Gravity Wave Beam by Wind Shear

With no wind and constant background stability, the solution is rather uninteresting. The beam shape is preserved with altitude and only its amplitude increases due to the decrease in density. Sponge layers are used at the right and upper boundaries in order to absorb the outward-propagating beam.

Although not used here, it is also possible to superimpose multiple beam solutions, as illustrated in the following images.

These cases are not as well resolved as in the case shown in Figure 1, and thus suffer from a bit of numerical dispersion that results in non-physical widening of the beam with increasing distance from the source.

The cases discussed above and those to be discussed below are all are initialized with the desired mean wind profile and no gravity wave components. The forcing terms then add wave energy at a fixed rate and this energy gradually propagates away from the source. The initial stages of this transient process are fairly dispersive since the abrupt start projects on to a full spectrum of wave frequencies. If the wind is constant (as in the above examples) the dispersion decreases with time as the non-conforming transient modes propagate out of the domain. Ultimately we are left with a non-dispersive, purely periodic solution with frequency equal to the forcing frequency. The transient process is shown in the following animation.

It is instructive to analyze the beam solution a bit more closely. Figure 3 shows a horizontal cut through the beam displayed in Figure 1 at an altitude of 10 km. By analyzing this plot one can deduce that the beam has a dominant horizontal wavelength of 800 m. What is not apparent is that a surprising collection of adjacent modes with rather disparate wavelengths are required to synthesize the beam. These seemingly unnecessary modes are required to cancel oscillations beyond the packet edges.

Density-compensated energy spectra at various altitudes from the simulation shown in Figure 1 are displayed in Figure 4. This plot shows that a fairly broad collection of modes are required to synthesize the packet. Measurements from this plot indicate that modes with wavelengths ranging from about 525 m to over 3000 m are required, giving a grand total of roughly 50 discrete Fourier modes for this case. Thus, although the packet has a central mode with wavelength close to 800 m, it is actually composed of a large collection of modes having rather different wavelengths. An understanding of this characteristic is essential for the proper interpretation of the refraction and reflection studies to be show next.

Here the term 1/(2*kH*)^{2} is negligible for even the longest
waves in the beam and thus may be neglected. Doing this and using the
requirement that *m*^{2} must be positive gives the following
condition on the mean wind

If this condition is not met, we expect the solution component at the
corresponding frequency ω and wavenumber *k* to reflect. The
beam solution is particularly simple since all solution components have
the same ground-based frequency. Thus the condition on reflection maps
to a single velocity for each wavenumber *k*. We also see that larger
values of *k* (shorter wavelengths) have more restrictive limits
on the allowable mean velocity. Finally, we see that the above condition
imposes a limit on the maximum allowable headwind. This is due to the
minus sign on the right hand side together with the fact that the beam
propagates in the direction of *k*.

Turning now to our particular beam simulation, we can use the above analysis to calculate the maximum headwind such that no modes in the packet are reflected. As noted earlier from Figure 4, the shortest wavelength in the packet is about 525 m. With the ratio τ_{w}/τ_{b} = 800/400 = 2, and
τ_{w}=800 we achieve the condition
*U*_{0} ≥ -0.66 m/s for no reflection. The results of a
simulation with a linear mean wind profile having its zero value at the
altitude of the source, and a minimum of -0.66 m/s at the top of the beam
(where the sponge near the upper boundary attenuates the solution) is
shown in Figure 5.

*Figure 5. Results from a linear mean wind profile having a minimum
value of -0.66 m/s at the top of the beam. *

As anticipated, no reflection occurs. This is evidenced by the positive phase slope for all positions. The packet disperses greatly under the refraction of all modes towards longer vertical wavelengths and shorter intrinsic frequencies.

If the headwind is steadily increased from -0.66 m/s then an increasing fraction of the modes contained in the packet will reflect. This effect is surprisingly sensitive to changes in the mean wind. For example, a headwind of only -1.1 m/s is sufficient to reflect the shorter wavelength half of the packet, and a headwind of -3.75 m/s should be sufficient to reflect all modes in the packet. An interesting intermediate case is a headwind of -2.5 m/s, which corresponds to two standard deviations of the of the distribution shown in Figure 4. This value is sufficient to reflect about 97.5% of the wave energy, leaving just a few modes at the longest wavelength end of the spectrum. Results from this case are shown in Figure 6.

*Figure 6. Results from a linear mean wind profile having a minimum
value of -2.50 m/s at the top of the beam. *

Here it is apparent that nearly the entire packet has been reflected as evidenced by the negative phase slopes on the left side of the figure. Careful measurements do show a small amount of upward-propagating wave energy at the longest wavelengths near the top of the domain. Due to symmetry, the downward-propagating, reflected modes refocus and pass exactly through the source region. Nearly all of the wave energy is then absorbed by the sponge layer used at the bottom of the domain. Although the reflected modes are readily identified by their negative phase slopes, a more striking visualization can be generated by looking at the momentum flux. This quantity changes sign upon reflection making the reflected modes unmistakable. Such a plot is shown in Figure 7.

