Significance of the fundamental mantle rotational relaxation mode in polar wander simulations

Geophys. J. Int. (1996) 127, F5-F9 FAST-TRACK PAPER Significance of the fundamental mantle rotational relaxation mode in polar wander simulations L. L. A. Vermeersen* and R. Sabadini Dipartimento di Scienze della Terra. Sezione Geojisica, Universitci degli Studi di Milano, Via L. Cicognara 7,20129 Milano, Italy Accepted 1996 September 5. Received 1996 August 5; in original form 1996 June 5 INTRODUCTION A number of authors have reported during the past decade that studies on glacially induced TPW allow in general for multiple solutions for the lower-mantle viscosity if the TPW is known and all the other rheological, elastic and constitutional parameters are fixed [Fig. 20 of Yuen et al. (1986), Fig. 5 of Spada et al. (1992), and, most recently, Fig. 1 of Milne & Mitrovica ( 1996)l. These multiple-branch solutions are also commonly found in other geophysical signatures related to glacially induced solid-earth deformations such as postglacial rebound, free-air gravity anomalies, and changes in the non-tidal acceleration of the Earth. In a recent study (Peltier & Jiang 1996, Fig. 7), such multi-branch solutions for the lower-mantle viscosity are conspicuously absent; only solutions in which the lower mantle is slightly more viscous than the upper mantle are allowed. Although the theories described by Sabadini, Yuen & Boschi (1982) and Wu & Peltier (1984) appear to have a number of differences, Sabadini, Yuen & Boschi (1984) have shown that the formulations are equivalent to some extent. The proof of the equivalence of eqs (16) and (17) of Sabadini et al. (1984) for the secular rotation term is an important result in this respect. Another result mentioned in Sabadini et al. (1984) is *Now at: Department of Geodetic Science, University Stuttgart, Keplerstrasse 11, 70174 Stuttgart, Germany. SUMMARY In some studies on glacially induced true polar wander (TPW), the tidal-effective relaxation of the fundamental mantle mode (MO) is lacking. We show that this is caused by the deletion of the Chandler wobble in an early stage of the theory development to facilitate the retrieval of the rotational relaxation modes. We derive an analytical approximation formula for the MO rotational relaxation mode (including the Chandler wobble), which can be of practical value for TPW simulations with realistically stratified earth models. However, we point out that the contribution of the MO rotational relaxation mode has, to a high approximation, the same effect on secular TPW as the contribution from an elastic term in models that do not have the MO rotational relaxation mode. The two model approaches lead to the same polar wander results whenever the Chandler wobble is filtered from models in which the MO rotational relaxation mode is retained. Key words: Chandler wobble, mantle viscosity, TPW (true polar wander). that each of the load relaxation modes has a corresponding rotational relaxation mode. This correspondence remains an important issue, since in Wu & Peltier s (1984) theoretical development one of the corresponding modes, the MO rotational relaxation mode, is lacking (see also Table 1 of Sabadini et al. 1984). Peltier & Jiang (1996) extend the analysis of Wu & Peltier (1984) to consider a basic state in which the two principal equatorial moments of inertia are distinct. The extension is, however, unimportant in predictions of TPW (Peltier & Jiang 1996). More importantly, the extended theory retains the Chandler wobble filtering procedure of Wu & Peltier (1984) and this raises the possibility that the different behaviour of the TPW curves in Peltier & Jiang (1996) is due to the neglect of the MO rotational relaxation mode. We show explicitly that the absence of the MO rotational relaxation mode cannot be the cause of this difference in TPW-rate behaviour. As support for this result, note that the TPW predictions of Milne & Mitrovica (1996; Fig. l), which agree with earlier analyses (e.g. Spada et al. 1992), are based on theory which is equivalent to Wu & Peltier s (1984) approach. The present paper is structured as follows. First, we show that Wu & Peltier s (1984) approach to deleting the Chandler wobble acts to remove the MO rotational mode. We then proceed by deriving a new analytical approximation formula for the MO rotational relaxation mode, which incorporates the Chandler wobble frequency for a stratified earth as the 0 1996 RAS F5

