On the elastodynamics of rotating planets

Equations of motion are derived for (visco)elastic, self-gravitating, and variably-rotating planets. The equations are written using a decomposition of the elastic motion that separates the body's elastic deformation from its net translational and rotational motion as far as possible. This separation is achieved by introducing degrees of freedom that represent the body's rigid motions; it is made precise by imposing constraints that are physically motivated and should be practically useful. In essence, a Tisserand frame is introduced exactly into the equations of solid mechanics. The necessary concepts are first introduced in the context of a solid body, motivated by symmetries and conservation laws, and the corresponding equations of motion are derived. Next, it is shown how those ideas and equations of motion can readily be extended to describe a layered fluid--solid body. A possibly new conservation law concerning inviscid fluids is then stated. Thereafter the equilibria and linearisation of the fluid--solid equations of motion are discussed, along with new equations for use within normal-mode coupling calculations and other Galerkin methods. Finally, the extension of these ideas to the description of multiple, interacting fluid--solid planets is qualitatively discussed

Particle relabelling transformations in elastodynamics

The motion of a self-gravitating hyperelastic body is described through a time-dependent mapping from a reference body into physical space, and its material properties are determined by a referential density and strain-energy function defined relative to the reference body. Points within the reference body do not have a direct physical meaning, but instead act as particle labels that could be assigned in different ways. We use Hamilton's principle to determine how the referential density and strain-energy functions transform when the particle labels are changed, and describe an associated ‘particle relabelling symmetry’. We apply these results to linearized elastic wave propagation and discuss their implications for seismological inverse problems. In particular, we show that the effects of boundary topography on elastic wave propagation can be mapped exactly into volumetric heterogeneity while preserving the form of the equations of motion. Several numerical calculations are presented to illustrate our results

On the parametrization of equilibrium stress fields in the Earth

A new method for parametrizing the possible equilibrium stress fields of a laterally heterogeneous earth model is described. In this method a solution of the equilibrium equations is first found that satisfies some desirable physical property. For example, we show that the equilibrium stress field with smallest norm relative to a given inner product can be obtained by solving a static linear elastic boundary value problem. We also show that the equilibrium stress field whose deviatoric component has smallest norm with respect to a given inner product can be obtained by solving a steady-state incompressible viscous flow problem. Having found such a solution of the equilibrium equations, all other solutions can be written as the sum of this equilibrium stress field and a divergence-free stress tensor field whose boundary tractions vanish. Given n divergence-free and traction-free tensor fields, we then obtain a simple n-dimensional parametrization of equilibrium stress fields in the earth model. The practical construction of such divergence- and traction-free tensor fields in the mantle of a spherically symmetric reference earth model is described using generalized spherical harmonics

Calculation of seismic displacement fields in self-gravitating earth models—applications of minors vectors and symplectic structure

An account is given of the minor vector method that allows for the stable numerical integration of the systems of linear ordinary differential equations occurring in a number of geophysical problems. In particular, new results are presented that allow for the application of the method to the solution of 6-D inhomogeneous boundary value problems, such as those encountered in the calculation of seismic displacement fields in spherically symmetric, self-gravitating earth models. In addition, the symplectic structure possessed by many of the ordinary differential equations of interest is described. It is shown how this structure can be used to simplify the numerical implementation of the minor vector method and also to concisely derive a number of theoretical results about the eigenfrequencies and eigenfunctions of a linearly anelastic earth model

Global dynamic topography observations reveal limited influence of large-scale mantle flow

Convective circulation of the Earth’s mantle maintains some fraction of surface topography that varies with space and time. Most predictive models show that this dynamic topography has peak amplitudes of about ±2km, dominated by wavelengths of 10⁴km. Here, we test these models against our comprehensive observational database of 2,120 spot measurements of dynamic topography that were determined by analysing oceanic seismic surveys. These accurate measurements have typical peak amplitudes of ±1km and wavelengths of approximately 10³km, and are combined with limited continental constraints to generate a global spherical harmonic model whose robustness has been carefully tested and benchmarked. Our power spectral analysis reveals significant discrepancies between observed and predicted dynamic topography. At longer wavelengths (such as 10⁴km), observed dynamic topography has peak amplitudes of about ±500m. At shorter wavelengths (such as 10³km), significant dynamic topography is still observed. We show that these discrepancies can be explained if short-wavelength dynamic topography is generated by temperature-driven density anomalies within a sub-plate asthenospheric channel. Stratigraphic observations from adjacent continental margins show that these dynamic topographic signals evolve quickly with time. More rapid temporal and spatial changes in vertical displacement of the Earth’s surface have direct consequences for fields as diverse as mantle flow, oceanic circulation and long-term climate change

A non-perturbative method for gravitational potential calculations within heterogeneous and aspherical planets

We present a numerically exact method for calculating the internal and external gravitational potential of aspherical and heterogeneous planets. Our approach is based on the transformation of Poisson’s equation into an equivalent equation posed on a spherical computational domain. This new problem is solved in an efficient iterative manner based on a hybrid pseudospectral/spectral element discretization. The main advantage of our method is that its computational cost reflects the planet’s geometric and structural complexity, being in many situations only marginally more expensive than boundary perturbation theory. Several numerical examples are presented to illustrate the method’s efficacy and potential range of applications

