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Geophysical internal equatorial waves of extreme form

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

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  • 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.


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  • 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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