Since the solutions of the SLE that are of interest insofar as their predictions of present-day rates of sea level change are concerned are clearly those based upon models of the radial viscoelastic structure of the planet that best reconcile the available data base of geological timescale observations, it is clearly important to first establish that the theory embodied in the SLE does indeed satisfy these long timescale constraints when its physical properties are appropriately selected.

The primary data base used to tune the global model of the GIA process that is embodied in the SLE consists of radiocarbon-dated records of the variation of the level of the sea relative to the deforming surface of the solid earth. In tuning the properties of the global model to best fit these data, the elastic properties of the model of the planetary interior, namely, the density p and the two elastic Lamé parameters À and ¡x, are assumed to be fixed, insofar as their variation with radius is concerned, by the results of free oscillation and body wave propagation seismology. I continue to base my analyses on an elastic structure defined by the PREM model of Dziewonski and Anderson (1981). Since the complete rheology of the interior is assumed to be adequately represented by a linear Maxwell solid, it is described by the Laplace transform domain stress-strain relation (see Peltier (1974) for discussion)

in which s is the Laplace transform variable, rv- is the stress tensor, and etj is the strain tensor. The compliances A(s) and fi(s) of the linear Maxwell model are just

Since the bulk modulus k = A + 2/a/3 = A(s) + 2fi(s)J3 is independent of s for this rheology, the model clearly has no bulk dissipation. The quantity TM = vI/m is called the "Maxwell time" and this is the timescale that must be exceeded in order to effect the transition from Hookean elastic to Newtonian viscous behavior in response to an applied shear stress that is embodied in the Maxwell model. With (p, A, ¡x) fixed by seismology, the only free parameter in the model is therefore the molecular viscosity v. For the purpose of all of the analyses herein, the model of the interior of the earth is assumed to be spherically symmetric both in elasticity and in viscosity. It is this basic assumption that leads to the simple mathematical expressions (3a) and (3b) for the Green functions and Gj. Since p(r), fx(r), and A(r) of this spherically symmetric model are very well constrained seismologically (as defined by PREM), we are at liberty to vary only v(r) in order to tune the model to fit the geological timescale observations. Given that the effective creep resistance of a solid (its viscosity) is strongly temperature dependent because the creep resistance of a solid is thermally activated, and because the mantle of the earth is assumed to be undergoing a process of thermally forced convection in which lateral variations of temperature are expected to be intense, it is clear that mantle viscosity must be a function of all three space coordinates. By employing a spherically symmetric model of the GIA process, we will therefore be testing the hypothesis that the aspherical viscosity structure that must characterize the actual mantle of the earth has no unambiguously observable effect. This is expected to be the case only if the actual lateral variations occur on sufficiently small horizontal scale, say over distances only on the order

of the thickness of the lithosphere. As we will show immediately, spherically symmetric models of the internal structure can reconcile most of the available geological observations exceptionally well. There are, of course, exceptions to this general rule in the form of relative sea level observations from sites undergoing active tectonic uplift or subsidence, but these appear to constitute local exceptions to the general rule that the best fitting spherically symmetric model is able to reconcile the vast majority of the observations very well.

A sequence of three viscosity models for which I will explicitly discuss RSL predictions herein is illustrated in Fig. 4.2, where the models are labeled VM1, VM2, VM3 with the letters VM simply representing "viscosity model" and the number affixed following the letters being employed to distinguish the models from one another. Given a viscosity model and a model 1(6, A, t) of the process of glaciation and déglaciation, we have all that is required to fix the inputs to Eq. (1) and thus to make predictions of postglacial relative sea level history. For all of the analyses discussed in this chapter, I will assume that the ICE-4G model described in Peltier (1994,1996) adequately describes the last déglaciation even of the current ice age. Northern Hemisphere isopacks from this model for six different instants of time are shown in Fig. 4.3 (see color plate). It is important to understand that, although it is expected that this model is close in form to the actual déglaciation event that began approximately 21,000 calendar years ago, it cannot be exact and is still being refined in the course of work to develop an improved ICE-5G follow-on model.

Radius (km)

Figure 4.2 The VM1, VM2, and VM3 radial profiles of mantle viscosity discussed in the text. These represent the range of smooth viscosity profiles that have been employed for this physical property of the spherically symmetric viscoelastic models of the glacial isostatic adjustment process.

Radius (km)

Figure 4.2 The VM1, VM2, and VM3 radial profiles of mantle viscosity discussed in the text. These represent the range of smooth viscosity profiles that have been employed for this physical property of the spherically symmetric viscoelastic models of the glacial isostatic adjustment process.

