Numeric developments in COSMO
SRNWP / EWGLAM-Meeting
Titelfoto
auf dem Titelmaster
einfügen
Dubrovnik,
08.-12.10.2007
Michael Baldauf1, Jochen Förstner1, Uli Schättler1
Pier Luigi Vitagliano2, Gabriella Ceci2 , Lucio Torrisi3,
Ronny Petrik4
1Deutscher
Wetterdienst, 2CIRA-Institute, 3USMA (Rome),
4Max-Plank-Institut Hamburg
Outlook
Dynamical cores in the COSMO model:
• Terrain following
• Leapfrog time integration
• Runge-Kutta time integration (COSMO Priority Project)
• ‚operational version‘
•
•
•
•
•
Stability considerations (Winter storm ‚Kyrill‘, ...)
p'T'-dynamics
Moisture advection
Deep / shallow atmosphere
Physics/Dynamics coupling
• alternatives (A. Gassmann)
• Semi-implicit (S. Thomas, ...)
• LM-Z (COSMO Priority Project)
SRNWP – 08.10.2007
2
The operational Model Chain of DWD:
GME, COSMO-EU and -DE
(since 16. April 2007)
COSMO-EU
GME
hydrostatic
parameterised convection
x  40 km
368642 * 40 GP
t = 133 sec., T = 7 days
SRNWP – 08.10.2007
(LME)
non-hydrostatic
parameterised convection
x = 7 km
665 * 657 * 40 GP
t = 40 sec., T= 78 h
3
COSMO-DE
(LMK)
non-hydrostatic
resolved convection
x = 2.8 km
421 * 461 * 50 GP
t = 25 sec., T = 21 h
COSMO - Working Group 2 (Numerics)
COSMO Priority Project 'LM-Z'
several improvements on the code:
• prevent decoupling of z-grid (dynamics) and tf-grid (physics) by
'nudging'
• implicit vertical advection  increase in time step
• tendencies of data assimilation are now also transformed to the
z-grid
Comparison of LM-Z and an older version of LM (COSMO-model)
(e.g. without prognostic precipitation)
--> report: end 2007
Collaboration with Univ. of Leeds started
SRNWP – 08.10.2007
4
COSMO - Working Group 2 (Numerics)
COSMO-Priority Project ‚Runge-Kutta‘:
1.
New Developments
1. NEW: Divergence damping in a 3D-(isotropic) version
2. NEW: DFI for RK
3. Advection of moisture quantities in conservation form
4. Higher order discretization in the vertical
5. Physics coupling scheme
6. Testing of alternative fast wave scheme
7.
8.
2.
3.
Development of a more conservative dynamics (planned)
Development of an efficient semi-implicit solver in combination with RK
time integration scheme (planned)
Developing diagnostic tools
1. Conservation inspection tool (finished)
2. Investigation of convergence
Known problems
1. Looking at pressure bias
2. Deep valleys
3. (Different filter options for orography) (finished)
SRNWP – 08.10.2007
5
Numerics and Dynamics - COSMO-DE developments
Grid structure
Prognostic Var.
Time integration
Fast modes
Advection
Other slow modes
Smoothing
SRNWP – 08.10.2007
horizontal: Arakawa C,
vertical:
Lorenz
cartesian components u, v, w, p’,T’ (LME: T)
time-splitting between fast and slow modes
- 3-timelevels: Leapfrog (+centered diff.) (Klemp, Wilhelmson, 1978)
- 2-timelevels: Runge-Kutta: 2. order, 3. order (Wicker, Skamarock, 1998, 2002)
(=sound waves, buoyancy, divergence filtering)
centered diff. 2. order, vertical implicit, (p’T’-Dyn.)
for u,v,w,p’,T’:
horizontal. adv.: upwind 3., 5. order / centered diff. 4. 6. order
vertical adv.: implicit 2. order
for qv, qc, qi, qr, qs, qg, TKE:
LME: qv, qc: centered diff. 2nd order
qi:
2nd ord. flux-form advection scheme
qr, qs: semi-lagrange (tri-linear interpol.)
