Steady Flow in a Rapidly Rotating Spheroid with Weak Precession

The steady flow in a precessing spheroid is studied in the strong spin and weak precession limit. The spin and precession axes are orthogonal to each other. we denote the spin and precession angular velocities by Ω_s and Ω_p, respectively, and define the Reynolds number Re=a²Ω_s/ν and the Poincare number Po=Ω_p/Ω_s, where a is the equatorial radius and ν is the kinematic viscosity of fluid. It is shown that in the above limit, Po<<δ<<1 (δ=1/Re^1/2), only the uniform vorticity flow (Poincare 1910), relative to the spinning spheroid, can develop as a steady flow in the inviscid case. The vorticity is 2(1+c²)Po/|1-c²| in magnitude, c being the aspect ratio of the polar and equatorial lengths of the spheroid, and points to (or against) the z-axis for an oblate (c<1) (or a prolate (c>1)) spheroid. The vorticity magnitude diverges to infinity for a sphere (c=1). This apparent singular behavior of the vorticity at c=1 is clarified by introducing viscosity and analyzing the flow for a spheroid close to a sphere |c-1|<< 1 in the limit Po<<Max(δ, |1-c|). It is found that the vorticity magnitude is expressed as 2Po/{(2.620δ)²+(0.2585δ+1-c)²}^1/2+O(Po³/δ³) and it makes an angle arctan[(0.2585δ+1-c)/2.620δ]+O(Po²/δ²) from the negative y-axis toward the z-axis (Busse 1968).