Article Contents
Article Contents

# Geophysical internal equatorial waves of extreme form

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

• 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.

Mathematics Subject Classification: Primary: 76B55, 76B15; Secondary: 35Q31.

 Citation:

• Figure 1.  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

Figure 2.  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

Figure 3.  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 $t = 0$

Figure 4.  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

Figure 5.  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

Figure 6.  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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