  The Rayleigh--Taylor instability of two immiscible fluids in the limit of
small Atwood numbers is studied by means of a phase-field description. In this
method the sharp fluid interface is replaced by a thin, yet finite, transition
layer where the interfacial forces vary smoothly. This is achieved by
introducing an order parameter (the phase field) whose variation is continuous
across the interfacial layers and is uniform in the bulk region. The phase
field model obeys a Cahn--Hilliard equation and is two-way coupled to the
standard Navier--Stokes equations. Starting from this system of equations we
have first performed a linear analysis from which we have analytically
rederived the known gravity-capillary dispersion relation in the limit of
vanishing mixing energy density and capillary width. We have performed
numerical simulations and identified a region of parameters in which the known
properties of the linear phase (both stable and unstable) are reproduced in a
very accurate way. This has been done both in the case of negligible viscosity
and in the case of nonzero viscosity. In the latter situation only upper and
lower bounds for the perturbation growth-rate are known. Finally, we have also
investigated the weakly-nonlinear stage of the perturbation evolution and
identified a regime characterized by a constant terminal velocity of
bubbles/spikes. The measured value of the terminal velocity is in perfect
agreement with available theoretical prediction. The phase-field approach thus
appears to be a valuable tecnhique for the dynamical description of the stages
where hydrodynamic turbulence and wave-turbulence enter into play.
