Transcript Geodynamics Flexure of the Lithosphere Tony Watts

Geodynamics
Flexure of the Lithosphere
Graduate Class: Scripps’s Institution of Oceanography
February 2005
Tony Watts
[email protected]
The aim of this part of the course is to discuss the observational evidence and the
constraints that the phenomena of flexure has provided on the mechanical
properties of the lithosphere. Particular emphasis is given to a consideration of
flexure’s role in understanding the evolution of the geological features in the
oceans and continents .
Outline
Lecture 1:
Plate flexure. Elastic plate model. Line loads. Oceanic lithosphere. Seamounts
and oceanic islands. River deltas. Trenches. Elastic thickness Vs. Load and Plate
age. Viscosity structure. Time scales of isostatic adjustment. Moat stratigraphy
and facies. Curvature and Yielding. Elastic and seismogenic layer.
The lithosphere as a filter.
Lecture 2:
Aggradation and the Steer’s head model. Progradation and the
clinoform break model. Continental lithosphere. Late glacial rebound.
Extensional tectonics and rifts. Compressional
tectonics and foreland basins. Wilson cycle. Inheritance. Spectral studies, elastic
thickness and terranes.
Resources
Books
Allen, P. A. & Allen, J. R. “Basin Analysis”. Blackwells. Chapter 6.
Turcotte, D. L. & Schubert, G. “Geodynamics”. John Wiley & Sons.
Chapters 3 and 4.
Watts, A. B. “Isostasy and Flexure of the Lithosphere”. Cambridge
University Press. Chapters 4 to 7.
Journal Articles
Web
http://topex.ucsd.edu/geodynamics
http://www.earth.ox.ac.uk/~tony/watts/INDEX.HTM
Plate flexure
Seamounts….River Deltas….Trenches….Late Glacial Rebound
Elastic plate (flexure) model
Assumptions :
Parameters:
1. Linear elasticity
2. Plane stress
3. Cylindrical bending
4. Thin plates (i.e. plate
thickness << radius
curvature)
5. Neutral surface,
fixed at the half depth
D = flexural rigidity
y = flexure
m = density of invsicid
substrate
infill = density of infill
E = Young’s modulus
Te = elastic thickness
v = Poissons ratio
d4 y
D 4 + (  m –  infill ) y g = 0
dx
E Te3
D=
12 (1 –  2)
Line loads
Continuous plate :
y=
Pb
e - x(Cos x + Sin  x)
2 (  m –  infill ) g
1/ = flexural parameter
( m –  infill ) g
=
4D
1/4
Discontinuous (ie broken) plate :
y=
2 Pb 
e – x Cos x
(  m –  infill ) g
Distributed loads can be modelled as one or more line loads
Hawaiian-Emperor seamount chain
Kauai
Plate
Mechanics
(flexure)
Plate
kinematics
Oahu
Molokai
Maui
Hawaii
Sandwell & Smith 1997 (offshore) +Woollard et al 1966 (onshore)
Gravity anomalies and crustal structure at Oahu/Molokai
Watts & ten Brink (1989)
Estimating Te
Te can be estimated by comparing the
amplitude and wavelength of the observed
gravity anomaly to the predicted anomaly
based on an elastic plate model.
The minimum in the RMS difference between
observed and calculated gravity anomaly
indicate a ‘best fit’ Te ~ 30 km.
Crustal structure and flexure along the Hawaiian Ridge
Watts and ten Brink (1989)
Sediment loading at the Amazon Cone
Amazon river
Peak bulge at coastline: Drainage divide
with subsidence seaward and uplift landward
Amazon Cone Flexure
Cochran (1973)
Te ~ 31 km
? stretched continental lithosphere ~105 Myr
after initiation of rifting
Topography seaward of the Kuril Trench
Distance to bulge ~ 120-140 km
Te ~ 30 km
Gravity anomalies and flexure
The gravity anomaly, Dgflexure, is given (approx.) by :
D g flexure = 2  G ( m –  w) h
D g flexure ~ 95 mGal / km
where G = Universal Gravitational Constant, w =
density of water, m = density of mantle, h = observed
bathymetry – “normal” bathymetry
Relationship between oceanic Te and plate and load age
Oceanic Te increases with age of the lithosphere at the time of loading but, decreases with load age.
