  We consider rotating flows in non-axisymmetric enclosures that are driven by
libration, i.e. by a small periodic modulation of the rotation rate. Thanks to
its simplicity, this model is relevant to various contexts, from industrial
containers (with small oscillations of the rotation rate) to fluid layers of
terrestial planets (with length-of-day variations). Assuming a multipolar
$n$-fold boundary deformation, we first obtain the two-dimensional basic flow.
We then perform a short-wavelength local stability analysis of the basic flow,
showing that an instability may occur in three dimensions. We christen it the
Libration Driven Multipolar Instability (LDMI). The growth rates of the LDMI
are computed by a Floquet analysis in a systematic way, and compared to
analytical expressions obtained by perturbation methods. We then focus on the
simplest geometry allowing the LDMI, a librating deformed cylinder. To take
into account viscous and confinement effects, we perform a global stability
analysis, which shows that the LDMI results from a parametric resonance of
inertial modes. Performing numerical simulations of this librating cylinder, we
confirm that the basic flow is indeed established and report the first
numerical evidence of the LDMI. Numerical results, in excellent agreement with
the stability results, are used to explore the non-linear regime of the
instability (amplitude and viscous dissipation of the driven flow). We finally
provide an example of LDMI in a deformed spherical container to show that the
instability mechanism is generic. Our results show that the previously studied
libration driven elliptical instability simply corresponds to the particular
case $n=2$ of a wider class of instabilities. Summarizing, this work shows that
any oscillating non-axisymmetric container in rotation may excite intermittent,
space-filling LDMI flows, and this instability should thus be easy to observe
experimentally.
