Mountain+Waves

=**Preliminary diagrams and a few specifications on Mountain Wave simulations obtained with a Variational Multiscale Finite Element Scheme (Q1 elements, explicit in time) for the solution of the compressible Euler equations.** References on mountain waves:= = =

[...] ... to be added ...

 * =**3D mountain waves with a finite element solver:**=


 * Bottom view of the w-velocity and corresponding waves that generate downstream of the mountain


 * Top view of the w-velocity and corresponding waves that generate downstream of the mountain


 * Bottom view of the u-velocity and corresponding waves that generate downstream of the mountain
 * Top view of the u-velocity and corresponding waves that generate downstream of the mountain


 * u-velocity 3D simulation, non hydrostatic flow around a mountain. Very low resolution (51 x 10 elements)




 * **Patterns with mixed distributions in terms of THETA. N may vary with altitude in the following few images**


 * Case 1
 * 1) Uin = 20.0 m/s
 * 2) Theta0 = T0 = 300 K
 * 3) N_bottom = 0.01 s^-1
 * 4) N_top = 0.02 s^-1
 * 5) Z-interface = 3142 m
 * 6) tfinal = 3 h
 * 7) dx = 1.5 km
 * 8) dz = 200 m
 * 9) domain size: [-90 km; 90km] x [0, 12km]
 * 10) Mountain: height Hm = 600 m, 1/2 amplitude: a = 10000 m




 * Case 2:
 * 1) Uin = 20.0 m/s
 * 2) Theta0 = T0 = 300 K
 * 3) N_bottom = 0.02 s^-1
 * 4) N_top = 0.01 s^-1
 * 5) Z-interface = 3142 m
 * 6) tfinal = 3 h
 * 7) dx = 1.5 km
 * 8) dz = 200 m
 * 9) domain size: [-90 km; 90km] x [0, 12km]
 * 10) Mountain: height Hm = 600 m, 1/2 amplitude: a = 10000 m




 * **Slightly non-Hydrostatic and Non linear : 2 consecutive mountains (ref case from Mayr and Gohm, 2000) :**
 * Specifications:
 * 1) Uin = 20.0 m/s
 * 2) Theta0 = T0 = 300 K
 * 3) N = 0.013 s^-1
 * 4) tfinal = 9 h
 * 5) dx = 1 km
 * 6) dz = 200 m
 * 7) domain size: [0km; 300km] x [0, 15km]
 * 8) Mountain: height Hm = 450 m, 1/2 amplitude: a = 2000 m, peak distance = 10*a_c


 * [[image:2agnesi-distanceLamda10_YVELO_900s0000.jpg width="715" height="286" caption="Z velo: contours and filled conoturs"]] ||
 * [[image:2agnesi-distanceLamda10_THETA_900s0000.jpg width="715" height="286" caption="Theta: vertical displacement"]] ||

Note: - The vetical axis is scaled to 10% for plotting purposes - The horizontal axis is scaled to 50% for plotting purposes


 * **slightly non-Hydrostatic and Non linear : 2 consecutive mountains (ref case from Mayr and Gohm, 2000) :**
 * Specifications:
 * 1) Uin = 20.0 m/s
 * 2) Theta0 = T0 = 300 K
 * 3) N = 0.013 s^-1
 * 4) tfinal = 9 h
 * 5) dx = 1 km
 * 6) dz = 200 m
 * 7) domain size: [0km; 300km] x [0, 15km]
 * 8) Mountain: height Hm = 800 m, 1/2 amplitude: a = 2000 m, peak distance = 5*a_c


 * [[image:2agnesi-distanceLamda5_YVELO0000.jpg width="715" height="531" caption="Z velocity: contours and filled contours"]] ||
 * [[image:2agnesi-distanceLamda5_YVELOandTHETA0000.jpg width="697" height="516" caption="Theta: vertical displacement"]] ||


 * **slightly non-Hydrostatic and Non linear 2 consecutive mountains: detail of vector field **
 * Specifications:
 * 1) Uin = 20.0 m/s
 * 2) Theta0 = T0 = 300 K
 * 3) N = 0.013 s^-1
 * 4) tfinal = 180 s ( still running when uploading this image! )
 * 5) dx = 1 km
 * 6) dz = 100 m
 * 7) domain size: [0km; 300km] x [0, 15km]
 * 8) Mountains: height Hm = 800 m, 1/2 amplitude: a = 2000 m
 * 9) Distance between mountain peaks: 5 times the half amplitude