F6 L. L. A. Vermeersen and R. Sabadini imaginary part. It will be shown, using numerical tests, that this approximation formula is extremely accurate. Finally, we show that the model approaches used in Sabadini et al. (1982) and Wu & Peltier (1984) lead to the same secular TPW results. OMISSION OF THE MO ROTATIONAL RELAXATION MODE We start with eq. (64) of Wu & Peltier (1984), reading in our notation with %(s) the complex-valued Laplace-transformed polar wander (s is the Laplace variable, which in general is complex-valued), i = fl, 6, the Chandler wobble frequency for a rigid earth, fdl the complex-valued Lapiace-transformed forcing, and M the total number of load relaxation modes j with (negative) inverse times sj. These sj are real-valued. The auxiliary variable xj is defined by in which kj' is the tidal-effective Love number associated with modej, and k; the fluid tidal-effective Love number. Note that the terms kj' have the same dimension as sj, while the fluid Love number k; is dimensionless, so that the terms xi have the same dimension as sj [note that our parameters kj' are the same as the parameters tj used in Wu & Peltier (1984)l. Inverse Laplace transformation of ( 1) requires finding the complex-valued roots of the denominator on the right-hand side of (1). At this stage, Wu & Peltier (1984) make the point that the unity term in the denominator of (1) may be neglected. This is only correct, however, if the imaginary parts of the roots have the same magnitude or are much smaller than the magnitude of the real parts. This is indeed the case for M - 1 roots, but it is not true for one root which has a much larger imaginary part than real part. This root turns out to be the rotational root that gives the relaxation of the fundamental mantle mode as the real part and the Chandler wobble as the imaginary part. The omission of this rotational root in Wu & Peltier (1984) is more apparent when we rewrite the term inside the square brackets of (1) as j=1 It is clear that if the first term of the numerator on the right-hand side of (3) were deleted, the numerator would be reduced from an expression of order M to an expression of order M - 1. This implies that one of the M load relaxation modes would have no rotational counterpart. Neglecting the first term of the numerator of (3) on the right-hand side of the equation is correct for the M - 1 roots for which the imaginary part is orders of magnitude smaller than the real part, as outlined by Wu & Peltier (1984). For these roots, the approximation k=l j#k is valid, being a purely real expression resulting in M - 1 real roots. These real roots constitute the M - 1 rotational inverse relaxation times associated with all modes except the MO mode. For a root with a large imaginary value, comparable in strength with the variables xj given by (2), an argument that the first term of the numerator on the right-hand sight of (3) is negligible with respect to the second term of this numerator is no longer valid. In fact, such a complex-valued root leads to a real part, being the MO rotational relaxation mode, which has the same order of magnitude as the other (real) M - 1 roots. This complex-valued mode, with the Chandler wobble frequency for a stratified earth as the imaginary part, has thus to be derived from the complete complex-valued equation M M M We will show in the next section that a highly accurate analytical approximation formula can be derived from (5) for the MO rotational relaxation mode. This approximation facilitates polar wander simulations on earth models with a large number of layers. Indeed, it is well known that complex root-finding procedures are numerically more difficult to apply and are less reliable than real-valued root-finding procedures. When an analytical formula can be obtained for the only complex root that has a non-negligible imaginary part, one can use root-finding procedures for real numbers using (4) instead of complex numbers using (5) in TPW calculations. Before deriving this formula, an example is given to illustrate the foregoing remarks. Table 1 gives the values for the densities and rigidities for an earth model consisting of an elastic lithosphere, viscoelastic shallow mantle, transition zone and lower mantle, and inviscid fluid core. The mantle has a uniform viscosity of loz1 Pa s. In Table 2, the nine inverse load relaxation times sj and the nine rotational relaxation roots aj are given for the five-layer earth model of Table 1. The complex rotational relaxation roots aj are determined by applying a complex root-finding procedure to (5). It is clear from Table 2 that eight roots have imaginary values that are negligible in strength, and that one root has a large imaginary value. This large imaginary value represents the Chandler wobble. The real value of this root is the MO rotational relaxation mode. This mode is not negligible at all; on the contrary, it is often the strongest relaxation mode, as is illustrated by the last two columns of Table 2. In Table 4 of Wu & Peltier (1984), where nine load relaxation and eight rotational relaxation roots are given, it is this MO rotational relaxation mode that is lacking. The small imaginary values of the other rotational relaxation roots in Table 2 indicate that these modes are also characterized by wobbles. One can easily prove from (5) that the imaginary Table 1. Density and rigidity values for the five-layer model. radial distance (km) density (kg m-3) rigidity (Pa) 6371-6250 6250-5951 5951-5701 5701-3480 3480-0 (4) 3184 6.02 x lo1' 3434 7.27 x 10'' 3857 1.06 x 10" 4878 2.19 x 10" 10 932 0

