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					  Governing Equations IV
   by Nils Wedi (room 007; ext. 2657)




Thanks to Anton Beljaars
    Governing Equations 4   Slide 1     ECMWF
Introduction

 Nonhydrostatic model NH - IFS
 Physics - Dynamics coupling




   Governing Equations 4   Slide 2   ECMWF
  Introduction – A history
 Resolution increases of the deterministic 10-day medium-range
  Integrated Forecast System (IFS) over ~25 years at ECMWF:
    1987: T 106 (~125km)

    1991: T 213 (~63km)
    1998: TL319 (~63km)
    2000: TL511 (~39km)
    2006: TL799 (~25km)
    2010: TL1279 (~16km)
    2015?: TL2047 (~10km)

    2020-???: (~1-10km) Non-hydrostatic, cloud-permitting, substan-
     tially different cloud-microphysics and turbulence parametrization,
     substantially different dynamics-physics interaction ?
     Governing Equations 4   Slide 3                    ECMWF
Ultra-high resolution global IFS simulations

 TL0799 (~ 25km) >>                 843,490 points per field/level
 TL1279 (~ 16km) >> 2,140,702 points per field/level
 TL2047 (~ 10km) >> 5,447,118 points per field/level
 TL3999 (~ 5km) >> 20,696,844 points per field/level (world
  record for spectral model ?!)




   Governing Equations 4   Slide 4                              ECMWF
                                    Max global altitude = 6503m
Orography – T1279




    Alps

  Governing Equations 4   Slide 5                 ECMWF
                                    Max global altitude = 7185m
Orography - T3999




   Alps

  Governing Equations 4   Slide 6                 ECMWF
Computational Cost at TL3999
hydrostatic vs. non-hydrostatic IFS



                                    GP_DYN          GP_DYN
                                    SP_DYN          SP_DYN
                                    TRANS           TRANS
                                    Physics         Physics
                                    other           other




NH TL3999                                     H TL3999



  Governing Equations 4   Slide 7              ECMWF
Nonhydrostatic IFS (NH-IFS)
 Bubnova et al. (1995); Benard et al. (2004), Benard et al.
 (2005), Benard et al. (2009), Wedi and Smolarkiewicz (2009),
 Wedi et al. (2009)


 Arpégé/ALADIN/Arome/HIRLAM/ECMWF nonhydrostatic
  dynamical core, which was developed by Météo-France and
  their ALADIN partners and later incorporated into the
  ECMWF model and adopted by HIRLAM.




   Governing Equations 4   Slide 8                    ECMWF
 Vertical coordinate


                                                 hybrid vertical coordinate
                                                  Simmons and Burridge (1981)
Denotes hydrostatic pressure in the context of a shallow, vertically
unbounded planetary atmosphere.

Prognostic surface pressure tendency:




        with
       coordinate transformation coefficient
      Governing Equations 4   Slide 9                                  ECMWF
Two new prognostic variables in the
nonhydrostatic formulation
                                                ‘Nonhydrostatic
                                                pressure departure’
                                                 ‘vertical divergence’
                                       Define also:
With residual residual




Three-dimensional divergence writes



    Governing Equations 4   Slide 10                      ECMWF
NH-IFS prognostic equations




                                       ‘Physics’




  Governing Equations 4   Slide 11   ECMWF
Diagnostic relations




    With




  Governing Equations 4   Slide 12   ECMWF
Auxiliary diagnostic relations




  Governing Equations 4   Slide 13   ECMWF
Numerical solution

 Advection via a two-time-level semi-Lagrangian numerical
  technique as before.
 Semi-implicit procedure with two reference states with respect
  to gravity and acoustic waves, respectively.
 The resulting Helmholtz equation is more complicated but can
  still be solved (subject to some constraints on the vertical
  discretization) with a direct spectral method as before.
                                           (Benard et al 2004,2005)




