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Fig. 2 Equipotential surfaces
and plumb lines
Geoid
W=W 0
Additionally to the centrifugal force F z another fictitious force is acting on a
moving body, the Coriolis force . It is proportional to the velocity of a moving mass
within a rotating reference system. In case of the Earth it can be observed at clouds.
As air moves from high to low pressure in the northern hemisphere, it is deflected
to the right by the Coriolis force, in the southern hemisphere to the left (Hofmann-
Wellenhof and Moritz 2005 ).
1.2.1 The Geometry of the Gravity Field
The gravity field can be described by surfaces of constant gravitational potential (see
Fig. 2 ), i.e. equipotential surfaces or level surfaces, where
W
(
x
,
y
,
z
) =
const
.
(17)
The surface of the oceans in a first approximation is part of such an equipoten-
tial surface. This particular surface was proposed by Carl Friedrich Gauss as the
“mathematical figure of the Earth” and was later named geoid by the mathematician
Johann Benedict Listing (Torge, 1989 ). Lines that intersect all equipotential surfaces
orthogonally are called lines of force or plumb lines. The tangent to such a plumb
line at any point equals the direction of the gravity vector at that point.
The differentiation of Eq. ( 16 ) leads to the gravity gradient tensor , also called
Eötvös tensor :
W xx W xy W xz
W yx W yy W yz
W zx W zy W zz
.
grad g
= Δ
W
=
(18)
The unit of the components is 1 Eötvös corresponding to 10 9 s 2 in SI-units.
Due to the irrotational nature of the gravity field,
rot g
=
rot grad W
=
0
,
(19)
and considering Poisson's differential in Eq. ( 8 ) and the centrifugal potential
Φ
in
Eq. ( 13 ),
2
Δ
W
=
W xx +
W yy +
W zz =−
4
π
G
ρ +
2
ω
,
(20)
the gravity gradient tensor Eq. ( 18 ) contains only five independent elements, meaning
the tensor is symmetric: W xy =
=
=
W yx , W xz
W zx and W yz
W zy . The third line of
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