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Model Predictability- Form Lorenz System to Operational Ocean and Atmospheric Models
Peter C Chu and Leonid IvanovNaval Ocean-Atmospheric Prediction Laboratory
Naval Postgraduate SchoolMonterey, California 93943, USA
[email protected]://www.oc.nps.navy.mil/~chu
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ReferencesChu, 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. GeophysicalResearch 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 predictabiltyof high-resolution regional ocean models. Nonlinear Processes in Geophysics, 11, 47-66.
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Physical Reality
Y
Physical Law: dY/dt = h(y, t)
Initial Condition: Y(t0) = Y0
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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’)
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Model Error
Z = X – Y
Initial: Z0 = X0 - Y0
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One Overlooked Parameter
Tolerance Level ε
Maximum accepted error
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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 ε).
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First-Passage Time
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Conditional Probability Density Function
Initial Error: Z0
(t – t0) Random Variable
Conditional PDF of (t – t0) with given Z0
P[(t – t0) |Z0]
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Two Approaches to Obtain PDF of VPP
Analytical (Backward Fokker-Planck Equation)
Practical
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Backward Fokker-Planck Equation
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Moments
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Extremely Long Predictability
Non-Gaussian Distribution with Long Tail Towards Large FPT Domain.
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Gulf of Mexico Forecast System
University of Colorado Version of POM1/12o ResolutionReal-Time SSH Data (TOPEX, ESA ERS-1/2) AssimilatedReal Time SST Data (MCSST, NOAA AVHRR) AssimilatedSix Months Four-Times Daily Data From July 9, 1998 for Verification
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Model Generated Velocity Vectors at 50 m on 00:00 July 9, 1998
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(Observational) Drifter Data at 50 m on 00:00 July 9, 1998
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Statistical Characteristics of VPP for zero initial error and 22 km tolerance level (Non-Gaussion)
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Probability Density Function of VPP calculated with different tolerance levels
Non-Gaussian distribution
with long tail toward large
values of VPP (Long-term
Predictability)
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Error Mean and Variance
Error Mean
Error Variance
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Exponential Error Growth
Classical Linear Theory
No Long-Term Predictability
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Power Law
Long-Term Predictability May Occur
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Scaling behavior of the
mean error (L1) growth
for initial error levels:
(a) 0
(b) 2.2 km
(c) 22 km
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Scaling behavior of the
Error variance (L2) growth
for initial error levels:
(a) 0
(b) 2.2 km
(c) 22 km
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Predictability Tube
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Conclusions(1) Nonlinear model predictability can be effectively represented by FPT.
(2) Backward Fokker-Planck equation is the theoretical framework for FPT.
(3) Nonlinear stochastic-dynamic modeling