In a previous paper, Oruba, Soward & Dormy (J.Fluid Mech., vol.818, 2017, pp.205-240) considered the primary quasi-steady geostrophic (QG) motion of a constant density fluid of viscosity $\nu$ that occurs during linear spin-down in a cylindrical container of radius $L$ and height $H$, rotating rapidly (angular velocity $\Omega$) about its axis of symmetry subject to mixed rigid and stress-free boundary conditions for the case $L=H$. Here, Direct Numerical Simulation (DNS) at large $L= 10 H$ and Ekman number $E=\nu/H^2\Omega=10^{-3}$ reveals structured inertial wave activity on the spin-down time-scale. The analytic study, based on $E\ll 1$, builds on the results of Greenspan & Howard (J.Fluid Mech., vol.17, 1963, pp.385-404) for an infinite plane layer $L\to\infty$. At large but finite distance $r^†$ from the symmetry axis, the meridional (QG-)flow, that causes the QG-spin down, is blocked by the lateral boundary $r^†=L$, which provides a QG-trigger for inertial waves. The true situation in the unbounded layer is complicated further by the existence of a secondary set of maximum frequency (MF) inertial waves (a manifestation of the transient Ekman layer) identified by Greenspan & Howard. Their blocking at $r^†=L$ provides a secondary MF-trigger for yet more inertial waves that we consider in a sequel (Part II). Here, for the QG-trigger, we solve a linear initial value problem by Laplace transform methods. The ensuing complicated inertial wave structure is explained analytically on approximating our cylindrical geometry at large radius by rectangular Cartesian geometry, valid for $L-r^\star=O(H)$ ($L\gg H$). Other than identifying small scale structure near $r^\star=L$, our main finding is that inertial waves radiated away from the outer boundary (but propagating towards it) reach a distance determined by the group velocity.

中文翻译：

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在快速旋转的硬币形圆柱体中自旋下降过程中的惯性波活动：简化模型

在之前的一篇论文中，Oruba、Soward 和 Dormy (J.Fluid Mech., vol.818, 2017, pp.205-240) 考虑了粘度为 $\nu 的恒定密度流体的主要准稳态地转 (QG) 运动在半径为 L 和高度为 H 的圆柱形容器中线性旋转期间发生的，绕其对称轴快速旋转（角速度 $\Omega$），受到刚性和无应力混合边界条件的约束情况 $L=H$。在这里，大 $L=10 H$ 和 Ekman 数 $E=\nu/H^2\Omega=10^{-3}$ 的直接数值模拟 (DNS) 揭示了自旋下降时间的结构化惯性波活动 -规模。基于 $E\ll 1$ 的分析研究建立在 Greenspan & Howard (J.Fluid Mech., vol.17, 1963, pp.385-404) 的无限平面层 $L\to\十美元。在距离对称轴的大但有限距离 $r^†$ 处，导致 QG 旋转向下的经向 (QG-) 流被横向边界 $r^†=L$ 阻挡，该边界为惯性波提供 QG 触发器。由于格林斯潘和霍华德确定的第二组最大频率 (MF) 惯性波（瞬态埃克曼层的表现）的存在，无界层的真实情况进一步复杂化。它们在 $r^†=L$ 处的阻塞为我们在续集（第二部分）中考虑的更多惯性波提供了次级 MF 触发。在这里，对于 QG 触发器，我们通过拉普拉斯变换方法解决了线性初值问题。随之而来的复杂惯性波结构通过矩形笛卡尔几何在大半径下近似我们的圆柱几何进行解析解释，对 $Lr^\star=O(H)$ ($L\gg H$) 有效。