There is therefore a “competition” between thermal cooling, which strengthens the lithosphere, and
a load-induced stress-relaxation which weakens it.
Watts & Zhong (2000)
Te and the Yield Strength Envelope
The elastic model implies that all stresses are supported
elastically and that the maximum stresses accumulate in
the uppermost and lowermost part of the plate.
M elastic =
–T e
2
Te
2
 x y f dy
2
1 = K = –M elastic 12 (1 –  )
r
E T3
e
In the YSE model, however, stresses are relieved by
brittle failure in the uppermost part of the plate and by
ductile flow in the lowermost part.
M YSE = M upper + M core + M lower
Te’ of an inelastic plate (ie one that yields) can be computed from MYSE by assuming it to have the same
curvature as an elastic plate.
Te , curvature and yielding
We can show using the YSE model that the amount of yielding in a flexed plate depends on the
curvature and, hence, size of an applied load.
These considerations show that because of curvature, the elastic thickness, Te’, will usually
be less than the mechanical thickness, Te.
Yielding on the seaward walls of deep-sea trenches
Kobayashi et al (1998)
Yield Strength Envelope
The Yield Strength Envelope (YSE) combines the brittle and ductile deformation laws of rock
mechanics into a single strength profile for the lithosphere.
The ductile flow law is given by:
. = Ap ( 1 –  3) n e – Rg T
Qp
where n is a positive integer, An is the power law stress constant, (1-3) is the stress difference, Qp is the
power law activation energy, Rg is the universal gas constant and T is temperature.
[Olivine : n=3, An =7.0 x 10-14 and Qp =520 kJ mol-1]
The area under the YSE is a measure of the integrated strength of the lithosphere. The YSE shows that
the thickness of the strong zone is greater than the elastic core and increases linearly with the
square root of age.
Viscosity structure of oceanic lithosphere
Creep Law - Olivine
 = A /m
n
b/d
m
e – (Q p + PV) / RgT
Karato & Wu (1993)
where  = shear stress, m = shear modulus, b and d are material properties, P = ambient
pressure, V = activation volume, Qp = activation energy, Rg = Universal Gas Constant and T =
temperature.
Weertman & Weertman (1975)
 =  . =  ref e (Q p + PV) / RgT
2
Comparison of predictions of simple plate (flexure) models with
the Zhong (1997) multilayered viscoelastic model
Watts & Zhong (2000)
Flexure and the time-scales of isostatic adjustment
Stratigraphy of flexural moats that flank oceanic volcanoes
Offlap due to load-induced stress
relaxation
Tenerife
Pre-existing Mesozoic sediments
Onlap as moat fills during Neogene
Tenerife
Pre-existing Mesozoic sediments
Collier & Watts (2001)
Te, the seismogenic layer thickness, and the strength of the
mantle
Watts & Burov (2003)
Oceans : earthquakes occur in the sub-oceanic mantle, but the mantle is also
involved in the support of long-term loads.
Continents : earthquakes are rare in the sub-continental mantle, but it is still
involved in the support of long-term loads
The Lithosphere as a filter
It is useful when considering the flexural response to consider the lithosphere as a time-invariant filter
which responds to loads in a way that takes into account both the amplitude and wavelength of loading.
First, consider the solution of the general equation for deflection of an elastic plate when subject to a
periodic load i.e.
4 y
D 4 + ( m –
x
 infill ) y g = ( c –  w) g h Cos(kx)
The solution is also periodic and of the form :
Flexural response
function
4
(  c –  w) h Cos(kx)
D
k
y=
+1
(  m –  infill )
(  m –  infill ) g
where k = wavenumber (2/,  = wavelength) and h = load height.
-1
Large flexure
under topographic
load. Hence,
small-amplitude
free-air gravity
anomalies
max
Small flexure
under topographic
load. Hence,
large-amplitude
free-air gravity
anomalies
The figure shows that the lithosphere is behaving as a filter in the way that it responds to loads;
suppressing the short-wavelength deformation associated with local models of isostasy (e.g. Airy)
and passing the long wavelengths associated with flexure.
Flexural response
function
The non-isostatic (flexural) contribution to the longwavelength gravity field
Continents
Te < ~90 km
max~ 2000 km
n ~ 20
Oceans
Te < ~ 25 km
max~ 1000 km
n ~ 40