Parameterization of the roughness length over the sea in forced and free convection

In the condition of free convection, the Charnock relation is inadequate. In this paper we extend the Charnock relation to include the effect of free convection on the roughness length. As a result, the singularity in the Monin–Obukhov similarity theory can be avoided. This paper shows two approaches to derive the roughness length formula in the forced and free convections. The first approach is based on the mixing length theory and the use of the observational data of the vertical velocity variance. We introduce a new vertical velocity scale based on the vertical velocity variance; this velocity variance is well behaved in the atmospheric boundary layer and easy to obtain from field experiments. The second approach is based on the theoretical framework of Sykes et al. (Q R Met Soc, 119: 409–421). From that framework, we develop a theory to obtain the roughness length formula. The results of these two approaches are in agreement with each other. In the past, a multiplication factor associated with free convection was considered to be a constant. This paper shows that this multiplication factor is, in fact, also dependent on the depth of the mixing height. In previous studies, experimental works were often conducted without taking into account the depth of the mixing height. Not taking into account the mixing height in the estimation of the roughness length in free convection would result in an inaccurate estimate of the roughness length and hence the drag coefficient. An erratum to this article can be found at