The rotational stability of a triaxial ice age Earth

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1 Click Here for Full Article JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115,, doi: /2009jb006564, 2010 The rotational stability of a triaxial ice age Earth I. Matsuyama, 1 J. X. Mitrovica, 2 A. Daradich, 2 and N. Gomez 2 Received 24 April 2009; revised 1 October 2009; accepted 20 November 2009; published 7 May [1] Mitrovica et al. (2005), following calculations by Nakada (2002), demonstrated that the traditional approach for computing rotation perturbations driven by glacial isostatic adjustment significantly overestimates present day true polar wander (TPW) speeds by underestimating the background oblateness on which the ice age loading is superimposed. The underestimation has two contributions: the first originates from the treatment of the hydrostatic form and the second from the neglect of the Earth s excess ellipticity supported by mantle convection. In Mitrovica et al. (2005), the second of these two contributions was computed assuming a biaxial nonhydrostatic form (i.e., the principal equatorial moments of inertia were assumed to be equal to their mean value). In this article we outline an extended approach that accounts for a triaxial planetary form. We show that differences in the TPW speed predicted using the Mitrovica et al. (2005) approach and our triaxial theory are relatively minor ( 0.1 /Myr) and are limited to Earth models with lower mantle viscosity less than Pa s. However, for this same class of Earth models, the angle of TPW predicted for a triaxial Earth is rotated westward (toward the axis of maximum equatorial inertia) by as much as 20 relative to the biaxial case. We demonstrate that these effects are a consequence of the geometry of the ice age forcing, which has a dominant equatorial direction that is intermediate to the axes defining the principal equatorial moments of inertia of the planet. We complete the study by computing updated Frechet kernels for the TPW speed datum, which provide a measure of the detailed depth dependent sensitivity of the predictions to variations in mantle viscosity. We show, in contrast to earlier efforts to explore this sensitivity based on the traditional rotation theory, that the datum does not generally have a sensitivity to viscosity that peaks near the base of the mantle. Citation: Matsuyama, I., J. X. Mitrovica, A. Daradich, and N. Gomez (2010), The rotational stability of a triaxial ice age Earth, J. Geophys. Res., 115,, doi: /2009jb Introduction [2] Predictions of ice age induced perturbations in the Earth s rotation vector date to the early 1980s [e.g., Sabadini and Peltier, 1981; Yuen et al., 1982; Wu and Peltier, 1984] and they were largely based on the viscoelastic Love number theory developed by Peltier and colleagues [Peltier, 1974; Wu, 1978; Peltier, 1985]. These predictions joined a suite of other Love number based calculations of the glacial isostatic adjustment (GIA) process, including relative sea level variations [e.g., Farrell and Clark, 1976;Peltier and Andrews, 1976; Wu and Peltier, 1983] and anomalies in the Earth s gravity field [Wu and Peltier, 1983], though the rotation predictions were characterized by unique, and largely unexplained, sensitivities [Sabadini and Peltier, 1981; Yuen et al., 1982; Peltier and Wu, 1983; Wu and Peltier, 1984; Spada et al., 1992; 1 Department of Earth and Planetary Science, University of California, Berkeley, California, USA. 2 Department of Physics, University of Toronto, Toronto, Ontario, Canada. Copyright 2010 by the American Geophysical Union /10/2009JB Peltier and Jiang, 1996; Vermeersen and Sabadini, 1996; Vermeersen et al., 1997; Mitrovica and Milne, 1998; Johnston and Lambeck, 1999]. [3] Figure 1 (solid line) shows a prediction of the presentday rate of true polar wander (TPW) as a function of the lower mantle viscosity (n lm ) of the adopted Earth model. The prediction also adopts an elastic lithosphere of thickness 100 km and a constant upper mantle viscosity of Pa s and it is based on the TPW theory described by Wu and Peltier [1984] (hereinafter referred to as the traditional rotation theory). The first unequivocal evidence for an inherent problem in this traditional TPW theory was discovered by Nakada [2002]. Specifically, his article showed that predictions of present day TPW speed obtained using a viscoelastic lithosphere did not, as they should, converge to the results for an elastic lithosphere of equal thickness even when the viscosity of the former was increased to extremely high values (e.g, Pa s; Figure 1, dashed line). Mitrovica et al. [2005] called this inconsistency Nakada s paradox, and they provided an explanation for the paradox, as well as a new rotation theory that circumvented an inaccuracy in the traditional predictions. The physical reasoning behind their 1of10

2 Figure 1. Predictions of present day true polar wander (TPW) speed versus the lower mantle viscosity for different versions of the equations governing ice age rotational stability. The solid line shows predictions generated using the traditional (biaxial) rotation formalism of Wu and Peltier [1984], in which the same Earth model used to compute the planetary deformation is used to compute the equilibrium background form (Figure 2a). This Earth model is characterized by an elastic lithosphere of thickness 100 km. The dashed line shows the same as the solid line, except that the elastic lithosphere is replaced by a viscoelastic lithosphere of viscosity Pa s (Figure 2b). The dotted line shows the prediction of TPW speed based on the (biaxial) rotation theory of Mitrovica et al. [2005]; in this case, the excess (i.e., nonhydrostatic) ellipticity term (second term, right hand side of equation (2)) is set to (An application of the Mitrovica et al. [2005] theory in which the nonhydrostatic ellipticity is set to zero yields results identical to the dashed line.) The dashed dotted line shows the prediction generated using a rotation theory valid for a triaxial Earth model (equation (13)), in which the nonhydrostatic components of k A f and k