_{Grav}_{it}_{a}_{ti}_{o}_{na}_{l }_{po}_{t}_{en}_{ti}_{a}_{l}
• Let’s assume:
– A particle of __u____n____it____ ____mas____s__ moving freely
– A body of mass *M*
• The particle is attracted by *M** *and moves toward it by a small quantity *d**r*.
• This displacement is the result of **w****or****k** *W** *exerted by the gravitational field generated by *M*:
Particle of unit mass *m*
^{dr}
^{r}^{ }_{r}
^{F}^{ }^{=}^{ }^{m}^{a}
*W ** *= *Fdr** *= *m** ** **a*
^{dr}^{ }^{=}^{ }^{a}^{ }^{ dr}^{ }^{ }*r*
! *W ** *= *G** *^{M }^{ }*dr *_{r}_{ }2
• The **p****otent****i****a****l** *U** *of mass *M** *is the amount of work necessary to bring the particle from infinity to a given distance *r*:
_{r}_{ }_{ }_{M}
_{U}_{ }_{=}_{ }_{*}_{ }_{G}
_{r}
_{dr}_{ }_{=}_{ }_{GM}_{ }_{*}
^{1 }^{ }_{dr}
_{=}_{ }_{GM}_{ }_{(}_{ }^{1}^{ }_{!}_{ }^{1}^{ }_{%}_{ }_{M}
_{)}_{ }_{r}_{ }2
_{2}_{ }_{ }_{&}_{ }_{#}
^{'}^{ }^{$}
^{)}
*r*
^{)}^{ }*r*
_{GM}
_{"}_{ }_{U}_{ }_{=}_{ }_{!}
^{r}
• **A****t**** d****i****stanc****e**__ __**r****,**__ __**th****e**** gravitat****i****ona****l**** potentia****l**** generate****d**** b****y ****M**__ __**i****s ****U**
• Earth’s gravitational acceleration *g** *exerts a work to move a unit mass particle from *U** *to *U**+**d**U** *(spherical homogeneous non-
rotating Earth):
*d**r*
_{U}_{ }_{+}_{ }_{d}_{U}
_{g}^{r}
^{U}
_{U}_{ }_{=}_{ }_{"}_{ }^{GM}
*r*
_{dU}_{ }_{ GM}_{ }_{E}
_{#}_{ }_{=}
_{dr}_{ }_{ }_{r}_{ }^{2}
# *dU** *= "*gdr*
_{#}_{ }_{g}_{ }_{=}_{ }_{"}_{ }^{dU}
^{dr}
= "*gra**d*_{(}*U** *_{)}
^{Ear}^{t}^{h}^{,}^{ }^{mas}^{s}^{ }^{M}_{E}
(think in terms of energy)
• Gravity potential of the Earth = - gradient of gravity
• Since gravity = gravitational attraction + centrifugal acceleration +
_{f}_{l}_{a}_{tt}_{en}_{i}_{ng}_{,}_{ }_{t}_{h}_{erefore}_{:}
*g** *= "*gra**d*(*U** *)
# *U** *= " ^{GM }*R*
_{GM}_{a}^{2}
_{+}
^{2}^{R}^{3}
^{2}
^{2}
*J** *_{(}3sin^{2}^{ }$ " 1_{)}_{ }" ^{1}^{ }% ^{2}^{ }*R*^{2}^{ }cos^{2}^{ }$
_{Equ}_{i}_{po}_{t}_{en}_{ti}_{a}_{l }_{surface}_{s}
• = surfaces on which the potential is constant
• *U** *= constant, recall that: *d**U** **=** -**g** **d**r*
⇒ *d**U *= zero on equipotential surfaces
⇒ *g** *not necessarily constant on
equipotential surfaces
• Non-rotating homogeneous Earth:
– Recall that: *U** **=** **G**M*_{E}*/**r*
^{– Therefore,}^{ }^{U =}^{ }^{constant}^{ }^{⇒}^{ }^{r =}
^{constant}^{ }^{⇒}^{ }^{equipotential}^{ }^{surfaces}^{ }^{=}
^{spheres centered}^{ }^{on}^{ }^{M}_{E}
• Practical use of equipotential surfaces:
– Definition of the vertical = direction of the gravity field = perpendicular to equipotential surfaces
– Equipotential surfaces = define the horizontal
Equipotential surfaces
_{g}r
_{g}r
_{g}r
_{g}r
Earth (homogeneous, non-rotating)
The geoid
• There is an infinity of equipotential surfaces
• There is a particular surface on the Earth that is
“easy” to locate: the mean sea level
• **Th****e**** Geoi****d**__ __**=**** th****e**** particula****r**** equipotentia****l** **surfac****e**** tha****t**** coincide****s**** wit****h**** th****e**** mea****n**** se****a**** leve****l**
• This is totally arbitrary.
• But is makes sense because the oceans are made of water (!): the surface of a fluid in equilibrium must follow an equipotential.
