# American Institute of Mathematical Sciences

August  2019, 39(8): 4471-4486. doi: 10.3934/dcds.2019183

## Geophysical internal equatorial waves of extreme form

 Department of Computing & Mathematics, Waterford Institute of Technology, Waterford, Ireland

The paper is for the special theme: Mathematical Aspects of Physical Oceanography, organized by Adrian Constantin

Received  August 2018 Revised  October 2018 Published  May 2019

The existence of internal geophysical waves of extreme form is confirmed and an explicit solution presented. The flow is confined to a layer lying above an eastward current while the mean horizontal flow of the solutions is westward, thus incorporating flow reversal in the fluid.

Citation: Tony Lyons. Geophysical internal equatorial waves of extreme form. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4471-4486. doi: 10.3934/dcds.2019183
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The rotating $\left(x,y,z\right)$-coordinate system fixed to the surface of the Earth. The $x$-axis points due-east, the $y$-axis points due-north and the $z$-axis points vertically upwards from the surface
A cross section of the flow in the equatorial plane $y = 0$ with a flow wavelength of $L = 200\,\mathrm{m}$. The average depth of the near surface layer is $60\,\mathrm{m}$, while the thermocline lies at an average depth of $120\,\mathrm{m}$. The transitional layer begins at $160\, \mathrm{m}$ approximately, while the motionless layer begins at $200\,\mathrm{m}$ beneath the free surface
The circular paths in the equatorial plane $y = 0$ traced by particles in the layer $\mathcal{M}(t)$ are circles with centre $(q,r-d_0)$ whose radius increases with depth. The wavelength of the solution shown is $L = 200\ \mathrm{m}$. The particle trajectories are counterclockwise and the particle positions shown in the figure are at time $t = 0$
A pictorial outline for the proof of Theorem 3.2, confirming the existence of a solution of the implicit relation $G(s,r) = G(0,0)$ of extreme form
The left-hand panel shows the graph $(s,r_0(s))$ when $\kappa = 0.5$ and $L = 200$ meters. The parameter $d_0 = 120$ ensures a mean equatorial depth of $120\,$m. In the right-hand panel the corresponding thermocline profile in the fluid domain, at the equatorial locations indicated in the figure
The thermocline surface when $L = 200\,$m, $r_0^* = 0\,$m and $d_0 = 120\,$m, with $\kappa = 0.5$. The surface is viewed from beneath and the value of $\kappa$ has been chosen to emphasise the extreme behaviour of the wave along the equator at the wave-troughs
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