B f are given by and , respectively (equations (3) and (4)). All other aspects of the Earth model, as well as the global sawtooth loading history, are described in the text. explanation and their extended theory is shown schematically in Figure 2. [4] The rotational stability of an ice age Earth (or, for that matter, any terrestrial planet subject to a surface mass load) is governed by two rather distinct issues that need to be addressed in any complete theoretical formalism. The first is the level of deformation experienced by a planet subject to both a surface mass load and a centrifugal potential (or rotational potential ) forcing that evolves as the rotation vector is perturbed. In general, a surface mass load acts to push the pole away, thus reorienting itself toward the equator, though this tendency is resisted by the (adjusting) equatorial bulge and is weakened as the isostatic compensation of the load increases. The second issue involves the so called background oblateness; that is, the flattening of the planetary form that serves as the background on which the surface loading is superimposed. The more oblate the planet, the more stable is its rotation vector in the presence of a surface loading. The traditional theory adopts a viscoelastic Love number formalism to predict the deformation of the planet (issue 1), and this aspect of the rotation theory is retained in the new formalism of Mitrovica et al. [2005]. It is the second issue, namely the treatment of the background oblateness, that introduces an inaccuracy in the traditional rotation theory and that is revised in the Mitrovica et al. [2005] approach. [5] Figure 2a illustrates the assumption inherent to the traditional theory. In particular, this theory assumes that the same Earth model used to compute the viscoelastic Love numbers necessary for the deformation calculation can also be used to compute the background oblateness. Let us say that the deformation equations adopt a planetary model with an elastic lithosphere of thickness LT (Figure 2a). In this case, the traditional theory assumes that the background oblateness is the same as would be achieved by spinning up the identical model with a rotation rate equal to the presentday rotation of the Earth and computing its equilibrium (i.e., infinite time) rotational form (Figure 2b). This equilibrium form, which is governed by the so called degree 2 fluid Love number, will exhibit a smaller flattening than a purely hydrostatic form since the lithosphere will be subject to permanent elastic stresses. [6] Nakada [2002] used the traditional formalism to compute rotation anomalies. In a subset of his models, he considered a viscoelastic lithosphere (Figure 2c). If this lithosphere is of sufficiently high viscosity, and of thickness LT, then the deformation predicted for such a model will be no different from the deformation predicted for the purely elastic lithosphere case in Figure 2a. However, the computed equilibrium background oblateness will be different, since the viscoelastic lithosphere will not retain any elastic strength at infinite time (no matter how high its viscosity is set). That is, the flattening, or equivalently the fluid Love number, is larger in Figure 2d (which is a purely hydrostatic form) than it is in Figure 2b. [7] As Mitrovica et al. [2005] argued, this inconsistency lies at the heart of Nakada s paradox. The two cases (an elastic lithosphere of thickness LT and a very high viscosity viscoelastic lithosphere of the same thickness) will not produce the same result when the traditional theory of Wu and Peltier [1984] is applied because the background oblateness assumed in the two calculations will be different. In particular, the latter case, since it leads to an increased background oblateness, will be characterized by a more stable rotation vector and thus smaller rates of present day TPW than the former case (Figure 1, solid and dashed lines). [8] So, which is the more accurate approach: adopting an elastic lithosphere or a viscoelastic lithosphere? On one level, one can argue that the traditional approach is clearly problematic since it makes the background oblateness a function of the adopted Earth model and therefore a physically unrealistic sensitivity is introduced into the calculations (why should the background form on which ice age loading is superimposed be model dependent?) More rigorously, one can ask a simple question: Which treatment produces a flattening that is closer to the observed flattening of the Earth? The Earth s form is, in fact, very close to a hydrostatic form, and for this reason the traditional TPW formalism, which leads to a progressively larger departure from this form as the thickness of the elastic lithosphere is increased, introduces obvious inaccuracies. One should note, in this regard, that the difference in the background oblateness between the two cases in Figure 1 (solid and dashed lines) is 1%. Yet, the governing equations are sufficiently 2of10

3 Figure 2. Schematic illustrating the distinct assumptions that underlie expressions for the background planetary oblateness in various theoretical treatments of the rotational stability of an ice age Earth. Top row: an initially nonrotating Earth (a) with an elastic lithosphere of thickness LT (denoted by the solid shaded region) is set spinning at the current rotation rate of the Earth and the background oblateness is given by the equilibrium (i.e., infinite time) rotational form achieved by this body. (b) Middle row: an initially nonrotating Earth (c) with a viscoelastic lithosphere of thickness LT (denoted by the hatched shaded region) is set spinning at the current rotation rate of the Earth and the background oblateness is given by the equilibrium form (d). The background oblateness in Figure 2d is greater than that in Figure 2b because the viscoelastic lithosphere is incapable of supporting elastic stresses in the long term (i.e., that shown in Figure 2d is a hydrostatic form). Bottom row: a fluid Earth has the equilibrium (hydrostatic) rotating form given in Figure 2e. This form is identical to that in Figure 2d. In Figure 