All Bent Out of Shape: The Dynamics of Thin Viscous Sheets
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All Bent Out of Shape: The Dynamics of Thin Viscous Sheets
Neil M. Ribe Institut de Physique du Globe, Paris
D. Bonn (U. of Amsterdam)M. Habibi (IASBS, Zanjan, Iran)M. Maleki (IASBS)H. Huppert (U. of Cambridge)M. Hallworth (U. of Cambridge)J. Dervaux (U. of Paris-7)E. Wertz (U. of Paris-7)E. Stutzmann (IPGP)Y. Ren (IPGP)N. Loubet (IPGP)Y. Gamblin (IPGP)R. Van der Hilst (MIT)
In collaboration with:
Photo courtesy of Mars, Inc.
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Modeling Earth’s lithosphere as a thin sheet
Analog experiments on subduction (Roma-III):
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Thin viscous sheets: two modes of deformation
1. Stretching :
2. Bending :
{
rate of change of curvature
viscousflexuralrigidity
{
Bending moment :
Stress resultant :
rate of stretching
Troutonviscosity
{ {
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Loaded viscous sheets: bending vs. stretching
Purestretching
Purebending
Partitioning depends on: (1) sheet shape (curvature) (2) loading distribution (3) edge conditions
Rule of thumb:
Example: periodic normal loading (no edges)
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Intermediate length scales in viscous sheet dynamics
Competition of bending and stretching
periodic buckling instabilities :
length scales that are intermediate between the
sheet’s thickness and its lateral dimension
Examples :
normally loaded spherical or cylindrical sheets :
stretching/bending transition occurs at load wavelength
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An analogous system: coiling of a viscous « rope »
Why coiling is simpler than folding:
folding is inherently unsteady
edge-on view: side view:
folding sheets contract in the transverse direction :
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Experimental setup
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Multistable coiling
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1 cm
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Coiling frequency vs. height: Experimental observations
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Steady coiling: mathematical formulation
19 boundary conditions
Numerical method: continuation (AUTO 97; Doedel et al. 2002)
17 governing equations (3 force balance; 3 torque balance; 7 geometric; 4 constitutive relations)
17 variables: coordinates of axis
Euler parameters
curvatures of axis
axial velocity
spin about axis
stress resultant vector
bending moment vector
2 unknown parameters: coiling frequency
filament length
17th-order nonlinear boundary value problem
(Ribe 2004, Proc. R. Soc. Lond. A)
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Coiling frequency vs. height: Experimental observations
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Comparison with numerical solutions
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Coiling frequency vs. height: Four regimes
« Pendulum » modes;
n=1
2
3
(Mahadevan et al., Nature 2000; Ribe, Proc. R. Soc. Lond. 2004;Maleki et al., Phys. Rev. Lett. 2004; Ribe et al., J. Fluid Mech. 2006; Ribe et al., Phys. Fluids 2006)
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Slow deformation of thin viscous sheets: governing equations (2D)
Evolution equationsfor the sheet’s shape:
Viscous stress resultant:
stretching bending
Force balance:viscous gravity forces
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Development of a buckling instability(numerical simulation; Ribe, J. Fluid Mech. 2002)
extension compression bending
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Multivalued folding
Frequencyvs. height :
Mode 1
Mode 2
First two« pendulum » modes :
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Two modes of slow (inertia-free) folding
Mode Amplitude Numerical simulation
Viscous (V)
Gravitational (G)
(Skorobogatiy &Mahadevan 2000)
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Composite scaling law:
2. stretching in the tail:
V
G
20
40
1. fold amplitude:
Folding amplitude: Universal scaling law
(Ribe, Phys. Rev. E 2003)
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Scaling law for folding amplitude: Experimental verification
Guillou-Frottier et al. 1995
(kg/m3) (Pa s) U0 (cm/s) h0 (cm) dip (deg.) (cm)
58 7 X 105 0.05 1.0 35 5.7
6.5 cm
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Scaling law for folding amplitude: Experimental verification
Guillou-Frottier et al. 1995
error: 2%
6.5 cm
(kg/m3) (Pa s) U0 (cm/s) h0 (cm) dip (deg.) (cm)
58 7 X 105 0.05 1.0 35 5.7
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• CMB ~2890 km depthCMB ~2890 km depth
Earth’s SurfaceEarth’s Surface
JapanJapanCentral AmericaCentral America
Izu BoninIzu BoninIndonesiaIndonesia
Fiji-TongaFiji-Tonga
Albarède and van der Hilst, Phil. Trans. R. Soc. Lond. A, 2002
Tomographic images of subducted slabs
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Case study : Central America
Regional seismictomography: (Ren et al. 2006)
(kg/m3) (Pa s) U0 (cm/yr) h0 (km) dip (deg.) (km)
65 1023 6.3 45 65 460Predicted foldingamplitude:
660
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Central America : predicted fold amplitude vs. observed width of tomographic anomalies
(Ribe et al., EPSL 2007)
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Effect of trench rollback on buckling
Numerical model of buckling with rollback
Buckling ceases when Buckling frequency vs. rollback speed
Parameters :
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Folding of subducted lithosphere at the CMB?
Experimental setup (IPGP)
Seismic observations of folding (?) beneath Central America
(Hutko et al. 2006)
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Rescaled folding frequencies:
Folding is unaffected by the ambient fluid
Limit 1: High viscosity contrast
1 cm
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Limit 2: Low viscosity contrast
1 cm
4 cm
side view: from above:edge-on view:
folding suppressed
small-amplitude waves propagate downward
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Boundary-integral formulation for free subduction
two coupled Fredholm integral equations for u(x) on the contours C1 and C2:
slab buoyancy
restoring forceon topography
double-layer integrals}
Advantages:
Reduction of dimensionality (3D 2D or 2D 1D)
No wall effects (unless desired)
Accurate interface tracking
True free surface
Closely matches typical experimental configurations
(e.g. Roma-III)
see alsoposter ofG. Morra et al.
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What holds the plate up?
Numericalmodel:
Laboratoryexperiments:
surface tension force (?)
buoyantrestoringforce
lubricationforce
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Example: 2D sheet subducting in a fluid half-space
Horizontal velocity of the free surface:
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Systematics :plate speed vs. slab dip and viscosity contrast
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Work in Progress
Comparison with laboratory experiments (Roma-III)
Extension of the numerical approach to :
Predictive scaling laws for key subduction parameters :
Plate speed Trench rollback speed Point of transition to back-arc extension State of stress in the slab Slab morphology (incl. buckling instabilities)
3D flow Mantle viscosity stratification
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Physical criterion for onset of buckling
Principle: (a) compression is required to amplify shape perturbations (Taylor 1968) (b) growth rate of perturbations must exceed thickening rate
Experimental observation:
Onset of buckling depends only on geometry,independently of viscosity and thickening rate
Mathematical analysis:
(1) Torque balance in the absence of body forces:
(2) Eigensolution for clamped ends:
(3) Growth rate exceeds thickening rate if