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