pure and applied geophysics

, Volume 117, Issue 4, pp 664–689

# The surface boundary layer as a part of the overlying convective layer

• James W. Telford
• John D. Presley
Article

## Abstract

This paper extends previous work developing a mechanistic theory of the convective boundary layer to the forced convective region between the base of the plumes and the surface. It is shown that a simple model based on specifying the entrainment between adjacent layers, gives quantitative relations between temperature at plume height and the surface temperature, and the shearing stress and the turbulence, thus completing the specifications of the surface boundary needed for the convective planetary boundary layer plume model, by describing the surface boundary layer without constants determined to match boundary layer measurements.

The formulae are deduced in terms of the entrainment constanta=1/12, and the turbulent decay constantA=1, by mechanistic reasoning, without the introduction of any adjustable empirical constants.

The formulae are,
$$i^2 = ({2 \mathord{\left/ {\vphantom {2 {aA}}} \right. \kern-\nulldelimiterspace} {aA}})^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} u*^2 ,$$
and when buoyant energy release is considered,
$$\begin{gathered} \frac{{\sigma _w }}{{u*}} = \left( {\frac{2}{{9Aa}}} \right)^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} \left( {1 - 4.54\frac{a}{k}\frac{{\sigma _w }}{{u*}}\frac{z}{L}} \right)^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} \hfill \\ = 1.28(1 - 0.945({{\sigma _w } \mathord{\left/ {\vphantom {{\sigma _w } {u*}}} \right. \kern-\nulldelimiterspace} {u*}})({z \mathord{\left/ {\vphantom {z L}} \right. \kern-\nulldelimiterspace} L}))^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} , \hfill \\ \end{gathered}$$
wherei is the total turbulence such thati2 u 2 + v 2 + w 2 , given that the σ are the standard deviations of the air velocity along each axis. Von Karman's constant isk=0.4, andL is the Obukhov scale length.

Rough surface conditions also allow the wind and the temperature excess at ten meters or so above the surface to be derived. In these conditionsv=(A/2a)1/2i, andTsurfaceTaverage at plume base=H/(ϱCpai). When the surface is not very rough, an additional roughness parameter is required to specify the number of layers needed to make the transition from the plume to the surface, and its function is examined. These formulae all compare well with published measured values.

It is shown by means of a fully descriptive theory, that the shear in the plume layer is very small.

## Key words

Boundary layer Convection

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