SRNWP Mini-workshop, Zagreb
5-6 December, 2006 1
Design of ALADIN NH with Vertical Finite Element Discretisation
Jozef Vivoda, SHMÚPierre Bénard, METEO FranceKarim Yessad, METEO FranceEmeline Larrieu Rosina, SHMÚ
SRNWP Mini-workshop, Zagreb
5-6 December, 2006 2
Content of presentation
1. Introduction2. VFE – basic features3. VFE derivative and integration operator4. Discretization of linear model5. Analysis of stability of 2TL ICI NESC scheme6. First 2D idealized tests7. Conclusions and future plans
SRNWP Mini-workshop, Zagreb
5-6 December, 2006 3
Introduction
Motivation: VFE scheme successfully implemented into HY model (Untch and
Hortal,2004) To extent the VFE scheme into NH model with HY model as a limit case VFE has two times higher accuracy than FD method on the same stencil
(eight order for cubic elements in the case of evenly spaced mesh) More accurate vertical velocities for SL scheme
Work done so far: Benard – study of accuracy of vertical integral operators and
of compatibility of VFE with existing model choices Vivoda – linear analysis of stability of VFE scheme Vivoda, Yessad – start of implementation of VFE into ALADIN code
SRNWP Mini-workshop, Zagreb
5-6 December, 2006 4
VFE scheme basic features
Starting point – Untch and Hortal,2004:
Basis functions – cubic B-splines with compact support No staggering – all variables are defined on model full levels Only integration/derivation is performed in FE space, the products of
variables are done in physical space In SL version of HY model (ECMWF or ARPEGE) only non-local
operations in the vertical are integrations. In NH version derivatives plays crucial role (structure equation contains vertical laplacian).
SRNWP Mini-workshop, Zagreb
5-6 December, 2006 6
FE derivative operator
ηη)f(x,
η)F(x,
We expand F and f in terms of chosen set of functions:
Rη)e(
(x)f̂)d((x)F̂ i
ii
iii
Truncation error R is orthogonalized (required to vanish in weighted
integral sense) on interval (0,1):
0ΨjR1
0
ij
ii
ijii d
d
dedd
1
0
1
0
f̂F̂
fBFA ˆˆ
DffBCTAF 11
Finally we incorporate the transforms from physical to FE space and back:
FTF ˆ
fCf 1ˆ
SRNWP Mini-workshop, Zagreb
5-6 December, 2006 7
FE derivative operator basis functions
•In order to cover the physical domain (eta=<0,1>) with the full set of basis functions the 4 additional functions must be determined, the two at the bottom and the two at the top (analogical approach as Untch and Hortal, 2004).
iii ed )(,)(,)( }4,1{ Li
surftop L-1 L
L+1
L+2
L+3
L+4
211 LLLLL 11 L
21010 00
Example for 35 unevenly spaced levels
SRNWP Mini-workshop, Zagreb
5-6 December, 2006 8
FE derivative operator accuracy
)6sin()( f
1)0( ff
Lff )1(
0)0(' f
0)1(' f
21
01
1
01)0( fff
1
21)0('
fff
111
1)1(
LL
LL
L
LL fff
1
1)1('
L
LL fff
FD
VFE
FD
VFE
The 4 additional assumptions:
SRNWP Mini-workshop, Zagreb
5-6 December, 2006 9
FE integration operator
0
')η'f(x,η)(x, dF
We expand F and f in terms of chosen set of functions:
R')'e((x)f̂)d((x)F̂0
ii
ii
ii
d
Truncation error R is orthogonalized (required to vanish in weighted
integral sense) on interval (0,1):
0ΨjR1
0
ijii
ijii ddedd
1
00
1
0
'f̂F̂
fBFA ˆˆII
JffCBTAF 11 II
Finally we incorporate the transforms from physical to FE space and back:
FTF ˆ
fCf 1ˆ
In: N values of fOut: N+1 values of integralN on full levels + integral fromModel top to surface
SRNWP Mini-workshop, Zagreb
5-6 December, 2006 10
FE integration operator basis functions
Untch and Hortal, 2004
ie )( }4,1{ Li
id )( }4,1{ Li
i)( }4,1{ Li
L L+1 L+2 L+3 L+4
01-1-3
10 ff 0'0 f LL ff 1 0' 1 Lf
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5-6 December, 2006 12
VFE scheme – linear isothermal NH model
DSdDC
C
t
q
qLTR
g
t
d
DNt
q
dDC
TR
t
T
qTRqIGTRTGRt
D
v
p
s
v
ss
)(ˆ
ˆ
)(
ˆˆ)(
2
Mass weighted vertical integral operators (already in HY model)
Mass weighted vertical laplacian operator (NH specific)
''
11
Xd
d
dXG
''
1
0
Xdd
dXS
''
1 1
0
Xdd
dXN
s
X
mmXL
1
Linear system Vertical operators
vD
q
w
mRT
gpv
mRT
pd
pq
ss
)ln(
)ln(ˆ
4
Prognostic var.
System describing small aplitude waves around reference state ),( TX
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5-6 December, 2006 13
Constraint C1 on vertical operators
0:1 ***** NGSSGC
4
2***22**2*2 1 dcGRTDDmCRTcI
DmcSRTLrH
ddLrH
cI 2
*2****
2*2
44*
2*
2*2 1
•When eliminating the linear system to obtain single variable Helmholtz solver the following two equations for (d4,D) are obtained:
Constrain C1 is 0 in continuous case:
•In discrete case C1 is NOT automatically 0. The FD vertical discretization is designed in order to satisfy C1. If C1 is not satisfied further elimination is not feasible.
•In order to be consistent with existing HY and NH code the C1 constrain shall be satisfied.
