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The Brillouin sphere is defined as the smallest sphere, centered at the origin of the geocentric coordinate system, that incorporates all the condensed matter composing the planet. The Brillouin sphere touches the Earth at a single point, and the radial line that begins at the origin and passes through that point is called the singular radial line. For about 60 years there has been a persistent anxiety about whether or not a spherical harmonic (SH) expansion of the external gravitational potential, V, will converge beneath the Brillouin sphere. Recently, it was proven that the probability of such convergence is zero. One of these proofs provided an asymptotic relation, called Costin's formula, for the upper bound, EN, on the absolute value of the prediction error, eN, of a SH series model, V N (θ , λ , r) , truncated at some maximum degree, N = n max . When the SH series is restricted to (or projected onto) a particular radial line, it reduces to a Taylor series (TS) in 1 / r . Costin's formula is E N ≃ B N − b (R / r) N , where R is the radius of the Brillouin sphere. This formula depends on two positive parameters: b, which controls the decay of error amplitude as a function of N when r is fixed, and a scale factor B. We show here that Costin's formula derives from a similar asymptotic relation for the upper bound, An on the absolute value of the TS coefficients, an, for the same radial line. This formula, A n ≃ K n − k , depends on degree, n, and two positive parameters, k and K, that are analogous to b and B. We use synthetic planets, for which we can compute the potential, V, and also the radial component of gravitational acceleration, g r = ∂ V / ∂ r , to hundreds of significant digits, to validate both of these asymptotic formulas. Let superscript V refer to asymptotic parameters associated with the coefficients and prediction errors for gravitational potential, and superscript g to the coefficients and predictions errors associated with gr. For polyhedral planets of uniform density we show that b V = k V = 7 / 2 and b g = k g = 5 / 2 almost everywhere. We show that the frequency of oscillation (around zero) of the TS coefficients and the series prediction errors, for a given radial line, is controlled by the geocentric angle, α, between that radial line and the singular radial line. We also derive useful identities connecting K V , B V , K g , and Bg. These identities are expressed in terms of quotients of the various scale factors. The only other quantities involved in these identities are α and R. The phenomenology of 'series divergence' and prediction error (when r < R) can be described as a function of the truncation degree, N, or the depth, d, beneath the Brillouin sphere. For a fixed r ⩽ R , as N increases from very low values, the upper error bound EN shrinks until it reaches its minimum (best) value when N reaches some particular or optimum value, N opt . When N > N opt , prediction error grows as N continues to increase. Eventually, when N ≫ N opt , prediction errors increase exponentially with rising N. If we fix the value of N and allow R / r to vary, then we find that prediction error in free space beneath the Brillouin sphere increases exponentially with depth, d, beneath the Brillouin sphere. Because b g = b V − 1 everywhere, divergence driven prediction error intensifies more rapidly for gr than for V, both in terms of its dependence on N and d. If we fix both N and d, and focus on the 'lateral' variations in prediction error, we observe that divergence and prediction error tend to increase (as does B) as we approach high-amplitude topography. [ABSTRACT FROM AUTHOR]

Greenland's main outlet glaciers have more than doubled their contribution to global sea level rise over the last decade. Recent work has shown that Greenland's mass loss is still increasing. Here we show that the ice loss, which has been well-documented over southern portions of Greenland, is now spreading up along the northwest coast, with this acceleration likely starting in late 2005. We support this with two lines of evidence. One is based on measurements from the Gravity Recovery and Climate Experiment (GRACE) satellite gravity mission, launched in March 2002. The other comes from continuous Global Positioning System (GPS) measurements from three long-term sites on bedrock adjacent to the ice sheet. The GRACE results provide a direct measure of mass loss averaged over scales of a few hundred km. The GPS data are used to monitor crustal uplift caused by ice mass loss close to the sites. The GRACE results can be used to predict crustal uplift, which can be compared with the GPS data. In addition to showing that the northwest ice sheet margin is now losing mass, the uplift results from both the GPS measurements and the GRACE predictions show rapid acceleration in southeast Greenland in late 2003, followed by a moderate deceleration in 2006. Because that latter deceleration is weak, southeast Greenland still appears to be losing ice mass at a much higher rate than it was prior to fall 2003. In a more general sense, the analysis described here demonstrates that GPS uplift measurements can be used in combination with GRACE mass estimates to provide a better understanding of ongoing Greenland mass loss; an analysis approach that will become increasingly useful as long time spans of data accumulate from the 51 permanent GPS stations recently deployed around the edge of the ice sheet as part of the Greenland GPS Network (GNET). Copyright © 2010 by the American Geophysical Union.

