This page provides several images and animations of gravity wave instability
structures for a ωi = N/√10 wave at non-dimensional
amplitude 0.9 in a tilted box with dimensions 3.225 x 2.106 x 0.949
λz . This is one of the cases discussed in Fritts et al.
2009(a),(b) (parts 1 and 2). In the part 2 paper, vortex rings were suggested
as the primary instability structure. Recent simulations of the same
configuration using the SAM code appear to show "hairpins" as the dominant
instability structure so additional analysis was performed.
Ling was able to provide the solution files for both the initial condition and for t~22 τb , the time visualized in the part 2 paper that appears to show vortex rings. A careful analysis of the initial condition indicates that the initial temperature white noise rms amplitude is 4.24e-3 (λz dT/dz), which gives it a relative rms amplitude of 4.188e-2 times the initial GW temperature rms. This is at huge odds with the relative rms amplitude of 1.0e-6 quoted in the part 1 paper. For what it is worth, the input file for the triple run does have a temperature noise level specification of 1.0e-6, but there must either be a misunderstanding of how this input is used, or a bug in the triple code. One possibility is that the triple input value is intended to be a variance value instead of rms. In any event, the character of the initial instability structures appears to be sensitive to the initial noise level when it is large and this is responsible for much of the disagreement between Ling's case and the recent SAM runs.
Beyond the sensitivity to the initial noise level, careful visualizations of Ling's results show that the primary instability structure is not quite a vortex ring but rather a "horseshoe" vortex interacting with the legs of a "hairpin" vortex. This same configuration appears in the lower initial noise amplitude cases, but one or more horseshoe vortices are spaced further from the head of the hairpin, giving less of an appearance of a vortex ring. Shown below is an image of λ2 formed from Ling's data file at a time of t~22 τb . The field has been shifted by (-0.4, 0.35, -0.3)λz in the (x, y, z) directions, respectively in order to minimize severing of the instability structures by the periodic boundaries.
In this top-down view, an apparent vortex ring is visible to the lower left. This structure is encased in a small rectangular box, which depict the subvolume that will be used below to scrutinize this structure. The subvolume has extent (-0.9:0.7, -1.4:-0.55, 0.4:1.1) λz in the (x, y, z) directions, respectively.
In addition to the apparent vortex ring structure, there is a prominent hairpin vortex near the top-center of the image and a second less obvious one at the lower right. The remaining structure at the top left appears to be another vortex ring.
As the image below indicates, a somewhat different impression is formed when the instability structure subvolume is viewed from a perspective angle.
In this view it is somewhat apparent that we are looking at a relatively weak hairpin vortex interacting with a much stronger horseshoe vortex below it.
More insight is available from the following video which shows the evolution of the apparent "ring" structure over the time interval 21.3-23.3 τb . The "ring" structure is also visualized in a small subvolume, which is slightly different than the subvolume used for the still image above. The fields were generated by evolving Ling's initial condition in the SAM code. Doing this also verifies that SAM and triple are giving essentially identical solutions.
The animation clearly shows that the structure begins as a "hairpin". Then, at a time of 21.5 τb , a vortex "smile" forms below and connects close to the head of the hairpin. Depending on the viewing angle, the resulting structure may or may not give the appearance of a vortex ring. The original hairpin legs remain throughout the evolution and additional vortical structures appear near the "ring" and wrap around it. The resulting structure is quite complex by the last frame in the animation and starts to loose its vortex ring appearance.
Ling's case shown above can be compared with an otherwise identical case but with lower initial noise. This case was run with the sam code an uses an initial random noise rms of 1.0e-04(λz dT/dz), which is a factor of 42 less than what Ling used. The animation below is in a subvolume containing just a single structure, but this structure is representative of others seen in the full domain.
Although there are many similarities to Ling's result, the lower noise case has several differentiating features. While he initial structure looks like a vortex pair, it is actually a hairpin with a diffuse head that does not image well using Lambda2. At a time of 27.4 a vortex "smile" forms below the hairpin in a manner similar to that in the higher noise case. The smile subsequently interacts with the legs of the horseshoe vortex and a second, weaker hairpin forms, this time with a visible head. The complex interacting pattern looks less like a vortex ring than in the high noise case since there is never closure of the vortex lines on the top side. At a time of 27.9 a second vortex "smile" appears and begins to interact with the complex structure above it. At this point the overall structure has more of a ring appearance.