2f, this hydrostatic form in Figure 2e is augmented by an excess oblateness driven by mantle convection (denoted by the circular shaded arrows). sensitive to this parameter that the error introduced reaches 80% for a lower mantle viscosity of Pa s. For this reason, Mitrovica et al. [2005] argued that whatever the Earth model adopted in the loading calculation, the background oblateness (or fluid Love number) should be computed using a model with no lithosphere (or equivalently a viscoelastic lithosphere; Figures 2d, 2e). [9] However, the Mitrovica et al. [2005] theory involves a second improvement to the traditional approach. As illustrated in Figure 2f, the Earth s shape departs from a pure hydrostatic form because of the dynamic effect of mantle convection [e.g., Goldreich and Toomre, 1969]. In particular, at spherical harmonic degree 2 (which governs the rotational stability) superplumes under Africa and the Pacific drive a so called excess flattening that has long been used as a constraint on models of the mantle convection process [e.g., Mitrovica and Forte, 2004]. This excess flattening, or oblateness, acts to further stabilize the planet relative to calculations that adopt an elastic lithosphere in the traditional rotation theory. [10] In effect, the Mitrovica et al. [2005] formalism replaces the fluid Love number adopted in the traditional theory with a Love number of the form k f ¼ 3G a 5 2 C A þ B ; ð1þ 2 where G is the gravitational constant and a and W are the radius and rotation rate of the Earth, respectively. A, B, and C are the principal moments of inertia of the Earth, with C representing the polar moment. [11] Tobemoreprecise,theMitrovica et al. [2005] approach is based on a slightly modified form of equation (1) in which the right hand side of the equation is decomposed into hydrostatic and nonhydrostatic contributions: k f ¼ k hyd f þ 3G a 5 2 C A þ B nhyd ; ð2þ 2 where the first term on the right hand side of the equation is the fluid Love number computed using an Earth model with 3of10

4 Figure 3. Schematics illustrating the shape and principal axis orientation of the triaxial Earth in both a (a) 3 D and (b) top view. The figures reflect, using an exaggerated scale, the relative sizes of the principal moments of inertia (A < B < C). The x 3 axis is coincident with the rotation vector, while x 1 and x 2 equatorial axes point toward E and E, respectively. no lithosphere (i.e., the hydrostatic case) and the second term is the excess (or nonhydrostatic) equatorial flattening of the planet (i.e., the observed equatorial flattening on the RHS of equation (1) minus the computed hydrostatic flattening). This latter term is, in turn, related to the nonhydrostatic component of the difference between the polar (C) and mean equatorial (A, B) (principal) moments of inertia. Within the Mitrovica et al. [2005] formalism, the hydrostatic term on the right hand side of equation (2) is determined using first order accurate viscous Love number theory. [12] Why replace the observed fluid Love number of equation (1) with equation (2) when describing the background oblateness? The form (2) is preferred because the first order accurate hydrostatic theory used to compute k f hyd (or the background oblateness) in the latter equation is consistent with the accuracy of the viscoelastic Love numbers used to predict planetary deformation within the full rotation theory. Nevertheless, the difference between these two choices is relatively small, and in Figure 1 we show predictions of the present day TPW speed (dotted line) when the background oblateness is computed using the expression (2) [Mitrovica et al., 2005]. As expected, the excess ellipticity supported by mantle convection acts to further stabilize the rotation pole. [13] The expression (2) assumes that the background ellipticity of the planet can be represented within the equations for rotational stability using the single number k f ; that is, it is assumed that this background form is biaxial or very nearly so (A B). In reality, the nonhydrostatic form is triaxial (C > B > A; Figure 3a) [Goldreich and Toomre, 1969]. Using satellite derived, spherical harmonic degree 2 geoid coefficients [Tapley et al., 2005], we derived the complete inertia tensor of the planet; diagonalizing the result yields the principal (equatorial) axis orientation shown in Figure 3b. Moreover, correcting these equatorial moments using the hydrostatic form predicted by Nakiboglu [1982] yields 3G a 5 2 ðc AÞnhyd ¼ 0:0115 3G a 5 2 ðc BÞnhyd ¼ 0:0052: This geometry and the principal moments are in agreement with an independent calculation of the Earth s equatorial figure by Soler and Mueller [1978, p. 44]. As Goldreich and Toomre [1969] first noted, the nonhydrostatic component of the planetary form is distinctly triaxial since C B B A.In applying their biaxial theory, Mitrovica et al. [2005] adopted a value of for the nonhydrostatic component of equation (2). This value is, as expected, the mean of the values in equations (3) and (4). [14] In this article we consider perturbations in the rotation vector arising from ice age loading on a triaxial Earth. Our goals are to (1) derive succinct expressions for TPW on a triaxial Earth that may be used in place of those derived by [Mitrovica et al., 2005]; (2) compare predictions of the present day TPW vector (magnitude and angle) based on this triaxial form with those generated using the biaxial theory of Mitrovica et al. [2005] and provide a physical explanation for these differences, and (3) quantify, using numerically determined Frechet kernels based on the triaxial theory, the detailed depth dependent sensitivity of the TPW speed predictions to variations in the radial profile of mantle viscosity. [15] We note that an effort to provide an extension of the Mitrovica et al. [2005] formalism to a triaxial Earth has ð3þ ð4þ 4of10

