1. Deutscher Wetterdienst 1FE 13 – 05.11.2015 Linear solutions of flow over mountains COSMO General Meeting WG2 'Numerics and Dynamics' 15.09.2008 Michael Baldauf Deutscher Wetterdienst, Offenbach, Germany

2. Deutscher Wetterdienst 2FE 13 – 05.11.2015 Development of a program to calculate a reference solution for convergence tests (task 5) Linear solutions of flow over mountain there exist a huge number of different linear solutions for flow over mountains: Queney (1947, 1948), Smith (1979, 1980),... reason: different goals: to get insight into the physical process: try to approximate as far as possible to get simple solutions/formulae to test numerical models: try to reduce the approximations as far as possible (perhaps by requiring more preconditions) to get an 'analytical' solution which fits best to the PDE- system (better say: a solution which can be calculated with higher numerical confidence, e.g. calculation of integrals or FFT's instead of solving PDE's)

3. Deutscher Wetterdienst 3FE 13 – 05.11.2015 Starting point: compressible Euler equations  COSMO-dynamical core Preconditions: no friction only adiabatic processes (in particular no phase changes) ideal gas law c p =const., c V =const., R=const. no Coriolis force all movements take place on a plane (no earth curvature) These preconditions can be fulfilled easily by a dynamical core. Only 2 approximations will be made: 1.linearisation  1/Fr = h N / U <<< 1  simulate very flat mountains at high inflow velocities 2.the assumption that k z =const (see below; not absolutely necessary)

4. Deutscher Wetterdienst 4FE 13 – 05.11.2015 'tracer'-parameter:  1 = 0/1 : hydrostatic / non-hydrostatic approximation  2 = 0/1 : incompressible / compressible model  3 = 0/1 : shallow / deep convection/atmosphere Compressible Euler equations (adiabatic, f=0,...) e.g. Smith (1979)

5. Deutscher Wetterdienst 5FE 13 – 05.11.2015 Base state: stationary, hydrostatic, only depends on z

6. Deutscher Wetterdienst 6FE 13 – 05.11.2015 Fouriertransformation: horizontal and temporal Perturbation equations: 4th 'tracer'-parameter introduced (Pichler, 1997):  4 =0/1 : small / big Mach-numbers

7. Deutscher Wetterdienst 7FE 13 – 05.11.2015 Def.: Heterogenity (Queney, 1947) Def: Stability parameter Def: Mach-number Def: Some abbreviations:

8. Deutscher Wetterdienst 8FE 13 – 05.11.2015 Stationary case (  =0 ) From perturbation equations: express u', v',  ' and p' by w'  equation 2nd order for w '( k x, k y, z ): with coefficient functions:

9. Deutscher Wetterdienst 9FE 13 – 05.11.2015 The whole derivation is in principle similar to many textbooks, but the only (!) approximation up to now is linearisation. Variable transformation (modified Bretherton (1960)-transformation): There are two special cases, which can be handled easily separately: 1.)  =0, k x =0, k y  0 2.)  =0, k x =0, k y =0

10. Deutscher Wetterdienst 10FE 13 – 05.11.2015 Boundary conditions: below: linearisation of the free-slip condition: above: assume k z =const. case k z ² < 0 : exponentially growing solution seems unphysical  chooseand omit term ~B case k z ² > 0 : no energy transport downwards (Smith, 1980)  choose and omit term ~B

11. Deutscher Wetterdienst 11FE 13 – 05.11.2015 Choice of the base state: N = const.  k z ~ const. (= 2nd approx.!) experience: k z =const. is fulfilled rather good Some features of 'Lin_Mountain': written in C/C++ Makefile for Linux available Binary output (with an additional.ctl-File for GrADS) Use of FFT (Numerical recipes, Press et al.) for estimation about the accuracy of the approximation ' k z =const.' max. height:~ 35 km