SRNWP Mini-workshop, Zagreb
5-6 December, 2006 14
Constraint C2 on vertical operators
*2*2******2:2 TcNGc
cS
c
cGSLgC
V
p
V
p
44*
2*2*4
2*
*2*2 dd
r
cN
rH
LcI
Τ
When C1 is satisfied we could eliminate D from the system. It gives the structure equation
With C2 constraint:
•In continuous case T* is an identity matrix. In FD case T* is the tridiagonal matrix.
•For stability reasons of SI time stepping algorithm we require:
• L* to have real and negative eigenvalues• T* to have positive and real eigenvalues
•In the case of C1 unsatisfied above conditions could be insufficient, we we still require them assuming that C1 is close to zero
SRNWP Mini-workshop, Zagreb
5-6 December, 2006 15
Laplacian term FE treatment
P
mmLP
1
)6sin()( πf
P
m
P
mmLP
221V1: V2:
•Boundary conditions the same as in FD model version
In linear and nonlinear model the laplacian has the same form (thanks to factthat we use vertical divergence related prognostic variable):
SRNWP Mini-workshop, Zagreb
5-6 December, 2006 16
Laplacian term FE treatment
P
mmLP
1
P
m
P
mmLP
221V1: V2:
Eigenvalues of laplacian for 100 levels
Green – eigenvalues of V1Red – eigenvalues of V2
Imaginary part multiplied by 100
Real part of eigenvalues
Imagin
ary
part
of
eig
envalu
es
SRNWP Mini-workshop, Zagreb
5-6 December, 2006 17
Stability of 2TL Non-Extrapolating Iterative Centered Implicit scheme
2.
2
)()( )0(*
)0( tOFO
tF
ttOF XX
BXRXR
t
XX
XBXBXBXMXR ...)()( **
2.
2
)()( )(*
)1()( tO
nFO
tF
ntO
nF XX
BXRXR
t
XX
Predictor:
Nth-corrector:
For the purpose of analysis of stability we linearise the nonlinear residual Raround resting, isothermal, hydrostatically balanced reference state:
We analyse the temperature instability due to fact that
The following analyses were computed for the wave with dx=5km, dt=600s, 35 vertical levels, T*=350K, T*a=100K, p*s=101325Pa (perfect pressure)
TT *
SRNWP Mini-workshop, Zagreb
5-6 December, 2006 18
C1 constrain – modification of G*
from C1 constrain we redefine the operator G*. It must be computed via iterative procedure.
In linear model only In linear and nonlinear model
SI N=1 N=2 N=3 Solid – VFE Dotted - FD
)()(:1 **** NSISGC
*
*
T
TT *
*
T
TT
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5-6 December, 2006 19
C1 constrain – modification of S* Directly from C1 constrain we can define operator S*
)()( **1** GNIGS
The time stepping is unstable, because eigenvalues of L*A2 are complex.
Stability – C1 satisfied in linear model only Stability – C1 satisfied for NL model as well
SI N=1 N=2 N=3 Solid – VFE Dotted - FD
*
*
T
TT *
*
T
TT
SRNWP Mini-workshop, Zagreb
5-6 December, 2006 20
C1 constrain unsatisfied
SI N=1 N=2 N=3 Solid – VFE Dotted - FD
Dt=600s Dt=999999s
•Solution of 2Lx2L soler for each spectral component – noncompatibility with existing model (LxL solver), problem with efficient introduction of map factor horizontal dependence (LSIDG resp. LESIDG key)•Yessad – proposal of iterative procedure, build from existing pieces of code, study of convergence required
Black – no C1Red - S* redefinition in lin and nonlin Green – G* redefinition in lin and nonlinImaginary part multiplied by 10Circle radius 0.4
Eigenvalues of C2 constraint
*
*
T
TT *
*
T
TT
SRNWP Mini-workshop, Zagreb
5-6 December, 2006 21
VFE scheme – first tests
CY30T1 Tested in 2D 2TL ICI NESC n=1 sponge
Ref FD, 100lev, dt=5s
Eta coordinate NLNH flow regime dx=200m V=10ms-1 Agnesi hill, h=1000m, L=1000m
Ref VFE, 100lev, dt=1s(X-term, Z-term FD)
v
gwmRT
pZ
v
mRT
pX
SRNWP Mini-workshop, Zagreb
5-6 December, 2006 22
VFE scheme – first tests
Ref FD, 35, dt=5sRef VFE, 35lev, dt=5s X-term, Z-term-FD
SRNWP Mini-workshop, Zagreb
5-6 December, 2006 23
VFE scheme – first tests
Ref VFE, 35lev, dt=5s X-term VFE, Z-term-FD
Ref VFE, 35lev, dt=5s X-term FD, Z-term-VFE
v
gwmRT
pZ
v
mRT
pX
SRNWP Mini-workshop, Zagreb
5-6 December, 2006 24
VFE scheme – conclusions
Analysis of stability in linear framework suggests that it is possible to extend HY model VFE scheme to NH model
From stability point of view the crucial is the definition of vertical laplacian operator (all eigenvalues must be real and negative)
The VFE scheme is stable in linear context when FD boundary conditions are used in VFE (we can adopt BBC and TBC from FD model), but oscillatory behaviour near model bottom is observed (this is not true for large number of levels !)
So far no acceptable stable solutions that satisfy C1 was found
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5-6 December, 2006 25
VFE scheme – future work
Non-oscillatory discretization of vertical laplacian
Extension of just described VFE discretization concept into full NL model
Coding of “draft” of VFE scheme in NH ALADIN with 2Lx2L solver (finished)
testing
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5-6 December, 2006 26
Thank you for your attention !Ďakujem za pozornosť !
Merci de votre attention !Hvala !