5 recently been described by Nakada [2009]. His governing equations incorporate time dependent, convection induced changes in the background form. We will assume, in contrast, that this contribution is static over the ice age time scale. The Nakada [2009] study did not include a comparison with the biaxial case and focused entirely on TPW speed. Furthermore, with the exception of a small suite of forward calculations, Nakada [2009] did not explore the sensitivity of the TPW results to mantle viscosity. 2. Angular Momentum Conservation [16] If we ignore external torques to consider Earth s TPW, angular momentum conservation in an Earth fixed frame that rotates with angular velocity w i can be written as 0 ¼ J ij ðtþ _w j ðtþþ J : ij ðtþw j ðtþþ ijk w j ðtþj kl ðtþw l ðtþ; ð5þ where J ij is the inertia tensor, ijk is the Levi Civita symbol, and the dot denotes a time derivative in the rotating reference frame. This angular momentum conservation equation is known as the Liouville equation. [17] We will consider perturbations in the rotating Earth system relative to an initial state prior to the onset of loading. Under the assumption that the loading involves small perturbations to both the rotation vector and inertia tensor relative to this initial state, we can derive a linearized form of the Liouville equations [e.g., Wu and Peltier, 1984]. The body reference frame is chosen with origin at the center of mass. In the unperturbed, initial state the inertia tensor is diagonal with principal moments J 11 = A, J 22 = B, J 33 = C, and the angular velocity is W = (0, 0, W) (Figure 3). In this case, we can write w i ðtþ ¼ ½ i3 þ m i ðtþš; J 11 ðtþ ¼ A þ I 11 ðtþ; J 22 ðtþ ¼ B þ I 22 ðtþ; J 33 ðtþ ¼ C þ I 33 ðtþ; J ij ðtþ ¼ I ij ðtþ for i 6¼ j; where we will assume m i 1 and I ij A, B, C. Retaining first order terms, the three components of equation (5) become _m 1 ðtþþ 1 m 2 ðtþ ¼ A I 23ðtÞ 1 A _ I 13 ðtþ; _m 2 ðtþþ 2 m 1 ðtþ ¼ B I 13ðtÞþ 1 B _ I 23 ðtþ; ð6þ ð7þ _m 3 ðtþ ¼ _ I 33 ðtþ C ; ð8þ where we introduce the Euler wobble frequencies ðc BÞ 1 A ðc AÞ 2 : B ð9þ To simplify these equations, we make two assumptions. First, we assume that the time scale over which the loading and planetary adjustment proceed is much longer than the period of rotation (i.e., 1 day). This is equivalent to ignoring the time derivative terms on the right hand side of equation (7). Second, we assume that these time scales are also much longer than the Euler wobble periods on a rigid Earth, which are inversely proportional to s 1 and s 2. In this case, the time derivative terms on the left hand side of equation (7) can be neglected. Thus, we ignore the time derivative terms in equation (7) to obtain 1 m 2 ðtþ ¼ A I 23ðtÞ; 2 m 1 ðtþ ¼ B I 13ðtÞ: ð10þ [18] Inertia tensor perturbations associated with the load (L) and load induced deformation (LD) may be written as I L;LD ij ðtþ ¼ ðtþþk2 L ðtþ *I L ij ðtþ ¼Iij L ðtþþkl 2 ðtþ*i ij L ðtþ; ð11þ where the asterisk denotes a time convolution and k 2 L is the viscoelastic surface load k Love number at spherical harmonic degree 2 [Peltier, 1974; Wu and Peltier, 1984]. The inertia tensor perturbations associated with the rotational driving potential (R) are I13 R ðtþ ¼ 2 a 5 3G kt 2 ðtþ*m 1ðtÞ; I23 R ðtþ ¼ 2 a 5 3G kt 2 ðtþ*m 2ðtÞ; ð12þ where k 2 T is viscoelastic tidal k Love number at spherical harmonic degree 2 [Peltier, 1974]. Using these relations, equation (10) becomes m 1 ðtþ ¼ I L 13 ðtþþkl 2 ðtþ*i L 13 ðtþ ðc AÞ m 2 ðtþ ¼ I L 23 ðtþþkl 2 ðtþ*i L 23 ðtþ ðc BÞ where k f A and k f B are parameters given by kf A ¼ 3G a 5 ðc AÞ; 2 kf B ¼ 3G a 5 ðc BÞ: 2 þ kt 2 ðtþ kf A *m 1 ðtþ; þ kt 2 ðtþ kf B *m 2 ðtþ; ð13þ ð14þ [19] In the results described below, we solve equation (13) using a slightly modified expression for equations (14). To understand this modification, let us first express the viscoelastic tidal k Love number in terms of the usual normal mode expansion [Peltier, 1985] k2 T ðlt; tþ ¼kT;E 2 ðtþþ XJ r j e sjt ; j¼1 ð15þ where the first term on the right hand side is the elastic component of the Love number and the second term is a 5of10

6 nonelastic component composed of J normal modes of pure exponential decay. The parameter LT makes explicit the dependence of the Love number on the thickness of the elastic lithosphere adopted in the deformation calculation. [20] Next, let us decompose equations (14) into terms associated with hydrostatic and nonhydrostatic forms: k A f ¼ k hyd f þ 3G a 5 ðc AÞnhyd 2 kf B ¼ k hyd f þ 3G a 5 2 ðc BÞnhyd : ð16þ The first term on the right hand side is the hydrostatic component of the fluid k Love number (which is generally computed using a second order accurate hydrostatic theory [Nakiboglu, 1982]), while the residual (i.e., nonhydrostatic) component is determined by simply taking the difference between the observed value of k f A or k f B [equation (14)] and the hydrostatic form. [21] The parameters k f A and k f B in equation (13) appear in ratios with the viscoelastic Love number k 2 T (t), which is computed using a first order accurate theory [Peltier, 1974, 1985]. Mitrovica et al. [2005] argued that, in order to maintain a consistent accuracy in the Liouville equation, the term k f hyd in equations (16) should be computed using the normal modes associated with the viscoelastic Love number k 2 T (LT, t) with LT set to zero [and all other aspects of the Earth model are retained from the k 2 T (LT, t) case]. Specifically, a hydrostatic form consistent with the accuracy of viscoelastic Love number theory can be determined by convolving (the LT = 0 case of) equation (15) with a Heaviside step function and then considering the infinite time response. This exercise yields: k hyd f ¼ k T;E 2 þ XJ j¼1 r j s j for LT ¼ 0: ð17þ To recap, our TPW predictions will be based on the timedependent Liouville equation (13) with the hydrostatic component of k f A and k f B [equation (16)] given by equation (17) and the nonhydrostatic components by equations (3) and (4), respectively. The viscoelastic tidal and surface load Love numbers, k 2 T (t) [equation (15)] and k 2 L (t), respectively, are computed using the standard approach described by Peltier [1974]. [22] Our Liouville equation for a triaxial Earth (13) collapses to the special (biaxial) case considered by Mitrovica et al. [2005] by setting the equatorial moments of inertia A and B to (A + B)/2. In this case, k f