Inspired by the need to theoretically understand the naturally occurring interactions between internal waves and mesoscale phenomena in the ocean, we derive a novel model equation from the primitive rotational Euler equations using the multi-scale asymptotic expansion method. By applying the classic balance $\epsilon =\mu ^2$ between nonlinearity (measured by $\epsilon$) and dispersion (measured by $\mu$), along with the assumption that variations in the transverse direction are of order $\mu$, which is smaller than those in the propagation direction, we arrive at terms from the classic Kadomtsev–Petviashvili equation. However, when incorporating background shear currents in two horizontal dimensions and accounting for Earth’s rotation, we introduce three additional terms that, to the best of the authors’ knowledge, have not been addressed in the previous literature. Theoretical analyses and numerical results indicate that these three terms contribute to a tendency for propagation in the transverse direction and an overall variation in wave amplitudes. The specific effects of these terms can be estimated qualitatively based on the signs of the coefficients for each term and the characteristics of the initial waves. Finally, the potential shortcomings of this proposed equation are illuminated.

We derive a higher order nonlinear evolution equation for a broader bandwidth three-dimensional capillary–gravity wave packet, in the presence of a surface current produced by an internal wave. Instead of a set of coupled equations, a single nonlinear evolution equation is obtained by eliminating the velocity potential for the wave-induced slow motion. Finally, the equation is expressed in an integro-differential equation form, similar to Zakharov’s integral equation. Using the evolution equation derived here, we show that the two sidebands of a surface capillary–gravity wave get excited as a result of resonance with an internal wave, all propagating in the same direction. It is also shown that surface waves can grow exponentially with time at the expense of the energy of the internal wave.