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Motivation Theoretical solution Modeling Fracture Zone Flexure Wenyuan Fan & Nick Mancinelli November 16, 2011 Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

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Page 1: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Fracture Zone Flexure

Wenyuan Fan & Nick Mancinelli

November 16, 2011

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 2: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Outline

1 MotivationOcean Floor TopographyReality and model

2 Theoretical solutionMathmatical DescriptionLithosphere FlexureStresses in the elastic lithosphere

3 ModelingAssumptionsResultsConclusion

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 3: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Ocean Floor TopographyReality and model

Ocean Floor Topography

Figure: From Geodynamics Turcotte & Schubert

Depth =2αρm(Tm − Ts)

(ρm − ρw )

(κx

πu0

)1/2

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 4: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Ocean Floor TopographyReality and model

Mendocino Fracture Zone

Figure: Mendocino Fracture Zone

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 5: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Ocean Floor TopographyReality and model

Relevent problem in reality

Sandwell and Schubert, 1982

Figure: What brings up the problem?

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 6: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Mathmatical DescriptionLithosphere FlexureStresses in the elastic lithosphere

Depth-Age Relation

Depth-Age Relation

d(t) = dref +2αρm(Tm − Ts)

(ρm − ρw )

(κtπ

)1/2

for ages less than about 70 Myr.

Parker and Oldenburg, 1973

Davis and Lister, 1974

Oxburgh and Turcotte, 1978

Parsons and Sclater, 1977

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 7: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Mathmatical DescriptionLithosphere FlexureStresses in the elastic lithosphere

Depth Difference

Sandwell and Schubert, 1982

d(t) = dref + 2αρm(Tm−Ts)(ρm−ρw )

(κtπ

)1/2

hA = d(tA′)− d(0)

= 2αρm(Tm−Ts)(ρm−ρw )

(κtA′π

)1/2

hB = d(tB′)− d(tB)

= 2αρm(Tm−Ts)(ρm−ρw )

(κπ

)1/2(t

1/2B′ − t

1/2B )

δB = hA − hB

= 2αρm(Tm−Ts)(ρm−ρw )

(κπ

)1/2

×(4t1/2 + t1/2B − t

1/2B′ )

4t = tA′ − 0 = tB′ − tB

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 8: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Mathmatical DescriptionLithosphere FlexureStresses in the elastic lithosphere

Model

Flexure Equation

Dd4w

dx4+ P

d2w

dx2+ g(ρm − ρw )w = qa(x)

D d4wdx4 Strength of plate

P d2wdx2 End load

g(ρm − ρw )w Restricting force

qa(x) Applied load

Flexure Equation

Dd4w

dx4+ g(ρm − ρw )w = 0

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 9: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Mathmatical DescriptionLithosphere FlexureStresses in the elastic lithosphere

Model

Flexure Equation

Dd4w

dx4+ P

d2w

dx2+ g(ρm − ρw )w = qa(x)

D d4wdx4 Strength of plate

P d2wdx2 End load

g(ρm − ρw )w Restricting force

qa(x) Applied load

Flexure Equation

Dd4w

dx4+ g(ρm − ρw )w = 0

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 10: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Mathmatical DescriptionLithosphere FlexureStresses in the elastic lithosphere

Model

Flexure Equation

Dd4w

dx4+ P

d2w

dx2+ g(ρm − ρw )w = qa(x)

D d4wdx4 Strength of plate

P d2wdx2 End load

g(ρm − ρw )w Restricting force

qa(x) Applied load

Flexure Equation

Dd4w

dx4+ g(ρm − ρw )w = 0

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 11: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Mathmatical DescriptionLithosphere FlexureStresses in the elastic lithosphere

Flexure Model

Figure: Model used to calculate the flexural topography

Governing Equation

D1d4w1dx4 + g(ρm − ρw )w1 = 0 x < x1

Did4widx4 + g(ρm − ρw )wi = 0 xi−1 < x < xi

DN+1d4wN+1

dx4 + g(ρm − ρw )wN+1 = 0 xN < x

where Di is the flexural rigidity of ith block.

