Precession of a Freely Rotating Rigid Body. Inelastic Relaxation in the Vicinity of Poles

When a solid body is freely rotating at an angular velocity ${\bf \Omega}$, the ellipsoid of constant angular momentum, in the space $\Omega_1, \Omega_2, \Omega_3$, has poles corresponding to spinning about the minimal-inertia and maximal-inertia axes. The first pole may be considered stable if we neglect the inner dissipation, but becomes unstable if the dissipation is taken into account. This happens because the bodies dissipate energy when they rotate about any axis different from principal. In the case of an oblate symmetrical body, the angular velocity describes a circular cone about the vector of (conserved) angular momentum. In the course of relaxation, the angle of this cone decreases, so that both the angular velocity and the maximal-inertia axis of the body align along the angular momentum. The generic case of an asymmetric body is far more involved. Even the symmetrical prolate body exhibits a sophisticated behaviour, because an infinitesimally small deviation of the body's shape from a rotational symmetry (i.e., a small difference between the largest and second largest moments of inertia) yields libration: the precession trajectory is not a circle but an ellipse. In this article we show that often the most effective internal dissipation takes place at twice the frequency of the body's precession. Applications to precessing asteroids, cosmic-dust alignment, and rotating satellites are discussed.Comment: 47 pages, 1 figur

Effects of pressure on diffusion and vacancy formation in MgO from non-empirical free-energy integrations

The free energies of vacancy pair formation and migration in MgO were computed via molecular dynamics using free-energy integrations and a non-empirical ionic model with no adjustable parameters. The intrinsic diffusion constant for MgO was obtained at pressures from 0 to 140 GPa and temperatures from 1000 to 5000 K. Excellent agreement was found with the zero pressure diffusion data within experimental error. The homologous temperature model which relates diffusion to the melting curve describes well our high pressure results within our theoretical framework.Comment: 4 pages, latex, 1 figure, revtex, submitted to PR

The elastic constants of MgSiO3 perovskite at pressures and temperatures of the Earth's mantle

The temperature anomalies in the Earth's mantle associated with thermal convection1 can be inferred from seismic tomography, provided that the elastic properties of mantle minerals are known as a function of temperature at mantle pressures. At present, however, such information is difficult to obtain directly through laboratory experiments. We have therefore taken advantage of recent advances in computer technology, and have performed finite-temperature ab initio molecular dynamics simulations of the elastic properties of MgSiO3 perovskite, the major mineral of the lower mantle, at relevant thermodynamic conditions. When combined with the results from tomographic images of the mantle, our results indicate that the lower mantle is either significantly anelastic or compositionally heterogeneous on large scales. We found the temperature contrast between the coldest and hottest regions of the mantle, at a given depth, to be about 800K at 1000 km, 1500K at 2000 km, and possibly over 2000K at the core-mantle boundary.Comment: Published in: Nature 411, 934-937 (2001

Tidal torques. A critical review of some techniques

We point out that the MacDonald formula for body-tide torques is valid only in the zeroth order of e/Q, while its time-average is valid in the first order. So the formula cannot be used for analysis in higher orders of e/Q. This necessitates corrections in the theory of tidal despinning and libration damping. We prove that when the inclination is low and phase lags are linear in frequency, the Kaula series is equivalent to a corrected version of the MacDonald method. The correction to MacDonald's approach would be to set the phase lag of the integral bulge proportional to the instantaneous frequency. The equivalence of descriptions gets violated by a nonlinear frequency-dependence of the lag. We explain that both the MacDonald- and Darwin-torque-based derivations of the popular formula for the tidal despinning rate are limited to low inclinations and to the phase lags being linear in frequency. The Darwin-torque-based derivation, though, is general enough to accommodate both a finite inclination and the actual rheology. Although rheologies with Q scaling as the frequency to a positive power make the torque diverge at a zero frequency, this reveals not the impossible nature of the rheology, but a flaw in mathematics, i.e., a common misassumption that damping merely provides lags to the terms of the Fourier series for the tidal potential. A hydrodynamical treatment (Darwin 1879) had demonstrated that the magnitudes of the terms, too, get changed. Reinstating of this detail tames the infinities and rehabilitates the "impossible" scaling law (which happens to be the actual law the terrestrial planets obey at low frequencies).Comment: arXiv admin note: sections 4 and 9 of this paper contain substantial text overlap with arXiv:0712.105

Thermal Nature of Mantle Upwellings Below the Ibero-Western Maghreb Region Inferred From Teleseismic Tomography

©2019. American Geophysical Union. All Rights Reserved. Independent models of P wave and S wave velocity anomalies in the mantle derived from seismic tomography help to distinguish thermal signatures from those of partial melt, volatiles, and compositional variations. Here we use seismic data from SW Europe and NW Africa, spanning the region between the Pyrenees and the Canaries, in order to obtain a new S-SKS relative arrival-time tomographic model of the upper mantle below Iberia, Western Morocco, and the Canaries. Similar to previous P wave tomographic results, the S wave model provides evidence for (1) subvertical upper-mantle low-velocity structures below the Canaries, Atlas Ranges, and Gibraltar Arc, which are interpreted as mantle upwellings fed by a common lower-mantle source below the Canaries; and (2) two low-velocity anomalies below the eastern Rif and Betics that we interpret as the result of the interaction between quasi-toroidal mantle flow induced by the Gibraltar slab and the mantle upwelling behind it. The analysis of teleseismic P wave and S wave arrival-time residuals and the conversion of the low-velocity anomalies to temperature variations suggest that the upwellings in the upper mantle below the Canaries, Atlas Ranges, and Gibraltar Arc system may be solely thermal in nature, with temperature excesses in the range ~100–350 °C. Our results also indicate that local partial melting can be present at lithospheric depths, especially below the Atlas Ranges. The locations of thermal mantle upwellings are in good agreement with those of thinned lithosphere, moderate to high heat-flow measurements, and recent magmatic activity at the surface

Stress, strain, and B‐type olivine fabric in the fore‐arc mantle: Sensitivity tests using high‐resolution steady‐state subduction zone models

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/94925/1/jgrb15022.pd

Bodily tides near spin-orbit resonances

Spin-orbit coupling can be described in two approaches. The method known as "the MacDonald torque" is often combined with an assumption that the quality factor Q is frequency-independent. This makes the method inconsistent, because the MacDonald theory tacitly fixes the rheology by making Q scale as the inverse tidal frequency. Spin-orbit coupling can be treated also in an approach called "the Darwin torque". While this theory is general enough to accommodate an arbitrary frequency-dependence of Q, this advantage has not yet been exploited in the literature, where Q is assumed constant or is set to scale as inverse tidal frequency, the latter assertion making the Darwin torque equivalent to a corrected version of the MacDonald torque. However neither a constant nor an inverse-frequency Q reflect the properties of realistic mantles and crusts, because the actual frequency-dependence is more complex. Hence the necessity to enrich the theory of spin-orbit interaction with the right frequency-dependence. We accomplish this programme for the Darwin-torque-based model near resonances. We derive the frequency-dependence of the tidal torque from the first principles, i.e., from the expression for the mantle's compliance in the time domain. We also explain that the tidal torque includes not only the secular part, but also an oscillating part. We demonstrate that the lmpq term of the Darwin-Kaula expansion for the tidal torque smoothly goes through zero, when the secondary traverses the lmpq resonance (e.g., the principal tidal torque smoothly goes through nil as the secondary crosses the synchronous orbit). We also offer a possible explanation for the unexpected frequency-dependence of the tidal dissipation rate in the Moon, discovered by LLR