3. Contributions of dissipative effects
Now we give estimations for the averaged contributions of the dissipation to and due to tides and core-mantle interaction. For a review of the major features concerning the evolution of the Earth-Moon system, one can refer to the book edited by Marsden and Cameron (1966). It helped us to delimit what was important for this study.
3.1. The body tides
One specific problem of modeling the dissipative tides on Earth is that they have two different origins: friction in the mantle and friction within shallow seas. The simplest way to overcome difficulties and large formulae is to link the global dissipation to one physical quantity. In particular, this leads to consider here the Earth, once again, as homogeneous.
The specific dissipation function Q (Munk and MacDonald, 1960) is often used. It is defined as the inverse of the ratio where is the energy dissipated during one period of tidal stress and the maximum of energy stored during the same period. MacDonald (1964) showed that
where is the phase lag of the deformation due to the stress.
Q is rather considered as a constant (see for example Kaula, 1964, Goldreich and Peale, 1967, Goldreich and Soter, 1969, Gold and Soter, 1969), what implies in particular that is independent of the speed of rotation. This point remains subject to controversy, especially for long time scales. Some others studies have also considered dependent of the tidal frequency: (Goldreich and Peale, 1966, 1970), (Lambeck, 1979), (Lago and Cazenave, 1979), (Dobrovolskis, 1980). Most of them use Fourier expansions of the tidal potential (Kaula, 1964) in which an arbitrary tidal phase lag has to be defined for each argument, and the way these phase lags are related to the frequency is not always clear. Moreover, relation (3) itself is subject to uncertainty as was pointed out by Zschau (1978).
As Touma and Wisdom (1994), we prefer here the simpler and more intuitive approach of Mignard (1979) where the torque resulting of tidal friction is proportional to the time lag that the deformation takes to reach the equilibrium. This time lag is supposed to be constant, and the angle between the direction of the tide-raising body and the direction of the axis of minimal inertia (i.e. the direction of the high tide), which is carried out of the former by the rotation of the Earth, is proportional to the speed of rotation. Such a model is called "viscous", and corresponds to the case for which is proportional to the tidal frequency.
Theories on tidal effects are generally based on the following assertion mainly due to Love at the beginning of the century (see Lambeck, 1988): the tidal potential due to the deformation induced by the differential gravitational attraction of a perturbing body (the Sun or the Moon) at from the Earth's center O holds, at any point P on its surface:
where of which the modulus is the planetary radius R, where the Love number and the spherical harmonic. As was done for the computation of the potential of precession, we restrict ourselves to the first term of the expansion, what seems to be sufficient for estimating secular variations. Hence
The potential V at any point outside the Earth of distance is the solution of the "Dirichlet's first boundary-value problem" (see Lambeck, 1988), so to say that it satisfies Laplace's equation
and its value on the boundary of the domain is known and given by Eq. (4). The unique solution to this problem is the function
Here stands for the perturbing body and for the interacting body at time t. would also be defined at time t if the Earth were perfectly elastic. But this is not the case since, due to internal friction, the deformation permanently takes a time to reach the equilibrium; the attribute referring to the perturbing body also means that the value is taken at time .
Assuming small compared to the diurnal period, Mignard gives the following approximation for the perturbing body:
where is the orbital velocity, the rotational velocity, and where all vectors in the right member are defined at time t. Then Mignard expands V at first order in and derives the force and torque undergone by the interacting body:
The determination of gives the contribution of the body tides to the variation of the spin:
where is the normal to the ecliptic.
We have computed with the help of the algebraic manipulator TRIP (Laskar, 1994b), writing all vectors in ecliptic coordinates and averaging both formula over the periods of mean anomaly, longitude of node and perigee of the perturbing body (and of the interacting body if it is not the same one). Taking second order truncations in eccentricities, we obtain:
the contributions of the solar tides :
where is the mean motion of the Sun around the Earth.
the contributions of the lunar tides :
the contributions of the "cross tides" and and , the cases where the Moon and the Sun are respectively the perturbing bodies accounting for the same quantity:
Let us have a look on the consequences of all these contributions. Neglecting the eccentricities, the variation of the obliquity due to the direct solar tides or the lunar ones with a low inclination of the Moon has the form:
where and k is a positive quantity. This implies that and are two instable positions of equilibrium and that is a stable position, but it is a relative stability because the braking of L makes it slowly moves down to . Furthermore, the obliquity increases when and decreases otherwise.
It is easy to see that the cross tides drive the equator towards the orbital plane. They are missing in Mignard's articles but Touma and Wisdom (1994) have pointed out their relative importance. Actually, the ratio of their magnitude with the one of the direct solar tides ( of the lunar one) is
As is proportional to , this contribution must be taken into account whenever the obliquity reaches high values.
We can derive now from the variations of the orbit of the interacting body induced by the tides. They can be obtained by determining the components and of in an osculating reference frame. These components write:
where is the so-called reduced mass of the system Earth-interacting body and the orbital angular momentum of the interacting body:
The orbital variations are given by the Lagrange equations (see Brouwer and Clemence, 1961):
where are respectively the true anomaly, the longitude of perigee and the inclination of the interacting body on the ecliptic . After expanding and averaging these equations and taking second order truncations in eccentricity, we find for the Moon:
and for the Sun:
but both last variations are negligible: about 3 meters per Myr for and per Myr for .
Two contributions are missing so far: the tides raised on the Moon
by the Earth and by the Sun. With the following assumptions:
the tide raised by the Moon on the Earth is obtained by exchanging Moon and Earth in Eqs. (5) with the above simplifications. We obtain the additional contributions:
If one takes suggested by the DE245 data, (Lambeck, 1980), and seconds, these contributions represent of the total and of the total for present conditions. As Mignard pointed it out, the terrestrial tides on the Moon have no substantial effect on the lunar orbit unless it is close to the Earth (at a few Earth's radii) and when the ratio is much greater than 1 (using DE245 data, we have ).