12. Deutscher Wetterdienst 12FE 13 – 05.11.2015 512 nx 2 ny 160 nz 200.0 dx 200.0 dy 100.0 dz 10.0 U_0 293.0 T_00 1.0e5 p_00 0.01 N_BV 1 delta1__0=hydrostat/1=nichthydrostat 1 delta2__0=inkompress/1=kompress 1 delta3__0=flach/1=tief 1 delta4__0=kleine/1=grosse_Machzahl topo.d oro_datei topo_ft.d oro_ft_datei Gaussdamm oro_typ 10000.0 oro_breite 10.0 oro_hoehe gauss_2D Ausgabekennung./ Ausgabepfad 100 ix_min 400 ix_max 1 ix_step 0 iy_min 0 iy_max 1 iy_step 0 iz_min 159 iz_max 1 FFT Input-File: (abc.inp)

13. Deutscher Wetterdienst 13FE 13 – 05.11.2015 Example 1: 2D-test case gaussian hill (for the Input-Data: slide before) w [m/s]

14. Deutscher Wetterdienst 14FE 13 – 05.11.2015 512 nx 2 ny 100 nz 200.0 dx 200.0 dy 100.0 dz 10.0 U_0 293.0 T_00 1.0e5 p_00 0.01 N_BV 1 delta1__0=hydrostat/1=nichthydrostat 1 delta2__0=inkompress/1=kompress 1 delta3__0=flach/1=tief 1 delta4__0=kleine/1=grosse_Machzahl topo.d oro_datei topo_ft.d oro_ft_datei Rippeldamm oro_typ 5000.0 oro_breite 250.0 oro_hoehe schaer Ausgabekennung./ Ausgabepfad 100 ix_min 400 ix_max 1 ix_step 0 iy_min 0 iy_max 1 iy_step 0 iz_min 99 iz_max 1 FFT Schaer et al. (2002) test case test case of Schaer et al (2002)

15. Deutscher Wetterdienst 15FE 13 – 05.11.2015 Example 2: 2D-test case from Schaer et al (2002) w [m/s]

16. Deutscher Wetterdienst 16FE 13 – 05.11.2015 512 nx 512 ny 20 nz 200.0 dx 200.0 dy 500.0 dz 10.0 U_0 293.0 T_00 1.0e5 p_00 0.01 N_BV 1 delta1__0=hydrostat/1=nichthydrostat 1 delta2__0=inkompress/1=kompress 1 delta3__0=flach/1=tief 1 delta4__0=kleine/1=grosse_Machzahl topo.d oro_datei topo_ft.d oro_ft_datei Gaussberg oro_typ 10000.0 oro_breite 10.0 oro_hoehe gauss_3D Ausgabekennung./ Ausgabepfad 100 ix_min 400 ix_max 1 ix_step 100 iy_min 400 iy_max 1 iy_step 0 iz_min 19 iz_max 1 FFT test case: 3D Gaussian hill

17. Deutscher Wetterdienst 17FE 13 – 05.11.2015 Example 3: 3D Gaussian Hill w [m/s]

18. Deutscher Wetterdienst 18FE 13 – 05.11.2015 Optimization of horizontal advection: up to COSMO 4.3: 'advection operators' = a subroutine acting on every single grid point  compiler has problems to optimize loops since COSMO 4.4: advection routines using 'field operations' (and additionally the DFI modifications by Lucio Torrisi) Efficiency gain for routine COSMO-DE at DWD (IBM): speedup of the horizontal advection alone: ~ 3 times faster overall reduction of model run time: ~ 1 Min. / 20 Min. ~ 5% Furthermore, some inconsistencies using metrical factors could be repaired acrlat(j,1)  acrlat(j,2) lent to an error of ~ -0.05% in the term v dw/dy

