New interpolation formula in stable
situation for the calculation of
diagnostic fields at measurement
height
László Kullmann
with supervision of
Jean-Francois Geleyn, Jean-Francois Mahfouf
Outline
• Description and of the current diagnostic formulas
– Paulson, Geleyn88, Canopy
• New interpolation formula
– determination of the new formula
– comparison with Geleyn88 and Canopy
• Application in DA
– Use the new formula as the TL/AD version of Canopy
• Conclusion
12-15 May 2009
ALADIN/HIRLAM WORKSHOP
Description of current diagnostic
schemes in SURFEX
• Paulson scheme
– fixed function, extrapolation from lowest model level
 Ts inconsistency
• Geleyn88 scheme
– fixed function with one parameter, interpolation between
surface, lowest model level
 underestimation of T2m in stable situation
• Canopy scheme (V. Masson)
– 6 SBL levels between lowest model level & surface, solving
prognostic equation (1d) with LS forcing + turbulence + drag
 T2m is prognostic (TL/AD problem)
12-15 May 2009
ALADIN/HIRLAM WORKSHOP
New interpolation formula
neutral
s s* 1
 z  z0 H 

h 

z  z  z0 H  L 
s*   z  z0 H
s ( z )  s (0)  ln 
   z0 H
stable/unstable

 z  z0 H 
z
  h 
  h  0 H
 L 
 L




z
1   ( )
 (  )  
d
L

• use prescribed  function (like in Paulson scheme)
• use the interpolation technique (JFG88)
() contains  parameter  determine  from s(zN)

s* 
  z N  z0 H 
 z N  z0 H 
 z0 H


s ( z N )  s (0) 
ln






 
h
h


z
L


 L
 
0H






s ( z N ) s (0)

bn

bH
 z  z0 H
h  N
L


 z 
  h  0 H   bn  bH
L 


12-15 May 2009

L







 f bH , bn 
ALADIN/HIRLAM WORKSHOP
New interpolation formula
Geleyn88 function:
 h ( )  1  
z
 z  z0 H 
z 
bH  bn 
h 
  h  0 H   
zN
 L 
 L 
Gratchev et al. function (proposed for stable condition):
 h ( )  1  ah
1 
1  ch   2
where ah=5, ch=3   too complicated...
simplified function (ch=2):
 z  z0 H
h 
 L

z
  h  0 H

 L
12-15 May 2009
 h ( )  1  ah

1  
bH bn


z  ah

e
 1
   ah ln 1 


 z N 

a
 
h 
ALADIN/HIRLAM WORKSHOP
z
bH  bn 
zN
New interpolation formula (2)
• The Gratchev formula for wind is too complicated
 use original formula which already gives good results
 determines the shape of the profile
z [m]
• New formula has a tuning parameter (ah)
• Determination of ah by fitting to observations
 big variance: ah  [2,20]
 use the proposed value ah=5
12-15 May 2009
ALADIN/HIRLAM WORKSHOP
ah
T [K]
Comparison of new/old formula
AROME forecast
04/12/2007-24/12/2007
only nature tile
T2m RMSE
T2m BIAS
Canopy
Geleyn88
New form.
RH2m RMSE
New formula gives similar results as Canopy
12-15 May 2009
ALADIN/HIRLAM WORKSHOP
RH2m BIAS
Tested in ALADIN (by Alena Trojakova)
RMSE
BIAS
ah = 
ah = 5
ah = 3
RMSE
12-15 May 2009
ALADIN/HIRLAM WORKSHOP
BIAS
Testing the TL/AD of Canopy
• Check if new formula is able to describe the TL of Canopy
• Canopy scheme is prognostic  change in TN does not
cause immediate change in T2m
– Checking the speed of relaxation
• Offline Surfex run
– random perturbation of forcing fields
– keep forcing fields constant for
relaxation period
– surface fields are kept constant
– Compare TL of new formula with the change in Canopy field:
T(t2-1)-T(t1-1)
– determine ah by fitting to Canopy profile
12-15 May 2009
ALADIN/HIRLAM WORKSHOP
T2m New
T2m
RH2m
wind 10m
ah: fitted
ah: fitted
ah: fitted
a h= 1
a h= 1
a h= 1
a h= 
a h= 
a h= 
12-15T
May 2009
Canopy
2m
ALADIN/HIRLAM
WORKSHOP
RH Canopy
2m
U10m Canopy
AROME 3dVar test
• Comparing results using TL/AD of old (G88)
and new formula
• 6h assimilation cycling
• Observations: TEMP, AIREP, AMSU-A,
AMSU-B, SYNOP (T2, RH2, U10)
• direct obs. operator is the Canopy scheme
(from guess)
• Ensemble B matrix
• Small differences on average, disappears
after few hours
12-15 May 2009
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time period: 05/12/2007 – 22/12/2007
rmse T2m
bias T2m
ah = 
ah = 5
ah = 1
rmse RH2m
12-15 May 2009
ALADIN/HIRLAM WORKSHOP
bias RH2m
Conclusion
• Determination of new interpolation formula
• The new formula gives similar result as Canopy or
Paulson in stable case
• TL of new formula approximates well the TL of
Canopy
• Using TL/AD of new formula in assimilation has
small impact compared to the old one
12-15 May 2009
ALADIN/HIRLAM WORKSHOP