Articles | Volume 15, issue 5
Research article
09 Mar 2015
Research article |  | 09 Mar 2015

Theory of the norm-induced metric in atmospheric dynamics

T.-Y. Koh and F. Wan

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.

Short summary
A metric measures the difference between two system states. So far, there is arbitrariness in its definition in terms of wind, temperature and surface pressure of the atmosphere. The choice of definition affects many applications: e.g. predictability studies, error growth analyses and ensemble forecasts. We construct a new metric based on vector space theory and energy conservation, and apply it to analytic models and atmospheric data. We discuss its advantages over a widely used older metric.
Final-revised paper