Status Report WG2 J. Steppeler, DWD Zurich 2005

Z-Coordinate Runge Kutta and Semi-Lagrangian methods Direct implicit solvers Courant number independent advection of moisture Other developments Suggested priority work Plan of Lecture

Realistic cases possible (Euler Version) SL version now 3-d Low stratus cases seem to have a problem after last modification of physics interface New QPF cases were run The QPF is improved significantly in all 4 available cases (1 Poland, 3 Greece) LM-Z Summary

The atmosphere at rest can be represented in Z- coordinates, but not in terrain following coordinates Stratiform clouds and low stratus are predicted better in LM-Z Mountain and valley winds are better with LM-z Precipitation amplitudes should be better with LM-Z, in particular maxima and minima near mountains Expected advantages of the Z-coordinate

Ilder Heinz-Werner 3-D Cloud-Picture 18 January1998 The mountain related bias of convectional clouds and precipitation is supposed to disappear with the Z-coordinate

The step-orography i - 1/2 j - 1/2 j + 1/2 j - 1/2 i + 1/2 j + 1/2 i, j Shaved elements The shaved elements are mathematically more correct than step boundaries By shaved elements the z- coordinate is improved such that the criticism of Gallus and Klemp (2000), Mon. Wea. Rev. 128, no longer applies

The Atmosphere at Rest Computed with the Z_LM Day Night

Low stratus

LM-Z OBS LM-tf to 6

LM_Z LM_tf

LM_Z LM_tf

MSG IR 08.7um :00 LM total cloud cover lLM_Z total cloud cover LM_Z medium cloud cover % cc LM high cloud cover LM low cloud cover LM_Z high cloud cover LM_Z low cloud cover © METEOSAT LM medium cloud cover

12h Accumulated Precipitation for March to 18 UTC. The initial conditions are from the Global Model of DWD (00 UTC analysis, 24hr forecast). LMLM_Z

March March March R: 12h forecasted accumulated precipitation from “Regular” LM is closer to observation Z: 12h forecasted accumulated precipitation from LM_Z is closer to observation Precipitation Scores March March March R R R ZZZ..

OBS LM LM_ Z March March March mm Station # Observed and Forecasted Precipitation Heights (06-18 UTC) from LM_Z and LM. Columns below 0 th horizontal correspond to no observed or forecasted precipitation. Stations with no precipitation shown correspond to traces of observed but no forecasted precipitation Station # mm

RK is a two time level 3rd order in time scheme, involving substepping for fast waves Spacial order is 3 or 5 (upstream differencing) Approximation conditions concern vert. coordiante and phys. interface Semi-lagrange: 2nd order in time, 3rd order in space, could be easier to achieve efficiency with large dt The Runge Kutta scheme (NCAR) t+dt t

RK (40s) compared to RK (72s)

Test of the dynamical core: linear, hydrostatic mountain wave Gaussian hill Half width = 40 km Height = 10 m U0 = 10 m/s isothermal stratification dx=2 km dz=100 m T=30 h analytic solution: black lines simulation: colours + grey lines w in mm/s RK 3. order + upwind 5. order

Test of the dynamical core: density current (Straka et al., 1993) RK2 + upwind 3. order RK3 + upwind 5. order  ‘ after 900 s. (Reference) by Straka et al. (1993)

What is the influence of different time-splitting schemes Euler-forward Runge-Kutta 2. order Runge-Kutta 3. order (WS2002) and smoothing (4. order horizontal diffusion) ? K smooth dt/dx 4 = 0 / 0.05 fast processes (with operatorsplitting) sound (Crank-Nic.,  =0.6), divergence-damping (vertical implicit, C div =0.1) no buoyancy slow process: upwind 5. order aspect ratio: dx/dz=10 dT/dt=12

no yes smoothing Euler-forwardRunge-Kutta 2. orderRunge-Kutta 3. order

What is the influence of divergence filtering ? fast processes (operatorsplitting): sound (Crank-Nic.,  =0.6), divergence damping (vertical implicit) no buoyancy slow process: upwind 5. order time splitting RK 3. order (WS2002-Version) aspect ratio: dx/dz=10 dT/dt=6

How to handle the fast processes with buoyancy? with buoyancy (C buoy =  a dt = 0.15, standard atmosphere) different fast processes: 1.operatorsplitting: ‘Sound -> Div. -> Buoyancy‘ 2.partial adding of tendencies: ‘(Sound+Buoyancy) -> Div.') 3.adding of all fast tendencies: ‘Sound+Div.+Buoyancy‘ different Crank-Nicholson-weights for buoyancy:  =0.6 / 0.7 RK3-scheme slow process: upwind 5. order aspect ratio: dx/dz=10 dT/dt=6

‘Sound -> Div. -> Buoyancy‘‘(Sound+Buoyancy) -> Div.')‘Sound+Div.+Buoyancy'  =0.6  =0.7

Conclusions from stability analysis of the 2-timelevel splitting schemes KW-RK2 allows only smaller time steps with upwind 5. order  use RK3 Divergence filtering is needed (C div,x = 0.1: good choice) to stabilize purely horizontal waves bigger dx:dz seems not to be problematic for stability increasing dT/dt does not reduce stability buoyancy in fast processes: better addition of tendencies than operator splitting (operator splitting needs purely implicit scheme for the sound)

RK exists in LM and is considered to be suitable for operational use It offers approximation order 3 which should be sufficient for practical purposes The scheme is a modification (shortcut) of the WRF scheme. Too little work concerning a deeper evaluation has been done (such as the investigation of conservation properties, physics interface). Runge Kutta conclusions

Implicit integration is the basis for increasing efficiency by enlarging the timestep Hydrostatic models reach savings of about a factor of 2 Non-Hydrostatic so far did not realise a similar saving By working on the implicit solution procedures LM could reach an efficiency gain of a factor of 2 Implicit Time Integration

1-d Shallow Water Equs. Periodic boundaries with bell shaped initial disturbance

2-d Shallow Water Equs. with Barrier

A direct si- method was proposed The method is based on a generalised Fourier Transform The generalised FT is potentially as efficient as the FT (fast FT) 1-d and 2-d tests have been performed Implicit Conclusions

Significant work has been done concerning LM_Z (Greece, Poland) and RK (Germany) Development work was not done as planned: LM_Z, evaluation and further development of physics interface; RK: Evaluation and further development Advanced numerics seminar within the LM-User Seminar 6 to 8 March 2006 Strength and deficiencies of 2005 work

LM_Z: Improvement of physics interface; and further evaluation; High order LM_Z Implicit time integration: save computer time by a factor of 2 Further development of RK: conservation properties; further evaluation of advantages Suggested Priority Projects