Figures 4.4 and 4.5 (see color plates) display comparisons of the present-day predicted rate of relative sea level rise as a function of the viscosity model (Fig. 4.4 shows the predictions of VM1 and VM2 together with their difference) and as a function of whether the influence of rotational feedback is included in the analysis (Fig. 4.5 presents results for viscosity model VM2 both including and excluding the rotational feedback effect as well as the difference between these two predictions). Inspection of these two figures will reveal all of the salient geographical characteristics of the global GIA process. Figure 4.4, for example, demonstrates that the regions that were ice-covered at LGM (Last Glacial Maximum) are regions in which relative sea level is currently falling as a consequence of ongoing postglacial rebound of the crust of the solid earth. Likewise, in the regions peripheral to those that were glaciated, relative sea level is predicted to be rising at present due to the collapse of the "glacial forebulge." The latter feature is a pronounced characteristic of the glaciated state and is a region in which the local radius of the planet is increased because of the mass that has been extruded from under the ice-loaded region as the earth isostatically adjusts to the weight of the surface load. When the ice load is removed and the rebound of the crust in the loaded region commences, the peripheral bulge of the solid earth begins to collapse. Beyond the region of forebulge collapse, in what I have previously referred to as the "far field" of the ice sheets, the rate of RSL change signal is characterized by a slow fall of RSL in ocean basin interiors and a region of similarly weak sea level rise confined to a "halo" around the coastline of each of the far field continents. The nature of the pattern of RSL rise/fall in the far field of the ice sheets may be understood as arising from the fact that, as the earth's shape relaxes following déglaciation, water is continuously "siphoned" from the central ocean basins in order to fill the depressions that are being created by proglacial forebulge collapse and by the "hydroisostatic tilting" of far field continental coastlines due to the weight applied to the surface by the offshore water load. Comparing the predictions of the VM1 and VM2 viscosity models demonstrates that the region in which the difference between them is largest is along the U.S. east coast. This region is therefore the one in which one would first look to discriminate between them.

Before discussing explicit analyses of this kind, it will be instructive to consider the similar set of predictions shown in Fig. 4.5, which has been designed to illustrate the impact of rotational feedback on the GIA-induced sea level signal. Results are shown in this figure for the VM2 model with and without this feedback included. The difference between the two predictions in the lower part of this figure shows that the influence of rotational feedback is very modest indeed, insofar as the amplitude of this signal is concerned. Furthermore, the pattern of the difference is almost entirely that of a degree 2 and order 1 Legendre polynomial, indicating on the basis of Eq. (10) that it is the polar wander component of the rotational response to GIA that feeds back most strongly onto sea level. It is important to note that the extremely weak influence of rotational feedback is at extreme odds with the contrary suggestion by Bills and James (1996), whose incorrect analysis of this problem led them to the conclusion that the strength of the rotational feedback was 1-2 orders of magnitude larger.

By way of demonstrating the way in which 14C-dated RSL histories may be employed to discriminate between competing models of the radial viscosity structure, Fig. 4.6 shows an intercomparison of predictions of ICE-4G (VM1) and ICE-4G (VM2) for a sequence of 16 locations along the east coast of the continental United States at which 14C-dated RSL histories are available. Also shown in this figure are RSL predictions for this set of locations for a model of Mitrovica and Forte (1997) which differs most extremely from both VM1 and VM2 in that it has a transition zone between 400 and 660 km depth in which the viscosity is extremely low (a feature that is balanced by the existence of a high-viscosity region in the middle of the lower mantle; see Peltier (1998a) for a figure explicitly comparing this model with VM2). Inspection of the results in Fig. 4.6 shows that the Mitrovica and Forte model is entirely excluded by the data because its "channel flow" characteristics predict the existence of raised beaches to the immediate south of the ice margin that are not observed. Channel flow models of this kind, in which the viscous flow of mantle material is confined to a low-viscosity near-surface region, have been known to be excluded by U.S. East coast RSL observations after the analyses presented by Cathles (1975) and Peltier (1974). Comparing the predictions of ICE-4G (VM1) and ICE-4G (VM2) demonstrates that at all sites along the northernmost part of the U.S. east coast the VM1 -based model drastically overpredicts the present-day rate of RSL rise whereas the VM2 based model essentially eliminates these misfits.

