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Next: Harmonics of Forcing Term Up: Terrestrial Ocean Tides Previous: Planetary Response

Laplace Tidal Equations

In the co-rotating frame, the linearized (in $ {\bf v}$ ) equations of motion of the ocean that covers the surface of the planet are (Fitzpatrick 2012)

$\displaystyle \nabla\cdot{\bf v}$ $\displaystyle = 0,$ (12.100)
$\displaystyle \frac{\partial{\bf v}}{\partial t} + 2\,$$\displaystyle \mbox{\boldmath$\Omega$}$$\displaystyle \times {\bf v}$ $\displaystyle = - \nabla\left(\frac{p}{\skew{3}\bar{\rho}} + {\mit\Phi} + \chi\right),$ (12.101)

where $ {\bf v}({\bf r})$ is the fluid velocity, and $ p({\bf r})$ the fluid pressure. Here, $ {\mit\Phi}({\bf r})$ is the total gravitational potential, $ \chi({\bf r})$ the centrifugal potential (due to planetary rotation), and $ 2\,$$ \Omega$ $ \times {\bf v}$ the Coriolis acceleration (likewise, due to planetary rotation). Furthermore, we have neglected the finite compressibility of the fluid that makes up the ocean. Let $ r$ , $ \theta $ , $ \phi$ be spherical coordinates in the co-rotating frame, and let $ z=r-a$ . It is helpful to define $ u=v_\theta$ , $ v=v_\phi$ , and $ w=v_r$ . Assuming that $ \vert z\vert\ll a$ , the previous equations of motion reduce to

$\displaystyle \frac{1}{a\,\sin\theta}\,\frac{\partial}{\partial\theta}\,(\sin\t...
...heta}\,\frac{\partial v}{\partial\phi}+ \frac{\partial w}{\partial z} \simeq 0,$ (12.102)

and

$\displaystyle \frac{\partial u}{\partial t} - 2\,{\mit\Omega}\,\cos\theta\,v$ $\displaystyle =- \frac{1}{a}\,\frac{\partial}{\partial\theta}\left(\frac{p}{\skew{3}\bar{\rho}} + {\mit\Phi} + \chi\right),$ (12.103)
$\displaystyle \frac{\partial v}{\partial t} + 2\,{\mit\Omega}\,\cos\theta\,u+2\,{\mit\Omega}\,\sin\theta\,w$ $\displaystyle = - \frac{1}{a\,\sin\theta}\,\frac{\partial}{\partial\phi}\left(\frac{p}{\skew{3}\bar\rho}+ {\mit\Phi} +\chi\right),$ (12.104)
$\displaystyle \frac{\partial w}{\partial t} - 2\,{\mit\Omega}\,\sin\theta\,v$ $\displaystyle =-\frac{\partial}{\partial z}\left(\frac{p}{\skew{3}\bar{\rho}}+{\mit\Phi}+\chi\right).$ (12.105)

In accordance with Equations (12.45), (12.47), and (12.53), the lower and upper surfaces of the ocean lie at

$\displaystyle z = \zeta_b(\theta,\phi,t),$ (12.106)

and

$\displaystyle z = d + \zeta_b(\theta,\phi,t)+\zeta(\theta,\phi,t),$ (12.107)

respectively, where

$\displaystyle \vert\zeta_b\vert,\, \vert\zeta\vert\ll d\ll a.$ (12.108)

It follows that

$\displaystyle \left.w\right\vert _{z=0}$ $\displaystyle =\frac{\partial\zeta_b}{\partial t},$ (12.109)
$\displaystyle \left. w\right\vert _{z=d}$ $\displaystyle = \frac{\partial(\zeta_b+\zeta)}{\partial t}.$ (12.110)

Let us assume that $ u=u(\theta,\phi,t)$ and $ v=v(\theta,\phi,t)$ : that is, the horizontal components of the fluid velocity are independent of $ z$ . Integration of Equation (12.105) from $ z=0$ to $ z=d$ , making use of the previous two boundary conditions, yields