_{Th}_{e }_{Geo}_{id = }_{Th}_{e }_{f}_{i}_{gur}_{e }_{o}_{f t}_{h}_{e }_{Ea}_{rth}
• __Over the____ ____ocean____s__, the geoid is the ocean surface (assuming no currents, waves, etc)
• __Over the____ ____continent____s__, the geoid is not the topographic surface (its location can be calculated from gravity measurements)
• __Geoid____ ____“____u____ndulation____s____”__ are caused by the distribution of mass in the Earth
geoid
ocean surface
topography
_{Th}_{e }_{E}_{lli}_{p}_{so}_{id}
• First evidence that the Earth is round: Erathostene (275-195 B.C.)
• First hypothesis that the Earth’s is flattened at the poles: Newton
• First measurement of the Earth’s flattening at the poles: Clairaut (1736) and Bouguer (1743)
• **The**__ __**shape of the Earth can be** **mathematically**__ __**represented**__ __**as an** **ellipsoid** defined by:
– Semi-major axis = equatorial radius = **a**
– Semi-minor axis = polar radius = **c**
– Flattening (the relationship between equatorial and polar radius): **f =**** (a-c)****/a**
– Eccentricity: **e**^{2}^{ }**=**** 2f-****f**^{2}
Comp arison between the WGS-84 ellip soid and a sp here of identical volume
The reference
_{e}_{lli}_{p}_{so}_{id}
• Many different reference ellipsoids have been defined and are in use!
• **Referenc****e**** ellipsoi****d**__ __**=**** th****e** **ellipsoi****d**** tha****t**** bes****t**** fit****s**** th****e** **geoi****d**.
• Totally arbitrary, but practical
• Reference ellipsoid = WGS-84
• Geoid undulations = differences, in meters, between the geoid reference ellipsoid
(= geoid “height”).
geoid
local ellipsoid
reference ellipsoid
Gravity on the reference ellipsoid
• It can be shown (Clairaut, ~1740) that the (theoretical) value of gravity on the rotating reference ellipsoid is:
^{g}^{ }^{=}^{ }^{g}_{o}
(1 +
*k** ** *sin ^{2}
^{!}^{ }^{"}^{ }^{k}_{2}
sin ^{2}
2!)
1
– *g*_{o}_{ }= gravity at the equator
^{– }^{k}_{1}_{ }^{and}^{ }^{k}_{2}_{ }^{=}^{ }^{constant}^{ }^{that}^{ }^{depend}^{ }^{on the}^{ }^{shape}^{ }^{and}^{ }^{rotation}^{ }^{of the}^{ }^{Earth}
^{– }^{g}_{o}^{,}^{ }^{k}_{1}^{,}^{ }^{and}^{ }^{k}_{2}_{ }^{are}^{ }^{estimated}^{ }^{from}^{ }^{actual}^{ }^{measurements.}^{ }^{For GRS-1967:}
^{• }^{g}_{o}_{ }^{=}^{ 97}^{8}^{ 031}^{.}^{84}^{6}^{ }^{m}^{Ga}^{ls}
• *k*_{1}_{ }= 0.005 302 4
^{• }^{k}_{2}_{ }^{=}^{ 0}^{.}^{00}^{0}^{ 00}^{5 8}
• = normal gravity formula
• According to this formula:
– *g *depends only on latitude, no longitude dependence.
– *g *does not vary linearly with latitude.
– g increases/decreases when latitude increases?
^{• }^{Th}^{is}^{ }^{formu}^{la}^{ }^{assume}^{s}^{ }^{a}^{n}^{ }^{homogeneou}^{s}^{ }^{Ear}^{t}^{h}^{:}^{ }^{he}^{t}^{erogene}^{iti}^{e}^{s}^{ }^{⇒}
^{dev}^{i}^{a}^{ti}^{on}^{s}^{ }^{fro}^{m}^{ }^{t}^{h}^{is}^{ }^{formu}^{la}^{ }^{⇒}^{ }^{grav}^{ity}^{ }^{a}^{noma}^{li}^{e}^{s}
Let’s be clear…
• Geoid = the equipotential surface of the Earth’s gravity field that best fits (in a least squares sense) the mean sea level
– Potential is constant on the geoid
– Gravity is not constant on the geoid
• Reference Ellipsoid = the ellipsoid that best fits the geoid
• Geoid = the (actual) figure of the
Earth
• Ellipsoid = the (theoretical) shape of the Earth
Geoid anomalies, or undulations = differences, in meters, between the geoid reference ellipsoid (= geoid “height”):
– Excess of mass ⇒ geoid (ocean surface) goes up (geoid = eqp surface ⇒ must
remain perpendicular to the gravity field direction!)
