Theory of the norm induced metric in atmospheric dynamics

Quantifying separation in phase space is fundamental in ensemble predictions, error growth analyses and predictability studies. Commonly used metrics are based on linearized flows about unrealistic reference states and hence may not be applicable to atmospheric flows. We introduce a new approach of defining a norm induced metric based on the physical law of energy conservation and the mathematics of Euclidean normed vector spaces. The metric is just as valid for linear and nonlinear flows and for small and large phase-space separations. The new metric is derived for models of various levels of sophistication (2-D barotropic model, shallow-water model and dry, compressible atmosphere) and is illustrated with analytic examples as well as global reanalysis data. The new metric differs importantly from the old by a factor of 2 in weighing temperature variations and can include a buoyant-to-kinetic energy conversion term that would impact the measurement of model error growth in strongly convecting systems like tropical cyclones.