Higher-order closure

Wyngaard, J. C., and Cote, O. R. (1974). The evolution of a convective planetary boundary layer—a higher order closure model study. Boundary Layer Meteorol. 7, 289-308. 

Katul, G.G., Albertson, J.D. (1998) An investigation of higher-order closure models for a forested canopy, Boundary-Layer Meteorol. 89, 47-74. 

Meyers, T.P., and Paw, U.K.T. (1987) Modelling the plant canopy micrometeorology with higher-order closure principles, J. Agric. Forest Meteorol. 41, 143-163. 

Wilson, N.R., and Shaw, R.H. (1977) A higher-order closure model for canopy flow, J. Appl. Meteorol. 16, 1197-1205. 

Katul, G. G., Leuning, R., Kim, )., Denmead, O. T., Miyata, A., and Hara-7.ono, Y. (2001). Estimating CO, source/sink distributions within a rice canopy using higher-order closure models. Boundary-Layer Meleorol. 98, 103- 125. 

Equation (25.28) is the simplest solution to the closure problem and is currently used in the majority of chemical transport models. Higher-order closure approximations have been developed but are computationally expensive. Some more recent formulations have shown promise of becoming computationally competitive with the commonly employed K theory (Pai and Tsang 1993). 

Integral equations provide a satisfactory formalism for the study of homogeneous and inhomogeneous fluids. If the usual OZ equation is used, the best results are obtained from semiempirical closures such as the MV and DHH closures. However, this empirical element can be avoided by using integral equations that involve higher-order distribution functions, but at the cost of some computational complexity. 

In what is one of the few examples of utilization of a higher order sigmatropic hydrogen shift in the synthesis of complex molecules, Eschenmoser, in studies directed toward the synthesis of Vitamin B12, found that an antarafacial [1,16] H-shift could be utilized to effect closure of secocorrin 63 to corrin 65 (Scheme 16)31. An intermediate biradical... 

R2Cu(CN)Li2 reaction with vie-epoxy mesylatesA higher-order cuprate reacts selectively with the epoxide group of the epoxy mesylate 1 to provide 2 with inversion at C3. Ring closure of 2 furnishes the epoxide 3, which reacts with a second equivalent of the higher-order cuprate to furnish meso-4, with inversion at both C, and C3. This two-step reaction provides a route to acyclic alcohols with useful stereocontrol at both adjacent centers. 

As discussed in Chapter 5, the complexity of the chemical source term restricts the applicability of closures based on second- and higher-order moments of the scalars. Nevertheless, it is instructive to derive the scalar covariance equation for two scalars molecular-diffusion coefficients ra and I, respectively. Starting from (1.28), p. 16, the transport equation for ((,) can be found following the same steps that were used for the Reynolds stresses. This process yields34... 

The closure problem thus reduces to finding general methods for modeling higher-order moments of the composition PDF that are valid over a wide range of chemical time scales. 

The failure of first-order moment closures for the treatment of mixing-sensitive reactions has led to the exploration of higher-order moment closures (Dutta and Tarbell 1989 Heeb and Brodkey 1990 Shenoy and Toor 1990). The simplest closures in this category attempt to relate the covariances of reactive scalars to the variance of the mixture fraction (I 2). The latter can be found by solving the inert-scalar-variance transport equation ((3.105), p. 85) along with the transport equation for (f). For example, for the one-step reaction in (5.54) the unknown scalar covariance can be approximated by... 

Obviously, conditional moments of higher order could also be modeled. However, as with moment closures, the unclosed terms in the higher-order transport equation are more and more difficult to close. 

Integral equations theories are another approach to incorporate higher order correlations, and consequently also lead to lowered osmotic coefficients. There are numerous variants of these theories around which differ in their used closure relations and accuracy of the treatment of correlations [36]. They work normally very well at high electrostatic coupling and high densities, and are able to account for overcharging, which was first predicted by Lozada-Cassou et al. [36] and also describe excluded volume effects very well, see Refs. [37] for recent comparisons to MD simulations. 

Another closure approximation relies on the use of higher order moments, but this is almost a deadlock as for only two reacting components, a 13-equation model is required (3). 

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