Present‐day secular variations in the zonal harmonics of Earth's geopotential
Authors: Mitrovica J. X. , Peltier W. R. 
Published: February 2008

We develop a new formalism for the prediction of secular variations in the gravitational potential field of a spherically symmetric, self‐gravitating, (Maxwell) viscoelastic planetary model subjected to an arbitrary surface load which may include a gravitationally self‐consistent ocean loading component. The theory is applied to generate the most accurate predictions to date, of the present‐day secular variations in the zonal harmonics of the geopotential (the so‐called
, for degree 𝓁) arising as a consequence of the late Pleistocene glacial cycles. In this respect, we use the very recent ICE‐3G reconstruction of the last late Pleistocene deglaciation event (Tushingham and Peltier, 1991). A comparison of these predictions with those generated using simplified disk models of the ice sheets, which have been used in all previous studies of the
, harmonics (𝓁 > 2), indicates that the disk model approximation introduces unacceptably large errors at all spherical harmonic degrees except perhaps 𝓁=2. Predictions have also been made using a eustatic loading approximation (also used in previous studies) in place of a gravitationally self‐consistent ocean loading component, and we have found that the resulting discrepancy is largest at degrees 2, 8 and 10. In the case of
the magnitude of the error incurred using the eustatic approximation can be as large as order 10–15% of the predicted value. We have attributed this discrepancy to the present day net flux of water away from the equatorial regions arising from the remnant present‐day adjustment associated with the late Pleistocene glacial cycles. The effect represents a heretofore unrecognized contribution to the
harmonics, or alternatively the nontidal acceleration of Earth's axial rate of rotation. In terms of the latter, the maximum anomaly in the length of day is approximately 1.7 μs/yr. We also consider the sensitivity of the
data to variations in the radial mantle viscosity profile by using a suite of forward calculations and an examination of Fréchet kernels. The theory required for the computation of those kernels is described herein. We find that the radial variation in sensitivity can be a strong function of the viscosity model used in the calculations. For models with a uniform upper mantle viscosity (νUM) of 1021 Pa s, forward predictions of the
harmonics exhibit a pronounced peak when a wide enough range of lower mantle viscosities (νLM) are considered (we denote the νLM value at this peak as
). At the lowest degrees (𝓁 ≤ 4), Fréchet kernels computed for a series of increasing νLM values (1021 Pa s ≤ νLM < 1023 Pa s) indicate a migration of the dominant sensitivity of the
data to variations in viscosity from regions below approximately 1200 km depth (for
) to regions above this depth in the lower mantle (for
). The sensitivity of the
data to variations in the viscosity profile in the shallowest parts of the lower mantle, for the case
, is also reflected in a set of forward calculations described herein. As an example,
, predictions made using Earth models in which the viscosity above 1200 km depth is constrained to be 1021 Pa s, do not exhibit the multiple solutions characteristic of the νUM= 1021 Pa s calculations. The same is true of Earth models in which the upper mantle viscosity is weakened an order of magnitude to 1020 Pa s. The theory described herein is also applied to compute the
signal (𝓁 ≤ 10) arising from the retreat of small ice sheets and glaciers described by Meier (1984) and also from any potential variations in the mass of the Antarctic and Greenland ice sheets. The present day
signal due to the late Pleistocene glacial cycles dominates the signal from Meier's sources at all degrees except 𝓁=3. In contrast, the
signal arising from mass variations in the Antarctic and Greenland ice sheets is potentially comparable to the former. A comparison of observational constraints on the
data with predictions of the postglacial rebound signal described in this paper, in order to infer mantle rheology, cannot proceed until constraints are placed on the present‐day mass flux of these large polar ice sheets. We show that the constraints required are weakest at degrees 𝓁=2 and 4. Finally, we outline a potentially important procedure for incorporating predictions of the
signal due to the late Pleistocene glacial cycles and Meier's sources, with an observational constraint on the
datum, to yield bounds on the present‐day net mass flux from the Antarctic and Greenland ice sheets. A rigorous application of this procedure must wait until observational constraints on
are reestablished in the literature.