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 12: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Mathmatical DescriptionLithosphere FlexureStresses in the elastic lithosphere

Physical Constrains(Boundary Conditions)

wi − wi+1 = δi

Finite at infinity far away

Continuous Slope

Continuous Moment

Continuous Shear force

limx→∞ w1,N < Const.dwidx = dwi+1

dx

Did2widx2 = Di+1

d2wi+1

dx2

Did3widx3 = Di+1

d3wi+1

dx3

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 13: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Mathmatical DescriptionLithosphere FlexureStresses in the elastic lithosphere

Physical Constrains(Boundary Conditions)

wi − wi+1 = δi

Finite at infinity far away

Continuous Slope

Continuous Moment

Continuous Shear force

limx→∞ w1,N < Const.

dwidx = dwi+1

dx

Did2widx2 = Di+1

d2wi+1

dx2

Did3widx3 = Di+1

d3wi+1

dx3

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 14: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Mathmatical DescriptionLithosphere FlexureStresses in the elastic lithosphere

Physical Constrains(Boundary Conditions)

wi − wi+1 = δi

Finite at infinity far away

Continuous Slope

Continuous Moment

Continuous Shear force

limx→∞ w1,N < Const.

dwidx = dwi+1

dx

Did2widx2 = Di+1

d2wi+1

dx2

Did3widx3 = Di+1

d3wi+1

dx3

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 15: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Mathmatical DescriptionLithosphere FlexureStresses in the elastic lithosphere

Physical Constrains(Boundary Conditions)

wi − wi+1 = δi

Finite at infinity far away

Continuous Slope

Continuous Moment

Continuous Shear force

limx→∞ w1,N < Const.dwidx = dwi+1

dx

Did2widx2 = Di+1

d2wi+1

dx2

Did3widx3 = Di+1

d3wi+1

dx3

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 16: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Mathmatical DescriptionLithosphere FlexureStresses in the elastic lithosphere

Physical Constrains(Boundary Conditions)

wi − wi+1 = δi

Finite at infinity far away

Continuous Slope

Continuous Moment

Continuous Shear force

limx→∞ w1,N < Const.dwidx = dwi+1

dx

Did2widx2 = Di+1

d2wi+1

dx2

Did3widx3 = Di+1

d3wi+1

dx3

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 17: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Mathmatical DescriptionLithosphere FlexureStresses in the elastic lithosphere

Physical Constrains(Boundary Conditions)

wi − wi+1 = δi

Finite at infinity far away

Continuous Slope

Continuous Moment

Continuous Shear force

limx→∞ w1,N < Const.dwidx = dwi+1

dx

Did2widx2 = Di+1

d2wi+1

dx2

Did3widx3 = Di+1

d3wi+1

dx3

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 18: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Mathmatical DescriptionLithosphere FlexureStresses in the elastic lithosphere

Physical Constrains(Boundary Conditions)

wi − wi+1 = δi

Finite at infinity far away

Continuous Slope

Continuous Moment

Continuous Shear force

limx→∞ w1,N < Const.dwidx = dwi+1

dx

Did2widx2 = Di+1

d2wi+1

dx2

Did3widx3 = Di+1

d3wi+1

dx3

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 19: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Mathmatical DescriptionLithosphere FlexureStresses in the elastic lithosphere

Physical Constrains(Boundary Conditions)

wi − wi+1 = δi

Finite at infinity far away

Continuous Slope

Continuous Moment

Continuous Shear force

limx→∞ w1,N < Const.dwidx = dwi+1

dx

Did2widx2 = Di+1

d2wi+1

dx2

Did3widx3 = Di+1

d3wi+1

dx3

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 20: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Mathmatical DescriptionLithosphere FlexureStresses in the elastic lithosphere

Differential Equations

Consider the nth order linear homogeneous differential equation:

L[y ] = any(n) + an−1y

(n−1) + · · ·+ a1y = 0

where y = y(x). Many years ago, one genius found that y = erx isa solution of the equation. Then we have:

(anrn + an−1r

n−1 + · · ·+ a1r)erx = 0

A polynomial of degree n has n zeros, say we have n rootsr1, · · · , rn. If er1x , · · · , ernx are linearly independent, the generalsolution of L[y ] = 0 is

y = c1er1x + c2e

r2x + · · ·+ cnernx

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 21: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Mathmatical DescriptionLithosphere FlexureStresses in the elastic lithosphere

