Transport Schemes for HighResolution Atmospheric Simulation

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Transport Schemes for High-Resolution
Atmospheric Simulation

• Bill Skamarock NCAR/MMM

24 h accumulated precip, valid 12 UTC 5 June
2005 blue gt 12 mm
Stage 4 analysis
ARW forecast, no PD adv.
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Moisture Transport in ARW
1D advection
overshoot
undershoot
ARW scheme is conservative, but not positive
definite nor monotonic. Removal of negative q
results in spurious source of q .
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ARW Transport
Continuous eqn. (PDE)
Spatial discretization ARW - divergence terms
2-6 order accuracy (for constant u, v, w).
Instantaneous fluxes defined on control volume
faces
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ARW Transport
Time integration - RK3 ODE solver
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ARW Transport
.
For constant u, v, ? - 3rd order accurate in
space and time Conservative Consistent with the
ARW continuity equation Not positive
definite Not monotonic
1D test Cr 0.5, 1 revolution (200 steps)
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Forward-in-Time Transport Schemes
Continuous eqn.
Integrate in time
Integrate in space over the control volume
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ARW-RK3 and Forward-in-Time Schemes
FIT schemes
ARW - RK3
Final RK3 step looks like an FIT scheme - hence
some PD and monotonic FIT corrections can be used
in RK3 integration.
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RK3 PD/monotonic limiter
Following Zalesak (1979, JCP)
Existing RK3 transport scheme
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RK3 PD/monotonic limiter
Existing RK3 transport scheme
replace with
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RK3 PD/monotonic limiter
(1) Decompose fluxes
(2) 1st order upwind update
(3) Preliminary update for min/max values.
(4) Renormalize if updates are not monotonic or PD
PD
Monotonic
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RK3 PD/monotonic limiter
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RK3 PD limiter examples
24 h accumulated precip, valid 12 UTC 5 June
2005 blue gt 12 mm
ARW forecast, no PD adv.
Stage 4 analysis
Skamarock and Weisman 2009 (thanks to Wei Wang
and Kevin Manning)
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RK3 PD limiter examples
24 h accumulated precip, valid 12 UTC 5 June
2005 blue gt 12 mm
ARW forecast, PD adv.
ARW forecast, no PD adv.
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RK3 PD limiter examples
24 h accumulated precip, valid 12 UTC 14 April
2007 blue gt 12 mm
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RK3 PD limiter examples
14 April 2007
5 June 2005
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RK3 PD limiter examples
Moisture budgets from the 5 June 2005 and 14
April 2007 cases.
Seasonal biases from 2005 and 2007
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RK3 monotonic limiter examples
Wang, Skamarock and Feingold (2009)
250 m grid Diagonal transport of a
cube Horizontal cross section T10 minutes
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RK3 monotonic limiter examples
Marine stratocumulus (DYCOMS II)
RK3
2 hours
4 hours
RK3 positive definite
RK3 monotonic
Wang et al. 2009
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RK3 monotonic limiter examples
Marine stratocumulus (DYCOMS II)
RK3
RK3 positive definite
RK3 monotonic
Wang et al. 2009
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Global ARW - Latitude-Longitude Grid

• Map factors - mx and my
• Polar boundary conditions
• Polar filtering

Converging gridlines severely limit timestep. The
polar filter removes this limitation.

• Filter procedure - Along a grid latitude circle
• Fourier transform variable.
• Filter Fourier coefficients.
• Transform back to physical space.

• Problems for PD and monotonic schemes
• Large Courant numbers near poles invalidates the
  limiters.
• Fourier filtering is not monotonic or PD.

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Global ARW - Latitude-Longitude Grid

• Problems for PD and monotonic schemes
• Large Courant numbers near poles invalidates the
  limiters.
• Fourier filtering is not monotonic or PD.

• Possible solutions
• Perform transport along latitude circles as part
  of the filtering process, using Large-Cr stable
  schemes with modified limiters.
• Use multiple applications of local filters for
  polar filters.

• Drawbacks
• Expense.
• Lack of filter control.

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Summary

• Positive-definite (PD) and monotonic transport
  options released with WRFV3.1
• PD transport decreases positive precipitation
  bias observed in convection-permitting scale
  forecasts.
• Monotonic transport improves moist LES
  simulations, and in chemistry/air quality
  applications.
• The PD and monotonic options do not work for
  global WRF (latitude-longitude grid). Other
  PD/monotonic approaches have problems.