A second case, with even lower initial noise amplitude is shown in the video below. This case is also for Re=10,000, but it is computed in a horizontal box and has an initial noise of T' = 1.0e-06(λz dT/dz). The full domain is imaged instead of the small subvolumes used above.
The pattern is quite complex in this case without any structures strongly resembling vortex rings.
Ling was able to provide the solution files for both the initial condition and for t~22 τb , the time visualized in the part 2 paper that appears to show vortex rings. A careful analysis of the initial condition indicates that the initial temperature white noise rms amplitude is 4.24e-3 (λz dT/dz), which gives it a relative rms amplitude of 4.188e-2 times the initial GW temperature rms. This is at huge odds with the relative rms amplitude of 1.0e-6 quoted in the part 1 paper. For what it is worth, the input file for the triple run does have a temperature noise level specification of 1.0e-6, but there must either be a misunderstanding of how this input is used, or a bug in the triple code. One possibility is that the triple input value is intended to be a variance value instead of rms. In any event, the character of the initial instability structures appears to be sensitive to the initial noise level when it is large and this is responsible for much of the disagreement between Ling's case and the recent SAM runs.
Beyond the sensitivity to the initial noise level, careful visualizations of Ling's results show that the primary instability structure is not quite a vortex ring but rather a "horseshoe" vortex interacting with the legs of a "hairpin" vortex. This same configuration appears in the lower initial noise amplitude cases, but one or more horseshoe vortices are spaced further from the head of the hairpin, giving less of an appearance of a vortex ring. Shown below is an image of λ2 formed from Ling's data file at a time of t~22 τb . The field has been shifted by (-0.4, 0.35, -0.3)λz in the (x, y, z) directions, respectively in order to minimize severing of the instability structures by the periodic boundaries.
In this top-down view, an apparent vortex ring is visible to the lower left. This structure is encased in a small rectangular box, which depict the subvolume that will be used below to scrutinize this structure. The subvolume has extent (-0.9:0.7, -1.4:-0.55, 0.4:1.1) λz in the (x, y, z) directions, respectively.
In addition to the apparent vortex ring structure, there is a prominent hairpin vortex near the top-center of the image and a second less obvious one at the lower right. The remaining structure at the top left appears to be another vortex ring.
As the image below indicates, a somewhat different impression is formed when the instability structure subvolume is viewed from a perspective angle.
In this view it is somewhat apparent that we are looking at a relatively weak hairpin vortex interacting with a much stronger horseshoe vortex below it.
More insight is available from the following video which shows the evolution of the apparent "ring" structure over the time interval 21.3-23.3 τb . The "ring" structure is also visualized in a small subvolume, which is slightly different than the subvolume used for the still image above. The fields were generated by evolving Ling's initial condition in the SAM code. Doing this also verifies that SAM and triple are giving essentially identical solutions.
The animation clearly shows that the structure begins as a "hairpin". Then, at a time of 21.5 τb , a vortex "smile" forms below and connects close to the head of the hairpin. Depending on the viewing angle, the resulting structure may or may not give the appearance of a vortex ring. The original hairpin legs remain throughout the evolution and additional vortical structures appear near the "ring" and wrap around it. The resulting structure is quite complex by the last frame in the animation and starts to loose its vortex ring appearance.
Ling's case shown above can be compared with an otherwise identical case but with lower initial noise. This case was run with the sam code an uses an initial random noise rms of 1.0e-04(λz dT/dz), which is a factor of 42 less than what Ling used. The animation below is in a subvolume containing just a single structure, but this structure is representative of others seen in the full domain.
Although there are many similarities to Ling's result, the lower noise case has several differentiating features. While he initial structure looks like a vortex pair, it is actually a hairpin with a diffuse head that does not image well using Lambda2. At a time of 27.4 a vortex "smile" forms below the hairpin in a manner similar to that in the higher noise case. The smile subsequently interacts with the legs of the horseshoe vortex and a second, weaker hairpin forms, this time with a visible head. The complex interacting pattern looks less like a vortex ring than in the high noise case since there is never closure of the vortex lines on the top side. At a time of 27.9 a second vortex "smile" appears and begins to interact with the complex structure above it. At this point the overall structure has more of a ring appearance.
A second case, with even lower initial noise amplitude is shown in the video below. This case is also for Re=10,000, but it is computed in a horizontal box and has an initial noise of T' = 1.0e-06(λz dT/dz). The full domain is imaged instead of the small subvolumes used above.
The pattern is quite complex in this case without any structures strongly resembling vortex rings.