Angular Momentum Balance Research Articles

On geometric discretization of elasticity

This paper presents a geometric discretization of elasticity when the ambient space is Euclidean. This theory is built on ideas from algebraic topology, exterior calculus, and the recent developments of discrete exterior calculus. We first review some geometric ideas in continuum mechanics and show how constitutive equations of linearized elasticity, similar to those of electromagnetism, can be written in terms of a material Hodge star operator. In the discrete theory presented in this paper, instead of referring to continuum quantities, we postulate the existence of some discrete scalar-valued and vector-valued primal and dual differential forms on a discretized solid, which is assumed to be a triangulated domain. We find the discrete governing equations by requiring energy balance invariance under time-dependent rigid translations and rotations of the ambient space. There are several subtle differences between the discrete and continuous theories. For example, power of tractions in the discrete theory is written on a layer of cells with a nonzero volume. We obtain the compatibility equations of this discrete theory using tools from algebraic topology. We study a discrete Cosserat medium and obtain its governing equations. Finally, we study the geometric structure of linearized elasticity and write its governing equations in a matrix form. We show that, in addition to constitutive equations, balance of angular momentum is also metric dependent; all the other governing equations are topological.

Decadal variations in gravity caused by a tilt of the inner core

SUMMARY A tilt of the geometric figure of the inner core with respect to the mantle results in a global internal mass displacement. This comprises two parts: the redistribution of mass from the rigid equatorial rotation of the elliptical inner core; and that from global elastic deformations that occur to maintain the mechanical equilibrium. This global mass reorganization leads to changes in the moment of inertia tensor and, equivalently, to changes in the degree 2 component of the gravitational field. In this work, we compute the predicted changes in both gravity and in the moment of inertia tensor as a function of inner core tilt. We show that the inclusion of elastic deformations increases the amplitude of the gravity change at the surface by a factor 1.97. The Stokes coefficients that are the most affected are C 21, S 21: a tilt angle of 0.05 ◦ leads to a change in these coefficients of ∼4 × 10 −11 , while leading to changes in other coefficients of degree 2 that are three orders of magnitude smaller. Observed changes in C 21, S 21 and in polar motion contain decadal variations of undetermined origin; in an effort to determine whether these could be caused by temporal changes in inner core tilt, we compute the changes in C 21, S 21 based on the observed polar motion and compare this prediction against observed variations as determined by satellite laser ranging (SLR) between 1985 and 2005. We show that observed decadal changes in C 21, S 21 and in polar motion suggest that both are predominantly driven by variations in the moment of inertia tensor. The source of these variations cannot be unambiguously determined, nor can we confirm whether they are of internal or surficial origin. Changes in inner core tilt are then not necessarily the cause of these variations, though if they are, our results show that motion in the fluid core must not play a significant role in the global angular momentum balance.

Inspiral, merger, and ringdown of unequal mass black hole binaries: A multipolar analysis

We study the inspiral, merger, and ringdown of unequal mass black hole binaries by analyzing a catalogue of numerical simulations for seven different values of the mass ratio (from q=M2/M1=1 to q=4). We compare numerical and post-Newtonian results by projecting the waveforms onto spin-weighted spherical harmonics, characterized by angular indices (l,m). We find that the post-Newtonian equations predict remarkably well the relation between the wave amplitude and the orbital frequency for each (l,m), and that the convergence of the post-Newtonian series to the numerical results is nonmonotonic. To leading order, the total energy emitted in the merger phase scales like η2 and the spin of the final black hole scales like η, where η=q/(1+q)2 is the symmetric mass ratio. We study the multipolar distribution of the radiation, finding that odd-l multipoles are suppressed in the equal mass limit. Higher multipoles carry a larger fraction of the total energy as q increases. We introduce and compare three different definitions for the ringdown starting time. Applying linear-estimation methods (the so-called Prony methods) to the ringdown phase, we find resolution-dependent time variations in the fitted parameters of the final black hole. By cross correlating information from different multipoles, we show that ringdown fits can be used to obtain precise estimates of the mass and spin of the final black hole, which are in remarkable agreement with energy and angular momentum balance calculations.

