The problem of optimal initial disturbances in thermal wind shear is revisited and extended to include non-hydrostatic effects. This systematic study compares transient and modal growth rates of submesoscale instabilities over a large range of zonal and meridional wave numbers, aspect ratios, and different Richardson number regimes. Selection criteria were derived to remove spurious and unresolved instability modes that arise from the eigenvalue problem and we generalize the study of the hydrostatic Eady problem by Heifetz and Farrell (2003, 2007, 2008) to thin fronts, characterized by large aspect ratios. Such fronts are commonly found at the early stages of frontogenesis, for example, in the ocean mesoscale eddies and near the eye wall of hurricanes. In particular, we show that transient energy growth rates are up to two orders of magnitude larger than modal counterparts for a wide range of Richardson number and that the effects of transient energy gain become even greater when non-hydrostatic effects become important and/or for large Richardson numbers. This study also compares the dominant energy pathways contributing to the energy growth at short and long times. For symmetric modes, we recover the inertia-gravity instability described in Xu et al. (2007). These mechanisms are shown to be the most powerful mediator of vertical transport when compared with the fastest growing baroclinic and symmetric modes. These results highlight the importance of transient processes in the ocean and the atmosphere.

中文翻译：

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地转 Eady 模型中的瞬态能量增长

重新审视了热风切变中的最佳初始扰动问题，并将其扩展到包括非流体静力效应。这项系统研究比较了大范围的纬向和经向波数、纵横比和不同理查森数制度下亚中尺度不稳定性的瞬态和模态增长率。推导出选择标准以消除由特征值问题引起的虚假和未解决的不稳定性模式，我们将 Heifetz 和 Farrell (2003, 2007, 2008) 对流体静力 Eady 问题的研究推广到以大纵横比为特征的薄前沿。这种锋通常出现在锋生的早期阶段，例如，在海洋中尺度涡旋和飓风眼壁附近。特别是，我们表明，对于大范围的理查森数，瞬态能量增长率比模态对应物大两个数量级，并且当非静水力效应变得重要和/或对于大理查森数时，瞬态能量增益的影响变得更大. 本研究还比较了促进短期和长期能量增长的主要能量途径。对于对称模式，我们恢复了 Xu 等人中描述的惯性重力不稳定性。(2007)。与增长最快的斜压和对称模式相比，这些机制被证明是垂直传输最强大的中介。这些结果突出了海洋和大气中瞬态过程的重要性。