Fundamental mantle rotational relaxation mode F7 Table 2. Inverse load relaxation times sj, inverse rotational relaxation times aj, load Love numbers kj" and tidal-effective Love numbers kj' for the nine relaxation modes of the five-layer model of Table 1, with a uniform mantle viscosity of 102' Pas. The Lam6 parameter A is infinite. The labelling of the modes agrees with the labelling of Table 4 of Wu & Peltier (1984). Note that the MO mode is the strongest mode for both load relaxation and tidal-effective relaxation. mode j -sj (kyr-') --a, (kyr-l) kj" (kyr-') kj' (kyr-') M2 2.90 x 7.32 x -9.64 x 10-loi -1.45 x 7.17 x M1 1.29 x 1.06 x - 1.73 x 10-6i -7.01 x 7.72 x LO 1.09 x lo-' 1.19 x lo-' - 1.89 x 10-6i - 1.69 x 7.84 x co 4.50 x lo-' 1.02-1.50 x 10-4i - 1.41 x lo-' 8.08 x MO 2.02 1.95-5.10 x 103i -3.32 x lo-' 4.21 x lo-' Tl 2.48 2.38-2.19 x 10-5i -7.82 x 8.13 x lo-' T2 2.84 2.62-2.45 x low5i -2.80 x lo-' 3.73 x lo-' T3 3.56 3.47-2.60 x lo-'i -9.95 x lo-' 1.13 x lo-' T4 4.00 3.89-2.94 x 10-5i - 1.15 x lo-' 1.79 x lo-' parts of these modes are not equal to zero, thus excluding the possibility that these imaginary values are the result of numerical inaccuracies. This is interesting from a physical point of view, although the amplitudes of these wobbles are too small to be of any immediate significance. Filtering the Chandler wobble out after the rotational relaxation roots have been found is not the same as deleting the mode in which the Chandler wobble will appear before (5) is solved. One might thus think that the omission of the MO rotational relaxation mode has major consequences for the TPW simulations. However, eq. (79) of Wu & Peltier (1984) contains the extra term Dlf( t) inside the square brackets, which is only created when the number of rotational relaxation modes is one less than the number of load relaxation modes. It will be shown that this term contains approximately the same contribution as is found from the relaxation of the MO rotational relaxation mode after the Chandler wobble is filtered out. This equivalence will be pointed out and discussed after the analytical formula for the MO rotational relaxation mode is derived. ANALYTICAL FORMULA FOR THE MO ROTATIONAL RELAXATION MODE As shown in the last section, the numerator on the right-hand side of (3) has M - 1 solutions s = aj (j = 1,..., M - 1) for which the imaginary part can be neglected. These M - 1 solutions can be found by applying a real-valued root-finder procedure to (4). The root that contains the Chandler wobble as the imaginary part and the MO rotational relaxation mode as the real part must be solved from (5). If we split this root s = amo into its real and imaginary parts as amo = ar + ia,, then it is clear from, for example, Table 2 that JaII >> larl and (a,( >> lskl for all M load relaxation modes k. With this, the first term of (5) can be approximated by and M k= 1 (9) k=l k=l j#k Eq. (8) yields the expression for the imaginary part of the root as while (10) in (9) leads to and this can be reduced to M From (10) and (12) we thus have as the Mth complex-valued root of eq. (5) The real part of this root gives the rotational inverse relaxation time of the fundamental mantle mode MO, while the imaginary part gives the Chandler wobble frequency cro of the stratified model [compare also with Wu & Peltier's (1984) eq. (68)]:.._ M 00% c x k, k= 1 or, with (2): j= 1 k=l while the second term of (5) can be approximated by M M 1 xk n ( k=l s-sj) j#k k=l k=l j#k The sum of (6) and (7) has terms which all contain either im or i'-'. Irrespective of the value of M, the terms must thus obey the relations For the five-layer model of Table 1, the complex root as given by (13) has the value amo = - 1.9467033513 -t 5096.9417% kyr-', while from a complex root-finder applied to (5) with quadrupole precision (COMPLEX* 16) the root is determined as having the value amo = - 1.9467033558 + 5096.942041 kyr-'. The analytical formula (13) gives thus an extremely accurate approximation of amo.

