# Presentation: 803912

## 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@nps.navy.mil](mailto:chu@nps.navy.mil)_
- http://www.oc.nps.navy.mil/~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:***     ***d***Y***/dt = ***h***(***y***, t)
- Initial Condition:  ***Y******(***t0) = ***Y***0

## Atmospheric Models 

-  ***X*****  **is the prediction of **Y**
- d ***X*****/** dt = ***f****(****X,**** *t) + q(t) ***X***
- Initial Condition:  ***X(***t0) = ***X***0
- Stochastic Forcing:
-            <q(t)> = 0
-             <q(t)q(t’)> = q2δ(t-t’)

## Model Error

-  ***  ***
***                ******Z = X – Y***
- Initial***:     Z******0****** = X******0****** -  Y******0***

## One Overlooked Parameter 

- Tolerance Level   
- 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:   ***Z***0
- (t – t0)     Random Variable
- Conditional  PDF of  (t – t0) with given  ***Z***0 
-  P[(t – t0) |***Z***0]

## 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) 

## Mean and Variance of VPP

## Analytical Solutions  

## Dependence of tau1 & tau2 on Initial Condition Error (           )

## Practical Approach

- Gulf of Mexico Ocean Prediction System

## Gulf of Mexico Forecast System

- University of Colorado Version of POM
- 1/12o Resolution
- Real-Time SSH Data (TOPEX, ESA ERS-1/2) Assimilated
- Real Time SST Data (MCSST, NOAA AVHRR) Assimilated
- Six Months Four-Times Daily Data From July 9, 1998 for Verification

## Model Generated Velocity Vectors at 50 m on 00:00 July 9, 1998

## (Observational) Drifter  Data at 50 m on 00:00 July 9, 1998

## Statistical Characteristics of VPP for zero initial error and 22 km tolerance level (Non-Gaussion)

## Conclusions

- (1) VPP (i.e.,  FPT) is  an effective prediction skill measure (scalar).
- (2) Theoretical framework for FPT (such as Backward Fokker-Planck equation) can be directly used for model predictability study.