The marker-in-cell
method
Core and Mantle Dynamics
Gregor J. Golabek
What is the marker-in-cell method?
2
Fixed (Eulerian) grid points
Mobile
(Lagrangian)
markers
Cell
[modified from Gerya, 2010]
What are grid points?
Temperature T
3
T  f x 
Distance x
Discretization of continous 1D function
4
T  f x 
Fixed grid points
Increase number of grid points
5
T  f x 
Higher resolution = Higher accuracy
6
T  f x 
[Gerya, 2010]
More complex 2D problem
7
  f P,T 



  0 1   T  T0   1  P  P0 
[Gerya, 2010]
Discontinous problems on a grid
8
numerical diffusion!
[Gerya, 2010]
Solution:
• Mobile markers transport physical properties
(e.g. composition, density, temperature, ...)
• Interpolation of marker properties on immobile grid points
• Solution of the constitutive equations (e.g. Stokes equation => velocities)
• Velocity field is used to advect the markers
With marker-in-cell method
9
[Gerya, 2010]
Interpolation from markers to nodes
10
N markers
e.g.
density
‘‘
‘
[modified from Gerya, 2010]
Averaging - Methods
11
N
N
h  N

1 1
1
1
  ...

Harmonic mean
1
1  2  ... N
a   i 
N i1
N
Arithmetic mean
i1  i
1
2
N
N
N
g  N  i  N 1  2  N
i1
Geometric mean
Averaging - Results
Velocity v
12
Higher grid resolution
[modified from
Schmeling et al., 2008]
Re-interpolation from nodes to markers
13
N markers
density
‘‘
‘
[modified from Gerya, 2010]
Marker advection – Euler scheme
14
Application: Corner flow problem
y
error
y(2)
y(1)
trajectory
x(1)
x
t 1
t 
x(2)
t 
 x  t v x
y
[Press et al.,
1997]
x(3)
 t 1
t 
t 
 y  t v y
Marker advection – Runge-Kutta 2nd order
15
Application: Corner flow problem
y
error
y(2)
y(1)
trajectory
smaller error
x(1)
x
t 1
trajectory
t 
 x  t v
x(2)
 1 
t  
 2 
x
[Press et al.,
1997]
x(3)
y
t 1
t 
 y  t v
 1 
t  
 2 
y
Marker advection – Runge-Kutta 4th order
16
A
x(t)
C
B
x(t+1)
D
x
t 1
[Press et al.,
1997]
 1 
 1 


t  
t  
1
t 
t 
 t 1
 2 
 2 
 x  t v xA  2v xB  2v xC  v xD 
6 



Marker advection – Runge-Kutta 4th order
17
A
x(t)
C
B
x(t+1)
D
x
t 1
[Press et al.,
1997]
 1 
 1 


t  
t  
1
t 
t 
 t 1
 2 
 2 
 x  t v xA  2v xB  2v xC  v xD 
6 



Marker advection – Runge-Kutta 4th order
18
A
x(t)
C
B
x(t+1)
D
x
t 1
[Press et al.,
1997]
 1 
 1 


t  
t  
1
t 
t 
 t 1
 2 
 2 
 x  t v xA  2v xB  2v xC  v xD 
6 



Marker advection – Runge-Kutta 4th order
19
A
x(t)
C
B
x(t+1)
D
x
t 1
[Press et al.,
1997]
 1 
 1 


t  
t  
1
t 
t 
 t 1
 2 
 2 
 x  t v xA  2v xB  2v xC  v xD 
6 



Geodynamical application
20
Geodynamics: More precise Runge-Kutta 4th order scheme used
STILL:
Accumulation of advection
errors after several overturns
 Formation of holes in
marker field
Always check your results!
21
Entrainment
Holes
[Schmeling et al., 2008]
Summary
22
• The marker-in-cell method is a powerful tool to advect strongly
discontinous fields in numerical models
• Mobile markers are advected through an immobile grid
• High order Runge-Kutta advection schemes preferred
• Non-diffusive markers store physical properties
BUT:
• Grid resolution has still to be sufficiently high for meaningful solution
• Sufficient number of markers in each cell for averaging to minimize interpolation errors
• Holes in the marker field can open after several overturns
• Marker refilling needed when cells are empty
The end
Lecture download:
http://perso.ens-lyon.fr/gregor.golabek/teaching.html
Numerical exercise: Nu-Ra relation
24

Nu ~  Ra
[Christensen,1984]
Reminder: Nu and Ra number
25
Nusselt number Nu:
qtot
Nu 
qcond
Rayleigh number:
 0Tgb
Ra 

3
[Turcotte and Schubert, 2002]
Numerical exercise: Nu-Ra relation
26

Nu ~  Ra
How to do that?
[Christensen,1984]
1. Vary the Ra number in your input file
2. Wait until steady-state is reached in the simulation
3. Read out the heat flux qsurf and compute corresponding Nu
4. Plot results and estimate parameters  and 