Improvements to stellar structure models, based on a grid of 3D   convection simulations. II. Calibrating the mixing-length formulation
  We perform a calibration of the mixing length of convection in stellar structure models against realistic 3D radiation-coupled hydrodynamics (RHD) simulations of convection in stellar surface layers, determining the adiabat deep in convective stellar envelopes.   The mixing-length parameter $\alpha$ is calibrated by matching averages of the 3D simulations to 1D stellar envelope models, ensuring identical atomic physics in the two cases. This is done for a previously published grid of solar-metallicity convection simulations, covering from 4200 K to 6900 K on the main sequence, and 4300-5000 K for giants with logg=2.2.   Our calibration results in an $\alpha$ varying from 1.6 for the warmest dwarf, which is just cool enough to admit a convective envelope, and up to 2.05 for the coolest dwarfs in our grid. In between these is a triangular plateau of $\alpha$ ~ 1.76. The Sun is located on this plateau and has seen little change during its evolution so far. When stars ascend the giant branch, they largely do so along tracks of constant $\alpha$, with $\alpha$ decreasing with increasing mass. 
Regner Trampedach, Robert F. Stein, J{\o}rgen Christensen-Dalsgaard,   {\AA}ke Nordlund and Martin Asplund
astro-ph.SR