Application of energy and angular momentum balance to gravitational radiation reaction for binary systems with spin-orbit coupling

We study gravitational radiation reaction in the equations of motion for binary systems with spin-orbit coupling, at order (v/c)7 beyond Newtonian gravity, or O(v/c)2 beyond the leading radiation reaction effects for non-spinning bodies. We use expressions for the energy and angular momentum flux at infinity that include spin-orbit corrections, together with an assumption of energy and angular momentum balance, to derive equations of motion that are valid for general orbits and for a class of coordinate gauges. We show that the equations of motion are compatible with those derived earlier by a direct calculation.

Geodynamic constraints on the steady and time-dependent inner core axial rotation

In this work, constraints on the steady and time-dependent rates of axial inner core rotation are established based on the angular momentum balance between the inner core, fluid core and mantle. It is shown that the rate of a steadily rotating inner core is limited by the torque on the mantle from surface forces at the core-mantle boundary (CMB). The rate of rotation of an oscillating inner core is constrained by the changes in mantle rotation induced by gravitational coupling, which must not exceed the observed changes in length of day. Assuming that the largest torque at the CMB is from electromagnetic forces, our results suggest that the maximum amplitude of an inner core oscillating at a period of 60 yr is ∼0.03° yr−1, while the maximum amplitude of a steadily rotating inner core is ∼0.3° yr−1. This implies that the largest inner core rotation rate compatible with the ensemble of seismic studies, 0.2° yr−1, may be explained by a steady rotation but cannot be explained by a decadal oscillation. Though a steady rotation of 0.2° yr−1 is possible, it constrains the conductance of the lower mantle to be in the range of 1.6 –2.5 × 109 S and the viscous relaxation timescale of the inner core to be in the range of 5–8 yr (i.e. a bulk inner core viscosity of 2.5 –4 × 1017 Pa s). Larger steady and oscillating rotation rates may be possible if the inner core can deform on a timescale shorter than ∼5 yr or if the conductance of the lower mantle is larger than ∼2.5 × 109 S, though this would then contradict constraints on these two quantities derived from other observations. Finally, we note that a steadily rotating inner core is compatible with the westward motions of magnetic field features at the CMB, as the latter provide the electromagnetic torque on the mantle that balances the gravitational torque from the rotating inner core. In addition, these westward motions must be predominantly due to advection by flow rather than to the propagation of hydromagnetic waves, as the electromagnetic torque at the CMB is otherwise too small to allow an inner core rotation rate as large as 0.2° yr−1.

Protoplanetary Disk Turbulence Driven by the Streaming Instability: Linear Evolution and Numerical Methods

We present local simulations that verify the linear streaming instability that arises from aerodynamic coupling between solids and gas in protoplanetary disks. This robust instability creates enhancements in the particle density in order to tap the free energy of the relative drift between solids and gas, generated by the radial pressure gradient of the disk. We confirm the analytic growth rates found by Youdin & Goodman (2005) using grid hydrodynamics to simulate the gas and, alternatively, particle and grid representations of the solids. Since the analytic derivation approximates particles as a fluid, this work corroborates the streaming instability when solids are treated as particles. The idealized physical conditions -- axisymmetry, uniform particle size, and the neglect of vertical stratification and collisions -- provide a rigorous, well-defined test of any numerical algorithm for coupled particle-gas dynamics in protoplanetary disks. We describe a numerical particle-mesh implementation of the drag force, which is crucial for resolving the coupled oscillations. Finally we comment on the balance of energy and angular momentum in two-component disks with frictional coupling. A companion paper details the non-linear evolution of the streaming instability into saturated turbulence with dense particle clumps.

Galerkin‐based Time Integrators for Geometrically Nonlinear Elasto‐Plastodynamics

AbstractComputational modelling often demands the incorporation of dynamic effects. The consideration of computational elastoplastodynamics requires in particular advanced numerical techniques to satisfy the classical balance laws like balance of linear and angular momentum or the laws of thermodynamics. Regarding the thermodynamical consistency of elasto‐plastodynamics, the conservation of the total energy in the elastic case and the fulfilment of the dissipation inequality in the plastic case are very important. In this contribution we deal with a mechanically and thermodynamically consistent Galerkin‐based time‐stepping scheme for finite elasto‐plastodynamics. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Modelling the formation of double white dwarfs