The Earth's largest earthquakes and tsunamis are usually caused by thrust-faulting earthquakes on the shallow part of the subduction interface between two tectonic plates, where stored elastic energy due to convergence between the plates is rapidly released (1,2). The tsunami that devastated the Samoan and northern Tongan islands on 29 September 2009 was preceded by a globally recorded magnitude-8 normal-faulting earthquake in the outer-rise region, where the Pacific plate bends before entering the subduction zone. Preliminary interpretation suggested that this earthquake was the source of the tsunami (3). Here we show that the outer-rise earthquake was accompanied by a nearly simultaneous rupture of the shallow subduction interface, equivalent to a magnitude-8 earthquake, that also contributed significantly to the tsunami. The subduction interface event was probably a slow earthquake with a rise time of several minutes that triggered the outer-rise event several minutes later. However, we cannot rule out the possibility that the normal fault ruptured first and dynamically triggered the subduction interface event. Our evidence comes from displacements of Global Positioning System stations and modelling of tsunami waves recorded by ocean-bottom pressure sensors, with support from seismic data and tsunami field observations. Evidence of the subduction earthquake in global seismic data is largely hidden because of the earthquake's slow rise time or because its ground motion is disguised by that of the normal-faulting event. Earthquake doublets where subduction interface events trigger large outer-rise earthquakes have been recorded previously (4), but this is the first well-documented example where the two events occur so closely in time and the triggering event might be a slow earthquake. As well as providing information on strain release mechanisms at subduction zones, earthquakes such as this provide a possible mechanism for the occasional large tsunamis generated at the Tonga subduction zone (5), where slip between the plates is predominantly aseismic (6)., The majority of slip in great subduction earthquakes occurs at 'asperities', that is, patches of the subduction interface that have remained tightly coupled together for some time while the surrounding [...]

Net westward rotation of the lithosphere relative to the underlying mantle is a controversial phenomenon first attributed to tidal effects, and later to the dynamics of mantle convection. In spite of a number of independent geological and geophysical arguments for westward tectonic drift, this phenomenon has received little recent attention. We suggest that this differential rotation is a combined effect of three processes: (1) tidal torques act on the lithosphere generating a westerly directed torque decelerating Earth's spin; (2) the downwelling of the denser material toward the bottom of the mantle and in the core slightly decreases the moment of inertia and speeds up Earth's rotation, only partly counterbalancing the tidal drag; (3) thin (3-30 km) layers of very low viscosity hydrate channels occur in the asthenosphere. It is suggested that shear heating and the mechanical fatigue self-perpetuate one or more channels of this kind, which provide the necessary decoupling zone of the lithosphere. Keywords: Earth's rotation, westward drift, lithosphere, asthenosphere viscosity, decoupling.

We derive exact asymptotic solutions for the static deformation due to a concentrated or point-like dislocation in a spherical, layered, elastic, isotropic and self-gravitating Earth. The exact asymptotic solutions are quite general and can provide the dislocation Love numbers on the Earth's surface, near the dislocation or 'source', and close to any layer interface or boundary. We also discuss the special case where both the source and field points are located on the Earth's surface. We compare our exact asymptotic solutions with previous results obtained from the analytical dual variable and position (DVP) method via curve fitting. Our comparison confirms that the analytical DVP results converge to the exact asymptotic solutions. These new exact asymptotic solutions are particularly helpful when evaluating slowly convergent series of Green's functions using a Kummer transformation, anywhere within the layered Earth, especially for field points located very close to the point dislocation or source. [ABSTRACT FROM AUTHOR]