Mode coupling analysis and differential rotation in a flow driven by a precessing cylindrical container

We present a theoretical weakly nonlinear analysis of the dynamics of an inviscid flow submitted to both rotation and precession of an unbounded cylindrical container, by considering the coupling of two Kelvin (inertial) waves. The parametric centrifugal instability known for this system is shown to saturate when one expands the Navier-Stokes equation to higher order in the assumed small precession parameter (ratio of precession to rotation frequencies) with the derivation of two coupled Landau equations suitable to describe the dynamics of the modes. It is shown that an azimuthal mean flow with differential rotation is generated by this modes coupling. The time evolution of the associated dynamical system is studied. These theoretical results can be compared with water experiments and also to some numerical simulations where viscosity and finite length effects cannot be neglected.