Scalar covariance spectrum

Likewise, the time-evolution equation for the scalar-covariance spectrum Eap , t) can be written as... 

Similar relationships exist for the scalar flux energy spectrum and the scalar covariance energy spectrum. 

The physics of this term can be understood using the scalar cospectrum Eap(ic, t), which, similar to the scalar spectrum, represents the fraction of the scalar covariance at wavenum-... 

Gap is the corresponding scalar-covariance source term, and Tap is the scalar-covariance transfer spectrum. In the following, we will relate the SR model for the scalar variance to (A.2) however, analogous expressions can be derived for the scalar covariance from (A.4) by following the same procedure. 

A key assumption in deriving the SR model (as well as earlier spectral models see Batchelor (1959), Saffman (1963), Kraichnan (1968), and Kraichnan (1974)) is that the transfer spectrum is a linear operator with respect to the scalar spectrum (e.g., a linear convection-diffusion model) which has a characteristic time constant that depends only on the velocity spectrum. The linearity assumption (which is consistent with the linear form of (A.l)) ensures not only that the scalar transfer spectra are conservative, but also that if Scap = Scr in (A.4), then Eap ic, t) = Eyy k, t) for all t when it is true for t = 0. In the SR model, the linearity assumption implies that the forward and backscatter rate constants (defined below) have the same form for both the variance and covariance spectra, and that for the covariance spectrum the rate constants depend on the molecular diffusivities only through Scap (i.e., not independently on Sc or Sep). 

The scalar-scalar transfer function Saoi(K q) appearing in the final term on the right-hand side of (A.6) denotes the contribution of scalar mode q to the scalar-variance transfer spectrum at k. (See Yeung (1994) and Yeung (1996) for examples of these functions extracted from DNS.) Similarly, the scalar-covariance transfer spectrum can be decomposed using... 

The scalar coherency spectrum is defined in terms of the variance and covariance spectra by... 

P(j+i)j for / = 1— 1 to be independent of Sc. This is the assumption employed in the SR model, but it can be validated (and modified) using DNS data for the scalar spectrum and the scalar-scalar transfer function. The linearity assumption discussed earlier implies that the rate constants will be unchanged (for the same Reynolds and Schmidt numbers) when they are computed using the scalar-covariance transfer spectrum. 

In linear algebra, I X, Y) is called as an inner product, or a scalar product of -dimensional vectors X and Y. Using the inner product, the 2D correlation spectrum can be constructed in the same way as in the case of covariance, and the Parseval s theorem can be straightforwardly applied without having to make the (sometimes invalid) assumption of vanishing mean values. Comparison between the covariance and the inner product has not been explored, and the question of whether the latter is superior or not for the NMR purposes is an open issue that require further studies. The idea of inner-product NMR spectroscopy is thus outside the scope of this review and will not be discussed further here. 

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