Recent work on pressure bias problem

1. Recent work on pressure bias problem Lucio TORRISI Italian Met. Service CNMCA – Pratica di Mare (Rome) l.torrisi@meteoam.it

2. Overview • Pressure bias problem: RK/LF • Dynamical bottom boundary condition for vertical pressure gradient in RK core • Pressure bias problem: domain size, model equation formulation • New reference atmosphere and consistent p0-h averaging (Zaengl, 2008) • Heat source term in pressure equation • Conclusion

3. Pressure bias problem (1) • This problem was pointed out in Torrisi (2005): “Sensitivity experiments with the Runge Kutta time integration scheme”, presentation at COSMO GM in Zurich 1) RK/LF core - Objective verification showed a difference in MSLP bias behaviour between 7km RK (red line) and LF (green line)runs. MSLP bias difference increases with forecast time. RK runs have a larger bias, typically positive, leading to a worse RMSE. RK LF

4. Old bottom bound. cond. • In COSMO model an extrapolated boundary condition for p*, which is based on the assumption of a constant vertical gradient at the lower boundary: = ke’ ke’ +1

5. “Pressure Bias Problem” Blue: Old Bottom Boundary Cond. Red: Dynamic Bottom Boundary Cond. Dynamical bottom bound. cond. • An improvement in RK core was obtained by the implementation of the Gassmann formulation (COSMO Newsletter No. 4) of the dynamic bottom boundary condition for metric pressure gradient term in equation for u- and v-component Old BBC DBBC

6. 7km RK/LF comparison LF RK (with DBBC)

7. RK/LF comparison Other work has been done on RK core (change in metric term discretization, change in vertical average on half levels, etc), but no positive impact was found (in some cases slight negative impact!) In summary: • 2.8/7km: the results of the previous experiment (7km, winter period) show that RK and LF MSLP bias are almost the same, but small differences (depending on season, domain, location, etc) are still found (RK larger bias) • 14 km: very big differences are found (RK has an increasing MSLP positive bias with forecast time)

8. Pressure bias problem (2) 2) Domain size - Objective verification showed a difference in MSLP bias behaviour in 7km RK (and LF) runs having a different domain size (smaller one - red line, larger one -green line). The larger domain, the greater MSLP bias Larger domain Smaller domain

9. Pressure bias problem (3) 3) Model formulation - Objective verification showed a difference in MSLP bias behaviour between 14km COSMO LF/RK (blue line) and HRM (red line) runs. HRM is the DWD regional hydrostatic model (LF time integration scheme) used in the CNMCA assimilation systems (3D-Var PSAS and LETKF). COSMO(LF) MSLP bias difference increases with forecast time (COSMO larger positive bias) EURO-HRM

10. Pressure bias problem • The increase of the MSLP bias with forecast time is a characteristics of COSMO model runs and it does not seem dependent only on dynamical core (point 1). This behaviour is evident using very large domain size (point 2) and particularly clear using a 14 km grid spacing (point 3). • The effects on the pressure bias of two changes in the model formulation will be addressed: • Zaengl (2008) proposed a new reference atmosphere to overcome the problem of limitation in vertical extent of the model domain using the default reference atmosphere • Gassmann and Herzog (2006) reconsidered the derivation of prognostic temperature and pressure equations to remove some inconsistencies in the formulation of these equations

11. New reference atmosphere Introducing a reference state reduces the computational error in the calculation of pressure gradient terms in the equation of motion for not too large deviations of pressure from reference pressure. The default reference atmosphere of COSMO model is based on assuming dT/d(logp)=const. The lapse rate dT/dz becomes more and more negative with height limiting the possible vertical extent of the model domain. To overcome this problem, Zaengl (2008) implemented in COSMO model a new reference atmosphere based on a temperature profile which starts with a prescribed sea-level temperature and exponentially approaches an isothermal stratosphere. An inconsistency in the full level reference pressure calculation was also removed.

12. New reference atmosphere 14 km 7 km Preliminary results!

13. Heat source term in p equation • Pressure bias increase has been found in experiments using a coarser resolution (14 km) and also enlarging the size of the domain (7km). This could be an indication of some inconsistencies in the formulation of T/p budget equ. Turbulent heat and Radiaton flux Diabatic heating due to cloud microphysical sources Cloud heat sources Turbulent flux for water constituents and Precipitation (gravitational diffusion) fluxes Turbulent flux for water constituents

14. Heat source term in p equation • The heat and moisture terms (QT and QM) are neglected in the COSMO model pressure equation • Gassmann and Herzog (2006) in their presentation at LM-User Meeting “reconsidered the derivation of prognostic temperature and pressure equations in the LM” “1. In pressure equation heat and moisture source terms neglected 2. dp/dt in T-equation eliminated after neglecting these terms 3. Formal addition of moist convection tendency, computational mixing, lateral and upper boundary relaxation terms in T/p equ.” “This operation is equivalent to the application of a wrong continuity equation producing a mass deficiency ………………………..” “The way to come to this result is wrong and leads to insufficient equations !” .

15. Heat source term in p equation The reformulated equations are: These new terms (except for QM) were added in the p/T equations and the saturation adjustment scheme was consistently adapt to the changes in p/T equations.

16. Heat source term in p equation • In a few real cases (winter) with a 7 km grid spacing: • domain averaged total precipitation is slightly decreased • MSLP bias is reduced (except for 12-15 UTC) LF RK 7 km Old Old Qh_Pe Qh_Pe

17. Heat source term in p equation LF 7 km RK

18. Heat source term in p equation LF 7 km

19. Heat source term in p-equation • In a summer period the effect of the heat source term on pressure bias seems to be different from the winter period results previously shown: • MSLP bias is reduced from 21 to 06 UTC and increased from 9-15 UTC Qh_Pe Old 7 km

20. Heat source term in p-equation 14 km Old Ref_atm Ref_atm+Qh_Pe Ref_atm+Qh_Pe Euro-HRM

21. Pressure bias problem • Impact of the new reference atmosphere on the pressure bias is very clear at 14km, but it quickly decreases with increasing resolution • Impact of heat source term in pressure equation on the pressure bias is apparent in 14 km runs. The improvement is not general using a 7 km grid spacing due to a slight deterioration around 12 UTC (larger in summer period). Two runs with 2.8 km grid spacing show no significant differences except for an enhancement of precipitation maxima. • More work is needed to understand deeply the effects of these changes!

22. Conclusion • The pressure bias problem is typical of COSMO model (not only of RK core!) • The pressure bias problem seems to be mainly related to the model equation formulation. The use of prognostic p equation does not guarantee an exact mass conservation • Problems arise in applications such as data assimilation systems, where the pressure accuracy is important • A more conservative dynamical core for COSMO model could be an option to tackle this problem

23. Thank you for your attention!

24. 7km RK/LF comparison

25. 7km RK/LF comparison

26. 7km RK/LF comparison