Iceland Simulation for Secondary Gravity Waves
Previously, a coarse simulation having 1 km isotropic resolution
over Iceland was performed in support of the PMC measurements taken
on 10 July 2018. As discussed in detail at
Iceland Time-Varying, this simulation revealed moderately strong mountain
waves that were extinguished at a critical level located at an altitude
of about 22 km. Since the critical level is so low, it was reasoned
that the coarse resolution was insufficient to capture the wave breaking
that was likely to occur in this region. Absent secondary waves, the
1 km resolution simulation only showed weak transient waves at PMC
altitudes.Here we present results from much finer simulations that are able to capture wave breaking and attendant secondary wave generation from the low-altitude critical level. The secondary waves strengthen as they propagate to PMC altitudes, but their amplitudes do not quite reach levels sufficient to produce a secondary wave breaking zone. Refinement studies having a maximum resolution of 150 x 150 x 100 m in the low-altitude breaking zone did not show significant increases in the secondary wave amplitudes. Running the simulations up to an additional 8 hours beyond the end of the 4 hour data segment provided by Steve Eckermann did allow the secondary waves to achieve significantly higher amplitudes, but still high-altitude breaking was not observed. The extended time simulations were performed by holding the wind constant at the last time provided by Steve.
NAVGEM Winds
The plots below show the temporal variation for the NAVGEM wind profiles
on the interval 02:00-06:00 UTC near the location of Iceland's highest peak
(64.5° N 17° W).
NOTE: Simulation extended in time an additional 8 hours using fixed wind profile at last time instance (06:00 UTC).
Wind Forcing
Forcing terms gradually introduce winds near the surface with the objective of achieving the wind profile within a two 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 thirty minutes.
NOTE: A somewhat rapid wind ramp was used since the NAVGEM data only span a four hour period. This feature results in transient waves that do not need to obey the critical level. Thus high altitude waves are seen at early times but these waves propagate out of the domain as seen in previous simulation.
Full Computational Domain
The mesh is uniform at 300 m spacing in both lateral directions in the refined zone over Iceland (white square). The vertical mesh has a 100 m uniform region between altitudes of 10 to 25 km to further resolve the turbulent region. Very gentle stretching is used from the high-resolution central region to the boundaries with intent on achieving uniform 1 km spacing at balloon location ( < 1 % horizontally ). In total, the mesh contains 1370 x 1504 x 360 points. Sponge layers are used on all outward boundaries in order to absorb outgoing waves.
Visualization plane locations are also shown in the figure. The location of the PMC-Turbo measurements is in the upper right hand corner. x-y planes are located at 14, 50, 84 km. An additional plane at 90 km added at 4 hr, and provides data for the remainder of the simulation.
Results for u' in a x-z Plane at y = -58.7 km (passing
through the high terrain on Iceland)
Wave breaking in the stratosphere is seen to begin at about 2 hr. The high-altitude response at this time is due to transient waves due to the wind ramp. The leading edge of the secondary wave packet is visible overhead Iceland starting at about 2.5 hr. The secondary waves then strengthen and become more widespread over the remainder of the simulation but no wave breaking is observed at the upper altitudes.
x-y Results and RMS Analysis
Results for u' in a x-y Plane in the turbulent region 14 km
overlaid on a view of the terrain
Wave breaking occurs at multiple sites containing high terrain. The turbulence spreads more widely between 2 and 4 hr. A semi-turbulent wake arises thereafter and a large area of retarded flow (blue color) develops due to wave momentum deposition. Curiously, a patch of accelerated flow detaches from the turbulent zone and advects downstream beginning at about 10 hr.
This plot shows a time history of the velocity rms values on the x-y plane at 14 km. The rms values begin to increase rapidly at a time of about 1 hr (half way through the wind ramp). Although turbulence begins at about 2 hr., there is no visible evidence of this in the rms plots. The rms values continue to increase with time, but show a tendency to saturate by the end of the simulation at 12 hr.
RMS of x-y Plane at an Altitude of 50 km
Here we see the signature of the secondary waves starting slightly after 2 hr. in the w_rms data. Rms values continue to increase out to about 8 hr., at which time they mainly saturate.
Results for u' in a x-y Plane
at an Altitude of 84 km
The response prior to 2 hours is completely due to transient waves. Starting at about 2.5 hr., Secondary waves begin to appear. Initially these waves form a bulls eye pattern over Iceland, but later form a much more complex pattern. The waves fill much of the domain by a time of 5 hr., but continue to increase in strength for the entire duration of the simulation.
The transient waves reach peak amplitude around 3 hr. and then begin to diminish as the secondary waves arrive and strengthen. This behavior results in local minimal in u_rms and v_rms around a time of 3.5 hr. From this point on, the rms values increase for the remainder of the simulation with a slight tendency to saturate at the latest times.
Results for u' in a x-y Plane
at an Altitude of 90 km
The xy-plane at 90 km was added with the restart at 4 hr., and thus does not show the earlier times where transient waves dominate. At 4 hr. the secondary waves have already produced a bulls eye pattern. The waves spread, become more complex, and strengthen at time progresses. Peak u' in excess of 30 m/s are observed but still there is no wave breaking.
The rms values at 90 km are similar to those at 84 km, displaying a modest tendency to saturate by the end of the simulation.
Further Grid Refinement
A subsection of the domain only encompassing Iceland was
used to test the effect of further grid refinement on the strength of
resulting secondary waves. The horizontal grid spacings were decreased
by a factor of 2, to 150 m spacing. Due to the high computational
cost, the simulation was only run to 2.5 hr. Here we are most interested
in the effect of increased resolution on the strength of the secondary
waves, so we compare rms values with the 300 m resolution simulation
on a plane at an altitude of 20.5 km. Data is also compared on a plane
at 9 km to assess any differences in the strength of the primary gravity
waves.
RMS for x-y at 9 km
RMS for x-y at 20.5 km
Although there are some small differences in the rms values between the 300 and 150 m simulations, the values are quite comparable. Perhaps the only exception is in w_rms at z=20.5 km where the 150 m simulation shows small, but consistently higher values beyond a time of 2 hr., where the secondary waves are present. The increase is less than 5%, however, and this is probably not enough to cause large differences in the amplitudes of the secondary waves at higher altitudes.