19. Deutscher Wetterdienst 19FE 13 – 05.11.2015 ngranularity: Each sample hit covers 4 bytes. Time: 16425.58 seconds called/total parents index %time self descendents called+self name index called/total children 53.46 12371.49 1/1.__start [2] [1] 75.6 53.46 12371.49 1.lmorg [1] 0.01 9450.25 8514/8514.organize_dynamics [3] 0.73 2529.11 8514/8514.organize_physics [6] 72.20 87.39 4256/4256.initialize_loop [32] 0.01 119.25 8515/8515.organize_diagnosis [39] 0.01 61.03 4258/8514.organize_data [38] 0.04 36.55 12770/12770.organize_assimilation [63] 10.19 0.82 4256/4256.near_surface [79] 0.02 3.36 4256/4256.exchange_runge_kutta [98]... -----------------------------------------------... ----------------------------------------------- 0.01 9450.25 8514/8514.lmorg [1] [3] 57.5 0.01 9450.25 8514.organize_dynamics [3] 262.68 9025.25 4256/4256.__src_runge_kutta_NMOD_org_runge_kutta [4] 113.02 49.28 4256/4256.__src_relaxation_NMOD_sardass [31] 0.02 0.00 1/1.init_dynamics [302] 0.00 0.00 1/1.__src_relaxation_NMOD_init_relaxation [641] 0.00 0.00 1/1.input_dynctl [1105] ----------------------------------------------- 262.68 9025.25 4256/4256.organize_dynamics [3] [4] 56.5 262.68 9025.25 4256.__src_runge_kutta_NMOD_org_runge_kutta [4] 3557.22 113.61 12768/12768.__fast_waves_rk_NMOD_fast_waves_runge_kutta [5] ===> 602.40 1593.07 12768/12768.__src_advection_rk_NMOD_advection_alt [7] 31.91 985.85 4256/4256.__src_advection_rk_NMOD_advection_pd [11] 930.86 0.00 12768/12768.__src_slow_tendencies_rk_NMOD_complete_tendencies_uvwtpp [13] ----------------------------------------------- 602.40 1593.07 12768/12768.__src_runge_kutta_NMOD_org_runge_kutta [4] [7] 13.4 602.40 1593.07 12768.__src_advection_rk_NMOD_advection_alt [7] 1574.75 0.00 18446744061366576640/18446744062714877440.__numeric_utilities_rk_NMOD_udsdx [8]... Old version with 'advection operator' (gprof - output)

20. Deutscher Wetterdienst 20FE 13 – 05.11.2015 ngranularity: Each sample hit covers 4 bytes. Time: 15485.44 seconds called/total parents index %time self descendents called+self name index called/total children 6.6s [1] 72.9 426.78 10867.61.lmorg [1] 0.01 8139.09 8514/8514.organize_dynamics [2] 0.56 2496.58 8514/8514.organize_physics [5] 0.02 120.22 8515/8515.organize_diagnosis [42] 0.02 60.68 4258/8514.organize_data [41] 0.00 38.48 12770/12770.organize_assimilation [65] 10.45 0.95 4256/4256.near_surface [81]... ----------------------------------------------- 0.01 8139.09 8514/8514.lmorg [1] [2] 52.6 0.01 8139.09 8514.organize_dynamics [2] 264.96 7711.97 4256/4256.__src_runge_kutta_NMOD_org_runge_kutta [3]... ----------------------------------------------- 264.96 7711.97 4256/4256.organize_dynamics [2] [3] 51.5 264.96 7711.97 4256.__src_runge_kutta_NMOD_org_runge_kutta [3] 3592.24 180.20 12768/12768.__fast_waves_rk_NMOD_fast_waves_runge_kutta [4] 31.61 1064.15 4256/4256.__src_advection_rk_NMOD_advection_pd [7] 1041.89 0.00 12768/12768.__src_slow_tendencies_rk_NMOD_complete_tendencies_uvwtpp [8] ===> 286.16 307.05 12768/12768.__src_advection_rk_NMOD_advection [14] 65.29 394.44 4256/4256.__hori_diffusion_NMOD_comp_hori_diff [17]... -----------------------------------------------... ----------------------------------------------- 286.16 307.05 12768/12768.__src_runge_kutta_NMOD_org_runge_kutta [3] [14] 3.8 286.16 307.05 12768.__src_advection_rk_NMOD_advection [14] 0.04 282.22 76608/76608.__src_advection_rk_NMOD_horiz_adv_driver [24] 0.07 18.80 25536/206951.__environment_NMOD_exchg_boundaries [35] 5.76 0.00 56179200/1404480000.__numeric_utilities_rk_NMOD_udsdx [38] New version with 'field operators' (gprof - output)