Time-dependent rotational stability of dynamic planets with viscoelastic lithospheres

Abstract

We extend previous work to derive a non-linear rotational stability theory governing true polar wander (TPW) of terrestrial planets with viscoelastic lithospheres. We demonstrate, analytically and using numerical examples, that our expressions are consistent with previous results in the limiting cases of low and infinite (i.e., purely elastic) viscosity lithospheres. To illustrate the stability theory, we compute TPW on Mars driven by a simple, prescribed mass loading. Our calculations demonstrate that on short time scales relative to the relaxation time of the viscoelastic lithosphere, the rotation axis follows a constrained path that reflects stabilization by remnant strength in the lithosphere, but that on long times scales this stabilization disappears and the load ultimately reaches the equator. Earlier work based on the assumption of a permanent remnant bulge in the case of a purely elastic lithosphere has suggested that Martian TPW would not persist for any significant time after a load is emplaced, and thus an equilibrium stability theory is sufficient to model long-term (order 1 Myr or longer) polar motion of the planet. Our results

A Full‐Maxwell Approach for Large‐Angle Polar Wander of Viscoelastic Bodies

“…It can be seen that the difference in the fluid Love number between M5 and M4 is almost the same as the mode strength ofT 1 in M4: k M4 f −k M5 f ≈ (k 1 ∕s 1 ) M4 . The remnant bulge is dealt with in previous studies by either those dealing with an elastic lithosphere (Cambiotti et al, 2010;Chan et al, 2014;Harada, 2012;Mitrovica et al, 2005) or viscoelastic lithosphere (Cambiotti et al, 2010;Moore et al, 2017) by isolating this part of the Love number and formulating the influence of it separately. Figure 7.…”

“…Here we show the effect when large‐angle TPW is considered. For this issue Moore et al () extend the theory in Chan et al () and consider the slowest mode(s) separately while applying the quasi‐fluid approximation to the rest of the modes. Apart from the bias introduced by the quasi‐fluid approximation as shown in Figure , another problem of this approach is that for a complex multilayered model like the SG6 Earth model with its lithosphere viscosity smaller than a certain value (e.g., 10 28 Pa s for SG6 model), the slowest mode might not be the mode from the lithosphere but one of the M modes ( M 2, M 3,…) which is generated by a density difference of the inner layers.…”

“…The remnant bulge is dealt with in previous studies by either those dealing with an elastic lithosphere (Cambiotti et al, 2010;Chan et al, 2014;Harada, 2012;Mitrovica et al, 2005) or viscoelastic lithosphere (Cambiotti et al, 2010;Moore et al, 2017) by isolating this part of the Love number and formulating the influence of it separately. One of the advantages of the full-Maxwell method is that we can choose any value for the initial loading time T h .…”

“…Another issue which is intensively studied in recent years is the effect of an elastic or highly viscous layer on TPW (Cambiotti et al, 2010;Chan et al, 2014;Harada, 2012;Harada & Xiao, 2015;Mitrovica et al, 2005;Moore et al, 2017;Willemann, 1984). The existence of such a layer can create a delayed readjustment of the equatorial bulge (often called remnant bulge) which significantly changes the behavior of TPW as discussed by Willemann (1984) and Mitrovica et al (2005).…”

“…As far as we know, this issue has not been discussed yet. Recent studies concerning the time-dependent solution of long-term large-angle TPW with an elastic or highly viscous lithosphere (Chan et al, 2014;Harada, 2012;Moore et al, 2017) are all based on the method developed by Ricard et al (1993) and adopt the quasi-fluid approximation.…”