For small heating magnitudes, numerical simulations of the interior-heat-source problem reproduce all of the well-known predictions from classical linear theory: the character and phase of the response change abruptly between tropics and midlatitudes; the land- and sea-breeze phases of the flow are exactly equal and opposite; and the mean circulation averaged over one oscillation cycle is zero. However, as the heat source increases the flow becomes increasingly dominated by higher-frequency motions and the latitude depencence of the response thus becomes progressively less pronounced. The solution also develops an asymmetry in the land- and sea-breeze phases of the flow, with the sea breeze dominating in the tropics and with a slightly stronger land breeze in midlatitudes. And perhaps most interestingly, the nonlinear solution features a distinct cycle-mean circulation, with descending flow and divergence from the coastline in the tropics and shore-parallel flow with anticyclonic cross-coastal shear in midlatitudes. A weakly nonlinear analysis is presented showing that each of these trends can be predicted from the next-order corrections to the classical linear theory.

The interior heating simulations are then compared to the corresponding boundary-forced problem in an effort to identify artifacts associated with arbitrarily specifying the vertical and temporal dependence of the heat source. Specifying a boundary flux is shown to introduce higher frequencies into the response as the vertical distribution of heat by the parameterized mixing is not strictly diurnal. Nonetheless, the nonlinear trends observed for the boundary-forced problem are similar to those described for interior heating, and for sufficiently large heating magnitudes the two problems are qualitatively indistinguishable.