A new horizontally explicit/vertically implicit (HEVI) time splitting scheme for atmospheric modelling is introduced, for which the horizontal divergence terms are applied within the implicit vertical substep. The new HEVI scheme is implemented in conjunction with a mixed mimetic spectral element spatial discretisation and semi-implicit vertical time stepping scheme that both preserve the skew-symmetric structure of the non-canonical Hamiltonian form of the equations of motion. Within this context the new HEVI scheme allows for the exact balance of all energetic exchanges in space and time. However since the choice of horizontal fluxes for which this balance is satisfied is not consistent with the horizontal velocity at the end of the time level the scheme still admits a temporal energy conservation error. Linearised eigenvalue analysis shows that similar to a fully implicit method, the new HEVI scheme is neutrally stable for all buoyancy modes, and unlike a second order trapezoidal HEVI scheme is stable for all acoustic modes below a certain horizontal CFL number. The scheme is validated against standard test cases for both planetary and nonhydrostatic regimes. For the planetary scale baroclinic instability test case, the new formulation exhibits a secondary oscillation in the potential to kinetic energy power exchanges, with a temporal frequency approximately four times that exhibited by a horizontally third order, vertically second order trapezoidal scheme. For the nonhydrostatic test case, the vertical upwinding of the potential temperature diagnostic equation is shown to reduce spurious oscillations without altering the energetics of the solution, since this upwinding is performed in an energetically consistent manner. For this test case, which is configured on an affine geometry, the exact balance of energy exchanges allows the model to run stably without any form of dissipation.
翻译:为大气建模引入一个新的水平直线/垂直隐含(HEVI)时间分割计划,为此在隐含的垂直子步骤中应用横向差异条件。新的HEVI计划与混合微米光谱元素空间分解和半隐含垂直时间踏步计划一起实施,这两种计划都保留了非卡尼汉密尔顿式运动方程式的双向对称结构。在此背景下,新的HEVI计划允许在空间和时间上的所有动态交流之间保持准确的平衡。然而,由于对平衡得到满足的横向通量的选择与时间级结束时的横向速度变化不一致,因此,该HEVI计划仍然承认了时间节能偏差。 线性和半隐含垂直时间跨度分析表明,与完全隐含的方法相似,新的HEVI计划对所有运动模式的隐性对等度结构都保持中性稳定,而对于低于某种水平直线的CFLLL数值的所有声调模式而言,该计划允许在新行星和非流直径直径直径的轨道系统系统进行标准测试案例中进行验证。对于在时间段直径直径直径直径直径直径直径的电动的电流的电流的电流的电流电流的电流电流电流的电压电流的电流的电流的电流的电流的电流的电流的电流平衡进行演演算法,从一个直向性电流的电流的电流的电流的电流的电流的电流的电流向向性电流变化动的电流流向曲线,从一个直流的电流的电流的电流的电流的电流的电流的电流向性电流向性电流向的电流的电流的电算法演演演演演进。