On Long's hypothesis of no upstream influence in uniformly stratified or rotating flow

Abstract

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 velocityU, 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 ε2somewhere 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 sinzand 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, beingO(ε2t−2) as timet→ ∞.The basic expansion in powers of ε will be invalid for times ∝ ε−1or greater, because of resonant-interactive instability of the lee waves.