   Governing Equations 4   Slide 14                  ECMWF
Hierarchy of test cases

 Acoustic waves
 Gravity waves
 Planetary waves
 Convective motion
 Idealized dry atmospheric variability and mean states
 Idealized moist atmospheric variability and mean states
 Seasonal climate, intraseasonal variability
 Medium-range forecast performance at hydrostatic scales
 High-resolution forecasts at nonhydrostatic scales


   Governing Equations 4   Slide 15             ECMWF
                                                                                                                                                                                           0.10



           Spherical acoustic wave
       Friday 15 Octob$r 2004 12UTC ECMWF For$cast t+10000 VT: Tu$sday 6 D$c$mb$r 2005 04UTC Mod$l L$v$l 91 **Exp$rim$ntal product

                       0.001
                           140°W                       120°W                          100°W     80°W                           60°W                          40°W
                                                                                                                                                                                                                                                         analytic
                                                                                                                                                                                           1.00
50°N


             explicit                                                                                                                                               50°N




                                                                                                                                                                         pr$ssur$
40°N                                                                                                                                                                40°N




30°N                                                                                                                                                                30°N
                                                                                                                                                                                          10.00

20°N                                                                                                                                                                20°N




10°N                                                                                                                                                                10°N
                                                                                                                                                                                                                                                          NH-IFS
                                                                                                                                                                                          100.00
                                          0.004




                                                                                                                                 -0.004




                                                                                                                                                     0.004
                                                                                                                                          -0.002
                                                           -0.004
                                                  -0.002
                                         0.004




                                                                                                                                                   0.004
  0°                                                                                                                                                                0°


                                                                                                                                                                                                                 Tim$ (   100.000)
10°S                                                                                                                                                                10°S                     0.01


20°S                                                                                                                                                                20°S
                                                                                                                                                                                     1000.00
                                                                                                                                                                                             -0.030    -0.018    -0.006   0.006      0.018    0.030
30°S                                                                                                                                                                30°S
       Friday 15 Octob$r 2004 12UTC ECMWF For$cast t+10 VT: Friday 15 Octob$r 2004 22UTC Mod$l L$v$l 91 **Exp$rim$ntal product                                                                                   pr$ss d$partur$
                                                                                                                                                                                             0.10
                       0.001
                          140°W                        120°W                          100°W     80°W                           60°W                          40°W
40°S                                                                                                                                                                40°S




50°S
50°N




40°N
                           140°W                       120°W
                                                                             horizontal
                                                                                      100°W     80°W                           60°W                          40°W
                                                                                                                                                                    50°S
                                                                                                                                                                    50°N




                                                                                                                                                                    40°N                     1.00
                                                                                                                                                                                                                vertical
30°N                                                                                                                                                                30°N
                                                                                                                                                                               pr$ssur$

                                                                                                             -0
                                                                                                                  .0
20°N                                                                                                                 08                                             20°N


                                                                                                                                                                                            10.00
10°N                                                                                                                                                                10°N
                                                                    -0.008




                                                                                                                      -0.002
                                                                             -0.002




  0°                                                                                                                                                                0°




10°S                                                                                                                                                                10°S
                                                                                                                                                                                           100.00                                                      C ~ 340m/s
20°S                                                                                                                                                                20°S

                                                                                                        08
                                                                                                 -0.0

30°S                                                                                                                                                                30°S



                                                                                                                                                                                          1000.00
40°S                                                                                                                                                                40°S
                                                                                                                                                                                              -0.030    -0.018    -0.006   0.006      0.018    0.030
                                                                                                                                                                                                                  pr$ss d$partur$
50°S
             implicit                                                                                                                                               50°S




                          140°W                        120°W

                                   Governing Equations 4
                                                                                      100°W     80°W                           60°W

                                                                                                                               Slide 16
                                                                                                                                                             40°W


