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

# Laplace Tidal Equations

In the co-rotating frame, the linearized (in ) equations of motion of the ocean that covers the surface of the planet are (Fitzpatrick 2012)

 (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 the Coriolis acceleration (likewise, due to planetary rotation). Furthermore, we have neglected the finite compressibility of the fluid that makes up the ocean. Let , , be spherical coordinates in the co-rotating frame, and let . It is helpful to define , , and . Assuming that , the previous equations of motion reduce to

 (12.102)

and

 (12.103) (12.104) (12.105)

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

 (12.111)

Equations (12.112)-(12.114) imply that

 (12.112)

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

 (12.114)

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

 (12.115)

It follows that

 (12.116)

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

 (12.118)

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

 (12.119) (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

 (12.121)

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

 (12.128) (12.129) (12.130)

We can write

 (12.131)

where [cf., Equation (12.37)]

 (12.132)

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

 (12.133)

Hence,

 (12.134)

where

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

 (12.135) (12.136) (12.137)

where . 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, , 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.

Table 12.1: The principal harmonics of the forcing term in the Laplace tidal equations.
 Classification 0 0 0 Equilibrium 0 Long Period 0 Long Period Diurnal Diurnal Semi-diurnal Semi-diurnal

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