This page shows the shear layer deformation when subjected to a N/5,
amplitude 0.15 gravity wave, superimposed on a N^2/N^2_0=4 duct. The
simulation is 2D but evolves correctly in time. The contour lines show
fixed values of v, which form material surfaces in the 2D representation.
The contour values are v = -0.5, 0.0, and 0.5.
The video below shows Richardson number contours at the levels 0.075, 0.10,
and 0.20. Note that the contours are somewhat distorted at time=0, as the
wave exerts a small influence on both the shear and on N^2. The wave
itself is most unstable in the middle of the domain and this effect is
visible in the initial set of contours, where the minimum Richardson number
is found at the domain center. As the flow evolves and the
shear layer distorts, the Richardson number becomes increasingly biased
by the shear layer thickness. This process is rather slow, however, and
the Richardson generally remains lower on the right third of the domain
as opposed to the left third. This observation appears to be at odds with
both the billow formation tendencies and with the analysis performed on
the 3D fields. Perhaps there is some self-induction due to the deformed
vortex sheet at play, or perhaps the billows respond much more to (dv/dz)^2
than to (du/dz)^2 + (dv/dz)^2, from which the Richardson number
in the animation is computed. This latter possibility was tested by
computing the Richardson number from just (dv/dz)^2 and the results were
nearly identical.