1. Michael Baldauf Deutscher Wetterdienst, Offenbach, Germany
COSMO Priority Project: Further developments of the Runge-Kutta Time Integration Scheme COSMO General Meeting Bukarest,
Michael Baldauf
Deutscher Wetterdienst, Offenbach, Germany

2. List of people contributing to the project: (alphabetical order)
Michael Baldauf (DWD, D)
Gabriella Ceci (CIRA, I)
Guy deMorsier (MeteoCH, CH)
Jochen Förstner (DWD, D)
Almut Gassmann (Univ. Bonn, D) (FTE not counted)
Paola Mercogliano (CIRA, I)
Thorsten Reinhardt (DWD, D)
Lucio Torrisi (CNMCA, I)
Pier Luigi Vitagliano (CIRA, I)
Klaus Stephan (DWD, D) (FTE not counted)
Matthias Raschendorfer (DWD, D) (FTE not counted)

3. Task 1: Looking at pressure bias (Torrisi, Förstner)
verifications of LM 7 km runs showed a higher positive pressure bias for the RK core than for the Leapfrog core, whereas other variables show comparable behaviour.
Work done:
5-day verifications were done for several model configurations
only little impact on PMSL by:
physics coupling
advection of qx (Bott, Semi-Lagrange)
most significant impact on PMSL:
new dynamical bottom boundary condition (DBBC) ( Task 6)
p‘T‘-dynamics in the RK-core
both measurements reduce the pressure bias
verification area

4. Task 1: Looking at pressure bias

5. Task 1: Looking at pressure bias
positive impact of p‘T‘-dynamics

6. Task 1: Looking at pressure bias
positive impact of DBBC

7. Status: verifications carried out
Task 1: Looking at pressure bias
Status: verifications carried out
Work to do: longer verifications periods should be inspected
Upper air verifications
Lateral boundary conditions
pressure bias improved by another fast waves solver?? ( --> task 10)

8. Task 2: Continue RK case studies (Torrisi, deMorsier)
extensive verification of the other tasks
Work done:
case study 4.-8.Dez (inversion in m)
was carried out: penetration of the stratus in Alpine valleys in 2km- sim. better performed by RK-core compared to LF-core
Status: test case carried out; verifications were made
Work to do: test cases should be continued

9. Task 3: Conservation (Baldauf)
Tool for inspection of conservation properties will be developed. Integration area = arbitrarily chosen cubus (in the transformed grid, i.e. terrain-following)
balance equation for scalar :
temporal change
flux
divergence
sources
/ sinks
Status:
integral over a volume (arbitrary square-stone): ready
Subr. init_integral_3D: define square-stone (in the transformed grid!), domain decomp.
Function integral_3D_total: calc. volume integral
Function integral_3D_cond: calc. vol. integral over individual processor
Work to do
flux integral over the surface

10. Task 4: Advection of moisture quantities in conservation form
(Förstner)
implementation of a Courant-number independent advection algorithm
for the moisture densities
Status: implemented schemes (Bott-2, Bott-4) behave well
(Semi-Lagrange-scheme as a testing tool is also available)
task finished

11. Transport of Tracer in a Real Case Flow Field
PP RK
Transport of Tracer in a Real Case Flow Field
Bott (2nd) “Flux Form - DIV” + Clipping
init
semi- Lagrange (tri-cubic) + Clipping
Bott (2nd) “Conserv. Form”

12. Task 5: investigation of convergence (Ceci, Vitagliano, Baldauf)
determination of the spatial and temporal order of convergence of the RK-scheme in combination with advection schemes of higher order.
Planned test cases:
linear mountain flows (2D, 3D)
nonlinear mountain flows (dry case)
nonlinear mountain flows with precipitation
Status: implementation of LM and test environment. First tests with linear mountain flow.
Work to do: determine L2, L – errors of KE, w, ..., dependent from x, t, ... for the tests cases

13. (Förstner, Torrisi, Reinhardt, deMorsier)
Task 6: deep valleys
(Förstner, Torrisi, Reinhardt, deMorsier)
detection of the reason for the unrealistic ‚cold pools‘ in Alpine valleys
Task 7: Different filter options for orography
(Förstner)
The reason for the cold pools was identified: metric terms of the pressure gradient
Dynamical Bottom boundary condition (DBBC) (A. Gassmann (2004), COSMO-Newsl.)
and a slope-dependent orography-filtering cures the problem to a certain extent.
Status: the orography filtering is now sufficiently weak for DWD-LMK applications (max. slopes 30% allowed)
Proposal for future work:
inspect the limitations of the terrain following coordinate for steeper slopes, e.g.
for application of aLMo 2 (MeteoCH) in Alpine region
for future LMK ~1 km horizontal resolution

14. cold pool – problem in narrow valleys
is essentially induced by pressure gradient term
T (°C)
starting point
after 1 h
after 1 h
modified version:
pressure gradient on z-levels, if
|metric term| > |terrain follow. term|

