(12.307) |

Hence, it follows that

(12.308) |

and

(12.309) |

where

(12.310) |

Likewise, the solutions to the eigenvalue problem (12.248) and (12.249), subject to the orthonormality constraint (12.250), is such that and , where

(12.311) |

and

(12.312) |

We also have (see Section 12.19)

(12.313) | ||

(12.314) | ||

(12.315) | ||

(12.316) |

which yields

(12.317) | ||

(12.318) | ||

(12.319) | ||

(12.320) |

where . However,

(12.321) |

Hence, if is even then the are zero. Otherwise, we obtain

(12.322) | ||

(12.323) | ||

(12.324) |

as well as

(12.325) |

where

(12.326) |

Let

(12.327) |

It follows, from symmetry, that when is odd. When is even (Wong 1998),

where denotes a

(12.328) |

and

(12.329) |

with an analogous definition for .

It can be shown that (Longuet-Higgins and Pond 1970)

(12.330) | ||

(12.331) | ||

(12.332) |

According to Equation (12.306), we can also write , where

(12.333) |

and

(12.334) |

Here, if is even, and

(12.335) |

otherwise. It follows from Equation (12.305) that , where

(12.336) |

Here, if is even, and

(12.337) |

otherwise. Now, if and are both even then

(12.338) |

if and are both odd then

(12.339) |

and if is odd then .

If we let and then Equations (12.311) and (12.312) become

and

respectively. Here,

(12.341) | ||

(12.342) | ||

(12.343) | ||

(12.344) | ||

(12.345) | ||

(12.346) | ||

(12.347) | ||

(12.348) |

By symmetry, and are only non-zero when is even, and is even; , , , and are only non-zero when is odd, and is odd; and and are only non-zero when is odd, and is even. It follows that all quantities appearing in Equations (12.347) and (12.348) are real. Once we have solved these equations to obtain the (which is a relatively straightforward numerical task), we can reconstruct the tidal elevation as follows:

(12.349) |

where

(12.350) | ||

(12.351) |

and

(12.352) |

Thus, the tidal amplitude at a given point on the ocean is

(12.353) |

It is easily demonstrated that

(12.354) |

where

(12.355) |

Here, is the oscillation period of the harmonic of the tide generating potential under consideration, is the time-lag between the peak tide at a given point on the ocean and the maximal tide generating potential at , and is the corresponding phase-lag.

Figures 12.4 and 12.5 show the amplitude and phase-lag of the ( ) long-period tide in a hemispherical ocean of mean depth (which corresponds to ), calculated assuming that and . The calculation includes all azimuthal harmonics up to . Note that only a quarter of the ocean is shown, because the amplitude is symmetric about the meridians and , whereas the phase-lag is symmetric about the meridian , and antisymmetric about the meridian . Here, . Given that (see Table 12.3), the maximum amplitude of the tide is about , and occurs at the poles. Moreover, it is clear from a comparison with Figure 12.1 that the tide is direct (i.e., it is in phase with the equilibrium tide). In fact, the tide in a hemispherical ocean is about four times larger in amplitude than that in a global ocean (i.e., an ocean that covers the whole surface of the Earth) of the same depth. Otherwise, the two tides have fairly similar properties.

Figures 12.6 and 12.7 show the amplitude and phase-lag of the ( ) diurnal tide in a hemispherical ocean of mean depth (which corresponds to ), calculated assuming that and . The calculation includes all azimuthal harmonics up to . Given that (see Table 12.3), the maximum amplitude of the tide is about , and occurs at mid-latitudes. This is very different to the case of a global ocean, where the amplitude of the tide is identically zero everywhere. (See Figure 12.2.) Note that the tidal phase-lag only exhibits a fairly weak dependence on the azimuthal angle, . In fact, the tide in a hemispherical ocean essentially oscillates in anti-phase with the equilibrium tide at the ocean's central longitudinal meridian ( ), except close to the poles, where it oscillates in phase. Again, this is very different to the case of a global ocean, where the tidal maximum lies on a meridian of longitude, and rotates steadily around the Earth from east to west.

Figures 12.8 and 12.9 show the amplitude and phase-lag of the (
)
semi-diurnal tide in a hemispherical ocean of
mean depth
(which corresponds to
), calculated assuming that
and
.
The calculation includes all azimuthal harmonics up to
.
Given that
(see Table 12.3), the maximum amplitude of the
tide is about
, and occurs at the poles. This is very different to the case of a global ocean, where the amplitude of the
tide is zero at the poles. (See Figure 12.3.)
Another major difference is that, in a hemispherical ocean, the tidal maxima circulate around points of zero tidal amplitude--such
points are known as *amphidromic points*.
Of course, in the case of a global ocean, the tidal maxima lie on opposite meridians of longitude, and rotate steadily around the
Earth from east to west. The systems of tidal waves, circulating around amphidromic points, that is evident in Figure 12.9,
are known as *amphidromic systems*, and are one of the the major features of tides in real oceans (Cartwright 1999).
Generally speaking, the sense of circulation of amphidromic systems in real oceans is counter-clockwise (seen from above) in the Earth's northern hemisphere, and clockwise
in the southern hemisphere.

In conclusion, our investigation of tides in a hemispherical ocean suggests that the impedance of the flow of tidal waves around the Earth, due to the presence of the continents, is likely to have a comparatively little effect on long-period tides, but a very significant effect on diurnal and semi-diurnal tides. In particular, for semi-diurnal tides, the impedance gives rise to the formation of amphidromic systems.