SRNWP Mini-workshop, Zagreb 5-6 December, 20061 Design of ALADIN NH with Vertical Finite Element...

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



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



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



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).



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



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



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:



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


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


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


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.


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


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):


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


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 *


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


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


*

*

T

TT *

*

T

TT


5-6 December, 2006 20

C1 constrain unsatisfied


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


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


5-6 December, 2006 22


Ref FD, 35, dt=5sRef VFE, 35lev, dt=5s X-term, Z-term-FD


5-6 December, 2006 23


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


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


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


5-6 December, 2006 26

Thank you for your attention !Ďakujem za pozornosť !

Merci de votre attention !Hvala !

SRNWP Mini-workshop, Zagreb 5-6 December, 20061 Design of ALADIN NH with Vertical Finite Element...

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Transcript of SRNWP Mini-workshop, Zagreb 5-6 December, 20061 Design of ALADIN NH with Vertical Finite Element...