An energetically balanced, implicit integrator for non-hydrostatic vertical atmospheric dynamics on the sphere is presented. The integrator allows for the exact balance of energy exchanges in space and time for vertical atmospheric motions by preserving the skew-symmetry of the non-canonical Hamiltonian formulation of the compressible Euler equations. The performance of the integrator is accelerated by a preconditioning strategy that reduces the dimensionality of the inner linear system. Here we reduce the four component velocity, density, density weighted potential temperature and Exner pressure system into a single equation for the density weighted potential temperature via repeated Schur complement decomposition and the careful selection of coupling terms. As currently formulated, the integrator is based on a horizontal-vertical spatial splitting that does not permit bottom topography. The integrator is validated for standard test cases for baroclinic instability and a non-hydrostatic gravity wave on the sphere and a rising bubble in a high-resolution planar geometry, and shows robust convergence across all of these regimes.

提出了一种用于球面上非静压垂直大气动力学的能量平衡隐式积分器。积分器通过保留可压缩的Euler方程的非规范哈密顿公式的偏对称性，可以实现垂直大气运动在空间和时间上能量交换的精确平衡。通过减少内部线性系统维数的预处理策略可以加速积分器的性能。在这里，我们通过反复进行Schur补数分解和仔细选择耦合项，将四个分量的速度，密度，密度加权势温度和Exner压力系统简化为密度加权势温度的单个方程式。按照目前的规定，积分器基于不允许底部地形的水平-垂直空间分裂。该积分器已针对斜压不稳定性和球体上的非静水重力波以及高分辨率平面几何形状中的上升气泡的标准测试用例进行了验证，并且在所有这些情况下均显示出稳健的收敛性。