Abstract It is well known that the primitive equations (the atmospheric equations of motion under the additional assumption of hydrostatic equilibrium for large scale motions) are ill posed when used in a limited area on the globe. Yet the atmospheric equations of motion for large scale motions are essentially a hyperbolic system that with appropriate boundary conditions should lead to a well posed system in a limited area. This apparent paradox was resolved by Kreiss through the introduction of the mathematical Bounded Derivative Theory (BDT) for any symmetric hyperbolic system with multiple time scales (as is the case for the atmospheric equations of motion). The BDT uses norm estimation techniques from the mathematical theory of symmetric hyperbolic systems to prove that if the norms of the spatial and temporal derivatives of the ensuing solution are independent of the fast time scales (thus the concept of bounded derivatives), then the subsequent solution will only evolve on the advective space and time scales (slowly evolving in time in BDT parlance) for a period of time. The requirement that the norm of the time derivatives of the ensuing solution be independent of the fast time scales leads to a number of elliptic equations that must be satisfied by the initial conditions and ensuing solution. In the atmospheric case this results in a 2D elliptic equation for the pressure and a 3D equation for the vertical component of the velocity. Utilizing those constraints with an equation for the slowly evolving in time vertical component of vorticity leads to a single time scale (reduced) system that accurately describes the slowly evolving in time solution of the atmospheric equations and is automatically well posed for a limited area domain. The 3D elliptic equation for the vertical component of velocity is not sensitive to small scale perturbations at the lower boundary so the equation can be used all of the way to the surface in the reduced system, eliminating the discontinuity between the equations for the boundary layer and troposphere and the problem of unrealistic growth in the horizontal velocity near the surface in the hydrostatic system.

中文翻译：

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用于大气流动的独特、精心设计的简化系统：存在小尺度表面不规则时的稳健性

摘要 众所周知，原始方程（大尺度运动静水平衡附加假设下的大气运动方程）在地球上的有限区域内是不适定的。然而，大尺度运动的大气运动方程本​​质上是一个双曲线系统，具有适当的边界条件应该导致有限区域内的适定系统。Kreiss 通过为任何具有多个时间尺度的对称双曲系统（如大气运动方程的情况）引入数学有界导数理论 (BDT) 解决了这个明显的悖论。BDT 使用对称双曲系统数学理论中的范数估计技术来证明，如果后续解的空间和时间导数的范数与快速时间尺度无关（因此是有界导数的概念），则后续解只会在一段时间内在平流空间和时间尺度上演化（用 BDT 的说法是在时间上缓慢演化）。后续解的时间导数范数与快速时间尺度无关的要求导致了许多椭圆方程必须满足初始条件和后续解。在大气情况下，这会导致压力的 2D 椭圆方程和速度的垂直分量的 3D 方程。将这些约束与涡度垂直分量在时间上缓慢演化的方程一起使用，可以得到一个单一的时间尺度（缩减）系统，该系统准确地描述了大气方程在时间上缓慢演化的解，并自动为有限区域域设定了良好的姿势。速度垂直分量的 3D 椭圆方程对下边界处的小尺度扰动不敏感，因此该方程可以一直用于简化系统中的表面，消除边界层方程和对流层和静水系统中地表附近水平速度不切实际增长的问题。