Courant-number-independent (CNI)-advection:
- Bott (1989) (2., 4. order), in conservation form
- Semi-Lagrange (tricubic interpol.)
(optional: complete Coriolis terms)
3D divergence damping
horizontal diffusion 4. order applied only in the boundary relaxation zone
slope dependent orographic filtering
6
Stability considerations
Winter storm ‚Kyrill‘, 18.01.2007
crash of all COSMO-DE (2.8 km)-runs from 03, 06, 09, ... UTC
two measures necessary:
• timestep:
• old: t = 30 sec. (winter storm ‚Lothar' could be simulated)
• new: t = 25 sec
• time integration scheme:
• old: TVD-RK3
(Shu, Osher, 1988) 
• new: 3-stage 2nd order RK3
(Wicker, Skamarock 2002) 
SRNWP – 08.10.2007
7
COSMO-DE (2.8 km), 18.01.2007
SRNWP – 08.10.2007
8
COSMO-DE (2.8 km), 18.01.2007
SRNWP – 08.10.2007
9
Von-Neumann stability analysis of a
2-dim., linearised Advection-Sound-Buoyancy-system
SRNWP – 08.10.2007
10
SRNWP – 08.10.2007
11
Crank-Nicholson-parameter for buoyancy terms in the p‘T‘-dynamics
=0.5 (‚pure‘ Crank-Nic.)
=0.6
=0.7
amplification
factor
=1.0 (pure implicit)
=0.9
Cadv = u T / x
=0.8
Csnd = cs t / x
 choose =0.7 as the best value
SRNWP – 08.10.2007
• RK3-scheme
(WS2002)
• upwind 5th order
• Sound: =0.6
• x/ z=10
• T/ t=6
12
What is the influence of divergence filtering ?
• fast processes (operatorsplitting):
• sound (Crank-Nic., =0.6),
• divergence damping (vertical implicit)
• no buoyancy
• slow process: upwind 5. order
• time splitting RK 3. order (WS2002-Version)
• aspect ratio: x / z=10
• T / t=6
--> Divergence damping is needed in this dynamical core!
SRNWP – 08.10.2007
13
Influence of Cdiv
stability limit by
long waves (k0)
Cdiv = div t/x2
in COSMO-model:
Cdiv = xkd * (cs * t/ x)2
~0.35
Cdiv=0
Cdiv=0.025
amplification
Cadv = u T / x
factor
Cdiv=0.05
Cdiv=0.1
Cdiv=0.15
Csnd = cs t / x
SRNWP – 08.10.2007
14
Advantages of p'T'-dynamics over p'T-dynamics
1. Improved representation of T-advection in terrain-following coordinates
Terms (a) and (b) cancel analytically, but not numerically
2. Better representation of buoyancy term in fast waves solver
Buoyancy term alone generates an oscillation equation:
using T:
 = g/cs
using T':
 = a = acoustic cut-off frequency
SRNWP – 08.10.2007
15
point 1.): 'improved T-advection' ...
Idealised test case:
Steady atmosphere with mountain
base state: T0, p0
deviations from base state: T', p'  0  introduces spurious circulations!
SRNWP – 08.10.2007
16
Runge-Kutta
old p*-T-dynamics
Leapfrog
contours: vertical velocity w
isolines: potential temperature 
SRNWP – 08.10.2007
17
Runge-Kutta
new p*-T*-Dynamik
Runge-Kutta
old p*-T-Dynamik
contours: vertical velocity w
isolines: potential temperature 
SRNWP – 08.10.2007
18
Climate simulations
 start: 1. july 1979 + 324 h (~2 weeks)
 results: accumulated precipitation (TOT_PREC) and PMSL
Problems:
 unrealistic prediction of pressure and precipitation distribution
 strong dependency from the time step
These problems occur in the Leapfrog and the (old) Runge-Kutta-Version
(both p'T-dynamics) but not in the semi-implicit solver or the RK-p'T'-dynamics.
