Scalar covariance conditional

The higher-order conditional scalar moments are defined in terms of the conditional fluctuations. For example, the conditional scalar covariances are defined by... 

The only new unclosed term that appears in the composition PDF transport equation is (Ui ijy>. The exact form of this term will depend on the flow. However, if the velocity and scalar fields are Gaussian, then the scalar-conditioned velocity can be expressed in terms of the scalar flux and the scalar covariance matrix ... 

The extension of the SR model to differential diffusion is outlined in Section 4.7. In an analogous fashion, the LSR model can be used to model scalars with different molecular diffusivities (Fox 1999). The principal changes are the introduction of the conditional scalar covariances in each wavenumber band 4> a4> p) and the conditional joint scalar dissipation rate matrix (e). For example, for a two-scalar problem, the LSR model involves three covariance components (/2, W V/A) and and three joint dissipation... 

Condition (1) follows directly from the definitions of the joint scalar dissipation rates and the covariance matrices, i.e.,... 

We recall that we can saturate the t Hooft anomaly conditions either with massless fermionic degrees of freedom or with gapless bosonic excitations. However in absence of Lorentz covariance the bosonic excitations are not restricted to be fluctuations related to scalar condensates but may be associated, for example, to vector condensates [51]. 

---