All Bent Out of Shape: The Dynamics of Thin Viscous Sheets

All Bent Out of Shape: The Dynamics of Thin Viscous Sheets Neil M. Ribe Institut de Physique du Globe, Paris In collaboration with: 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) Photo courtesy of Mars, Inc.

Analog experiments on subduction (Roma-III): Modeling Earth’s lithosphere as a thin sheet

1. Stretching : Stress resultant : { { Trouton viscosity rate of stretching 2. Bending : Bending moment : { { viscous flexural rigidity rate of change of curvature Thin viscous sheets: two modes of deformation

Pure bending Pure stretching Rule of thumb: Loaded viscous sheets: bending vs. stretching Partitioning depends on: (1) sheet shape (curvature) (2) loading distribution (3) edge conditions Example:periodic normal loading (no edges)

 length scales that are intermediate between the sheet’s thickness and its lateral dimension • normally loaded spherical or cylindrical sheets :  periodic buckling instabilities : stretching/bending transition occurs at load wavelength Intermediate length scales in viscous sheet dynamics Competition of bending and stretching Examples :

An analogous system: coiling of a viscous « rope » • folding sheets contract in the transverse direction : edge-on view: side view: Why coiling is simpler than folding:  folding is inherently unsteady

Experimental setup

1 cm Multistable coiling

Coiling frequency vs. height: Experimental observations

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 Steady coiling: mathematical formulation (Ribe 2004, Proc. R. Soc. Lond. A) 17 governing equations(3 force balance; 3 torque balance; 7 geometric; 4 constitutive relations) 19 boundary conditions  Numerical method:continuation(AUTO 97; Doedel et al. 2002)

Coiling frequency vs. height: Experimental observations

Comparison with numerical solutions

« Pendulum » modes; 3 2 n=1 (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) Coiling frequency vs. height: Four regimes

Force balance: viscous gravity forces Viscous stress resultant: stretching bending Evolution equations for the sheet’s shape: Slow deformation of thin viscous sheets: governing equations (2D)

Development of a buckling instability (numerical simulation; Ribe, J. Fluid Mech. 2002) extension compression bending

Frequency vs. height : Mode 1 First two « pendulum » modes : Mode 2 Multivalued folding

Mode Amplitude d Numerical simulation Viscous (V) Gravitational (G) (Skorobogatiy & Mahadevan 2000) Two modes of slow (inertia-free) folding

Folding amplitude: Universal scaling law (Ribe, Phys. Rev. E 2003) 1.fold amplitude: 2. stretching in the tail: V G 20 40 Composite scaling law:

Scaling law for folding amplitude: Experimental verification  6.5 cm  Guillou-Frottier et al. 1995

Scaling law for folding amplitude: Experimental verification  6.5 cm  Guillou-Frottier et al. 1995 error: 2%

Japan Central America Izu Bonin Indonesia Earth’s Surface Fiji-Tonga • CMB ~2890 km depth Tomographic images of subducted slabs Albarède and van der Hilst, Phil. Trans. R. Soc. Lond. A, 2002

Case study : Central America Regional seismic tomography: (Ren et al. 2006) 660 Predicted folding amplitude:

Central America : predicted fold amplitude vs. observed width of tomographic anomalies (Ribe et al., EPSL 2007)

Parameters : Numerical model of buckling with rollback Buckling frequency vs. rollback speed Buckling ceases when  Effect of trench rollback on buckling

Seismic observations of folding (?) beneath Central America Experimental setup (IPGP) (Hutko et al. 2006) Folding of subducted lithosphere at the CMB?

Limit 1: High viscosity contrast Rescaled folding frequencies: 1 cm Folding is unaffected by the ambient fluid

Limit 2: Low viscosity contrast edge-on view: from above: side view: 4 cm 1 cm +folding suppressed + small-amplitude waves propagate downward

see also poster of G. Morra et al.  two coupled Fredholm integral equations for u(x) on the contours C1 and C2: slab buoyancy restoring force on topography } double-layer integrals • Closely matches typical experimental configurations (e.g. Roma-III) Boundary-integral formulation for free subduction Advantages:  Reduction of dimensionality (3D  2D or 2D  1D)  True free surface  No wall effects (unless desired)  Accurate interface tracking

surface tension force (?) Laboratory experiments: lubrication force buoyant restoring force Numerical model: What holds the plate up?

Example: 2D sheet subducting in a fluid half-space Horizontal velocity of the free surface:

Systematics : plate speed vs. slab dip and viscosity contrast

Work in Progress Extension of the numerical approach to :  3D flow  Mantle viscosity stratification 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) Comparison with laboratory experiments (Roma-III)

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 n and thickening rate D Mathematical analysis: (1) Torque balance in the absence of body forces: (2) Eigensolution for clamped ends: (3) Growth rate l exceeds thickening rate D if

All Bent Out of Shape: The Dynamics of Thin Viscous Sheets