                                                                                                                                                                                                                                                       ECMWF
Orographic gravity waves H - IFS




  Governing Equations 4   Slide 17   ECMWF
Orographic gravity waves – NH - IFS




  Governing Equations 4   Slide 18   ECMWF
                                                                                                                                      50

                                                                                                                                      40


               “Scores”                                                                                                               30
                                                                                                                                             0                  1                  2    TL1279 L91 ~ 16 km
                                                                                                                                                                                                       3                    4    5     6   7       8         9   10
                                                                                                                                                                                                                            For$cast Day




                                         M$an curv$s
                                                                                                             f35d nh-ifs       NH                                            M$an curv$s
                                                                                                                                                                                                                                               f35d nh-ifs
                                     500hPa G$opot$ntial                                                                                                                 500hPa G$opot$ntial
                                Anomaly corr$lation for$cast
                       S.h$m Lat -90.0 to -20.0 Lon -180.0 to 180.0
                        Dat$: 20070301 12UTC to 20081101 12UTC
                            M$an calculation m$thod: standard
                                                                                                             f354 h-ifs
                                                                                                                               H                                  Root m$an squar$ $rror for$cast
                                                                                                                                                           S.h$m Lat -90.0 to -20.0 Lon -180.0 to 180.0
                                                                                                                                                            Dat$: 20070301 12UTC to 20081101 12UTC
                                                                                                                                                                M$an calculation m$thod: standard
                                                                                                                                                                                                                                               f354 h-ifs

Population: 45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45 (av$rag$d)                                               Population: 45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45 (av$rag$d)

110                                                                                                                                 120

100
                                                                                                                                    100
  90

                                                                                                                                      80
  80

  70                                                                                                                                  60

  60
                                                                                                                                      40
  50

                                                                                                                                      20
  40

  30                                                                                                                                    0
         0                  1                  2                   3                    4    5     6    7        8         9   10            0                  1                  2                   3                    4    5     6   7       8         9   10
                                                                                        For$cast Day                                                                                                                        For$cast Day



                                                                                                             f35d nh-ifs
                                         M$an curv$s
                                     500hPa G$opot$ntial
                              Root m$an squar$ $rror for$cast
                       S.h$m Lat -90.0 to -20.0 Lon -180.0 to 180.0
                                                                                                             f354 h-ifs
                        Dat$: 20070301 12UTC to 20081101 12UTC
                            M$an calculation m$thod: standard
Population: 45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45 (av$rag$d)




                                                                                                                                                                                                                                     ECMWF
120
                                    Governing Equations 4                                         Slide 19
100
Physics – Dynamics coupling
 ‘Physics’, parametrization: “the mathematical procedure
  describing the statistical effect of subgrid-scale processes on the
  mean flow expressed in terms of large scale parameters”,
  processes are typically: vertical diffusion, orography, cloud
  processes, convection, radiation
 ‘Dynamics’: “computation of all the other terms of the Navier-
  Stokes equations (eg. in IFS: semi-Lagrangian advection)”
 The „Physics‟ in IFS is currently formulated inherently
  hydrostatic, because the parametrizations are formulated as
  independent vertical columns on given pressure levels and
  pressure is NOT changed directly as a result of sub-gridscale
  interactions !
 The boundaries between „Physics‟ and „Dynamics‟ are “a moving
  target” …



   Governing Equations 4   Slide 20                    ECMWF
Different scales involved




                                     NH-effects visible
  Governing Equations 4   Slide 21                        ECMWF
Single timestep in two-time-level-scheme




   Governing Equations 4   Slide 22   ECMWF
Cost partition of a single time-step



                                      T799L91 CPU time spent (D. Salmond)

                                                  wave       spectral
                                                  model     transforms
                                                   4%       9.5%      spectral
                                                                       comm
                                                                        17%
                                       physics
                                        40%
                                                                  dynamics
                                                 dynamics
                                                                    22%
                                                  comm
  Note: Increase in CPU time                      6.5%
   substantial if the time step
   is reduced for the ‘physics’
   only.


