Finite Difference and Finite Volume Concepts for ZCoordinates

Finite Difference and Finite Volume Concepts for Z-Coordinates J. Steppeler, DWD Ljubljana 2005 Offenbach 11. 3. 2003

• • Z-Coordinate Concepts Vertical Velocities Cloud and Precipitation forecasts Conclusions Offenbach 11. 3. 2003

Shaved Elements j + 1/2 i + 1/2 j + 1/2 i - 1/2 i, j j - 1/2 i - 1/2 j - 1/2 i + 1/2 Offenbach 11. 3. 2003

The finite volume method for the treatment of the fast waves in LM-Z • The computation of the fast waves is based on the evaluation of fluxes into a cell • The evaluation of the fluxes requires weights associated with the cell surfaces, which depend on their open part • Advection terms (slow waves) are computed by finite difference methods Offenbach 11. 3. 2003

The finite volume method for the treatment of the fast waves in LM-Z • A-Grid Finite Difference Z-Discretisation Offenbach 11. 3. 2003

The step-orography • Advantage: simle boundary conditions with z-coordiante • Disadvantages: No proven convergence; In CFD not used (“legoland topography”; Problems with precipitation forecasts; Problems pointed out by Klemp • The disadvantages are avoided by representing the orography by a linear spline rather than steps Offenbach 11. 3. 2003

Mountain generated waves: W (left), U (right) Eta - Model Analytic solution LM_Z Offenbach 11. 3. 2003

Expected advantages of the Zcoordinate • The Violation of an Approximation condition is corrected, both for high and low wind situations • Stratiform clouds and low stratus are predicted better in LM-Z • Mountain and valley winds are better with LM-z • Precipitation should be better with LM-Z, in particular near mountains Offenbach 11. 3. 2003

Offenbach 11. 3. 2003

Day Night The Atmosphere at Rest Computed with the Z_LM Offenbach 11. 3. 2003

Low stratus Offenbach 11. 3. 2003

Z-coordinate Offenbach 11. 3. 2003

28039700 6 to 6 • LM-Z OBS Offenbach 11. 3. 2003 LM-tf

28039700+12 • LM_Z • LM_tf Offenbach 11. 3. 2003

• LM_Z 28039700+12 Offenbach 11. 3. 2003 LM

LM LM_Z Cross sections east-west for vertical velocity Offenbach 11. 3. 2003

Cloud water East west cross section LM LM_Z Offenbach 11. 3. 2003

LM w, North-South Offenbach 11. 3. 2003 LM_Z

Qc for north south cross section LM LM_Z Offenbach 11. 3. 2003

MSG IR 08. 7 um 06. 03. 2005 12: 00 % cc © METEOSAT lm total cloud cover lm medium cloud cover lm high cloud cover lm_z medium cloud cover lm_z high cloud cover Offenbach 11. 3. 2003 lm_z total cloud cover lm low cloud cover lm_z low cloud cover

LM LM_Z 12 h Accumulated Precipitation for March 6 2005 06 to 18 UTC. The initial conditions are from the Global Model of DWD (00 UTC analysis, 24 hr forecast). Offenbach 11. 3. 2003

March 6 2005 March 7 2005 March 8 2005 R: 12 h forecasted accumulated precipitation from “Regular” LM is closer to observation Z: 12 h forecasted accumulated precipitation from LM_Z is closer to observation Precipitation Scores . Z R March 6 2005 Offenbach 11. 3. 2003 March 7 2005 . R Z March 8 2005

Total and average observed and forecasted precipitation height (mm) March 06 March 07 March 08 Observed: Total Average 10. 52 <0. 18> 152. 00 <2. 82> 78. 01 <1. 37> LM: Total Average 145. 99 <2. 52> 304. 56 <5. 64> 266. 06 <4. 67> LM_Z: Total Average 36. 87 <0. 64> 167. 82 <3. 11> 155. 57 <2. 73> Offenbach 11. 3. 2003

March 06 2005 mm 0 mm March 07 2005 Station # 0 Station # mm March 08 2005 0 LM_Z LM OBS Station # • Observed and Forecasted Precipitation Heights (06 -18 UTC) from LM_Z and LM. • Columns below 0 th horizontal correspond to no observed or forecasted precipitation. • Stations with no precipitation shown correspond to traces of observed but no forecasted precipitation. Offenbach 11. 3. 2003

Conclusions • Results concerning the vertical velocity are very different • The cases computed so far show a substantial improvement of precipitation forecasts Offenbach 11. 3. 2003