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)

$$\nabla\cdot{\bf v} = 0,$$ (12.100)

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

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

and

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

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

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

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

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

and

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

respectively, where

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

It follows that

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

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

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

Equations (12.112)-(12.114) imply that

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

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

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

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

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

It follows that

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

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

Hence, we deduce that

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

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

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

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

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

Hence, Equation (12.124) becomes

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

$$u \sim \frac{\delta{\mit\Phi}}{a\,{\mit\Omega}},$$ (12.124)

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

$$\zeta_b = \sum_{n=1,\infty}\left[h_2\,\skew{5}\bar{\zeta}_2\,\delta_{n\,2} + h_n'\,\alpha_n\,\zeta_n\right],$$ (12.126)

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

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

$$\frac{\partial \zeta}{\partial t} = - \frac{d}{a\,\sin\theta}\left[\frac{\partial}{\partial\theta}\,(\sin\theta\,u)+\frac{\partial v}{\partial\phi}\right],$$ (12.128)

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

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

We can write

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

where [cf., Equation (12.37)]

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

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

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

Hence,

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

$$\frac{\partial \zeta}{\partial t} = - \frac{d}{a\,\sin\theta}\left[\frac{\partial}{\partial\theta}\,(\sin\theta\,u)+\frac{\partial v}{\partial\phi}\right],$$ (12.135)

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

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