Upstream influence of a dipole in rotating flow

Abstract

McIntyre (1972) has demonstrated that transient nonlinear self-interactions among the lee waves downstream of an obstacle in an axisymmetric, inviscid, rotating flow yield a columnar disturbance that moves upstream of the obstacle. This disturbance is calculated for a dipole, the moment of which increases slowly from 0 to a³, in a tube of radius R as a function of δ = a/R and K = 2ΩR/U (Ω and U being angular and axial velocities in basic flow). The asymptotic limit δ ↓ 0 with k ≡ K δ fixed, which is relevant for typical laboratory configurations and for the abstraction of unbounded flow, yields: (a) an upstream velocity with an axial peak of u₀ = 0·011k⁵ δ U and a first zero at a radius of 2·8U/Ω; (b) an upstream energy flux of roughly $- \frac{3}{2}\pi\rho U^2 R^2 u_0$; (c) an upstream impulse flux of 0·20D (D = wave drag on dipole). The results (a) and (b), albeit based on the hypothesis of small disturbances, suggest that nonlinear self-interactions among the lee waves could be responsible for upstream blocking. The results (a) and (c) imply that, although upstream influence is absent from an unbounded (δ = 0) flow in the sense that the axial velocity vanishes like δ, it is present in the sense that approximately one-fifth of the impulse flux associated with the wave drag appears in the upstream flow.