Convection Convection Matt Penrice Astronomy 501 University of Victoria

Outline Convection overview Mixing Length Theory (MLT) Issues with MLT Improvements on MLT Conclusion

Conditions for convection Radiation temperature gradient Convective temperature gradient Convection condition

Mixing length theory (MLT) Assume groups of convective elements which have same properties at given r Each element travels on average a distance know as the mixing length before mixing with the surrounding matter The are assumed to have the same size and velocity at a given r, respectively

MLT 11 Assume complete pressure equilibrium Assume an average temperature T(r) which is the average of all elements at a given r at an instant in time Therefore elements hotter then T will be less dense and rise because of the assumed pressure equilibrium and vice versa for cooler elements

MLT 111 What we are really interested in is the convective flux C p =Specific heat at constant pressure =Mixing length =The distance over which the pressure changes by an appreciable fraction of itself

MLT 1V =The average temperature gradient of all matter at a given radius =The temperature gradient of the falling or rising convective elements

Issues with MLT Neglecting turbulent pressure Neglecting asymmetries in the flow Clear definition of a mixing length Failure to describe the boundaries

Improvements to MLT Arnett, Meakin, Young (2009) Convective Algorithms Based on Simulations (CABS) Creates a simple physical model based on fully 3D time-dependant turbulent stellar convection simulation

Kinetic Energy Equation MLT does not deal with KE loss due to turbulence Turbulent energy will cascade down from large scales to small (large scale being the size of the largest eddy) Energy is dissipated through viscosity at small scales (Kolmogorov micro scales)

Fluxes Pressure perturbation (sound waves) Convective Turbulent motions

Boundaries Redefine a convective zone as a region in which the stratification of the medium is unstable to turbulent mixing Defined using the Bulk Richardson number The Bulk Richardson number is the ratio of thermally produced turbulence and turbulence produced by vertical shear u is the rms velocity of the fluid involved with the shear and l is the scale length of the turbulence

Boundaries 11 = The change in buoyancy across a layer of thickness N=The Burnt-Vaisala frequency or the frequency at which a vertically displaced parcel will oscillate in a statistically stable environment

Arnett, Meakin &Young 2009

Conclusion Mixing Length Theory provides a simple description of convection but has numerous draw backs CABS introduces a way to take into account loss of KE due to turbulence as well as a dynamic definition of the boundary layers