Alison Rust & Neil Balmforth

1. Convection, degassing and rheology of crystal-rich magma Alison Rust & Neil Balmforth

2. The Porridge Problem “Any Scottish cook knows that if porridge is cooked in a single saucepan and not stirred it will burn at the bottom. It can still be poured – it is still liquid, but at a certain stage of stickiness convection currents can be prevented even when the bottom is some hundreds of degrees hotter than the top.” Sir Harold Jeffreys Why prevents porridge convection? Can magma be like porridge?

3. Outline Rheology of magma Convection in crystal-rich magma chambers Theory: Classical (Newtonian) Rayleigh-Bénard convection Convection with a yield strength Experiments Applications and conclusions

4. Silicate melts are Newtonian With enough crystals, magmas are viscoplastic Basalt lava flows

5.  = Crystals increase magma viscosity Dilute crystal-melt suspensions are Newtonian • deflected flow lines 20% crystals Stress  • no-slip boundaries pure melt 0 Shear rate, Viscosity =

6. log10viscosity (Pa s) Enstatite spherulites crystallized from Mg3Al2Si3O12 0% 40% 60% 100% Crystal Content Newtonian Non-Newtonian after Lejeune and Richet (1995)

7. Yield strength If crystals form a touching framework then the magma has a yield strength (Y) and there is no flow for <Y stress  Y 0 Shear rate Bingham model: .  =  The yield strength increases with the abundance of crystals Saar et al., 2001

8. Yield strength (kPa) Percent crystals Experiments with basalt Philpotts & Carrol, 1996 Basalt cube Melt 40% 30% 25% Crystal content after partial melting

9. There is a link between basalt surface morphology and crystal content pahoehoe a’a’ % crystals Folley [1999]

10. Rhyolite melt + crystals (+ bubbles) ~no crystals (obsidian flow) almost all crystals ~50% crystals y~105 Pa

11. Rheology of dome samples Levallée et al 2007 60 40 20 Power-law rheology .  = c 0 . = c

12. How important is yield strength for magma chamber dynamics? What happens at low stresses?  { Rigid, elastic, very viscous? 0

13. Evidence for convection of magma of current eruption of Mt. St. Helens The chemistry of zones in crystals indicate large increases and decreases of pressure Rutherford, in press 2008

14. Could the large pressure changes recorded be related to explosive eruptions of overlying magma? Mount St. Helens May-June 1980 Plinian Subplinian Gas flux Decompressed, non-erupted magma Repressurisation Distinctive band related to May 18, 1980 eruption

15. Post-May 18th 1980 Mount St. Helens glasses have elevated CO2 Melt inclusion Matrix glasses (dense clasts): ppm CO2 ppm CO2 wt% H2O wt% H2O Enrichment in CO2might be explained by cycles in decompression-recompression (which causes cycle: vesiculation-permeability development and gas flux-foam collapse-resorption)

16. A decrease in XH2O (increase in XCO2) of the vapour can drive crystallisation even at constant Ptotal Crystallisation at constant Ptotal

17. Many silicic lavas with high crystal contents >40% show a range of disequilibrium textures that indicate a heating event prior to eruption. Convection suggested for Souffriere Hills, Montserrat Couch et al., 2001 Alternative explanation (for Montserrat and Fish Canyon Tuff) Gas sparging through rigid mush Bachmann and Bergantz (2003, 2006)

18. Classical Rayleigh-Bénard convection T2<T1 A Newtonian fluid between two plates is heated from below. d T1 Thermal expansion of fluid near the bottom plate produces buoyancy-driven convection. Two important dimensionless parameters: g = gravity acceleration = thermal expansion coefficient T = T1-T2  = kinematic viscosity  = thermal diffusivity

19. Classical Rayleigh-Bénard convection Turbulent Time-dependent patterns increasing Ra Ra Steady convection cells Ra* No motion Pr Krishnamurti for free-slip boundaries Ra*≈658 for no-slip boundaries Ra*=1708

20. Linear stability analysis Determine the basic state of the system Add small perturbations to the basic state and evaluate whether perturbations grow (unstable) or decay (stable) Ball will sit at top of hill but if perturbed slightly to the right or left, it will roll away (unstable) If perturbed, the ball at bottom of gully will return towards basic state (stable)

21. T2 d Newtonian fluid z T1 x Linear stability analysis for the onset of thermal convection in a Newtonian fluid d Basic state: Fluid is stationary, u=0 Steady conductive temperature: T = T1-  T1-T2) z/d z 0 T2 T1 T Under what conditions will perturbations (e.g., in T or u) grow? Under what conditions will the system convect rather than simply transferring heat by conduction?

