Abstrakt
Understanding the feedback between the glacial isostatic adjustment (GIA) and the Earth's rotation is important for an accurate prediction of sea level changes induced by climate and tectonic processes. Here we consider a simple, four-layer incompressible Earth model, recently used for a benchmark of GIA codes to estimate the accuracy of the linearized Liouville equation (LE) and to demonstrate that models with an incomplete or missing rotational feedback violate the principle of energy conservation.
First, we compute GIA on a rotating Earth by solving the equation of motion coupled with LE in its full nonlinear form. By comparing the nonlinear LE solution with the traditional linearized one, we find that the radial component of the angular velocity vector is inaccurate in the latter case, with an error exceeding 10 per cent already after 1 kyr of evolution.
To understand the cause of this discrepancy, we investigate the time evolution of different kinds of energy involved in the process. While the rotational, elastic and dissipative energies are straightforward to compute, the formula for the gravitational energy contains an integral that requires a careful, higher-order accurate evaluation of the gravitational potential perturbation.
We circumvent this problem by transforming the integral into a different one, formulated in terms of displacement instead of potential. We find that the solution of the linearized LE equation does not conserve the energy, since, in the linearized case, the rate of change of the rotational energy is not equal to the power of the centrifugal force.
We also compute the energy balance of GIA on a constantly rotating Earth, and demonstrate the importance of the rotational feedback in the equation of motion. The formalism derived in this study allows a detailed examination of the energy balance for a rotating, incompressible planetary body deformed by a surface load.
As such, it may not only serve as a reliable tool for a posteriori testing of GIA numerical solutions but it can also be used in different planetary science applications.