  We develop a weakly nonlinear model to study the spatiotemporal manifestation
and the dynamical behavior of surface waves in the presence of an underlying
interfacial solitary wave in a two-layer fluid system. We show that interfacial
solitary-wave solutions of this model can capture the ubiquitous broadening of
large-amplitude internal waves in the ocean. In addition, the model is capable
of capturing three asymmetric behaviors of surface waves: (i) Surface waves
become short in wavelength at the leading edge and long at the trailing edge of
an underlying interfacial solitary wave. (ii) Surface waves propagate towards
the trailing edge with a relatively small group velocity, and towards the
leading edge with a relatively large group velocity. (iii) Surface waves become
high in amplitude at the leading edge and low at the trailing edge. These
asymmetric behaviors can be well quantified in the theoretical framework of
ray-based theories. Our model is relatively easily tractable both theoretically
and numerically, thus facilitating the understanding of the surface signature
of the observed internal waves.