22. Will the hot buoyant perturbation grow? or decay by thermal diffusion? T2 T1

23. Add temperature perturbation: T = T1-  T1-T2) z/d +T, T = T1-T2) T Boussinesq approximation:  constant except for buoyancy related to temperature Continuity for incompressible fluid: Conservation of momentum: T = temperature d = layer thickness  = density = thermal expansion coefficient u =velocity g = gravity acceleration  = thermal diffusivity p = pressure Thermal conduction and advection: Use stream function define as:

24. For stress-free boundary conditions, look for solutions of form: where n is an integer For high Pr, the most unstable mode has n=1, k=k* d Ra unstable Ra* stable k, wavenumber g = gravity acceleration = thermal expansion coefficient T = T1-T2  = kinematic viscosity  = thermal diffusivity k*

25. How does yield strength affect convection? Consider simplest rheology with a yield strength: Bingham    Rigid for  < y Onset problem - Linear stability theory indicates:

26. Weakly nonlinear stability theory: Bifurcation diagram for the onset of convection of Newtonian fluid 0.3 convection 0.2 stable 0.1 conduction Amplitude 0 stable unstable -0.1 -0.2 -0.3 600 650 700 750 Ra Ra* Pitchfork bifurcation

27. Weakly nonlinear stability theory: Perturb about the Newtonian problem - Bingham with smally Bifurcation diagram for weakly nonlinear steady convection Ra* Saddle-node bifurcation

28. Laboratory experiments with Carbopol (~hair gel) d=4-11cm T1 increases with time no-slip boundaries Density and thermal properties same as water Rheology changes very little with heating 0.1 wt% 0.08 wt% Stress (Pa) 0.07 wt% 0.06 wt% 0.045 wt% Shear rate (s-1) Roberts and Barnes 2001

29. 1 cm No convection even for T1=80oC + major perturbation Convection Stable but will convect if kicked hard enough Increasing Carbopol concentration and yield strength Weak yield strength - see convection Strong yield strength - no convection 0.05% Carbopol 0.1% Carbopol y< 0.1 Pa y~10 Pa  > 10 Pa s > 600 Pa s  ~ 0.6 Pa s  ~ 6 Pa s

30. 1 hour later new plumes form Plumes form; T reduced Thermocouples 0 to 4 cm from base; d=11 cm

31. Same conditions: similar T required for plumes Time t+8minutes Time t+24 minutes Time t d=9cm 0.05% Carbopol Plumes at T1=57 C

32. Laboratory experiments with Carbopol Strong yield strength - no convection Weak yield strength - see convection Can we rationalize this case as finite amplitude convection? What happens below the yield stress? 1) Viscous deformation but very large viscosity? 2) Elastic deformation? Return to linear stability analysis

33.  1) High viscosity at low stress - Regularized Bingham At low shear rates: Ra* is same as for viscous Newtonian case (e.g., ~658 for free-slip) Very large viscosity means Ra tends to be small and the system is stable except under extreme conditions: convection for

35. for 2) Add elastic deformation at low stresses Combine Maxwell (viscoelastic) and Bingham models. Constitutive equation:  is relaxation time , Maxwell: for for , Bingham: Viscous relaxation time too short to excite oscillatory instability

36. For the strong yield strength Carbopol experiments - no convection Ra~10^2 For the dilute Carbopol experiments - see convection but no oscillatory instability Ra~10^4

37. Convection suggested for Souffriere Hills, Montserrat Couch et al., 2001 Assumed =106 Pa s and calculated d required to exceed Ra=658 i.e., assumed Newtonian (yield stress not important)

38. Couch et al. 2001 T=100 oC =2600 kg/m3 =2x10-4 K-1 =8x10-7m2/s =106 to 107Pa s Found need d=1-2 m for Ra≈658 d y dome lavas ~105 Pa Stress balance for d~200m

39. A Ra Convection of crystal-rich magmas? Finite amplitude convection may be possible - The “kick” needed to initiate convections is lower for smaller yield stress and higher Rayleigh number. - Once convection begins, yield stress becomes less significant and convection likely resembles Newtonian counterpart. Interesting twist: rheology will change with time. If convection begins while a dilute suspension, could continue to convect despite significant yield strength increase from crystallization. Heating a “locked” magma from below could lead to convection of hotter, lower portion of magma, overlain by locked magma.

40. The Porridge Problem Jeffreys attributed the lack of convection to high viscosity. However, porridge has a yield strength (Dejongh and Steffe, 2004) and the non-Newtonian rheology seems to be important.

41. On the topic of convection of the Earth: “I think, however, that the introduction of a non-zero strength will completely alter the solution.”Sir Harold Jeffreys Griggs (1939), Vening Meinesz (1947) and Jeffreys (1952) suggested that if the mantle has a finite “strength” then many conclusions about convection based on the assumption of Newtonian viscosity are quantitatively and qualitatively unfounded. Orowan (1965) showed considerable insight: “Naturally convection cannot start if the mantle… has a finite yield stress; however, if temperature differences of sufficient magnitude and extent are present, hot masses can rise if the yield stress is low enough.”