Abstract:
Data assimilation of turbulent signals is an important challenging problem because of the extremely complicated large dimension of the signals and incomplete partial noisy observations which usually mix the large scale mean flow and small scale fluctuations. Due to the limited computing power in the foreseeable future, it is desirable to use multi-scale forecast models which are cheap and fast to mitigate the curse of dimensionality in turbulent systems; thus model errors from imperfect forecast models are unavoidable in the development of a data assimilation method in turbulence. Here we propose a suite of multi-scale data assimilation methods which use stochastic superparameterization as the forecast model. Superparameterization is a seamless multi-scale method for parameterizing the effect of small scales by cheap local problems embedded in a coarse grid. The key ingredient of the multi-scale data assimilation methods is the systematic use of conditional Gaussian mixtures which make the methods efficient by filtering a subspace whose dimension is smaller than the full state. The multi-scale data assimilation methods proposed here are tested on a six dimensional conceptual dynamical model for turbulence which mimics interesting features of anisotropic turbulence including two way coupling between the large and small scale parts, intermittencies, and extreme events in the smaller scale fluctuations. Numerical results show that suitable multi-scale data assimilation methods have high skill in estimating the most energetic modes of turbulent signals even with infrequent observation times. This is a joint work with Andy J. Majda. |