Flux Diagnosis in the COSMO model and their problems

Transcription

1 Flux Diagnosis in the COSMO model and their problems Ronny Petrik 1 Michael Baldauf 2 Almut Gassmann 1 1 Max-Planck-institute for meteorology 2 German Weather Service 2nd March 2008

2 Outline Motivation Idea of the diagnostic method the idealized advection world - seeing problems Apply tool to COSMO Conclusions and Outlook

3 Outline Motivation Idea of the diagnostic method the idealized advection world - seeing problems Apply tool to COSMO Conclusions and Outlook

4 Motivation 1 nonhydrostatic equation system of COSMO deals with various approximations, different numerical schemes and artificial sources (like horizontal diffusion) energy and mass consistency not fulfilled in this forecast system Does it play only an essential role in theory? What is the magnitude of this error? perhaps a diagnostic tool determining the conservation properties can help to answer this questions...

5 Motivation 1 nonhydrostatic equation system of COSMO deals with various approximations, different numerical schemes and artificial sources (like horizontal diffusion) energy and mass consistency not fulfilled in this forecast system Does it play only an essential role in theory? What is the magnitude of this error? perhaps a diagnostic tool determining the conservation properties can help to answer this questions...

6 Motivation 1 nonhydrostatic equation system of COSMO deals with various approximations, different numerical schemes and artificial sources (like horizontal diffusion) energy and mass consistency not fulfilled in this forecast system Does it play only an essential role in theory? What is the magnitude of this error? perhaps a diagnostic tool determining the conservation properties can help to answer this questions...

7 Motivation 2 Is the formal violation of conservation properties negligible for weather forecast? Are there artificial sources and sinks due to physical parametrisations in water mass? (related to the hydrological cycle and the quantitative precipitation forecast) northwesterly flow in a broad warm sector interacting with the Erzgebirge provided by Zimmer (university Leipzig) 10 mm 20 mm 30 mm provided by Zimmer (university Leipzig) FIGURE 1: , 24h precipitation in LMK-model and as radar product. Janek Zimmer, university Leipzig

8 Motivation 3 investigation of fluxes in non- and convection-resolved model setups beside energy and mass - calculate budgets of quantities like ertels potential vorticity or helicity gives the opportunity to look for a local consistency on a temporal resolution of one time step monitoring and analysing - among verification techniques - the actual model and new stages in model devolopment (for example the Nambu-Bracket-approach)

9 Outline Motivation Idea of the diagnostic method the idealized advection world - seeing problems Apply tool to COSMO Conclusions and Outlook

10 Concept of diagnostic - the simple part evaluate the budget equation for a quantity ψ (like ψ = ρq v) ψ + f ψ = q ψ integral Z ZZ Z ψdv + fφ ds = t t V F V (with the fluxes f ψ and the sources q ψ ) q ψ dv e r e ϕ e λ negative flux (boundary west) wind v w ds λ,ζ positive flux (boundary north) ds λ,ϕ FIGURE 2: Gridbox in COSMO with wind example u ds ζ,ϕ calculating the fluxes over the boundaries with surface integrals the technique becomes a finite volume method in contrast to the advective form of prognostic equations in COSMO: ψ t = u i ψ x i +... a volume integral is discretised with second order accuracy (summing up values of the mass midpoints) and the fluxes...

11 Concept of diagnostic - the tricky part beware of sloping ( dz a grid box, dz dλ dφ ) surface elements at the bottom and the top of the devil is in the flux f φ = ρq v v interpolation of scalars is needed to recontruct cell face values from grid points our wish for interpolation: gradient free flow patterns FIGURE 3: flow pattern in COSMO output our reality - weather: pertubated patterns of the atmosphere how to define ρq vu or even worse ρq vw for convective cases?

