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Shallow katabatic flow on a non-uniformly cooled slope
Environmental Fluid Mechanics ( IF 2.2 ) Pub Date : 2022-07-27 , DOI: 10.1007/s10652-022-09887-w
Richard Hewitt, Jay Unadkat, Anthony Wise

We examine katabatic flow driven by a non-uniformly cooled slope surface but unaffected by Coriolis acceleration. A general formulation is given, valid for non-uniform surface buoyancy distributions over a down-slope length scale \(L\gg \delta _0\), where \(\delta _0=\nu /(N\sin \alpha )^{1/2}\) is the slope-normal Prandtl depth, for a kinematic viscosity \(\nu\), buoyancy frequency N and slope angle \(\alpha\). We demonstrate that the similarity solution of Shapiro and Fedorovich (J Fluid Mech 571:149–175, 2007) can remain quantitatively relevant local to the end of a non-uniformly cooled region. The usefulness of the steady similarity solution is determined by a spatial eigenvalue problem on the L length scale. Broadly speaking, there are also two modes of temporal instability; stationary down-slope aligned vortices and down-slope propagating waves. By considering the limiting inviscid stability problem, we show that the origin of the vortex mode is spatial oscillation of the buoyancy profile normal to the slope. This leads to vortex growth in a region displaced from the slope surface, at a point of buoyancy inflection, just as the propagating modes owe their existence to an inflectional velocity. Non-uniform katabatic flows that detrain fluid to the ambient are shown to further destabilise the vortex mode whereas entraining flows lead to weaker vortex growth rates. Rayleigh waves dominate in general, but the vortex modes become more significant at small slope angles and we quantify their relative growth rates.



中文翻译:

非均匀冷却斜坡上的浅下垂流

我们检查了由非均匀冷却的坡面驱动但不受科里奥利加速度影响的下沉流。给出了一个通用公式,适用于下坡长度尺度\(L\gg \delta _0\)上的非均匀表面浮力分布,其中\(\delta _0=\nu /(N\sin \alpha )^ {1/2}\)是斜率法线普朗特深度,对于运动粘度\(\nu\)、浮力频率N和倾斜角\(\alpha\)。我们证明了 Shapiro 和 Fedorovich (J Fluid Mech 571:149–175, 2007) 的相似性解决方案在非均匀冷却区域的末端可以保持局部的定量相关性。稳定相似性解的有用性由空间特征值问题决定L长度刻度。广义上讲,时间不稳定性也有两种模式;静止的下坡对齐涡流和下坡传播波。通过考虑限制无粘稳定性问题,我们证明了涡流模式的起源是垂直于斜坡的浮力剖面的空间振荡。这导致在浮力拐点处偏离斜坡表面的区域中的涡流增长,正如传播模式的存在归因于拐点速度一样。非均匀下流将流体驱散到环境中,这表明会进一步破坏涡流模式,而夹带流会导致涡流生长速率变弱。瑞利波通常占主导地位,但涡流模式在小倾斜角处变得更加显着,我们量化了它们的相对增长率。

更新日期:2022-07-27
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