On Long's hypothesis of no upstream influence in uniformly stratified or rotating flow
On Long's hypothesis of no upstream influence in uniformly stratified or rotating flow
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关于均匀分层或旋转流中没有上游影响的朗假设
DOI：
10.1017/s0022112072001387
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发表时间：
1972
影响因子：
3.7
通讯作者：
M. McIntyre
中科院分区：
文献类型：
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作者：
M. McIntyre
本文研究了均匀分层Boussinesq流体中细长障碍物以恒定基本水平速度U流过障碍物时引起的二维弱非线性扰动问题，其扰动振幅ε为二阶。类似的旋转问题也被处理。特别注意的是明确计算柱扰动强度的上游和下游的障碍，无论是在分层和旋转的问题，以讨论的真理或其他龙的假设（LH）。柱扰动是否被发现远离上游，违反LH，取决于，除其他外，是否流是外部有界的刚性水平面（或由管或环，在旋转问题），并在该问题是否是确定通过一个'无粘瞬态'的提法，或通过一个'粘性'的。无粘、瞬态、有界问题，对于无初始扰动状态下的背风波随时间的发展，总是在流体中的某个地方出现ε2以下的柱状扰动。它们不是在障碍物附近产生的，而是在背风波列的“尾部”或瞬态终端区产生的。柱扰动强度在很大程度上独立于流动是如何从最初的未受干扰的状态。在所有情况下，除了一个例子之外，在远上游，效果是非零的。唯一的例外是单亚临界分层（或窄间隙旋转）情况，其中激励具有模态结构sin（2z），流体区域为0 [les ] z [les ] π。在这种情况下，唯一可以穿透上游的柱状扰动具有结构sinz，因此不被激励。一个完全不同的结果适用于“粘性”公式的未分离，有界的regimes（与稳定的背风波空间衰减的小分子扩散的影响）。所有柱状扰动的强度，上游和下游，消失在小扩散率的限制。在无粘性、瞬态、无界问题中，上游影响同样是消失的，当时间t → ∞时为O（ε2t−2）。由于背风波的共振相互作用不稳定性，ε的幂的基本展开式在ε ε−1或更大的时候将是无效的。
The weakly nonlinear, two-dimensional problem for the disturbance due to a slender obstacle in a uniformly stratified, Boussinesq fluid moving past the obstacle with constant basic horizontal velocity U, is considered up to second order in the amplitude ε of the disturbance. Analogous rotating problems are also treated. Particular attention is given to calculating explicitly the columnar-disturbance strengths upstream and downstream of the obstacle, both in the stratified and in the rotating problems, with a view to discussing the truth or otherwise of Long's hypothesis (LH). Whether or not columnar disturbances are found far upstream, violating LH, depends, interalia, on whether or not the flow is externally bounded by rigid horizontal planes (or by a tube or annulus, in the rotating problem), and on whether the problem is made determinate by means of an ‘inviscid transient’ formulation, or by means of a ‘viscous’ one. The inviscid, transient, bounded problem, for time-development of lee waves from a state of no initial disturbance, always exhibits columnar disturbances oforder ε2 somewhere in the fluid. They are generated, not near the obstacle, but in the ‘tails’ or transient terminal zones of the lee-wave trains. The columnar-disturbance strengths are largely independent of how the flow is set up from an initially undisturbed state. I n all but one instance the effect is non-zero far up-stream. The exception is the singly-subcritical stratified (or narrow-gap rotating) case, in which the excitation has modal structure sin(2z), the fluid region being 0 [les ] z [les ] π in this case the only columnar disturbance that can penetrate up-stream has structure sinz and so is not excited. A completely different result holds for ‘viscous’ formulations for unseparated, bounded régimes (with steady lee waves spatially attenuated by effects of small molecular diffusion). The strengths of all columnar disturbances, upstream and downstream, vanish in the limit of small diffusivity. In the inviscid, transient, unbounded problem, the upstream influence is, likewise, evanescent, being O(ε2t−2) as time t → ∞. The basic expansion in powers of ε will be invalid for times ∝ ε−1 or greater, because of resonant-interactive instability of the lee waves.