3.12 Connections with mixing length recipes

The simplest model of turbulent convection is called “Mixing Length Theory” (MLT; see Böhm-Vitense, 1958Jump To The Next Citation Point). It is extensively used in stellar evolution calculations where the convection zone must be modeled many times during the course of a star’s evolution. MLT is based on an unphysical picture of moving fluid parcels in pressure equilibrium with their surroundings. In a convectively unstable region the entropy increases inward. A fluid parcel moving upward adiabatically will have higher entropy than the surrounding mean atmosphere and so has a higher temperature and lower density than the mean stratification, so it is buoyant and is accelerated upward. Conversely, a fluid parcel moving downward adiabatically will have lower entropy than its surroundings and so has a lower temperature and higher density than the mean stratification and is pulled down by gravity. After moving some characteristic distance ℓ (the mixing length) the fluid parcel dissolves back into its surroundings. The mixing length, ℓ, determines the magnitude of the temperature fluctuations, flow velocities, and consequently the energy flux. Typically this mixing length parameter is taken to be some multiple of the pressure scale height or as the distance to the boundary of the convectively unstable region.

In the first place, this is an unphysical picture. Turbulent convection is best pictured as upflows and downdrafts that can extend over many pressure and density scale heights. MLT is a local theory, where the flow velocities and temperature fluctuations are determined by local conditions. In reality, the entropy and temperature fluctuations are determined primarily by radiative cooling in the thin surface thermal boundary layer.

The free parameter, the mixing length ℓ, is not determined within the theory and varies across the Hertzsprung–Russell diagram according to 3D convection simulations (Abbett et al., 1997Ludwig et al., 1999Freytag et al., 1999). Hence, numerical convection simulations are needed to calibrate the mixing length parameter. Furthermore, standard MLT does not allow for overshoot of convective motions into the surrounding stable layers (Deng and Xiong, 2008, but see). It can not, for instance, account for phenomena such as reverse granulation or the observed destruction of lithium in the Sun by mixing below the convection zone. Finally, even with a mixing length that produces the correct interior adiabat, MLT produces a different mean atmosphere structure than given by the numerical simulations. This leads to disagreements between calculated and observed p-mode oscillation frequencies (Rosenthal et al., 1999Jump To The Next Citation Point). However, the mixing length scaling of temperature fluctuations and velocity with the convective flux,

2 ( )2∕3 ΔT--∼ ⟨uz⟩ ∼ Fconv , (41 ) T c2s ρc3s
does hold fairly accurately with the proportionality factor depending on viscosity and radiative energy exchange (Brandenburg et al., 2005). Other works that attempt a connection between mixing-length theory and numerical simulations are the ones by Kim et al. (1996) and Robinson et al. (2003).


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