15. Dynamic Bottom Boundary...
“(Positive) Pressure Bias Problem”
blue: Old Bottom Boundary Cond.
red: Dynamic Bottom Boundary Cond. (Figures by Torrisi, CNMCA Rom)
Dynamic Bottom Boundary...
... Condition...
... for metric pressure gradient term in equation for u- and v-component. Gaßmann (COSMO Newsletter No. 4)

18. Task 8: Higher order discretization in the vertical for RK-scheme
(Baldauf)
Improved vertical advection for the dynamic var. u, v, w, T (or T‘), p‘
motivation: resolved convection
vertical advection has increased importance => use scheme of higher order (compare: horizontal adv. from 2. order to 5. order)
=> bigger w (~20 m/s) => Courant-crit. is violated => implicit scheme or CNI-explicit scheme
up to now: implicit (Crank-Nicholson) advection 2. order (centered differences)
new: implicit (Crank-N.) advektion 3. order  LES with 5-banddiagonal-matrix
but: implicit adv. 3. order in every RK-substep; needs ~ 30% of total computational time!
 planned: use outside of RK-scheme (splitting-error?, stability with fast waves?)
Status: implicit scheme of 3. order implemented (5-banddiagonal solver, ...)
Work to do: best combination with time integration scheme?
test suite; verification

19. Idealized 1D advection test
Task 8: Improved vertical advektion for
dynamic var. u, v, w, T, p‘
analytic sol.
implicit 2. order
implicit 3. order
implicit 4. order
C=1.5
80 timesteps
Idealized 1D advection test
C=2.5
48 timesteps

20. case study ‚25.06.2005, 00 UTC‘ total precipitation sum after 18 h
Task 8: Improved vertical advektion for
dynamic var. u, v, w, T, p‘
case study ‚ , 00 UTC‘
total precipitation sum after 18 h
with vertical advection 2. order
difference total precpitation sum after 18 h
‚vertical advection 3. order – 2. order‘

21. Task 9: Physics coupling scheme (Förstner, Stephan, Raschendorfer)
original task: problems with reduced precipitation could be due to a nonadequate coupling between physics scheme and dynamics
Problems in new physics-dynamics coupling (NPDC):
Negative feedback between NPDC and operational moist turbulence parameterization (not present in dry turbulence parameterization)
2-z - structures in the specific cloud water field (qc)
2-z - structures in the TKE field, unrealistic high values, where qc > 0
Status: NPDC-scheme analogous to WRF was implemented. Problems occuring  reduced variant is used now
Work to do:
what are the reasons for the failure of the WRF-PD-scheme in LM? (turbulence scheme?)
 test tool (Bryan-Fritsch-case) is developed in PP ‚QPF‘, task 4.1

22. Physics-Dynamics-Coupling
n = (u, v, w, pp, T, ...)n
Descr. of Advanced Research WRF Ver. 2 (2005)
Physics (I)
Radiation
Shallow Convection
Coriolis force
Turbulence
‚Physics (I)‘-Tendencies: n(phys I)
+ n-1(phys II)
Dynamics
Runge-Kutta [ (phys) + (adv)  fast waves ]
* = (u, v, w, pp, T, ...)*
- n-1(phys II)
Physics (II)
Cloud Microphysics
‚Physics (II)‘-Tendencies: n(phys II)
n+1 = (u, v, w, pp, T, ...)n+1

23. Task 10: Testing of alternative fast wave scheme
(Gassmann, Förstner, Baldauf)
p‘T‘-RK-scheme
‚shortened-RK2‘-scheme (Gassmann)
this allows the use of the ‚radiative upper boundary condition‘ (RUBC)
Status:
p‘T‘-RK-scheme is already tested and is used now in LMK
‚shortened RK2‘-scheme works
RUBC is tested in idealized and one real test case
Work to do:
implement ‚shortened RK2‘ version in official LM version
experiments especially with RUBC
tests both versions in CLM-application (dx=18 km)

24. contours: vertical velocity isolines: potential temperature
Runge-Kutta new p*-T*-dynamics
Runge-Kutta old p*-T-dynamics
contours: vertical velocity isolines: potential temperature

25.  choose =0.7 as the best value
Choose CN-parameters for buoyancy in p‘T‘-dynamics from stability analysis
=0.5 (‚pure‘ Crank-Nic.)
=0.6
=0.7
=0.8
=0.9
=1.0 (purely implicit)
 choose =0.7 as the best value

26. LMK (Lokal-Modell Kürzestfrist)
grid length: x = 2.8 km
direct simulation of the coarser parts of deep convection
interactions with fine scale topography
timestep t=30 sec.
421 x 461 x 50 grid points ~ 1200 * 1300 * 22 km³ lowest layer in 10 m above ground
forecast duration: 18 h started at 0, 3, 6, 9, 12, 15, 18, 21 UTC
center of the domain 10° E, 50° N
boundary values from LME (x = 7 km)
in pre-operational mode at DWD since