*Figure 7. As in figure 6 but showing the momentum flux.*

This plot illustrates the reflection quite dramatically and also shows that negligible momentum flux propagates to higher altitudes. The momentum flux also has the advantage that is compensates for the wave amplitude increase with altitude due to decreasing density. Thus while u' increases with altitude due to density changes (as evidenced in Figure 1), the momentum flux will remain constant. This makes the momentum flux more useful here since the decrease in wave energy due to increasing reflection at the upper altitudes is partially masked in the u' field due to the density amplification effect.

It is interesting to note from either Figure 6 or Figure 7 that a headwind of -0.66 m/s occurs precisely at the apex of the void between the upward and downward propagating modes at an altitude of 11.4 km. It is also interesting to note from Figure 7 that the apparent extinction of the packet above an altitude of 24 km occurs at a position where 96% of the wave energy should have been reflected.## Headwind Jet

The second set of experiments consider headwind jets with maxima well above
the beam source. The first of these has a maximum headwind of -1.1 m/s,
located at an altitude of 20 km (see Figure 8). As deduced above, a headwind
of -1.1 m/s is sufficient to reflect half of the modes in the packet.

Turning now to our particular beam simulation, we can use the above analysis to calculate the maximum headwind such that no modes in the packet are reflected. As noted earlier from Figure 4, the shortest wavelength in the packet is about 525 m. With the ratio τ

As anticipated, no reflection occurs. This is evidenced by the positive phase slope for all positions. The packet disperses greatly under the refraction of all modes towards longer vertical wavelengths and shorter intrinsic frequencies.

If the headwind is steadily increased from -0.66 m/s then an increasing fraction of the modes contained in the packet will reflect. This effect is surprisingly sensitive to changes in the mean wind. For example, a headwind of only -1.1 m/s is sufficient to reflect the shorter wavelength half of the packet, and a headwind of -3.75 m/s should be sufficient to reflect all modes in the packet. An interesting intermediate case is a headwind of -2.5 m/s, which corresponds to two standard deviations of the of the distribution shown in Figure 4. This value is sufficient to reflect about 97.5% of the wave energy, leaving just a few modes at the longest wavelength end of the spectrum. Results from this case are shown in Figure 6.

Here it is apparent that nearly the entire packet has been reflected as evidenced by the negative phase slopes on the left side of the figure. Careful measurements do show a small amount of upward-propagating wave energy at the longest wavelengths near the top of the domain. Due to symmetry, the downward-propagating, reflected modes refocus and pass exactly through the source region. Nearly all of the wave energy is then absorbed by the sponge layer used at the bottom of the domain. Although the reflected modes are readily identified by their negative phase slopes, a more striking visualization can be generated by looking at the momentum flux. This quantity changes sign upon reflection making the reflected modes unmistakable. Such a plot is shown in Figure 7.

This plot illustrates the reflection quite dramatically and also shows that negligible momentum flux propagates to higher altitudes. The momentum flux also has the advantage that is compensates for the wave amplitude increase with altitude due to decreasing density. Thus while u' increases with altitude due to density changes (as evidenced in Figure 1), the momentum flux will remain constant. This makes the momentum flux more useful here since the decrease in wave energy due to increasing reflection at the upper altitudes is partially masked in the u' field due to the density amplification effect.

It is interesting to note from either Figure 6 or Figure 7 that a headwind of -0.66 m/s occurs precisely at the apex of the void between the upward and downward propagating modes at an altitude of 11.4 km. It is also interesting to note from Figure 7 that the apparent extinction of the packet above an altitude of 24 km occurs at a position where 96% of the wave energy should have been reflected.

Results for this case are shown in Figures 9 and 10.

As expected, a certain fraction of the beam is reflected and the remainder propagates to higher altitudes. This latter effect is better seen in Figure 9 since the density effect amplifies the velocity perturbations near the upper boundary. This case has several similarities with the linear wind profile results shown in Figures 6 and 7. The downward-propagating reflected modes again refocus to form an image of the upward-propagating beam. Symmetry is absent with the jet however, and thus the reflected beam does not pass through the source region. The apex of the void between the upward and downward propagating modes is again precisely at the altitude where the mean headwind reaches -0.66 m/s. The reflection is also seen to end at an altitude of 20 km, which corresponds to the headwind maximum.

Additional insight is gained by looking at the streamwise spectra taken at several different altitudes. Such a plot showing spectra at stations between 15 and 40 km is shown in Figure 11.

The spectrum at z=15 km is somewhat anomalous since it contains both upward and downward-propagating modes. However, it does illustrate that the shorter wavelengths (higher wavenumber components) are present at this altitude. The spectrum at z=20 km is interesting since it is taken at the position of maximum headwind. The shorter wavelength half of the beam is entirely absent as these modes have all been reflected. There is an energy pile-up near the shortest penetrating mode, which is mainly due to large increases in the vertical perturbation velocity as the vertical wavenumber approaches zero for the modes in this vicinity. This effect quickly subsides at higher altitudes and is not detectable by and altitude of 25 km. The spectra for 25-40 km collapse quite nicely indicating that the both energy and spectral content is preserved in the region where the headwind is subsiding.