Test Case 2: Mountain-triggered meso-scale flow phenomena

Description

This test case investigates the generation of meso-scale dynamics in the lee of a mountain. Two variants are presented, which differ only in the shape of the topography: 

• Case 2a models a mountain chain with a gap. This leads to an acceleration of the zonal wind through the gap and the generation of lee vortices around the topography.

• Case 2b contains a symmetrical mountain offset from the equator. This induces vortex shedding due to a difference in velocities between the North and South faces.

Both tests use a simple setup of an isothermal atmosphere, constant static stability (Brunt–Väisälä frequency) and a zonal wind of u = u0 cos(lat), which is a solid body rotation. To be able to model more realistic topographies without resoting to high horizontal resolution, a small Earth is used (Klemp et al. 2015), which sees the radius reduced by a factor of X=20. For consistency, the Coriolis force is also increased by a factor of X. Smaller time steps are required for a smaller Earth, but as the dynamics evolves faster, the number of timesteps is the same as in an equivalent full Earth simulation.

A key nondimensional number in this test case is the Froude number, or specifically its inverse,

Fr_inv = N h0 / U,

N is the Brunt-Vaisalla frequency, h0 is the mountain height, U is a characteristic flow velocity (u0 at the equator).

• Fr_inv < 1 (or Fr >> 1) is the 'flowover regime', where the dominant dynamics is linear with gravity waves propagating upwards from the mountain.

• Fr_inv > 1 (or Fr << 1) is the 'flowaround regime', where the flow primarily moves around the mountain and leads to nonlinear dynamics in the lee of the topography.

Both the gap and vortex cases are in the nonlinear flowaround regime, which leads to the interesting downstream dynamics. Varying the mountain height and flow velocity will modify the strength of this nonlinear response.

These tests are also run with and without the Coriolis force (scaled for the small Earth), which leads to interesting distinctions in the resulting dynamics.

Case 2a: Gap flow

Description

Gaps in high topographical features, such as mountain chains, can lead to a strong acceleration of winds through the gap. There are many examples of gap flow around the world, including:

• Chivela pass in Mexico, with a gap of approximately 100 km, which leads to strong winds in the Gulf of Tehuantepec.

• Juan de Fuca strait between the United States and Mexico, approximately 18-27 km wide.

• Shelikov Strait in Alaska, 40-48 km wide.

• Cook Strait in New Zealand, which is 20 km wide at the narrowest point.

To simulate a gap in a mountain chain, a topography of the following form is constructed:

The parameters of e1, e2, e3 govern the shape of the mountain and d1, d2, d3 the size of the mountain chain and gap. The d parameters are computed from length scales of x_lat, x_lon, x_gap. The default for x_gap is 50 km on the small Earth. 

Modifications: All dynamical cores

1. Change the wind speed, u0.

2. Change the isothermal temperature T0 (changes the Brunt-Vaisala frequency N). 

3. Note that both u0 and T0 & N variations impact the Froude number Fr = u0/(N h). Changing the Fr number changes the character of the flow (e.g. you might lose the vortex shedding signatures, try it out).

4. Increase the horizontal resolution from ~1 degree/5.5 km on the small Earth (ne30, C96, mpasa120) to ~0.5 degrees/2.75 km on the small Earth (ne60, C192, mpasa60).

Modifications: Dycore specific

SE and FV3 (not MPAS)

1. Change the height of the topography, h0.

2. The response through the gap varies on the geometry of the gap; this can be classified as a long gap (x_gap << x_lon) or a short gap (x_gap >= x_lon) (Stull 2015). The default setup has xgap = 50 km, x_lon = 40 km. Try making the gap shorter, with xgap=100km and x_lon = 40 km, or making this a long gap, with xgap=50km and x_lon=100 km.

3. Move the gap at a different latitude, i.e. not at the equator. 

4. Include the Coriolis force: What changes are seen with and without the Coriolis force?

5. Change the vertical resolution; Other grids are stored in ~/glade/u/home/timand/vertical_grids. Currently, a stretched grid is used, starting with a spacing of dz = 300 m . Instead, try cam_vcoords_horiz_mount_flow_stretch_dzlow150m_top30km_L48.nc with a minimum spacing of dz=150 m, or cam_vcoords_horiz_mount_flow_stretch_dzlow100m_top30km_L57.nc has a minimum spacing of dz=100 m.

6. Change the characteristics of the Rayleigh friction sponge layer.

SE

See the SE namelist options here.

