Interactions between the cryosphere and the solid Earth

1. Interactions between the cryosphere and the solid Earth Glenn Milne, Durham University, UK Karthaus 2007

2. Overview Fig. 3.14(b) from Wilson et al. Fig. 3.1 from Wilson et al.

3. GLACIAL ISOSTATIC ADJUSTMENT Surface Mass Redistribution Earth Earth Response • Relative sea level • Geopotential • Rotation vector • 3D solid surface deformation Model Surface load + Rotational potential Rheological Earth model Constraints on surface mass redistribution Better understanding of GIA process Constraints on Earth rheology

4. Key Elements of a GIA Model Earth Forcing Earth Model Rotational potential Surface loading Geometry Rheology Euler equations Spherical/Flat Internal structure: 1D & 3D Viscoelastic Linear and non-linear viscous deformation Ice Model Ocean Model Multidisciplinary approach Sea-level equation

5. General Outline Part 1: Modelling the response of the solid Earth to surface loads. Part 2: Modelling sea-level changes driven by growth/melting of cryosphere. Part 3: Constraining past and present cryospheric changes using geophysical observations.

6. Part 1: Modelling the Response of the Earth to Surface Loads

7. Isostasy Surface loading history Earth density structure Earth rheology structure http://gsc.nrcan.gc.ca/geodyn/rebound_e.php#load

8. Learning About the Rheology of Rocks • There are three general approaches: • (1) Theoretical (mineral physics). • (2) Experimental (lab deformation experiments on rock and mineral samples). • (3) Geophysical modelling of surface observables. • Each approach has specific strengths and weaknesses, but they are all complementary.

9. Bulk rheology governed by processes at grain and sub-grain scale Means, Stress and Strain, Fig. 1.2

10. Processes at lattice-scale govern bulk rheological response Knipe, J. Struct. Geol., 11, pp127-146, 1989.

11. Bulk Rheological Response of Rocks Is it possible to simulate this general (plastic) response using simple rheologies?

12. Combining simple macro rheologies Ranalli, p83

13. Material Failure is a Key Concept Ranalli, Rheology of the Earth (Second Edition)

14. Brittle and Ductile Failure • Brittle failure is described by the Coulomb-Navier failure criterion: • Ductile failure can be estimated via the Dorn equation:

15. Strength Temperature increasing strength with depth due to increased overburden pressure crust Depth upper mantle lithosphere Decrease in strength due to increase in strain rate with higher temperature Courtesy of Kurt Lambeck

16. Strength Different mineralogies? Temperature crust Depth upper mantle lithosphere Courtesy of Kurt Lambeck

17. Some Example Strength Profiles Ranalli

18. Application of Strength Profiles: Fibre Stress and Effective Elastic Thickness Ranalli See also Ranalli & Murphy, Tectonophysics, 132, 281-295, 1987.

19. Common surface deformation formulations • Elastic or viscoelastic half-space, with or without depth-dependent parameters. • Suitable for relatively small loads on short time scales. • Elastic layer over a viscous mantle • Suitable for small to medium sized loads on long time scales. • Spherical and fully gravitational viscoelastic Earth models • Suitable for small to large sized loads over a range of time scales.

20. Elastic Layer Over a Viscous Mantle • Amplitude of deformation governed by properties of elastic plate. • Viscous substrate governs the rate of deformation through a specified relaxation time. • Key parameters are the flexural rigidity and the relaxation time. • Some useful references: Turcotte and Schubert (2002); Watts (2001).

21. Spherically Symmetric, Self-Gravitating, Visco-Elastic Earth Models (I) • Consider the response of Earth model to an impulse forcing (Peltier, 1974). • Simultaneously solve three equations: (i) momentum equation, (ii) continuity equation, (iii) Poisson’s equation. Response of model to an impulse forcing. • Apply the correspondence principle. • Solution usually expressed in terms of Love numbers.

