An Equation- Free Reduced Order Modeling Approach to Tropic Pacific Simulation

The “equation-free” (EF) method is based on a procedure, which is often used in complex, multi-scale problems, given an initial macroscopic variable, by lifting, evolve in Direct Numerical Simulations (DNS), restriction, and projective integration. In this paper, we apply the Equation-free method to the reduced modeling of a large-scale upper ocean circulation in the tropic Pacific domain. We carry out a series of experiments to discuss its convergence and error, and we also discuss some factors that affect the results, such as the number of snapshots and basis functions based on proper orthogonal decomposition (POD) mode, large-scale time step, short time step and the number of iterations during the course of the DNS running in a large-scale time step. Compared with POD method, we do not need availability of explicit equation or right-hand-side (RHS), and we may obtain better results for horizontal velocity components of depth-averaged currents than those obtained in POD case. The results from the equation-free based on different POD modes are compared with results of POD and DNS. We also illustrate the convergence and error of the EF method. The main findings are: we reduce the computation work enormously in contrast to DNS, for instance, the computational cost of the equation-free POD-assisted method is less than 10% of that for DNS. The method can capture the main variability by a low dimensional system based on equation-free POD mode. We also attained a RMS error for the upper ocean layer thickness that is less than 1m i.e. less than 1% of the average thickness and the correlations between the upper layer thickness are around 0.99 based on equation-free POD method, while the convergence and the error are illustrated through numerical experiments.