Numerical solutions of the precession-driven flows inside a sphere are presented by means of a previously proposed spectral method based on helical wave decomposition, and flow properties are investigated in helical wave spectral space. Four different flow states can be generated under precession, including the steady, periodic, quasi-periodic, and turbulent ones. Flow fields are decomposed into two components of opposite polarity by the sign of the helicity of each helical wave. It is found that the flows in the steady and periodic states are polarity-symmetric, while the quasi-periodic and turbulent states are polarity-asymmetric, regarding the kinetic energy distribution for each polarity. The two components of opposite polarity for the quasi-periodic case have exactly the same frequency spectra with respect to the kinetic energy, differing from the turbulent case. At high Reynolds numbers, the helical wave energy spectra show a scaling of λ − 7 / 3, which is different from the scaling of k − 2 for the homogeneous turbulence under precession. The helical wave spectral dynamic equation is derived for the investigation of the mechanism of the turbulent flows. The energy to sustain the precession-driven flows comes from the boundary motion, which is equivalent to a body force being enforced on all scales in spectral space. The energy is concentrated on the largest scales and transferred to smaller scales through the nonlinear interaction. In contrast, the Coriolis force gives rise to an inverse cascade that transfers energy from small to large scales.
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