CERI 7211/8211 Global Geodynamics

Review

I What is the international reference ellipsoid? I What is the (polar) ﬂattening? What is the Earth’s ﬂattening and its implication for the internal structure? I What is the McCullagh’s formula? I What is the meaning of the two terms? I Why do we usually ignore the thir or higher terms? I What is the geopotential? Normal Gravity

I How to define “latitude” on a spheroid? I How do we define the “vertical” and “horizontal” direction at a point on the reference ellipsoid? I Gravity, by definition, perpendicular to an equipotential surface. I “Vertical” is not equal to “radial”. I Geocentric latitude vs. geographic latitude. Normal Gravity cont’d

aG g

gravity =g= + G aa c Fig. 2-23 (Lowry, 2007) Normal Gravity cont’d I As a part of an Earth model like WGS84, the theoretical value of gravity on the rotating ellipsoid is calculated according to the normal gravity formula as the derivative of the geopotential:

2 2 gn = ge(1 + β1 sin λ + β2 sin 2λ,

where GE 3 27 g = 1 + f − m + f 2 − f m e a2 2 14 5 15 17 β = m − f + m2 − f m 1 2 4 14 1 5 β = f 2 − f m 2 8 8

−2 −3 and ge = 9.780318ms , β1 = 5.3024 × 10 , and −6 β2 = −5.87 × 10 . Normal Gravity cont’d I Note that f and m are roughly 1/300. So the quadratic terms, f 2, m2, and f m are all all about 300 times smaller than f and m. I Dropping all the quadratic terms, we get for λ = 90◦, g − g 5 p e = m − f ge 2

I Called Clairaut’s theorem −2 −2 I gp = 9.832177ms andge = 9.780318ms . I Corresponding to increase in gravity by 5.186 × 10−2ms−2. I What causes this increase in gravity? I distance to the center of mass of the Earth. I centrifugal force that vanishes at the poles. I extra mass under the equator due to the equatorial bulge. I Think about the sign and magnitude of each contribution. Geoid: The “Real” Shape of the Earth

I Reference ellipsoid and the normal gravity formula are idealizations for convenience. I There can be numerous propositions about what is the “real shape” of the Earth and how to represent it. I Geoid: The equipotential surface that coincides with the mean sea level over the ocean. Doesn’t have to be an ellipsoid! Geoid cont’d

(a) hill local (b) gravity geoid N ocean ellipsoid N geoid

plumb ellipsoid -line

mass excess Fig. 2-24 (Lowry, 2007)

I Geoid is represented by the spherical harmonics:

∞ m=n E X R X U = −G (C cos mφ+S sin mφ)P (cos θ). r r nm nm nm n=0 m=0

I Coefﬁcients, Cnm and Snm are calculated based on satellite geodesy and published. Anyone can construct U at desired locations (r, θ, φ) using them. Geoid cont’d I Geoid undulation or geoid anomaly height I deﬁned as (geoid radius - ellipsoid radius) I can be related to anomaly in gravitational potential:

gn∆N = ∆U = (U − Ug),

where ∆N (m) is the geoid undulation, gn is the normal gravity, and Ug the geopotential.

https://en.wikipedia.org/wiki/Earth_Gravitational_Model Geoid cont’d EGM96 Geoid: 5000 km < λ < 15,000 km

(“Geoid and topography over subduction zones: The effect of phase transformations”, King, JGR, 2002) Geoid cont’d EGM96 Geoid: λ < 5,000 km

(“Geoid and topography over subduction zones: The effect of phase transformations”, King, JGR, 2002)