Differential Equations

Consider the nth order linear homogeneous differential equation:

L[y ] = any(n) + an−1y

(n−1) + · · ·+ a1y = 0

where y = y(x). Many years ago, one genius found that y = erx isa solution of the equation. Then we have:

(anrn + an−1r

n−1 + · · ·+ a1r)erx = 0

A polynomial of degree n has n zeros, say we have n rootsr1, · · · , rn. If er1x , · · · , ernx are linearly independent, the generalsolution of L[y ] = 0 is

y = c1er1x + c2e

r2x + · · ·+ cnernx

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 22: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Mathmatical DescriptionLithosphere FlexureStresses in the elastic lithosphere

Analytical Solution 1

Governing Equation

Did4wi

dx4+ g(ρm − ρw )wi = 0

4th polynomial

Di r4i + g(ρm − ρw ) = 0

Roots

ri1 =(g(ρm−ρw )

Di

)1/4e iπ/4 ri2 =

(g(ρm−ρw )

Di

)1/4e i3π/4

ri3 =(g(ρm−ρw )

Di

)1/4e i5π/4 ri4 =

(g(ρm−ρw )

Di

)1/4e i7π/4

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 23: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Mathmatical DescriptionLithosphere FlexureStresses in the elastic lithosphere

Analytical Solution 1

Governing Equation

Did4wi

dx4+ g(ρm − ρw )wi = 0

4th polynomial

Di r4i + g(ρm − ρw ) = 0

Roots

ri1 =(g(ρm−ρw )

Di

)1/4e iπ/4 ri2 =

(g(ρm−ρw )

Di

)1/4e i3π/4

ri3 =(g(ρm−ρw )

Di

)1/4e i5π/4 ri4 =

(g(ρm−ρw )

Di

)1/4e i7π/4

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 24: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Mathmatical DescriptionLithosphere FlexureStresses in the elastic lithosphere

Analytical Solution 1

Governing Equation

Did4wi

dx4+ g(ρm − ρw )wi = 0

4th polynomial

Di r4i + g(ρm − ρw ) = 0

Roots

ri1 =(g(ρm−ρw )

Di

)1/4e iπ/4 ri2 =

(g(ρm−ρw )

Di

)1/4e i3π/4

ri3 =(g(ρm−ρw )

Di

)1/4e i5π/4 ri4 =

(g(ρm−ρw )

Di

)1/4e i7π/4

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 25: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Mathmatical DescriptionLithosphere FlexureStresses in the elastic lithosphere

Analytical Solution 2

General Solution

wi (x) = ci1er1x + ci2e

r2x + ci3er3x + ci4e

r4x

Wavelength of the flexure λ and flexture rigidity are given as:{λ = 2π( 4D

g(ρm−ρw ) )1/4

D = Eh3e

12(1−ν2)

where he is the effective elastic thickness as:

he = 2(κt)1/2erfc−1

(Tm − Te

Tm − Ts

)

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 26: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Mathmatical DescriptionLithosphere FlexureStresses in the elastic lithosphere

Analytical Solution 3

Roots

ri1 = 2πλi

(1 + i) ri2 = 2πλi

(−1 + i)

ri3 = 2πλi

(−1− i) ri4 = 2πλi

(1− i)

Flexure Topography

wi (x) = ci1er1x + ci2e

r2x + ci3 + ci4er4x

= e−2πx/λi(Ai sin 2πx

λi+ Bi cos 2πx

λi

)+e2πx/λi

(Ci sin 2πx

λi+ Di cos 2πx

λi

)with help of e iθ = cos θ + i sin θ. Ai ,Bi ,Ci ,Di are determined byphysical constrains.

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 27: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Mathmatical DescriptionLithosphere FlexureStresses in the elastic lithosphere

Analytical Solution 3

Roots

ri1 = 2πλi

(1 + i) ri2 = 2πλi

(−1 + i)

ri3 = 2πλi

(−1− i) ri4 = 2πλi

(1− i)

Flexure Topography

wi (x) = ci1er1x + ci2e

r2x + ci3 + ci4er4x

= e−2πx/λi(Ai sin 2πx

λi+ Bi cos 2πx

λi

)+e2πx/λi

(Ci sin 2πx

λi+ Di cos 2πx

λi

)with help of e iθ = cos θ + i sin θ. Ai ,Bi ,Ci ,Di are determined byphysical constrains.

