Physics of Convection

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Physics of Convection

• Motivation
• Convection is the engine that turns heat
  into
• motion.
• Examples from Meteorology, Oceanography and
  Solid Earth Geophysics
• Basic Equations, stationary convection,
  time-dependence, influence of mechanical inertia,
  volumetric effects ..

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Atmospheric phenomena - Large scale
Headly-cells gt horizontal transport -
Thermals which result in Cumulus and
Cumulo-Nimbus clouds gt vertical transport from
surface to the Tropospause - characteristic
Inertia Coriolis forces
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Oceanographic processes - Large scale water
exchange Arctics-Tropics - El Nino - Double
Diffusive Convection (e.g. Polynoyas) -
characteristic density determined by temp.
salinity
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Solid Earth Planets - Convection in the
Earth mantle - MHD - convection in the Earth
core generating mag. field - Magama chambers
-characteristic no inertia(mantle),
multicomponent
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Basic scenario
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Non dimensional equation for time-dependent
convection in a constant-property Boussinesq
fluid
with
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scaled by
where
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How to solve the equations
- Problem coupled system i.e v depends on T
and T
depends on v - Analytic
-linearize equation -see if
infinitesimal disturbance gets amplified gt
critical value for Ra 600, independent of Pr
- first instablities have a roll pattern -
other patterns also exist like square patter,
hexagon pattern, cross-roll pattern ... - no
extrema principal
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Higher Rayleigh numbers
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Numerical Simulation
Solve the equations by a numerical method (e.g.
finite element, fd, spectral, fv...) variables
are available at any point in space high
viscosity, rotation, spherical geometry are
easily realized - long 3D timeseries are still
expensive - small-scale features can not be
resolved
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Rayleigh
Prandtl
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Time-dependent convection
- onset of time-dependence from boundary layer
theory - At high Pr. large scale coherent
structures with superimposed boundarie layer
instabilities (BLI's) which are drifting with
the main flow - with incrasing Ra the strength
of the major up- and downwelling decreases
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Influence of the Prandtl number
- The Prandtl number measures the ratio of
mechanical inertia - Typical values are Pr(Water)
7., Pr(Air) 0.7 Pr(EarthMantle) 1024
, Pr(OuterCore) 0.04
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Pr 0.025
Pr0.7
Pr100.
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Temperature - Depth profiles for different
Prandtl numbers
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Percentage of vertical vorticity
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The influence of volumetric heating
- Decay of U, Th, and K lead to a volumetric
heating of the Earth mantle
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Volumetric heating leads to - break of symmetry
between up-and down wellings - 'passive'
upwellings with no distinct temperature
signature - cylindrical shape of down-wellings -
no large scale coherent structures - no different
scales for the downwelling
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Temperature and Pressure dependent viscosity

• Investigations of material properties for the
  Earth?s mantle indicate a strong dependence on
  both temperature and pressure.

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Thermochemical Convection
The density is not only a function of the
temperature but also of a second component
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Examples of 'fingers'
Experiment sugar-salt system
Numerical simulation
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Layer formation
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Effects observed - motion can be observed in
hydrostatic stable systems - potential energy is
converted in kinematic energy - formation of
well mixed convection layers - dynamics
strongly dependent on the diffusivity difference
between the two components
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Effects of Rotation
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What has not been talked about ... - effect of
pressure dependent thermal expansivity -
non-Newtonian rehologie - effects of
non-Cartesian geometry - effects due to
rotation -.....
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Conclusion

• Convection is THE important transport mechanism
  in geophysical systems
• for moderate heat differences systems exhibit a
  stationary flow
• depending on the magnitude of the Prandtl number
  the flows are becoming time-dependent
• for low-Pr. flow the velocity fields have a
  strong toroidal component
• effect like volumetric heating break the symmetry
  between up- and down-wellings
• most geophysical flows are in a regime where the
  flows are chaotic