A Numerical Study of Mass and Energy Transfer of Surface, Surface Boundary Layer and Planetary Boundary Layer

Abstract

The horizontal inhomogeneity is a common occurrence in the lower layer of the atmosphere. For example, in land-lake interface problems, and in flows over finite areas of irrigated land, abrupt changes of surface conditions, i. e., roughness, temperature and humidity occur simultaneously, and the profiles are no longer in equilibrium. The principal purpose of this paper is to present these phenomena quantitatively from a theory.In this study, we used the same numerical method as Estoque and Bhumralkar (1970) describing a finite-difference procedure for numerical integrating the planetary boundary-layer equations. According to the similarity theory expressed as those equations, the thermal stability was assessed in terms of Richardson number (Ri). The evaluation of the Ri is open to question because it is based on the doubtful assumption that the stability condition of the planetary boundary layer is equal with height. If the surface of the earth is heterogeneous as mentioned above, the flow pattern, temperature and humidity field must change with its surface characters. In this case, one should have a more the appropriate parameter in describing the internal boundary layer.In order to take the varying stability conditions with height into consideration, Hayakawa et al. (1979) suggested to use a gradient Richardson number (Rig) for the stability parameter. The flow pattern with the Rig shows that the prevailing flow accelerates on encountering the windward edge of a localized heat source (LHS) and decelerates on leaving the leeward edge. As a consequence of the horizontal velocity variations there is a downwind above LHS and an upward motion in most part of the leeward area. The characteristics of this flow pattern with the Rig is completely contrary to Hayakawa's results (1978) based on Rib. It can be that the reason for these serious differences are due to the different diffusion coefficient caused by a different definition of Ri.The profiles of the diffusion coefficient (K) obtained by use of Rig show its peaks at reasonable levels, and the values of K and the peak height increase with the scale of LHS. These characteristics agree qualitatively well with the observation results of the planetary boundary layer.The perturbation of potential temperature based on the Rig can not develop in an altitude so high and horizontally so extensive as the case with Rib.By an analysis of the condition that the perturbation of potential temperature vanishes in the atmospheric boundary layer, we have reached to the conclusion that the inertial term is more important in the “cross-over” phenomenon than the bouyant force.We define ZH, which is decided from the potential temperature profiles (Fig. 6) and Ri profiles (Fig. 2), as a thickness of the atmospheric boundary layer, and have examined the growth of the internal boundary layer thickness with fetch (x). We have compared these values of thickness obtained from the simulation experiment with those obtained from analytic expressions by other researchers. As a result, the best linear fit was obtained from Kimura's expression (ZH∝R-1/6·x), and Sutton's expression (ZH∝K1/2·x1/2·U-1/2) followed.When air moves from a cool area to a hot one, the horizontal temperature gradient ∂T/∂x becomes positive above the hot area, then the gradient of vertical sensible heat flux ∂H/∂z becomes positive above that area. On the other hand, when air moves from a hot area to a cool one, ∂T/∂x and ∂H/∂z become negative above the cool area.