A finite volume solution method for thermal convection in a spherical shell with strong temperature- und pressure-dependent viscosity

CIG Workshop 2005Boulder, Colorado

K. Stemmer, H. Harder and U. Hansen

stemmer@uni-muenster.de

Münster University, GermanyDepartment of Geophysics

Outline

Motivation: Importance of mantle rheology

Basic principles of thermal convection with variable viscosity Mathematical model Numerical model

Simulation results: Thermal convection in a spherical shell Temperature-dependent viscosity Temperature- and pressure-dependent viscosity

Conclusions

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MotivationImportance of mantle rheology

Laboratory experiments of mantle material: viscosity is temperature-, pressure- and stress-dependent

Many models have constraints: Cartesian isoviscous / depth-dependent viscosity

High numerical and computational effort for lateral variable viscosity mode coupling sophisticated numerical methods

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Thermal convectionmathematical model

Rayleigh-Bénard convection

0=⋅∇ ur

continuity equation

equation of motion

02 =−∇−∇+∂∂

RaRa

TTutT Qr

heat transport equation

( )[ ] 0)( =∇−+∇+∇⋅∇ peRaTuu r

T rrrη

Rayleigh numberref

TdgRa

κηαρ 3Δ

=ref

Q k

QdgRa

ηκαρ 5

=

Arrhenius equation ( ) ( )))(ln())(ln(exp, 0 refprefT TrRTTpT −−Δ+−Δ= ηηη

.

pηΔTηΔ

viscosity contrast with pressure

viscosity contrast with temperature

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Thermal convection with lateral variable viscositynumerical model

Implemented methods: Discretization with Finite Volumes (FV) Collocated grid Equations in Cartesian formulation Primitive variables Spherical shell topologically divided in 6 cube surfaces Massive parallel, domain decomposition (MPI)

Time stepping: implicit Crank-Nicolson method Solver: conjugate gradients (SSOR) Pressure correction: SIMPLER and PWI

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control volume

Thermal convection with lateral variable viscositynumerical model

grid generation lateral grid

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Advantages of this spatial discretization: Efficient parallelization No singularities at the poles Approximately perpendicular grid lines Implicit solver (finite volumes)

discretization of the viscous term

Problem: required: derivatives of velocities in x-,y- und z-direction

available: curved gridlines (not in x-,y- und z-direction)

( )[ ]Tuu )(rr

∇+∇⋅∇ η

Solution:transformation of the viscous termapplying Gauß / Stokes theorem and lokal CV coordinate systems simplification of integrals

( ) ( )uurr

×∇×∇+∇⋅∇ ηη2

CV: control volume

Thermal convection with lateral variable viscositynumerical model

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SduIVS

rr∫ ∂=

∇= η21

( )∫ ∂=×∇×=

VSuSdIrr

η2

Gauss integral theorem

known Laplacian solution

applying Stokes theoremchange to local orthonormal basis

to simplify notation: ( ) ( )i

i

i S

S

S hcgbfa

c

b

a

urrrr

++=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=×∇ :

iSlocal orthonormal basis of the CV surface( )hgfrrr,,

∫ ∫ ∂=⋅=⋅∇

V VSSdudVurrr

Thermal convection with lateral variable viscositydiscretisation of the viscous term

( )[ ] ( ) ( )dVudVudVuuV VV

T ∫ ∫∫ ×∇×∇+∇⋅∇=∇+∇⋅∇rrrr

ηηη 2)(

stress tensor 1I 2I

viscous term:

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( ) i

SS

iiS SVS

dS

c

b

a

n

n

n

uSdI

ii

ii

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛×

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=×∇×= ∑∫∫

=∂=

6

3

2

1

2 ηηrr

solution of integral :

( )TSinnn 321 ,,normal vector

many terms are vanishing due to the use of local coordinates

( ) ( ) SSS SN

NN N dShafcdShafc ∫∫ +−+−++rrrr

ηη

( ) ( ) BBB BT

TT T dSgafbdSgafb ∫∫ −+++−+rrrr

ηη

( ) ( ) WWW WE

EE E dShbgcdShbgc ∫∫ −+++−=rrrr

ηη

remains the calculation of the curl of velocities on the CV surfaces

Thermal convection with lateral variable viscositydiscretisation of the viscous term

2I

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integration along selected paths

Thermal convection with lateral variable viscositydiscretisation of the viscous term

linear approximation of line integrals

Calculation of the curl of velocities on the CV surfaces:

applying Stokes theorem

( ) dzwdyvdxuwdzvdyudxrdu ++≈++=⋅∫ ∫rr

i

ii

Sc

b

a

Sj gf

j hf

j hg

Sh

g

f

cS

bS

aS

rdu

rdu

rdu

rdu

rdu

rdu

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

∑ ∫∑ ∫∑ ∫

∫∫∫

=

=

=

4

1 ,

4

1 ,

4

1 ,

rr

rr

rr

rr

rr

rr

a,b,c

∫∫ ⋅=×∇LS

rduSdurrrr

( )

( )

( )32,31,333,3

23,21,222,2

13,12,111,1

1

1

1

RvTuTDT

w

RwTuTDT

v

RwTvTDT

u

PPP

PPP

PPP

−−−=

−−−=

−−−=

entries adjacent R

term driving D

weightcentral T

i

i

ji ,

coupling of velocity components central weight is a vector

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Thermal convection with lateral variable viscositypressure weighted interpolation (PWI)

Solution: mathematical principle [Rhie and Chow, 1983] small regularizing terms are added that excludes spurious modes perturb the continuity equation with pressure terms regulating pressure terms do not influence the accuracy of the discretisation

( ) 2

1

11 42

1 +

+− Δ++=n

P

pressurenP

nP

nP p

cw

auuu

pressure of tiondiscretiza central :p

operator diffusion of weightcentral :cw

Δ

pressure is defined to an intermediate time levelpressure correction: fluxes are pertubated with pressure terms

Problem: insufficient coupling )()1( 211 tgcu jj

j+−= 21)1(2

jjj cp +−=

checkerboard oscillations

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Thermal convection in a spherical shelltemperature-dependent viscosity, basal heating 5102/1 =Ra

residualtemperaturedt = +/- 0.1

010=ΔT

η 310=ΔT

η 610=ΔT

η

temperatureisosurfacesand slices

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5102/1 =Ra010=Δ

Tη 310=Δ

Tη 610=Δ

Tη

T=0.25T=0.60 T=0.83

Thermal convection in a spherical shelltemperature-dependent viscosity, basal heating

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Three regimes: 1) mobile lid2) transitional (sluggish)3) stagnant lid

velocities minimum with depth of lateral velocities

Thermal convection in a spherical shelltemperature-dependent viscosity, basal heating

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100,10,10 54

0 =Δ=Δ= pTRa ηη

„highviscosityzone“

Thermal convection in a spherical shelltemperature- and pressure-dependent viscosity

Temperature dependence and pressuredependence of viscosity compete each other!

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0=QRa 4105⋅=QRa

isosurfaces:

100,10,10 54

0 =Δ=Δ= pTRa ηη

slices:

red = high viscousblue = low viscous

„high viscosity zones“

Thermal convection in a spherical shelltemperature- and pressure-dependent viscosity

)ln(η

1.0/−+=Tδ

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Conclusions

Mantle Convection Importance of spherical shell geometry Importance of mantle rheology ... ?

…thanks for your attention!

High numerical and computational effort for lateral variable viscosity

BUT: temperature-dependent viscosity has a strong effect on…

…convection pattern

…heat flow

…temporal evolution

Münster University, GermanyDepartment of Geophysics

stemmer@uni-muenster.de