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 28: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Mathmatical DescriptionLithosphere FlexureStresses in the elastic lithosphere

2 Blocks situation

Flexure Topography

w1 = e2πx/λ1

(C1 sin 2πx

λ1+ D1 cos 2πx

λ1

)x1 < 0

w2 = e−2πx/λ2

(A2 sin 2πx

λ2+ B2 cos 2πx

λ2

)x2 > 0

Based on the four physical constrains, we could solve the fourunknown parameter.

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 29: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

Mathmatical DescriptionLithosphere FlexureStresses in the elastic lithosphere

Stress and Moment

Moment

Mi (x) = −Did2wi (x)

dx2

Bending stress

σxxi (x) =6Mi (x)

h2e

=−Ehe

2(1− ν2)

d2wi (x)

dx2

Shear stress(Average)

σxzi (x) = −−Di

he

d3wi (x)

dx3

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 30: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

AssumptionsResultsConclusion

Time to Deal with Real World

Model Assumptions

FZ scarp fossilized (no slip on plane)

No lateral heat conduction

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 31: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

AssumptionsResultsConclusion

Model Results

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 105

−400

−300

−200

−100

0

100

200

300

400

w(x) − Deflection Due to Flexure

Distance across FZ (m)

Fle

xu

ral T

op

og

rap

hy (

m)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 105

−5000

−4900

−4800

−4700

−4600

−4500

−4400

−4300

−4200

−4100

Depth Profile − No Slip

Distance across FZ (m)

Actu

al T

op

og

rap

hy (

m)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 105

−6

−4

−2

0

2

4

6x 10

7 Bending Stress − σxx

Distance across FZ (m)

Be

nd

ing

Str

ess (

Pa

)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 105

−4

−2

0

2

4

6

8

10

12x 10

6 Shear Stress − σxz

Distance across FZ (m)

Sh

ea

r S

tre

ss (

Pa

)

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 32: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

AssumptionsResultsConclusion

Model Results

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 105

−400

−300

−200

−100

0

100

200

300

400

w(x) − Deflection Due to Flexure

Distance across FZ (m)

Fle

xu

ral T

op

og

rap

hy (

m)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 105

−5000

−4900

−4800

−4700

−4600

−4500

−4400

−4300

−4200

−4100

Depth Profile − No Slip

Distance across FZ (m)

Actu

al T

op

og

rap

hy (

m)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 105

−6

−4

−2

0

2

4

6x 10

7 Bending Stress − σxx

Distance across FZ (m)

Be

nd

ing

Str

ess (

Pa

)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 105

−4

−2

0

2

4

6

8

10

12x 10

6 Shear Stress − σxz

Distance across FZ (m)

Sh

ea

r S

tre

ss (

Pa

)

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 33: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

AssumptionsResultsConclusion

Model Results

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 105

−400

−300

−200

−100

0

100

200

300

400

w(x) − Deflection Due to Flexure

Distance across FZ (m)

Fle

xu

ral T

op

og

rap

hy (

m)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 105

−5000

−4900

−4800

−4700

−4600

−4500

−4400

−4300

−4200

−4100

Depth Profile − No Slip

Distance across FZ (m)

Actu

al T

op

og

rap

hy (

m)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 105

−6

−4

−2

0

2

4

6x 10

7 Bending Stress − σxx

Distance across FZ (m)

Be

nd

ing

Str

ess (

Pa

)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 105

−4

−2

0

2

4

6

8

10

12x 10

6 Shear Stress − σxz

Distance across FZ (m)

Sh

ea

r S

tre

ss (

Pa

)

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 34: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

AssumptionsResultsConclusion

Model Results

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 105

−400

−300

−200

−100

0

100

200

300

400

w(x) − Deflection Due to Flexure

Distance across FZ (m)

Fle

xu

ral T

op

og

rap

hy (

m)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 105

−5000

−4900

−4800

−4700

−4600

−4500

−4400

−4300

−4200

−4100

Depth Profile − No Slip

Distance across FZ (m)

