Interpretations of the total energy and rotational energy norms applied to determination of singular vectors

The interpretation of the commonly-used energy norm is examined in the context of a simple vertically-discrete model. The norm is shown to include expressions for kinetic and available potential energy in addition to an expression for a portion of unavailable potential energy. Another norm is then introduced that only includes the rotational-mode contribution to these. The characterization of the two norms in terms of corresponding covariance functions is shown to be quite different, with that for the tatter norm looking more like prior error statistics used in synoptic-scale data assimilation, The leading singular vectors are determined for both norms. Those computed for the new norm have slower associated growth. Their corresponding structures are similar at the initial time, however, with some notable differences, but after 24 hours their shapes are almost identical. The new norm has advantages over the old norm for some applications; e.g. for effectively filtering ageostrophic, convectively-driven singular vectors and for being more consistent with a spatially and dynamically correlated error norm.