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

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

Uin = 20.0 m/s

Theta0 = T0 = 300 K

N_bottom = 0.01 s^-1

N_top = 0.02 s^-1

Z-interface = 3142 m

tfinal = 3 h

dx = 1.5 km

dz = 200 m

domain size: [-90 km; 90km] x [0, 12km]

Mountain: height Hm = 600 m, 1/2 amplitude: a = 10000 m

Nbott = 0.01, N_top = 0.02

Case 2:

Uin = 20.0 m/s

Theta0 = T0 = 300 K

N_bottom = 0.02 s^-1

N_top = 0.01 s^-1

Z-interface = 3142 m

tfinal = 3 h

dx = 1.5 km

dz = 200 m

domain size: [-90 km; 90km] x [0, 12km]

Mountain: height Hm = 600 m, 1/2 amplitude: a = 10000 m

Nbott = 0.02, Ntop = 0.01

Slightly non-Hydrostatic and Non linear: 2 consecutive mountains (ref case from Mayr and Gohm, 2000) :

Specifications:

Uin = 20.0 m/s

Theta0 = T0 = 300 K

N = 0.013 s^-1

tfinal = 9 h

dx = 1 km

dz = 200 m

domain size: [0km; 300km] x [0, 15km]

Mountain: height Hm = 450 m, 1/2 amplitude: a = 2000 m, peak distance = 10*a_c

Z velo: contours and filled conoturs

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:

Uin = 20.0 m/s

Theta0 = T0 = 300 K

N = 0.013 s^-1

tfinal = 9 h

dx = 1 km

dz = 200 m

domain size: [0km; 300km] x [0, 15km]

Mountain: height Hm = 800 m, 1/2 amplitude: a = 2000 m, peak distance = 5*a_c

Z velocity: contours and filled contours

Theta: vertical displacement

slightly non-Hydrostatic and Non linear 2 consecutive mountains: detail of vector field

Specifications:

Uin = 20.0 m/s

Theta0 = T0 = 300 K

N = 0.013 s^-1

tfinal = 180 s (still running when uploading this image!)

dx = 1 km

dz = 100 m

domain size: [0km; 300km] x [0, 15km]

Mountains: height Hm = 800 m, 1/2 amplitude: a = 2000 m

Distance between mountain peaks: 5 times the half amplitude

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:

Uin = 20.0 m/s

Theta0 = T0 = 300 K

N = 0.013 s^-1

tfinal = 9 h

dx = 1 km

dz = 200 m

domain size: [0km; 300km] x [0, 15km]

Mountain: height Hm = 800 m, 1/2 amplitude: a = 10000 m

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

Vertical velocity (filled contour) and potential temperatur (solid lines)

Complex topography (Schar et al) (coarse mesh)

Specifications:

Uin = 10 m/s

Theta0 = T0 = 288 K

N = 0.01

tfinal = 2.5h (9000 s)

dx = 400 m

dz = 300 m

domain size: [0km; 40km] x [0, 20km]

Mountain: height Hm = 250m, 1/2 amplitude: a = 5000 m, lambda = 4000 m

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

Nonhydrostatic, Nonlinear, without breaking wake

Specifications:

Uin = 13.28 m/s

Theta0 = T0 = 273 K

N = 0.02 --> f = 0.013 s^-1

tfinal = 2.5h (9000 s)

dx = 400 m

dz = 200 m

domain size: [0km; 40km] x [0, 20km]

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

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:

Uin = 13.28 m/s

Theta0 = T0 = 273 K

N = 0.02 --> f = 0.013 s^-1

tfinal = 3 h (10800 s)

dx = 200 m

dz = 100 m

domain size: [0km; 40km] x [0, 20km]

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:

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Agnesi with SIGMA vertical. NHNL, 100x20 elements, Hm=900m

Hydrostatic and Linear :

Specifications:

Uin = 20.0 m/s

Theta0 = T0 = 273 K

N = 0.0179 --> f = 0.002 s^-1

tfinal = 12 h (43200 s)

dx = 2 km

dz = 250 m

domain size: [0km; 80km] x [0, 30km]

Mountain: height Hm = 1 m, 1/2 amplitude: a = 10000 m

x-velo Domain=(-20:20)km x (0:20)km

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

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:Patterns with mixed distributions in terms of THETA. N may vary with altitude in the following few imagesSlightly non-Hydrostatic and Non linear: 2 consecutive mountains (ref case from Mayr and Gohm, 2000) :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) :slightly non-Hydrostatic and Non linear 2 consecutive mountains: detail of vector fieldHydrostatic and Non linear (ref case from Doyle and Smith, 2008) :Complex topography (Schar et al) (coarse mesh)Nonhydrostatic, Nonlinear, without breaking wakeVector field:Contour lines of the vertical velocity field: contours between -3.8 m/s and 3.5 m/sContours of potential temperature:Nonhydrostatic, Nonlinear, with breaking wake downstream of a 900m mountain: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:

||NEWSame as above but with coarser mesh and using a Sigma vertical coordinate:||

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