A semi-geostrophic model incorporating well-mixed boundary layers


  • Robert J. Beare,

    Corresponding author
    1. College of Engineering, Mathematics and Physical Sciences, University of Exeter, UK
    • School of Engineering, Mathematics and Physical Sciences, University of Exeter, EX4 4QF, UK.
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  • Michael J. P. Cullen

    1. Met Office, Exeter, UK
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    • The contribution of Michael J. P. Cullen was written in the course of his employment at the Met Office, UK and is published with the permission of the Controller of HMSO and the Queen's Printer for Scotland


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.