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In this paper, we consider an incompressible viscous fluid in an infinitelydeep ocean, being bounded above by a free moving boundary. The governingequations are the gravity-driven incompressible Navier-Stokes equations withvariable density and no surface tension is taken into account on the freesurface. After using the Lagrangian transformation, we write the main equationsin a perturbed form in a fixed domain. In the first part, we describe aspectral analysis of the linearized equations around a hydrostatic equilibrium$(\rho_0(x_3), 0, P_0(x_3))$ for a smooth increasing density profile $\rho_0$.Precisely, we prove that there exist multiple normal modes to the linearizedequations by following the operator method initiated by Lafitte and Nguyen\cite{LN20}. In the second part, we study the nonlinear Rayleigh-Taylorinstability around the above profile by constructing a wide class of initialdata for the nonlinear perturbation problem departing from the equilibrium,based on the finding of multiple normal modes. Our nonlinear result follows theprevious framework of Guo and Strauss \cite{GS95} and also of Grenier\cite{Gre00} with a refinement.

Nonlinear Rayleigh-Taylor instability of the viscous surface wave in an infinitely deep ocean