The quasi-geostrophic theory of the thermal shallow water equations
The quasi-geostrophic theory of the thermal shallow water equations
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DOI：
10.1017/jfm.2013.101
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发表时间：
2013-04
影响因子：
3.7
通讯作者：
E. Warneford;P. Dellar
中科院分区：
文献类型：
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作者：
E. Warneford;P. Dellar
摘要浅水热力方程提供了流体层运动的深度平均描述，允许材料性质的水平变化。它们通常通过两层系统的等效正压近似产生，由于活动层中不断变化的温度场而具有空间变化的密度对比度。我们形式化以前的准地转（QG）理论推导这些方程，通过执行小Rossby数的直接渐近展开。然后，我们提出了第二个推导作为小Rossby数限制的平衡模型，项目的高频动态由于惯性重力波。后一种推导具有更广泛的有效性，不限于中纬度$\beta $ -飞机。我们还从热浅水赝能量守恒方程的QG极限导出了它们的局部能量守恒方程。该推导涉及对首阶地转速度的非地转修正，该修正在QG位涡闭合演化方程的通常推导中被消除。最后，我们推导出的非正则哈密顿结构的热QG方程的Rossby数分解的赝能和泊松括号的热浅水方程。
Abstract The thermal shallow water equations provide a depth-averaged description of motions in a fluid layer that permits horizontal variations in material properties. They typically arise through an equivalent barotropic approximation of a two-layer system, with a spatially varying density contrast due to an evolving temperature field in the active layer. We formalize a previous derivation of the quasi-geostrophic (QG) theory of these equations, by performing a direct asymptotic expansion for small Rossby number. We then present a second derivation as the small Rossby number limit of a balanced model that projects out high-frequency dynamics due to inertia-gravity waves. This latter derivation has wider validity, not being restricted to mid-latitude $\beta $ -planes. We also derive their local energy conservation equation from the QG limit of a thermal shallow water pseudo-energy conservation equation. This derivation involves the ageostrophic correction to the leading-order geostrophic velocity that is eliminated in the usual derivation of a closed evolution equation for the QG potential vorticity. Finally, we derive the non-canonical Hamiltonian structure of the thermal QG equations from a decomposition in Rossby number of a pseudo-energy and Poisson bracket for the thermal shallow water equations.