Mass Conserving Courant Number Independent Eulerian Advection of the Moisture Quantities for the LMK

Transcription

1 Mass Conserving Courant Number Independent Eulerian Advection of the Moisture Quantities for the LMK Jochen Förstner, Michael Baldauf, Axel Seifert Deutscher Wetterdienst, Kaiserleistraße 29/35, Offenbach am Main J. Förstner - 6 March

2 Grid structure: Time integration: modes order TVD Aktionsprogramm 2003 horizontal: Arakawa-C staggered vertical: Lorenz staggered time-splitting (Klemp-Wilhelmson-splitting) of slow and fast 2-timelevels: Runge-Kutta Kutta: 2 nd order, 3 rd order, 3 rd (7km LM: 3-timelevels: Leapfrog (+ centered differences)) Advection: for u, v, w, p*, T: horizontal adv.: upwind 3 rd or 5 th order, centered 4 th or 6 th order vertical adv.: implicit centered 2 nd order, explicit upwind 3 rd or 5 th order (J.Förstner), implicit upwind 3 rd order (M. Baldauf) for q v, q c, q i, q r, q s, q g, TKE: Courant-number-independent (CNI) advection: Motivation: no constraint for w (deep convection!) Euler forward (J.Förstner): - van Leer-type scheme - Bott-scheme (2 nd, 4 th order) in Conservation Form Semi-Lagrange (M. Baldauf): - tri-linear: q r, q s, q g tri-cubic: q v, q c, q i, TKE (clipping) - tri-cubic: q v, q c, q i, q r, q s, q g, TKE (multiplicative filling) Smoothing: divergence damping 4 th order horizontal diffusion (to control instabilities near damping layers) J. Förstner - 6 March

3 Advective Form Flux Form 3D Wind Divergence Conservation Form diagnostic prognostic or 3D Strang-Splitting J. Förstner - 6 March

4 Courant Number Independent Formulation: Standard Courant Number Integer Flux Integer Courant Number Fractional Courant Number Fractional Flux Positive Definite Bott - Scheme: J. Förstner - 6 March

5 Solid Body Rotation of a tracer cone with an initial maximum of 1 Aktionsprogramm 2003 MAX( INT(Cr) ) = 2 (nearly 3) 80 time steps for one turn J. Förstner - 6 March

6 xz - Advection (u = 150 m/s, w = -10 m/s) of a tracer cone with an initial maximum of 1 Aktionsprogramm 2003 vertically stretched grid: MAX( INT(Cr) ) = 1 (near the bottom boundary) J. Förstner - 6 March

7 Aktionsprogramm 2003 southwesterly flow Observation Bott (2nd order) Flux Form - DIV (vqx) - qx v semi-lagrange qv, qc, qi: tri-cubic qr, qs, qg: tri-linear 07/08/2004 Mass Conserv. CNI Eulerian Advection of qx for the LMK LM-User 2006 J. Förstner - 6 March

8 southwesterly flow Observation Aktionsprogramm 2003 Bott (2 nd order) Conservation Form (ρvq x ) = (vρ x ) moist turbulence with (subscale) cloud diagnostic (only) scale clouds 07/08/2004 J. Förstner - 6 March

9 (north)westerly flow Observation Aktionsprogramm 2003 Bott (2 nd order) Flux Form - DIV (vq x ) - q x v semi-lagrange q v, q c, q i : tri-cubic q r, q s, q g : tri-linear 07/24/2004 J. Förstner - 6 March

10 Transport of Tracer in a Real Case Flow Field init Bott (2 nd ) Flux Form - DIV + Clipping semi- Lagrange (tri-cubic) + Clipping Bott (2 nd ) Conserv. Form

11 Bott (2 nd order) semi-lagrange J. Förstner - 6 March

12 ϕ n = (u, v, w, pp, T,...) n Physics-Dynamics Dynamics-Coupling Descr. of Advanced Research WRF Ver. 2 (2005) Physics (I) Radiation Shallow Convection Coriolis force Turbulence Physics (I) -Tendencies: Δϕ n (phys I) + Δϕ n-1 (phys II) Dynamics Runge-Kutta [ Δϕ (phys) + Δϕ (adv) fast waves ] ϕ * = (u, v, w, pp, T,...) * - Δϕ n-1 (phys II) Physics (II) Cloud Microphysics Physics (II) -Tendencies: Δϕ n (phys II) ϕ n+1 = (u, v, w, pp, T,...) n+1 J. Förstner - 6 March

13 Problems with New Physics-Dynamics-Coupling (NPDC) Negative feedback between NPDC and operational moist turbulence parameterization (... not present in dry turbulence parameterization) 2-Δz - structures in the specific cloud water field (q c ) 2-Δz - structures in the TKE field, unrealistic high values, where q c > 0 Measurements (K. Stephan, J. Förstner and M. Raschendorfer) Turbulence Parameterization Bug - correction in formulation of explicit correction No upper value for q c (scale) + q c (subscale) in SUBROUTINE cloud_diag (statistical scheme for cloud diagnostic) No vertical smoothing in calculation of explicit correction M. 1 (wichfakt = 0.0 in PHYCTL namelist) NPDC Latent heat conversions in SUBROUTINE satad (saturation adjustment) directly after Runge-Kutta dynamics are not taken into account M. 2 J. Förstner - 6 March

14 neither M. 1 nor M. 2 J. Förstner - 6 March

15 only M. 1 J. Förstner - 6 March

16 M. 1 and M. 2 J. Förstner - 6 March

17 Dynamic Bottom Boundary Condition... (Positive) Pressure Bias Problem blue: Old Bottom Boundary Cond. red: Dynamic Bottom Boundary Cond. (Figures by Torrisi, CNMCA Rom)... for metric pressure gradient term in equation for u-u and v-v component. Gaßmann (COSMO Newsletter No. 4) J. Förstner - 6 March

18 J. Förstner - 6 March

19 J. Förstner - 6 March

20 Dynamic BBC 4-Δ Low-Pass-Filter of Orography (whole Domain) Old BBC... plus 5-Δ extra Smoothing (for Δh 375 m) Dynamic BBC... plus 5-Δ extra Smoothing (for Δh 375 m) Old BBC... plus 6-Δ extra Smoothing (for Δh 250 m) Dynamic BBC... plus 6-Δ extra Smoothing (for Δh 250 m) J. Förstner - 6 March

21 LMK LME J. Förstner - 6 March

22 LMK LME J. Förstner - 6 March

23 LMK LME J. Förstner - 6 March

24 Summary and Outlook Aktionsprogramm 2003 Completely Revised Conservative Advection of Moisture Quantities: Clipping of negative values of q x (semi-lagrangian) and different diffusivity of the schemes: Model sensitivity with respect to patterns, (too small) mean and (too big) maximum values of precipitation. Monotone Bott schemes (slightly more diffusive than PD) necessary? Further Tests of the Various Options in LMK: semi-lagrangian (tri-cubic) with multiplicative filling / Eulerian advection of q x? New Coupling of Physics and Dynamics: Include (complete) diabatic forcing due to latent heat in RK scheme? Pressure Bias Problem solved? yes Corrected Moisture Convergence = d t q v Advection + d t q v Diffusion no... work with Lucio Torrisi in COSMO project Merge Different Turbulence Parameterizations (work with M. Raschendorfer has started) documentation, configuration of vertical layers and filter for orography, conservation properties, divergence damping, convergence tests, tuning,... J. Förstner - 6 March