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:

[1] Bonaventura 2000, JCP 158 and ref. therein

[2] Skamarock and Klemp 1994, MWR 122 and ref. therein

[3] Skamarock webpage on standard benchmarks for new mesoscale dynamic models

[...] ... 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
YVELO-bott.gif

  • Top view of the w-velocity and corresponding waves that generate downstream of the mountain
YVELO-top.gif

  • Bottom view of the u-velocity and corresponding waves that generate downstream of the mountain
XVELO-bott.gif
  • Top view of the u-velocity and corresponding waves that generate downstream of the mountain
Xvelo-top.gif

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


XVELOview1.gif







  • 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

N001002-3142m-10ksec0000.jpeg
Nbott = 0.01, N_top = 0.02


  • 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

N002001-3142m-10ksec0000.jpeg
Nbott = 0.02, Ntop = 0.01









  • 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

2agnesi-distanceLamda10_YVELO_900s0000.jpg
Z velo: contours and filled conoturs

2agnesi-distanceLamda10_THETA_900s0000.jpg
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

2agnesi-distanceLamda5_YVELO0000.jpg
Z velocity: contours and filled contours

2agnesi-distanceLamda5_YVELOandTHETA0000.jpg
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
2agnesi_300x75el_180s_THandYVELO_vectors_DETAIL0000.jpg
Velocity vectors, THETA (bold isolines) and YVELOCITY (full contours). The x-scale is scaled to 1% for plotting only



  • 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

doyleSmith2003_300x75el_6000s_THonly.jpg
THETA. 15 isolines: (300 (ground) - 384.6) K

doyleSmith2003_300x75el_6000s_bw.jpg
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

YVELO_SCHAR.jpg
Y-velo: contours from -1.88 m/s (blue) to 1.9 m/s (yellow)







  • 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


Vector field:
movie_450m_vectors_from_negative4to4ms.jpg0138.jpeg




Contour lines of the vertical velocity field: contours between -3.8 m/s and 3.5 m/s
BW-YVELO-h450m_scharVertical0000.jpeg
Yvelo NHNL Agnesi, H=450m -3.8 < V < 3.5






Contours of potential temperature:

tempe_isolines_ataCertainTime_450m0000.jpeg




  • 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

movie_900m_vectors.jpg0009.jpeg



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:


tempe_isolines_ataCertainTime0000.jpeg


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

||
agnesi_SIGMAvertical_Hm900_NHNL_10ks_100x20el-0.jpg
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

xvelo_skamarock_HSLIN.jpg
x-velo Domain=(-20:20)km x (0:20)km
----* Semi-analytic solution of the Linear mountain wave test case:dEta_Linear_analytic.jpg
dEtadx_Linear_analytic.jpg