2. Theoretical Preliminaries


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2. Theoretical Preliminaries

The key to the understanding of postglacial sea level and geoidal perturbations is embodied in what has come to be referred to as the sea level equation (SLE). This construct of first order perturbation theory is an integral equation of Fredholm type, a primary input to which is the glaciation history of the continents represented by the ice thickness which is a component of the complete history of surface mass loading that has the composite form:

, (1)

in which and are the densities of ice and water respectively and in which “S” is the space-time variation of relative sea level, that is sea level measured relative to the evolving local radius of the solid Earth. The independent variables and t in (1) represent co-latitude, longitude and time respectively. In terms of these fields, the SLE may be written as:

(2)

In equation (2) the function “C” is the so-called “ocean function” of Munk and MacDonald (1960) which is unity over the oceans and zero over the continents. The Green functions and are those which, when convolved with surface mass loading and tidal potential loading (but see below for a necessary qualification concerning the second of the terms in the integrand), respectively, translate these loads into a impact upon the potential field measured with respect to the deforming surface of the solid Earth, an equi-potential surface of which defines the surface of the equilibrium ocean, the geoid of classical geodesy. The angle in the surface mass load Green function is the angular separation between the source point and the field point. This simple dependence is a consequence of the assumption that the Earth model is spherically symmetric in its viscoelastic properties. The field .is the space and time dependent variation in the centrifugal potential associated with the GIA process itself, whereas is a (large, see Section 6 in Text S5) time dependent but space independent correction that must be included in (2) in order to ensure that the dynamical system conserves mass in the sense that the mass lost by land ice melting translates precisely into a corresponding and equal mass gain by the oceans and vice-versa. The solution of equation (2) answers the question as to where in the ocean basins the water that is produced by land ice-melting must be distributed, in order that the surface of the equilibrium ocean (sea level) remains an equipotential surface. Following Dahlen (1976), the space and time dependent expression for the GIA driven variation in the centrifugal potential may be written, accurate to first order in perturbation theory, as:

, (3)

with

(4)

In equations (4), theare the variations in the Cartesian components of Earth’s angular velocity vector induced by the GIA process, variations which are computed using the theory presented in Peltier (1982) and Wu and Peltier (1984). In order to solve (2) in such a way as to include the full influence of rotational feedback onto sea level that is captured by the second of the convolution integrals in (2), we must proceed iteratively as discussed in Peltier (1998). We first neglect this impact of rotational feedback and solve (2), using the complete ice and ocean loading history that this solution delivers, to predict the using the Laplace transform domain methodology discussed in Peltier (1982) and Wu and Peltier (1984). We then compute the time variation in the centrifugal potential and use it to construct the second term in the integrand of equation (2). This enables us to include the influence of rotational feedback in a further solution to the integral equation. This iterative process is continued until convergence is achieved. The method that is employed to solve equation (2) is a semi-spectral method. An initial version of this method that did not include the rotational feedback effect was presented in Mitrovica and Peltier (1991). In this method the fields are expanded in complex spherical harmonics that are not fully normalized and which satisfy the orthogonality condition: . (5)

We take the individual spherical harmonics to be defined as:

for , and . (6)

Theare the usual Associated Legendre Polynomials and “*” here denotes complex conjugation. Careful comparison will show that the expressions for the individual spherical harmonic coefficients in the expansion (4) for the centrifugal potential that we employ differ slightly from those in equations (116) and (117) of Dahlen (1976). This is solely a consequence of the fact that Dahlen’s analyses were based upon the use of fully normalized spherical harmonics which differ from the un-normalized forms by a factor of.

It will be important for the appreciation of what is to follow to recognize that the previously published analyses of the SLE that include the rotational feedback term involving the changing centrifugal potential defined in equation (3) have suggested this to be important to the explanation of Holocene relative sea level histories in the near field of each of the 4 extrema of the degree 2 and order 1 pattern that characterizes the influence of rotational feedback. The first unequivocal suggestion of the importance of this effect was provided in Peltier (2002) based upon the large data base of relative sea level histories compiled in Rostami et al (2000) for the east coast passive continental margin of the South American continent. Much more modest effects had previously been suggested to be possible in the papers of Milne and Mitrovica (1996) and Peltier (1998). The data presented in Rostami et al (2000) demonstrated that there existed along this coast a “mid-Holocene high-stand” of the sea that increased significantly in its height above present sea level as a function of increasing southern latitude along the coast. The analysis published in Peltier (2002) showed that this phenomenon was apparently explicable as a consequence of rotational feedback. This was demonstrated to be a consequence of the fact that one of the four extrema of the degree two and order one pattern is located on the southern tip of the continent. Later analysis in Peltier (2007) and PL09 extended this analysis to include sea level histories from the other three centres of action. Although the present analyses will lead us to somewhat reduce the values of the degree two and order one Stokes coefficients predicted by the global model of the GIA process, this adjustment will not eliminate the importance of rotational feedback insofar as the understanding of relative sea level histories in these critical regions is concerned. Our interest here, however, is in geoid height time dependence. In this analysis we will be making use of the expression for the time dependence of geoid height that follows from the Sea Level equation (2), namely:



