Zonal flow in a resonant precessing cylinder
Résumé
A cylinder undergoes precession when it rotates around its axis and this axis itself rotates around another direction. In a precessing cylinder full
of fluid, a steady and axisymmetric component of the azimuthal flow is generally present. This component is called a zonal flow. Although zonal
flows have been often observed in experiments and numerical simulations, their origin has eluded theoretical approaches so far. Here,
we develop an asymptotic analysis to calculate the zonal flow forced in a resonant precessing cylinder. We find that the zonal flow originates from three different sources: (1) the nonlinear interaction of the inviscid Kelvin mode with its viscous correction; (2) the steady and axisymmetric
response to the nonlinear interaction of the Kelvin mode with itself; and (3) the nonlinear interactions in the end boundary layers. In a precessing
cylinder, two additional sources arise due to the equatorial Coriolis force and the forced shear flow. However, they cancel exactly. The study thus
generalises to any Kelvin mode, forced by precession or any other mechanism. The present theoretical predictions of the zonal flow are
confirmed by comparison with numerical simulations and experimental results. We also show that the zonal flow is always retrograde in a
resonant precessing cylinder. Although this result seems intuitive, to our knowledge it is the first proof. A retrograde zonal flow is also produced
for resonant Kelvin modes of azimuthal wavenumbers m=2, 3, and presumably higher.