POWER SERIES PROOFS FOR LOCAL STABILITIES OF KÄHLER AND BALANCED STRUCTURES WITH MILD -LEMMA

By use of a natural map introduced recently by the first and third authors from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the small differentiable deformation of this manifold, we will give a power series proof for Kodaira–Spencer’s local stability theorem of Kähler structures. We also obtain two new local stability theorems, one of balanced structures on an n-dimensional balanced manifold with the $(n-1,n)$ th mild $\partial \overline {\partial }$ -lemma by power series method and the other one on p-Kähler structures with the deformation invariance of $(p,p)$ -Bott–Chern numbers.

Dynamically informed covariance modelling in data assimilation

<div>Data assimilation systems are progressively getting better, resulting in improved analyses and forecasts. One important reason for this is thought to be the improved representation of the multivariate PDF of a-priori errors seen by the assimilation. This means that observations can influence the trajectory/ies of the numerical model in more physically meaningful ways. While some improvement is gained by modelling deviations of the PDF from Gaussianity, and by statistical modelling of Gaussian covariances with ensembles, there is still scope to improve the structure of the `B-matrix' used in pure and hybrid versions of 3D/4D-Var.</div><div>Our hypothesis is that a good B-matrix for geophysical data assimilation applications should have multivariate structure functions that reflect the dynamics of the underlying physical system. So, if the underlying system is close to some balanced manifold, then the assimilation should not disturb that property. Existing practice is to impose any balances explicitly, but this is difficult when the balances are weak or difficult to determine, such as in convective-scale or tropical applications, etc. In this talk we look at how such covariances can be modelled, including an approach that uses the normal modes of the underlying dynamics.</div>