A Sensitivity Study of Integration TimeStep Size

1
A Sensitivity Study of Integration Time-Step
Size in Heavy Rainfall Simulations
Dong-Kyou Lee and Kyung-Ho Lee
School of Earth and Environmental Sciences, Seoul
National University, Korea
1. Introduction
24-h precipitation (mm) for Case 2 (00 UTC 24
00 UTC 25 Aug.)

• 24-h rainfall amount (mm) at maximum rainfall
  points

• In the integration of a non-hydrostatic system,
  computational efficiency and accuracy are
  significantly dependent on the integration time
  step. In general, most numerical models have used
  a time splitting method for computational
  efficiency. This method is also applied to the
  Weather and Research Forecast (WRF) model using
  the third-order Runge-Kutta scheme for time
  integration and a high order spatial discrete
  scheme for the advection term (Klemp et al.,
  2000). Wicker and Skamarock (2002) showed that it
  results in good computational efficiency and high
  accuracy in solution with high order spatial
  differentiation.
• In heavy rainfall simulations, maximum rainfall
  amount and location highly depend on a number of
  factors such as initial and boundary condition,
  model resolution, and physical parameterization.
  An integration time-step in high resolution
  mesoscale models also plays an important role to
  precipitation prediction (Wu et al., 2001). We
  investigate the sensitivity of heavy
  precipitation simulation to time-step sizes, and
  find an optimal range of time-step sizes between
  accuracy and efficiency in the WRF model
  integration of heavy rainfall cases over East
  Asia.

OBS
EXP90
EXP120

• 24-h area-mean rainfall amount (mm) in Area B

167.1mm
201.8mm
232.9mm
192.7mm

• At 24-h maximum rainfall points

2. Model Description and Experiments Setup
EX150
EX180
Model Configuration
140.6mm
168.2mm

• In the test advection of the 1-d wave equation,
  Wicker and Skamarock (2002) used the Runge-Kutta
  3rd order scheme for a time step and the 5th
  order spatial difference, and confirmed that
  smaller Courant numbers have higher accuracy of
  wave advection.

• All simulated 24-h rainfall amount at maximum
  rainfall points is overestimated. The rainfall
  bias at the maximum points is smallest in EX150.
• Area-mean rainfall amount bias from observation
  increases as the time-step increases. The
  area-mean rainfall amount bias is smallest in
  EX90.
• Simulated precipitation results mostly from
  non-convective rain in this case.
• Differences in non-convective rain between
  time-steps are relatively large, while those in
  convective rain are small in terms of area- mean
  rainfall.

144.9mm
167.1mm

• The 24-h total rainfall at the maximum rain
  points are quite different between time-step
  sizes. Total and hourly maximum rainfall amount
  in EX150 150 s is closer to observation.
• Two major maximum rainfall points are captured in
  all simulations, but rainfall patterns around the
  maximum points are different between time-steps.
• Distance error between observed and simulated
  maximum rainfall points is smallest in EX90 90
  s.

• Time series of hourly rainfall at 24-h maximum
  rainfall points

Experiments setup

• Four time-step sizes are considered to test the
  sensitivity of heavy rainfall simulation to
  time-step size. A time step of 3 ?X (in seconds
  with ?X in kilometers) and greater is recommended
  for use with WRF.
• ?t 90 seconds is equivalent to the recommended
  ?t 3 ?X.

Temporal variation of area-mean model variables
Case1 (12UTC 24 12 UTC 25 July, Area A)
Domain

• Areas A and B indicate heavy rainfall areas of
  Case 1 (24 July) and Case 2 (23 August),
  respectively.

B
A

• In this case, temporal variations of the
  variables are not sensitive to time step.

Case2 (00UTC 24 00 UTC 25 August, Area B)
3. Case Description
W500
Case 2 (23-25 August 2003)
Case 1 (24-25 July 2003)

• In this case, temporal variations of the
  variables are significantly sensitive to
  time-step sizes during the intensive
  precipitation period. During the intensive
  precipitation period, the trends of total rain
  and 500 hPa vertical velocity are similar.

• Heavy rainfall from MCSs developing alonga
  frontal rain-band over central Korea.

• Heavy rainfall from MCSs developing rapidly over
  central and southern Korea.

24-h area-mean vertical profile of hydrometeors
resolved by the explicit moisture scheme
4. Simulation Results

• 24-h rainfall amount (mm) at maximum rainfall
  points

24-h precipitation (mm) for Case 1 (12 UTC 24
12 UTC 25 July)
Case 1 (12UTC 24 - 12UTC 25 July)
63.5mm
61.0mm

• 24-h area-mean rainfall amount (mm) in Area A

Case 2 (00 UTC 24 00 UTC 25 Aug)
68.4mm
71.0mm

• At 24-h maximum rainfall points

75.4mm
65.4mm

• In case 2, there are significant differences in
  vertical profiles of hydrometeors between
  time-step sizes.

• The24-h rainfall amount at maximum rainfall
  points is less simulated than observation.
  Rainfall amount error at the maximum point is
  largest in EX90, while area-mean rainfall amount
  bias is smallest in EX90.
• As the time step size increases, bias of
  area-mean rainfall amount from observation
  increases in the experiments with the initial
  time, 00 UTC 24 July.
• Simulated precipitation results mostly from
  convective rain in this case.
• Differences in convective rain between time-steps
  are relatively small.
• Hourly rainfall at the maximum points is
  sensitive to time-step sizes.

5. Summary and Discussion
71.3mm
77.4mm

• As the time-step size increases from the
  recommended size, computational efficiency almost
  linearly increases.
• In heavy rainfall simulations over Korea, the
  sensitivity of time-step size to spatial pattern
  and temporal variation of precipitation is
  relatively small, while maximum rainfall points,
  and total and hourly rainfall amount are very
  sensitive to time-step size. In WRF time-step
  sizes, ?t 3 to 5 ?X, can be used to simulate
  the maximum rainfall points and amount of heavy
  rainfall over Korea.
• In the simulation of heavy rainfall resulted from
  intense convection such as MCSs, the differences
  in rainfall amount between time-step sizes are
  relatively small. However, extreme rainfall
  amount forced by large-scale environment (e.g. a
  stationary front) which is resulted from the
  explicit moisture scheme are quite sensitive to
  time-step size.
• During the time integration period except for
  precipitation duration, model variables are not
  sensitive to time-step sizes. This means that
  truncation error is not sensitive to time-step
  size. However, there are significant differences
  in model variables between time-step sizes during
  intensive precipitation duration. Thus, physics
  schemes related to precipitation process are very
  sensitive to time-step size.
• In this study, the explicit moisture scheme is
  more sensitive to time-step size than the cumulus
  parameterization scheme. One of the possible
  reasons is that the explicit moisture scheme uses
  a time-dependent formulism. Therefore, simulated
  hydrometeors resolved in the explicit moisture
  scheme are different between time-step sizes.

• Maximum rainfall points are quite different
  between time step sizes.
• Differences in rainfall amount and spatial
  pattern between time-steps are small.

• Time series of hourly rainfall at 24-h maximum
  rainfall points