ABSTRACT Optimal balance is a near-optimal computational algorithm for nonlinear mode decomposition of geophysical flows into balanced and unbalanced components. It was first proposed as “optimal potential vorticity balance” by Viúdez and Dritschel [J. Fluid Mech., 2004, 521, 343] in the specific setting of semi-Lagrangian potential vorticity-based numerical codes. Later, it was recognised as an instance of the more general principle of adiabatic invariance of fast degrees of motion under slow perturbations. From this point of view, the system is slowly deformed from a linearised configuration to the full nonlinear dynamics. In the former, linear analysis yields an exact separation of balanced and unbalanced flow. In the latter, a given base-point coordinate, e.g. the height or potential vorticity field, can be matched. This formulation leads to a boundary value problem in time. In this paper, we show that this more general viewpoint leads to practical implementations of optimal balance on top of a primitive variables (here, velocity-height variables) numerical code. We identify preferred choices for several design parameters. The most critical choices concern the linear projector onto the slow modes at the linear-end boundary and the choice of base-point coordinate at the nonlinear end. We find that, even though the evolutionary model is formulated in primitive variables, potential vorticity based end-point conditions are advantageous. In particular, the only universally robust linear projector is the oblique projector onto the Rossby modes along the gravity-wave modes, which can be interpreted as the distinct non-orthogonal projector onto the Rossby modes that preserves the linear potential vorticity. Hence, the projector can be formulated as an elliptic partial differential equation which holds promise for using the method to produce an accurate nonlinear mode decomposition for more general models without the need to resort to asymptotic analysis.

中文翻译：

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在原始变量中旋转浅水的最佳平衡

摘要 最优平衡是一种近似最优的计算算法，用于将地球物理流非线性模式分解为平衡和不平衡分量。它首先被 Viúdez 和 Dritschel 提出为“最佳势涡平衡”[J. Fluid Mech., 2004, 521, 343] 在基于半拉格朗日位涡的数值代码的具体设置中。后来，它被认为是慢扰动下快速运动程度绝热不变性的更一般原理的一个实例。从这个角度来看，系统从线性配置缓慢变形为完全非线性动力学。在前者中，线性分析产生平衡和不平衡流的精确分离。在后者中，可以匹配给定的基点坐标，例如高度或势涡场。这个公式会及时导致边界值问题。在本文中，我们展示了这种更一般的观点导致在原始变量（此处为速度-高度变量）数字代码之上的最佳平衡的实际实现。我们确定了几个设计参数的首选。最关键的选择是线性投影到线性端边界的慢模上，以及非线性端基点坐标的选择。我们发现，即使进化模型是在原始变量中制定的，基于潜在涡度的端点条件也是有利的。特别是，唯一普遍稳健的线性投影仪是沿重力波模式投射到罗斯比模式上的斜投影仪，这可以解释为保留线性势涡度的罗斯比模式上的独特非正交投影。因此，投影仪可以表述为椭圆偏微分方程，该方程有望使用该方法为更一般的模型生成准确的非线性模式分解，而无需求助于渐近分析。