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Numerical Modelling of Time-Dependent Thermal Convection in MantlePetar Glisovic, Alessandro Forte, Robert Moucha

Université du Québec à Montréal

Thermal convection is a well-known phenomenon of hydrodynamic instability.It is the result of gravitational forces acting on thermally generated density anomalies in a layer of fluid heated from below or within, or cooled from above. Convecting systems are characterized by the ascension of hot and therefore light materials and the sinking of cold ones.

The most important difficulty in the understanding of the dynamics of mantle convection is due to the complexity of the rheology; the viscosity is expected to vary by several orders of magnitude as a result of its strong dependence on temperature and pressure.

case 1 => constant viscosity = 95x10e21 Pas;case 2 => realistic variations of viscosity(derived form Grand) but amplitude x 10 (mean viscosity 95x10e21 Pas).Characteristics common to both cases:- initial temperature field (t=0) derived on the basis of Grand's seismic-tomography model;- isothermal boundaries:surface temperature = 300K, CMB temperature = 4000K;- internal heating: Q = 5.1 x 10e-12 Wkg-1;

1 Thermal convection

10-4 GPa

23 GPa

135 GPa

330 GPa

360 GPa

10-4 GPa

23 GPa

135 GPa

330 GPa

360 GPa

2 Plate tectonics and mantle convectionThe hypothesis that thermal convection occurs in the Earth's mantle emerged from the first ideas of continental drift. The long-standing observations of isostatic support of topographic relief and the detailed studies of post-glacial rebound of the Earth's surface in response to the rapid melting of ice caps reveals that the lithosphere lies on a viscous fluid.

Alps

Persia - Tibet - Burma

Ninety East - Sumatra

Philippines

Laptev Sea

western Aleutians

Alaska - Yukon

Gorda-California-Nevada

Rivera-Cocos

west central Atlantic

Peru

Puna-SierrasPampeanas

New Hebrides - Fiji

Africa (AF)

Arabia (AR)

Eurasia (EU)

India (IN)

Somalia (SO)

Antarctica (AN)

Australia (AU)

Sunda (SU)

Philippine Sea (PS)

Caroline (CL)

Pacific (PA)

Yangtze (YA)

Amur (AM)

Okhotsk (OK)

Eurasia (EU)

Pacific (PA)

Pacific (PA)

Pacific (PA)

Pacific (PA)

Antarctica (AN)

MS

BS

BH

MO

WL SB

SSTI

ONOkinawa

Mariana

MA

CR

BR

NH

FT

NI

TO

Tonga

Kermadec

KE

Aegean Sea

AS

AT

Anatolia

BU

Burma

NB

Manus (MN)

Juan de Fuca

JF

North America (NA)

Caribbean (CA)

Cocos (CO)

RiveraRI

Galápagos (GP)North Andes

ND

PA

Panama

Nazca (NZ)

Easter

EA

JZJuan Fernandez

Antarctica (AN)Scotia (SC)

ShetlandSL

SW

Sandwich

South America (SA)

Altiplano

AP

Eurasia (EU)

Africa (AF)

14

15

37

21

7

11 13

29

20

26

13

13

12

59

36

14

14

10

15

48

54

71

69

84

51

39

87

14

9270 96

58

70

6968

10

66

56

78

62

55

10083

86

26

102

92

13

18

16

59

90

103

62119

44

82

14

14

102

51

5183

95

69

25

26

19

22

10

67

40

19

51

53

14

25

3131

34

26

32

27

11

24

19

158

5

47

34

6

11

46

17

57

44

15

76

18

23

10

14

10

96

70

14

14

32

Equator

OCEAN

PACIFIC

OCEAN

ATLANTIC

INDIANOCEAN

AUSTRAL OCEAN AUSTRAL OCEAN

Alps

30

continental/oceanic convergent boundary

continental/oceanic transform fault

continental rift boundary/oceanic spreading ridge

subduction zone

velocity with respect to Africa (mm/y)

orogeny

The energy that drives mant le convect ion is believed to originate from two sources. The first is the heat transfer at the core-mantle boundary (CMB) associated with the freezing of the outer core and the growth of the innercore, and the second one is the decay of radioactive isotopes present in the mantle itself.

