M. Baldauf Deutscher Wetterdienst, Offenbach
description
Transcript of M. Baldauf Deutscher Wetterdienst, Offenbach
![Page 1: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/1.jpg)
M. BaldaufDeutscher Wetterdienst, Offenbach
COSMO-General Meeting05.09.2011, Rome
Development of a new fast waves solverfor the Runge-Kutta scheme
![Page 2: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/2.jpg)
Motivation
• Internal DWD project 'COSMO-DE L65' (increase # of vertical levels 50 65):(goal: better representation of convection and its initiation)But several numerical problems occur (lower BC for w, vertical advection scheme, tracer advection scheme, ...)Speculation: discretizations of fast processes not yet entirely consistent
• DWD-'Extramurale Forschung' project 'Numerische Raumdiskretisierungsverfahren hoher Ordnung für das COSMO Modell' (A. Will/J. Ogaja)Prerequisite: use of only centered finite difference and centered averaging operators
![Page 3: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/3.jpg)
D = div v
‚Fast waves‘ processes (p'T'-dynamics):
fu, fv, ... denote advection, Coriolis force and all physical parameterizations
sound buoyancy artificialdivergence damping
integration procedure: horizontally forward-backward, vertically implicit
![Page 4: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/4.jpg)
Integration procedure:
horizontally: forward-backward (Mesinger (1977) Contr. Phys. Atm.)vertically: implicit(Klemp, Wilhelmson (1978) MWR,Wicker, Skamarock (1998, 2002) MWR)
With the possibility of a 3D divergence damping, stability of the whole RK3-scheme (p'T'-dynamics) was shown in Baldauf (2010) MWR
![Page 5: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/5.jpg)
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)
![Page 6: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/6.jpg)
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
![Page 7: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/7.jpg)
3. Isotropic divergence damping
Gassmann, Herzog (2007) MWR recommend the use of the complete (=isotropic) 3D divergence damping more realistic dispersion relation for sound and gravity waves
The following tests comparethe current FW solver (fast_waves_rk.f90, version COSMO 4.18)with the new FW solver (fast_waves_sc.f90)
![Page 8: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/8.jpg)
3.a Linear flow over mountains
FW new
test case definition in: Schär et al. (2002) MWRLinear analytic solution (black contours) : Baldauf (2008) COSMO-Newsl.
current FW
U0=10 m/s, N=0.01 1/s, hmax=25 m
![Page 9: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/9.jpg)
3.b Nonlinear flow over mountains
FW new
U0=10 m/s, N=0.01 1/s, hmax=750 m
current FW
![Page 10: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/10.jpg)
3.b Nonlinear flow over mountains
FW new
U0=10 m/s, N=0.01 1/s, hmax=900 m
model abort after 6 h
current FW
![Page 11: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/11.jpg)
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.
![Page 12: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/12.jpg)
Real simulations ...
... at the beginning caused several problems,
mostly in connection with the divergence damping near the bottom.
![Page 13: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/13.jpg)
Quasi-3D - divergence damping in terrain following coordinates
Stability criterium:
in particular near the bottom (x, y >> z) 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
![Page 14: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/14.jpg)
… additionally necessary for stability:
• don't apply divergence damping in the first small time step!(J. Dudhia, 1993 (?))
• boundary treatment in the term (d/dt) / of the divergence in ‚strong conservation‘ form:the obvious d/dt|ke+1=0 leads to a divergence which is (near the ground) about a factor 10 larger than a numerical approximation of this term.
![Page 15: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/15.jpg)
Boundary condition for the Euler equations
= free slip condition at the bottom (and at the top)
1.) Extrapolation of u, v to the ground (ke+1/2):this improved the pressure bias in COSMO-EU Options now: extrapolation 0th order (= 'true' free slip BC),1st or 2nd order.
2.) Discretisation of dh/dx, dh/dy:up to now: analogous to the order of the advection operator (upwind 5th order)now: centred diff. 4th order necessary
ke
ke+1/2
ke-1/2
w
w
u
u
![Page 16: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/16.jpg)
'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.
![Page 17: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/17.jpg)
pressure bias in COSMO-EU (12.01.2011, 0 UTC run, stand alone)
FW_current
FW_new
![Page 18: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/18.jpg)
BC for w: extrapolation of u, v to the lower half level (=surface) does not have any influence to the mean surface pressure
FW_current
FW_new0., 1., 2. orderExtrapolation
pressure bias in COSMO-EU (12.01.2011, 0 UTC run, stand alone)
![Page 19: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/19.jpg)
'dynamical bottom BC for p' helps in producing the correct surface pressure (at least the mean value) in the current FW solver
FW_current
FW_new
FW_currentldyn_bbc=F
pressure bias in COSMO-EU (12.01.2011, 0 UTC run, stand alone)
![Page 20: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/20.jpg)
Boundary treatment of p'/ (or of p'/z) at the lower boundary:
1.) one sided finite difference (G. Zängl):
p'/ = 0 p'(ke) + 1 p'(ke-1) + 2 p'(ke-2)
2.) ‚dynamical bottom BC‘ (A. Gassmann, 2004, COSMO-Newsl.)From
and
derive a condition for p'/ .
![Page 21: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/21.jpg)
'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)
![Page 22: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/22.jpg)
influence of the FW-solver to 'indirect' variables like precipitation is quite smallexample: COSMO-DE, 12.01.11, 0 UTC
FW new FW currentDiff.
