On the Validity of Reynolds Assumptions for Running-Mean Filters in the Absence of a Spectral Gap

Abstract

A running-mean operator is used to average predefined signals in order to calculate subgrid-scale (SGS) contributions. The role of the Leonard and cross terms (l and C, respectively) in the definition of the SGS is studied as a function of the spectral characteristics of the signal and of the width of the averaging interval. In particular the properties of these two terms are analyzed in relation to the width of the spectral gap, which is the separation in the power spectrum between ?small-? and ?large-scale? atmospheric energy distribution. The study of theoretical cases indicates that in the absence of a well-marked scale separation the neglect of these terms can lead to considerable errors in the estimate of the SGS contribution when a running mean is used. Other than the separation between the energy contributions, the intensity of the signal plays a relevant role in determing the importance of l and C compared to the currently used Reynolds turbulent terms [fluctuations (co)variances]. The absence of a scale separation and the properties of l and C do not allow for the straightforward application of the so-called Reynolds assumption [f(x, t) = f(x, t)]. This may have implications on the formal definition of averaging filters in a model as well as for the treatment of atmospheric measurements. By means of predefined signals the consequences of the generally assumed assumption [?f(xj, t)/?xj = ?f(xj, t)/?xj] in relation to grid stretching are also analyzed. The results show that a correction term has to be accounted for and that its magnitude becomes significant even for stretching factors commonly adopted, for example, in mesoscale models.