Abstract
Abstract. We suggest that some metrics for quantifying distances in phase space are based on linearized flows about unrealistic reference states and hence may not be applicable to atmospheric flows. A new approach of defining a norm-induced metric based on the total energy norm is proposed. The approach is based on the rigorous mathematics of normed vector spaces and the law of energy conservation in physics. It involves the innovative construction of the phase space so that energy (or a certain physical invariant) takes the form of a Euclidean norm. The metric can be applied to both linear and nonlinear flows and for small and large separations in phase space. The new metric is derived for models of various levels of sophistication: the 2-D barotropic model, the shallow-water model and the 3-D dry, compressible atmosphere in different vertical coordinates. Numerical calculations of the new metric are illustrated with analytic dynamical systems as well as with global reanalysis data. The differences from a commonly used metric and the potential for application in ensemble prediction, error growth analysis and predictability studies are discussed.
Highlights
1.1 The contextIn predictability studies, the sensitivity of numerical models to initial conditions is an important topic
These metrics should overcome the limitations of having unrealistic reference states and the need to linearize the flow about those states
When an inner product is defined for a vector space, the inner product of the difference between two vectors with itself yields the square of the norm-induced metric
Summary
The sensitivity of numerical models to initial conditions is an important topic. Some definitions look similar to wave energy (Bannon, 1995; Zou et al, 1997; Kim et al, 2011), while others use quadratic expressions that resemble kinetic and available potential energy (Buizza et al, 1993; Zhang et al, 2003; Leutbecher and Palmer, 2008; Rivière et al, 2009). None of these metrics are truly energy or energy differences, as already noted by some authors We shall first review an often used metric as a concrete illustration of the problem
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