Two common approximations to the full Shallow Water Equations (SWEs) are non-divergence and quasi-geostrophy, and the degree to which these approximations lead to biases in numerical solutions are explored using the test bed of barotropic instability. Specifically, we examine the linear stability of strong polar and equatorial jets and compare the growth rates obtained from the SWEs along with those obtained from the Non-Divergent barotropic vorticity (ND) equation and the Quasi-Geostrophic (QG) equation. The main result of this paper is that the depth over which a layer is barotropically unstable is a crucial parameter in controlling the growth rate of small amplitude perturbations and this dependence is completely lost in the ND equation and is overly weak in the QG system. Only for depths of 30 km or more are the growth rates predicted by the ND and QG systems a good approximation to those of the SWEs, and such a convergence for deep layers can be explained using theoretical considerations. However, for smaller depths, the growth rates predicted by the SWEs become smaller than those of the ND and QG systems and for depths of between 5 and 10 km they can be smaller by more than . For polar jets, and for depths below 2 km the mean height in geostrophic balance with the strong zonal jet becomes negative and hence the barotropic instability problem is ill-defined. While in the SWEs an equatorial jet becomes stable for layer depths smaller than 3–4 km, in the QG and ND approximations it is unstable for layer depths down to 1 km. These result may have implications for the importance of barotropic instability in Earth's upper stratosphere and perhaps also other planets such as Venus.

中文翻译：

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球体上纬向射流的正压不稳定性：从非发散到准地转到浅水

完整浅水方程 (SWE) 的两种常见近似是非发散和准地转，并且使用正压不稳定性试验台探索这些近似导致数值解偏差的程度。具体来说，我们检查了强极地和赤道急流的线性稳定性，并将从 SWE 获得的增长率与从非发散正压涡度 (ND) 方程和准地转 (QG) 方程获得的增长率进行比较。本文的主要结果是，层正压不稳定的深度是控制小幅度扰动增长速率的关键参数，这种依赖性在 ND 方程中完全消失，在 QG 系统中过于弱。只有对于 30 公里或以上的深度，ND 和 QG 系统预测的增长率才能很好地近似于 SWE 的增长率，并且可以使用理论考虑来解释深层的这种收敛。然而，对于较小的深度，SWE 预测的增长率变得小于 ND 和 QG 系统的增长率，并且对于 5 到 10 公里之间的深度，它们可以小超过 。对于极地急流，对于 2 公里以下的深度，强纬向急流的地转平衡平均高度变为负值，因此正压不稳定问题不明确。虽然在 SWE 中，赤道急流在小于 3-4 公里的层深度时变得稳定，但在 QG 和 ND 近似值中，它对于低至 1 公里的层深度是不稳定的。这些结果可能对地球正压不稳定的重要性产生影响。