"In this proposal we outline a framework for studying interfacial wave dynamics between gas-liquid and liquid-liquid flows using the nonlinear Parabolized Stability Equations (PSE). Previous studies have shown that the nonlinear PSE is an efficient and accurate method for modeling convectively traveling disturbance waves in a slowly evolving mean flow. By combining the PSE approach with an appropriate level set technique, the method can then track the growth and evolution of interfacial waves in two phase flows. The envisaged PSE framework is superior to the linear stability approach because it incorporates the effects of nonlinear mean flows, finite interfacial deformations, and nonlinear modal interactions. In addition, the streamwise marching based solution procedure in the proposed PSE-level set method is also more computationally efficient when compared to higher fidelity Direct Numerical Simulations (DNS). The proposed framework will be applied to a variety of two phase flow problems, including problems of both fundamental and practical importance. In order to validate the method, the PSE-level set framework will be tested on the canonical Kelvin-Helmholtz shear flow instability, and the results compared with linear stability and DNS calculations. Once verified, the approach will be applied to various challenging problems, including the capillary wave breakup of liquid jets, liquid film instabilities in annular two phase flow, wind-ocean interactions in geophysical flows, and bypass transition phenomena in two phase boundary layers."