1 Reformulation of the LM fast- waves equation part including a radiative upper boundary condition Almut Gassmann and Hans-Joachim Herzog (Meteorological Institute of Bonn University, DWD Potsdam)

2  Time integration is divided into 2 parts 1.Fast waves (gravity and sound waves) 2.Slow tendencies (including advection) Short review… n+1n* F(n*) n F(n)  Fast waves and slow tendencies  improper mode separation  improper combination in case of the Runge-Kutta- variants  Further numerical shortcomings  divergence damping  vertical implicit weights: symmetry: buoyancy term other implicit terms  lower boundary condition

3 2-time-level scheme (KW-RK2-short) Comments on the Murthy-Nanundiah-test (Baldauf 2004) The test relies on the equation whose stationary solution is known as  g . Baldauf claimed that the KW-RK2-short-scheme was not suitable since the splitting into fast forcing and slow relaxation does not yield the correct stationary solution. BUT: The splitting into slow forcing and fast relaxation does. What is our approch alike? -> slow forcing and fast relaxation Forcing: physical and advective processes, nonlinear ones Relaxation: wave processes, linear ones

4 2-dimensional linear analysis of the fast wave part Vertical advection of background pressure and temperature These terms are essential for wave propagation and energy consistency Which is the state to linearize around? LM basic state (current) or State at timestep „n“, slow mode background Brunt-Vaisala-Frequency for the isothermal atmosphere scale height variables are scaled to get rid of the density

5 Time scheme for fast waves horizontal explicit – vertical implicit divergence damping symmetric implicitness (treatment as in other implicit terms) vertical temperature advection Remark: Acoustic and gravity waves are not neatly separable!

6 Divergence damping Relative phase change Phase speeds of gravity waves are distorted. With divergence dampingWithout divergence damping

7 Symmetric implicitness Amplification factor unsymmetric symmetric

8 Vertical advection of temperature Relative phase change Phase speeds are incorrect. The impact in forecasts can hardly be estimated. Without T-advection (nonisothermal atmosphere) With T-advection

9 Conclusions from linear analysis No divergence damping! Symmetric implicit formulation! Vertical temperature advection belongs to fast waves as well as vertical pressure advection!  Further conclusion: state to linearize around is state at time step „n“ and not the LM base state! Further conclusion: state to linearize around is state at time step „n“ and not the LM base state!

10 Appropriate splitting with slow tendenciesfast waves in vertical advection for perturbation pressure or temperature

11 prescribe Neumann boundary conditions with access to which is also used to derive surface pressure „fast“ LBC prescribe w(ke1) via prescribe metrical term in momentum equation via (Almut Gassmann, COSMO Newsletter 4, 2004, ) Lower boundary condition fast waves „slow and fast“ LBC In that way we avoid any computational boundary condition. slow tendencies „slow“ LBC prescribe via out of fast waves

12 Gassmann, Meteorol Atmos Phys (2004):“An improved two-time-level split-explicit integration scheme for non-hydrostatic compressible models“ Crank-Nicolson-method is used for vertical advection. Runge-Kutta-method RK3/2 is used for horizontal advection only and should not be mixed up with the fast waves part. Splitting errors of mixed methods (Wicker-Skamarock-type) are larger. Splitting slow modes and fast waves Gain of efficiency: No mixing of slow-tendency computation with fast waves No mixing of vertical advection with Runge-Kutta-steps

Background profile

14 Mountain wave with RUBC w-field isothermal background and base state

15 Strong sensitvity of surface pressure at the lee side of the Alps, if different formulations of metric terms in the wind divergence are used Conservation form (not used in the default LM version), Direct control over in- and outflow across the edges Alternative representation (used in the default LM version) Divergence and metric terms p u u

16 Southerly flow over the Alps 12UTC, 3. April 2005, Analysis

17 Pressure problem at the lee side of the Alps – Reference LM

18 Pressure problem at the lee side of the Alps – 2tls ALM

19 Cross section: pressure problem

20 Significance?

21 Potential temperature

22 Potential temperature

23 Northerly wind over the Alps

24 Northerly wind over the Alps

25 Moisture profiles in Lindenberg with different LM-Versions (Thanks to Gerd Vogel) 7-day mean with oper. LM version and new version, dx=2.8 km

26 Numerical error in the first time step

28 Conclusions and plans Conclusions The presented split-explicit algorithm is fully consistent and proven by linear analysis. It needs no artificial assumptions and thus overcomes intuitive ad hoc methods. Divergence formulation in terrain following is a very crucial point. Plans Higher order advection and completetion for more prognostic variables Further realistic testing Comparison with Lindenberg data