1. SRNWP Mini-workshop, Zagreb 5-6 December, 20061 Design of ALADIN NH with Vertical Finite Element Discretisation Jozef Vivoda, SHMÚ Pierre Bénard, METEO France Karim Yessad, METEO France Emeline Larrieu Rosina, SHMÚ

2. SRNWP Mini-workshop, Zagreb 5-6 December, 20062 Content of presentation 1. Introduction 2. VFE – basic features 3. VFE derivative and integration operator 4. Discretization of linear model 5. Analysis of stability of 2TL ICI NESC scheme 6. First 2D idealized tests 7. Conclusions and future plans

3. SRNWP Mini-workshop, Zagreb 5-6 December, 20063 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

4. SRNWP Mini-workshop, Zagreb 5-6 December, 20064 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).

5. SRNWP Mini-workshop, Zagreb 5-6 December, 20065 VFE scheme – redefinition of A and B  In current ALADIN the functions A and B are determined on half levels (deeply hardcoded feature)  VFE requires definition of derivatives of A and B function on full levels Starting from half level A and B we correct and in a manner that for VFE integral operator the mass is conserved Than using VFE integral operator we get full level A and B

6. SRNWP Mini-workshop, Zagreb 5-6 December, 20066 FE derivative operator We expand F and f in terms of chosen set of functions: Truncation error R is orthogonalized (required to vanish in weighted integral sense) on interval (0,1): Finally we incorporate the transforms from physical to FE space and back:

7. SRNWP Mini-workshop, Zagreb 5-6 December, 20067 FE derivative operator basis functions In order to cover the physical domain (eta= ) 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). surf top L-1 L L+1 L+2 L+3 L+4 Example for 35 unevenly spaced levels

8. SRNWP Mini-workshop, Zagreb 5-6 December, 20068 FE derivative operator accuracy FD VFE FD VFE The 4 additional assumptions:

9. SRNWP Mini-workshop, Zagreb 5-6 December, 20069 FE integration operator We expand F and f in terms of chosen set of functions: Truncation error R is orthogonalized (required to vanish in weighted integral sense) on interval (0,1): Finally we incorporate the transforms from physical to FE space and back: In: N values of f Out: N+1 values of integral N on full levels + integral from Model top to surface

10. SRNWP Mini-workshop, Zagreb 5-6 December, 200610 FE integration operator basis functions Untch and Hortal, 2004 L L+1 L+2L+3 L+4 0 1 -3

11. SRNWP Mini-workshop, Zagreb 5-6 December, 200611 VFE scheme – design Development in the framework of linear isothermal resting atmosphere with small perturbations with respect to  Stability (2TL ICI NESC scheme with 0,1,2,3 iterations) With respect to SHB temperature instability  Compatibility of existing solution (feasibility of elimination) Generalisation of operators in NL framework Implementation and debugging Testing (2D -> 3D)

12. SRNWP Mini-workshop, Zagreb 5-6 December, 200612 VFE scheme – linear isothermal NH model Mass weighted vertical integral operators (already in HY model) Mass weighted vertical laplacian operator (NH specific) Linear system Vertical operators Prognostic var. System describing small aplitude waves around reference state

13. SRNWP Mini-workshop, Zagreb 5-6 December, 200613 Constraint C1 on vertical operators 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.

14. SRNWP Mini-workshop, Zagreb 5-6 December, 200614 Constraint C2 on vertical operators 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

15. SRNWP Mini-workshop, Zagreb 5-6 December, 200615 Laplacian term FE treatment V1: V2: Boundary conditions the same as in FD model version In linear and nonlinear model the laplacian has the same form (thanks to fact that we use vertical divergence related prognostic variable):

16. SRNWP Mini-workshop, Zagreb 5-6 December, 200616 Laplacian term FE treatment V1: V2: Eigenvalues of laplacian for 100 levels Green – eigenvalues of V1 Red – eigenvalues of V2 Imaginary part multiplied by 100 Real part of eigenvalues Imaginary part of eigenvalues

17. SRNWP Mini-workshop, Zagreb 5-6 December, 200617 Stability of 2TL Non-Extrapolating Iterative Centered Implicit scheme Predictor: Nth-corrector: For the purpose of analysis of stability we linearise the nonlinear residual R around 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)

18. SRNWP Mini-workshop, Zagreb 5-6 December, 200618 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

19. SRNWP Mini-workshop, Zagreb 5-6 December, 200619 C1 constrain – modification of S*  Directly from C1 constrain we can define operator S* 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

20. SRNWP Mini-workshop, Zagreb 5-6 December, 200620 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 C1 Red - S* redefinition in lin and nonlin Green – G* redefinition in lin and nonlin Imaginary part multiplied by 10 Circle radius 0.4 Eigenvalues of C2 constraint

21. SRNWP Mini-workshop, Zagreb 5-6 December, 200621 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)

22. SRNWP Mini-workshop, Zagreb 5-6 December, 200622 VFE scheme – first tests Ref FD, 35, dt=5s Ref VFE, 35lev, dt=5s X-term, Z-term-FD

23. SRNWP Mini-workshop, Zagreb 5-6 December, 200623 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

24. SRNWP Mini-workshop, Zagreb 5-6 December, 200624 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

25. SRNWP Mini-workshop, Zagreb 5-6 December, 200625 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

26. SRNWP Mini-workshop, Zagreb 5-6 December, 200626 Thank you for your attention ! Ďakujem za pozornosť ! Merci de votre attention ! Hvala !