Application to the Convective Boundary Layer

Abstract

For most applications, the assumption of Gaussian turbulence in deriving the several stochastic Lagrangian models in chapters 6–9 is an acceptable approximation. A major exception is the convective boundary layer (CBL) because the persistent updrafts and downdrafts dominate the vertical flow. The resultant trajectory of a passive tracer cannot, therefore, be regarded as a superposition of independent increments as in Eqs. (9.8c) and (9.10b), for example. For this reason the Eulerian eddy diffusivity model, Eq. (8.44b), is not applicable everywhere in the CBL. For an area source at the top of a horizontally homogeneous CBL, the effective eddy diffusivity, K eff , is well-behaved for downward diffusive transport. On the other hand, the eddy diffusivity model fails for an area source at the bottom of the same CBL. There is a singularity in the value of K eff (from K eff ≫ 0 to K eff ≪ 0) about halfway up and a countergradient flux (meaning negative K eff ) in the upper part of the CBL. The situation is more complex for vertical diffusion with horizontal advection from a steady source (point or horizontal line) near the surface in the CBL. In this case, K eff is ill-behaved (negative) in some portions of the plume (for example, see the appendix in Deardorff and Willis 1975). Wyngaard (1987) proposed that the skewness of the vertical velocity field is responsible for this diffusion phenomenon. Numerical simulations by Wyngaard and Weil (1991) suggest that this transport asymmetry is caused by the interaction between the skewness of the turbulence and the gradient of the scalar flux. There have been attempts to reformulate this model for use in the CBL. For example, Holtslag and Moeng (1991) proposed a modification that includes countergradient terms and separate parameterizations for top-down and bottom-up diffusion. Their model is well behaved over most of the CBL, but gives negative values of K eff in the upper part of the CBL under some conditions. This failure of the eddy diffusivity model [and the equivalent Lagrangian random displacement model, Eq. (6.23)] provides the principal motivation for applying the Lagrangian Langevin equation model to the CBL.