12 Outline Motivation Idea of the diagnostic method the idealized advection world - seeing problems Apply tool to COSMO Conclusions and Outlook

13 Motivation Idea of the diagnostic method the idealized advection world - seeing problems Apply tool to COSMO Conclusions and Outlook Shapes for ideal advection assume an ideal advection algorithm and shift different patterns on a discrete grid - assuming a physical meaning of density ideal advection algorithm means a shifting of a known time-dependent analytic function

14 Diagnostic calculations use the advection equation ρ ρ = u a and t x a proof the integral budget form, here 1D: t imax X i=imin ρ i = (flux ρ imax+1/2 flux ρ imin 1/2 ) first guess: mass change and flux cancel each other with an exact advection scheme, otherwise calculate two error indices Control V a relative error (RE) and a volume error (VE) M n+1 M n + flux(ρ) t t RE = M t (CV) M n+1 M n t + flux(ρ) t ControlVolume average this errors when using different shapes and CFL-criteria

15 Results with exact advection scheme 1 Control Volume normalized error [kg/m] (from shifting analytic solutions) 1D-Test for parabel, triangel and sinus with 3 CFL-numbers average density in control volume ~ 7.8 kg/m, Timesteps ~ 93 0,080 0,070 0,060 0,050 0,040 0,030 0,020 0,010 0,000 Flux_Upwind1 Flux_Centered2 Flux LaxWen Flux_3order Flux_LaxWenlim Flux_5order Flux_Upwind1 (t=n+1)

16 Results with exact advection scheme 2 relative error [%/100] for different CV-sizes (from shifting analytic solutions) 1D-Test for parabel, triangel and sinus with CFL=1/3 Timesteps ~ 120 0,0060 0,0050 0,0040 0,0030 0,0020 0,0010 0,0000 Volume error, 25GP Volume error, 8GP Volume error, 15GP Volume error, 5GP

17 Results with exact advection scheme 3 Analytical advection of a cone with a upstream flux diagnosis 5e+07 4e+07 3e+07 2e+07 1e e+07-2e+07-3e+07-4e+07-5e+07 mass divergence mass change residuum Zeitschritt FIGURE 4: analytical advection of a cone function φ with CFL = 0.5 a CFL-criteria 1 cause budget deficit for every flux interpolation method (see figure for upwind 1order method) due to grid point space and not advection errors occur a base noise is in the budget diagnostic - minimise this with Lax-Wendroff scheme - the greater the control volume the smaller the budget error: ratio of surface area divided by volume size 1/x - in addition to interpolation problems an volume error exist during shifting in grid space the same investigation with an numerical advection scheme is recommended

18 Results with Runge-Kutta Upwind5 scheme 0,080 0,070 0,060 0,050 0,040 0,030 0,020 0,010 0,000 Control Volume error [kg/m] from Runge-Kutta 3stage Upwind 5 forecast 1D-test for parabel, triangel and sinus with 3 CFL-numbers average density in control volume ~ 7.8 kg/m, timesteps ~ 93 Flux_Upwind1 Flux_Centered2 Flux_3Order Flux_5Order Flux LaxWen Flux_LaxWenlim Flux_UDS t=n+1 Flux_RK3Up5 Flux_RK3Up5Noise

19 Outline Motivation Idea of the diagnostic method the idealized advection world - seeing problems Apply tool to COSMO Conclusions and Outlook

20 The mass and energy budget in COSMO mass budget following (Wacker2003) for a cloudy atmosphere with mean mass weighted velocity of system air-mixture v - six categories: vapour, cloud water and ice, rain water, snow and graupel are allowed: t Z ZZ ρ gq k dv = V F vρ gq k + S r,s,g e z F T v,c,i ez d F + Z σ k dv V options for testing whole mass or cloudy mass or total water mass total energy budget in the atmosphere (Herzog2007): ZZZ ZZ ZZ ρe gdv = ρ v r E gdf v r p df + radiation + heatfluxes +... t V F investigation could be a hard task in case of full physics use a Lax Wendroff method for COSMO runs as first reference choose a control volume far away from the non-phyiscal dynamics (Rayleigh-Damping, relaxation zones) prefer cell faces in unspectacular zones F Control Volume

21 the most easiest configuration, SBCAPE 2450J, EL = 12000m test case of temperature anomaly (Weisman-Klemp profile) as a first application without physics, in particular rain and no condensation (case 1) problems in the 1D-splitting-BOTT advection scheme fluxes and residuum [kg/s] total water mass rho*qx in a 3670km 3 volume, no condensation 1e e+11 mass change Bott mass mass divergence mass change SL precipitation flux residuum 1.68e+11 5e+06 Splitting 0-5e e e e+11 mass[kg] ave. density: 4.57 * 10-3 kg/m 3 absolute Vol error: 3.52 * 10-6 kg/m 3 Vol error per steps: 3.52 * 10-6 kg/m 3 /TS Relative error: 7,69 *10-4 %/100-1e e # Timesteps

