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Next: Useful Lemma Up: Terrestrial Ocean Tides Previous: Global Ocean Tides


Non-Global Ocean Tides

Suppose that the surface of the Earth is covered by an ocean of uniform depth $ d$ that extends from $ \theta=0$ to $ \theta=\pi$ , and from $ \phi=\phi_-$ to $ \phi=\phi_+$ , where $ \vert\phi_+-\phi_-\vert< 2\pi$ . For consistency with the previous analysis, we must imagine that the remaining surface is covered by a frozen (i.e., $ \zeta=0$ ) ocean of uniform depth $ d$ . The motion of the ocean under the action of the tide generating potential is governed by the Laplace tidal equations, (12.153)-(12.155), which are written

$\displaystyle \frac{1}{\sin\theta}\left[\frac{\partial}{\partial\theta}\,(\sin\theta\,\xi)+ \frac{\partial\eta}{\partial\phi}\right] + \zeta$ $\displaystyle = 0,$ (12.181)
$\displaystyle \skew{5}\ddot{\xi} -2\,{\mit\Omega}\,\cos\theta\,\skew{3}\dot{\eta}$ $\displaystyle =- \frac{g\,d}{a^{\,2}}\,\frac{\partial }{\partial\theta}\,(\zeta-{\cal F}),$ (12.182)
$\displaystyle \skew{3}\ddot{\eta} + 2\,{\mit\Omega}\,\cos\theta\,\skew{5}\dot{\xi}$ $\displaystyle =- \frac{g\,d}{a^{\,2}\,\sin\theta}\,\frac{\partial}{\partial\phi}\,(\zeta-{\cal F}),$ (12.183)

where

$\displaystyle {\cal F}(\theta,\phi,t)=(1+k_2-h_2)\,\skew{5}\bar{\zeta}_2(\theta...
...)+\oint G_F(\theta,\phi;\theta',\phi')\,\zeta(\theta',\phi',t)\,d{\mit\Omega}'.$ (12.184)

The finite extent of the ocean in the $ \phi$ direction introduces the additional constraint

$\displaystyle \eta(\theta,\phi_\pm,t) = 0$ (12.185)

into the problem.


next up previous
Next: Useful Lemma Up: Terrestrial Ocean Tides Previous: Global Ocean Tides
Richard Fitzpatrick 2016-03-31