  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 resulting from a Kelvin mode 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.
