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.

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