Finally, the tides raised on the Moon by the Sun can also be neglected because the ratio of the magnitudes of solar and terrestrial tides on the Moon is
3.2. The core-mantle friction
Here we basically rely on Rochester's model (1976).
The inner Earth is composed of a mantle and a core separated into a central rigid part and a fluid one. We neglect interactions between both last parts because they are supposed to be strongly coupled by pressure forces (Hinderer, 1987). The core and the mantle have different dynamical ellipticities, so they tend to have different precession rates. This trend produces a viscous friction at the core-mantle boundary (CMB). Thus, there are motions in the outer liquid core inducing electric currents which generate a torque of electromagnetic friction because of magnetization of the deepest layer of the mantle.
Rochester showed that the magnetic friction has only a faint effect on the Earth's long term rotational dynamics. If the coefficient of viscous friction due to the viscosity of the liquid metal of the outer core is thought weaker that the magnetic one, some turbulences and inhomogeneïties in the outer core could make the viscous friction far more efficient (Lumb and Aldridge, 1991), (Williams, 1993), so that we will suppose that the friction is solely viscous. We will consider here that an effective viscosity can account for a weak laminar friction, as well as a strong turbulent one which thickens the boundary layer.
Two additional torques account for the coupling: the inertial torque due to pressure forces at the CMB (which is not spherical due to the Earth's rotation) , and the topographic torque (Hide, 1969) due to likely "bumps" of this boundary. tends to attach strongly the core and the mantle but its effect is reduced if the boundary layer beneath the CMB is thick. Although irregularities of the CMB increase the surface of friction, this topographic torque (which is not taken into account in Rochester's model) would rather have a conservative effect, the bumps acting as notches; in this way it can be considered as the irregular part of . Besides, it is still too poorly known to estimate its long term contributions (Jault, private communication). We will also ignore it.
The angular momentum theorem applied to the core (subscript c) and to the mantle (subscript m) then gives:
where , are the precession torques, and the frictional torque. and denote the moment of inertia with . In first approximation, one can write:
where is the effective coefficient of friction. Rochester gives the following approximation to the solution of system :
where is the elasticity correcting factor of the mantle and the Earth's rate of precession. is the dynamical ellipticity of the core which, as the whole body one, is assumed to be proportional to the square of the speed of rotation. It should not differ from the value corresponding to the hydrostatic equilibrium by more than a few percents (Hinderer, 1987). Hence and . where the subscript in denotes the initial value. As pointed out by Yoder (1995), such approximations would not be valid any more for a slow rotation, for which the non-hydrostatic parts of the ellipticities can dominate.
This solution is valid only when the inertial torque is non-zero. This happens when the ellipticity of the CMB exceeds the ratio of the rotation period to the precession period (Poincaré, 1910, see also Peale, 1976). Assuming that this ellipticity is roughly in hydrostatic equilibrium, this condition is equivalent to:
where is the density of the outer core. The rate being proportional to (see Sect. 2), such a condition is satisfied since exceeds a given constant; this is the case for a fast-rotating planet like the Earth.
We determine as a function of the effective viscosity with the help of Goldreich and Peale (1967, 1970). First, we compute the "spin-up" time which corresponds to the time that the core needs to adjust its rotation to the one of the mantle in absence of any external force. This is the characteristic time of the following system:
The solution is
Thus, according to Greenspan and Howard (1963),
The contributions of the core-mantle friction to and are obtained thanks to the fact that is very close to because of the action of . Then the precession torques and belonging to the orbital plane of normal (this clearly appears when the torques are expressed in a vectorial form; see for example Goldreich 1966), the addition of both equations of and the scalar product by of the resulting equation leads to
This quantity being always negative, the core-mantle friction (CMF) tends to slow down the rotation and to bring the obliquity down to if and up to otherwise, what contrasts with the effect of the tides. Furthermore, one can see that, despite the strong coupling by pressure forces, there can be a substantial contribution to the variation of the spin for high viscosities and moderate speeds of rotation. It must be pointed out that this contribution has no sense for an arbitrary high viscosity (which would attach the core to the mantle) and for an arbitrary small core dynamical ellipticity (for which vanishes).
Finally, the equation has also the following remarkable consequence: since
This simple relation is independent of the variations of and strongly constraints the possible evolution of .
3.3. The atmospheric tides
The Earth's atmosphere also undergoes some torques which can be transmitted to the surface by friction: a torque caused by the gravitational tides raised by the Moon and the Sun, a magnetic one generated by interactions between the magnetosphere and the solar wind. Both effects are negligible; see respectively Chapman and Lindzen (1970) and Volland (1988).
Finally, a torque is produced by the daily solar heating which induces a redistribution of the air pressure, mainly driven by a semidiurnal wave, hence the so-called thermal atmospheric tides (Chapman and Lindzen, 1970). The axis of symmetry of the resulting bulge of mass is permanently shifted out of the direction of the Sun by the Earth's rotation. As for the body tides, this loss of symmetry is responsible for the torque which, at present conditions, tends to accelerate the spin.
Volland (1988) showed that this effect is not negligible, reducing the Earth's despinning by about 7.5 . But, although the estimate of their long term contributions surely deserves careful attention, we have not taken the atmospheric tides into account in the computations presented in the next sections, assuming that, as the uncertainty on some of the other factors is still important, the global results obtained here will not differ much when taking this additional effect into consideration.
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998