A = k f B = k f as given in equation (2) [with the hydrostatic contribution computed from equation (17)]. This is the background form given in Figure 2f. (The case shown in Figure 2e or 2d is furthermore obtained by setting the nonhydrostatic form to zero.) [23] To obtain the traditional rotation theory of Wu and Peltier [1984] we again set the equatorial moments of inertia A and B to (A + B)/2 in equation (13) and k f A = k f B = k f. Moreover, in this case, equation (2) for the k fluid Love number is replaced by the expression: k f ¼ k T;E 2 þ XJ j¼1 r j s j for LT 6¼ 0; ð18þ that is, the background form is computed using normal modes obtained from the same model used to compute the viscoelastic load and tide Love numbers, i.e., an Earth model with a lithosphere having nonzero elastic thickness (Figure 2b). [24] Finally, we note that the derivation of our triaxial theory is based on a geocentric Cartesian reference frame that coincides with the principal axis orientation of the Earth rather than the usual geographic coordinate system; that is, the equatorial x 1 axis is directed toward E instead of the Greenwich meridian (Figure 3b). In the numerical implementation of our theory, one has to be careful to define the loading geometry in the same coordinate system. 3. Results and Discussion [25] The results in this section are based on spherically symmetric, self gravitating Maxwell viscoelastic Earth models with elastic and density structure prescribed by the seismic model PREM [Dziewonski and Anderson, 1981]. Following Wu and Peltier [1982], the density variation is assumed to be nonadiabatic. The radial viscosity structure is comprised of a purely elastic lithosphere of thickness 100 km and constant upper mantle viscosity (i.e., below the lithosphere and down to 670 km depth) of Pa s. The lower mantle viscosity, n lm, serves as the free parameter of the modeling. [26] We consider two distinct surface mass loads that are distinguished on the basis of their spatial geometry but that share a simplified, sawtooth loading history. This history will include eight identical glacial cycles of duration 100 kyr, each with a 90 kyr glaciation phase followed by a 10 kyr deglaciation event, ending at 6 kyr before present. Our primary ice load geometry, henceforth described as the global load model, is taken from Mitrovica et al. [2005], following Mitrovica and Milne [1998]. These authors considered the difference between the maximum and minimum states of the ICE 3G deglaciation model [Tushingham and Peltier, 1991] and, to conserve mass, augmented this ice load with a geographically uniform change in ocean height. The load was characterized by inertia tensor perturbations of I 13 = kg m 2, I 23 = kg m 2, and I 33 = kg m 2. This global load model was used to compute the results in Figure 1. [27] To highlight the physics of TPW on a triaxial Earth model, we will also consider a simple disc load geometry shown schematically in Figure 4a. In this case, the geometry is composed of a single, circular ice disk with parabolic vertical cross section centered at a colatitude of 30 together with a complementary (global) ocean load of uniform thickness. The longitude of the load center,, will be varied. At glacial maximum, when = 0, the relevant inertia tensor perturbations for this model are I 13 = kg m 2, I 23 = 0, and I 33 = kg m 2. [28] In Figures 4b and 4c we show predictions of presentday TPW speed and angle, respectively, generated by varying the center longitude of disk load model from 0 to 360. For this calculation, n lm was set to (dashed line) Pa s, (dotted line) Pa s or (dashed dotted line) Pa s. For the purpose of comparison, the solid lines on each frame are results obtained when we assume a biaxial rotation theory of Mitrovica et al. [2005] where the excess (nonhydrostatic) 6of10

7 Figure 4. (a) Schematic illustrating the geometry of a simple disk load ice age forcing applied to a triaxial Earth. The figure of the Earth reflects, using a highly exaggerated scale, the relative sizes of the principal moments of inertia (A < B < C). Predictions of the present day ice age induced TPW speed (b) and angle (c) generated using the simplified disk loading geometry shown in Figure 4a as a function of the adopted longitude of the load center ( in Figure 4a). The sawtooth load history is specified in the text. The dashed, dotted, and dashed dotted predictions are based on the rotational stability theory valid for a triaxial Earth (equations (13) and (16)) with nonhydrostatic inertia differences given by equations (3) and (4). The Earth model has an elastic lithospheric thickness of 100 km, constant upper mantle viscosity of Pa s, and lower mantle viscosity of (dashed) Pa s, (dotted) Pa s, or (dashed dotted) Pa s. In contrast, the solid lines are predictions generated using a biaxial rotation theory of Mitrovica et al. [2005] where the excess (i.e., nonhydrostatic) ellipticity term (second term, right hand side of equation (2)) is set to In this case, the Earth model has a lower mantle viscosity of Y Pa s, where Y is specified next to each solid line in Figure 4b. The three biaxial lines lie on top of each other in Figure 4c. ellipticity term [second term, right hand side of equation (2)] is set to In the biaxial case, the value of the adopted lower mantle viscosity is specified next to each solid line on Figure 4b. The biaxial results all lie on the same solid line in Figure 4c. [29] As one would expect, the TPW longitude predicted for the biaxial case is identical to the center longitude of the disk load regardless of the adopted Earth model. In this case, since the equatorial moments of inertia (A and B) are assumed to be equal, there is no preferential direction for the polar motion and the TPW must lie along the longitudinal great circle that includes the center of the load. In contrast, the triaxial model results do have a preferred direction. In the triaxial case, the TPW direction coincides with the load longitude only when this longitude is coincident with one of the principal axes (x 1 or x 2 ) of the triaxial form. This coincidence is expected on the grounds of