physics of convection " motivation: convection is the engine that turns heat into motion....
TRANSCRIPT
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 ..
Atmospheric phenomena: - Large scale Headly-cells => horizontal transport - Thermals which result in Cumulus and Cumulo-Nimbus
clouds = > vertical transport from surface to the Tropospause- characteristic: Inertia & Coriolis forces
Oceanographic processes:
- Large scale water exchange Arctics-Tropics
- El Nino - Double Diffusive
Convection (e.g. Polynoyas)
- characteristic: density determined by temp. & salinity
Solid Earth & Planets: - Convection in the Earth mantle - MHD - convection in the Earth core generating mag. field - Magama chambers -characteristic: no inertia(mantle), multicomponent
Basic scenario:
Non dimensional equation for time-dependent convection in a constant-property Boussinesq fluid:
with:
scaled by:
where:
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
=> 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
Higher Rayleigh numbers:
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
Rayleigh
Prandtl
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
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) = 10**24 , Pr(OuterCore) = 0.04
Pr = 0.025 Pr=0.7
Pr=100.
Temperature - Depth profiles for different Prandtl numbers
Percentage of vertical vorticity:
The influence of volumetric heating:- Decay of U, Th, and K lead to a volumetric heating of the Earth mantle
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
Temperature and Pressure dependent viscosity
Investigations of material properties for the Earths mantle indicate a strong dependence on both temperature and pressure.
Thermochemical Convection:
The density is not only a function of the temperature but also of a second component:
0
1 TT0 CC
0
Examples of 'fingers':
Experiment: sugar-salt system
Numerical simulation
Layer formation:
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
Effects of Rotation
What has not been talked about ...
- effect of pressure dependent thermal expansivity- non-Newtonian rehologie- effects of non-Cartesian geometry- effects due to rotation-.....
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