1. Alternative Vertical Coordinates

4. Consider scalar A as a 4-d variable, using either z or s.

5. Consider

6. Note: now have “s” terms and “z” terms on opposite sides.

7. Then d/dt becomes

8. But

9. So then we can relate the vertical speeds

10. Horizontal momentum

11. But

12. So, horizontal momentum becomes

13. Pressure coordinate

14. Then with s=p,

15. Or

16. Continuity Equation

17. Change to p-coord. Key factor:

18. So

19. Mass continuity is then

20. Using pT = constant p top of atmosphere, and pH = surface pressure:
Other coordinates
Using pT = constant p top of atmosphere, and pH = surface pressure: