(12.100) | ||

(12.101) |

where is the fluid velocity, and the fluid pressure. Here, is the total gravitational potential, the centrifugal potential (due to planetary rotation), and

and

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

(12.106) |

and

(12.107) |

respectively, where

(12.108) |

It follows that

(12.109) | ||

(12.110) |

Let us assume that and : that is, the horizontal components of the fluid velocity are independent of . Integration of Equation (12.105) from to , making use of the previous two boundary conditions, yields

Equations (12.112)-(12.114) imply that

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

(12.113) |

where 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 -dependence of the perturbing potential, because the variation lengthscale of this potential is , rather than , so that

where . Let us assume that the ocean is in approximate vertical force balance: that is,

(12.115) |

It follows that

where is the (uniform and constant) pressure of the atmosphere. However, pressure balance at the ocean's upper surface requires that

(12.117) |

Hence, we deduce that

to first order in small quantities [i.e., , , , and ]. Thus, Equations (12.106), (12.107), and (12.115) yield

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

where we have assumed that and . However, Equations (12.121)-(12.123) yield

(12.122) |

Hence, Equation (12.124) becomes

(12.123) |

which is equivalent to Equation (12.119).

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

(12.124) | ||

(12.125) |

where . In a similar manner, it can be shown that .

According to Equations (12.95) and (12.100),

(12.126) | ||

(12.127) |

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

We can write

(12.131) |

where [cf., Equation (12.37)]

(12.132) |

and the are arbitrary complex constants. Now, (Abramowitz and Stegun 1965)

Hence,

(12.134) |

where

is termed the

where . Equations (12.139)-(12.141) are known collectively as the

The Laplace tidal equations are a closed set of equations for the perturbed ocean depth, , and the polar and azimuthal components of the horizontal ocean velocity, , and , respectively. Here, is the planetary radius, the mean gravitational acceleration at the planetary surface, the mean ocean depth, the planetary angular rotation velocity, and , , , 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, . Finally, the Farrell Green's function parameterizes the self-gravity of the ocean.