(7)

In this equation the geoid Green function for the surface load is:

(8)

in which “a” is Earth’s radius and Me is Earth’s mass. The corresponding Green function for the impact upon the geoid associated with the changing rotation is, correspondingly:

(9)

in which “g” is the acceleration of gravity at Earth’s surface. The parameters and in equations (8) and (9) are the viscoelastic surface mass load and tidal potential load Love numbers which enable the above Green functions, when convolved with the surface mass load and tidal potential load respectively, to translate these applied “forcings” into the impacts upon the surface potential field (the geoid) due to both the instantaneous effect of the forcing and the induced internal redistributions of mass that it causes (see below, however, for discussion of a problem that arises due to the incorporation of explicit space dependence in the tidal Green function (9)). Insofar as the geoid is concerned, however, these variations of potential are now measured with respect to the center of mass of the planet rather than with respect to the surface of the solid Earth. The only difference between a solution for relative sea level history from equation (2) and a solution for geoidal history from (7) is therefore a shift in the local datum from the surface of the solid Earth in the former case to the center of mass in the latter case. The Green functions that appear in (2) differ from those of (8) and (9) only in that their expansions include additional contributions to the degree dependent amplitudes that involve the negative of the appropriate Love number for radial displacement (see below). The theory required to construct the time dependent Love numbers was first articulated in Peltier (1974) and Pettier and Andrews (1976) In the Green function expression for the surface load, the addition theorem for spherical harmonics (eg. Mathews and Walker, 1987) may be invoked to write

(10)

The issue as to whether the space independent but time-dependent conservation of mass factor , which is clearly required in the SLE, should also appear in the geoid equation (7) is an issue that has been raised in C10 and so this term has been placed in brackets in equation (7). This point will be further discussed in the penultimate section of the paper.

For the purpose of understanding the arguments to be presented in the following sections it will be important for the reader to appreciate that the methodology that has been developed to compute the time series of the Cartesian components of the angular velocity vector that appear in equations (4) above are highly accurate. These elements of the solution are determined on the basis of a solution of the Liouville equation that relates them to the so-called products of inertia. Because this demonstration of accuracy is laborious mathematically, the details have been provided in the auxiliary material as Text S6. In what follows we will refer to the equations in Text S6 as equations (S1), (S2), etc., and the figures as Figures S3–S5.
References

Mathews, J. and R. L. Walker (1987) Mathematical Methods of Physics, 2nd edn., Benjamin, New York.

Milne, G. A. and J. X. Mitrovica (1996) Post-glacial sea level change on a rotating Earth: first results from a gravitationally self-consistent sea-level equation. Geophys. J. Int. 126, F13-F20.

Mitrovica, J. X. and W.R. Peltier (1991) On postglacial sea level over the equatorial ocean. J. geophys. Res. 96, 20053-20071.

Munk, W.H. and G.J.F. Macdonald (1960) The Rotation of the Earth: A geophysical discussion. Cambridge University Press, Cambridge, UK.

Peltier, W.R. (1974) The impulse response of a Maxwell Earth. Rev. Geophys. Space Physics. 12, 649-669.

Peltier, W. R. (1994) Ice-age paleotopography. Science 265, 195-201.

Peltier, W. R. (1996) Mantle viscosity and ice-age ice-sheet topography. Science 273, 1359-1364.

Peltier, W.R. (1998) Postglacial variations in the level of the sea: implications for climate dynamics and solid-Earth geophysics. Rev. Geophys. 36, 603-689.

Peltier, W.R. (2002) Global glacial isostatic adjustment: paleo-geodetic and space-geodetic tests of the ICE-4G (VM2) model. J. Quat. Sci. 17, 491-510.

Peltier, W.R. (2004) Global glacial isostasy and the surface of the ice-age Earth: the ICE-5G (VM2) model and GRACE. Ann. Rev. Earth and Planet. Sci. 32, 111-149.

Peltier, W.R. (2007) History of Earth rotation. In Volume 9 of “The Treatise on Geophysics”, G. Schubert Ed., pp 243-293, Elsevier Press, Oxford, UK.

Peltier, W. R. and J. T. Andrews (1976) Glacial isostatic adjustment I: the forward problem. Geophys. J. R. Astronom. Soc. 46, 605-646.

Rostami, K., W.R. Peltier and M. Mangini (2000), Quaternary marine terraces, sea level changes and uplift history of Patagonia, Argentina: Comparisons with predictions of the ICE-4G (VM2) model of the global process of glacial isostatic adjustment. Quat. Sci. Rev. 19, 1495-1525.

Wu, P. and W.R. Peltier (1984), Pleistocene deglaciation and the Earth’s rotation: A new analysis. Geophys. J.R. astr. Soc. 76, 202-242.

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