Geoid height
– Deficit of mass ⇒ geoid (ocean surface) goes down
_{+ }^{-}
• Contribution to gravity of hot rising mantle:
– Elevated surface over hot spot => positive anomaly (mass excess due to extra topography)
– Hot, buoyant (less dense) mantle material => negative anomaly (mass deficit)
– Positive > negative => overall positive anomaly
• Contribution of cold sinking lithospheric material:
– Depressed topography => negative anomaly
– Cold, dense material enters mantle =>
positive anomaly
– Negative > positive => overall negative anomaly
_{-}_{ }_{ }^{(a)}
_{-}_{E}
^{.}-^{r;}
^{-}^{.}^{:.::}
_{a.}
^{Q)}
**0**
_{Temperature}
_{600}_{ }_{b============-11111111111111}
^{-600}^{ }^{ }^{0}
**Distance** **(km)**
600
_{G}_{l}_{oba}_{l }_{geo}_{id }_{undu}_{l}_{a}_{ti}_{o}_{n}_{s}
Global geoid undulations
The geoid of the Conterminous
United States
Geoid undulations (= heights ) for the conterm inous United States and s urrounding areas
(__h____tt____p____://____www____.____ng____s____ ____.____noaa____.____go____v____/G____E____OI____D__).
Geoid heights range from a low of -51.6 m eters in the Atlantic (m agenta) to a high of -7.2 m eters (red) in the
Rockies .
Example: The geoid of the
Conterminous United States
• The geoid high, seen as a red spot, is located over the "Yellowstone hot spot". While high elevations do contribute to a portion of the geoid signal here, some geophysicists feel that the geoid high is evidence of a thermal mantle plume.
• The isolated ridge (visible more from the light/shadowing than from a different color) seen running from Minnesota to Iowa is associated with the "mid-continent gravity high". This gravity high (and subsequently, the geoid high) is the result of dense masses which lie close to the surface. These masses lie in an old rift in the North American plate.
• The brightly lit slope in the geoid, off of the east coast, is the effect of a 4000 meter (~13,000 feet) drop in the bathymetry, demarking the edge of the continental shelf, a passive continental margin that formed some
120 million years ago
• The few noticeably large bumps in the Pacific ocean are caused by massive seamounts up to 3000 meters
(~10,000 feet) in height above the sea floor.
• Details of the topographic anomalies of the Western Rockies can be seen superimposed upon this anomaly, although with much less magnitude.
• The Great San Joaquin Valley of California, formed through the tectonics of the earlier subduction of the Pacific plate by North America is outlined in detail in the Geoid. Comparison with this feature can be made with those smaller yet similar Geoid lows to the north in Oregon and Washington state.
• At the very top of the figure on the right can be seen the outline of the most recently formed feature of Geoid of North America. This is the postglacial Geoid low caused by the depression of the continent under the ice load from the last Ice Age some 20,000 years ago. Because of the viscous nature of the Earth's Mantle this feature will slowly disappear until the end of the next Ice Age when the process will repeat itself again.
• From: http://www.ngs.noaa.gov/GEOID
_{A }_{ver}_{y }_{prac}_{ti}_{c}_{a}_{l }_{us}_{e }_{o}_{f t}_{h}_{e }_{geo}_{id}
• GPS positioning → height above or below a reference ellipsoid (WGS-84) = **ellipsoidal**__ __**height**.
• Topographic maps, markers, etc.: height above or below **mean sea level**__ __**=**__ __**orthometric** **height**.
• Transformation from ellipsoidal height to orthometric height requires to know the geoid height.
_{• In the}_{ }_{conterminous}_{ }_{US, this}
ellipsoid ocean surface geoid
topography/
bathymetry
h
H N
⇒ *H** **=** **h +** **N*
transformation implies a correction ranging from 51.6 m (Atlantic) to 7.2 m (Rocky Mountains)
*h** *= ellipsoidal height
*N** *= geoid height
*H** *= orthometric height
= height above mean sea level
_{Ellipsoid}
^{.}^{·}^{ }^{· }^{-}^{.}^{.}_{...............}_{.}_{,}
N
_{······}_{·}^{·········}^{ }^{·····························}^{·}_{....}_{ }_{··•}_{·}_{·}_{·}_{·}_{•}_{·}_{·}
Po
''G_{90}_{1}"d''
OCEAN
h (Ellipsoid Height)= Distance along ellipsoid normal (Q to P) N (Geoid Height)= Distance along ellipsoid normal (Q to P_{0}_{ })
^{H}^{ }^{(Orthometric}^{ }^{Height)=}^{ }^{Distance}^{ }^{along}^{ }^{Plumb}^{ }^{line}^{ }^{(Pa }^{ }^{to}^{ }^{P}^{)}
What have we learned?
• The gravitational field generated by a mass *M** *is associated to a potential.
• There is a particular equipotential surface called the geoid, defined to best fit the mean sea level.
• The geoid undulations are expressed in meters above the reference ellipsoid.
• The reference ellipsoid is a mathematical representation of the shape of the Earth.
• The geoid undulations reflect rock density variations, topography, mantle processes, etc.
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