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This talk proposes a mathematical theory explaining the sharp transition between tropics and extra-tropics in terms of the diurnal cycle of thermal forcing by the sun. This transition, at a latitude of 30 degrees, coincides with the outer edge of the Hadley cells, and is marked by a steep jump in the height of the troposphere, from around fifteen kilometers in the tropics to about nine in the mid and high latitudes. The tropics, equatorwards of 30 degrees, are characterized by easterly surface winds -the Trades- and a strong diurnal signal in the wind, pressure and temperature, often marked by regular daily storms in the rainy season. Polewards of 30 degrees, the winds are westerly, and the weather systems have longer spatio-temporal scales.

This change of behavior can be explained in terms of diurnal waves, created by thermal forcing and trapped equatorwards of 30 degrees by the Coriolis effect \cite{MilewskiTabak}. These waves organize the convective activity, leading to more active mixing and vertical transport in the tropics. This can be illustrated in simple mathematical models, presently ranging from forced linear oscillators to nonlinear conservation laws with entraining shock waves, accounting for the entrainment into the troposphere of air from the surface boundary layer.