5:15 PM - 6:45 PM
[PPS07-P19] Numerical Simulation of Rotational Flattening of Viscoelastic bodies and Implications for the fossil bulge of the Moon
Keywords:Moon, fossil bulge, Maxwell's viscoelasticity
The term "fossil bulge" refers to an equatorial bulge of the Moon that is significantly larger than the predicted one from hydrostatic theory. This is interpreted to be the relic of a hydrostatic shape acquired in the past when the Moon's orbital radius was smaller than it is today. In this study, motivated by the earlier work by Qin et al. (2018), we are developing a numerical technique to trace the temporal changes in the shapes of rocky planetary bodies owing to the effects of rotational and tidal forces. Our ultimate goal is to deepen the understanding of the evolution of the Moon-Earth system and that of the Moon itself, by making the best use of the overall shape of the Moon such as the fossil bulge.
The numerical program developed here solves for the deformation of the interior and interfaces (surface, CMB) of a rocky planetary body which is caused by the rotational and tidal forces under self-gravitation. The planet is assumed to be composed of two layers whose internal structure is characterized by the size and density of the inviscid core and the density and rheology (viscosity or Maxwell viscoelasticity) of the mantle. The basic equations are solved by the separation of variables, using the spherical harmonic expansion for the horizontal direction and the finite difference method for the vertical direction. A major innovation of our program lies in the solution method for the boundary value problems in the vertical direction. The set of fundamental equations is recast into a single fourth-order ordinary differential equation for the displacement velocity in the vertical direction, and the simultaneous linear equations obtained by discretization are solved by a direct solution method. Unlike the well-known propagator matrix method, our method is free from iterative calculations and is easy to improve the accuracy.
As a first step to validate the program developed here, we performed a series of calculations assuming that the mantle deforms only in a viscous manner (without elasticity), particularly focusing on the variations in the time constant of the planetary deformation changes depending on the internal viscosity structure. We found that the time constant increases in proportion to the viscosity of the entire mantle or to that in a thin highly viscous surface layer mimicking the "crust". It turned out however that a viscosity of as high as 10^27 Pa-s is required to yield the time constant up to about several billion years.
We next carried out several preliminary simulations where the effects of Maxwellian viscoelasticity are taken into account. The calculations for the simple cases in the absence of the temporal changes in the external forces (centrifugal and tidal forces) or the spatial variations in viscosity enabled us to verify that the effects of viscoelasticity were correctly modeled. That is, the deformation takes place mostly due to elasticity for a short period of time after the external force was applied, while the viscous deformation becomes dominant after a sufficiently long time, resulting in the equilibrium equatorial bulge given by the hydrostatic theory regardless of the value of elasticity. Furthermore, the comparison of the results of the viscous and viscoelastic cases showed that the elasticity helps increase the timescales to achieve equilibrium bulge increase, but only slightly (up to about 10 times). This may suggest the importance of spatial variations in the viscosity of the mantle, particularly the high viscosity near the surface at low temperatures which highlights elastic dominance deformation, in order to preserve the fossil bulge over a long period of billions of years.
The numerical program developed here solves for the deformation of the interior and interfaces (surface, CMB) of a rocky planetary body which is caused by the rotational and tidal forces under self-gravitation. The planet is assumed to be composed of two layers whose internal structure is characterized by the size and density of the inviscid core and the density and rheology (viscosity or Maxwell viscoelasticity) of the mantle. The basic equations are solved by the separation of variables, using the spherical harmonic expansion for the horizontal direction and the finite difference method for the vertical direction. A major innovation of our program lies in the solution method for the boundary value problems in the vertical direction. The set of fundamental equations is recast into a single fourth-order ordinary differential equation for the displacement velocity in the vertical direction, and the simultaneous linear equations obtained by discretization are solved by a direct solution method. Unlike the well-known propagator matrix method, our method is free from iterative calculations and is easy to improve the accuracy.
As a first step to validate the program developed here, we performed a series of calculations assuming that the mantle deforms only in a viscous manner (without elasticity), particularly focusing on the variations in the time constant of the planetary deformation changes depending on the internal viscosity structure. We found that the time constant increases in proportion to the viscosity of the entire mantle or to that in a thin highly viscous surface layer mimicking the "crust". It turned out however that a viscosity of as high as 10^27 Pa-s is required to yield the time constant up to about several billion years.
We next carried out several preliminary simulations where the effects of Maxwellian viscoelasticity are taken into account. The calculations for the simple cases in the absence of the temporal changes in the external forces (centrifugal and tidal forces) or the spatial variations in viscosity enabled us to verify that the effects of viscoelasticity were correctly modeled. That is, the deformation takes place mostly due to elasticity for a short period of time after the external force was applied, while the viscous deformation becomes dominant after a sufficiently long time, resulting in the equilibrium equatorial bulge given by the hydrostatic theory regardless of the value of elasticity. Furthermore, the comparison of the results of the viscous and viscoelastic cases showed that the elasticity helps increase the timescales to achieve equilibrium bulge increase, but only slightly (up to about 10 times). This may suggest the importance of spatial variations in the viscosity of the mantle, particularly the high viscosity near the surface at low temperatures which highlights elastic dominance deformation, in order to preserve the fossil bulge over a long period of billions of years.