Actu

al T

op

og

rap

hy (

m)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 105

−6

−4

−2

0

2

4

6x 10

7 Bending Stress − σxx

Distance across FZ (m)

Be

nd

ing

Str

ess (

Pa

)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 105

−4

−2

0

2

4

6

8

10

12x 10

6 Shear Stress − σxz

Distance across FZ (m)

Sh

ea

r S

tre

ss (

Pa

)

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 35: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

AssumptionsResultsConclusion

Topography

Crustal ages: 46 Myr x < 0 and 54 Myr x > 0

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 105

−5000

−4900

−4800

−4700

−4600

−4500

−4400

−4300

−4200

−4100

Depth Profile − No Slip

Distance across FZ (m)

Actu

al T

op

og

rap

hy (

m)

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 36: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

AssumptionsResultsConclusion

Deflection Due to Flexture

Flexural wavelengths: 342km(x < 0) and 358km(x > 0)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 105

−400

−300

−200

−100

0

100

200

300

400

w(x) − Deflection Due to Flexure

Distance across FZ (m)

Fle

xu

ral T

op

og

rap

hy (

m)

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 37: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

AssumptionsResultsConclusion

Deflection Due to Flexture

Flexural rigidities: 4.99× 1010 Pa× km4 and 6.00× 1010 Pa× km4

Effective elastic thicknesses: 20.5 km and 21.8 km

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 105

−6

−4

−2

0

2

4

6x 10

7 Bending Stress − σxx

Distance across FZ (m)

Be

nd

ing

Str

ess (

Pa

)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 105

−4

−2

0

2

4

6

8

10

12x 10

6 Shear Stress − σxz

Distance across FZ (m) S

he

ar

Str

ess (

Pa

)

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 38: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

AssumptionsResultsConclusion

Model VS. Reality

Age of block

Age of Left Block 42.3 Myr & Age of Right Block 23.3 Myr

−8 −6 −4 −2 0 2 4 6 8

x 104

−5500

−5000

−4500

−4000

Mendocino Fracture Zone − Model Prediction

Distance across FZ (m)

Actu

al T

opogra

phy (

m)

23.3 Myr

42.3 Myr

39.6 39.8 40 40.2 40.4 40.6−5.5

−5

−4.5

−4

depth

to s

eafloor

(km

)

latitude (degrees N)

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 39: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

AssumptionsResultsConclusion

Model VS Reality 2

−8 −6 −4 −2 0 2 4 6 8

x 104

−5500

−5000

−4500

−4000

Mendocino Fracture Zone − Model Prediction

Distance across FZ (m)

Actu

al T

op

og

rap

hy (

m)

23.3 Myr

42.3 Myr

39.6 39.8 40 40.2 40.4 40.6−5.5

−5

−4.5

−4

depth

to s

eafloor

(km

)

latitude (degrees N)

Comparisons between theoretical and actual

bathymetric profiles for Mendocino-Pioneer FZ

Pair (Sandwell and Schubert):

Model fits multiple FZs fairly well

Asymmetric flexure

Elastic coupling (apparent tilting)

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 40: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

AssumptionsResultsConclusion

Conclusion

1 No significant slip on the fossil fault planes of the Mendocinoand Pioneer FZs.

2 The lithosphere bends in the vicinity of a FZ as a consequenceof the frozen-in scarp and the differential subsidence of thelithosphere far from the FZ.

3 Good fits can be obtained by modeling this flexure as a thinelastic plate with age dependent effective elastic thickness.

4 We can use this model to calculate Te , the isotherm whichdefines the base of the elastic lithosphere. ( T = 450C )

5 Maximum bending stress ( 100 MPa) much less than stressesencountered in subduction zones.

6 In the case of multiple fracture zones, there is elastic couplingwhen two adjacent FZs are spaced less than a flexuralwavelength apart. This is true for the Mendocino-Pioneer FZpair.

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 41: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

AssumptionsResultsConclusion

Conclusion

1 No significant slip on the fossil fault planes of the Mendocinoand Pioneer FZs.

2 The lithosphere bends in the vicinity of a FZ as a consequenceof the frozen-in scarp and the differential subsidence of thelithosphere far from the FZ.

