## 1 Introduction

Redistributions of mass on a planet, whether driven by internal processes such as thermochemical convection, external processes such as the growth and ablation of ice sheets, or some combination of the two (e.g., volcanism), will drive reorientations of the rotation axis relative to the surface geography over a broad spectrum of timescales. Long-term, secular components of this reorientation are known as true polar wander (henceforth, TPW).

The analysis of the rotational stability of terrestrial planets, and in particular TPW, is a classic problem that dates, in the geophysical literature, to the canonical midtwentieth century study of *Gold* [1955]. *Gold*'s conceptual model of load-induced TPW is illustrated schematically in Figure 1a. We begin with a rotating planet whose gravitational figure has fully relaxed to an imposed centrifugal potential, i.e., a planet with a hydrostatic figure (Figure 1a1). If the planet is subject to a positive load, then the load will be thrown outward toward the equator, or, for an observer fixed to the planet, the pole will migrate (or wander) away from the load (Figure 1a2). This wander will be opposed by the rotational bulge, since it represents an excess mass that will resist displacement off the equator. However, *Gold* [1955] argued that this resistance is transient, since the Earth will viscously deform in response to the perturbed centrifugal potential so as to bring the bulge back onto the equator (Figure 1a3). Once this adjustment takes place, the load is free to move farther away from the rotation axis and the process will continue in incremental steps (Figure 1a4) until the load has reached the equator (Figure 1a5). In the conceptual model of *Gold* [1955], planetary rotation is inherently unstable because a load of any size will ultimately migrate to the equator.

We can extend these ideas to the hypothetical case of a planet with an elastic lithosphere, the initial figure of which is not hydrostatic (Figure 1b1). This case is consistent with a nonrotating, spherical planet (Figure 1b0) with an elastic lithosphere that is spun up to a final form in which all viscous stresses have relaxed. As in Figure 1a2, loading this planet will drive TPW, and this polar motion will meet transient resistance from a rotational bulge that will ultimately reorient perfectly to any new rotational state (Figure 1b3). The process will continue and, once again, the load will reach the equator (Figure 1b5). The key point in Figures 1a and 1b is that the rotation axis is inherently unstable whenever the rotational bulge can relax so that the initial form of the planet is reestablished around the new rotational state, whether this initial form is hydrostatic or not. In this case, in the terminology of *Gold* [1955], the rotating planet is said to lose all memory of previous rotational states.

*Goldreich and Toomre* [1969] provided a theoretical framework for the conceptual model of *Gold* [1955] and considered rotational stability in the case of multiple loads. They demonstrated, for slow changes in the shape of a quasi-rigid body, that the angle between the angular momentum vector and the principal axes is an adiabatic invariant. In the context of TPW, this invariance means that the rotation axis will remain aligned with the principal axis of inertia whenever changes in the shape of the planet are slow.

It is important to note that *Gold*'s conceptual model predicts the final orientation of the rotation axis (i.e., Figures 1a5 and 1b5), but not the timescale of TPW (i.e., Figures 1a1–1a4 and 1b1–1b4). A variety of methodologies have been developed within the geophysical literature to estimate the time dependence of TPW driven by mantle convection, including both nonlinear [*Ricard et al.*, 1993; *Steinberger and O'Connell*, 2002; *Tsai and Stevenson*, 2007] and linearized [*Chan et al.*, 2011; *Cambiotti et al.*, 2011] treatments of the governing equations. *Ricard et al.* [1993] made use of the adiabatic invariance described by *Goldreich and Toomre* [1969] and incorporated perturbations in the moment of inertia tensor associated with: (i) time-dependent mass redistribution, including boundary deformations, associated with internal convective forcing, and (ii) the time-dependent, viscous response of the rotational bulge to the perturbed centrifugal potential. They expressed the latter contribution in terms of a long timescale, asymptotic expansion of the viscoelastic tidal Love number at spherical harmonic degree 2.

