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**Dynamic Balance of Cloud Vertical Velcoty**

Yuanfu Xie FAB/GSD/ESRL

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**A variational balance (LAPS)**

Written in a Lagrangian function,

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**Poisson Equation for λ (McGinley 1987)**

Take a perturbation of the Lagrangian function in terms of u, v, ω,

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**Poisson Equation for λ (cont’)**

Integration by part yields Or,

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**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.

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**Evidence of Incorrect vertical finite difference formulation**

New adjustment of wind LAPS balance

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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

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**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.

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**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.

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Forecast at Z ETS Bias

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Forecast at Z ETS Bias

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Forecast at Z ETS Bias

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

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**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!

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**Evidence of inconsistency between analysis anf WRF forecast 0**

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**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.

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