22. Spherically Symmetric, Self-Gravitating, Visco-Elastic Earth Models (II) • Love numbers depend on elastic, viscous and density structure of a given model. • Compute the Earth model response to a general load by convolving the load history with the appropriate Green’s function. • Key references: Peltier (1974), Wu and Peltier (1982).

23. Seismic Body Waves Fig. 2.1 from Brown & Mussett, The Inaccessible Earth (2nd edition)

24. Body Wave Ray Paths and Seismographs Fig. 1.1-3 from Stein and Wysession

25. Body Wave Travel Time Data Fig. 1.1-3 From Stein and Wysession

26. Some 1-D Earth Models Shearer Stein and Wysession

27. Typical Earth Model Structure • Elastic and density structure from seismic models • Viscosity structure commonly inferred via geophysical modelling. • Elastic lithosphere simulated by adopting high values of viscosity in outer layer. • Hl = 50-120 km • h2 = (1-10)×1020 Pa s • h3 = (1-100)×1021 Pa s Courtesy of Kurt Lambeck

28. Example of a Viscosity Inference Mitrovica & Forte, Earth and Planetary Science Letters, 225, pp177-189, 2004.

29. Time Dependence of Vertical Deformation Courtesy of Kurt Lambeck

30. Most Appropriate Bedrock Model for Ice Sheet Modellers? • Elastic plate over viscous substrate found to give sufficiently accurate results for simulations of the Antarctic ice sheets (Le Meur and Huybrechts, 1996). • However, this model has been found to give poor results for simulations of the Eurasian ice sheets (Van den Berg et al., 2007).

31. Comparison for Eurasian Ice Sheets A Van den Berg et al. (2007) B C

32. Comparison for Eurasian Ice Sheets Van den Berg et al. (2007)

33. Models of 3-D Viscoelastic Structure Lateral variations in Earth properties can have a significant influence on the vertical deformation in some areas. Whitehouse et al. (2006)

34. Part 2: Modelling Sea-Level Changes Driven by Growth/Melting of the Cryosphere

35. Processes Contributing to Sea-Level Change

36. Glacio-Eustasy

37. Sea-Level Change at Barbados Bard et al. (1990)

38. Deviation From Eustatic

39. Processes Causing Deviation From Eustatic • Geoid perturbation, • Gravity field perturbed directly by surface mass redistribution and changing rotational potential and indirectly by earth deformation caused by these forcings. • Solid surface perturbation, • Vertical solid surface deformation associated with surface mass redistribution and changing rotational potential. • Ice-water mass conservation, • Volumetric sea-level change must be consistent with mass lost/gained by ice sheets.

40. A Theory of Sea-Level Change Associated With Ice-Ocean Mass Exchange • Represented by the sea-level equation (Farrell and Clark 1976). • An integral equation that considers the influence of ice-ocean loading and, more recently, perturbations in Earth rotation on sea level. • Commonly solved using a pseudo-spectral approach (Mitrovica and Peltier 1991). • The most recent version of the sea-level equation is given by Mitrovica and Milne (2003).

41. Key Elements of a GIA Model Earth Forcing Earth Model Rotational potential Surface loading Geometry Rheology Euler equations Spherical/Flat Internal structure: 1D & 3D Viscoelastic Linear and non-linear viscous deformation Ice Model Ocean Model Multidisciplinary approach Sea-level equation

42. Changes in Earth Rotation www.iers.org/MainDisp.csl?pid=95-89

43. Pole Tide Effect • Global scale perturbation to rotational potential. • Motion of pole is on the order of 10 km over a glacial cycle. • Perturbs both the gravity field and the solid Earth.

44. Isostatic Response of the Solid Earth

45. Gravitational Effects ICE OCEAN Ice melt ICE OCEAN Tamisiea et al. (2003)

46. Ice-Induced Syphoning Mitrovica & Milne (2002)

47. Ocean-Induced Syphoning Mitrovica & Milne (2002)

48. Deviation From Eustatic