symmetry, and it occurs when = E (+x 2 direction), E ( x 1 ), E ( x 2 ), and E (x 1 ). As one would also expect, the highest rates of TPW are predicted when the load is aligned with the x 2 axis since this direction is characterized by the smallest equatorial flattening and minimum rotational stability (Figure 3b). In contrast, the smallest TPW rates occur when the load lies along the x 1 axis, which is the direction of greatest equatorial flattening and hence maximum rotational stability. Thus, relative to the biaxial case, polar motion in the triaxial calculations is increased in the x 2 direction and decreased in the x 1 direction; accordingly, the angle of TPW in the triaxial case is always deflected toward the local x 2 axis relative to the biaxial case (Figure 4c). [30] As the lower mantle viscosity is increased, the error incurred by adopting a biaxial form decreases. This result is consistent with Figure 1, where the sensitivity of the predicted TPW rate to the inclusion of any amount of excess equatorial flattening is reduced as n lm is increased (compare the dashed and dotted lines). Nevertheless, the results in Figure 4 indicate that the triaxiality of the planetary form can, in general, have a significant impact on both the TPW speed and angle. As an example, for n lm =10 21 Pa s, the predicted TPW speed on the triaxial Earth model varies between 0.18 and 0.50 /Myr, depending on the orientation of the disk load relative to the principal axes. The biaxial case, when the excess ellipticity term is set to 0.008, yields a TPW rate close to the mean of these values (0.35 /Myr). Clearly, the biaxial model provides an accurate approximation of the TPW rate on a triaxial Earth only when (i) the 7of10

8 Figure 5. Predictions of the present day ice age induced TPW generated using the global loading geometry and the sawtooth loading history. The predictions are based on a rotational stability theory valid for a triaxial Earth [equations (13) and (16)] with nonhydrostatic inertia differences given by equations (3) and (4). Moreover, the calculations adopt an Earth model with an elastic lithospheric thickness of 100 km, upper mantle viscosity of Pa s, and a lower mantle viscosity of Y Pa s, where Y is specified by the label adjacent to the arrowhead of each line. Also shown, for the sake of comparison, is the direction of TPW associated with all predictions based on the biaxial rotation theory of Mitrovica et al. [2005] (dotted line). lower mantle viscosity is sufficiently high ( Pa s) or (ii) the load is oriented in an equatorial direction that is intermediate between the two equatorial principal axes. However, even if condition (ii) holds, the error in the TPW angle may not be small. As an example, for a load longitude of 120 E (which is halfway between the x 2 and x 1 principal axes; Figure 3b) and n lm =10 21 Pa s, the triaxial calculations yield a TPW rate that is very close the biaxial results ( 0.37 /Myr; Figure 4b); however, the TPW angle in the triaxial case is rotated 25 toward the x 2 axis relative to the biaxial case (or, equivalently, relative to the load longitude; Figure 4c). [31] Next we turn to the global ice model case, where the orientation of the load is no longer a free parameter but is rather specified by the geometry of the Late Pleistocene ice cover. The dashed dotted line in Figure 1 gives the results of the TPW predictions as a function of the lower mantle viscosity based on the triaxial theory. Figure 5 shows the predicted TPW vectors for five specific values of n lm extending from Pa s to Pa s. Figure 5 also shows the direction of TPW predicted in any biaxial calculation. This latter direction, 286 E, also represents, as discussed above, a measure of the mean orientation of the ice age load relative to the principal axes of the solid Earth. This orientation is 30 and 60 from the x 2 and x 1 axes, respectively. That is, it is closer to the direction of the minimum rotational stability, and this explains why the triaxial calculations in Figure 1 are characterized by larger present day TPW rate than the analogous biaxial calculations (Figure 1, dotted line). In the case of n lm =10 21 Pa s, the biaxial prediction (0.344 /Myr) is 20% less than the triaxial case (0.426 /Myr); the error drops to 6% and 3% for predictions based on models with n lm = Pa s and n lm = Pa s, respectively. This difference is small relative to the large error associated with either the traditional rotation theory of Wu and Peltier [1984] (Figure 1, solid line) or a biaxial theory in which the background form is hydrostatic (i.e., the excess ellipticity is set to 0; Figure 1, dashed line); however, it is not insignificant for models with n lm Pa s. [32] As we observed for the disk load model, the triaxial prediction of the TPW angle is displaced in the direction of the x 2 axis relative to the biaxial case. Moreover, this displacement increases as the lower mantle viscosity is decreased (Figure 5); it is 8 for n lm = Pa s and 20 for n lm =10 21 Pa s. [33] The results in Figures 1 and 5 provide a rather coarse measure of the sensitivity of the TPW predictions to variations in mantle viscosity. To further elucidate the physics of ice age induced TPW it will be instructive to refine this measure of sensitivity. To this end, following Mitrovica and Milne [1998], we can define Frechet kernels for predictions of TPW speed as Z 1 d ¼ FK½ðrÞ; ršlog 10 ðrþdr; ð19þ CMB=a where d represents a perturbation, d is a prediction of TPW speed, n(r) is the radial profile of mantle viscosity, and r is the radius (nondimensionalized by the surface radius, a). The Frechet kernel provides a detailed, depth dependent measure of the sensitivity of a TPW speed prediction to variations in the viscosity profile. That is, any radial region where the kernel has a relatively large amplitude is a zone in which a perturbation in the viscosity profile will lead to a relatively large change in the prediction of TPW speed. Since the problem is nonlinear, this sensitivity will be a function of the viscosity profile, as indicated by the form of the equation. [34] Mitrovica and Milne [1998] used