Lois de conservation Exemple: Formation du noyau Rotation de la terre Variations periodiques (cycles de Milankovicic) Chandler wobble Marees et Variations seculaires de la rotation Figure d’equilibre de la Terre Ellipsoide de reference et geoide Anomalies de gravite Isostasie Rebond post-glaciaire

Angular momentum is a vector Direction and length must be conserved

Core formation (conservation laws) Gravitational potential energy decreases when core forms Moment of inertia decreases Angular velocity of rotation increases Rotational energy increases Increase in energy of rotation < Decrease in gravitational potential energy Total energy must be conserved Difference goes into heat Estimates: Core formation -> K temperature increase

Order of magnitudes Gravitational potential energy of uniform Earth (Note that this is the amount of energy available at accretion) Moment of inertia (0.33 Mr 2 ) Present energy of rotation Change in rotation energy due to core formation

The acceleration of gravity of the sun and the moon is equal to the centrifugal (inertia) force. Sun T= 1 year, r = km Moon T= 28 days, r = 400,000km (note: the acceleration of moon to earth is equal to that of earth to the moon)

Definitions Ecliptic plane Inclination Eccentricity (ellipse) Perihelion Variations in orbital parameters cause the Milankovitch (Milancovicic in French) cycles. Theorie astronomique du climat.

Tides

Earth shape is that of a self gravitational rotating fluid Condition of equilibrium Gravity pseudo-potential Flattening

Equilibrium of a fluid if gravity equi-potential surfaces are isobaric surfaces Gravity potential must include the Centrifugal acceleration First approximation earth equipotentials are ellipsoids Reference ellipsoid Geoid

Gravity anomalies: Difference between observed and predicted gravity (on the reference ellipsoid) Free Air: measured gravity is corrected to account for the elevation above the reference ellipsoid. Free Air correction (dg/dr = mgal/m) Bouguer: a correction is made to account for the additional masses between the observation level and the reference ellipsoid. Bouguer correction = 2 π G ρ H (~ mgal/m) Geoid anomaly: difference between the reference ellipsoid and the actual sea level. Geoid anomaly is proportional to the variation of the potential on the reference ellipsoid (it is an anomaly because predicted potential is constant on the elipsoid). Geoid elevated = potential is higher. Comment (gravity is gradient of potential. U ~ 1/R; g ~ 1/R 2

Bouguer anomaly is systematically negative over mountain ranges. Free air is almost not correlated with topography. There is a mass deficit at depth just equal to mass of the rocks above the reference surface. This leads to the idea of isostasy: it implies that the total mass/unit area of any column of rock above some level inside the earth is constant. The pressure being constant implies hydrostatic equilibrium below this level. The concept was proposed by Pratt and Airy to explain anomalies in the geodetic survey in India.

It is now recognized that isostasy is not local but regional. It is achieved by the flexure of the lithospheric plate when a load is applied at its surface (internal loads can also cause flexure). Because the plate has some strength, only the long wavelength (wide) loads will cause flexure. This provide an estimate of the strength of the plate.

Isostatic compensation is more or less achieved independently of age or erosion level. The Earth must respond to any changes in the loading and maintain the isostatic equilibrium. For instance, Appalachians rebound as they are being eroded. Observation of the rebound following the retreat of glacier shows that the response is very rapid and provides estimate of viscosity.

Crude estimate of time constant Viscosity µ has units Pa s Pressure = g ρ L (L = length scale) We get a time constant τ = µ / g ρ L With proper b.c., we should get µ = g ρ λ τ / 4 π