Isostasy
• Distribution of elevations: two preferred elevations => fundamental difference between ocean and continents
• Continents ~ granite (2.67), oceans ~ basalts (3.3) => idea that Earth’s elevations are supported hydrostatically
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basic knowledge on isostacy and their theory
Typology: Lecture notes
2017/2018
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Isostasy
- • Distribution of elevations: two preferredContinents ~ granite (2.67), oceans ~ basalts (3.3) => idea that elevations => fundamental difference between ocean and continents Earth’s elevations are supported hydrostatically
• Problem: In spite of the additional terrainIsostasy
- volume, mountains are associated with^ negative Airy root that compensates for the relief (1854): Mountains have a crustal^ Bouguer^ anomalies…
- • Pratt lateral variations of temperature or composition)In both models, mountains (1855): Density varies laterally ( “float” on e.g.
- denser mantle in equilibrium = isostatic^ equilibrium, orIsostasy condition: the weight of columns of rock, at some depth called the depth of^ isostasy
compensation, is everywhere equal.
2.67 2.62 2. crust density = 2.
2.52 2.59 2.67 2.
mantle density = 3.
0 0
Partt’s model 100 km Airy’s model^ compensation depth compensation depth
Isostasy
http://atlas.geo.cornell.edu/education/student/isostasy.html
• Why are mountains high?Think about this…
- • • • Why do mountains have roots?How to produce a root?What are the effects of erosion?Are mountain roots permanent features?
- • • How to produce a positive topography in isostatic equilibrium assuming an Airy model? A Pratt model?What could disturb isostatic equilibrium?Can you think of another kind of equilibrium (besides isostatic
equilibrium) that would be called the processes that dynamic equilibrium may involve. “dynamic equilibrium”? Describe
- Buoyancy force: vertical, opposes loadIsostasy^ and buoyancy forces
- In reality, buoyancy force results in additional horizontal force: – – Load (Fl) acts downward on lithospheric columnRestoring buoyancy (Fb) acts upward on
- – lithospheric columnAs a result, lithospheric column experiences vertical compression (green arrows)Which is associated with horizontal
- As a result, state of horizontal stress: – – extensional (blue arrows)Extensional region => drivesCompressional at its at the center of the elevated “gravitational collapse edges => drives ” shortening
load lithosphere asthenosphere F^ Fbl
- AlpsBuoyancy forces in action?
- Tibet
• Earth’s lithosphere can be approximated as a thin elastic plate:Flexure
w(x) x q(x) F = distance= constant horizontal force per unit length == deflection vertical force per unit length (load)
• Causes for flexure of oceanic lithosphere:Flexure
- – – Seamount loadingOceanic plateau loadingSediment loading
• Causes for loading of continental lithosphere:^ – –^ Plate bending entering a subduction zoneSediment loading
- – IcesheetsThrust sheets
• Deformation of oceanic depression => isostatic equilibrium perturbed: restring buoyancy force? lithosphere under vertical load => depression + water fillsFlexure
water, oceanic lithosphere, ρw ρm hw h fluid mantle, ρm^ w
load Compensation depth^ w !
" w g ( hw + w ) + " m gh !
- Weight per unit area of column:Net hydrostatic force is the difference =^ " w^ ghw^ +^ " m^ g ( h^ +^ w )^ weight after - weight before:Weight per unit area of column: !
( " m # " w ) gw
• Further assumptions: Flexure
- – No horizontal force => F = 0Line load: • • AtAt x=0at x≠, load =0, load = 0 qo
- • For xWith a solution for x>0:≠0, the flexure equation becomes:
- Important parameters and length scales in this solution: – – α 2 πα= flexural parameter = flexural wavelength
- xo = 3πα /4 = distance to the first zero crossing.
D d dx^4 w 4 + ( " m # " w ) gw = 0 !
w = q 8 o^ " D^3 e #^ x^ "^ (cos " x + sin " x )
" = % & ' ( # m^4 $ D # w ) g^ ( ) *^14
Bathymetry and free-air gravity anomaly along a N-S line centered on the Hawaiian island of Ohau 1. 2. 3. => Combination of 3 effects:Topography => short wavelengths gravity signalWeight of the volcano => flexure of the plate => long-wavelength negative anomalyUprising asthenospheric plume => very long-wavelength positive anomaly
Plate flexure under line load: e.g., Hawai
Plate flexure under line load: e.g., Hawai After removal of the very long wavelength (mantle (a) plume) signal:Best-fitting flexural models using conventional, two-dimensional techniques. The dotted curve is the response of a 25- km-thick elastic plate while the solid curve is the response of a variable thickness plate where the thickness ranges from 35 km away from the load to 25 km beneath the (b) load.Gravity predictions based on the models in Figure 6a. Within the uncertainties of the data, both models provide reasonable fits, although the constant thickness model fits (Wessel^ slightly better et al., 1993).
Flexure - subduction
- In addition to load of overriding plate: – – SedimentsNon-elastic response
• Recall that: Flexure^ ⇔^ Isostasy
- Let – – – ’Plate is very thin, or has ~zero strength or zero flexural rigidity => D = 0No horizontal forces acting => H = 0Load is due to topography h(x) =>s assume that :
- Then the flexure equation reduces to:
- • Which is Airy isostasy … Airy is a special case of flexure when D isostasy! ( w here is hr in our Airy equation)→ 0
!
D d dx^4 w 4 + H d dx^2 w 2 + ( " m # " c ) gw = V ( x )
V ( x ) = " c gh ( x )
!
( " m # " c ) gw = " c gh ( x ) !
w = ( "^ " mc^ h #( x ") c )