Status report of WG2 - Numerics and Dynamics COSMO General Meeting 05.-09.09.2011, Rome Michael...
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Transcript of Status report of WG2 - Numerics and Dynamics COSMO General Meeting 05.-09.09.2011, Rome Michael...
![Page 1: Status report of WG2 - Numerics and Dynamics COSMO General Meeting 05.-09.09.2011, Rome Michael Baldauf Deutscher Wetterdienst, Offenbach, Germany.](https://reader030.fdocuments.net/reader030/viewer/2022032414/56649eec5503460f94bfd742/html5/thumbnails/1.jpg)
Status report of
WG2 - Numerics and Dynamics
COSMO General Meeting05.-09.09.2011, Rome
Michael BaldaufDeutscher Wetterdienst, Offenbach, Germany
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Topics
• Avoid Theta-peaks• New fast waves solver• Bott-advection scheme with Strang-splitting near the surface layer• Changes of the small time steps in the RK sub stepping• Idealised test cases• Discontinuous Galerkin project (DFG) • Convergence studies (‚extramurale Forschung‘)
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The COSMO-model produces in connection with small-scale, thermically driven circulations strange effects like „Ө-peaks“ in Alpine valleys or grid point storms at mountains or at the coast.
U. Blahak (DWD)Avoid Ө-Peaks in COSMO
© Routine-Wachhund (B. Ritter)
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Explanation by an idealised study
2D-simulation (dry), at t=0: V = 0, N = 0.01 s-1, overheated surface
~ 15 m/s !
U is staggered. grid boxes are shifted by 1/2 dx
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Adv X , T ≈ − 12
{ U i−1 /2 Δ upwind (T i , sign(U i−1 /2)) +
U i+ 1/2 Δ upwind (T i , sign(U i+ 1 /2)) }
Although correct for the center of the column, it is notrepresentative for the grid box averaged horizontal advection of T (and p') !... whereas vertical adv. and divergence terms are representatively estimated!
too few lateral inflow of cool air into the column
artificial heat source !!!
Adv X , T = −UdTdx
≈ − (U i−1 /2 + U i+ 1 /2
2 )⏟Δ upwind (T i , sign(U ))
Ad-hoc correction:
~ 0 !!U
Explanation by an idealised study
2
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~ 3 m/s !
U ist gestaggert. Darstellung der Gittersäulen um 1/2 Box verschoben
Explanation by an idealised study
2D-simulation (dry), at t=0: V = 0, N = 0.01 s-1, overheated surface
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Experiment: 1.6. - ~20.7.2011
© B. Ritter
Reduction of „Theta-peaks“ in July 2011 (COSMO-DE)
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• the proposed method cures the most of the Theta-peaks in the interior of the domain
• influence on the verification scores is nearly neutral (slightly positive for pressure)
• additionally in COSMO-EU runs the maximum w is reduced
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Main changes towards the current solver:
1. improvement of the vertical discretization: use of weighted averaging operators for all vertical operations
2. divergence in strong conservation form
3. optional complete 3D divergence damping
additionally some 'technical' improvements;hopefuly a certain increase in code readability
new version fast_waves_sc.f90 (based on COSMO 4.18)
Development of a new fast waves solverfor the Runge-Kutta scheme
M. Baldauf (DWD)
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Results of idealised test cases see: COSMO-user Seminar, March 2011
SRNWP-workshop Bad Orb, May 2011
• Sound wave expansion• Linear Gravity wave in a channel (Skamarock, Klemp, 1994)• Linear flow over mountains (compare with analytic solution)• Non-linear flow over a mountain• mountain in a steady atmosphere• moist warm bubble test (Weisman, Klemp, 1982)• dry cold bubble (Straka et al.,1993)
All these idealised tests are simulated with either similar accuracyor slightly better.
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Quasi-3D - divergence damping in terrain following coordinates
Stability criterium:
in particular near the bottom (Δx, Δy >> Δz) a strong reduction of div is necessary!
This violates the requirement of not too small div in the Runge-Kutta-time splitting scheme (xkd~0.1 (Wicker, Skamarock, 2002),in Baldauf (2010) MWR even xkd~0.3 is recommended).
Otherwise divergence damping is calculated as an additive tendency (no operator splitting) a certain 'weakening' of the above stability criterium is possible
possible consequence on the COSMO-1 development at MeteoCH
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'COSMO-EU, 05.01.2011, 0 UTC', PMSL after 78 h (Exp. 8230)
during the simulation a negative pressure bias of about -0.5 hPa/3d develops(remark: the operational COSMO-EU was nearly bias free in Jan. 2011)
Exp. 8230 RoutineDiff.
