source term side (essentially three constituents: the divergency of the displacement
field, the projection of the displacement field onto the gradient of the reference mass
density function and finally a "mass condensation term" appearing at discontinuity
interfaces) .
The second aim is to present a special solution of the incremental gravitational field
equations of a deformable body for the incremental Eulerian potential, given in terms
of the reference configuration with boundary discontinuities. Emphasis is on the spherical harmonic expansion of the incremental Eulerian gravitational potential in terms
of dilatation multipoles, transport displacement multipoles and those multipoles which
are generated by mass condensation on the reference surface of the body, for instance
Sk+ j their spheroidal and toroidal components are identified. The term of degree one
expresses as "the dipole moment" the displacement of the mass centre of the deformable body, an effect which overlaps with a similar effect in the post- Newtonian
approximation of Einstein gravitation (General Relativity).
For a deformable body which is characterized by a spherically symmetric constitution in its reference configuration case studies are offered, namely representing the
response of the gravitational field to active volume and surface forces of type tidal
force, loading forces and stress forces. We succeed to outline the functional relation
between Fourier-/ Laplace- transformed Love-Shida numbers hi, k/, 11 and h~, kf, l~,
which we numerically test with respect to a particular Earth model. On the boundary surface level Sk+ we document geoidal undulations of the order of ±15cm due
to the deformation of the terrestrial gravitational field, an effect we cannot neglect
for precise geoid computations.
The paper is essentially based on the lengthy manuscript "The spacetime gravitational field of a deformable body" by E. W. Grafarend, J. Engels and P. Varga submitted
for publication to "Geophysical Journal International".
References
A.E.H. Love: The yielding of the earth to disturbing forces, Proc. R. Soc. London
A82 (1909) 73-88
T. Shida: Horizontal pendulum observations of the change of plumbline at Kamigano,
Kyoto, Mem. ColI. Sci. Eng. Tokyo 4 (1912) 23-174
175
field, the projection of the displacement field onto the gradient of the reference mass
density function and finally a "mass condensation term" appearing at discontinuity
interfaces) .
The second aim is to present a special solution of the incremental gravitational field
equations of a deformable body for the incremental Eulerian potential, given in terms
of the reference configuration with boundary discontinuities. Emphasis is on the spherical harmonic expansion of the incremental Eulerian gravitational potential in terms
of dilatation multipoles, transport displacement multipoles and those multipoles which
are generated by mass condensation on the reference surface of the body, for instance
Sk+ j their spheroidal and toroidal components are identified. The term of degree one
expresses as "the dipole moment" the displacement of the mass centre of the deformable body, an effect which overlaps with a similar effect in the post- Newtonian
approximation of Einstein gravitation (General Relativity).
For a deformable body which is characterized by a spherically symmetric constitution in its reference configuration case studies are offered, namely representing the
response of the gravitational field to active volume and surface forces of type tidal
force, loading forces and stress forces. We succeed to outline the functional relation
between Fourier-/ Laplace- transformed Love-Shida numbers hi, k/, 11 and h~, kf, l~,
which we numerically test with respect to a particular Earth model. On the boundary surface level Sk+ we document geoidal undulations of the order of ±15cm due
to the deformation of the terrestrial gravitational field, an effect we cannot neglect
for precise geoid computations.
The paper is essentially based on the lengthy manuscript "The spacetime gravitational field of a deformable body" by E. W. Grafarend, J. Engels and P. Varga submitted
for publication to "Geophysical Journal International".
References
A.E.H. Love: The yielding of the earth to disturbing forces, Proc. R. Soc. London
A82 (1909) 73-88
T. Shida: Horizontal pendulum observations of the change of plumbline at Kamigano,
Kyoto, Mem. ColI. Sci. Eng. Tokyo 4 (1912) 23-174
175