Jean-Pierre Pinty (LA) Starting paper (to read) Klemp, Skamarock and Fuhrer, JAS, 2003

1. Consistency in the discretization of metric terms in MésoNH when increasing the order of the transport schemes Jean-Pierre Pinty (LA) Starting paper (to read) Klemp, Skamarock and Fuhrer, JAS, 2003 Final report SOLSTICE, December 2010

2. Problem: Transport of the buoyancy (variable “Θ” in MésoNH) includes the transport of the mean value “Θ”. The transport is computed in generalized coordinatesZ=H ×(z-zsurf)/H-zsurf), with advective velocities corresponding to the so called contravariant components of the windspeed vector (u,v and ω). The atmosphere is adiabatic:with on Z surfacesand so must cancel with Θ~Θ Θ~Θ ADVection term «CONTRAV» Disc. Metric Term The discretization order of ADV-Θ must coïncide with the one of DMT Final report SOLSTICE, December 2010

3. Test case made with the orography profile of Schär et al (2002) taken from Klemp et al., JAS, 2003 ”Schär"orography h(x)=h exp(-(x/a)2) cos2(πx/λ) Dx=500m,Dy=300m,10s U=10ms-1,N2=10-4s-2 ADV2 DMT2 ADV2 DMT2 ADV4 DMT4 ADV4 DMT4 Boussineq analytical sol. WRFnumerical sol. ADVn: nth ord. advection scheme DMTn: nth ord. discrete metric term WRF simulates the inconsistency when orders of ADV and DMT differ ADV4 DMT2 ADV4 DMT2 Final report SOLSTICE, December 2010

4. Test case made with space (and time) centered transport schemes of MesoNH (CENxxx + numerical diffusion) ADV2 DMT2 ADV2 DMT2 CEN2ND ADV4 DMT4 ADV4 DMT4 MesoNH numerical sol. WRFnumerical sol. ADVn: nth ord. advection scheme DMTn: nth ord. discrete metric term MésoNH retrieves the inconsistency between ADV and DMT schemes CEN4TH ADV4 DMT2 ADV4 DMT2 CEN4TH Final report SOLSTICE, December 2010

5. Test case made with the orography profile of Schär et al (2002) • What to do with the PPM scheme ? Unkown order of PPM to compensate the term in the Discrete Metric Term of «CONTRAV» ! • Is the WENO scheme sensitive to the problem ? PPM+WENO are expected to replace the « time & space » centered schemes in a future version of MesoNH ! • Are the SLEVE coordinates providing a better solution ? The purpose of the SLEVE vertical coordinate is to attenuate perturbations induced by the « terrain-following » coordinate aloft, so the SLEVE coordinate should minimize the « Metric Term » effect ! Final report SOLSTICE, December 2010

6. Test case made with upstream scheme (PPM)Wind is transported with CEN4TH ADV2 DMT2 space-centered  upstream advection scheme (PPM): MésoNH still captures the inconsistency due to ADV_ Θ with PPM and any DMT scheme No clean solution can be obtained ! ADV? DMT4 ADV4 DMT4 MesoNH numerical sol. WRFnumerical sol. ADVn: nth ord. advection scheme DMTn: nth ord. discrete metric term PPM CEN4TH ADV? DMT2 ADV4 DMT2 PPM CEN4TH Final report SOLSTICE, December 2010

7. Test case made with upstream scheme (PPM)Wind is transported with WENOx (instead of CEN4TH) CEN4TH (wind)  WENOx schemes: MésoNH still captures the inconsistency due to ADV_ Θ with PPM and the DMT scheme Solution with WENO3 or WENO5 is just stabilized ADV? DMT4 ADV2 DMT2 PPM WENO3 ADV? DMT4 ADV4 DMT4 MesoNH numerical sol. WRFnumerical sol. ADVn: nth ord. advection scheme DMTn: nth ord. discrete metric term PPM WENO5 ADV? DMT2 ADV4 DMT2 PPM Final report SOLSTICE, December 2010

8. Test case made with upstream scheme (PPM)…same as before but with SLEVE coordinates of Schär Using SLEVE coordinates and upstream advection schemes: MésoNH still captures the inconsistency between ADV_ Θwith PPM and DMT scheme Solution with WENO3 or WENO5 is nearly PERFECT ADV? DMT4 ADV2 DMT2 PPM WENO3 SLEVE ADV? DMT4 ADV4 DMT4 MesoNH numerical sol. WRFnumerical sol. ADVn: nth ord. advection scheme DMTn: nth ord. discrete metric term Still an extra wave pattern here PPM WENO5 SLEVE ADV? DMT2 ADV4 DMT2 PPM WENOx SLEVE Final report SOLSTICE, December 2010

9. Conclusion • Performing the « Schär» test case is mandatory to assess the full compatibility of high level transport schemes with a terrain-following coordinate over orography (Klemp et al., 2003) • Revision of the discrete form of the metric terms in «CONTRAV» is necessary when using high order (CEN4TH) or highly performant non-linear (PPM) transport schemes for Θ • The exact form of the discrete metric terms in «CONTRAV» has been derived for finite difference schemes only (so results are not formally applicable to the PPM scheme) • However associating WENO (wind components) and PPM (Θ) schemes remains a viable solution when the SLEVE vertical coordinate suggested by C.Schär is employed additionally Final report SOLSTICE, December 2010

10. Test case made with upstream scheme (PPM)…same as before but with SLEVE coordinates of Schär Using SLEVE coordinates and upstream advection schemes: MésoNH still captures the inconsistency between ADV_ Θwith PPM and DMT scheme Solution with WENO3 or WENO5 is nearly PERFECT ADV? DMT4 ADV2 DMT2 PPM WENO3 SLEVE ADV? DMT4 ADV4 DMT4 MesoNH numerical sol. WRFnumerical sol. ADVn: nth ord. advection scheme DMTn: nth ord. discrete metric term Still noise here PPM WENO5 SLEVE ADV4 DMT2 PPM CEN4TH SLEVE Final report SOLSTICE, December 2010