GEODYNAMICS

The most precise way of estimating the dissipation of tidal energy in the oceans is by evaluating the rate at which work is done by the tidal forces and this quantity is completely described by the fundamental harmonic in the ocean tide expansion that has the same degree and order as the forcing function. The contribution of all other harmonics to the work integral must vanish. These harmonics have been estimated for the principal M 2 tide using several available numerical models and despite the often significant difference in the detail of the models, in the treatment of the boundary conditions and in the way dissipating forces are introduced, the results for the rate at which energy is dissipated are in good agreement. Equivalent phase lags, representing the global ocean-solid Earth response to the tidal forces and the rates of energy dissipation have been computed for other tidal frequencies, including the atmospheric tide, by using available tide models, age of tide observations and equilibrium theory. Orbits of close Earth satellites are periodically perturbed by the combined solid Earth and ocean tide and the delay of these perturbations compared with the tide potential defines the same terms as enter into the tidal dissipation problem. They provide, therefore, an independent estimate of dissipation. The results agree with the tide calculations and with the astronomical estimates. The satellite results are independent of dissipation in the Moon and a comparison of astronomical, satellite and tidal estimates of dissipation permits a separation of energy sinks in the solid Earth, the Moon and in the oceans. A precise separation is not yet possible since dissipation in the oceans dominates the other two sinks: dissipation occurs almost exclusively in the oceans and neither the solid Earth nor the Moon are important energy sinks. Lower limits to the Q of the solid Earth can be estimated by comparing the satellite results with the ocean calculations and by comparing the astronomical results with the latter. They result in Q > 120. The lunar acceleration n , the Earth’s tidal acceleration O T and the total rate of energy dissipation E estimated by the three methods give astronomical based estimate —1.36 —28±3 —7.2 ± 0.7 4.1±0.4 satellite based estimate —1.03 —24 ±5 — 6.4 ± 1.5 3.6±0.8 numerical tide model — 1.49 —30 ±3 —7.5± 0.8 4.5±0.5 The mean value for O T corresponds to an increase in the length of day of 2.7 ms cy -1 . The non-tidal acceleration of the Earth is (1.8 ± 1.0) 10 -22 s ~2 , resulting in a decrease in the length of day of 0.7 ± 0.4 ms cy -1 and is barely significant. This quantity remains the most unsatisfactory of the accelerations. The nature of the dissipating mechanism remains unclear but whatever it is it must also control the phase of the second degree harmonic in the ocean expansion. It is this harmonic that permits the transfer of angular momentum from the Earth to the Moon but the energy dissipation occurs at frequencies at the other end of the tide’s spatial spectrum. The efficacity of the break-up of the second degree term into the higher modes governs the amount of energy that is eventually dissipated. It appears that the break-up is controlled by global ocean characteristics such as the ocean-continent geometry and sea floor topography. Friction in a few shallow seas does not appear to be as important as previously thought: New estimates for dissipation in the Bering Sea being almost an order of magnitude smaller than earlier estimates. If bottom friction is important then it must be more uniformly distributed over the world's continental shelves. Likewise, if turbulence provides an important dissipation mechanism it must be fairly uniformly distributed along, for example, coastlines or along continental margins. Such a global distribution of the dissipation makes it improbable that there has been a change in the rate of dissipation during the last few millennium as there is no evidence of changes in ocean volume, or ocean geometry or sea level beyond a few metres. It also suggests that the time scale problem can be resolved if past ocean-continent geometries led to a less efficient breakdown of the second degree harmonic into higher degree harmonics.

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Space Debris – The Short Term Orbital Evolution in the Earth Gravity Field

International Astronomical Union Colloquium ◽

10.1017/s0252921100046388 ◽

1997 ◽

Vol 165 ◽

pp. 71-78

Author(s):

Edwin Wnuk

Keyword(s):

Space Debris ◽

Orbital Evolution ◽

Gravity Potential ◽

Short Term ◽

Near Earth Space ◽

Earth Gravity ◽

The Earth ◽

Geopotential Models ◽

Theory Of Motion

AbstractTwo aspects of the orbital evolution of space debris – the long-term evolution and the short-term one – are of interest for an exploration of the near- Earth space. The paper presents some results concerning the estimation of the accuracy of predicted positions of Earth-orbiting objects for the short-term: a few revolutions or a time-span interval of a few days. Calculations of predicted positions take into account the influence of an arbitrary number of spherical coefficients of the Earth gravity potential. Differences in predicted positions due to differences in the best contemporary geopotential models (JGM-2, JGM-3 and GRIM4-S4) are estimated with the use of an analytical theory of motion and a numerical integration.

