Forced nutations of the Earth: Influence of inner core dynamics: 2. Numerical results and comparisons

Abstract

We apply the theory developed in Paper 1 (Mathews et al., this issue), which includes the solid inner core explicitly in the dynamical equations, to obtain the eigenfrequencies and other characteristics of the Earth's nutational normal modes as well as the amplitudes of forced nutations at various tidal frequencies, for two commonly used Earth models, 1066A and the Preliminary Reference Earth Model (PREM). We also make an evaluation of various procedures for taking account of known deviations of the Earth from models, notably in the dynamical ellipticity e for which the two models yield values which are over 1% smaller than the value e* deduced from the precession constant. On adopting the procedure of simply replacing e by e* in the equations of our theory, the values obtained for some of the nutation amplitudes for model 1066A differ significantly from the corresponding results of Wahr (1981b). The largest of the differences, which occur in the prograde semiannual, retrograde 18.6 year, and retrograde annual nutation terms, amount to −0.59, 0.35, and −0.25 milliarcseconds (mas), respectively, while the standard errors in the very long baseline interferometry (VLBI) determinations are now only about 0.04 mas except in the long period terms. The difference in the procedures used to take account of the discrepancy between e and e* contributes −0.56, 0.81, and −0.17 mas, respectively, to the above‐noted differences. For the purpose of comparison with VLBI‐observed data, we use the results for a “modified PREM,” defined by a set of Earth model parameters which differ from those of PREM only in having e* for the dynamical ellipticity of the Earth as a whole and a modified value for the dynamical ellipticity eƒ of the fluid outer core. The amplitudes computed for this model, with corrections applied for the effects of ocean tides and mantle anelasticity, are in generally satisfactory agreement with observed values, when the modified eƒ is determined by matching the theoretical and observed values for the retrograde annual term. (The modified eƒ is 0.002665, about 4.6% higher than in PREM, equivalent to an increase, relative to PREM, of about 430 meters in the difference between the equatorial and the polar radii of the core‐mantle boundary. We find that contributions from inner‐core dynamics to the prograde semiannual and annual, and the retrograde 18.6 year and annual terms, recomputed for modified PREM, amount to −0.09, 0.03, −0.36, and −0.09 mas, respectively.) The largest residual remaining, other than in the long‐period terms which still have an uncertainty of about 1 mas, is −0.25 mas in the prograde fortnightly amplitude. Consideration of possible sources of the discrepancies is facilitated by a resonance expansion of the amplitude of forced nutations, as a function of frequency, normalized relative to that for a rigid Earth model. We also provide tables which exhibit the sensitivities of various relevant quantities (the eigenfrequencies and the coefficients which appear in the resonance expansion, as well as t...

Consistent estimates of the dynamic figure parameters of the earth

“…where Case I: G = 6.67384 × 10 −11 m 3 kg −1 s −2 and e = 3.2845479 × 10 −3 are adopted Case II: G = 6.67384 × 10 −11 m 3 kg −1 s −2 and e = 3.2845161 × 10 −3 are adopted Case III: G = 6.67259 × 10 −11 m 3 kg −1 s −2 and e = 3.2845479 × 10 −3 are adopted Case VI: G = 6.67259 × 10 −11 m 3 kg −1 s −2 and e = 3.2845161 × 10 −3 are adopted The four cases are the same as those in Table 8 All the listed values of MHB2000 Earth model are from Mathews et al (1991) except for those of e and e f , which are fitted from the VLBI nutation data by Mathews et al (2002) Using Eqs. (21) and (22), as well as the potential coefficients in Table 2, we can derive the orientations of the principal axes as listed in Table 7.…”

Shapes of two‐layer models of rotating planets

“…However, the observed shape of the coremantle boundary includes a nonhydrostatic component of about 500 m excess equatorial radius. Also, the observed ellipticity of the core is a dynamical ellipticity [Mathews et al, 1991] not an actual geometric or shape ellipticity. The two ellipticities are approximately equivalent, at least for a constant density core.…”

The Earth’s variable Chandler wobble

“…Mathews et al (2002) predicts a resonance due to the inner core with a theoretical period of ∼2400 days but with an amplitude expected to be insignificant for our purpose. The resonant frequency σ c , or eigenfrequency, can be expressed in terms of the whole Earth's dynamical ellipticity e = (C − A)/C and the Love number k, with additional contributions from anelasticity, and ocean loading and currents to the mantle deformability (see, e.g., Sasao et al 1980;Mathews et al 1991Mathews et al , 2002. The theoretical computation of σ c leads to a period of 430.3 days and a quality factor Q = 88 (Mathews et al 2002).…”