Dias nummer 1

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A cell-integrated semi-Lagrangian semi-implicit
integration scheme Basic hydrostatic
formulations and preliminary tests in the HIRLAM
system.
Eigil Kaas, University of Copenhagen (from 1st
April). Peter Hjort Lauritzen, NCAR, Boulder
CO. Karina Lindberg and Bennert Machenhauer, DMI
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LAURITZEN, P.H., KAAS, E., MACHENHAUER, B.
(2006). A mass-conservative semi-implicit,
semi-Lagrangian limited-area shallow water model
on the sphere. Mon. Wea. Rev., 134 (4), 11961212.
LAURITZEN, P.H., LINDBERG, K., KAAS, K.,
MACHENHAUER, B. (2006b). A locally
massconservative Semi-Lagrangian, Semi-implicit,
Limited-Area Gridpoint Model of the Primitive
Equations. Part I Forecast Model. Mon. Wea.
Rev., in prep.
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To show up in Handbook on Numerical Analysis.
Elsevier
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• Motivations for development of the new scheme
• Inherent local mass conservation
• Avoid CFL criterion
• High order of accuracy and shape conservation
• More consistent estimate of the vertical velocity
  and the energy conversion term
• Applications
• Improved synoptic scale weather forecasting
• Consistent/improved transport of multiple tracers
  incl. cloud constituents i.e. avoid the
  wind-mass inconsistency in off-line tracer
  models.
• General applications in fully integrated
  meteorological / chemistry models

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Flux form formulation of the continuity equation
with
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Time and space discretisation
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DCISL (Departure Cell-Integrated Semi-Lagrangian
scheme)
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• Different variants of CISL or finite volume
  schemes
• Traditional operator splitted flux based schemes
  (e.g. Bott (1989, 1992))
• Conservative operator splitted schemes also
  conserving a constant in a non-divergent flow
  (e.g. Lin and Rood (1996), Lin (2004), Leonard
  (1996), Adcroft et al. (2005))
• Non-split flux based schemes (e.g. Holm (1995))
• Lagrangian type (DCISL) schemes (Rancic, (1990),
  Machenhauer and Olk (1997), Nair and Machenhauer
  (2002), Lauritzen et al. (2006), Zerroukat et al.
  (2002).
• The CFL problem or not !

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DCISL (Nair and Machenhauer (2002)
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Equations of motion. Primitive equations with
standard ?(p,ps) vertical coordinate
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Continuity equation, vertical winds, trajectories
and vertical remappings
Horizontal accelerations taken into account in
estimation of horizontal trajectories
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Staggered hybryd sigma p vertical coordinate
(SIMMONS and BURRIDGE 1981)
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Semi-implicit formulation
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The semi-implicit formulation of DCISL (1)
CISL explicit forecast
Ideal semi-implicit CISL forecast
Elliptic equation too complicated !
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The semi-implicit formulation of DCISL (2)
The most direct approach
Inconsistent !
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Various options in the new dynamical core
Consistent omega in thermodynamic equation
(COP) Accuracy of horizontal trajectories and
extrapolation filters Application of filters on
upstream representation (monotonicity, positive
definiteness). Cascade interpolation as e.g. in
Nair et al. (2002) (CASC) Modification of
vertical diffusion (VDIFF). General tuning of
physics.
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Case study 18 hour forecast valid on 3th
December 1999 (extratropical storm over Denmark)
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Observed
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HIRLAM
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CISL
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CISL - CO
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CISL - COP - CASC
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CISL - COP - CASC, 0.1vdiff on all variables
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CISL - COP - CASC, 0.1vdiff on T and winds
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CISL - COP - CASC, 0.1vdiff on q only
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HIRLAM
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CISL
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CISL - COP - CASC
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CISL - COP - CASC, 0.1vdiff on all variables
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CISL - COP - CASC, 0.1vdiff on T and winds
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CISL - COP - CASC, 0.1vdiff on q only
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• A new hydrostatic dynamical core of HIRLAM has
  been developed
• High order of accuracy.
• It is possible to apply possitive definite and
  monotonic filters.
• The departure area is defined with high accuracy,
  i.e. high degree of local mass conservation
• It is generally possible to use long time steps,
  i.e. no CFL criterion
• Non-hydrostatic version should be possible to
  formulate
• Tuning of vertical diffusion needed (the scheme
  seems to have very little inherent numerical
  dispersion)
• Disadvantages of DCISL schemes
• The degree of conservation of a constant field in
  a non-divergent flow depends critically on the
  accuracy of the calculation of trajectories.
• To do
• Tuning of vertical diffusion optimise the code
  test the different options.
• Formulate conservation of enstrophy and total
  energy

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Hej Eigil,Jeg opdagede faktisk at jeg havde
kørt diverse vertikal diffusionscases med
dt300s (hvor de andre kørsler jeg sendte
tidligere var med dt360s - det var en
forglemmelse fra da jeg testede med at køre uden
horisontal diffusion fra i går). Minimum mslp
for de forskellige kørsler bliver lidt
anderledes, men trenden er stadig den samme, men
jeg sender dig de opdaterede filer her - samt
dem for 6 timers nedbør.Beklager
forvirringen.Min mslp for følgende Standard
HIRLAM 968.3 hPa 0.1 momentum og heat vdiff
967,52hPa 0.1 q vdiff 970,83 hPa0.1 alle vdiff
965, 44 hPa- 0.1qdiff- extensions på figurnavne
er hvor kun q og andre tracers er mult. med
0.1.- 0.1vtdiff - er når momentum og heat er
mult. med 0.1- 0.1vdiff - er når alle er mult.
med 0.1
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