![Page 23: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/23.jpg)
Diff.
influence of the FW-solver to 'indirect' variables like precipitation is quite smallexample: COSMO-DE, 28.04.11, 0 UTC
FW new FW current
![Page 24: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/24.jpg)
COSMO-DE, 12.01.2011, 0 UTC, pmsl
![Page 25: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/25.jpg)
very strong inversion in an Alpine valley
COSMO-DE, 12.01.2011, 0 UTC run, ldyn_bbc=.TRUE.
current FW solver new FW solver
![Page 26: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/26.jpg)
COSMO-DE, 12.01.2011, 0 UTC run, ldyn_bbc=.FALSE.
current FW solver new FW solver
![Page 27: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/27.jpg)
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
![Page 28: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/28.jpg)
crash with the operational COSMO-DE at 12. July 2011, 6 UTC
unrealistic high value of qr in one grid box;strongly deformed wind field
model crash with thecurrent FW
stable simulation with the new FW
![Page 29: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/29.jpg)
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
![Page 30: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/30.jpg)
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
![Page 31: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/31.jpg)
Summary
New fast waves solver with• improved vertical discretizations• strong conservation form of divergence• (optional: 3D isotropic divergence damping)
• 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 whole 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
![Page 32: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/32.jpg)
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
![Page 33: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/33.jpg)
Ende
![Page 34: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/34.jpg)
Neu gegenüber dem bisherigen fast waves -Löser:
1. Verbesserung in der vertikalen Diskretisierung: abstandsgewichtete Mittelungsoperatoren
2. Divergenz in strong conservation form
3. optional volle 3D Divergenzdämpfung (d.h. in allen 3 Bewegungsgleichungen)
zusätzlich einige 'technische' Verbesserungen;Erhöhung der Lesbarkeit
neue Version fast_waves_sc.f90 (basierend auf COSMO 4.17)
![Page 35: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/35.jpg)
Eine bessere analytische Lösung für die lineare Schwerewelle im Kanal
Analytische Lösung von Skamarock, Klemp (1994) MWRist Boussinesq-approximiertkeine perfekte Übereinstimmungzwischen analytischer und simulierter Lösung möglich.
Eine bessere analytische Lösung als Referenz wäre auch bei Modellvergleichen wünschenswert und wird z.Z. im Rahmen von
durchgeführt
![Page 36: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/36.jpg)
Eine bessere analytische Lösung für die lineare Schwerewelle im Kanal
Eine analytische Lösung für kompressible, nicht-hydrostatische Euler-Gleichungen kann für eine isotherme (T0=const) Hintergrund-Atmosphäre gefunden werden!
Anfangszustand: ruhend (bzw. u=const.) , p‘=0, vorgegebene T‘- (oder ‘-) Verteilung z.B. der Form:
![Page 37: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/37.jpg)
Die zeitliche Entwicklung der Fouriertransformierten Felder lautet
(analog für u, T‘, …) und sind gegebene Funktionen von kx, kz, T0, g, cp, cv
![Page 38: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/38.jpg)
Die zeitliche Entwicklung der Fouriertransformierten Felder lautet
(analog für u, T‘, …) und sind gegebene Funktionen von kx, kz, T0, g, cp, cv
![Page 39: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/39.jpg)
1. Sound wave expansion test
current FW FW new
Isothermal atmosphere (T=250 K)
![Page 40: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/40.jpg)
1. Sound wave expansion test
FW new
?
current FW
![Page 41: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/41.jpg)
1. Sound wave expansion test
FW new
?
current FW
![Page 42: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/42.jpg)
1. Sound wave expansion test
FW new
?
current FW
Artificial stationary solution (?)
![Page 43: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/43.jpg)
2. Linear gravity wave
test definition and (anelastic approx.) analytic solution: Skamarock, Klemp (1994) MWR
FW newcurrent FW
![Page 44: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/44.jpg)
2. Linear gravity wave
test definition and (anelastic approx.) analytic solution: Skamarock, Klemp (1994) MWR
FW newcurrent FW
![Page 45: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/45.jpg)
2. Linear gravity wave
test definition and (anelastic approx.) analytic solution: Skamarock, Klemp (1994) MWR
FW newcurrent FW
![Page 46: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/46.jpg)
2. Linear gravity wave
test definition and (anelastic approx.) analytic solution: Skamarock, Klemp (1994) MWR
FW newcurrent FW
![Page 47: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/47.jpg)
4. Mountain in a steady atmosphere with vertical grid stretching
FW newcurrent FW
![Page 48: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/48.jpg)
4. Mountain in a steady atmosphere(b) with vertical grid stretching
FW newcurrent FW
![Page 49: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/49.jpg)
4. Mountain in a steady atmosphere(b) with vertical grid stretching
FW newcurrent FW
![Page 50: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/50.jpg)
5. Testcase Weisman, Klemp (1982)
FW new
Umax=30 m/s, qv,0,max=13 g/kg
current FW
![Page 51: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/51.jpg)
FW new
5. Testcase Weisman, Klemp (1982)
current FW
Umax=30 m/s, qv,0,max=13 g/kg
![Page 52: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/52.jpg)
FW new
5. Testcase Weisman, Klemp (1982)
current FW
Umax=30 m/s, qv,0,max=13 g/kg
![Page 53: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/53.jpg)
FW new
5. Testcase Weisman, Klemp (1982)
current FW
Umax=30 m/s, qv,0,max=13 g/kg
![Page 54: M. Baldauf Deutscher Wetterdienst, Offenbach](https://reader036.fdocuments.in/reader036/viewer/2022081507/56815af5550346895dc8afe6/html5/thumbnails/54.jpg)
5. Testcase by Weisman, Klemp (1982) MWR
Max vertical velocity w:
New FW:small deviationsbetween the anisotropic (quasi-2D)and theisotropic (3D)divergence damping
Current FW
60 min.30 min.