The current practical numerical weather forecasting involves General Circulation Model where the prognostic model is discretized in space and time while the unresolved processes are parametrized. Thus, making accurate predictions with partial observation of this high dimensional complex dynamical system becomes a major issue in operations.
Many ensemble Kalman filter approaches have been successfully developed for these extreme complex system, however, new mathematical issues arise in practical application of these filtering strategies. One challenge is that it requires many realizations that involve expensive calculations for propagating each ensemble member forward in time.
Back to Small Scales and Extreme Events: The Hurricane
In this talk, I will discuss recent theoretical work of Majda and Grote (PNAS 2006, 2007), that is, a possibility of using a larger time step that sometimes violates the classical CFL-stability condition for an explicit difference scheme. With this approach, then one reduces the computational time in propagating the large ensemble size required for filtering but still obtaining stable and statistically accurate filtering. I will also show results from the ongoing joint work with Emilio Castronovo and Andy Majda in developing and implementing new test criteria for these issues. Specifically, I will discuss results of filtering an Ornstein-Uhlenbeck process, an advection-diffusion equation, and finally a 40-dimensional chaotic Lorenz-96 model that mimics the midlatitude weather pattern.