We investigate the formation of the ten double-lined double white dwarfs that have been observed so far. A detailed stellar evolution code is used to calculate grids of single-star and binary models and we use these to reconstruct possible evolutionary scenarios. We apply various criteria to select the acceptable solutions from these scenarios. We confirm the conclusion of Nelemans et al. (2000) that formation via conservative mass transfer and a common envelope with spiral-in based on energy balance or via two such spiralins cannot explain the formation of all observed systems. We investigate three different prescriptions of envelope ejection due to dynamical mass loss with angular-momentum balance and show that they can explain the observed masses and orbital periods well. Next, we demand that the age difference of our model is comparable to the observed cooling-age difference and show that this puts a strong constraint on the model solutions. However, the scenario in which the primary loses its envelope in an isotropic wind and the secondary transfers its envelope, which is then re-emitted isotropically, can explain the observed age differences as well. One of these solutions explains the DB-nature of the oldest white dwarf in PG1115+116 along the evolutionary scenario proposed by Maxted et al. (2002a), in which the helium core of the primary becomes exposed due to envelope ejection, evolves into a giant phase and loses its hydrogen-rich outer layers.

Atmospheric angular momentum balance for the southern hemisphere during the polar vortex break-up of September 2002

Atmospheric angular momentum balance for the southern hemisphere during the polar vortex break-up of September 2002

On the vortex created by multiple blades joined at a hub

AbstractExperimental studies often use several blades, joined on the centreline, to generate a vortex in a wind tunnel. The circulation profile of this vortex is non-trivial, and a rapid prediction method will facilitate the design of such devices. Prompted by an experiment of Beninati and Marshall, such a model was derived. It assumes a large number of narrow blades and small turning angles, and rests on a balance of angular momentum and on simple aerofoil function. It is claimed to be valid even close to the centreline, can be completed analytically, contains no adjustable constants, and agrees well with two experiments although the turning angle reached 16° and the devices only had two or four blades. With typical geometries, the interference between blades makes the circulation profile quite different from that behind an elliptically-loaded wing, prompting some doubt over the relevance of such a vortex to those trailing real wings.

An h-adaptive spacetime-discontinuous Galerkin method for linear elastodynamics

We present an h-adaptive version of the spacetime-discontinuous Galerkin (SDG) finite element method for linearized elastodynamics (Abedi et al., 2006). The adaptive version inherits key properties of the basic SDG formulation, including element-wise balance of linear and angular momentum, complexity that is linear in the number of elements and oscillationfree shock capturing. Unstructured spacetime grids allow simultaneous adaptation in space and time. A localized patch-by-patch solution process limits the cost of reanalysis when the error indicator calls for more refinement. Numerical examples demonstrate the method’s performance and shock-capturing capabilities.

Atmospheric angular momentum balance for the southern hemisphere during the polar vortex break-up of September 2002

The evolution of the axial component of the atmospheric angular momentum vector (AAM) is investigated for September 2002, using ECMWF analyses. The components of the conservation equation of AAM for southern polar cap volume are calculated in order to examine the importance of their contributions to the AAM tendency during the unexpected break-up of the southern polar vortex in the last week of September.The AAM tendency for the mean polar cap (90◦–65◦ S) oscillates with a period of 5 d until the break-up of the polar vortex, after that it is negative for about 9 d. The polar friction torque induced by katabatic winds is weakly positive and nearly constant. For this mean polar cap, the mountain torque at the surface is mainly positive and larger than the friction torque. The strong changes of the mountain torque are caused by shifts of surface pressure patterns induced by a Rossby wave train touching the orography of Antarctica. The fluxes of AAM are northwards and reduce the AAM of the polar cap volume. The changes of AAM fluxes are mainly determined by the evolution of momentum fluxes due to transient Rossby waves in the upper troposphere and lower stratosphere. The vortex break-up is associated with a strong decrease of the amount of mountain torque and an increase in the convergence of relative AAM fluxes.

Atmospheric and oceanic contributions to Chandler wobble excitation determined by wavelet filtering

Fluctuations of Earth rotation are associated with the redistribution and motion of mass elements in the Earth system. On seasonal to interannual timescales, the largest effects are due to mass redistributions within atmosphere and oceans. In order to study the Earth's reaction on geophysical excitations, the dynamic Earth system model DyMEG has been developed. It is based on the balance of angular momentum in the Earth system. The model is forced by consistent atmospheric and oceanic angular momenta from reanalysis data and a global ocean circulation model. As rotational deformations of the Earth are regarded, forced variations of Earth rotation due to atmospheric and oceanic excitations influence the free polar motion (Chandler wobble) of DyMEG. The comparison of the numerical model results with geodetic observations reveals a good agreement in both the annual and the Chandler frequency band. In order to support the presumption that the excitation energy of atmospheric and oceanic angular momenta within a spectral band around the Chandler frequency is sufficient to reproduce the observed Chandler wobble, the excitation series are band‐pass filtered. For consistent filtering and signal analysis, wavelet techniques based on the Morlet function are applied. When DyMEG is forced with the filtered excitations, the resulting polar motion resembles the actually observed Chandler oscillation, which is determined from geodetic observations applying the same filtering method. Between 1980 and 2002 the correlation coefficient between the unconstrained model result and the observed Chandler wobble amounts to 0.99 and the root‐mean‐square difference is 16 milliarc seconds.

