  This paper is devoted to the study of the long wave approximation for water
waves under the influence of the gravity and a Coriolis forcing. We start by
deriving a generalization of the Boussinesq equations in 1D (in space) and we
rigorously justify them as an asymptotic model of the water waves equations.
These new Boussinesq equations are not the classical Boussinesq equations. A
new term due to the vorticity and the Coriolis forcing appears that can not be
neglected. Then, we study the Boussinesq regime and we derive and fully justify
different asymptotic models when the bottom is flat : a linear equation linked
to the Klein-Gordon equation admitting the so-called Poincar{\'e} waves; the
Ostrovsky equation, which is a generalization of the KdV equation in presence
of a Coriolis forcing, when the rotation is weak; and finally the KdV equation
when the rotation is very weak. Therefore, this work provides the first
mathematical justification of the Ostrovsky equation. Finally, we derive a
generalization of the Green-Naghdi equations in 1D in space for small
topography variations and we show that this model is consistent with the water
waves equations.
