### 3.12 Connections with mixing length recipes

The simplest model of turbulent convection is called “Mixing Length Theory” (MLT;
see Böhm-Vitense, 1958). 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 (the mixing length) the fluid parcel dissolves back into its surroundings. The mixing
length, , 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 , is not determined within the theory and varies across the
Hertzsprung–Russell diagram according to 3D convection simulations (Abbett et al., 1997; Ludwig
et al., 1999; Freytag 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., 1999).
However, the mixing length scaling of temperature fluctuations and velocity with the convective flux,

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).