3 Good fits can be obtained by modeling this flexure as a thinelastic plate with age dependent effective elastic thickness.

4 We can use this model to calculate Te , the isotherm whichdefines the base of the elastic lithosphere. ( T = 450C )

5 Maximum bending stress ( 100 MPa) much less than stressesencountered in subduction zones.

6 In the case of multiple fracture zones, there is elastic couplingwhen two adjacent FZs are spaced less than a flexuralwavelength apart. This is true for the Mendocino-Pioneer FZpair.

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 42: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

AssumptionsResultsConclusion

Conclusion

1 No significant slip on the fossil fault planes of the Mendocinoand Pioneer FZs.

2 The lithosphere bends in the vicinity of a FZ as a consequenceof the frozen-in scarp and the differential subsidence of thelithosphere far from the FZ.

3 Good fits can be obtained by modeling this flexure as a thinelastic plate with age dependent effective elastic thickness.

4 We can use this model to calculate Te , the isotherm whichdefines the base of the elastic lithosphere. ( T = 450C )

5 Maximum bending stress ( 100 MPa) much less than stressesencountered in subduction zones.

6 In the case of multiple fracture zones, there is elastic couplingwhen two adjacent FZs are spaced less than a flexuralwavelength apart. This is true for the Mendocino-Pioneer FZpair.

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 43: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

AssumptionsResultsConclusion

Conclusion

1 No significant slip on the fossil fault planes of the Mendocinoand Pioneer FZs.

2 The lithosphere bends in the vicinity of a FZ as a consequenceof the frozen-in scarp and the differential subsidence of thelithosphere far from the FZ.

3 Good fits can be obtained by modeling this flexure as a thinelastic plate with age dependent effective elastic thickness.

4 We can use this model to calculate Te , the isotherm whichdefines the base of the elastic lithosphere. ( T = 450C )

5 Maximum bending stress ( 100 MPa) much less than stressesencountered in subduction zones.

6 In the case of multiple fracture zones, there is elastic couplingwhen two adjacent FZs are spaced less than a flexuralwavelength apart. This is true for the Mendocino-Pioneer FZpair.

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 44: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

AssumptionsResultsConclusion

Conclusion

1 No significant slip on the fossil fault planes of the Mendocinoand Pioneer FZs.

2 The lithosphere bends in the vicinity of a FZ as a consequenceof the frozen-in scarp and the differential subsidence of thelithosphere far from the FZ.

3 Good fits can be obtained by modeling this flexure as a thinelastic plate with age dependent effective elastic thickness.

4 We can use this model to calculate Te , the isotherm whichdefines the base of the elastic lithosphere. ( T = 450C )

5 Maximum bending stress ( 100 MPa) much less than stressesencountered in subduction zones.

6 In the case of multiple fracture zones, there is elastic couplingwhen two adjacent FZs are spaced less than a flexuralwavelength apart. This is true for the Mendocino-Pioneer FZpair.

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 45: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

AssumptionsResultsConclusion

Conclusion

1 No significant slip on the fossil fault planes of the Mendocinoand Pioneer FZs.

2 The lithosphere bends in the vicinity of a FZ as a consequenceof the frozen-in scarp and the differential subsidence of thelithosphere far from the FZ.

3 Good fits can be obtained by modeling this flexure as a thinelastic plate with age dependent effective elastic thickness.

4 We can use this model to calculate Te , the isotherm whichdefines the base of the elastic lithosphere. ( T = 450C )

5 Maximum bending stress ( 100 MPa) much less than stressesencountered in subduction zones.

6 In the case of multiple fracture zones, there is elastic couplingwhen two adjacent FZs are spaced less than a flexuralwavelength apart. This is true for the Mendocino-Pioneer FZpair.

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure

Page 46: Fracture Zone Flexure - topex.ucsd.edutopex.ucsd.edu/geodynamics/HW5_2011_presentations/... · Figure:From Geodynamics Turcotte & Schubert ... 1=4 ei3ˇ=4 r i3 = g ... Wenyuan Fan

MotivationTheoretical solution

Modeling

AssumptionsResultsConclusion

Questions?

Wenyuan Fan & Nick Mancinelli Fracture Zone Flexure