m. baldauf deutscher wetterdienst, offenbach
DESCRIPTION
Development of a new fast waves solver for the Runge-Kutta scheme. M. Baldauf Deutscher Wetterdienst, Offenbach. COSMO-General Meeting 05.09.2011, Rome. Motivation. - PowerPoint PPT PresentationTRANSCRIPT
M. BaldaufDeutscher Wetterdienst, Offenbach
COSMO-General Meeting05.09.2011, Rome
Development of a new fast waves solverfor the Runge-Kutta scheme
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
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
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
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)
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
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)
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
3.b Nonlinear flow over mountains
FW new
U0=10 m/s, N=0.01 1/s, hmax=750 m
current FW
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
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.
Real simulations ...
... at the beginning caused several problems,
mostly in connection with the divergence damping near the bottom.
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
… 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.
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
'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.
pressure bias in COSMO-EU (12.01.2011, 0 UTC run, stand alone)
FW_current
FW_new
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)
'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)
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'/ .
'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)
influence of the FW-solver to 'indirect' variables like precipitation is quite smallexample: COSMO-DE, 12.01.11, 0 UTC
FW new FW currentDiff.
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
COSMO-DE, 12.01.2011, 0 UTC, pmsl
very strong inversion in an Alpine valley
COSMO-DE, 12.01.2011, 0 UTC run, ldyn_bbc=.TRUE.
current FW solver new FW solver
COSMO-DE, 12.01.2011, 0 UTC run, ldyn_bbc=.FALSE.
current FW solver new FW solver
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
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
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
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
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
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
Ende
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)
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
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:
Die zeitliche Entwicklung der Fouriertransformierten Felder lautet
(analog für u, T‘, …) und sind gegebene Funktionen von kx, kz, T0, g, cp, cv
Die zeitliche Entwicklung der Fouriertransformierten Felder lautet
(analog für u, T‘, …) und sind gegebene Funktionen von kx, kz, T0, g, cp, cv
1. Sound wave expansion test
current FW FW new
Isothermal atmosphere (T=250 K)
1. Sound wave expansion test
FW new
?
current FW
1. Sound wave expansion test
FW new
?
current FW
1. Sound wave expansion test
FW new
?
current FW
Artificial stationary solution (?)
2. Linear gravity wave
test definition and (anelastic approx.) analytic solution: Skamarock, Klemp (1994) MWR
FW newcurrent FW
2. Linear gravity wave
test definition and (anelastic approx.) analytic solution: Skamarock, Klemp (1994) MWR
FW newcurrent FW
2. Linear gravity wave
test definition and (anelastic approx.) analytic solution: Skamarock, Klemp (1994) MWR
FW newcurrent FW
2. Linear gravity wave
test definition and (anelastic approx.) analytic solution: Skamarock, Klemp (1994) MWR
FW newcurrent FW
4. Mountain in a steady atmosphere with vertical grid stretching
FW newcurrent FW
4. Mountain in a steady atmosphere(b) with vertical grid stretching
FW newcurrent FW
4. Mountain in a steady atmosphere(b) with vertical grid stretching
FW newcurrent FW
5. Testcase Weisman, Klemp (1982)
FW new
Umax=30 m/s, qv,0,max=13 g/kg
current FW
FW new
5. Testcase Weisman, Klemp (1982)
current FW
Umax=30 m/s, qv,0,max=13 g/kg
FW new
5. Testcase Weisman, Klemp (1982)
current FW
Umax=30 m/s, qv,0,max=13 g/kg
FW new
5. Testcase Weisman, Klemp (1982)
current FW
Umax=30 m/s, qv,0,max=13 g/kg
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.