Equatorial magnetohydrodynamic shallow water waves in the solar tachocline.

The influence of toroidal magnetic field on the dynamics of shallow water waves in the solar tachocline is studied. Sub-adiabatic temperature gradient in the upper overshoot layer of the tachocline causes significant reduction of surface gravity speed, which leads to the trapping of the waves near the equator and to the increase of Rossby wave period up to the time scale of solar cycles. Dispersion relations of all equatorial MHD shallow water waves are obtained in the upper tachocline conditions and solved analytically and numerically. It is found that the toroidal magnetic field splits equatorial Rossby and Rossby-gravity waves into fast and slow modes. For reasonable value of reduced gravity, global equatorial fast magneto-Rossby waves (with the spatial scale of equatorial extent) has the periodicity of 11 yrs matching the time scale of activity cycles. The solutions are confined around the equator between 20-40 latitudes coinciding with sunspot activity belts. Equatorial slow magneto-Rossby waves have the periodicity of 90-100 yrs resembling observed long-term modulation of cycle strength i.e. Gleissberg cycle. Equatorial magneto-Kelvin and slow magneto-Rossby-gravity waves have the periodicity of 1-2 yrs and may correspond to observed annual and quasi-biennial oscillations. Equatorial fast magneto-Rossby-gravity and magneto-inertia-gravity waves have the period of hundreds of days and might be responsible for observed Rieger-type periodicity. Consequently, the equatorial MHD shallow water waves in the upper overshoot tachocline may capture all time scales of observed variations in solar activity, but detailed analytical and numerical studies are necessary to make firm conclusion towards the connection of the waves to solar dynamo.