10 th COSMO General Meeting, Krakow, 15-19 September 2008 Recent work on pressure bias problem Lucio TORRISI Italian Met. Service CNMCA – Pratica di Mare

10th COSMO General Meeting, Krakow, 15-19 September 2008

Recent work on pressure bias problem

Lucio TORRISI

Italian Met. Service CNMCA – Pratica di Mare (Rome)

[email protected]


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


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


• In COSMO model an extrapolated boundary condition for p*, which is based on the assumption of a constant vertical gradient at the lower boundary:

Old bottom bound. cond.

=ke’ +1 ke’


“Pressure Bias Problem”Blue: Old Bottom Boundary Cond.Red: Dynamic Bottom Boundary 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

Dynamical bottom bound. cond.

Old BBC

DBBC


7km RK/LF comparison

LF

RK (with DBBC)


RK/LF comparison

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)

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!)


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


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).

MSLP bias difference increases with forecast time (COSMO larger positive bias)

COSMO(LF)

EURO-HRM


Pressure bias problemThe 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


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.


New reference atmosphere

7 km 14 kmPreliminary results!


• 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.

Heat source term in p equation

bRbbCMT

MCTT

vd

TTzTTMMQc

p

td

Td )(

v

bRbbCMpvdpd ppzppMpcc

td

pd )(/ v

f

pd

sl

pd

v

pdT S

c

LS

c

L

cQ RH

1

Turbulent heat and Radiaton flux

Diabatic heating due tocloud microphysical sources

flflv PPFFF

pd

dfl

pd

v

d

v

pd

dM c

TRSS

c

TR

R

R

c

TRQ

1

Turbulent flux forwater constituents

Turbulent flux for water constituents and Precipitation (gravitational diffusion) fluxes

Cloud heat sources


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 !” .



1/

1 1( )

MC CM CMpd vd T M T T p

vd pd

b b b R b bpd pd

d T pc c Q Q M M M

d t c c

T T p p z T T p pc c

v

bRbb

CMpMpdvdpd

MCTTpdvdpdvdpd

ppzppMQccc

MQcccpcctd

pd

)(/

1//

v

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.

The reformulated equations are:



• 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 RK7 km

OldQh_Pe

OldQh_Pe


LF

RK

7 km




7 km

LF


Heat source term in p-equation

7 km

• 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

Old Qh_Pe


Heat source term in p-equation

14 km

Ref_atm

Ref_atm+Qh_Pe

Old

Ref_atm+Qh_Pe

Euro-HRM


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!


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


Thank you for your attention!







Documents

10 th COSMO General Meeting, Krakow, 15-19 September 2008 Recent work on pressure bias problem Lucio TORRISI Italian Met. Service CNMCA – Pratica di Mare