Reciprocity and sensitivity kernels for sea level fingerprints

Reciprocity theorems are established for the elastic sea level fingerprint problem including rotational feedbacks. In their simplest form, these results show that the sea level change at a location x due to melting a unit point mass of ice at x' is equal to the sea level change at x' due to melting a unit point mass of ice at x. This identity holds irrespective of the shoreline geometry or of lateral variations in elastic Earth structure. Using the reciprocity theorems, sensitivity kernels for sea level and related observables with respect to the ice load can be readily derived. It is notable that calculation of the sensitivity kernels is possible using standard fingerprint codes, though for some types of observable a slight generalisation to the fingerprint problem must be considered. These results are of use within coastal hazard assessment and have a range of potential applications within studies of modern-day sea level change.Comment: Paper submitted to Geophysical Journal Internationa

Quantifying the sensitivity of post-glacial sea level change to laterally varying viscosity

We present a method for calculating the derivatives of measurements of glacial isostatic adjustment (GIA) with respect to the viscosity structure of the Earth and the ice sheet history. These derivatives, or kernels, quantify the linearised sensitivity of measurements to the underlying model parameters. The adjoint method is used to enable efficient calculation of theoretically exact sensitivity kernels within laterally heterogeneous earth models that can have a range of linear or non-linear viscoelastic rheologies. We first present a new approach to calculate GIA in the time domain, which, in contrast to the more usual formulation in the Laplace domain, is well suited to continuously varying earth models and to the use of the adjoint method. Benchmarking results show excellent agreement between our formulation and previous methods. We illustrate the potential applications of the kernels calculated in this way through a range of numerical calculations relative to a spherically symmetric background model. The complex spatial patterns of the sensitivities are not intuitive, and this is the first time that such effects are quantified in an efficient and accurate manner.NERC studentship for the first autho

An extended ice-age sea-level equation: incorporating water flux across sills

We present a generalized theory governing gravitationally self-consistent, spatio-temporal sea-level changes within an ocean-plus-lake system that is intermittently connected by water mass flux across a sill. Our expressions for the change in sea level (defined as the difference in height of the sea-surface equipotential relative to the solid surface) hold for any Earth model, and easily incorporate effects of viscoelastic deformation of the solid Earth and perturbations in both the gravitational field and rotation vector (as is now standard in ice-age sea-level calculations). In its most general form, the theory also includes an exact treatment of the evolving shoreline position in both water bodies. Our formalism involves three cases: (1) one global ocean, in which mass transfer may occur between ice sheets and the global ocean; (2) an ocean and lake separated by an exposed sill, in which mass transfer may occur between ice sheets and the global ocean, and between the ocean and lake via evaporative flux and (3) transitional phases between these two states, when the ocean surface reaches the height of the sill from below (i.e. the sill is breached) or above (the sill is exposed). We illustrate the new theory using examples from the Black Sea flooding during the last deglacial phase (∼10 ka) and sea-level fall in the Mediterranean Sea during the Messinian Salinity Crisis (5.96–5.33 Ma). These examples demonstrate the importance of including the geophysical feedbacks associated with sea-level change in an isolated basin in the dynamics of flooding and desiccation

Percutaneous mitral annuloplasty through the coronary sinus: An anatomic point of view

ObjectiveWe assessed the anatomic relationships among the mitral annulus, coronary sinus, and circumflex artery in human cadaver hearts.MethodsPercutaneous posterior mitral annuloplasty has been proposed to treat functional mitral regurgitation on the basis of the proximity of the coronary sinus to the mitral annulus. However, concern remains about the ability to perform a trigone-to-trigone posterior annuloplasty and the potential for compromise of the circumflex coronary artery. Ten hearts were studied after injection of expansible foam into the coronary sinus and circumflex artery. The mitral annulus perimeter, posterior intertrigonal (T1–T2) and intercommissural (C1–C2) distance, and coronary sinus projection on the native annulus (S1–S2) were measured. The spatial geometry of the coronary sinus was correlated with the circumflex artery route and the distance with the native mitral annulus.ResultsThe projection of coronary sinus annuloplasty achieves at best a commissure-to-commissure annuloplasty 14.5 (6–24) mm behind each trigone: T1–T2: 74 (56–114) mm, C1–C2: 62.2 (48–80) mm, S1–S2: 59.5 (40–80) mm. The coronary sinus was distant from the native annulus (8–14 mm at the coronary sinus ostium, 13.7–20.4 mm at the middle of the coronary sinus, 6.9–14 mm at the level of the great coronary vein). The circumflex artery was located between the coronary sinus and the mitral annulus in 45.5% of cases.ConclusionsThis anatomic study highlights the 3-dimensional structure of the coronary sinus and its distance from the native mitral annulus and fibrous trigones. Human anatomic studies are mandatory for the further development of percutaneous mitral repair technology