Since the 1970's, there has been a growing recognition, supported by
recent work on coherent structures, that large eddies
(of size on the order of PBL depth) carry most of the turbulent fluxes within
the bulk of the convective PBL. Thus, relating vertical
flux of a quantity to its local vertical gradient as if transport were by
simple diffusion (e.g., K theory in which e.g., the
momentum flux is expressed as 
,
where U is the mean wind speed)
is not strictly correct. It is more correct to consider the
transport within the convective PBL to be nonlocal.
This recognition has led to development of numerous
nonlocal closure models, e.g., Pleim and Chang [1992],
Stull [1993], and Ahmed et al. [1993] for convective PBL.
In the mass-flux modeling approach, scalar fluxes are expressed by the
differences of mean concentrations between updrafts and
downdrafts, through a mass flux velocity.
Businger and Oncley (1990)
suggested that this mass flux velocity is proportional
to the standard deviation of vertical velocity fluctuations.
Wyngaard and Moeng [1992] further suggested that
the proportionality factor is determined uniquely by the joint
probability density of vertical velocity and scalar fluctuations.
Another alternative to conventional K theory is to assume that the flux can still be related to the vertical gradient, if one adds a correction term allowing for countergradient flux. Building upon Troen and Mahrt's K-profile model (Troen and Mahrt 1986), Holtslag and Moeng [1991] added a countergradient term that is proportional to the surface heat flux, and also generalized it to include top-down and bottom-up asymmetry. Hamba [1993] developed a term for countergradient transport, which is proportional to the second derivative of the mean scalar, based on the two-scale direct interaction approximation.