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