The conceptual model of *Gold* [1955] was extended by *Willemann* [1984] [see also *Matsuyama et al.*, 2006] to incorporate stabilization associated with an elastic lithosphere, as illustrated in Figure 1c. Consider an initially hydrostatic planet characterized by an unstressed elastic lithosphere (Figure 1c1). This situation may arise in several ways. First, an elastic lithosphere may gradually cool out of a protoplanet that, at least initially, had no lithosphere. In this case, the creation of an unstressed elastic lid would not alter the gravitational figure of the planet (i.e., the evolution from Figure 1c0 to Figure 1c1). Alternatively, one can imagine a planet with a rotation vector that has been stable over a timescale long enough that the viscous stresses in even a high-viscosity lithosphere would have relaxed. The planets in Figures 1b1 and 1c1 have the same rotation rate; the important distinction between them is that the lithosphere in the former is stressed in the preloaded state and, as a consequence, the gravitational figure of the planet is not hydrostatic. As discussed by *Willemann* [1984] and *Matsuyama et al.* [2006], the distinction has profound implications for rotational stability [see also *Daradich et al.*, 2008].

Consider a scenario in which the planet in Figure 1c1 is subject to a load. As before, the pole will move away from the load, the rotational bulge will act to resist this motion, and this resistance will gradually weaken as the rotational bulge readjusts to the new orientation of the rotation axis. However, in contrast to the previous scenarios, elastic stresses induced in the lithosphere by TPW will prevent the rotational bulge from adjusting perfectly to the new rotational state. That is, the rotational bulge remains misaligned with the geographic equator in Figure 1c3. Nevertheless, the (imperfect) adjustment of the rotational bulge will permit further TPW and the process will continue until elastic stresses in the lithosphere balance the forcing associated with the load (Figure 1c5). In this case, the final position of the load will not be the equator. The stabilization associated with TPW-induced elastic stresses is known as the remnant bulge [*Willemann*, 1984] (Figure 1c5).

*Willemann* [1984] concluded that the remnant bulge stabilization is independent of the elastic thickness of the lithosphere. An extension and minor correction of his derivation by *Matsuyama et al.* [2006] demonstrated, in contrast, that the magnitude of TPW would be dependent on the lithospheric thickness; however, this dependence is only significant for planets with relatively thin lithospheres. As in *Gold* [1955] and *Goldreich and Toomre* [1969], *Willemann* [1984] and *Matsuyama et al.* [2006] were concerned with equilibrium theories that predicted the final state of the rotation axis in response to a loading.

*Chan et al.* [2011] derived a linearized (valid for small-amplitude TPW), time-dependent treatment of convection-driven TPW that incorporated stabilization by both the remnant bulge and the delayed viscous adjustment of the rotational bulge. In this paper, we derive a nonlinear, time-dependent rotational stability theory that incorporates both of these stabilization mechanisms. In our derivation, the first of these stabilization mechanisms is incorporated using the method first outlined by *Ricard et al.* [1993]. The derivation is valid for both internal and external loading, and we will describe the minor modifications necessary to move from one application to the other. In this regard, one might interpret this manuscript as either an extension of the *Ricard et al.* [1993] study to incorporate remnant bulge stabilization, or an extension of the equilibrium theory of *Matsuyama et al.* [2006] to treat time-dependent rotational stability. A preliminary discussion of this extended theory, in which the governing equation was provided without derivation, may be found in *Creveling et al.* [2012].

In the next section, we begin with a detailed mathematical derivation of the governing equations. Following this, we present a series of illustrative numerical simulations that adopt model parameters consistent with the Earth and Mars. Recent finite element modeling has shown that the Earth's broken lithosphere has an effective elastic thickness more than an order of magnitude smaller than the mean plate thickness when considering its impact on rotational stability [*Creveling et al.*, 2012]. In the simulations described below, we generally adopt a lithospheric thickness of 15 km, which is above the upper bound of 10 km suggested by *Creveling et al.* [2012]; however, to illustrate the physics of the stabilization we consider the sensitivity of the results to variations in this parameter over the range of 5–25 km. The scenario in Figure 1c1 is more directly appropriate for Mars, which has an elastic lithosphere with a thickness of several hundred kilometers, but a form that is very close to hydrostatic [*Daradich et al.*, 2008].