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 = 0.025Pr=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 Earth ‘ s 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:

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