22 fluxes and residuum [kg/s] 4e+06 2e e+06-4e+06 the non-condensation configuration 2 total water mass rho*qx in a 10*10*10 gridpoint volume, lower atmoshere ave. density: 0.74 kg/m 3 mass change mass divergence precipitation flux residuum absolute Vol error: * 10-5 kg/m 3 Vol error per steps: 1.77 * 10-7 kg/m 3 /TS Relative error: 8.61 * 10-5 [%/100] mass change and flux [kg/s] 0-5e+06 ave. density: kg/m 3 absolute Vol error: 1.98 * 10-4 kg/m 3 Vol error per steps: 5.50 * 10-7 kg/m 3 /TS Relative error: [%/100] # Timesteps total mass in a 3669 km 3 volume 1e+07 mass change mass divergence flux upwards residuum 5e+06 wet mass divergence -1e # Timesteps

23 condensation allowed, SBCAPE fluxes and residuum [kg/s] 400J, EL = 6600m total water mass rho*qx in a 3669 km 3 volume, 0-10 km 3e+06 mass change residuum SL 2e+06 mass divergence precipitation flux residuum ave. density: * 10-3 kg/m 3 1e e+06-2e+06 absolute Vol error: * 10 1 J/m 3 Vol error per steps: 4.7 * 10-2 J/m 3 /TS Relative error: 9.89 * 10-5 [%/100] 1e e+12-2e+12-3e+12-4e+12 absolute Vol error: 2.98 * 10-6 kg/m 3 Vol error per steps: 8.27 * 10-9 kg/m 3 /TS Relative error: 1.31 * 10-3 [%/100] -3e # Timesteps total energy in a 3669 km 3 volume 4e+12 energy change ave. density: 3e+12 energy divergence workflow 1.72 * 10 5 J/m 3 2e+12 residuum energy change and flux [J/s] # Timesteps

24 Results for configurations with physics 1 test case with full physics, especially rain and vertical turbulent mixing, but no radiation take evaporation and precipitation fluxes into account relative error in water mass 0.5 percent and 1 percent for whole mass 4e+06 total water mass rho*(qx) in a 30*30*32 gridpoint volume, mass change mass divergence precipitation flux evaporation flux residuum precipitation u p evapo u p 6e+06 4e+06 fluxes and residuum [kg/s] 2e e+06 2e e+06-4e+06-4e+06-6e # Timesteps

25 Outline Motivation Idea of the diagnostic method the idealized advection world - seeing problems Apply tool to COSMO Conclusions and Outlook

26 Conclusions the diagnostic tool works online in the COSMO in first tests - carefully exposure for reasonable results even if a exact advection scheme is available an error in diagnostic scheme will occur - actual choose Lax Wendroff a lokal investigation only practical with more than 5 3 GP good performance in mass and energy budgets of the Weisman-Klemp case - reasonable budget analysis in relation to physical processes: intake of wet air (growing phase) and gust and anvil outflows performance loss during condensation phase (especially SL-advection) and precipation events (especially whole mass) visualisation of features like splitting errors in 1D BOTT-Advection scheme (jerky leaps) and adjustment processes to given initial states different behaviour in conservation for various vertical model level and turbulence settings

27 Outlook investigate the mountain flow with this tool (terrain-following coordinates) - more realistic initialisations for convection (heating the ground to overcome CIN) applicate this tool in realistic weather situations far away dream: an output of the model fluxes with an later analysis of arbitrary control volume try implementing terms of the Nambu-brackets theory, i.e. v, M, H and n v, h o a, H, that deals with 3D-vortex quantity study vortex quantities like helicity or the thermodynamical Ertels potential vorticity in the COSMO to verify Nambu-technique verify energetics and nonlinear interaction with power spectra missing thermodynamic terms in COSMO with new saturation adjustment scheme?

28 Thank Thank you for your time and your attention