Governing Equations of atmospherc flow

1. Governing Equations of atmospherc flow 1. Equation of state, gas law 2. Mass conservation, continuity equation 3. Equation of motion, Newton law, momentum equation Hydrostatic balance

2. 4. Equation of energy conservation, first law of thermodynamics

3. 5. Conservation of moisture Neglecting viscous force, we have total nine equations for nine unknowns

4. Mean governing equations in turbulent flow 1. Mean equation of state 2. Mean continuity equation

5. 3. Mean momentum equation

6. Take average

7. 4. Mean equation of energy conservation 5. Mean equation of moisture conservation

8. Mean governing equations in turbulent flow 1. Mean equation of state 2. Mean continuity equation 3. Mean momentum equation 4. Mean equation of energy conservation 5. Mean equation of moisture conservation

9. How can we solve this problem?

10. First-order equations ----> second-order moments Second-order equations ----> third-order moments ………. This is the notorious problem of turbulence!

11. Einstein Summation convention Second order equations Einstein wrote to his friend “I have made a great discovery in mathematics; I have suppressed the summation sign every time that the summation must be made over an index which occurs twice..."

12. TKE budget equation A. Local change term D. Buoyancy production term E. Transport term B. Advection term C. Shear production term F. Pressure correlation term G. Dissipation For horizontal homogeneous condition, x direction along the mean wind direction, mean vertical velocity is zero.

13. A. Local change term B. Advection term C. Shear production term Dynamic stability

14. D. Buoyancy production term Static stability

15. E. Transport term It does not generate or dissipate TKE, it only transports TKE within the boundary layer. F. Pressure correlation term This term is difficult to measure. G. Dissipation Turbulent energy eventually dissipated due to viscosity.

16. Static stability and instability The atmosphere is unstable if a parcel at equilibrium is displaced slightly upward and finds itself warmer than its environment and therefore continues to rise spontaneously away from its starting equilibrium point. The atmosphere is stable if a parcel at equilibrium is displaced slightly upward and finds itself colder than its environment andtherefore sink back to its original equilibrium point.

17. Stable equilibrium Stable Unstable

18. Static stability For convective boundary layer, It is necessary to look at entire layer How to solve the problem? Using buoyancy flux Dynamic stability Under unstable condition, both buoyancy and shear tend to produce turbulence Under stable condition, buoyancy tends to suppress turbulence while shear tends to produce turbulence

19. 1. Flux Richardson number Shear production dominates buoyancy suppression if it’s static stable Buoyancy suppression dominatesshear production

20. 2. Gradient Richardson number z 1 2 U Turbulent flow Non-turbulent flow Bulk Richardson number