A space–time discontinuous Galerkin method for linearized elastodynamics with element-wise momentum balance

We present a new space–time discontinuous Galerkin finite element method for linearized elastodynamics that delivers exact balance of linear and angular momentum over every space–time element. The method is formulated for use with fully unstructured space–time grids and uses displacement basis functions that are discontinuous across all inter-element boundaries. We introduce a new space–time formulation of continuum elastodynamics that uses differential forms and the exterior calculus on manifolds to generate a system of space–time field equations and jump conditions. Then we invoke a Bubnov–Galerkin weighted residuals procedure to formulate the finite element method. We describe an implementation on patch-wise causal meshes that features linear complexity in the number of elements and special per-pixel accurate visualization. Numerical examples confirm an a priori error estimate and demonstrate the method’s shock-capturing capabilities.

Comments on the Culture of the Force

Comments on the Culture of the Force

Reducing Climatology Bias in an Ocean–Atmosphere CGCM with Improved Coupling Physics

Abstract The cold tongue in the tropical Pacific extends too far west in most current ocean–atmosphere coupled GCMs (CGCMs). This bias also exists in the relatively high-resolution SINTEX-F CGCM despite its remarkable performance of simulating ENSO variations. In terms of the importance of air–sea interactions to the climatology formation in the tropical Pacific, several sensitivity experiments with improved coupling physics have been performed in order to reduce the cold-tongue bias in CGCMs. By allowing for momentum transfer of the ocean surface current to the atmosphere [full coupled simulation (FCPL)] or merely reducing the wind stress by taking the surface current into account in the bulk formula [semicoupled simulation (semi-CPL)], the warm-pool/cold-tongue structure in the equatorial Pacific is simulated better than that of the control simulation (CTL) in which the movement of the ocean surface is ignored for wind stress calculation. The reduced surface zonal current and vertical entrainment owing to the reduced easterly wind stress tend to produce a warmer sea surface temperature (SST) in the western equatorial Pacific. Consequently, the dry bias there is much reduced. The warming tendency of the SST in the eastern Pacific, however, is largely suppressed by isopycnal diffusion and meridional advection of colder SST from south of the equator due to enhanced coastal upwelling near Peru. The ENSO signal in the western Pacific and its global teleconnection in the North Pacific are simulated more realistically. The approach as adopted in the FCPL run is able to generate a correct zonal SST slope and efficiently reduce the cold-tongue bias in the equatorial Pacific. The surface easterly wind itself in the FCPL run is weakened, reducing the easterly wind stress further. This is related with a weakened zonal Walker cell in the atmospheric boundary layer over the eastern Pacific and a new global angular momentum balance of the atmosphere associated with reduced westerly wind stress over the southern oceans.

Modelling of complex elastic crystals by means of multi-spin micromorphic media

The aim of the paper is to model essentially magnetic crystals having complicated structure. These crystals have interesting magnetoacoustic properties. Each cell (a point-body) of such a crystal consists of many atoms, combined in clusters (subparticles) responsible for the magnetic properties of a crystal. The magnetic moment of clusters is related to the angular momentum (including moment of momentum and spin). Subparticles are considered as rotating point-bodies, interacting with their neighbours, possessing rotational symmetry. We apply two approaches: phenomenological and microstructural. The phenomenological approach is based on the statement of balance laws. We obtain strain tensors for an inhomogeneous elastic polar medium (Kelvin medium) and the associated constitutive equations. To write down the stress and couple tensor for a point-body corresponding to a cell we have to integrate the laws of balance of momentum and angular momentum over a cell. Thus, the strain energy has a complicated structure depending on various deformations and various internal parameters such as the relative rotation of subparticles in the natural configuration. For the microstructural consideration we model the interaction between subparticles as a potential interaction of sufficiently general kind between rigid bodies both of force and torque nature, and calculate the force and the torque acting from the neighbourhood upon a subparticle. We sum the laws of the balance of linear and angular momenta over all subparticles of a cell. To pass from the discrete to the continuum consideration we expand these laws in two space co-ordinates: a “micro”-gradient corresponds to the translations in the neighbourhood of the subparticle, and a “macro”-gradient corresponds to the translation by a basic vector of a crystal lattice.

