Stable boundary-layer relative humidity profiles and the conditions for onset of radiation fog over land

Fog is a physically complex problem, with substantial impacts motivating further advances in prediction. We focus here on radiation fog over land, and the typical development of an optically thin initial phase which can be regarded as a continuation of the stable boundary layer. Textbook analytical models exist for quasi-steady stable boundary layer models, but these have hitherto not been extended to treat the relative humidity appropriate to the onset conditions for fog. Such extension is complicated by the nonlinearity of the saturation curve, identified in Taylor's classic work as a key term due to the high level of cancellation between cooling and drying fluxes. Here we show how, by idealization of the saturation curve, the nonlinear mixing mechanism can be combined with analytical stable boundary layers so as to provide solutions for two canonical cases. To complement these solutions, a simple numerical model is presented, illustrating key aspects of behaviour and the potential impact of additional physics. A non-dimensional parameter is derived as a ratio of time-scales governing the magnitude of nonlinear mixing in the boundary-layer problem. Boundary-layer properties, surface transfer and water properties, saturation nonlinearity, clear-air radiation and sedimentation are all supported as important ingredients in the general fog problem. The findings suggest that a hierarchical approach can be developed in a manner traceable to the analytic solutions for the simpler cases.