assumption: point 2.) 'treatment of the buoyancy term' improves this case
(simulations: U. Schättler, in cooperation with the CLM-community)
SRNWP – 08.10.2007
19
Leapfrog – t = 90s
Leapfrog – t = 75s
SRNWP – 08.10.2007
20
RR
(mm/h)
RK (p*-T) – t = 180s
RK (p*-T) – t = 150s
SRNWP – 08.10.2007
21
RR
(mm/h)
LF (semi-implizit) – t = 75s
SRNWP – 08.10.2007
LF (semi-implizit) – t = 90s
22
RR
(mm/h)
RK (p*-T*) – t = 180s
RK (p*-T*) – t = 150s
SRNWP – 08.10.2007
23
RR
(mm/h)
Advection of moisture quantities qx
• implementation of the Bott (1989)-scheme into the Courant-number
independent advection algorithm for moisture densities (Easter, 1993,
Skamarock, 2004, 2006)
• ‚classical‘ semi-Lagrange advection with 2nd order backtrajectory and
tri-cubic interpolation (using 64 points) (Staniforth, Coté, 1991)
SRNWP – 08.10.2007
24
Problems found with Bott (1989)-scheme in the meanwhile:
1.) Directional splitting of the scheme:
Parallel Marchuk-splitting of conservation equation for density can lead to a
complete evacuation of cells
Solution: Easter (1993), Skamarock (2004, 2006), mass-consistent splitting
2.) Strang-splitting ( 'x-y-z' and 'z-y-x' in 2
time steps) produces 2*dt oscillations
Solution: proper Strang-Splitting ('x-y-2z-y-x')
in every time step solves the problem, but
nearly doubles the computation time
3.) metric terms prevent the scheme to be properly mass conserving
<-- Schär–test case of an unconfined jet and ‚tracer=1‘ initialisation
(remark: exact mass conservation is already violated by the 'flux-shifting' to make the
Bott-scheme Courant-number independent)
SRNWP – 08.10.2007
25
COSMO-ITA 2.8 km: comparison
RK+Bott / RK+Semi-Lagrange
RK+SL for light precipitation:
TS is larger, whereas FBI is smaller
than that for RK+Bott.
Moreover, RK+SL has slightly less
domain-averaged precipitation and
larger maximum prec. values than RK.
L. Torrisi
SRNWP – 08.10.2007
26
Semi-Lagrangian advection in COSMO-model
‚classical‘ semi-Lagrange advection (Staniforth, Coté, 1991) with 2nd order
backtrajectory and tri-cubic interpolation (using 64 points)
SL is not positive definite  clipping necessary
'multiplicative filling' (Rood, 1987) combines clipping with global conservation
problem: global summation is not ‚reproducible‘ (dependent from domain
decomposition) -> solution: REAL -> INTEGER mapping
Moisture transport in COSMO model:
DWD:
COSMO-DE: Bott-scheme used
COSMO-EU: SL scheme planned operationally
MeteoCH: COSMO-S2 and COSMO-S7: SL scheme used pre-operationally
CNMCA:
COSMO-ITA 2.8: SL-scheme used pre-operationally
SRNWP – 08.10.2007
27
Deep / shallow atmosphere
Momentum equations for deep atmosphere (spherical coordinates):
shallow atmosphere approximation:
• r~a
• neglect terms in advection and Coriolis force
additionally:
• introduce a hydrostatic, steady base state
• transformation to terrain following coordinates
deep atmosphere terms are implemented in COSMO 3.21
• diploma thesis R. Petrik, Univ. Leipzig
• White, Bromley (1995), QJRMS
• Davies et al. (2005), QJRMS
SRNWP – 08.10.2007
28
Test case Weisman, Klemp (1982):
wmax
warm bubble in a base flow with vertical
velocity shear + Coriolis force
dx= 2 km
precipitation distribution
‚deep‘ (shaded), ‚shallow‘ (isolines)
SRNWP – 08.10.2007
RR
RRdeep- RRshallow (shaded)
29
Case study ‚12.08.2004‘
(Diploma thesis R. Petrik)
summary for precipitation forecast in ‚deep‘, convection resolving models:
• additional advection terms: not relevant
• additional Coriolis terms:
• have a certain influence, but don't seem to be important for COSMO-DE application
• could be important for simulations near the equator
SRNWP – 08.10.2007
30
Physics coupling scheme
original idea: 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
Work to do:
• what are the reasons for the failure of the WRF-PD-scheme in LM?
