Existing r-adaptive moving mesh methods in atmospheric dynamics are predominantly based on finite element methods. They suffer from inherent conservation degradation and are highly sensitive to mesh distortion, making it difficult to meet the stringent requirements for long-term mass and energy conservation in atmospheric simulations. Furthermore, applying r-adaptive methods to three-dimensional (3D) structured hexahedral meshes faces significant challenges, including non-coplanar geometries and the loss of conservation during physical variable reconstruction. To address these issues, this paper proposes a conservative r-adaptive moving mesh method based on a finite volume scheme, specifically designed for atmospheric dynamical cores. By introducing a moving mesh equation to prevent mesh tangling, and employing a hexahedral element decomposition and block-integration reconstruction strategy, the method strictly guarantees mass and energy conservation even under severe geometric deformation. Validated through two-dimensional (2D) and 3D rising thermal bubble and flow-over-mountain test cases, the results demonstrate that the proposed method significantly enhances the resolution of strong-gradient, small-scale regions using fewer total grid points. It also maintains robust mass and energy conservation while achieving favorable parallel efficiency and scalability. This study provides a novel and efficient numerical solution for multi-scale atmospheric dynamical simulations.

Data assimilation integrates irregularly distributed observations into model grid points to provide initial values for numerical models. Once a data assimilation system is established on a chosen grid type, it cannot be easily adapted to another grid type. In this paper, we introduce a gridless method for the three-dimensional variation assimilation (3DVar) system. Unlike grid-based methods, the gridless method uses arbitrarily distributed points for calculation and does not require pre-defined grid cells; thus, it can switch to any grid distribution, namely the data assimilation system based on a gridless method can be adapted to most model grid structures without the need to add new codes. In the data assimilation system based on the gridless method presented here, the Cressman analysis technique is adopted as the observation operator and the physical transformation matrix is handled by using the Taylor expansion. Idealized experiments based on the Rankine vortex are conducted to demonstrate the 3DVar system based on the gridless method, and it is validated that the system can handle structured, unstructured, and mixed (structured and unstructured) grids. Furthermore, we demonstrate that the gridless data assimilation method can perform data assimilation on grids of different resolutions and structural types simultaneously using a single cost function.