  We develop a theory of three-dimensional slow Rossby waves in rotating
spherical density stratified convection. The Rossby waves with frequencies
which are much smaller than the rotating frequency, are excited by a
non-axisymmetric instability from the equilibrium based on the developed
convection. These waves interact with the inertial waves and the density
stratified convection. The density stratification is taken into account using
the anelastic approximation for very low-Mach-number flows. We study long-term
planetary Rossby waves with periods which are larger than two years. We suggest
that these waves are related to the Southern Oscillation and El Ni\~{n}o. The
El Ni\~{n}o is an irregularly periodical variation in winds and sea surface
temperatures over the tropical Pacific Ocean, while the Southern Oscillation is
an oscillation in surface air pressure between the tropical eastern and the
western Pacific Ocean. The strength of the Southern Oscillation is
characterised by the Southern Oscillation Index (SOI). The developed theory is
applied for the interpretation of the observed periods of the SOI. This study
demonstrates a good agreement between the theoretical predictions and the
observations.
