Physical interpretation of the adjoint functions for sensitivity analysis of atmospheric models

The adjoint functions for an atmospheric model are the solution to a system of equations derived from a differential form of the model's equations. The adjoint functions can be used to calculate efficiently the sensitivity of one of the model's results to variations in any of the model's parameters. This paper shows that the adjoint functions themselves can be interpreted as the sensitivity of a result to instantaneous perturbations of the model's dependent variables. This interpretation is illustrated for a radiative convective model, although the interpretation holds equally well for general circulation models. The adjoint functions are used to reveal the three time scales associated with 1) convective adjustment, 2) heat transfer between the atmosphere and space, and 3) heat transfer between the ground and atmosphere. Calculating the eigenvalues and eigenvectors of the matrix of derivatives occurring in the set of adjoint equations reveals similar physical information without actually solving for the adjoint functions. The sensitivities given by the adjoint functions are verified by comparison with sensitivities obtained directly from recalculations. Despite sharp changes in the adjoint functions arising from convective adjustment switching on and off during a diurnal cycle, a first-order numerical scheme to solve the adjoint equations gives agreement with direct recalculations to three significant figures. For a model with N time steps, this comparison has also shown that the adjoint method is at least N/2 times more efficient than recalculation. Such an efficient method of calculating the sensitivity of a simulated synoptic state to all previous synoptic states is valuable not only to identify the time scales of various physical processes but also to assimilate data for initialization.