$\displaystyle \frac{\partial \zeta}{\partial t}= - \frac{d}{a\,\sin\theta}\left...
...rtial}{\partial\theta}\,(\sin\theta\,u)+\frac{\partial v}{\partial\phi}\right].$ (12.111)

Equations (12.112)-(12.114) imply that

$\displaystyle w \sim \left(\frac{d}{a}\right)u,\,\left(\frac{d}{a}\right)v \ll u, v.$ (12.112)

In accordance with the analysis of Section 12.8, we can write

$\displaystyle {\mit\Phi} + \chi = \left(g-\frac{2}{3}\,{\mit\Omega}^{\,2}\,a\right)(z-d) + \delta{\mit\Phi}(\theta,\phi,t),$ (12.113)

where $ \delta{\mit\Phi}$ is the total perturbing potential due to tidal, rotation, and ocean self-gravity, effects. We have neglected an unimportant constant term. We have also neglected the $ z$ -dependence of the perturbing potential, because the variation lengthscale of this potential is $ a$ , rather than $ d$ , so that

$\displaystyle {\mit\Delta}(\delta{\mit\Phi})\sim d\,\frac{\partial \delta{\mit\Phi}}{\partial z} \sim \frac{d}{a}\,\delta{\mit\Phi}\ll \delta{\mit\Phi},$ (12.114)

where $ {\mit\Delta} (\delta{\mit\Phi})= \left.\delta{\mit\Phi}\right\vert _{z=d}-\left.\delta{\mit\Phi}\right\vert _{z=0}$ . Let us assume that the ocean is in approximate vertical force balance: that is,

$\displaystyle \frac{\partial}{\partial z}\left(\frac{p}{\skew{3}\bar{\rho}}+{\mit\Phi}+\chi\right)\simeq 0.$ (12.115)

It follows that

$\displaystyle \frac{p-p_{\rm atm}}{\skew{3}\bar{\rho}}+{\mit\Phi}+\chi = P(\theta,\phi,t),$ (12.116)

where $ p_{\rm atm}$ is the (uniform and constant) pressure of the atmosphere. However, pressure balance at the ocean's upper surface requires that

$\displaystyle \left.p\right\vert _{z=d+\zeta_b+\zeta} = p_{\rm atm}.$ (12.117)

Hence, we deduce that

$\displaystyle P(\theta,\phi,t) = ({\mit\Phi}+\chi)_{z=d+\zeta_b+\zeta} \simeq g\,(\zeta_b+\zeta) + \delta{\mit\Phi},$ (12.118)

to first order in small quantities [i.e., $ \zeta_b/a$ , $ \zeta/a$ , $ \delta{\mit\Phi}/(g\,a)$ , and $ {\mit\Omega}^{\,2}\,a/g$ ]. Thus, Equations (12.106), (12.107), and (12.115) yield

$\displaystyle \frac{\partial u}{\partial t} - 2\,{\mit\Omega}\,\cos\theta\,v$ $\displaystyle = - \frac{1}{a}\,\frac{\partial}{\partial\theta}\left[g\,(\zeta_b+\zeta)+ \delta{\mit\Phi}\right],$ (12.119)
$\displaystyle \frac{\partial v}{\partial t} + 2\,{\mit\Omega}\,\cos\theta\,u$ $\displaystyle = - \frac{1}{a\,\sin\theta}\,\frac{\partial}{\partial\phi}\left[g\,(\zeta_b+\zeta)+ \delta{\mit\Phi}\right].$ (12.120)

Let us justify our previous assumption that the ocean is in approximate vertical force balance. Equations (12.108), (12.115), and (12.119) imply that

$\displaystyle \frac{p-p_{\rm atm}}{\skew{3}\bar{\rho}}+{\mit\Phi}+\chi = P(\theta,\phi,t) + {\cal O}(d\,{\mit\Omega}\,v),$ (12.121)

where we have assumed that $ \partial/\partial t\sim{\cal O}({\mit\Omega})$ and $ \partial/\partial z\sim d^{\,-1}$ . However, Equations (12.121)-(12.123) yield

$\displaystyle P\sim g\,\zeta\sim a\,{\mit\Omega}\,v\gg d\,{\mit\Omega}\,v.$ (12.122)

Hence, Equation (12.124) becomes

$\displaystyle \frac{p-p_{\rm atm}}{\skew{3}\bar{\rho}}+{\mit\Phi}+\chi = P(\theta,\phi,t),$ (12.123)

which is equivalent to Equation (12.119).

