A k-  model for turbulently thermal convection in solar like and RGB stars Li Yan Yunnan Astronomical Observatory, CAS

1. Turbulently thermal convection in stars 2. Basic equations and the mixing length theory 3. k-  model for stellar turbulent convection 4. Local solution of the k-  model 5. Applications to the solar model 6. Applications to RGB stars

1. Turbulently thermal convection in stars  Thermal convection ： In a gravitationally stratified fluid, the temperature gradient results in buoyancy to drive hot fluid moving upward and cold fluid moving downward. The structure of thermal convection is characterized by rolling cells, acting as thermal engines to transform heat into kinetic energy.

1. Turbulently thermal convection in stars  Effects of turbulently thermal convection in stars ： transfer of heat mixing of materials

 Equation of mass continuity ：  Navier-Stokes equations ：  Equation of energy conservation ： 2. Basic equations and the mixing length theory

 Mixing length theory ： Convection cells move in average a mixing length l. Equation of momentum ： Equation of energy conservation ： Equation of heat flux ： 2. Basic equations and the mixing length theory

 Equation of mixing length theory ： Defining a heat transfer efficiency of convection ： we obtain the famous cubic equation of the MLT ： where  Question: What is the effect of rolling cell’s structure How to determine the mixing length l Solution is unavailable in stably stratified region 2. Basic equations and the mixing length theory

 Equation of turbulent kinetic energy ：  Equation of dissipation rate of turbulent kinetic energy ： where,  Shear production rate ：  Buoyancy production rate ： 3. k-  model for stellar turbulent convection

 Stellar structure:  Shear of convective rolling cells:  Temperature difference of convective rolling cells: 3. k-  model for stellar turbulent convection

 The general solution is:  The size of a rolling cell is:  Averaged shear of rolling cells and shear production rate: 3. k-  model for stellar turbulent convection

 For the solar model ： typical length times typical velocity  T times typical velocity 3. k-  model for stellar turbulent convection

 Model of convective heat flux ：  Buoyancy frequency ：  Buoyancy production rate ： 3. k-  model for stellar turbulent convection

 In fully equilibrium state, the k-  model reduces to: where,  Fully equilibrium state appears in the unstably stratified region when c’  > 1, and in the stably stratified region when c’  < 0 4. Local solution of the k-  model

 In order to ensure positive shear production, we choose ：  Fully equilibrium condition for shear production results in: if  Turbulent kinetic energy should be proportional to the heat carried by convective rolling cells: 4. Local solution of the k-  model

 For convective cells moving adiabatically, we may assume: In fully equilibrium state, it can be derived that: It shows that the macro-length of turbulence is proportional to local pressure scale height.  For convective cells in general, we assume: 4. Local solution of the k-  model

In fully equilibrium state, it can be modeled as: We use this macro-length model of turbulence not only in the convection zone but also in the overshooting region.  In fully equilibrium state, we obtain: where, 4. Local solution of the k-  model

 Compared with the MLT, the local solution of the k-  model show similar asymptotic behavior in the limiting cases. 4. Local solution of the k-  model

 The sound speed of turbulent solar model is almost identical below the convection zone and higher in the convection zone than that of the MLT solar model. 5. Applications to the solar model

 Comparisons of the typical velocity and typical length scale between the k-  model and MLT. 5. Applications to the solar model

 Comparisons of local and general solutions of the k-  model 5. Applications to the solar model

Comparison of turbulent diffusivity resulted from the general solution of the k-  model and from the MLT. 5. Applications to the solar model

 For stars with different masses, the k-  model results in bluer RGB sequences than the MLT does. 6. Applications to RGB stars

 Comparisons of turbulent velocity and typical length for 1M ⊙ star in RGB bump stage 6. Applications to RGB stars

 Comparisons of temperature gradient and turbulent diffusivity for 1M ⊙ star in RGB bump stage 6. Applications to RGB stars

 Comparisons of turbulent velocity and typical length for 3M ⊙ star at the top of RGB stage 6. Applications to RGB stars

 Comparisons of temperature gradient and turbulent diffusivity for 3M ⊙ star at the top of RGB stage 6. Applications to RGB stars

THANKS ！