Transformation of Laplace Tidal Equations

Let $X=\xi$ , $Y=\eta$ , $P={\mit\Phi}$ , and $Q={\mit\Psi}$ in Equations (12.191)-(12.196). It follows that [cf., Equation (12.161)]

$$D\,{\mit\Phi} =\zeta,$$ (12.202)

$$D\,{\mit\Psi} = \frac{1}{\sin\theta}\left[\frac{\partial}{\partial\theta}\,(\sin\theta\,\eta) -\frac{\partial\xi}{\partial \phi}\right],$$ (12.203)

where

$$\frac{1}{\sin\theta}\,\frac{\partial{\mit\Phi}(\theta,\phi_\pm,t)}{\partial \phi} = 0,$$ (12.204)

$${\mit\Psi}(\theta,\phi_\pm,t) =0.$$ (12.205)

Here, use has been made of Equations (12.186) and (12.190). Furthermore [cf., Equations (12.159) and (12.160)],

$$\xi = -\frac{\partial{\mit\Phi}}{\partial \theta} - \frac{1}{\sin\theta}\,\frac{\partial{\mit\Psi}}{\partial\phi},$$ (12.206)

$$\eta = -\frac{1}{\sin\theta}\,\frac{\partial{\mit\Phi}}{\partial\phi} +\frac{\partial{\mit\Psi}}{\partial\theta}.$$ (12.207)

Let $X=\cos\theta\,\eta$ , $Y=-\cos\theta\,\xi$ , $P={\mit\Phi}'$ , and $Q={\mit\Psi}'$ in Equations (12.191)-(12.196). It follows that

$$D\,{\mit\Phi}' =-\frac{1}{\sin\theta}\left[\frac{\partial}{\partial\theta}\,(\si... ...eta\,\cos\theta\,\eta)-\frac{\partial}{\partial\phi}\,(\cos\theta\,\xi)\right],$$ (12.208)

$$D\,{\mit\Psi}' = -\frac{1}{\sin\theta}\left[\frac{\partial}{\partial\theta}\,(\s... ...a\,\cos\theta\,\xi) +\frac{\partial}{\partial \phi}\,(\cos\theta\,\eta)\right],$$ (12.209)

where

$$\frac{1}{\sin\theta}\,\frac{\partial{\mit\Phi}'(\theta,\phi_\pm,t)}{\partial \phi} =\cos\theta\,\xi(\theta,\phi_\pm,t),$$ (12.210)

$${\mit\Psi}'(\theta,\phi_\pm,t) =0.$$ (12.211)

Furthermore,

$$\cos\theta\,\eta = -\frac{\partial{\mit\Phi}'}{\partial\theta} -\frac{1}{\sin\theta}\,\frac{\partial{\mit\Psi}'}{\partial\phi},$$ (12.212)

$$-\cos\theta\,\xi = -\frac{1}{\sin\theta}\,\frac{\partial{\mit\Phi}'}{\partial\phi} +\frac{\partial{\mit\Psi}'}{\partial\theta}.$$ (12.213)

Substitution of Equations (12.211), (12.212), (12.217), and (12.218) into Equations (12.187) and (12.188) yields

$$\frac{\partial A}{\partial\theta} + \frac{1}{\sin\theta}\,\frac{\partial B}{\partial\phi} = 0,$$ (12.214)

$$\frac{\partial B}{\partial\theta} -\frac{1}{\sin\theta}\,\frac{\partial A}{\partial\phi} = 0,$$ (12.215)

where

$$A = \skew{5}\ddot{\mit\Phi} -2\,{\mit\Omega}\,\skew{5}\dot{\mit\Phi}' - \frac{g\,d}{a^{\,2}}\,(\zeta-F),$$ (12.216)

$$B =\skew{5}\ddot{\mit\Psi} - 2\,{\mit\Omega}\,\skew{5}\dot{\mit\Psi}'.$$ (12.217)

Equations (12.219) and (12.220) can be combined to give

$$D\,A = 0,$$ (12.218)

$$D\,B =0.$$ (12.219)

Moreover, it follows from Equations (12.188), (12.190), (12.209), (12.210), (12.215), and (12.216) that

$$\frac{1}{\sin\theta}\,\frac{\partial A(\theta,\phi_\pm,t)}{\partial \phi} = 0,$$ (12.220)

$$B(\theta,\phi_\pm,t) =0.$$ (12.221)

We have already seen that the solution of Equation (12.205), subject to the boundary condition (12.204), is $P'={\rm constant}$ . It follows that the solution of Equation (12.223), subject to the boundary condition (12.225), is $A={\rm constant}$ . Analogous arguments reveal that the solution of Equation (12.224), subject to the boundary condition (12.226), is $B=0$ . Hence, we deduce that the Laplace tidal equations, (12.186)-(12.188), are equivalent to the following set of equations:

$$D\,{\mit\Phi} =\zeta,$$ (12.222)

$$\skew{5}\ddot{\mit\Phi} -2\,{\mit\Omega}\,\skew{5}\dot{\mit\Phi}' - \frac{g\,d}{a^{\,2}}\,(\zeta-{\cal F}) ={\rm constant},$$ (12.223)

$$\skew{5}\ddot{\mit\Psi} - 2\,{\mit\Omega}\,\skew{5}\dot{\mit\Psi}' = 0,$$ (12.224)

where ${\mit\Phi}$ , ${\mit\Psi}$ , ${\mit\Phi}'$ , ${\mit\Psi}'$ , and ${\cal F}$ are defined in Equations (12.211), (12.212), (12.217), (12.218), and (12.189), respectively.