Modelling of local length‐scale dynamics and isotropizing deformations

The correlation length‐scale which characterizes the shape of the correlation function is often used to parametrize correlation models. This article describes how the length‐scale dynamics can be employed to estimate a spatial deformation (coordinate transformation). Of particular interest is the isotropizing deformation, which transforms anisotropic correlation functions into quasi‐isotropic ones. The evolution of the length‐scale field under a simple advection dynamics is described in terms of the local metric tensor. This description leads to a quadratic constraint satisfied by the isotropizing deformation and from which a system of Poisson‐like partial differential equations is deduced. The isotropizing deformation is obtained as the solution of a coupled system of Poisson‐like partial differential equations. This system is then solved with a pseudo‐diffusion scheme, where the isotropizing deformation is the steady‐state solution. The isotropization process is illustrated within a simulated 2D setting. The method is shown to provide an accurate estimation of the original deformation used to build the anisotropic correlations in this idealized framework. Applications in data assimilation are discussed. First, the isotropization procedure can be useful for background‐error covariance modelling. Secondly, the length‐scale dynamics provides a way to simulate the dynamics of covariances for the transport of passive scalars, as encountered in chemical data assimilation.