Covariant equations for geophysical fluids with an asymmetric gravitational potential and their hydrostatic limit

The method of the geopotential parameters determination with the use of the gradiometry data is considered. The second derivative of the gravitational potential in the correction equation on the rectangular coordinates x, y, z is used as a measured variable. For the calculated value of the measured quantity required for the formation of a free member of the correction equation, the the Cunningham polynomials were used. We give algorithms for computing the second derivatives of the Cunningham polynomials on rectangular coordinates x, y, z, which allow to calculate the second derivatives of the geopotential at the rectangular coordinates x, y, z.Then we convert derivatives obtained from the Cartesian coordinate system in the coordinate system of the gradiometer, which allow to calculate the free term of the correction equation. Afterwards the correction equation coefficients are calculated by differentiating the formula for calculating the second derivative of the gravitational potential on the rectangular coordinates x, y, z. The result is a coefficient matrix of the correction equations and corrections vector of the free members of equations for each component of the tensor of the geopotential. As the number of conditional equations is much more than the number of the specified parameters, we go to the drawing up of the system of normal equations, from which solutions we determine the required corrections to the harmonic coefficients.

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Evaluation of the Second, Fourth and Sixth Harmonics in the Earth's Gravitational Potential

Nature ◽

10.1038/187490b0 ◽

1960 ◽

Vol 187 (4736) ◽

pp. 490-491 ◽

Cited By ~ 5

Author(s):

D. G. KING-HELE

Keyword(s):

Gravitational Potential

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Gravitational Interaction in the Chimney Lattice Universe

Universe ◽

10.3390/universe7040101 ◽

2021 ◽

Vol 7 (4) ◽

pp. 101

Author(s):

Maxim Eingorn ◽

Andrew McLaughlin ◽

Ezgi Canay ◽

Maksym Brilenkov ◽

Alexander Zhuk

Keyword(s):

Helmholtz Equation ◽

Gravitational Potential ◽

Fourier Expansion ◽

Gravitational Interaction ◽

Alternative Solution ◽

The Third ◽

Present Universe ◽

Delta Functions ◽

Yukawa Potentials ◽

The Universe

We investigate the influence of the chimney topology T×T×R of the Universe on the gravitational potential and force that are generated by point-like massive bodies. We obtain three distinct expressions for the solutions. One follows from Fourier expansion of delta functions into series using periodicity in two toroidal dimensions. The second one is the summation of solutions of the Helmholtz equation, for a source mass and its infinitely many images, which are in the form of Yukawa potentials. The third alternative solution for the potential is formulated via the Ewald sums method applied to Yukawa-type potentials. We show that, for the present Universe, the formulas involving plain summation of Yukawa potentials are preferable for computational purposes, as they require a smaller number of terms in the series to reach adequate precision.

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Gravitational potential: a thought experiment

Physics Education ◽

10.1088/0031-9120/50/4/397 ◽

2015 ◽

Vol 50 (4) ◽

pp. 397-398 ◽

Cited By ~ 1

Author(s):

William H Baird

Keyword(s):

Gravitational Potential ◽

Thought Experiment

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The Ballistic Particle Model

Symposium - International Astronomical Union ◽

10.1017/s0074180900032629 ◽

1983 ◽

Vol 100 ◽

pp. 133-134

Author(s):

Frank N. Bash

Keyword(s):

Radial Velocity ◽

Gravitational Potential ◽

Molecular Clouds ◽

Milky Way ◽

Terminal Velocity ◽

Particle Model ◽

The Sun ◽

Giant Molecular Clouds ◽

Spiral Pattern ◽

Armed Spiral

Bash and Peters (1976) suggested that giant molecular clouds (GMC's) can be viewed as ballistic particles launched from the two-armed spiral-shock (TASS) wave with orbits influenced only by the overall galactic gravitational potential perturbed by the spiral gravitational potential in the arms. For GMC's in the Milky Way, the model predicts that the radial velocity observed from the Sun increases with age (time since launch). We showed that the terminal velocity of CO observed from l ≃ 30° to l ≃ 60° can be understood if all GMC's are born in the spiral pattern given by Yuan (1969) and live 30 × 106 yrs. Older GMC's were predicted to have radial velocities which exceed observed terminal velocities.

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Learning dominant physical processes with data-driven balance models

Nature Communications ◽

10.1038/s41467-021-21331-z ◽

2021 ◽

Vol 12 (1) ◽

Cited By ~ 2

Author(s):

Jared L. Callaham ◽

James V. Koch ◽

Bingni W. Brunton ◽

J. Nathan Kutz ◽

Steven L. Brunton

Keyword(s):

Nonlinear Optics ◽

Unsupervised Learning ◽

History Of Science ◽

Traditional Approach ◽

Data Driven ◽

Mechanistic Models ◽

Physical Processes ◽

History Of ◽

Separation Of Scales ◽

Geophysical Fluids

AbstractThroughout the history of science, physics-based modeling has relied on judiciously approximating observed dynamics as a balance between a few dominant processes. However, this traditional approach is mathematically cumbersome and only applies in asymptotic regimes where there is a strict separation of scales in the physics. Here, we automate and generalize this approach to non-asymptotic regimes by introducing the idea of an equation space, in which different local balances appear as distinct subspace clusters. Unsupervised learning can then automatically identify regions where groups of terms may be neglected. We show that our data-driven balance models successfully delineate dominant balance physics in a much richer class of systems. In particular, this approach uncovers key mechanistic models in turbulence, combustion, nonlinear optics, geophysical fluids, and neuroscience.

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