collectively, our results indicate a strong connection between longitudinally propagating MHD Rossby waves in the tachocline and surface manifestations in the form of similarly propagating coronal holes and patterns of bright points.