Dynamic Balance of Cloud Vertical Velocity

1. Dynamic Balance of Cloud Vertical Velocity Yuanfu Xie and Steve Albers FAB/GSD/ESRL

2. A variational balance (LAPS) Written in a Lagrangian function,

3. Poisson Equation for λ (McGinley 1987) Take a perturbation of the Lagrangian function in terms of u, v, ω,

4. Poisson Equation for λ (cont’) Integration by part yields Or,

5. Issues with the Poisson Eqn • Need a efficient 3D solver; • It is complicated for vertical finite difference as vertical levels are non-uniform (McGinley 1987 uses a uniform grid; unfortunately, LAPS balance uses a uniform grid even it is not (e.g. LAPS_ci domain)!! See qbalpe.f line 2878-2881); • LAPS uses a relaxation method: • A relaxation is convergent but extremely slow; • Most iterations tries to push shorter waves except the first few iterations; • It usually results in high frequency noise (balance package shows a lot of noise in divergence). • STMAS multigrid is a designed scheme for this equation; • A simple trick to see if the minimization is right.

6. Evidence of Incorrect vertical finite difference formulation New adjustment of wind LAPS balance

7. A simple trick • After LAPS/STMAS analysis, the balance package adds cloud ω to wind. Instead of a 3D Poisson solver, a simple scheme is used to adjust wind

8. An Approximation of the Balance • It keeps all wind information from previous analysis except adding vertical gradient of cloud omega to wind divergence; • Fish package can be easily and efficiently used for solving these 2D Poisson equations; • A quick verification for improving the forecast scores; • A simple quick fix of the balance before a more sophisticated implementation.

9. Experiments at the CI Domain • There are two STMAS runs set on the convective initiation domain, (stmas_ci and stmas_ci_cyc) for experimenting; • The two runs are identical except the cloud and omega adjustment • cloud omega ‘vvmax’ a linear function of reflectivity (along with cloud depth); • Smoothed cloud omega upon input to balance; • Added cloud omega to the horizontal wind. • LAPS uses the linear increment and smoothed cloud omega and is better than STMAS HWT forecasts without adding the cloud omega; • More experiments are needed to evaluate the simple scheme comparing to the linear increment cloud omega and smoothed scheme (test each of three improvements separately).

10. Forecast at 2011-09-04 00Z ETS Bias

11. Forecast at 2011-09-04 00Z ETS Bias

12. Forecast at 2011-09-04 00Z ETS Bias

13. Comparison with add_omega

14. Divergence, cloud omega and reflectivity plots: 2011-09-04 06Z

15. Preliminary Conclusions • LAPS balance uses an incorrect finite difference formula in Poisson equation (vertical) if the vertical is non-uniform; • The simple trick shows the minimization of balance package indeed improves ETS; but this simple trick brings in too strong reflectivity forecasts; • It adjusts wind only but not other fields and is too simple; • It shows that the relaxation scheme in the balance package can be improved; • A more sophisticated multigrid minimization technique should be considered in STMAS development. • The wide forecast bias may also warn us on the cloud omega computation; some sensitivity study is needed; • A correct 3D Poisson equation and multigrid will be applied in STMAS for improving the balance!

16. Evidence of inconsistency between analysis and WRF forecast 0

17. Difference between analysis and 0h forecast

18. Vertical Finite Difference • When a uniform vertical grid is used, a center finite difference is correct, such as for a stagger grid, for example, a first derivative, • This is incorrect if the vertical is not uniform. • What is LAPS balance currently using for ?