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Local earth gravity/potential modeling using ASCH

Arabian Journal of Geosciences ◽

10.1007/s12517-014-1767-2 ◽

2015 ◽

Vol 8 (10) ◽

pp. 8681-8685 ◽

Cited By ~ 1

Author(s):

Ghadi Younis

Keyword(s):

Gravity Potential ◽

Earth Gravity

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Numerical Analysis of Flow Evolution in a Helium Jet Injected Into Ambient Air

Volume 2, Parts A and B ◽

10.1115/ht-fed2004-56811 ◽

2004 ◽

Cited By ~ 1

Author(s):

Rajani P. Satti ◽

Ajay K. Agrawal

Keyword(s):

Grid Method ◽

Ambient Air ◽

Staggered Grid ◽

Earth Gravity ◽

Helium Jet ◽

Entrainment Process ◽

Numerical Formulation ◽

Flow Evolution ◽

The Stability ◽

Velocity Diffusion

A computational model to study the stability characteristics of an evolving buoyant helium gas jet in ambient air environment is presented. Numerical formulation incorporates a segregated approach to solve for the transport equations of helium mass fraction coupled with the conservation equations of mixture mass and momentum using a staggered grid method. The operating parameters correspond to the Reynolds number varying from 30 to 300 to demarcate the flow dynamics in oscillating and non-oscillating regimes. Computed velocity and concentration fields were used to analyze the flow structure in the evolving jet. For Re = 300 case, results showed that an instability mode that sets in during the evolution process in Earth gravity is absent in zero gravity, signifying the importance of buoyancy. Though buoyancy initiates the instability, below a certain jet exit velocity, diffusion dominates the entrainment process to make the jet non-oscillatory as observed for the Re = 30 case. Initiation of the instability was found to be dependent on the interaction of buoyancy and momentum forces along the jet shear layer.

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Automated Orbital Transfer and Hovering Control Using Artificial Potential

Mathematical Problems in Engineering ◽

10.1155/2019/6186283 ◽

2019 ◽

Vol 2019 ◽

pp. 1-16

Author(s):

Xun Wang ◽

Zhaokui Wang ◽

Yulin Zhang

Keyword(s):

Control Method ◽

Great Influence ◽

Solar Radiation Pressure ◽

Lyapunov Theory ◽

Relative Distance ◽

Attractive Potential ◽

Orbital Transfer ◽

Earth Gravity ◽

Relative Dynamics ◽

The Stability

On-orbit servicing for GEO targets has attracted great attention due to particular significance. In this study, the GEO nearby flying is considered, which denotes that the servicing spacecraft approaches GEO targets within tens of meters. Based on the analysis of differential spherical Earth gravity, J2 perturbation, third-body perturbation, and solar radiation pressure (SRP) between two spacecraft, a relative dynamics of GEO nearby flying is built, in which the differential SRP has great influence on relative motion. Therein, considering the differential SRP, an analytical solution to relative dynamics is obtained, which is an extension to CW equation’s solution. Based on the derived relative dynamics, a novel control method utilizing feedback compensation and artificial potential is developed for spacecraft orbital transfer and hovering at the desired position. During orbital transfer, a bounded space is constrained with maximum and minimum relative distance between two spacecraft. An improved repulsive potential based on Gauss function is designed to drive the servicer off the bound and guarantee the potential value converge to zero at the desired position. Meanwhile, a novel attractive potential about relative distance that drives the servicer to the desired position and to hover with high accuracy against perturbations is designed. Besides, a velocity potential with variable gain is designed to ensure that the servicer achieve the desired position without overshoot. The stability of the system is proved with Lyapunov theory, and the feasibility of the proposed control law is verified through numerical simulations with obvious advantages.

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Some Interpolation Formulas for Approximating the Solution of the Heat Equation.

DAIMI Report Series ◽

10.7146/dpb.v2i17.6436 ◽

1973 ◽

Vol 2 (17) ◽

Author(s):

Bruce D. Shriver

Keyword(s):

Heat Conduction ◽

Heat Equation ◽

Heat Conduction Problem ◽

Degree Harmonic ◽

The Relationship ◽

Harmonic Interpolation ◽

Second Degree

Interpolation formulas of the form u(x_*1 , ... ,x_(* n),t_*) ~= Sum_{i=1}^{N} A_i u(v_i1,...,v_in, t_i)are presented. These formulas are based on the heat polynomials of Appell. The point (x_{* 1}, ... x_{* n}t_*) is in the interior of an (n+1) dimensional region, R_n+1, and the points (v_i1,...,v_in, t_i) are on the boundary of R_n+1. These formulas can be used to approximate the solution of the heat conduction problem in R_n+1. The relationship between formulas of the above type of deqree 2 in R_n+1 and the second degree harmonic interpolation formulas of Stroud, Chen, Wang, and Mao in R_n is persented. Some higher degree formulas for special regions in R_2 are also developed.

