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Status report of WG2 - Numerics and Dynamics COSMO General Meeting 10-13 Sept. 2012, Lugano

Status report of WG2 - Numerics and Dynamics COSMO General Meeting 10-13 Sept. 2012, Lugano. Michael Baldauf Deutscher Wetterdienst, Offenbach, Germany. Topics New Fast Waves Solver A new idealized test case for non-hydrostatic compressible Models

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Status report of WG2 - Numerics and Dynamics COSMO General Meeting 10-13 Sept. 2012, Lugano

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  1. Status report of WG2 - Numerics and Dynamics COSMO General Meeting 10-13 Sept. 2012, Lugano Michael Baldauf Deutscher Wetterdienst, Offenbach, Germany M. Baldauf (DWD)

  2. Topics • New Fast Waves Solver • A new idealized test case for non-hydrostatic compressible Models • COSMO-DG: Discontinuous Galerkin dynamical core for COSMO • outlook • CONSOL: development of an implicit finite volume solver see CDC presentation M. Baldauf (DWD)

  3. Development of a new fast waves solver for the Runge-Kutta scheme M. Baldauf (DWD) • Main changes towards the current solver: • improvement of the vertical discretization: use of weighted averaging operators for all vertical operations • divergence in strong conservation form • optional: complete 3D (=isotropic) divergence damping • optional: Mahrer (1984) discretization of horizontal pressure gradients • additionally some 'technical' improvements; • hopefully a certain increase in code readability • overall goal: improve numerical stability of COSMO M. Baldauf (DWD)

  4. Improved stability in real case applications • COSMO-DE 28.06.2011, 6 UTC runs (shear instability case)must be cured by Smagorinsky diffusionoperational deterministic and several COSMO-DE-EPS runscrashed after ~16 h • COSMO-DE 12.07.2011, 6 UTC run (‚explosion‘ of qr in Alps)only stable with Bott2_Strang • COSMO-DE L65, 15.07.2011, 12 UTC runmodel crash after 215 timesteps (~ 1.5 h) • COSMO-2, 16.06.2011, 0 UTC run (reported by O. Fuhrer)model crash after 1190 timesteps (~6.6 h) • ‚COSMO-1, 24.08.2011‘, resolution 0.01° (~ 1.1km) 1700 * 1700 grid points (reported by A. Seifert) model crash after 10 time steps • ... • COSMO-DE L65 run stable for 2 months ('1.7.-31.8.2012') M. Baldauf (DWD) 4

  5. Discretization error in stretched grids; Divergence Divergence with weighted average of u (andv) to the half levels: s·dz dz 1/s ·dz Divergence with only arithmetic average of u (and v) to the half levels: not a consistent discretization if s1 !

  6. Discretization error in stretched grids; Buoyancy (T'/T0) buoyancy term with weighted average of T' (T0 exact): s·dz dz 1/s ·dz buoyancy term with arithmetic average of T'(T0 exact): buoyancy term with weighted average for T' and T0:

  7. Winter case experiment COSMO-DE New / Old fast waves solver M. Baldauf (FE13)

  8. Summer case experiment COSMO-DE New / Old fast waves solver M. Baldauf (FE13)

  9. M. Baldauf (FE13)

  10. upper air verification of COSMO-DE L65 with new FW solver compared with Parallel-Routine COSMO-DE L50

  11. Summary • New fast waves solver cures crashes in several realistic test cases • this is partly due to the fact that it allows steeper slopes • dynamic bottom BC probably not necessary • problems with geopotential bias are solved due to improved buoyancy formulation • efficiency: the new fast waves solver needs about 30% more computation time than the current one on the NEC-SX9  about 6% more time in total for COSMO-DE • new fast_waves_sc.f90 is now available in COSMO 4.24(if irefatm=1 is used  use int2lm 1.20 with lanalyt_calc_p0T0=.TRUE.) M. Baldauf (DWD) 06.-08. March 2012 11

  12. Outlook • Work on • lateral radiation BC • alternative upper damping layer (Klemp et al., 2008) • lw_freeslip • is finished  COSMO 5.1 • Documentation in COSMO Sci. Doc. Part I(currently new fast waves doc. only available in German) • Currently new FW runs in Parallelroutine at DWD • further investigation of the Mahrer-discretization;benefits from Zängl (2012) MWR ? • Other stuff: • Consolidation on dynamical bottom BC • Consolidation and inspection of 3D divergence damping M. Baldauf (DWD)

  13. An analytic solution for linear gravity waves in a channel as a test case for solvers of the non-hydrostatic, compressible Euler equations M. Baldauf (DWD) For development of dynamical cores (or numerical methods in general) idealized test cases are an important evaluation tool • Existing analytic solutions: • stationary flow over mountains linear: Queney (1947, ...), Smith (1979, ...), Baldauf (2008)non-linear: Long (1955) for Boussinesq-approx. atmosphere • non-stationary, linear expansion of gravity waves in a channelSkamarock, Klemp (1994) for Boussinesq-approx. atmosphere • Most of the other idealized tests only possess 'known solutions' gained • by other numerical models. There exist even less analytic solutions which use the exact equations, i.e. in a sense that a numerical model would converge to this solution. One exception is presented here: linear expansion of gravity/sound waves in a channel

  14. Small scale test with a basic flow U0=20 m/s f=0 Initialization similar to Skamarock, Klemp (1994) Black lines: analytic solution Shaded: COSMO

  15. Analytic solution of the fully compressible, non-hydrostatic Euler equations for the Fourier transformed vertical velocity w analogous expressions for ub(kx, kz, t), ... The frequencies ,  are the gravity wave and acoustic branch, respecticely, of the dispersion relation for compressible waves in a channel with height H; kz = ( / H)  m m=2    cs / g m=1 m=0  kx cs2 / g

  16. Convergence properties of COSMO

  17. A new dynamical core based on Discontinuous Galerkin methods Project ‘Adaptive numerics for multi-scale flow’, DFG priority program ‘Metström’ • cooperation between • DWD, • Univ. Freiburg, • Univ. Warwick • goals of the DWD: new dynamical core for COSMO • high order • conservation of mass, momentum and energy / potential temperature • scalability to thousands of CPUs. • ➥use of discontinuous Galerkin methods • HEVI time integration (horizontal explicit, vertical implicit) • terrain following coordinates • coupling of the physical parameterisations • reuse of the existing parameterisations D. Schuster, M. Baldauf (DWD) M. Baldauf (DWD)

  18. COSMO-RK test case linear mountain overflow COSMO-DG Test case of: Sean L. Gibbons. Impacts of sigma coordinates on the Euler and Navier-Stokes equations using continuous Galerkin methods. PhD thesis, Naval Postgradudate School, March 2009. M. Baldauf (DWD)

  19. test case density current (Straka et al., 1993) ‘ after 900 sec. COSMO-DG-simulation without diffusion M. Baldauf (DWD)

  20. Outlook, Milestones • Further consolidation of the explicit DG solver in COSMO • Use of implicit solver and HE-VI • Coupling of Turbulence scheme DFG bundle project: 'Gravity waves' together with the group of U. Achatz (Univ. Frankfurt) • Coupling of Microphysics : more realistic test simulations • Coupling to remaining physics: COSMO-DG ready for realistic simulations M. Baldauf (DWD)

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