Yumeng Chen

Yumeng Chen, Hilary Weller, Stephen Pring et al.

Dimensionally split advection schemes are attractive for atmospheric modelling due to their efficiency and accuracy in each spatial dimension. Accurate long time-steps can be achieved without significant cost using the flux-form semi-Lagrangian technique. The dimensionally split scheme used here is constructed from the one-dimensional Piecewise Parabolic Method and extended to two dimensions using COSMIC splitting. The dimensionally split scheme is compared with a genuinely multi-dimensional, method of lines scheme with implicit time-stepping which is stable for large Courant numbers. Two-dimensional advection test cases on Cartesian planes are proposed that avoid the complexities of a spherical domain or multi-panel meshes. These are solid body rotation, horizontal advection over orography and deformational flow. The test cases use distorted meshes either to represent sloping terrain or to mimic the distortions of a cubed sphere. Such mesh distortions are expected to accentuate the errors associated with dimension splitting, however, the dimensionally split scheme is very accurate on orthogonal meshes and accuracy decreases only a little in the presence of large mesh distortions. The dimensionally split scheme also loses some accuracy when long time-steps are used. The multi-dimensional scheme is almost entirely insensitive to mesh distortions and asymptotes to second-order accuracy at high resolution. As is expected for implicit time-stepping, phase errors occur when using long time-steps but the spatially well resolved features are advected at the correct speed and the multi-dimensional scheme is always stable. An estimate of computational cost reveals that the implicit scheme is the most expensive, particularly for large Courant numbers. If the multi-dimensional scheme is used instead with explicit time-stepping, the cost becomes similar to the dimensionally split scheme.

Ivo Pasmans, Yumeng Chen, Tobias Sebastian Finn et al.

Popular (ensemble) Kalman filter data assimilation (DA) approaches assume that the errors in both the a priori estimate of the state and those in the observations are Gaussian. For constrained variables, e.g. sea ice concentration or stress, such an assumption does not hold. The variational autoencoder (VAE) is a machine learning (ML) technique that allows to map an arbitrary distribution to/from a latent space in which the distribution is supposedly closer to a Gaussian. We propose a novel hybrid DA-ML approach in which VAEs are incorporated in the DA procedure. Specifically, we introduce a variant of the popular ensemble transform Kalman filter (ETKF) in which the analysis is applied in the latent space of a single VAE or a pair of VAEs. In twin experiments with a simple circular model, whereby the circle represents an underlying submanifold to be respected, we find that the use of a VAE ensures that a posteri ensemble members lie close to the manifold containing the truth. Furthermore, online updating of the VAE is necessary and achievable when this manifold varies in time, i.e. when it is non-stationary. We demonstrate that introducing an additional second latent space for the observational innovations improves robustness against detrimental effects of non-Gaussianity and bias in the observational errors but it slightly lessens the performance if observational errors are strictly Gaussian.