  We consider an Einstein-aether type Lorentz-violating theory of gravity in
which the aether vector field $V_{\mu }$ is represented as the gradient of a
scalar field $\phi $, $V_{\mu }=\nabla _{\mu }\phi $. A self interacting
potential for the scalar aether field is considered, as well as the possibility
of a coupling between the hydrodynamic matter flux and the aether field, with
the imposition of the timelike nature of the aether vector. The gravitational
field equations and the equation of motion of the scalar field are derived by
varying the action with respect to the metric and $\phi $. In the absence of
matter flux and scalar field coupling the effective energy-momentum tensor of
the scalar aether is conserved. The matter flux-aether coupling generates an
extra force acting on massive test particles and consequently the motion
becomes non-geodesic. The Newtonian limit of the theory is investigated and the
generalized Poisson equation for weak gravitational fields is obtained. The
cosmological implications of the theory is also considered and it is shown that
in the framework of the Scalar Einstein-aether theory both decelerating and
accelerating cosmological models can be constructed.
