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Mixing in Convection
- Introduction - Numerical Methods - 2D
Investigations - stationary -
timedependent - 3D Investigation -
stationary - timedependent - Active
chemical components
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Numerical Methods for Investigating Mixing
- Mixing often studied on a case-by case
approach - Experiments - limited to
2D (mostly) - hard to quantify
- parameter range limited - Kinematic
flow fields - ABC flow, switching
vortex, ... very accurate
- good for identifying fundamental
processes -
not the 'real thing'
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Numerical Methods for Investigating Mixing
- Mixing often studied on a case-by case
approach - Experiments - limited to
2D (mostly) - hard to quantify
- parameter range limited - Kinematic
flow fields - ABC flow, switching
vortex, ... very accurate
- good for identifying fundamental
processes -
not the 'real thing'
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- Numerically solving the whole set of
equations - numerical errors
- easy to analyze - closer to
the 'real' thing - Three different
approaches - field
approach - tracer
approach - marker line
approach
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Field approach
- solve a transport equation on a eulerian grid
using a standart method FD, FE, FV, spectral
.... - for convection problems equivalent to
temperature problem -
problem numerical diffusion
Values are given at grid-points
Inbetween values are approximated by function
-normally T is already the bottleneck
But good for active components
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Tracer Method
- points are advanced using the precalulated
veolcity field v 3 steps 1st find
grid-cell the tracer is in - trivial for
equidistant grids or when the position of the
nodes is given by a function (e.g. Chebychev
pol.)
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else (e.g. FE grid) - loop over
gridcells to check if tracer is inside
or Advanved search using Tree (quad-,
oct-tree)
or when timestepping fulfills CFL criteria
search only neighborhood
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2nd Interpolate velocities within gridcells
- bi- (tri-) linear - higher order - splines -
depending of the spatial accuracy of the method
used for calculating v - for splines derivatives
can be precalculated at gridpoints good when
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3rd Integrate tracers in Time
- Euler, Runge-Kutta (2nd , 4th order), ... -
when calculating tracers by postprocessing
velocity data interpolate velocities in
Time
gt DONE
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How many tracers ? gt depends on the
statistics for 2D 10e5 .. 10e6 for 3D
.. more - but parallelization is trivial
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Active tracers
2 possibilities - calculate stokes drag and
add velocity v to velocity field - good
for problems like crystal setteling -
tracer tend to glue together - calculate
concentratioin field from the tracer distribution
and use in solving equation of motion
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3rd Marker chain method
- Advance tracer - if or check derivative gt
insert new tracer using spline - subdivision
scheme in 3D - calculating concentration field
using a 'IsInside' type method initially
very fast very accurate - for chaotic
flows
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Comparison between methods
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Different numerical methods - Conclusion
- discussed three different methods - all
methods give comparable results - using a field
approach is the fastest method - it suffers
from numerical diffusion - tracer approach is the
most flexibel - in order to reduce
statistical noise tracer count per element
has to be high - maker chain method is accurate
and computational cheap for short time-length
- for chaotic flows however the required time
increases exponentially in time
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Mixing in Convection
Steady 2D convection - Hamiltonian system -
streamline is Hamiltonian
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Steady 2D flows - surface area preserved - shear
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Mixing in 2D time-periodic flows Aref (1984)
Blinking vortex flow
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- even simple 2D time-periodic flows can exhibit
lagrangian chaos
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2D time-dependent flows
- maintaines large-scale structure -
super-imposed boundary-layer-instabilities
(BLI's)
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-mixes fast within one cell (c) -transport over
cell boundaries
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Experiments Kinematic approach (SolomonGollub)

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Higher Rayleigh number
- influence of the major up- and down-wellings
get less - more drop like instabilities - still
a large scale cirulation (shear flow) gt looking
at the Earth mantle with Ra gt 106 it
seems impossible to preserve chemical
heterogeneities over a long period of
time gt Earth mantle must have lateral
distincted reservoirs
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3D Convective flows
- different stationary patterns - rolls -gt 2D -
square or spoke pattern
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Multiple tracers. ..
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- tracer is confined to one quarter of the box -
they move on the surface of differnt tori -
oszillate fast around the large diameter and slow
around the small diameter of the torus (approx.
110) - if the ratio - is a natural number the
streakline is closed - is a rational number the
streakline is flosed after n orbits - is a real
number the streakline is surface filling gt
analog to 2D flows ... but material is moving
from the cell interior to the boundary layers gt
no chaotic regions
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Hexagonal patterns
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Spherical geometry
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Comparison with ABC flowsdefined by
For C0 - integrable
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Conlucsion 3D statonary convection
Ferrachat Ricard
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Conclusion Stationary convection
In the absence of a toroidal flow component -
tracer move on the surface of torii - the
symmetry planes of the temperature field confine
the torii - material is communicated between the
boundaries and the interia of the cell A
torodial flow component - can be generated by a
finite Prandtl number or by boundary
conditions - this leads to the coexistence of
regular and chaotic mixing regions - the
mixing efficiency is greatly enhanced by these
regions
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Time-dependent convection
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High Raylaeigh numbers
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Conclusion time-dep. Convection
- efficient mixing within a cell - cross cell
mixing on a much longer time scale - trajectories
have torus-like stucture - mixing efficiency much
reduced compared with 2D simulations
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More realistic parameters- varable viscosity
The influence of plate motion on the
mixing process
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Conclusion time-dep. convection with variable
viscosity

• Two different processes
• efficient mixing within a cell
• -large scale transport due to plate motion
  causing
• cell reorganization
• - enhanced mixing in direction of the plate
  motion
• - could eventually explain observed geochemical
• differences in MORB
• BUT
• very simple model which with many problems