Using Adjoint Models for Stability and Predictability Analysis

Abstract

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.