Andes Gaussian Mountain Approximation Simulations

This page describes the results of a resolution study for the numerical simulation of gravity waves over the Andes Southern Peak, Cerro Roma, Argentina. The simulations are 2D and a Gaussian approximation was used to simplify the terrain.

Terrain Approximation Compared to Terrain Data

The two images below show the real terrain and a Gaussian approximation to it. Both grids use 500x500 m uniform resolution.

Actual terrain

Gaussian approximation

Several different uniform horizontal mesh spacings were used for analysis. Namely, 250m, 500m, and 1000m. In the regions far ahead and behind the peak the mesh is stretched at ~1.6% in the x direction. The domain extends to an altitude of 144 km and uses uniform vertical spacing equal to what is used in the x-direction.

Wind and Thermodynamic Profiles

The mean winds and temperature profile are taken from one of Han Li's runs for wintertime at 50 S. These profiles extend to only an altitude slightly less than 140 km. The winds were extended above this altitude by smoothing to constant values. Note that the meridional winds are zero for the following 2-D simulations as the computational domain is not rotated with respect to lines of constant latitude.

Wind Condition

The mean winds near the surface are increased in time. Forcing terms gradually introduce winds near the surface with the objective of achieving the wind profile within a twelve hour period. A hyperbolic tangent function is used in order to produce gentle acceleration of the wind near the beginning and end of the forcing period. The maximum forcing rate is equivalent to that of a linear ramp with a duration of four hours.

Results

The panels below show u' at 7.0 hr for the three different mesh spacings.

250m

The panel below shows the result from a 3-D run with 500 m grid spacing and using the real terrain. It compares quite favorably with the 500 m 2-D Gaussian terrain result shown above. The good agreement justifies the use of Gaussian terrain and also suggests that the actual wavefield is reaonably 2-D in the plane of the peak (since the peak is part of an extended north-south running massif).

Full 3-D simulation plane. (RerunAndes07/old_code/run01)

Analysis

Moving back up to the 2D results, there are several important differences related to the mesh resolution. While the 250 and 500 m results are quite similar below the near critical level at an altitude of 83 km, they differ noticeably above this level. The 250 m resolution case shows a larger increase in wave strength near the critical level and much more pronounced wave activity above. These effects are likely due to the differences in the vertical resolution, which is also 250 and 500 m. The 250 m resolution case is better able to resolve the compressed vertical GW scales that arise due to the diminishing zonal wind in this altitude range. The larger wave amplitudes in this altitude range then produce stronger transmitted waves above.
Unlike the other two cases, the 1000 m resolution case shows much more pronounced differences at lower altitudes. Both the horizontal and vertical wavelengths are increased and the waves are weaker near the surface. The behavior changes, however, above the first reflection zone (between about 40 and 65 km where the wave fronts are nearly vertical). Since the horizontal wavelength is overpredicted in the 1000 m resolution case, there is less tendency for the waves to reflect. With less reflection there is a marked increase in wave strength above 65 km in the 1000 m resolution case. The same thing happens in the second reflection zone between about 90 and 105 km. The 1000 m resolution case shows strong waves above the second reflection zone whereas the 500 m resolution case shows practically none.
The foregoing analysis can be made more quantitative by looking at the streamwise velocity spectra. The following plot shows the x-spectra of u' at an altitude of 20 km, which is below the first filtering region.

There is a clear resolution dependence in the spectral content at this altitude. As deduced from the color contour plots above, coarsening the mesh results in a shift of the spectral peak to lower wavenumbers (longer wavelengths). In addition, the the wave strength is reduced as the mesh is made progressively more coarse. The shift to longer horizontal wavelengths as the grid is coarsened makes the resultant wave fields more resiliant to reflection as the winds increase above and altitude of 30 km. The following plot, displaying spectra computed at an altitude of 50 km show that the situation has now reversed with the 1000m case having the most energy. The energy peak for this resolution is still at a much lower wavenumber than for the higher resolution cases.

The trend continues at higher altitudes and the plot below for an altitude of 80 km shows that the 1000 m case incorrectly has more than an order of magnitude more wave energy than the finer two cases.

One good way to spot reflected waves is to image the momentum flux. Waves with energy propagating up and to the left will have negative momentum flux, but the sign will switch when the waves reflect and propagate downward. The panel below shows the momentum flux for the 250 m resolution case. The upward-propagating waves are shown in blue whereas the downward-propagating reflected waves are shown in red. The reflected waves interfere with the upward-propagating waves at low altitudes but are clearly visible. There also some red waves above the turbulent zone, but these are most likely right-running upward-propagating waves, which also have positive momentum flux.

Further Exploration

Another unpredicted effect observed was an apparent time shift or wave propagation speed for varying mesh resolution. Further exploration is required to draw significant conclusions, however initial results indicate coursing the mesh resolution lends to waves propagating through the filtering region at earlier times. In addition, finer resolutions display some tendency for waves to travel downstream of the mountain peak faster.