A generalization of Lorenz's model for the predictability of flows with many scales of motion

In a seminal paper, E. N. Lorenz proposed that flows with many scales of motion in which smaller-scale error spreads to larger scales and in which the error-doubling time decreases with decreasing scale have a finite range of predictability. Although the Lorenz theory of limited predictability is widely understood and accepted, the model upon which the theory is based is less so. The primary objection to the model is that it is based on the two-dimensional vorticity equation (2DV) while simultaneously emphasizing results using a basic turbulent flow with a "-5/3" energy spectrum in the atmospheric synoptic scale instead of those using a more theoretically and observationally consistent "-3" spectrum. The present work generalizes the Lorenz model so that it may apply to the surface quasigeostrophic equations (SQGs), which are mathematically very similar to 2DV but are known to have a -5/3 kinetic energy spectrum downscale from a large-scale forcing. This generalized Lorenz model is applied here to both 2DV (with a -3 spectrum) and SQG (with a -5/3 spectrum), producing examples of flows with unlimited and limited predictability, respectively. Comparative analysis of the two models allows for the identification of the distinctive attributes of a many-scaled flow with limited predictability.