a numerical perturbation scheme discussed in detail by Mitrovica and Peltier [1991] to compute Frechet kernels for TPW speed for a suite of radial viscosity profiles in which n lm was varied, while the upper mantle viscosity and lithospheric thickness were set to Pa s and 100 km, respectively. Peltier and Jiang [1996] computed the kernel for the specific case of n lm = Pa s. However, both these previous analyses of sensitivity were based on the traditional ice age theory for TPW [Wu and Peltier, 1984]. It is clear from Figure 1 that the large error associated with the traditional theory would significantly affect these calculations. To update the sensitivity analysis, we have used the Mitrovica and Peltier [1991] approach to compute Frechet kernels for TPW speed based on the triaxial theory described above. The results, for a sequence of viscosity profiles in which n lm is increased from to Pa s, are shown in Figure 6. 8of10

9 Figure 6. (a c) Frechet kernels, defined by equation (19), for predictions of present day TPW speed generated using the global loading geometry (and sawtooth loading history) and based on a rotational stability theory valid for a triaxial Earth (equations (13) and (16)) with nonhydrostatic inertia differences given by equations (3) and (4). The predictions adopt an Earth model with an elastic lithospheric thickness of 100 km, upper mantle viscosity of Pa s, and a lower mantle viscosity n lm of Y Pa s, where Y is specified by the label adjacent to each line. (d) Reproduction of the triaxial (dashed dotted line) results in Figure 1, where the dots on the line indicate the specific n lm values adopted in Figures 6a 6c. [35] For this class of Earth models, the TPW speed predictions are relatively insensitive to variations in upper mantle viscosity. In contrast, the sensitivity to viscosity below 670 km depth is larger, though the detailed depthdependent sensitivity within this region is a strong function of the radial viscosity profile. For a model with uniform mantle viscosity of Pa s, the peak amplitude (i.e., sensitivity) occurs at the base of the mantle, above the core mantle boundary (CMB). However, for even a moderate increase in lower mantle viscosity, to values between Pa s, the kernels shift upwards within the lower mantle, and the region of peak sensitivity migrates to the shallowest portions of this region. This is an important transition because models with n lm Pa s have been preferred in some analyses of GIA data sets [e.g., Tushingham and Peltier, 1991]. Frechet kernels for TPW speed predictions computed using the traditional rotational theory [Wu and Peltier, 1984] and based on this viscosity model [e.g., Peltier and Jiang, 1996; Mitrovica and Milne, 1998] are characterized by a peak sensitivity to variations in the radial viscosity profile at the base of the lower mantle; our updated calculations indicate that the peak sensitivity actually resides in the shallowest regions of the lower mantle. [36] For a lower mantle viscosity of Pa s, the integrated kernel is of nearly equal amplitude and opposite sign in the top and bottom halves of the lower mantle. As a consequence, these sensitivities cancel when n lm is varied close to this model and the result is a local maximum in Figure 6d (and Figure 1, dashed dotted line). Frechet kernels computed for models with n lm >10 22 Pa s show a sensitivity that is relatively uniform throughout the lower mantle, although this sensitivity monotonically diminishes with increasing stiffness (as suggested by the slope of the forward predictions in Figure 6d). As one would expect from Figure 1, these kernels are in close agreement with those computed using the traditional rotation formalism of Wu and Peltier [1984; Mitrovica and Milne, 1998]. 4. Final Remarks [37] We have extended the biaxial rotational stability theory of Mitrovica et al. [2005] to incorporate triaxiality in the background shape of the planet. We conclude that predictions of ongoing, ice age induced TPW speed based on the triaxial equations are reasonably well approximated by a biaxial theory. Specifically, the latter (which overestimates the former) has a peak error of 20% for a uniform mantle viscosity of Pa s. The error is less than 3% for n lm > Pa s. The agreement between the biaxial and triaxial results for models with relatively high lower mantle viscosity is to be expected, since such models become progressively less sensitive to the background form adopted in the ice age calculations (Figure 1; Mitrovica et al. [2005]). In contrast, the relatively close agreement for weaker viscosity models is somewhat fortuitous (see, e.g., Figure 4b), since it is in part the result of an ice age forcing with a dominant equatorial direction that is intermediate to the orientation of the principal equatorial moments of inertia of the Earth. However, this geometry does lead to errors in the predicted angle of TPW. In the n lm =10 21 Pa s case, incorporating triaxiality in the rotation theory yields a TPW angle that is rotated westward (i.e., toward the negative x 2 9of10

10 axis that defines the equatorial moment of inertia B) by 20 relative to the biaxial solution. [38] The TPW datum has been used in a large suite of geophysical analyses, most recently in efforts to infer present day melting from polar ice complexes associated with global warming [e.g., Mitrovica et al., 2006]. This application requires a correction of the datum for the signal due to GIA. Clearly, the correct treatment of the background form in the GIA prediction, as described by Mitrovica et al. [2005], is of fundamental importance in analyses of presentday ice flux based on TPW speed (compare the solid and dotted lines in Figure 1). While a triaxial treatment of the background form is less important when focussing on TPW speed than the improvement discussed by Mitrovica et al. [2005] (compare the dotted and dashed dotted lines in Figure 1), triaxiality is potentially important in obtaining an accurate GIA correction for the TPW vector (Figure 5). [39] TPW speed has also been a key datum in GIA based inferences of mantle viscosity [e.g., Sabadini and