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

$$\frac{1}{\sin\theta}\left[\frac{\partial}{\partial\theta}\,(\sin\theta\,\xi)+ \frac{\partial\eta}{\partial\phi}\right] + \zeta = 0,$$ (12.181)

$$\skew{5}\ddot{\xi} -2\,{\mit\Omega}\,\cos\theta\,\skew{3}\dot{\eta} =- \frac{g\,d}{a^{\,2}}\,\frac{\partial }{\partial\theta}\,(\zeta-{\cal F}),$$ (12.182)

$$\skew{3}\ddot{\eta} + 2\,{\mit\Omega}\,\cos\theta\,\skew{5}\dot{\xi} =- \frac{g\,d}{a^{\,2}\,\sin\theta}\,\frac{\partial}{\partial\phi}\,(\zeta-{\cal F}),$$ (12.183)

where

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

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

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

into the problem.