F8 L. L. A. Vermeersen and R. Sabadini UNIFICATION OF THE TWO APPROACHES Formulas (13) and (15) are not only useful in the model approach following the method of Sabadini et al. (1982), but also prove helpful in establishing the equivalence with the model approach of Wu & Peltier (1984) for secular TPW. If we consider eq. (79) of Wu & Peltier (1984), then the formulation of Sabadini et al. (1982) alters this equation by the following four points (J. X. Mitrovica, personal communication). (1) The term D, of Wu & Peltier s eq. (79) becomes zero. This term D, is an elastic term which arises from the first term (being 1) on the right-hand side of Wu & Peltier s eq. (75). The term D, is a direct consequence of the fact that there is one rotational relaxation mode less than the number of load relaxation modes. This term of unity would be absent if there were N rotational relaxation modes corresponding with the N load relaxation modes inside the square brackets of the last line in Wu & Peltier s eq. (74). (2) The term D2 becomes - ia,d, in Wu & Peltier s eq. (79), but at the same time the term D2 is divided by the extra root -amo of our eq. (13) in Wu & Peltier s eq. (80). As the imaginary part of eq. (13) is orders of magnitude larger than the real part, the effect on Wu & Peltier s eq. (80) is that D, is to a high approximation divided by -h0, so that the total effect on Wu & Peltier s eq. (79) is that the original term D2 remains unchanged (note that our roots ai have the opposite sign of Wu & Peltier s corresponding roots Ai). (3) The terms Ei in Wu 8c Peltier s eq. (79) become - iaoei, but at the same time the terms Ei in Wu & Peltier s eq. (80) are to a high approximation divided by the extra term - ia, which is again a consequence of the fact that the extra rotational relaxation mode has an imaginary term in our eq. (13) being orders of magnitude larger than the real part. The net effect is thus, just as in point (2), that the terms Ei remain to a high approximation unchanged in Wu & Peltier s eq. (79). (4) The final change in Wu & Peltier s eq. (79) concerns the addition of the extra term E,f* exp(a,,t). This term causes the wobble. It turns out (J. X. Mitrovica, personal communication) that if this extra term is averaged over time, i.e. when the wobble is filtered out, then the contribution that remains is numerically equal to D, to a high approximation. The net effect of points (1)-(4) is thus that the elastic term D1 in Wu 8c Peltier s (1984) theory contains the signal that is contributed by Sabadini et al. s (1982) MO rotational relaxation mode when the Chandler wobble is filtered out. Together with points (2) and (3) one thus can conclude that, to a high approximation, the theoretical developments of Sabadini et al. (1982) and Wu & Peltier (1984) lead to the same results for secular TPW simulations. A corollary of this result is that the discrepancy in the glacially induced TPW results between Peltier & Jiang (1996) and other authors, which was briefly discussed in the Introduction, is due to the application of the theory. TPW INDUCED BY PLEISTOCENE GLACIAL CYCLES Fig. 1 shows the TPW rate as a function of lower-mantle viscosity for three cases: a five-layer earth model with parameters given in Table 1, a 31-layer model and a 56-layer model. All three models have volume-averaged properties of PREM (Dziewonski & Anderson 1981) and a value for the I I I I I I LOG,,( Lower mantle viscosity) Figure 1. TPW velocity as a function of the lower-mantle viscosity. The horizontal band marks the error bounds of the present-day TPW datum given by McCarthy & Luzum (1996). 0 1996 RAS, GJI 127, FSF9

