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Notes on “What is an adjoint model?” by Errico (1997, BAMS)


An adjoint model enables to study directly the sensitivity of the model to the initial conditions. Sensitivity fields are actual outputs of the model and can be used for data-assimilation, parameter assimilation, stability analysis (see Farrell and Ioannou 1996a,b; Molteni and Palmer 1993) and synoptic studies.

The traditional way to study the sensitivity of a model is to perform a control experiment as well as a second experiment where one component has been modified (it can be the resolution, the level of viscosity, etc). The differences between the two experiments is a quantification of the model to the perturbation. “The problem with this previous type of sensitivity analysis is that, in fact, all that has been determined is the impact of one specific” perturbation. Another perturbation of same type and strength may trigger a different response and a different interpretation would be drawn. “In order to more generally answer the question of sensitivity using this method, an ensemble of perturbations must be examined”.

Another way to study sensitivity of a model is to calculate the gradient of the diagnostic used to study the sensitivity with respect to the perturbation of the initial conditions. The adjoint model enables to relate this gradient to another but calculable gradient, that of the diagnostic with respect to the perturbations of the output. The Jacobian (or linearized model operator, or propagator) of the model is the quantity that enables to link perturbation of the output with respect to the input. “[T]he adjoint operates backward in the sense that it determines a gradient with respect to input from a gradient with respect to output.”.

When normal modes are non-orthogonal, growing solutions that are not normal modes can exist and can be revealed by the eigenvectors (of the adjoint?); see Farrell and Ioannou 1996a,b.

Limitations: “results are useful only when the linearized approximation is valid”. For instance, the linearized approximation will fail when a perturbation has grown substantially. This is why the adjoint is most useful for short forecasts, the shortness of the period being dependent on the growth rate of the perturbation considered. Another limitation is when the diagnostic considered is not differentiable.


Can we use a model adjoint to study the sensitivity of the solution to the resolution of the model? From what I understand, probably not.