Detached shear layers in a rotating fluid

Abstract

The occurrence of detached shear layers should, according to straightforward theoretical arguments, often characterize hydrodynamical motions in a rapidly rotating fluid. Such layers have been produced and studied in a very simple system, namely a homogeneous liquid of kinematical viscosity v filling an upright, rigid, cylindrical container mounted coaxially on a turn-table rotating at Ω₀ rad/s about a vertical axis, and stirred by rotating about the same axis at Ω₁ rad/s a disk of radius a cm and thickness b’ cm immersed in the liquid with its plane faces parallel to the top and bottom end walls of the container. By varying Ω₀, Ω₁ and a, ranges of Rossby number, the modulus of ε ≡ (Ω₁ + Ω₀)/½ (Ω₁ + Ω₀), from 0·01 to 0·3, and Ekman number, E ≡ 2v/a²(Ω₁ + Ω₀), from 10⁻⁵ to 5 × 10⁻⁴ were attained. Although the apparatus was axisymmetric, only when |ε| did not exceed a certain critical value, |ε_T|, was the flow characterized by the same property of symmetry about the axis of rotation. Otherwise, when |ε| > |ε_T|, non-axisymmetric flow occurred, having the form in planes perpendicular to the axis of rotation of a regular pattern of waves, M in number, when ε was positive, and of a blunt ellipse when ε was negative.

The axial flow in the axisymmetric detached shear layer, and the uniform rate of drift of the wave pattern characterizing the non-axisymmetric flow when ε is positive, depend in relatively simple ways on ε and E. The dependence of|ε_T| on E can be expressed by the empirical relationship |ε_T| = AEⁿ, where A = 16·8 ± 2·2 and n = 0·568 ± 0·013 (= (4/7) × (1·000 − (0·005 ± 0·023))!), standard errors, 25 determinations. M does not depend strongly on E but generally decreases with increasing ε.