Fundamental mantle rotational relaxation mode F9 upper-mantle viscosity of 10 Pas. The ice model that has been used is based on ICE-3G of Tushingham & Peltier (1991). The ICE-3G model gives the deglaciation of the last Pleistocene cycle by a set of decrements, starting at 18 kyr before present and ending at 5 kyr before present. After 5 kyr before present the ice loads remain constant until present. We have used ICE-3G for times between 18 and 5 kyr before present, and extended it with seven glacial sawtooth pre-cycles and a linear glaciation phase, which ends at 18 kyr before present. Each cycle is connected to its previous and following one, and the end of the seventh cycle is the beginning of the linear glaciation phase which ends at 18 kyr before present. The seven precycles each consist of a 90kyr linear growth phase and a 10 kyr linear decay phase. The minimum amount of ice is the same as in ICE-3G at 5 kyr before present, while the maximum amount of ice is the same as in ICE-3G at 18 kyr before present. The TPW rates are determined 5 kyr after ICE3G ends, i.e. the rates are determined at present. The most recent observed value of the present-day TPW comes from McCarthy & Luzum (1996). They determine a mean TPW velocity during the period from 1899 to 1994 of 0.925 0.022 deg Myr-l. The area between the error bounds of this TPW velocity datum is shaded in Fig. 1. The form of the curves of Fig. 1, which exhibit a clear nonmonotonic behaviour, are consistent with independent calculations by Yuen et al. (1986), Spada et a/. (1992) and Milne & Mitrovica (1996). They are, in contrast, in disagreement with the recent predictions of Peltier & Jiang (1996). Fig. 1 illustrates several interesting points which we will discuss more fully in a companion article (Vermeersen, Fournier & Sabadini 1996). As an example, the number of layers used to discretize the viscoelastic earth model can influence the accuracy of the predictions. However, the fact that the 31- and 56-layer models virtually cover each other confirms the results on saturated limit behaviour for stratified models in Vermeersen & Sabadini (1996b). Also, the curves of Fig. 1 show TPW rates which are significantly higher than those of Spada et al. (1992). The reason for this is that Spada et al. (1992) adopt a five-layer fixed boundary contrast model rather than the volume-averaged models we use here (see also Vermeersen & Sabadini 1996b). In the case of lower-mantle viscosities lower than lo2 Pa s, our predictions are significantly higher than those of Milne & Mitrovica (1996). This can be attributed to the fact that we have employed incompressible rheologies to produce Fig. 1, whereas Milne & Mitrovica (1996) have used compressible rheologies (Vermeersen & Sabadini 1996a; J. X. Mitrovica, personal communication). Apart from these differences in modelling, there are a number of reasons that make it difficult to establish a unique relationship between the viscosity of the lower mantle and TPW. For example, uncertainties in lithospheric thickness and uppermantle viscosity will corrupt any such inference and, furthermore, it is highly questionable whether glacial processes are the sole driver of the present-day TPW. For example, tectonic processes such as subduction (e.g. Ricard, Sabadini & Spada 1992) and mountain building (e.g. Vermeersen et al. 1994) have been shown to be potentially effective contributors to the present-day TPW. CONCLUDING REMARK A first step towards unifying the approaches by Sabadini et al. (1982) and Wu & Peltier (1984) was taken in Sabadini et a/. (1984). The present paper completes this unification, in the sense that the theories are shown to be equivalent to a high approximation whenever the Chandler wobble is filtered out. The complex-valued approximation formula (13) for the strength of the MO rotational relaxation mode can be helpful in models that follow the procedures outlined in Sabadini et at. (1982), either with the Chandler wobble included or with the Chandler wobble filtered out. ACKNOWLEDGMENTS This work was financially supported by the European Space Agency by contract PERS/mp/4178 and by the Italian Space Agency by grant AS1 95-RS-153. Alexandre Fournier is acknowledged for assistance in producing the figure. We are grateful to Jerry Mitrovica for many discussions, benchmark comparisons, and a thorough review of the original manuscript. Detlef Wolf is thanked for thoughtful remarks. REFERENCES Dziewonski, A.M. & Anderson, D.L., 1981. Preliminary reference Earth model, Phys. Earth planet. Inter., 25, 297-356. McCarthy, D.D. & Luzum, B.J., 1996. Path of the mean rotational pole from 1899 to 1994, Geophys. J. Int.. 125, 623-629. Milne, G.A. & Mitrovica, J.X., 1996. 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