3 Complexity of mantle dynamics

The viscosity is an essential parameter used to describe a convecting system because it is one of the principal parameters which control the vigor of the flow. The dynamics of the mantle is also affected by the presence of phase transitions throughout its depth which are inherently accompanied by discontinuities in the density profile.

The major discontinuity, located at 670 km depth in the Preliminary Reference EarthModel (PREM) of Dziewonski and Anderson (1981), delimits the horizon between upper and lower mantle. This discontinuity has been at the heart of a long-standing controversy on the style of convection in the mantle; long debates have opposed the idea of whole-mantle versus layered convection.

4 Seismic-tomography-based modelsSeismic-tomography models provide a unique tool for mapping the thermal heterogeneities of the mantle. In spite of the great progress, seismic-tomography models provide only smooth images of the Earth's interior. Also, the amplitude of the heterogeneities is not well constrained and its determination is biased by thechoice of the numerical and theoretical treatments used in the inversion of the data

Numerical simulations of time-dependent convection

(Forte, 2000).

Seismic Tomography

3-D Mantle Flow Convection 'Signals'

dV (r, q, j)

V (r)

Density Contrasts

dr (r, q, j)

r (r)

u (r, q, j)

Viscous Flow Theory(mantle rheology)

- Non-hidrostatic Geoid (free-air gravity anomalies)- Testonic Plate Motions- CMB Topography- Surface Topography- Surface Stress- Seismic Anisotropy (Preferred Orientation)

dlnr

dlnV

The equations to be solved are derived from the principles of conservation of mass, momentum and energy in a viscous medium, and are solved numerically.

5

Non-dimensional Thermal Convection Equations

convection of mass

convection of momentum

convection of energy

Scaling parameters

Non-dimensional parameters

- mixed boundary conditions at the surface (free-slip and no-slip) describe the coupling of plates with mantle flow.

Spectral Green Function Solution of Mass and Momentum Conservation Equations- dr obtained from seismic-tomography modelsis substituted into equations which are then solved for the velocity field;- the component of the velocity field and the stress tensor are expressed on the form of spectral Green function;- spectral Green function are computed ONCEby solving a system of propagator equations;

Pseudo-spectral Time Integration of Energy Equation- lateral expansion in terms of spectral harmonics;- diffusive term: Crank-Nicolson sheme;- non-linear term: Predictor + Corrector;

Importance of Depth-Dependent Viscosity Profile

Equatorial cross-section of the

temperature field for Model 1 (isoviscous)

Equatorial cross-section of the temperature field for Model 2 (viscosity profile inferred from geodynamic data)

Conclusion6We have developed a geophysically constrained, pseudo-spectral model of time dependent thermal convection in Earth's mantle based on a robust spectral Green function solution of the mass and momentum conservation equations. We have applied this formalism towards tomography-based thermal convection simulations which incorporate geophysically constrained depth-dependent variations of mantle viscosity. These simulations reveal the major impact of large increases in viscosity in the lower mantle on the stability and longevity of large-scale hot upwelling plumes. This tomography-based convection model has also been applied towards the reconstruction of the time evolution of lateral temperature variations in the geologic past. Good accuracy reconstruction are possible over the past 30 to 50 million years bt backward advecting along a adiabatic geotherm.

http://en.wikipedia.org/wiki/Image:Tectonic_plates_boundaries_detailed-en.svg

Mantle Viscosity from Occam Inversions of Global Convection Data (free-air gravity anomalies, tectonic plate motions, excess CMB ellipticity)

Mantle Density Anomalies from Occam Inversions of Global Convection Data(free-air gravity anomalies, tectonic plate motions, CMB ellipticity, surface topography)

Acknowledgments and further information7We thank Aude Espesset for her master thesis Numerical Modelling of Time-Dependent Thermal Convection in Earth's Mantle Constrained by Seismic Tomography and Geodynamic Data.

Please contact: Petar Glisovic - , Alessandro Forte - and Robert Moucha -

pglisovic@gmail.comforte60@gmil.com rmoucha@gmail.com