Numerical simulations of nonhydrostatic atmospheric flow, based on linearly decoupled semi-implicit or fully-implicit techniques, usually solve linear systems by a pre-conditioned Krylov method without preserving the skew-symmetry of convective operators. We propose to perform atmospheric simulations in such a fully-implicit manner that the difference operators preserve both the skew-symmetry and the tightly nonlinear coupling of the differential operators. We demonstrate that a symmetry-preserving Jacobian-free Newton-Krylov (JFNK) method mimics a balance between convective transport and turbulence dissipation. We present a wavelet method as an effective symmetry preserving discretization technique. The symmetry-preserving JFNK method for solving equations of nonhydrostatic atmospheric flows has been examined using two benchmark simulations of penetrative convection – a) dry thermals rising in a neutrally stratified and stably stratified environment, and b) urban heat island circulations for effects of the surface heat flux $H_0$ varying in the range of $25\le H_0\le 930$ W m⁻². The results show that an eddy viscosity model provides the necessary dissipation of the subgrid-scale modes, while the symmetry-preserving JFNK method provides the conservation of mass and energy at a satisfactory level. Comparisons of the results from a laboratory experiment of heat island circulation and a field measurement of potential temperature also suggest the modelling accuracy of the present symmetry-preserving JFNK framework.

基于线性解耦半隐式或全隐式技术的非静水压大气流数值模拟通常通过预处理 Krylov 方法求解线性系统，而不会保留对流算子的偏斜对称性。我们建议以一种完全隐式的方式进行大气模拟，使得差分算子保持偏斜对称性和差分算子的紧密非线性耦合. 我们证明了保持对称性的无雅可比牛顿 - 克雷洛夫 (JFNK) 方法模拟了对流传输和湍流耗散之间的平衡。我们提出了一种小波方法作为一种有效的保持对称性的离散化技术。已使用穿透对流的两个基准模拟检查了用于求解非静水压大气流动方程的对称保持 JFNK 方法——a) 在中性分层和稳定分层环境中上升的干热气流，以及 b) 城市热岛环流对地表的影响热通量$H_0$ 范围不同 $25\le H_0\le 930$W  m ^-2 。结果表明，涡粘性模型提供了亚网格尺度模式的必要耗散，而保持对称性的 JFNK 方法提供了令人满意的质量和能量守恒。热岛环流实验室实验结果与位温现场测量结果的比较也表明了当前保持对称性的 JFNK 框架的建模精度。