Gravitational radiation reaction in the equations of motion of compact binaries to 3.5 post-Newtonian order

We compute the radiation-reaction force on the orbital motion of compact binaries to the 3.5 post-Newtonian (3.5PN) approximation, i.e. one PN order beyond the dominant effect. The method is based on a direct PN iteration of the near-zone metric and equations of motion of an extended isolated system, using appropriate ‘asymptotically matched’ flat-spacetime retarded potentials. The formalism is subsequently applied to binary systems of point particles, with the help of the Hadamard self-field regularization. Our result is the 3.5PN acceleration term in a general harmonic coordinate frame. Restricting the expression to the centre-of-mass frame, we find perfect agreement with the result derived in a class of coordinate systems by Iyer and Will using the energy and angular-momentum balance equations.

On the physical meaning of the dynamical equations governing geometrically-exact multilayer shells

In this paper, the physical meaning of the equations governing geometrically-exact multilayer shells is provided through a totally new derivation by means of dynamic equilibrium consideration, starting from a 3-D continuum setting in curvilinear coordinates. The theory is valid for large deformation and large overall motion, as characteristic of geometrically exact formulations. The dynamic equilibrium derivation offers a clear physical insight into the meaning of the terms in the equations of motion; such insight was not afforded by previous derivations based on the principle of virtual power. In particular, we obtain directly the balance of angular momentum in true resultant couples, without the need to use the constitutive restriction as in our previous papers. Moreover, we also obtain here new expressions for the inertia operators that will greatly simplify the computational formulation, when compared to our previous work. In addition, an analytical justification of the inertial operators is also provided. The present formulation is independent of any kinematic assumption along the shell director, and is in particular valid for the case of shells with variable-length director (e.g., for modeling through-the-thickness deformation) and higher-order kinematic assumptions.

Jet rotation: Launching region, angular momentum balance and magnetic properties in the bipolar outflow from RW Aur

Using STIS on board the HST we have obtained a spectroscopic map of the bipolar jet from RW Aur with the slit parallel to the jet axis and moved across the jet in steps of . After applying a velocity correction due to uneven slit illumination we find signatures of rotation within the first 300 AU of the jet (15 at the distance of RW Aur). Both lobes rotate in the same direction (i.e. with different helicities), with toroidal velocities in the range 5–30 km s-1 at 20 and 30 AU from the symmetry axis in the blueshifted and redshifted lobes, respectively. The sense of rotation is anti-clockwise looking from the tip of the blue lobe (PA 130° north to east) down to the star. Rotation is more evident in the [OI] and [NII] lines and at the largest sampled distance from the axis. These results are consistent with other STIS observations carried out with the slit perpendicular to the jet axis, and with theoretical simulations. Using current magneto-hydrodynamic models for the launch of the jets, we find that the mass ejected in the observed part of the outflow is accelerated from a region in the disk within about 0.5 AU from the star for the blue lobe, and within 1.6 AU from the star for the red lobe. Using also previous results we estimate upper and lower limits for the angular momentum transport rate of the jet. We find that this can be a large fraction (two thirds or more) of the estimated rate transported through the relevant portion of the disk. The magnetic lever arm (defined as the ratio between the Alfvèn and footpoint radii) is in the range 3.5–4.6 (with an accuracy of 20–25%), or, alternatively, the ejection index is in the range 0.025–0.046 (with similar uncertainties). The derived values are in the range predicted by the models, but they also suggest that some heating must be provided at the base of the flow. Finally, using the general disk wind theory we derive the ratio of the toroidal and poloidal components of the magnetic field at the observed location (i.e. about 80–100 AU above the disk). We find this quantity to be at 30 AU from the axis in the red lobe and at 20 AU from the axis in the blue lobe (assuming cylindrical coordinates centred on the star and with positive z along the blue lobe). The toroidal component appears to be dominant, which would be consistent with magnetic collimation of the jet. The field appears to be more tightly wrapped on the blue side.