Using Adjoint Models for Stability and Predictability Analysis

  • Andrew M. Moore
  • Brian F. Farrell
Part of the NATO ASI Series book series (volume 19)


The primary purpose of data assimilation in meteorology and oceanography is to initialise numerical models so that they can be used to forecast the state of the atmosphere or ocean at some time in the future. The reliability of the resulting model prediction depends upon a number of factors. These include the accuracy of the model and the quality of the assimilated observational data, both of which are reflected in uncertainties in the model initial conditions produced by the initialisation procedure. The predictability of the system will also depend upon its initial state in that certain configurations of the system are dynamically more unstable than others when subjected to perturbations arising from errors in the initial conditions or model physics. The model may also have inaccuracies which may not reflected in the initial conditions but which will also influence the predictability of the system.

Lorenz (1965) first suggested that the growth of forecast errors in a model of the atmosphere could be expressed in terms of the growth of the singular vectors of the forecast error norm. In this paper we will show how the adjoint of a numerical model can be used to find the singular vectors, which are sometimes referred to as the “optimal perturbations”, of a forecast error norm. The structure and growth of the optimal perturbations will be compared to that of the normal modes of the system which result from more traditional stability analyses.

Aside from the dynamical significance of optimal perturbations and their relation to development of disturbances, studies of such perturbations can be used to answer questions relating to the predictability of the ocean and atmosphere. Work along these lines is already in progress at a number of Numerical Weather Prediction centres and recent examples will be presented. The study of optimal perturbations should also yield valuable information about the structure and growth of model forecast errors, information which can be used to improve existing data assimilation schemes.

A great deal of effort is required to construct the adjoint of a complex numerical model. While adjoint models are currently being developed primarily for data assimilation purposes, stability analysis is another very powerful application of adjoint models which complements data assimilation, and which further justifies the effort involved in setting up the model.


Data Assimilation Forecast Error Gulf Stream Forecast Skill Ensemble Forecast 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Blumenthal MB (1991) Predictability of a coupled ocean-atmosphere model. J Climate 4:766–784CrossRefGoogle Scholar
  2. Booth AD (1955) Numerical Methods. Butterworth Scientific Publications, 195ppGoogle Scholar
  3. Borges MD, Hartmann DL (1992) Barotropic instability and optimal perturbations of observed non-zonal flows. J Atmos Sci 49:335–354CrossRefGoogle Scholar
  4. Chen WY (1989) Estimate of dynamical predictability from NMC DERF experiments. Mon Wea Rev 117:1227–1236CrossRefGoogle Scholar
  5. Farrell BF (1982) The initial growth of disturbances in a baroclinic flow. J Atmos Sci 39:1663–1686CrossRefGoogle Scholar
  6. Farrell BF (1989) Optimal excitation of baroclinic waves. J Atmos Sci 46:1193–1206CrossRefGoogle Scholar
  7. Farrell BF (1990) Small error dynamics and the predictability of flows. J Atmos Sci 47:2409–2416CrossRefGoogle Scholar
  8. Farrell BF, Moore AM (1992) An adjoint method for obtaining the most rapidly growing perturbation to oceanic flows. J Phys Oceanogr 22:338–349CrossRefGoogle Scholar
  9. Golub GH, Van Loan CF (1990) Matrix Computations. The Johns Hopkins University Press, Baltimore, 642ppGoogle Scholar
  10. Hoffman RN, Kalnay E (1983) Lagged average forecasts. Tellus 35:100–118Google Scholar
  11. Kalnay E, Toth Z (1993) Ensemble forecasting at NMC: The generation of perturbations. Bull Amer Met Soc SubmittedGoogle Scholar
  12. Lacarra JF, Talagrand O (1988) Short range evolution of small perturbations in a barotropic model. Tellus 40A: 81–95CrossRefGoogle Scholar
  13. Lorenz EN (1965) A study of the predictability of a 28-variable atmospheric model. Tellus 17:321–333CrossRefGoogle Scholar
  14. Lorenz EN (1982) Atmospheric predictability with a large numerical model. Tellus 34:505–513CrossRefGoogle Scholar
  15. Molteni F, Palmer TN (1993) Predictability and finite-time instability of the northern winter circulation. Quart J R Met Soc 119:269–298CrossRefGoogle Scholar
  16. Moore AM, Farrell BF (1993) Rapid perturbation growth on spatially and temporally varying oceanic flows determined using an adjoint method: Application to the Gulf Stream. J Phys Oceanogr 23:1682–1702CrossRefGoogle Scholar
  17. Mureau R, Molteni F, Palmer TN (1993) Ensemble prediction using dynamically - conditioned perturbations. Quart J R Met Soc 119:299–323CrossRefGoogle Scholar
  18. Palmer TN (1988) Medium and extended range predictability and stability of the Pacific. North American mode. Quart J Roy Met Soc 114:691–713CrossRefGoogle Scholar
  19. Pedlosky J (1987) Geophysical fluid dynamics. Springer-Verlag, New York, 710ppCrossRefGoogle Scholar
  20. Robinson AR, Spall MA, Walstad LJ, Leslie WG (1989) Data assimilation and dynamical interpolation in GULFCAST experiments. Dynamics of Atmos and Oceans 13:301–316CrossRefGoogle Scholar
  21. Vukicevic T (1991) Nonlinear and linear evolution of initial forecast errors. Mon Wea Rev 119:1602–1611CrossRefGoogle Scholar
  22. Wallace JM, Gutzler DS (1981) Teleconnections in the geopotential height field during the Northern hemisphere winter. Mon Wea Rev 109:784–812CrossRefGoogle Scholar
  23. Zebiak SE, Cane MA (1987) A model of El Nino-Southern Oscillation. Mon Wea Rev 115:2262–2278CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Andrew M. Moore
    • 1
  • Brian F. Farrell
    • 1
    • 2
  1. 1.Bureau of Meteorology Research CentreMelbourneAustralia
  2. 2.Division of Applied Science, Pierce HallHarvard UniversityCambridgeUSA

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