  Shadowing trajectories are model trajectories consistent with a sequence of
observations of a system, given a distribution of observational noise. The
existence of such trajectories is a desirable property of any forecast model.
Gradient descent of indeterminism is a well-established technique for finding
shadowing trajectories in low-dimensional analytical systems. Here we apply it
to the thermally-driven rotating annulus, a laboratory experiment intermediate
in model complexity and physical idealisation between analytical systems and
global, comprehensive atmospheric models. We work in the perfect model scenario
using the MORALS model to generate a sequence of noisy observations in a
chaotic flow regime. We demonstrate that the gradient descent technique
recovers a pseudo-orbit of model states significantly closer to a model
trajectory than the initial sequence. Gradient-free descent is used, where the
adjoint model is set to $\lambda$I in the absence of a full adjoint model. The
indeterminism of the pseudo-orbit falls by two orders of magnitude during the
descent, but we find that the distance between the pseudo-orbit and the
initial, 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 to
noise reduction but not to finely-tuned convergence towards a model trajectory.
We find that $\lambda=0.25$ gives optimal results, and that candidate model
trajectories begun from this pseudo-orbit shadow the observations for up to 80
s, about the length of the longest timescale of the system, and similar to
expected shadowing times based on the distance between the pseudo-orbit and the
truth. There is great potential for using this method with real laboratory
data.