Let us justify our previous assumption that $ u$ only depends weakly on $ z$ . It follows from Equations (12.117) and (12.122) that

$\displaystyle u$ $\displaystyle \sim \frac{\delta{\mit\Phi}}{a\,{\mit\Omega}},$ (12.124)
$\displaystyle {\mit\Delta} u$ $\displaystyle \sim \frac{{\mit\Delta} (\delta{\mit\Phi})}{a\,{\mit\Omega}}\sim \frac{d}{a}\,\frac{\delta {\mit\Phi}}{a\,{\mit\Omega}}\sim \frac{d}{a}\,u\ll u,$ (12.125)

where $ {\mit\Delta} u = \left.u\right\vert _{z=d}-\left.u\right\vert _{z=0}$ . In a similar manner, it can be shown that $ {\mit\Delta} v\ll v$ .

According to Equations (12.95) and (12.100),

$\displaystyle \zeta_b$ $\displaystyle = \sum_{n=1,\infty}\left[h_2\,\skew{5}\bar{\zeta}_2\,\delta_{n\,2} + h_n'\,\alpha_n\,\zeta_n\right],$ (12.126)
$\displaystyle - g^{\,-1}\,\delta{\mit\Phi}$ $\displaystyle = \sum_{n=1,\infty}\left[(1+k_2)\,\skew{5}\bar{\zeta}_2\,\delta_{n\,2} + (1+k_n')\,\alpha_n\,\zeta_n\right].$ (12.127)

Hence, Equations (12.114), (12.122), and (12.123) give

$\displaystyle \frac{\partial \zeta}{\partial t}$ $\displaystyle = - \frac{d}{a\,\sin\theta}\left[\frac{\partial}{\partial\theta}\,(\sin\theta\,u)+\frac{\partial v}{\partial\phi}\right],$ (12.128)
$\displaystyle \frac{\partial u}{\partial t} - 2\,{\mit\Omega}\,\cos\theta\,v$ $\displaystyle = - \frac{g}{a}\,\frac{\partial}{\partial\theta}\left[\zeta-(1+k_...
...skew{5}\bar{\zeta}_2 -\sum_{n=1,\infty}(1+k_n'-h_n')\,\alpha_n\,\zeta_n\right],$ (12.129)
$\displaystyle \frac{\partial v}{\partial t} + 2\,{\mit\Omega}\,\cos\theta\,u$ $\displaystyle = - \frac{g}{a\,\sin\theta}\,\frac{\partial}{\partial\phi}\left[\...
...skew{5}\bar{\zeta}_2 -\sum_{n=1,\infty}(1+k_n'-h_n')\,\alpha_n\,\zeta_n\right].$ (12.130)

We can write

$\displaystyle \zeta(\theta,\phi)=\sum_{n=1,\infty} \zeta_n(\theta,\phi),$ (12.131)

where [cf., Equation (12.37)]

$\displaystyle \zeta_n(\theta,\phi) = \sum_{m=0,n}c_{n}^{\,m}\,P_n^{\,m}(\cos\theta)\,{\rm e}^{\,{\rm i}\,m\,\phi},$ (12.132)

and the $ c_n^{\,m}$ are arbitrary complex constants. Now, (Abramowitz and Stegun 1965)

$\displaystyle \int_0^\pi P_n^{\,m}(\cos\theta)\,P_{n'}^{\,m}(\cos\theta)\,\sin\...
...\,d\theta = \left(\frac{2}{2\,n+1}\right)\frac{(n+m)!}{(n-m)!}\,\delta_{n\,n'}.$ (12.133)

