The 14th FDR prize

Abstract

Steady flow in a rapidly rotating spheroid with weak precession: I Shigeo Kida Vol. 52, No. 1, 015513 (2020) The flow in a rotating spheroidal container that executes precession has long attracted attention as a model for pursuing the origin of the dynamos of celestial bodies including the geodynamo. In recent years, research has been made from the viewpoint of generating a compact turbulence and utilizing it for efficient mixer (Goto et al 2007). The first key result is a steady inviscid solution heuristically derived by Poincaré (1910), which is a uniform vorticity state with the velocity field linear in coordinates; a spheroidal container filled with an inviscid fluid rapidly rotating around the axis of symmetry (referred to as the x axis) precesses around another axis (referred to as the z axis) perpendicular to the symmetric axis. The vorticity, relative to the spinning spheroid, is parallel to the z axis for an oblate spheroid but is antiparallel for a prolate spheroid. In between, there is no special direction for a spherical container, for which the magnitude of the vorticity diverges. To rescue this difficulty, Busse (1968) made an attempt at incorporating the effect of the boundary layer on the vessel by using the integral balance of torque, but his treatment turned out to be incomplete. This paper has resolved the problem of singularity by making a precise perturbation analysis of the boundary layer. There are three parameters in this problem, the aspect ratio c= b/a of the spheroid with the axis length 2 b and the equatorial length a, the Poincaré number Po=Ωp/Ωs, being the ratio of the precession angular velocity Ωp to the angular velocity Ωs of the main spin, and the Reynolds number Re= aΩs/ν, with ν being the kinematic viscosity of fluid. For a spheroid close to a sphere |c− 1| ≪ 1, a boundary-layer solution is constructed in the region of Po≪Max(Re−1/2, |c− 1|) (the obtained solution requests |c− 1| ≪ Re−1/2). Introducing ellipsoidal coordinates in the ‘rigidly rotating system’ with the X axis parallel to the full angular velocity vector ω̄, a solution is constructed in the form of perturbations in powers of a small parameter ε̄, a measure of the magnitude of the deviation flow, in such a way that the boundary-layer solution smoothly matches to the solution in the inviscid region. Unknown parameters are the three components of the perturbation of the angular velocity due to precession. For the spherical case (c= 1), these remain undetermined. The balance of the total torque is invoked, consisting of the integrals of the pressure, the precession-induced Coriolis force, and the viscous stress. Busse (1968) carried out this procedure to O(ε̄), which is insufficient for obtaining the correction term c̄ of the x component of the angular velocity. This paper has gained the axisymmetric component c̄ for the first time by including, in the torque, the circumferential average of the nonlinear boundary-layer solution of O(ε̄2). Without this term, the total angular velocity |ω̄| is not obtained correctly. The correction c̄ takes a significant value only when the container is close to a sphere (|c− 1| ≈ Re−1/2), and is indispensable for describing how the perturbation vorticity vector flips from negative to positive z direction when the spheroidal shape changed from prolate to oblate. Moreover, a mathematical proof is given for the uniqueness of the stationary inviscid solution.