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Gravity field models derived from the second degree radial derivatives of the GOCE mission: a case study

Annals of Geophysics ◽

10.4401/ag-7049 ◽

2017 ◽

Vol 59 (6) ◽

Author(s):

Alexander N. Marchenko ◽

Dmitriy A. Marchenko ◽

Alexander N. Lopushansky

Keyword(s):

Gravity Field ◽

Field Model ◽

The Black Sea ◽

Gravity Gradients ◽

Reference Field ◽

Satellite Mission ◽

Gravity Field Model ◽

Goce Satellite ◽

Second Degree

The GOCE satellite mission is one of the main achievements of the satellite geodesy for the Earth’s gravitational field recovery. Three different approaches have been developed for the estimation of harmonic coefficients from gradiometry data measured on board of GOCE-satellite. In this paper a special version of the space-wise method based on the second method of Neumann for fast determination of the harmonic coefficients Cnm, Snm of the Earth’s gravitational potential is given based on the radial gravity gradients of the EGG_TRF_2 product, except of two polar gaps filled by radial gradients from the EGM2008 gravity model. In the pre-processing stage GOCE-based second degree radial derivatives were averaged to the regular grid through Kalman static filter with additional Gaussean smoothing of residual radial derivatives. All computations are made by iterations. As the first step the determination of the preliminary NULP-01S model up to degree/order 220 derived from the Gaussean grid of the GOCE radial derivatives with respect to the WGS-84 reference field was developed based only one of the radial gradients EGG_TRF_2 in the EFRF-frame. In the second iteration the same algorithm is applied to build the NULP-02S gravity field model up to degree/order 250 using the same Gaussean grid with respect to the NULP-01S reference field. The NULP-02S model was verified by means of applying various approaches for the construction of the gridded gravity anomalies and geoid heights in the Black sea area using processing of datasets from six altimetry satellite missions. Comparison of different models with GNSS-levelling data in the USA area demonstrates the independent verification of achieved accuracy of the constructed NULP-02S Earth’s gravity field model.

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Open discussion

Symposium - International Astronomical Union ◽

10.1017/s0074180900029533 ◽

1982 ◽

Vol 99 ◽

pp. 605-613

Author(s):

P. S. Conti

Keyword(s):

Buenos Aires ◽

Large Mass ◽

List Type ◽

Common Phenomenon ◽

Open Discussion ◽

The Stability ◽

Ring Nebulae

Conti: One of the main conclusions of the Wolf-Rayet symposium in Buenos Aires was that Wolf-Rayet stars are evolutionary products of massive objects. Some questions:–Do hot helium-rich stars, that are not Wolf-Rayet stars, exist?–What about the stability of helium rich stars of large mass? We know a helium rich star of ∼40 MO. Has the stability something to do with the wind?–Ring nebulae and bubbles : this seems to be a much more common phenomenon than we thought of some years age.–What is the origin of the subtypes? This is important to find a possible matching of scenarios to subtypes.

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Symmetric Multistep Methods Revisited: II. Numerical Experiments

International Astronomical Union Colloquium ◽

10.1017/s0252921100031602 ◽

1999 ◽

Vol 173 ◽

pp. 309-314 ◽

Cited By ~ 3

Author(s):

T. Fukushima

Keyword(s):

Multistep Methods ◽

Computational Time ◽

Integration Time ◽

Implicit Methods ◽

Symmetric Methods ◽

Celestial Bodies ◽

The Stability ◽

Predictor Corrector ◽

Symmetric Multistep Methods ◽

Integration Errors

AbstractBy using the stability condition and general formulas developed by Fukushima (1998 = Paper I) we discovered that, just as in the case of the explicit symmetric multistep methods (Quinlan and Tremaine, 1990), when integrating orbital motions of celestial bodies, the implicit symmetric multistep methods used in the predictor-corrector manner lead to integration errors in position which grow linearly with the integration time if the stepsizes adopted are sufficiently small and if the number of corrections is sufficiently large, say two or three. We confirmed also that the symmetric methods (explicit or implicit) would produce the stepsize-dependent instabilities/resonances, which was discovered by A. Toomre in 1991 and confirmed by G.D. Quinlan for some high order explicit methods. Although the implicit methods require twice or more computational time for the same stepsize than the explicit symmetric ones do, they seem to be preferable since they reduce these undesirable features significantly.

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Uniformity of Magnification from Different Electron Microscopes

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100060179 ◽

1967 ◽

Vol 25 ◽

pp. 32-33

Author(s):

Godfrey C. Hoskins ◽

V. Williams ◽

V. Allison

Keyword(s):

Diffraction Grating ◽

The Other ◽

Carbon Replica ◽

Electron Microscopes ◽

The Stability

The method demonstrated is an adaptation of a proven procedure for accurately determining the magnification of light photomicrographs. Because of the stability of modern electrical lenses, the method is shown to be directly applicable for providing precise reproducibility of magnification in various models of electron microscopes.A readily recognizable area of a carbon replica of a crossed-line diffraction grating is used as a standard. The same area of the standard was photographed in Phillips EM 200, Hitachi HU-11B2, and RCA EMU 3F electron microscopes at taps representative of the range of magnification of each. Negatives from one microscope were selected as guides and printed at convenient magnifications; then negatives from each of the other microscopes were projected to register with these prints. By deferring measurement to the print rather than comparing negatives, correspondence of magnification of the specimen in the three microscopes could be brought to within 2%.

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