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Shadowing trajectories are model trajectories consistent with a sequence ofobservations of a system, given a distribution of observational noise. Theexistence of such trajectories is a desirable property of any forecast model.Gradient descent of indeterminism is a well-established technique for findingshadowing trajectories in low-dimensional analytical systems. Here we apply itto the thermally-driven rotating annulus, a laboratory experiment intermediatein model complexity and physical idealisation between analytical systems andglobal, comprehensive atmospheric models. We work in the perfect model scenariousing the MORALS model to generate a sequence of noisy observations in achaotic flow regime. We demonstrate that the gradient descent techniquerecovers a pseudo-orbit of model states significantly closer to a modeltrajectory than the initial sequence. Gradient-free descent is used, where theadjoint model is set to $\lambda$I in the absence of a full adjoint model. Theindeterminism of the pseudo-orbit falls by two orders of magnitude during thedescent, but we find that the distance between the pseudo-orbit and theinitial, true, model trajectory reaches a minimum and then diverges from truth.We attribute this to the use of the $\lambda$-adjoint, which is well suited tonoise reduction but not to finely-tuned convergence towards a model trajectory.We find that $\lambda=0.25$ gives optimal results, and that candidate modeltrajectories begun from this pseudo-orbit shadow the observations for up to 80s, about the length of the longest timescale of the system, and similar toexpected shadowing times based on the distance between the pseudo-orbit and thetruth. There is great potential for using this method with real laboratorydata.

Shadowing the rotating annulus. Part II: Gradient descent in the perfect model scenario