Dynamic Balance of Cloud Vertical Velocity

Yuanfu Xie and Steve AlbersFAB/GSD/ESRL

A variational balance (LAPS)

Written in a Lagrangian function,

Poisson Equation for λ (McGinley 1987)

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

Poisson Equation for λ (cont’)

Integration by part yields

Or,

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.

Evidence of Incorrect vertical finite difference formulation

New adjustment of wind LAPS balance

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

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.

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).

Forecast at 2011-09-04 00Z

ETS Bias

Forecast at 2011-09-04 00Z

ETS Bias

Forecast at 2011-09-04 00Z

ETS Bias

Comparison with add_omega

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

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!

Evidence of inconsistency between analysis and WRF forecast 0

Difference between analysis and 0h forecast

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 ?