lecture 7 flexure and gia - ernetaghosh/teaching/lecture7_gia.pdflecture 7 flexure and gia flexure...
TRANSCRIPT
Lecture 7
Flexure and GIA
Flexure Flexure
• Earth’s lithosphere can be approximated as a thin elastic plate:
w(x) = deflection
x = distance
q(x) = vertical force per unit length (load)
F = constant horizontal force per unit length
V(x)
D d4w/dx4 = V(x) – H d2w/dx2
Flexure
• Bending of elastic plate as a function of distance x is given by:
• D is the flexural rigidity of the plate, defined by:
(= force couple required to bend a rigid structure to a unit curvature)
• Example values for Te and D:
– Appalachians: Te = 105 km, D = 10600x1021 Nm
– Appenines: Te = 11.5 km, D = 14x1021 Nm
!
q(x) = Dd4w
dx4
+ Fd2w
dx2
+ R
!
D =Eh
3
12(1"# 2)
w(x) = deflection
x = distance
q(x) = vertical force per unit length (load)
F = constant horizontal force per unit length
D = flexural rigidity of the plate
R = other restoring forces
E = Young’s modulus
h = plate thickness
σ = Poissons’ ratio
Verticalload
Resistance + end load + other
Flexural rigidity
H
Flexure
• Deformation of oceanic lithosphere under vertical load => depression + water fillsdepression => isostatic equilibrium perturbed: restring buoyancy force?
hw
h
water, ρw
oceanic lithosphere, ρm
fluid mantle, ρm
w
load
wCompensation depth
!
"wg(hw + w) + "mgh
!
"wghw + "mg(h + w)
Weight per unit area of column: Weight per unit area of column:
• Net hydrostatic force is the difference = weight after - weight before:
!
("m # "w )gw
D d4w/dx4 = V(x) – (ρm – ρw) gw
Flexure
• Further assumptions:
– No horizontal force => F = 0
– Line load:
• At x=0, load = qo
• At at x≠0, load = 0
• For x≠0, the flexure equation becomes:
• With a solution for x>0:
• Important parameters and length scales in thissolution:
– α = flexural parameter
– 2πα = flexural wavelength
– xo = 3πα /4 = distance to the first zero crossing.
!
Dd4w
dx4
+ ("m # "w )gw = 0
!
w =qo"
3
8De#x
" (cosx
"+ sin
x
")
!
" =4D
(#m $ #w )g
%
& '
(
) *
1
4
Flexure
• Further assumptions:
– No horizontal force => F = 0
– Line load:
• At x=0, load = qo
• At at x≠0, load = 0
• For x≠0, the flexure equation becomes:
• With a solution for x>0:
• Important parameters and length scales in thissolution:
– α = flexural parameter
– 2πα = flexural wavelength
– xo = 3πα /4 = distance to the first zero crossing.
!
Dd4w
dx4
+ ("m # "w )gw = 0
!
w =qo"
3
8De#x
" (cosx
"+ sin
x
")
!
" =4D
(#m $ #w )g
%
& '
(
) *
1
4
flexural parameter
V
Flexure, infinite plate, line load
e.g., oceanic island chain
!
w =qo"
3
8De#x
" (cosx
"+ sin
x
")
Flexural forebulgeZero crossing
Flexure - subduction
• In addition to load of overriding plate:
– Sediments
– Non-elastic response
Fowler: The Solid Earth
Credit: Researchgate
A special case of flexure and isostasy…
11,000 years ago, large parts of
N. Europe and N. America were
covered by ice sheets up to 3 km
thick.
Ice sheets melted rapidly ~10,000
years ago as a result of global
climate change.
Isostatic rebound
E. Calais notes
colorado.edu
• Both elastic response (instantaneous) of lithosphere and viscous response (delayed) of mantle
• Measured from sealevel changes
• GPS
measurements
Glacio-isostatic adjustment (GIA)
http://www.antarcticglaciers.org
In North America…
Calais et al., 2006
• Morphological and gravity observations
in Scandinavia:– Total uplift ~ 275 m
– Current uplift: up to ~ 1 cm/yr
– Negative Bouguer anomaly (mass deficit
because the lithosphere is still rising)
GPS data in Scandinavia
GIA is happening today…
GPS data in North America
GIA can tell us about the absolute viscosity of the mantle