recent work on pressure bias problem

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

Recent work on pressure bias problem

Lucio TORRISIItalian Met. Service CNMCA – Pratica di Mare (Rome)

[email protected]


Overview• The pressure bias problem (RK/LF)• Dynamics bottom boundary condition• RK/LF comparison• The pressure bias problem (domain

size, model equation formulation)• New reference atmosphere impact• Heat source term in pressure

equation• Conclusion


- Runs using mostly COSMO V4.2+- Domain dependent on grid spacing used- Objective verification of forecast parameters against European Temp/Synop network

Configuration


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.


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

Dynamic bottom boundary 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

Dynamic bottom boundary cond.


7km RK/LF comparison


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


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)


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 a constant rate of increase of temperature with the logarithm of pressure 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 (about 29km with the current default settings).


New reference atmosphereZaengl (2008) implemented a new reference atmosphere based on a temperature profile which starts with a prescribed sea-level temperature and exponentially approaches an isothermal stratosphere. In this way there is no limitation on vertical extent of model domain.


New reference atmosphere7 km


New reference atmosphere14 km


• 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

TTzTTMMQcp

tdTd

)(

v

bRbbCMpvdpd ppzppMpcc

tdpd )(/ v

f

pd

sl

pd

v

pdT S

cLS

cL

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

TRSScTR

RR

cTRQ

1

Turbulent flux forwater constituents

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

Cloud heat sources


Heat source term in p equation• An adequate approximation of pressure equation is

to neglect the moisture source term QM

• Dudhia (1993) has pointed out, that the neglection of diabatic heating term (QT) in the pressure equation might be even advantageous for models with a rigid upper boundary.


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 !” .


Heat source term in p equationThey concluded:“- ……- It is shown that the well-known heat- and moisture source/sink terms are necessary to be taken into account in the pressure equation in order to allow in small-scale simulations the direct influence of thermal and moisture effects on the pressure field - and so on the wind field too.- Ignoring these terms or having some inconsistent approximation of them in the pressure equation is equivalent to a hidden mass budget error in the model.- Furthermore, even the heating terms in the final prognostic temperature equation appear to be not correct, if for the elimination of the individual pressure tendency a pressure equation is applied where the diabatic terms are already ignored or incomplete.- …..”

From this work the motivation to investigate in real cases the effects of the heat source term in pressure equation (moisture source term neglected)



1/

1 1( )

MC CM CMpd vd T M T T p

vd pd

b b b R b bpd pd

d T p c c Q Q M M Md t c c

T T p p z T T p pc c

v

bRbb

CMpMpdvdpd

MCTTpdvdpdvdpd

ppzppMQccc

MQcccpcctdpd

)(/

1//

v

These new terms 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:



2.8 km

RK+Qh_PERK 2.8 km



2.8 km

RK+Qh_PERK

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

RK


7 km


Heat source term in p equation• In a few real cases (winter) with a 7 km grid spacing:

– domain averaged near upper boundary temperature is not significant changed (very very … slightly decreased)

– domain averaged total precipitation is slightly decreased– MSLP bias is reduced (except for 12-15 UTC)

LF RK

7 km


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

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


14 km

LF


Heat source term in p-equation14 km


Pressure bias problem

• Other work has been done (divergenge and pressure gradient in conservation form, Gassmann time splitting scheme, etc), but no positive impact was found

• 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 in pressure equation on the pressure bias is apparent in 14 km runs. The improvement is not general using a 7 km grid spacing (slight deterioration around 12 UTC). Two runs with 2.8 km show an enhancement of maximum of precipitation. More work is needed!


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, which is equivalent to introduce artificial sources or sinks in the continuity equation

• Problems arise in applications such as data assimilation systems, where the pressure accuracy is important

• A more conservative dynamical core is needed for COSMO model


Thank you for your attention!

recent work on pressure bias problem

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