Applying finite‐element vertical discretization to a mass‐based non‐hydrostatic kernel has proved difficult due to the constraints of vertical operators. This article proposes a novel hybrid finite‐element vertical discretization method for a semi‐implicit mass‐based non‐hydrostatic kernel, which integrates a finite‐differential scheme and a finite‐element scheme. In the hybrid method, the finite‐differential scheme which satisfies the set of constraints is applied to the linear part, while a cubic finite‐element scheme with high‐order accuracy is applied to the non‐linear part. Furthermore, to improve the accuracy of the linear part, an enlarged set of vertical levels is applied to the differential scheme. This set of vertical levels is only used to solve semi‐implicit equations, and has no impact on the grid point calculation and spectral transformations. A series of 2D idealized test cases are conducted to verify the stability and the accuracy of our new method.

The prediction of chaotic time series systems has remained a challenging problem in recent decades. A hybrid method using Hankel Alternative View Of Koopman (HAVOK) analysis and machine learning (HAVOK-ML) is developed to predict chaotic time series. HAVOK-ML simulates the time series by reconstructing a closed linear model so as to achieve the purpose of prediction. It decomposes chaotic dynamics into intermittently forced linear systems by HAVOK analysis and estimates the external intermittently forcing term using machine learning. The prediction performance evaluations confirm that the proposed method has superior forecasting skills compared with existing prediction methods.

In this paper, we proposed a high accurate and stable Legendre transform algorithm, which can reduce the potential instability for a very high order at a very small increase in the computational time. The error analysis of interpolative decomposition for Legendre transform is presented. By employing block partitioning of the Legendre-Vandermonde matrix and butterfly algorithm, a new Legendre transform algorithm with computational complexity O(Nlog²N /loglogN) in theory and O(Nlog³N) in practical application is obtained. Numerical results are provided to demonstrate the efficiency and numerical stability of the new algorithm.

Meteorological satellite remote sensing is important for numerical weather forecasts, but its accuracy is affected by many things during observation and retrieval, showing that it can be improved. As a standard way to measure wind from space, atmospheric motion vectors (AMVs) are used. They are separate pieces of information spread out in the troposphere, which gives them more depth than regular surface or sea surface wind measurements. This makes rectifying problems more difficult. For error correction, this research builds a deep-learning model that is specific to AMVs. The outcomes show that AMV observational errors are greatly reduced after correction. The root mean square error (RMSE) drops by almost 40% compared to ERA5 true values. Among these, the optimization of solar observation errors exceeds 40%; the discrepancies at varying atmospheric pressure altitudes are notably improved; the degree of optimization for data with low QI coefficients is substantial; and there remains potential for enhancement in data with high QI coefficients. Furthermore, there has been a significant enhancement in the consistency coefficient of the wind’s physical properties. In the assimilation forecasting experiments, the corrected AMV data demonstrated superior forecasting performance. With more training, the model can fix things better, and the changes it makes last for a long time. The results show that it is possible and useful to use deep learning to fix errors in meteorological remote-sensing data.