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'dynamical bottom BC for p' also improves pressure bias in the new fast waves - solver
FW_new with ldyn_bbc=T
FW_newwith ldyn_bbc=F
pressure bias in COSMO-EU (12.01.2011, 0 UTC run, stand alone)
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very strong (unrealistic!?) inversion in an Alpine valley
COSMO-DE, 12.01.2011, 0 UTC run, ldyn_bbc=.TRUE.
current FW solver new FW solver
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COSMO-DE, 12.01.2011, 0 UTC run, ldyn_bbc=.FALSE.
current FW solver new FW solver
more realistic temperature fields without dynamic bottom BC for p
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Which benefits are currently visible ?
The original intention to develop the new fast wave solverwas to produce more consistent dynamic fields.Indeed in some situations the stability seems to be slightly higher, examples:
• a model crash of COSMO-2 at 16.06.2011, 0 UTC runs stable with the new FW solver
• a model crash of COSMO-DE at 12. July 2011, 6 UTC run couldbe repaired by the use of Bott2_Strang, but also alternatively by theuse of the new FW solver
• a simulation with high resolution of 0.01° only runs stable with thenew FW solver
• the new solver also runs without crash during 01.-31. Jan. 2011in both a COSMO-EU and a COSMO-DE setup
But: of course the fundamental difficulty of split explicit schemes with steep orography remain
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from Axel Seifert (DWD)
simulated radar reflectivityCOSMO-run with a resolutionof 0.01° (~ 1.1km)1700 * 1700 grid points
model crash after 10 time steps with the current fast waves solver
stable simulation with the new FW
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Summary New fast waves solver
• improved vertical discretizations• strong conservation form of divergence• Idealised test cases (stationary/unstationary, linear/nonlinear,
with/without orography) are simulated with either similar accuracyor slightly better
• runs stable in all inspected cases (COSMO-EU, COSMO-DE duringthe Jan. 2011, and in selected cases)
• for both FW solvers holds: dynamical bottom boundary condition necessary to reduce pressure bias in COSMO-EU, but can have detrimental influence on COSMO-DE (?)
• satisfying optimization for NEC SX9 achieved:needs ~30% more than current FW total model run time ~5% greater(but not yet optimized for cahce based machines)
• extensive verification needed• further efficiency optimization for Cache based machines
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M. Baldauf (DWD)
Changes of the small time steps in the RK sub stepping
This can increase the calculation time a bit (e.g. about 2% in COSMO-DE), but
• it is the preferred method known from literature (Wicker, Skamarock, 2002, MWR)
• seems to be more stable from stability analysis (Baldauf, 2010, MWR) • it could repair a model abort of COSMO-2 at 16.06.2011, 0 UTC run• it could improve several Taifun events in climatological runs
(reported by A. Will)
This is now the standard setting in COSMO 4.20 (internal switch icalc_version=0)
use the same dtsmall in each of the three RK substeps use 2 / 3 / 6 small steps in each RK substep (or 4/6/12, ...)
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Stability analysis of the Runge-Kutta time splitting scheme;(2D Euler equations with sound, buoyancy and horizontal advection)Maximum amplification factor
(stability theory is described in Baldauf (2010) MWR)
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Bott-advection scheme with Strang-splitting
1. Old scheme (J. Förstner): odd timestep: x-y-z,
even timestep: z-y-x,sometimes unstable
2. Strang-splitting (G. deMorsier): every timestep: ½ z – ½ y – x – ½ y - ½ z
more stable, but also more expensive
3. New idea (G. Zängl):observation: splitting problems always occur near the bottom boundary
use x - z - y in levels k=1 ... ke - 5 y - z - x
use ½ x – ½ y – z – ½ y - ½ x in levels k=ke - 5 ... Kegood compromise between stability and computer time (?)
