  The present work is devoted to the derivation of a fully well-balanced and
positivepreserving numerical scheme for the shallow water equations with
Coriolis force. The first main issue consists in preserving all the steady
states, including the geostrophic equilibrium. Our strategy relies on a
Godunov-type scheme with suitable source term and steady state discretisations.
The second challenge lies in improving the order of the scheme while preserving
the fully well-balanced property. A modification of the classical methods is
required since no conservative reconstruction can preserve all the steady
states in the case of rotating shallow water equations. A steady state detector
is used to overcome this matter. Some numerical experiments are presented to
show the relevance and the accuracy of both first-order and second-order
schemes.