 * **Hydrostatic and Non linear (ref case from Doyle and Smith, 2008) :**
 * Specifications:
 * 1) Uin = 20.0 m/s
 * 2) Theta0 = T0 = 300 K
 * 3) N = 0.013 s^-1
 * 4) tfinal = 9 h
 * 5) dx = 1 km
 * 6) dz = 200 m
 * 7) domain size: [0km; 300km] x [0, 15km]
 * 8) Mountain: height Hm = 800 m, 1/2 amplitude: a = 10000 m


 * [[image:doyleSmith2003_300x75el_6000s_THonly.jpg width="697" height="516" caption="THETA. 15 isolines: (300 (ground) - 384.6) K"]] ||
 * [[image:doyleSmith2003_300x75el_6000s_bw.jpg width="715" height="531" caption="Vertical velocity (filled contour) and potential temperatur (solid lines)"]] ||




 * **__Complex topography (Schar et al) (coarse mesh)__**
 * Specifications:
 * 1) Uin = 10 m/s
 * 2) Theta0 = T0 = 288 K
 * 3) N = 0.01
 * 4) tfinal = 2.5h (9000 s)
 * 5) dx = 400 m
 * 6) dz = 300 m
 * 7) domain size: [0km; 40km] x [0, 20km]
 * 8) Mountain: height Hm = 250m, 1/2 amplitude: a = 5000 m, lambda = 4000 m




 * **__Nonhydrostatic, Nonlinear, without breaking wake__**
 * Specifications:
 * 1) Uin = 13.28 m/s
 * 2) Theta0 = T0 = 273 K
 * 3) N = 0.02 --> f = 0.013 s^-1
 * 4) tfinal = 2.5h (9000 s)
 * 5) dx = 400 m
 * 6) dz = 200 m
 * 7) domain size: [0km; 40km] x [0, 20km]
 * 8) Mountain: height Hm = 450m, 1/2 amplitude: a = 1000 m
 * 1) domain size: [0km; 40km] x [0, 20km]
 * 2) Mountain: height Hm = 450m, 1/2 amplitude: a = 1000 m


 * Vector field:**


 * Contour lines of the vertical velocity field: contours between -3.8 m/s and 3.5 m/s**
 * [[image:BW-YVELO-h450m_scharVertical0000.jpeg width="818" height="569" caption="Yvelo NHNL Agnesi, H=450m -3.8 < V < 3.5"]] ||


 * Contours of potential temperature:**




 * **__Nonhydrostatic, Nonlinear, with breaking wake downstream of a 900m mountain:__**
 * Specifications:
 * 1) Uin = 13.28 m/s
 * 2) Theta0 = T0 = 273 K
 * 3) N = 0.02 --> f = 0.013 s^-1
 * 4) tfinal = 3 h (10800 s)
 * 5) dx = 200 m
 * 6) dz = 100 m
 * 7) domain size: [0km; 40km] x [0, 20km]
 * 8) Mountain: height Hm = 900m, 1/2 amplitude: a = 1000 m




 * Contour lines of vertical velocity: contours between -13 m/s and +10 m/s. Note the wake at low altitudes downstream of the mountain. There are clear oscillation on top of the mountain that may be due to the lack of absorbing layer on the top boundary:**




 * ** NEW ** Same as above but with coarser mesh and using a Sigma vertical coordinate:


 * [[image:agnesi_SIGMAvertical_Hm900_NHNL_10ks_100x20el-0.jpg caption="Agnesi with SIGMA vertical. NHNL, 100x20 elements, Hm=900m"]] ||
 * [[image:agnesi_SIGMAvertical_Hm900_NHNL_10ks_100x20el-0.jpg caption="Agnesi with SIGMA vertical. NHNL, 100x20 elements, Hm=900m"]] ||


 * **Hydrostatic and Linear :**
 * Specifications:
 * 1) Uin = 20.0 m/s
 * 2) Theta0 = T0 = 273 K
 * 3) N = 0.0179 --> f = 0.002 s^-1
 * 4) tfinal = 12 h (43200 s)
 * 5) dx = 2 km
 * 6) dz = 250 m
 * 7) domain size: [0km; 80km] x [0, 30km]
 * 8) Mountain: height Hm = 1 m, 1/2 amplitude: a = 10000 m

* **Semi-analytic solution of the Linear mountain wave test case:**