Hence,

$\displaystyle \sum_{n=1,\infty} (1+k_n'-h_n')\,\alpha_n\,\zeta_n = \int_{-\pi}^...
...eta,\phi;\theta',\phi')\,\zeta(\theta',\phi')\,\,\sin\theta'\,d\theta'\,d\phi',$ (12.134)

where

\begin{multline}
G_F(\theta,\phi;\theta',\phi')=\sum_{n=1,\infty}\sum_{m=0,n}\fr...
...
)\,P_n^{\,m}(\cos\theta')\,{\rm e}^{\,{\rm i}\,m\,(\phi-\phi')}.
\end{multline}

is termed the Farrell Green's function (Farrell 1972). It follows that

$\displaystyle \frac{\partial \zeta}{\partial t}$ $\displaystyle = - \frac{d}{a\,\sin\theta}\left[\frac{\partial}{\partial\theta}\,(\sin\theta\,u)+\frac{\partial v}{\partial\phi}\right],$ (12.135)
$\displaystyle \frac{\partial u}{\partial t} - 2\,{\mit\Omega}\,\cos\theta\,v$ $\displaystyle =- \frac{g}{a}\,\frac{\partial}{\partial\theta}\left[\zeta-(1+k_2-h_2)\,\skew{5}\bar{\zeta}_2 -\oint G_F\,\zeta\,d{\mit\Omega}'\right],$ (12.136)
$\displaystyle \frac{\partial v}{\partial t} + 2\,{\mit\Omega}\,\cos\theta\,u$ $\displaystyle = - \frac{g}{a\,\sin\theta}\,\frac{\partial}{\partial\phi}\left[\zeta-(1+k_2-h_2)\,\skew{5}\bar{\zeta}_2 -\oint G_F\,\zeta\,d{\mit\Omega}'\right],$ (12.137)

where $ d{\mit\Omega}'\equiv \sin\theta'\,d\theta'\,d\phi'$ . Equations (12.139)-(12.141) are known collectively as the Laplace tidal equations, because they were first derived (in simplified form) by Laplace (Lamb 1993).

The Laplace tidal equations are a closed set of equations for the perturbed ocean depth, $ \zeta(\theta,\phi,t)$ , and the polar and azimuthal components of the horizontal ocean velocity, $ u(\theta,\phi,t)$ , and $ v(\theta,\phi,t)$ , respectively. Here, $ a$ is the planetary radius, $ g$ the mean gravitational acceleration at the planetary surface, $ d$ the mean ocean depth, $ {\mit\Omega}$ the planetary angular rotation velocity, and $ h_n$ , $ h_n'$ , $ k_n$ , $ k_n'$ are Love numbers defined in Equations (12.97), (12.98), (12.101), and (12.102), respectively. Amongst other things, the Love numbers determine the elastic response of the planet to the forcing term, $ \skew{5}\bar{\zeta}_2$ . Finally, the Farrell Green's function parameterizes the self-gravity of the ocean.


Table 12.1: The principal harmonics of the forcing term in the Laplace tidal equations.
$ j$ $ \skew{5}\tilde{\zeta}_j$ $ m_j$ $ \sigma_j$ Classification
         
         
0 $ -(\zeta_1 + \zeta_2)$ 0 0 Equilibrium
$ 1$ $ -3\,e\,\zeta_2$ 0 $ \omega_1-\omega_2$ Long Period
$ 2$ $ +(3/2)\,\sin^2\delta\,\zeta_2$ 0 $ 2\,\omega_1$ Long Period
$ 3$ $ -\sin\delta\,\zeta_2$ $ 1$ $ {\mit\Omega}-2\,\omega_1$ Diurnal
$ 4$ $ -\sin\delta\,\zeta_2$ $ 1$ $ {\mit\Omega}$ Diurnal
$ 5$ $ +(7\,e/4)\,\zeta_2$ $ 2$ $ 2\,{\mit\Omega}-3\,\omega_1+\omega_2$ Semi-diurnal
$ 6$ $ +(1/2)\,\zeta_2$ $ 2$ $ 2\,{\mit\Omega}-2\,\omega_1$ Semi-diurnal



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Next: Harmonics of Forcing Term Up: Terrestrial Ocean Tides Previous: Planetary Response
Richard Fitzpatrick 2016-03-31