The most commonly-used surface flux formula is based on the M-O similarity theory, which relates the surface fluxes to the difference of wind (or temperature) between the roughness height and any height within the surface layer. However, because the M-O theory is not directly applicable to free convection (zero wind) conditions, various surface-flux formulae have been proposed for such conditions. Recently a new formula was proposed by Sykes et al. [1993] using a scaling argument, in which the surface-layer thickness and surface-friction velocity can vary with the roughness length and the PBL depth. Their results were shown to agree well with their LES data.
The ``bulk'' similarity theory extends the M-O theory
to relate the surface fluxes to the difference of
wind (or temperature) between the skin surface and the mixed layer.
The similarity functions are then assumed to depend on 
(as oppose to 
used in the M-O theory),
where 
is the height of the PBL, 
is the M-O length, 
the friction velocity,
the thermal coefficient,
the von Karman constant, and 
the buoyancy flux.
Brutsaert and Sugita [1991] derived
functions for the bulk similarity theory from FIFE data. Recently,
Stull [1994] proposed a similar bulk theory: a convective transport
theory, in which the surface
fluxes are again related to the skin surface condition
and the mixed-layer values, but now through a ``buoyancy velocity scale.''
This velocity scale is assumed to be a function of the potential
temperature difference between the skin surface and the mixed layer.
The transport coefficients were determined empirically.
Stull's scheme can apply to zero wind conditions because the heat
and moisture fluxes no longer depend on the mean wind speed.
The bulk theory ignores the surface layer, and thus is independent
of the surface roughness.