Nonhydrostatic Pollard-like internal geophysical waves

Figure 1.  The nonhydrostatic model and its flow regions for fixed $ y $ and $ \phi $. The thermocline separates two layers of ocean water with different but constant densities $ \rho_+>\rho_0 $ in a stable stratification [15,18,43]. The thermocline is described by a trochoid propagating with the phase speed $ c $. The transitional layer $ \cal{T}(t) $ provide a transition from the wave motion region to the motionless abyssal deep-water region of the ocean. The schematic is presented for fixed $ \tilde{f} = f/\hat{f} = \tan\phi $

Figure 2.  The angle of the inclination of particles' orbits is $ \arctan(-d/a) $ [14,35] and is increasing with the latitude resulting in the three-dimensional profile of the internal water wave [35] (see figure 3). The upper and lower interface of the transitional layer becomes also inclined at the angle $ \phi $ with respect to the local meridional axis. The inclination of the orbits and the interfaces is the result of the Earth's constant rotation

Figure 3.  The schematic three-dimensional profile of the internal water waves. The cross-wave tilt is a result of Earth's rotation [14,35,38]. At the equator the wave profile is in the vertical plane [3]

Figure 4.  The plot of the polynomial P(X) evaluated at 45$ ^\circ $ N for $ (\rho_+-\rho_0)/\rho_0 = 4\times 10^{-3} $, $ k = 6.28\times 10^{-2} $ $ m^{-1} $. The upper plot shows two roots of order $ O(1) $, whereas the lower plot presents two roots of order $ O(\epsilon) $