Useful Lemma

Let $X(\theta,\phi,t)$ , $Y(\theta,\phi,t)$ , $P(\theta,\phi,t)$ , and $Q(\theta,\phi,t)$ be well-behaved functions of $\theta$ and $\phi$ . Suppose that

$$D\,P = -\frac{1}{\sin\theta}\left[\frac{\partial}{\partial\theta}\,(\sin\theta\,X) +\frac{\partial Y}{\partial\phi}\right],$$ (12.186)

where

$$\frac{1}{\sin\theta}\,\frac{\partial P(\theta,\phi_\pm,t)}{\partial\phi}=-Y(\theta,\phi_\pm,t),$$ (12.187)

Suppose, further, that

$$D\,Q =\frac{1}{\sin\theta}\left[\frac{\partial}{\partial\theta}\,(\sin\theta\,Y) - \frac{\partial X}{\partial\phi}\right],$$ (12.188)

where

$$Q(\theta,\phi_\pm,t) = 0.$$ (12.189)

We wish to demonstrate that

$$X = -\frac{\partial P}{\partial\theta} - \frac{1}{\sin\theta}\,\frac{\partial Q}{\partial\phi},$$ (12.190)

$$Y =-\frac{1}{\sin\theta}\,\frac{\partial P}{\partial\phi} + \frac{\partial Q}{\partial\theta}.$$ (12.191)

Let

$$X' =X + \frac{\partial P}{\partial\theta} + \frac{1}{\sin\theta}\,\frac{\partial Q}{\partial\phi},$$ (12.192)

$$Y' = Y + \frac{1}{\sin\theta}\,\frac{\partial P}{\partial\phi} - \frac{\partial Q}{\partial\theta}.$$ (12.193)

Note, from Equations (12.192) and (12.194), that

$$Y'(\theta,\phi_\pm,t) = 0.$$ (12.194)

It follows from Equations (12.191), (12.193), (12.197), and (12.198) that

$$\frac{\partial}{\partial\theta}\,(\sin\theta\,X') +\frac{\partial Y'}{\partial\phi} = 0,$$ (12.195)

$$\frac{\partial}{\partial\theta}\,(\sin\theta\,Y')-\frac{\partial X'}{\partial\phi} =0.$$ (12.196)

Equations (12.199) and (12.201) imply that

$$X' =\frac{\partial P'}{\partial\theta},$$ (12.197)

$$Y' =\frac{1}{\sin\theta}\,\frac{\partial P'}{\partial\phi},$$ (12.198)

and

$$\frac{1}{\sin\theta}\,\frac{\partial P'(\theta,\phi_\pm,t)}{\partial\phi} = 0.$$ (12.199)

Furthermore, Equation (12.200) yields

$$D\,P'=0.$$ (12.200)

Multiplying by $P'$ , integrating over the ocean, and making use of the boundary condition (12.204), we obtain

$$\int_{\mit\Omega}\left[\left(\frac{\partial P'}{\partial\theta}\r... ...}{\sin\theta}\,\frac{\partial P'}{\partial\phi}\right)^2\right]d{\mit\Omega}=0.$$ (12.201)

Here, ${\mit\Omega}$ is the surface of the Earth that is covered by ocean, and $d{\mit\Omega}=\sin\theta\,d\theta\,d\phi$ . It follows that $P'$ is a constant. Thus, Equations (12.202) and (12.203) imply that $X'=Y'=0$ , and, hence, that Equations (12.195) and (12.196) are valid.