(turbulence scheme?)
• Test different sequences of dynamics and physics (especially physics after
dynamics)
 test tool (Bryan-Fritsch-case) is developed in PP ‚QPF‘, task 4.1
SRNWP – 08.10.2007
31
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
SRNWP – 08.10.2007
32
Plans (RK-core, short, medium range)
• 3D- (isotropic) divergence filtering in fast waves solver
• implicit advection of 3. order in the vertical
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?)
• Efficiency gains by using RK4?
• Development of a more conservative dynamics (rho’-Theta’-dynamics?)
• diabatic terms in the pressure equation (up to now neglected, e.g.
Dhudia, 1991)
• radiation upper boundary condition (non-local in time )
• continue diagnostics:
• convergence (mountain flows)
• conservation: mass, moisture variables, energy
SRNWP – 08.10.2007
33
Stability limit of the ‚effective Courant-number‘ for advection schemes
Ceff := C / s,
Euler
LC-RK2
LC-RK3
LC-RK4
LC-RK5
LC-RK6
LC-RK7
s= stage of RK-scheme
up1
cd2
up3
cd4
up5
cd6
1
0.5
0.419
0.348
0.322
0.296
0.282
0
0
0.577
0.707
0
0
0.252
0
0.437
0.542
0.436
0.391
0.385
0.369
0
0
0.421
0.515
0
0
0.184
0
0
0.478
0.433
0.329
0.311
0.323
0
0
0.364
0.446
0
0
0.159
Baldauf (2007), submitted to J. Comput. Phys.
SRNWP – 08.10.2007
34
Plans (long range)
Higher order discretization on unstructured grids using Discontinuous Galerkin
methods
Univ. Freiburg: Kröner, Dedner, NN., DWD: Baldauf
In the DFG priority program 'METSTROEM' a new dynamical core for the COSMOmodel will be developed. It will use Discontinuous Galerkin methods to achieve higher
order, conservative discretizations. Currently the building of an adequate library is
under development. The work with the COSMO-model will start probably at the end of
2009. This is therefore base research especially to clarify, if these methods can lead to
efficient solvers for NWP.
start: 2007, start of implementation into COSMO: 2009
SRNWP – 08.10.2007
35
SRNWP – 08.10.2007
36
Investigation of convergence
solution with a damping layer of 85 levels
and nRΔt=200.
SRNWP – 08.10.2007
Analytical solution (Klemp-Lilly (1978) JAS)
37
CONVERGENCE OF VERTICAL VELOCITY w
HYDROSTATIC FLOW
10-2
L1 = average of errors
10-3
DW
L = maximum error
10-4
10
-5
10
-6
10
Convergence slightly less
than 2. order.
(2. order at smaller scales?)
L1
L0
2nd order
-2
10
-1
10
0
10
1
DX
SRNWP – 08.10.2007
38
10
2
NON LINEAR HYDROSTATIC FLOW
NON LINEAR HYDROSTATIC FLOW
100
DW
10-1
10-2
10
-3
10
-4
L1
L0
2nd order
10
-2
10
-1
10
0
10
1
DX
Convergence of vertical velocity w
L1 = average of absolute errors
L = maximum error
Stable and stationary solution of
this non-linear case!