1. Change the timestepping method

FV3

See the FV3 namelist options here.

1. Modify the choice of horizontal transport scheme, using the hord namelist parameters.

2. Vorticity damping is required to keep some of these simulations stable. What is the impact of changing the strength of vorticity damping from vtdm4 = 0.05?

3. Modify the diffusion settings, such as the order of divergence damping, nord.

4. Change to the equiangular grid (as used by SE) instead of the equi-edge grid. With nord=2, this requires a smaller coefficient for divergence damping, of d4_bg < 0.148.

MPAS

See the MPAS namelist options here.

1. Modify the diffusion settings

Case 2b: Vortex shedding

This test case is motivated by von Kármán vortex streets that can be observed from the atmosphere, such as those behind the Madeira Island, e.g. Gao et al. (2023), and the Canary Islands. To replicate this behaviour in an idealised test, we define a Gaussian mountain and set this at a nonzero latitude. The shedding is instigated by an imbalance of the flow between the two sides of the mountain. In this case, the flow velocity is different past the Northern and Southern slopes, which is sufficient to generate a vortex street. This does not occur when the mountain is placed at the equator.

An interesting nondimensional parameter for vortex shedding is the Strouhal number,

St = f_S L / U

with f_S the frequency of the vortex shedding, and L and U are characteristic length and velocity scales. Assuming St remains constant, f_S will change as a function of the mountain width and flow velocity.

Modifications: All dynamical cores

1. Change the wind speed, u0.

2. Increase the horizontal resolution from ~1 degree/5.5 km on the small Earth (ne30, C96, mpasa120) to ~0.5 degrees/2.75 km on the small Earth.

3. Change the vertical resolution.

4. (Expert) This test is sensitive to reflections from the model top. To mitigate this, a Rayleigh friction layer is added, which is 15 km thick and has a timescale of 0.1 days. How does modifying these choices affect the flow at a much lower altitude?

Modifications: Dycore specific

SE and FV3 (not MPAS)

1. Change the height of the topography, h0

2. Increase the width of the topography through the half-width, d. This changes the height/diameter aspect ratio which is a decisive parameter for this test case (see lecture 4 from Monday)

3. Change to an oval mountain shape

4. Include the Coriolis force: how does this change the dynamics?

5. Change the vertical resolution; Other grids are stored in ~/glade/u/home/timand/vertical_grids. Currently, a stretched grid is used, starting with a spacing of dz = 300 m . Instead, try cam_vcoords_horiz_mount_flow_stretch_dzlow150m_top30km_L48.nc with a minimum spacing of dz=150 m, or cam_vcoords_horiz_mount_flow_stretch_dzlow100m_top30km_L57.nc has a minimum spacing of dz=100 m.

6. Move the mountain to a different latitude; this will modify the difference between velocities between the Northern and Southern flow around the mountain.

   1. In particular, try moving the mountain center to the equator and see if the model maintains symmetry

SE

1. Change the timestepping method

FV3

1. Modify the horizontal transport scheme, hord_xx.

2. Vorticity damping is required to keep some of these simulations stable. What is the impact of changing the strength of vorticity damping from vtdm4 = 0.05?

3. Modify the diffusion settings, such as the order of divergence damping, nord.

4. Change to the equiangular grid (as used by SE) instead of the equi-edge grid. With nord=2, this requires a smaller coefficient for divergence damping, of d4_bg < 0.148.

5. (Expert) Derive and apply a non-zero initial condition for the nonhydrostatic vertical velocity, using appendix D of Kent et al. (2014). What difference does this make?

MPAS:

1. Vary the diffusion settings.

2. (Expert) Derive and apply a non-zero initial condition for the nonhydrostatic vertical velocity, using appendix D of Kent et al. (2014). What difference does this make?

References:

Gao, Q., Zeman, C., Vergara-Temprado, J., Lima, D. C., Molnar, P., & Sch¨ar, C. (2023). Vortex streets to the lee of Madeira in a kilometre-resolution regional climate model. Weather and Climate Dynamics, 4 (1), 189–211.

Kent, J., Ullrich, P. A., & Jablonowski, C. (2014). Dynamical core model intercomparison project: Tracer transport test cases. Quarterly Journal of the Royal Meteorological Society, 140 (681), 1279–1293

Stull, R. B. (2015). Practical meteorology: An algebra-based survey of atmospheric science. University of British Columbia.