Planetary Response

and subject to the incompressibility constraint

Here, is the stress tensor, the identity tensor,

The planet's equation of elastic motion in the co-rotating frame is (Fitzpatrick 2012)

where is the totally antisymmetric tensor (Riley 1974), and (Fitzpatrick 2012)

Note that is a solid harmonic of degree . The second term on the left-hand side of Equation (12.62) is the Coriolis acceleration (due to planetary rotation), whereas the final term on the right-hand side is the centrifugal acceleration (likewise, due to planetary rotation). The first two terms on the right-hand side are the forces per unit mass due to internal stresses and gravity, respectively.

Equations (12.60), (12.61), and (12.62) can be combined to give

Let us assume that

(12.67) |

that is, the typical timescale on which the tide generating potential varies (in the co-rotating frame) is much longer than the transit time of an elastic shear wave through the interior of the planet. In this case, we can neglect the left-hand side of Equation (12.66), and write

which is equivalent to saying that the interior of the planet always remains in an equilibrium state.

Let

where

is the sum of the tide generating potential and the fictitious centrifugal potential due to planetary rotation. Note that is a solid harmonic of degree 2. It follows from Equations (12.50), (12.63), and (12.68) that

and

Taking the divergence of the previous equation, and making use of Equations (12.61), we find that , which implies that is a solid harmonic (of degree ).

Equation (12.71) can be integrated to give

where use has been made of Equations (12.55) and (12.64), as well as

Here, is the atmospheric pressure at sea level. The previous boundary condition ensures that the mean pressure at the surface of the planet is able to support the mean weight of the ocean, as well as the weight of the atmosphere. Incidentally, it is assumed that , and we have neglected terms of order with respect to terms of order .

The elastic displacement in the interior of the planet satisfies Equations (12.61) and (12.72). It is helpful to define the radial component of the displacement

(12.75) |

as well as the stress acting (outward) across a constant surface,

where use has been made of Equation (12.60). The radial displacement at is equivalent to the displacement of the planet's surface. Hence, according to Equation (12.45),

The stress at any point on the surface must be entirely radial (because a fluid ocean cannot withstand a tangential stress), and such as to balance the weight of the column of rock and ocean directly above the point in question. In other words,

It follows from Equation (12.57), (12.69), (12.70), (12.74), (12.76), and (12.78) that

where

is a surface harmonic of degree . Here, , which has the dimensions of length, parameterizes the perturbation due to tidal and rotational effects at the planet's surface. Moreover, is a Kronecker delta symbol (Riley 1974). Hence, we need to solve Equations (12.61) and (12.72), subject to the boundary conditions (12.77) and (12.79).

Let us try a solution to Equations (12.61) and (12.72) of the form

(12.80) |

where (Love 1927)

Here, and are spatial constants, and is an arbitrary solid harmonic of degree . It follows that

(12.82) |

where use has been made of Equation (12.40). Moreover,

(12.83) |

and

(12.84) |

Thus, the boundary conditions (12.77) and (12.79) become

and

respectively. Suppose that is such that

(12.86) |

In this case, the boundary conditions (12.86) and (12.87) reduce to

and

respectively.

The expression for given in Equation (12.82) satisfies Equations (12.61) and (12.72) provided that

(12.88) | ||

(12.89) |

respectively, where use has been made of Equations (12.40)-(12.42). It follows that

(12.90) | ||

(12.91) |

Hence, the boundary conditions (12.89) and (12.90) yield

where

Here, the parameterize the self-gravity of the ocean. Let

(12.96) |

This quantity can be recognized as the total perturbing potential at the planet's surface due to the combination of tidal, rotational, and ocean self-gravity, effects. It follows from Equation (12.57), (12.80), and (12.95) that

where

Here, the dimensionless quantities , , , and are known as