SRNWP – 08.10.2007
39
10
2
Operational timetable
of the
DWD model suite
GME, COSMO-EU, COSMO-DE
and WAVE
SRNWP – 08.10.2007
40
Equation system of LM/LMK in spherical coordinates
additionally:
• introduce a hydrostatic, steady base state
• Transformation to terrain-following coordinates
• shallow/deep atmosphere
SRNWP – 08.10.2007
41
(from spatial discretization of advection operator)
SRNWP – 08.10.2007
42
How to handle the fast processes with buoyancy?
with buoyancy (Cbuoy = adt = 0.15, standard atmosphere)
• different fast processes:
1. operatorsplitting (Marchuk-Splitting): ‘Sound -> Div. -> Buoyancy‘
2. partial adding of tendencies:
‘(Sound+Buoyancy) -> Div.')
3. adding of all fast tendencies:
‘Sound+Div.+Buoyancy‘
• different Crank-Nicholson-weights for buoyancy:
=0.6 / 0.7
•
•
•
•
SRNWP – 08.10.2007
RK3-scheme
slow process: upwind 5. order
aspect ratio: dx/dz=10
dT/dt=6
43
‘Sound -> Div. -> Buoyancy‘
‘(Sound+Buoyancy) -> Div.')
‘Sound+Div.+Buoyancy'
Cadv = u T / x
=0.6
=0.7
Csnd = cs t / x
curious result: operator splitting of the fast processes is not the best choice,
better: simple addition of tendencies.
SRNWP – 08.10.2007
44
amplification
factor
Task 3: Conservation
(Baldauf)
Tool for inspection of conservation properties will be developed.
balance equation for scalar :
temporal
change
flux
divergence
sources
/ sinks
integration area = arbitrarily
chosen cuboid (in the transformed
grid, i.e. terrain-following)
Status: available in LM 3.23:
• Subr. init_integral_3D: define cuboid (in the transformed grid!), prepare domain decomp.
• Function integral_3D_total: calc. volume integral
V ijk Vijk
• Subr. surface_integral_total: calc. surface integrals
V jijk * Aijk
• prelimineary idealised tests were carried out
• report finished; will be published in the next COSMO-Newsletter Nr. 7 (2007)
Task is finished
(Study of conservation properties will be continued in collaboration with MPI-Hamburg, see WG2 Task 2.10.1)
SRNWP – 08.10.2007
45
Task 3:
Weisman-Klemp (1982)-test case
without physical parameterisation
(only advection &
condensation/evaporation)
total moisture mass
M =  x dV
Semi-Lagrange-Adv. for qx
with multiplicative filling
x :=  (qv + qc )
(Mn-Mn-1) / t
Res.
violation in moisture conservation (?)
total surface flux
timestep
SRNWP – 08.10.2007
46
Task 3:
Weisman-Klemp (1982)-test case
with warmer bubble (10 K)
without physical parameterisation,
without Condensation/Evap.
total moisture mass
M =  x dV
Semi-Lagrange-Adv. for qx
with multiplicative filling
x :=  (qv + qc )
Residuum  0
 advection seems to be
‚conservative enough‘
(Mn-Mn-1) / t
possible reasons for conservation violation:
Res.
saturation adjustment conserves
specific mass (and specific energy)
but not mass (and energy) itself !
total surface flux
Baldauf (2007), COSMO-Newsletter Nr. 7
SRNWP – 08.10.2007
timestep
47
COSMO-ITA: RK+SL / RK+new Bott
SL
Bott
RK+new Bott has a larger FBI for all
precipitation thresholds than RK+SL
(= COSMO-ITA operational run).
Moreover, RK+new Bott has a
deterioration in MSLP bias and
RMSE after T+12h.
SRNWP – 08.10.2007
48
Verbesserte Vertikaladvektion für
dynamische 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
SRNWP – 08.10.2007
49