Titelfoto auf dem Titelmaster einfügen Numeric developments in COSMO SRNWP / EWGLAM-Meeting Dubrovnik, 08.-12.10.2007 Michael Baldauf 1 , Jochen Förstner 1 , Uli Schättler 1 Pier Luigi Vitagliano 2 , Gabriella Ceci 2 , Lucio Torrisi 3 , Ronny Petrik 4 1 Deutscher Wetterdienst, 2 CIRA-Institute, 3 USMA (Rome), 4 Max-Plank-Institut Hamburg
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Titelfoto auf dem Titelmaster einfügen Numeric developments in COSMO SRNWP / EWGLAM-Meeting Dubrovnik, 08.-12.10.2007 Michael Baldauf 1, Jochen Förstner.
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Titelfoto auf dem Titelmaster einfügen
Numeric developments in COSMO
SRNWP / EWGLAM-MeetingDubrovnik, 08.-12.10.2007
Michael Baldauf1, Jochen Förstner1, Uli Schättler1
Pier Luigi Vitagliano2, Gabriella Ceci2 , Lucio Torrisi3, Ronny Petrik4
non-hydrostaticresolved convectionx = 2.8 km421 * 461 * 50 GPt = 25 sec., T = 21 h
SRNWP – 08.10.2007 4
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 5
COSMO-Priority Project ‚Runge-Kutta‘:
1. New Developments1. NEW: Divergence damping in a 3D-(isotropic) version2. NEW: DFI for RK3. Advection of moisture quantities in conservation form4. Higher order discretization in the vertical5. Physics coupling scheme 6. Testing of alternative fast wave scheme7. Development of a more conservative dynamics (planned)8. Development of an efficient semi-implicit solver in combination with RK
time integration scheme (planned)2. Developing diagnostic tools
1. Conservation inspection tool (finished)2. Investigation of convergence
3. Known problems1. Looking at pressure bias2. Deep valleys 3. (Different filter options for orography) (finished)
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 8
COSMO-DE (2.8 km), 18.01.2007
SRNWP – 08.10.2007 9
COSMO-DE (2.8 km), 18.01.2007
SRNWP – 08.10.2007 10
Von-Neumann stability analysis of a 2-dim., linearised Advection-Sound-Buoyancy-system
SRNWP – 08.10.2007 11
SRNWP – 08.10.2007 12
Crank-Nicholson-parameter for buoyancy terms in the p‘T‘-dynamics=0.5 (‚pure‘ Crank-Nic.) =0.6 =0.7
=0.8 =0.9 =1.0 (pure implicit)
choose =0.7 as the best value Csnd = cs t / x
Ca
dv
= u
T
/
x
amplification
factor
• RK3-scheme(WS2002)
• upwind 5th order• Sound: =0.6 x/ z=10 T/ t=6
SRNWP – 08.10.2007 13
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=10T / t=6
--> Divergence damping is needed in this dynamical core!
SRNWP – 08.10.2007 14
Cdiv=0.025
Cdiv=0.05 Cdiv=0.1 Cdiv=0.15
Influence of Cdiv
Cdiv = xkd * (cs * t/ x)2
~0.35
stability limit by long waves (k0)
Cdiv=0
Csnd = cs t / x
Ca
dv
= u
T
/
x
amplification
factor
Cdiv = div t/x2
in COSMO-model:
SRNWP – 08.10.2007 15
Advantages of p'T'-dynamics over p'T-dynamics
1. Improved representation of T-advection in terrain-following coordinates
2. Better representation of buoyancy term in fast waves solver
Terms (a) and (b) cancel analytically, but not numerically
using T:
Buoyancy term alone generates an oscillation equation:
= g/cs
= a = acoustic cut-off frequencyusing T':
SRNWP – 08.10.2007 16
Idealised test case:Steady atmosphere with mountain
base state: T0, p0
deviations from base state: T', p' 0 introduces spurious circulations!
point 1.): 'improved T-advection' ...
SRNWP – 08.10.2007 17
LeapfrogRunge-Kutta
old p*-T-dynamics
contours: vertical velocity w isolines: potential temperature
SRNWP – 08.10.2007 18
contours: vertical velocity w isolines: potential temperature
Runge-Kuttaold p*-T-Dynamik
Runge-Kuttanew p*-T*-Dynamik
SRNWP – 08.10.2007 19
Climate simulations
start: 1. july 1979 + 324 h (~2 weeks)
results: accumulated precipitation (TOT_PREC) and PMSL
(simulations: U. Schättler, in cooperation with the CLM-community)
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
SRNWP – 08.10.2007 20
Leapfrog – t = 75s Leapfrog – t = 90s RR(mm/h)
SRNWP – 08.10.2007 21
RK (p*-T) – t = 150s RK (p*-T) – t = 180s RR(mm/h)
SRNWP – 08.10.2007 22
LF (semi-implizit) – t = 75s LF (semi-implizit) – t = 90s RR(mm/h)
SRNWP – 08.10.2007 23
RK (p*-T*) – t = 150s RK (p*-T*) – t = 180s RR(mm/h)
SRNWP – 08.10.2007 24
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 25
Problems found with Bott (1989)-scheme in the meanwhile:
2.) Strang-splitting ( 'x-y-z' and 'z-y-x' in 2 time steps) produces 2*dt oscillationsSolution: proper Strang-Splitting ('x-y-2z-y-x') in every time step solves the problem, but nearly doubles the computation time
1.) Directional splitting of the scheme:Parallel Marchuk-splitting of conservation equation for density can lead to a complete evacuation of cellsSolution: Easter (1993), Skamarock (2004, 2006), mass-consistent splitting
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)
• 3D- (isotropic) divergence filtering in fast waves solver
• implicit advection of 3. order in the verticalbut: 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 )
Stability limit of the ‚effective Courant-number‘ for advection schemes
Ceff := C / s, s= stage of RK-scheme
Baldauf (2007), submitted to J. Comput. Phys.
SRNWP – 08.10.2007 35
Higher order discretization on unstructured grids using Discontinuous Galerkin methodsUniv. Freiburg: Kröner, Dedner, NN., DWD: Baldauf
In the DFG priority program 'METSTROEM' a new dynamical core for the COSMO-model 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
Plans (long range)
SRNWP – 08.10.2007 36
SRNWP – 08.10.2007 37
Analytical solution (Klemp-Lilly (1978) JAS)
Investigation of convergence
solution with a damping layer of 85 levels and nRΔt=200.
SRNWP – 08.10.2007 38
CONVERGENCE OF VERTICAL VELOCITY w
DX
DW
10-2 10-1 100 101 10210-6
10-5
10-4
10-3
10-2
L1L02nd order
HYDROSTATIC FLOW
L1 = average of errors
L = maximum error
Convergence slightly less than 2. order.(2. order at smaller scales?)
SRNWP – 08.10.2007 39
NON LINEAR HYDROSTATIC FLOW
Stable and stationary solution of this non-linear case!
DX
DW
10-2 10-1 100 101 10210-4
10-3
10-2
10-1
100
L1L02nd order
NON LINEAR HYDROSTATIC FLOW
Convergence of vertical velocity wL1 = average of absolute errors
L = maximum error
SRNWP – 08.10.2007 40
Operational timetableof the
DWD model suiteGME, COSMO-EU, COSMO-DE
and WAVE
SRNWP – 08.10.2007 41
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 42
(from spatial discretization of advection operator)
SRNWP – 08.10.2007 43
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