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Title: Approximate analytical solution to diurnal atmospheric boundary-layer growth under well-watered conditions

item Rigby Jr, James
item YIN, J - Duke University
item ALBERTSON, JOHN - Duke University
item PORPORATO, AMILCARE - Duke University

Submitted to: Boundary Layer Meteorology
Publication Type: Peer Reviewed Journal
Publication Acceptance Date: 3/5/2015
Publication Date: 3/31/2015
Citation: Rigby Jr, J.R., Yin, J., Albertson, J.D., Porporato, A. 2015. Approximate analytical solution to diurnal atmospheric boundary-layer growth under well-watered conditions. Boundary Layer Meteorology. 156:73-89 doi: 10.1007/s10546-015-0018-8.

Interpretive Summary: The atmospheric boundary-layer (ABL) is the region above the surface where turbulent eddies govern the exchange of heat and moisture between the upper atmosphere and the ground surface. Understanding this physical development and dynamics of this region is a key component in understanding surface effects on climate, local evapotranspiration, and the initiation of convective precipitation. Analytical solutions to such problems have the advantage of providing insight into the interactions of components of the system that eludes numerical investigation. Further, as every climate model requires a representation of the ABL at each grid point in order to balance heat and moisture budgets, analytical relations provide the most computationally efficient representation. This work extends work on an existing model representation of the ABL to provide approximate analytical solutions which are highly accurate across a wide range of parameter sets. This work has potential for improving our understanding of local agro-meteorological interactions as well as large-scale climate representations. It also provides a step toward analytical understanding of more complex phenomena such as the initiation of convective precipitation.

Technical Abstract: The system of governing equations of a simplified slab model of the uniformly-mixed, purely convective, diurnal atmospheric boundary layer (ABL) is shown to allow immediate solutions for the potential temperature and specific humidity as functions of the ABL height and net radiation when expressed in integral form. By employing a linearized saturation vapor relation, the height of the mixed layer is shown to obey a nonlinear ordinary differential equation with quadratic dependence on ABL height, which may be further reduced to a particular form of Abel's differential equation of the second kind. While the Abel equation yields analytic solutions only for special cases of the radiative forcing, an approximate analytic solution is presented for the diurnal evolution of the ABL which is accurate on average to within 2% of the numerical solution for a randomized sample of parameter values and arbitrary forcing. In addition an expression is derived for an effective constant Bowen ratio which only overestimates the ABL height on average by 5% across a wide range of parameter values. These solutions allow the diurnal evolution of the height, potential temperature, and specific humidity (i.e., also vapor pressure deficit) of the mixed layer to be expressed analytically for arbitrary radiative forcing functions.