1. Dynamic Balance of Cloud Vertical Velcoty
Yuanfu Xie
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 );
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;
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;
An 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
Linear increment for cloud omega analysis;
Smoothed cloud omega;
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.

10. Forecast at Z
ETS
Bias

11. Forecast at Z
ETS
Bias

12. Forecast at Z
ETS
Bias

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

14. 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!

15. Evidence of inconsistency between analysis anf WRF forecast 0

16. 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.