   Governing Equations 4   Slide 23                             ECMWF
dynamics-physics coupling


                                        
  
F F 1     0
      G G  R  P
           0   1/ 2 1/ 2

  t  2


P (t  , g  2 )  P (t  t , g )  P (t , g   )  O((t ) )
 1/ 2          1 
             t
             2
                    2
                                       0
                                             
                                                            2



P is not a black box !!!
           1 0           1                   
 P  Prad convcloud  Prad convcloud  Pvdif  gwdrag
     1/ 2

           2             2

      Governing Equations 4   Slide 24            ECMWF
Noise in the operational forecast
eliminated through modified coupling




Governing Equations 4   Slide 25       ECMWF
Wrong equilibrium ?


T
     D  P , P(T )  gT , g  const.
 t
correct steady state solution:
    D
T
     g
  Governing Equations 4   Slide 26   ECMWF
Compute D+P(T) independant


       T         T n 1  T n
    1.     D                   Dn
       t              t
       T             T n 1  T n
    2.       gT                   g  T n 1  (1   )T n 
                                                                
        t                 t
    add 1.  2. together and seek steady state solution:
                             D
     explicit (γ  0 ) : T          n

                             g
                             D
     implicit (γ  1 ) : T  (1  g t ), wrong!
                          n

                             g

  Governing Equations 4   Slide 27                       ECMWF
Compute P(D,T)


       T         T n 1  T n
    1.     D                   Dn
       t              t
       T                 T n 1  T n
    2.      D  gT                    D n  g  T n 1  (1   )T n 
                                                                        
        t                     t
    seek steady state solution of 2. :
                             D
     explicit (γ  0 ) : T      n

                             g
                             D
     implicit ( γ  1 ) : T  , correct!
                           n

                             g


  Governing Equations 4   Slide 28                              ECMWF
      Sequential vs. parallel split of 2 processes                                                              A. Beljaars
      vdif + dynamics




                                                                                                  parallel split

                      (90 W, 60 S) T159 forecasts 2002011512, dt=60 min
          15




          10
U (m/s)




           5
                                                    parallel split (ej4k)
                                                    sequential split (ej4n)
                                                    bad sequential split (ej4x)
                                                    sequential split, dt=5 min (ej4m)
                                                                                             sequential split
           0
            12   15     18         21        24         27        30          33        36
                                    Forecast step (hours)


                      Governing Equations 4                        Slide 29                             ECMWF
Negative tracer concentration – Vertical diffusion

 Negative tracer concentrations noticed despite a quasi-
  monotone advection scheme
(Anton Beljaars)




   Governing Equations 4   Slide 30              ECMWF
   Physics-Dynamics coupling
   Vertical diffusion



  Single-layer                             (Kalnay and Kanamitsu, 1988)
    problem




dynamics positive definite




        Governing Equations 4   Slide 31                ECMWF
 Physics-Dynamics coupling
 Vertical diffusion



Two-layer
 problem



                                       Not positive
dynamics                               definite
positive                               depends on  !!!
definite



    Governing Equations 4   Slide 32     ECMWF
                                                         72.3N/2.5E
                        4e-14

                                   old time level
                                   after dynamics only                         (D+P)t
                        3e-14      new time level


                                                                              (D+P)t+t
Aerosol concentration




                                                                                              = 1.5
                        2e-14



                        1e-14
                                                                                Dt+t
                            0

                                                                            Negative tracer concentration
                        -1e-14
                              50                            55
                                                         Model level
                                                                       60   with over-implicit formulation
                                                         72.3N/2.5E
                        4e-14

                                   old time level
                                   after dynamics only
                        3e-14      new time level

                                                                                              =1
Aerosol concentration




                        2e-14



                        1e-14

                                                                                          Anton Beljaars
                             0




                        -1e-14
                              50                            55         60
                                                         Model level

				
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