Earlier studied suggested that secondary waves can be radiated from an
evolving packet prior to instability and turbulence. The packets in
these prior studies ultimately transitioned to turbulence, however, thus
allowing skeptics to claim that the secondary waves were due to turbulence
and the attendant wave momentum transfer. In order to unambiguously
demonstrate that secondary waves can be emitted from an evolving packet
in the absence of turbulence, we designed a simulation where the packet
becomes non-linear, but does not break. The key to doing this is to
tailor the wind profile so that refraction opposes the self-acceleration
effect.
The packet is generated by prescribing a time-varying mean wind blowing over an idealized mountain with Gaussian shape. While the simulation discussed here is two-dimensional, an analogous procedure could be used in 3D. The Gaussian mountain has a standard deviation of 10 km (FWHM=23.6 km) and a height of 1 km. The mean wind is initially damped to zero near the surface and then is forced with a Gaussian function in time. The Gaussian function in time results in a packet that is approximately shaped with a Gaussian function in the vertical. The standard deviation of the forcing function is 30 minutes (FWHM=71 mins). The wind profile is shown below.
The green curve depicts the initial and final wind profile, where the red curve shows the profile at a time of 1.75 hrs when the surface wind is at its maximum. Several different wind profiles were tested prior to selecting this one. One might expect that the upper level winds should not drop below zero since this is a critical level for steady-state mountain waves. The packet is a transient object, however, and the self-acceleration effect also works to increase the wind speed. Upper level wind speeds of -10 and -20 m/s reduced the tendency for self-acceleration instability but did not eliminate it. The chosen wind speed of -50 m/s may be higher than necessary, but does prohibit wave breaking.
The simulation results are shown in the video below. The blinking at early times is due to discrete changes in the color scale, which allow the entire evolution to be vizualized.
Acoustic noise is seen initially but the packet starts to emerge at the surface beginning at about 1 hr. The packet then grows and largely detaches from the surface at about 2.5 hrs. The packet strengthens as it propagates upward and transient changes to the mean wind are seen starting at about 3 hrs. The leading edge of the packet also starts to become visibly refracted at this time and weak secondary waves at the highest altitude emerge. As time goes on, the secondary waves strengthen whereas the packet weakens. Since the upper level winds are opposite the surface winds, the secondary waves propagate to the right.
The packet is generated by prescribing a time-varying mean wind blowing over an idealized mountain with Gaussian shape. While the simulation discussed here is two-dimensional, an analogous procedure could be used in 3D. The Gaussian mountain has a standard deviation of 10 km (FWHM=23.6 km) and a height of 1 km. The mean wind is initially damped to zero near the surface and then is forced with a Gaussian function in time. The Gaussian function in time results in a packet that is approximately shaped with a Gaussian function in the vertical. The standard deviation of the forcing function is 30 minutes (FWHM=71 mins). The wind profile is shown below.
The green curve depicts the initial and final wind profile, where the red curve shows the profile at a time of 1.75 hrs when the surface wind is at its maximum. Several different wind profiles were tested prior to selecting this one. One might expect that the upper level winds should not drop below zero since this is a critical level for steady-state mountain waves. The packet is a transient object, however, and the self-acceleration effect also works to increase the wind speed. Upper level wind speeds of -10 and -20 m/s reduced the tendency for self-acceleration instability but did not eliminate it. The chosen wind speed of -50 m/s may be higher than necessary, but does prohibit wave breaking.
The simulation results are shown in the video below. The blinking at early times is due to discrete changes in the color scale, which allow the entire evolution to be vizualized.
Acoustic noise is seen initially but the packet starts to emerge at the surface beginning at about 1 hr. The packet then grows and largely detaches from the surface at about 2.5 hrs. The packet strengthens as it propagates upward and transient changes to the mean wind are seen starting at about 3 hrs. The leading edge of the packet also starts to become visibly refracted at this time and weak secondary waves at the highest altitude emerge. As time goes on, the secondary waves strengthen whereas the packet weakens. Since the upper level winds are opposite the surface winds, the secondary waves propagate to the right.