Semi-geostrophic theory has proved a powerful framework for understanding the dynamics of mid-latitude weather systems. However, one limitation is the lack of a realistic boundary-layer representation. Semi-geostrophic theory can be modified to include an atmospheric boundary layer by replacing the geostrophic wind with the ‘geotriptic’ (or Ekman-balanced) value in the substantive derivative and appropriately approximating the momentum diffusion term– the so-called semi-geotriptic theory. However, until now, solutions of the semi-geotriptic equations using predictor– corrector methods have not been possible for the important case of well-mixed boundary layers. Existing predictor– corrector methods require a Brunt– Väisälä frequency greater than zero to be solvable.
Here we describe a method of incorporating well-mixed boundary layers into semi-geotriptic theory. We modify the hydrostatic relationship by including a small horizontal diffusion of vertical velocity. This enables the formation of a well-posed predictor– corrector method. Given well-mixed boundary layers are a ubiquitous feature of the lower atmosphere, the modification increases the usability of the model. Calculations are also performed at much higher vertical resolution than before.
The revised semi-geotriptic model is compared with a hydrostatic primitive-equation model for a test case of a two-dimensional idealized diurnal cycle of a sea breeze. The performance of the revised semi-geotriptic model in the growth phase of the sea breeze is improved, as a well-mixed boundary layer is now permitted. The additional vertical resolution captures the capping inversion and the sea-breeze circulation better. The hydrostatic primitive-equation model is shown to produce inertial oscillations that persist beyond the evening decay of the boundary layer until the following morning. In contrast, the semi-geotriptic model decays following the boundary-layer state in a more realistic way. The semi-geotriptic model thus demonstrates its usefulness as a critical tool in understanding boundary-layer dynamics coupling issues in operational models. Copyright © 2010 Royal Meteorological Society and Crown Copyright.