# American Institute of Mathematical Sciences

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Dispersion relations for steady periodic water waves of fixed mean-depth with two rotational layers
September  2019, 39(9): 5171-5183. doi: 10.3934/dcds.2019210

## Nonhydrostatic Pollard-like internal geophysical waves

 School of Mathematical Sciences, University College Cork, Western Road, Cork, Ireland

Received  August 2018 Revised  March 2019 Published  May 2019

We present a new exact and explicit Pollard-like solution describing internal water waves representing the oscillation of the thermocline in a nonhydrostatic model. The derived solution is a modification of Pollard's surface wave solution in order to describe internal water waves at general latitudes. The novelty of this model consists in the embodiment of transitional layers beneath the thermocline. We present a Lagrangian analysis of the nonlinear internal water waves and we show the existence of two modes of the wave motion.

Citation: Mateusz Kluczek. Nonhydrostatic Pollard-like internal geophysical waves. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5171-5183. doi: 10.3934/dcds.2019210
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##### References:
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 $\tilde{f} = f/\hat{f} = \tan\phi$
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
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]
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 $O(\epsilon)$
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