Münster University, Germany
Department of Geophysics
A finite volume solution method for thermal convection in a spherical shell with strong temperature- und pressure-dependent viscosity
CIG Workshop 2005
Boulder, Colorado
K. Stemmer, H. Harder and U. Hansen stemmer@uni-muenster.de

Münster University
 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

Münster University
Motivation
Importance 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

.
Münster University
Thermal convection mathematical model
RayleighBénard convection continuity equation equation of motion heat transport equation
 
 u

0
 




 u

(

 u ) T
 


RaT e r

T
 t
 u


T
 
2 T
  p

0

Ra
Q
Ra

0
Rayleigh number
Arrhenius equation
 
T , p

Ra


 g
 
Td exp
 ln(
 ref
 
T
3
)( T ref
Ra
Q

 g

Qd 5
 k
 ref

T )
 ln(
  p
)( R
0
 r

T ref
)

 
T
  p viscosity contrast with temperature viscosity contrast with pressure

Münster University
Thermal convection with lateral variable viscosity numerical 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

Münster University
Thermal convection with lateral variable viscosity numerical model
Advantages of this spatial discretization:
 Efficient parallelization



No singularities at the poles
Approximately perpendicular grid lines
Implicit solver (finite volumes) grid generation control volume lateral grid

Münster University
Thermal convection with lateral variable viscosity numerical model discretization of the viscous term
 




 u

(

 u ) T
  required: derivatives of velocities in x-,y- und z-direction
Problem:
Solution: available: curved gridlines (not in x-,y- und z-direction) transformation of the viscous term 2
 
  
 u

  
   
 u
 applying Gauß / Stokes theorem and lokal CV coordinate systems simplification of integrals
CV: control volume

Münster University
Thermal convection with lateral variable viscosity discretisation of the viscous term viscous term:

V
 



 u


(

 u )
T
  dV
 
V
2
 
   u
  dV
 
V
 
   
 u
 dV stress tensor
Gauss integral theorem

V
 
 u dV
 
I
S u


1


V d S

I
1

2

S
 
V
 
 u d S
 known Laplacian solution
I
2
I
2
 
S
 
V d S


   
 u
 applying Stokes theorem change to local orthonormal basis to simplify notation:

 
 u

S i
 f

,
 g ,
 h
 a
:
 
 b c


S i

 a f


 b g

 c h
S i
 local orthonormal basis of the CV surface S i

Münster University
Thermal convection with lateral variable viscosity discretisation of the viscous term
I
2
 
S
 
V
I
2 d S


   
 u

 n
1 i
6 
 i
S i

S i

 n
2 n
3


S i a
 
 b c


S i dS i normal vector
 n
1
, n
2
, n
3

T
S i




T

E

N



T
E
N






 c g c f
 b f




 b h a h

 a g
T

E

N
 dS
E dS dS
T
N




B

W

S


W

B
S





 c f
 b f

 c g



 b h a h

 a g


B
S

W dS
W dS dS
B
S many terms are vanishing due to the use of local coordinates remains the calculation of the curl of velocities on the CV surfaces

Münster University
Thermal convection with lateral variable viscosity discretisation of the viscous term
Calculation of the curl of velocities on the CV surfaces: applying Stokes theorem linear approximation of line integrals integration along selected paths










 f u d
 d g u h u
 d

S
 u

 d r

 u d S

 r
 r
 r






S i










L u
 udx
4
4
4



1

1

1 d r
 vdy g u
,

 h d f u h d

, f u
, g d r
 r
 r






 wdz
S i




 u dx aS a bS b cS c


S i
 v dy
 w dz a,b,c u
P

1
T
1 , 1

D
1

T
1 , 2 v
P

T
1 , 3 w
P

R
1
 v
P

1
T
2 , 2

D
2

T
2 , 1 u
P

T
2 , 3 w
P

R
2
 w
P

1
T
3 , 3

D
3

T
3 , 1 u
P

T
3 , 2 v
P

R
3



T i , j central weight
D i driving term
R i adjacent entries coupling of velocity components central weight is a vector

Münster University
Thermal convection with lateral variable viscosity pressure weighted interpolation (PWI)
Problem: insufficient coupling u j
 c
1
(

1 ) j
1
 j
2 g ( t ) checkerboard oscillations p j
 c
2
(

1 ) j
1
 j
2
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 u
P n 
1
2
 u n
P

1
 u
P n

1

 a pressure  p n

1
2
4 cw
P cw : central weight of diffusion operator
 p : central discretiza tion of pressure pressure is defined to an intermediate time level pressure correction: fluxes are pertubated with pressure terms

Münster University
Thermal convection in a spherical shell temperature-dependent viscosity, basal heating
Ra
1 / 2

10
5 temperature isosurfaces and slices
 
T

10 0
 
T

10 3
 
T

10 6 residual temperature dt = +/- 0.1

Münster University
Thermal convection in a spherical shell temperature-dependent viscosity, basal heating
 
T

10 0
 
T

10 3
 
T
Ra
1 / 2

10 6

10
5
T=0.25
T=0.60
T=0.83

Münster University
Thermal convection in a spherical shell temperature-dependent viscosity, basal heating
Three regimes:
1) mobile lid
2) transitional (sluggish)
3) stagnant lid velocities minimum with depth of lateral velocities

Münster University
Thermal convection in a spherical shell temperature- and pressure-dependent viscosity
Ra
0

10 4 ,
 
T

10 5 ,
  p

100
„high viscosity zone“
Temperature dependence and pressure dependence of viscosity compete each other!

Münster University
Thermal convection in a spherical shell temperature- and pressure-dependent viscosity
Ra
Q

0 Ra
Q

5

10 4 „high viscosity zones“ slices: ln(

) red = high viscous blue = low viscous
Ra
0

10 4 ,
 
T

10 5 ,
  p

100 isosurfaces:

T
 
/

0 .
1

Münster University, Germany
Department of Geophysics
High numerical and computational effort for lateral variable viscosity
BUT: temperaturedependent viscosity has a strong effect on…
…convection pattern
…heat flow
…temporal evolution
Mantle Convection
 Importance of spherical shell geometry


Importance of mantle rheology
... ?
…thanks for your attention!
stemmer@uni-muenster.de