G. Zängl, M. Baldauf (DWD), G. deMorsier (MeteoCH)
y_scalar_advect="Bott2_Strang_new"
y_scalar_advect="Bott2"
y_scalar_advect="Bott2_Strang"
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Remarks:• 'Bott2_Strang' needs about 60% more computing time than 'Bott2'
the whole COSMO needs about 10% more time• 'Bott2_Strang_new' needs about 8% more computing time than 'Bott2'
the whole COSMO needs only about 1% more time• 'Bott2_Strang_new' needs z-advection 'in the middle'
advantage: x- and y- are treated symmetricallydisadvantage: this doubles the time step for the z-advection possible reduction in stability (!)(this can be cured by the use of ' ½ z - ½ z ' instead of ' z ', but this needs about 30% more computing than 'Bott2')
Other:some inconsistencies/bugs could be found and will be fixed:- inconsistent time level for w- calculation of d/dt should be the same as in other parts of the model
option 'Bott2_Strang_new' could be available in 4.21,but this version has to be extensively tested first!
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New version of src_artifdata.f90
U. Blahak (DWD), with contributions from O. Fuhrer (MeteoCH), M. Baldauf (DWD)
There is only one src_artifdata.f90 to generate initial and boundary data for all idealized test cases
very extensive list of NAMELIST-Parameters to configure such idealised runs: define
• orography, • stratifications, • wind fields, • warm/cold bubbles, • ... • in a model consistent way (numerical hydrostatic balance, ...)
Periodic boundary conditionsin x- or y- direction now are properly working lperi_x, lperi_y
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A new dynamical core based on Discontinuous Galerkin methodsProject ‘Adaptive numerics for multi-scale flow’, DFG priority program ‘Metström’
PhD student (financed by DFG (german research community) for 4 years)
• DG-RK method in a toy model implemented• currently: implementation of DG solver in the
COSMO model (explicit (RK integration), flat terrain)
shallow water equations:
D. Schuster, M. Baldauf (DWD)
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from: Brdar, Baldauf, Klöfkorn, Dedner:Comparison of dynamical cores for NWP models,submitted to Theor. Comp. Fluid Dynamics
A new dynamical core based on Discontinuous Galerkin methods
Comparison between COSMO and the DUNE library:test case linear gravity wave (Skamarock, Klemp (1994) MWR)
higher order DG methods have the potential to be more efficient if the accuracy requirements are high
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A. Will, J. Ogaja (Univ. Cottbus)'Extramurale Forschung' of DWD
Convergence studies of COSMO
L2-error of w for linear flow over a mountain
results:• the convergence rate is
less than 2nd order• it even decreases if the
order of the advection scheme is increased!
consistent discretization of all dynamic processes
is recommended
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Updated Version of
COSMO scientific documentation, part I: Dynamics and Numerics
available.
New chapter 8 describing the Runge-Kutta scheme:
8.1 Slow processes and the Runge-Kutta time integration scheme8.12 horizontal advection schemes8.13 Coriolis terms
8.2 Fast processes8.2.1 fast waves solver (to be extended)
Boundary treatment8.3 Tracer advection
8.3.1 Semi-Lagrangian scheme (to be extended)8.3.2 Bott advection and related schemes
8.4 Damping mechanisms
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D = div v
‚Fast waves‘ processes (p'T'-dynamics):
fu, fv, ... denote advection, Coriolis force and all physical parameterizations
sound buoyancy artificialdivergence damping
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1. Improvement of the vertical discretization
Averages from half levels to main level:
Averages from main levels to half level with appropriate weightings (!):
centered differences (2nd order if used for half levels to main level)
G. Zängl could show the advantages of weighted averages in the explicit parts of the fast waves solver.New: application to all vertical operations (also the implicit ones)
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2. 'Strong conservation form' of the divergence operator
Divergence operator used up to now:
Strong conservation form:
Discretization of metric terms
more compact expressions in the strong conservation form
Doms, Schättler (2002) COSMO Sci. Doc. (I), Prusa, Smolarkiewicz (2003) JCP
~ d/dt proper BC
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Efficiency:
on NEC - SX9: new fast_waves_sc: reaches ~20 GFlops and needs about 30% more computation time than the current fast waves solver (~18 GFlops)
a COSMO-EU run needs about 5% more time
but it is not yet optimized for Intel processors (cache based): it takes about 80% more computing time (reason?) higher computation time can be expected due to- more vertical weightings - exchange of p' and div v
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Outlook• extensive verification• efficiency:
• on NEC-SX9: further increase probably only with help of NEC specialists• on Intel/cache based: better inspection tools necessary (valgrind, ...)
• add currently available features:lateral radiation BC; p'-dynamics solver; lower BC for w via 'RK advection of height', ...
• application to the COSMO-DE L65 setup • Closer examination of the influence of (an-)isotropic divergence damping