Sensitivity experiments with the Runge Kutta time integration

Sensitivity experiments with the Runge Kutta time integration scheme Lucio TORRISI CNMCA – Pratica di Mare (Rome) l. torrisi@meteoam. it

Introduction • A new dynamical core has been developed in LM (Forstner and Doms, 2004). It is based on a TVD variant of 3 rd-order Runge Kutta time integration scheme (RK) using a 5 th-order spatial discretization of advection. • The RK core should be more accurate than the standard Leap-Frog/2 nd-order advection scheme (LF) and it will be used for very detailed short range forecasts. • RK core needs to be tested and evaluated, before it can be operationally implemented.

Overview • LM configuration and verification method. • RK compared to LF. • RK sensitivity to the: - integration time step (72 s, 48 s); - interval between two calls of some parameterizations (convection, turbulence); - turbulence parameterization scheme (diagnostic and prognostic TKE); - domain size; - moisture variables advection scheme (eulerian, semi-lagrangian); - moisture variables transport formulation. • Summary and conclusion.

LM configuration (v. 3. 16+) Domain size 465 x 385 (Euro. LM) Grid spacing 0. 0625 (7 km) Number of layers 35 Time step 72 sec Forecast range 24 hrs Initial time of model run 00 UTC Lateral boundary conditions and initial state Op. IFS (preproc. with CNMCA-IFS 2 LM) L. B. C. update frequency 3 hrs Orography Filtered (eps = 0. 1) Prognostic precipitation On Rayleigh damping scheme Relaxation to LM filtered fields Interval between two calls of turbulence 1 time step Turbulence parameterization Prognostic TKE R. damping: filter iteration number 10 R. damping: filter length 1

Objective verification method Statistical verification through comparison of LM forecasts with lowland station observations in the period 24 – 28 March 2005 (5 runs). Nearest grid point is used as LM forecast. Only land stations with h<700 m and height mismatch with model topography less than 100 m were used. About 3500 fc-obs pairs were used to calculate the mean error and RMSE of the surface variables forecast. They are enough to make statistical comparisons between different configurations of LM.

LF (40 s) compared to RK (72 s)

LF (40 s) compared to RK (72 s)

LF (40 s) compared to RK (72 s)

LF (40 s) compared to RK (72 s)

LF (40 s) compared to RK (72 s)

LF (40 s) compared to RK (72 s)

LF (40 s) compared to RK (72 s) RK performs worse than LF for MSLP forecasts due to a large bias (also small differences in other surface variables and wind vector). RK has a slightly smaller upper level temperature RMSE. RK and LF use different time steps that determine a different time interval between two calls of physics. One experiment to find out the cause of the MSLP deficiency in RK could be to decrease the time step from 72 s to 40 s (time step for LF), in order to have the same parameterizations calling frequency for LF and RK.

RK (40 s) compared to RK (72 s)

RK (40 s) compared to RK (72 s) RK with 40 s time step has a slightly smaller MSLP bias than RK with 72 s time step at T+18 h and T+24 h. This result seems to be due to the higher accuracy associated with the smaller time step. To totally exclude the influence of the interval between two calls of parameterization schemes, some experiments are useful. One experiment is to decrease the convection calling frequency nincconv from 10 to 5. Another experiment is to increase the interval between two calls of the prognostic TKE turbulence scheme ninctura from 1 to 2 (every two time steps instead of every time step).

RK (40 s) with ninctura = 1 , 2 or with nincconv = 10 , 5

RK (40 s) with ninctura = 1 , 2 or with nincconv = 10 , 5 RK is not significantly sensitive to the calling frequency of the convection and prognostic TKE turbulence scheme. What is the behaviour of the diagnostic TKE turbulence scheme compared to the prognostic one?

RK with new and old turbulence

RK with new and old turbulence

RK with new and old turbulence RK with the old turbulence scheme seems to perform better (smaller bias and standard deviation) than RK with the prognostic TKE turbulence parameterization. The improvement in the MSLP bias is related to the low level positive temperature bias of RK with old turbulence. The prognostic TKE turbulence parameterization seems to be one of the likely candidate to justify the MSLP forecast deficiency in RK. However, a large bias is still present! How does RK with old turbulence perform compared to the LF?

LF and RK with old turbulence

LF and RK with old turbulence

LF and RK with old turbulence Using the old turbulence scheme the large MSLP bias difference between RK and LF is slightly reduced. A positive MSLP bias (except for T+12 h) is present in RK, but a slightly smaller standard deviation than in LF is also found. Some tuning of the RK+physics core is necessary to reduce the large bias in MSLP forecast (cold bias in upper levels), but the slight improvement in the standard deviation seems to be an indication of the higher-order accuracy of the RK compared to the LF.

RK with different domain sizes

RK with different domain sizes The enlargement of the domain size seems to have a negative impact (larger RMSE forecast times greater than T+6 h) on the MSLP forecast. A similar result was obtained for a longer period using the LF (Torrisi, 2005). The increase of the standard deviation could be related to the improvement of the intrinsic variability of the model associated with the enlargement of the domain (BC are slightly affecting the forecast).

RK with Eulerian and SL moisture variables advection

RK with Eulerian and SL moisture variables advection

RK with Eulerian and SL moisture variables advection The SL moisture advection scheme does not show any significant difference in MSLP forecast compared to the Eulerian one (slightly larger standard deviation after T+18 h balanced by a slightly smaller bias), but it seems to have a slightly better skill for 6 h accumulated precipitation.

RK with conservation form of moisture variables transport

RK with conservation form of moisture variables transport

RK with conservation form of moisture variables transport The conservative form of the moisture variables transport has a larger MSLP bias than the default formulation. A slight improvement after T+18 h is obtained switching on the prognostic advection of density.

Summary and Conclusion • The comparison of LF and RK schemes was performed for a 5 days period using the Euro. LM configuration. • Statistical verification results showed that RK performance for surface variables was slightly better than LF one. A large MSLP bias was typical of the RK runs. • Some sensitivity studies were performed on RK to determine the cause of the MSLP forecast deficiency. RK did not show any sensitive to the calls of the prognostic TKE turbulence and convection schemes. • An improvement in the MSLP forecast was obtained using the old turbulent scheme, but a larger bias was found again. • The impact of the domain size and different moisture variables transport formulations was also evaluated. • More work, especially investigations on the numericsphysics interaction, are needed to improve the RK core.