Peltier, 1981; Yuen et al., 1982; Wu and Peltier, 1984; Peltier and Jiang, 1996; Johnston and Lambeck, 1999]. Future analyses of this kind must adopt a rotational theory that incorporates a correct treatment of the background form [Mitrovica et al., 2005]. In this regard, the Frechet kernels presented herein (Figure 6) indicate that a proper treatment of this background form, whether based on a biaxial or triaxial theory, will alter the detailed depth dependent constraint on mantle viscosity inferred from earlier analyses based on the traditional rotation theory. A clear indication of this impact is illustrated by the Frechet kernel computed using n lm = Pa s. Earlier analyses based on this viscosity model have concluded that the TPW speed datum is sensitive to variations in the viscosity at the base of the lower mantle. Our updated calculation of the kernel for this model indicates that the sensitivity actually peaks at the top of the lower mantle. [40] Acknowledgments. We became aware of the manuscript Nakada [2009] while preparing this article for submission. We thank Prof. Nakada for allowing us to make reference to his submitted work. We also thank Prof. Roberto Sabadini and an anonymous reviewer for their constructive comments. The authors acknowledge funding and support from the Miller Institute for Basic Research (IM), NSERC (JXM, NG) and Harvard University (NG, JXM). References Dziewonski, A. M., and D. L. Anderson (1981), Preliminary reference Earth model (PREM), Phys. Earth Planet. Inter., 25, Farrell, W. E., and J. T. Clark (1976), On postglacial sea level, Geophys. J. R. Astron. Soc., 46, Goldreich, P., and A. Toomre (1969), Some remarks on polar wandering, J. Geophys. Res., 74, Johnston, P., and K. Lambeck (1999), Postglacial rebound and sea level contributions to changes in the geoid and the Earth s rotation axis, Geophys. J. Int., 136, Mitrovica, J. X., and A. M. Forte (2004), A new inference of mantle viscosity based upon joint inversion of convection and glacial isostatic adjustment data, Earth Planet. Sci Lett., 225, Mitrovica, J. X., and G. A. Milne (1998), Glaciation induced perturbations in the Earth s rotation: A new appraisal, J. Geophys. Res., 103 (B1), Mitrovica, J. X., and W. R. Peltier (1991), A complete formalism for the inversion of post glacial rebound data: Resolving power analysis, Geophys. J. Int., 104, Mitrovica, J. X., C. Beaumont, and G. T. Jarvis (1989), Tilting of continental interiors by the dynamical effects of subduction, Tectonics, 8, Mitrovica, J. X., J. Wahr, I. Matsuyama, and A. Paulson (2005), The rotational stability of an ice age earth, Geophys. J. Int., 161, Mitrovica, J. X., J. Wahr, I. Matsuyama, A. Paulson, and M. E. Tamisiea (2006), Reanalysis of ancient eclipse, astronomic and geodetic data: A possible route to resolving the enigma of global sea level rise, Earth Planet. Sci. Lett., 243, Nakada, M. (2002), Polar wander caused by the Quaternary glacial cycles and the fluid Love number, Earth Planet. Sci. Lett., 200, Nakada, M. (2009), Polar Wander of the Earth associated with the Quaternary glacial cycle on a convecting mantle, Geophys. J. Int., 179, Nakiboglu, S. M. (1982), Hydrostatic theory of the Earth and its mechanical implications, Phys. Earth Planet. Inter., 28, Peltier, W. R. (1974), The impulse response of a Maxwell Earth, Rev. Geophys., 12, Peltier, W. R. (1985), The LAGEOS constraint on deep mantle viscosity: Results from a new normal mode method for the inversion of visco elastic relaxation spectra, J. Geophys. Res., 90, Peltier, W. R., and J. T. Andrews (1976), Glacial isostatic adjustment: I. The forward problem, Geophys. J. R. Astron. Soc., 46, Peltier, W. R., and X. Jiang (1996), Glacial isostatic adjustment and Earth rotation: Refined constraints on the viscosity of the deepest mantle, Geophys. J. Int., 101, Peltier, W. R., and P. Wu (1983), Continental lithospheric thickness and deglaciation induced true polar wander, Geophys. Res. Lett., 10, Sabadini, R., and W. R. Peltier (1981), Pleistocene deglaciation and the Earth s rotation: Implications for mantle viscosity, Geophys. J. R. Astron. Soc., 66, Soler, T., and I. I. Mueller (1978), Global plate tectonics and the secular motion of the Pole, Bull. Godsique, 52, Spada, G., R. Sabadini, D. A. Yuen, and Y. Ricard (1992), Effects on postglacial rebound from the hard rheology in the transition zone, Geophys. J. Int., 109, Tapley, B., et al. (2005), GGM02: An improved Earth gravity field model from GRACE, J. Geod., 79, Tushingham, A. M., and W. R. Peltier (1991), Ice 3G: A new global model of late Pleistocene deglaciation based upon geophysical predictions of post glacial relative sea level change, J. Geophys. Res., 96, Vermeersen, L. L. A., and R. Sabadini (1996), Significance of the fundamental mantle relaxation mode in polar wander simulations, Geophys. J. Int., 127, F5 F9. Vermeersen, L. L. A., A. Fournier, and R. Sabadini (1997), Changes in rotation induced by Pleistocene ice masses with stratified analytical Earth models, J. Geophys. Res., 102(B12), Wu, P. (1978), The response of a Maxwell Earth to applied surface mass loads: Glacial isostatic adjustment, M.S. thesis, Dep. of Phys., Univ. of Toronto, Toronto, Ontario, Canada. Wu, P., and W. R. Peltier (1982), Viscous gravitational relaxation, Geophys. J. R. Astron. Soc., 70, Wu, P., and W. R. Peltier (1983), Glacial isostatic adjustment and the free air gravity anomaly as a constraint on deep mantle viscosity, Geophys. J. R. Astron. Soc., 74, Wu, P., and W. R. Peltier (1984), Pleistocene deglaciation and the Earth s rotation: A new analysis, Geophys. J. R. Astron. Soc., 76, Yuen, D. A., R. Sabadini, and E. Boschi (1982), Viscosity of the lower mantle as inferred from rotational data, J. Geophys. Res., 87, 10,745 10,762. A. Daradich, N. Gomez, and J. X. Mitrovica, Department of Physics, University of Toronto, 60 St. George St., Toronto, ON M5S 1A7, Canada. (amy.daradich@gmail.com; ngomez@physics.utoronto.ca; jxm@ physics.utoronto.ca) I. Matsuyama, Department of Earth and Planetary Science, University of California, 307 McCone Hall, Berkeley, CA 94720, USA. (isa@berkeley.edu) 10 of 10