Backward Fokker.ppt

How Long Can an Atmospheric Model Predict? Peter C Chu and Leonid Ivanov Naval Ocean-Atmospheric Prediction Laboratory Naval Postgraduate School Monterey, California 93943, USA chu@ /~chu References Chu, P.C., L.M. Ivanov, T. M. Margolina, and O.V. Melnichenko, On probabilistic stability of an atmospheric model to various amplitude perturbations. Journal of the Atmospheric Sciences, 59, 2860-2873. Chu, P.C., L.M. Ivanov, C.W. Fan, 2002: Backward Fokker-Planck equation for determining model valid prediction period. Journal of Geophysical Research, 107, C6, 10.1029/2001JC000879. Chu, P.C., L. Ivanov, L. Kantha, O. Melnichenko, and Y. Poberezhny, 2002: Power law decay in model predictability skill. Geophysical Research Letters, 29 (15), 10.1029/2002GLO14891. Chu, P.C., L.M. Ivanov, L.H. Kantha, T.M. Margolina, and O.M. Melnichenko, and Y.A, Poberenzhny, 2004: Lagrangian predictabilty of high-resolution regional ocean models. Nonlinear Processes in Geophysics, 11, 47-66. Physical Reality Y Physical Law: dY/dt = h(y, t) Initial Condition: Y(t0) = Y0 Atmospheric Models  X is the prediction of Y d X/ dt = f(X, t) + q(t) X Initial Condition: X(t0) = X0 Stochastic Forcing: q(t) = 0 q(t)q(t’) = q2δ(t-t’) Model Error Z = X – Y Initial: Z0 = X0 - Y0 One Overlooked Parameter Tolerance Level e Maximum accepted error Valid Predict Period (VPP) VPP is defined as the time period when the prediction error first exceeds a pre-determined criterion (i.e., the tolerance level ?). First-Passage Time Conditional Probability Density Function Initial Error: Z0 (t – t0) ? Random Variable Conditional PDF of (t – t0) with given Z0 ? P[(t – t0) |Z0] Two Approaches to Obtain PDF of VPP Analytical (Backward Fokker-Planck Equation) Practical Backward Fokker-Planck Equation Moments Example: Maximum Growing Maniford of Lorenz System (Nicolis, 1992)