One-Layer Turbulent Geophysical Flows

Statistical bounds controlling the total fluctuations in mean and variance about a basic steady 6 state solution are developed for the truncated barotropic flows over topography. Statistical ensemble predic7 tion is an important topic in weather and climate research. Here the evolution of an ensemble of trajectories 8 is considered in the statistical instability analysis and is compared and contrasted with the classical deter9 ministic instability for the growth of perturbations in one pointwise trajectory. The maximum growth of the 10 total statistics in fluctuations is derived relying on the statistical conservation principle of the pseudo-energy. 11 The saturation of the statistical mean fluctuation and variance in the unstable regimes with non-positive12 definite pseudo-energy is achieved by linking with a class of stable reference states and minimizing the stable 13 statistical energy bounds. Two cases with dependence on initial statistical uncertainty and on external forc14 ing and dissipation are compared and unified with a consistent statistical stability framework. The flow 15 structures and statistical stability bounds are illustrated and verified by numerical simulations among a 16 wide range of dynamical regimes, where subtle transient statistical instability exists in general with positive 17 short-time exponential growth rate in the statistical covariance even when the pseudo-energy is positive18 definite. In the various scenarios illustrated below, there are strong forward and backward cascades of energy 19 between large and small flow scales which are estimated by the rigorous statistical bounds. 20