Impact of a small ellipticity on the sustainability condition of developed turbulence in a precessing spheroid

We experimentally investigate the flow transition to developed turbulence in a precessing spheroid with a small ellipticity. Fully developed turbulence appears through a subcritical transition when we fix the Reynolds number (the spin rate) and gradually increase the Poincare number (the precession rate). In the transitional range of the Poincare number, two qualitatively different turbulent states (i.e., fully developed turbulence and quiescent turbulence with a spin-driven global circulation) are stable and they are connected by a hysteresis loop. This discontinuous transition is in contrast to the continuous transition in a precessing sphere, for which neither bistable turbulent states nor hysteresis loops are observed. The small ellipticity of the container makes the global circulation of the confined fluid more stable, and it requires much stronger precession of the spheroid, than a sphere, for fully developed turbulence to be sustained. Nevertheless, once fully developed turbulence is sustained, its flow structures are almost identical in the spheroid and sphere. The argument [Lorenzani and Tilgner, J. Fluid Mech. 492, 363 (2003); Noir et al., Geophys. J. Int. 154, 407 (2003)] on the basis of the analytical solution [Busse, J. Fluid Mech. 33, 739 (1968)] of the steady global circulation in a weak precession range well describes the onset of the fully developed turbulence in the spheroid.