As discussed most recently in Peltier (1998b), the viscosity model VM2 was in fact deduced on the basis of the application of the formal theory of Bayesian inference to a small subset of GIA data. In this procedure, model VM1 was employed as a starting model and the Frechet kernels for the observed data were used to refine this model to optimally reduce the misfits of the starting model to the observations. None of the U.S. east coast data were employed in this process; rather they were withheld to provide a definitive check on the improvement to the model delivered by the automated inversion procedure. The data used for inversion actually consisted of the set of wavenumber-dependent relaxation times inferred by McConnell (1968) to characterize the postglacial recovery of Fennoscandia (the validity of which was recently reconfirmed by Weiczerkowski etal., 1999), a set of approximately 23 relaxation times inferred on the basis of exponential fits to 14C-dated RSL curves from both Canada and Fennoscandia, and the observed rate of nontidal acceleration of planetary rotation. VM2 serves as a good zeroth-order model of the radial variation of viscosity based upon numerous posteriori tests of its predictions against the very large set of observations that were not employed in its construction (see Peltier 1998a). Given this zeroth-order acceptable

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Figure 4.6 Predicted histories of relative sea level change at 16 sites along the eastern coast of the North American continent based upon use of the ICE-4G model of the déglaciation history. Results are shown for three different models of the radial variation of viscosity, respectively VM1 (dotted lines) and VM2 (solid lines) and a third model MF (dashed lines) recently obtained by Mitrovica and Forte (1997) by the simultaneous inversion of both glacial isostatic adjustment and aspherical geoid constraints. It will be clear that model MV2 entirely eliminates the misfits

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of model VM1 to these data (which were not employed to constrain VM2) and that model MF grossly misfits the observations at essentially all locations along this coast. As discussed in Peltier (1998a), model MF is a "channel flow" model in which the upper mantle and transition zone have such low viscosity that adjustment takes places through motions that are essentially confined to this region. Models of this kind have been known not to fit the GIA constraints since the earliest work of Cathles (1975) and Peltier (1974).

model for the global GIA process, we will proceed in the following section to examine the extent to which the GIA process is expected to contaminate tide gauge observations of secular sea level trends.

Before we pursue this line of argument, however, it will prove interesting to consider the nature of the global sea level signal that an artificial earth satellite designed to measure the time dependence of Earth's gravitational field might be expected to observe. Since this signal will be measured relative to the center of mass of the planet rather than relative to the surface of the solid earth, it will be clear that it need look nothing at all like the fields shown on Figs. 4.4 or 4.5. If we denote by RSL (0, A, t) the rate of relative sea level change with respect to the surface of the solid earth, and by RAD (0, A, t) the rate of change of the local radius of the planet measured with respect to the center of mass, then it will be clear that ABS (0, A, i), the rate of change of absolute sea level measured with respect to the center of mass, is just

ABS (0, A, i) = RSL (0, A, t) + RAD (0, A, t). (17)

Figures 4.7 and 4.8 (see color plates) show both of the component parts of ABS together with ABS itself for models which respectively exclude (Fig. 4.7) and include (Fig. 4.8) the influence of rotational feedback, both of which employ the ICE-4G (VM2) model as input. Figure 4.9 (see color plate) shows the predictions with and without rotational feedback together with the difference between the predictions. The difference is clearly identical to the degree 2 and order 1 pattern shown previously in Fig. 4.5 as must, of course, be the case.

These results for absolute sea level, which constitute predictions of what the GRACE and CHAMP satellites should see if the GIA process were the only process currently influencing large-scale absolute sea level (geoid height) variability in the Earth System (see Peltier, 1999, for further discussion), demonstrate that the impact of rotational feedback on this observation will not be negligible even though it is negligible insofar as its impact upon sea level history relative to the surface of the solid earth is concerned. It will be extremely exciting over the next decade to keep careful watch on the outcomes of these ultra-high-resolution satellite experiments. They promise to provide a wealth of new knowledge concerning Earth system form and process. One useful prediction which we can make at present, however, by averaging the ABS signals shown in Figs. 4.7 and 4.8 over the area of the oceans covered by TOPEX/POSEIDON, is that the global rate of RSL rise estimated on the basis of this measurement system must be corrected upward by approximately 0.30 mm/yr in order to correct for the ongoing influence of the glacial isostatic adjustment process. This is a significant upward revision of the number of 0.08 mm/yr previously cited in Peltier (1998) in reference to a first attempt by Rapp et al. (personal communication) to compute the magni tude of the GIA contamination based upon the ICE-4G ( VM2) model predictions.

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