Spontaneous instability in internal solitary-like waves

The onset and growth of shear instability in large-amplitude internal solitary-like waves propagating in a quasi-two-layer stratification is studied using high-resolution direct numerical simulations with a spectral collocation method. These waves have a minimum Richardson numbers of approximately 0.08 and a length ratio $L_{\mathrm{Ri}}/L_{\text{wave}}$ between 0.80 and 0.88, where $L_{\mathrm{Ri}}$ is the length of high shear region with a local Richardson number $\mathrm{Ri}<0.25$ and $L_{\text{wave}}$ is the half-width of the wave. In the wave with $L_{\mathrm{Ri}}/L_{\text{wave}}\approx 0.88$, the onset of instability occurs spontaneously without interacting with any externally imposed physical noise. When $L_{\mathrm{Ri}}/L_{\text{wave}}\lesssim 0.86$, the onset of instability is limited by viscous effects, so that criteria for the onset also include the Reynolds number of the flow field and the amplitude of the external noise. In the wave with $L_{\mathrm{Ri}}/L_{\text{wave}}\approx 0.85$, the spontaneous instability is possible only when the Reynolds number is sufficiently large, whereas in the wave with $L_{\mathrm{Ri}}/L_{\text{wave}}\approx 0.80$, instability does not grow spontaneously but must be triggered by perturbations of finite amplitude. For instabilities triggered by externally imposed noise, the amplitude of the noise has a crucial influence on their growth, whether such noise is in the form of random perturbations or normal-mode disturbances. On the other hand, the importance of non-normal growth decreases as the length ratio $L_{\mathrm{Ri}}/L_{\text{wave}}$ increases and as the amplitude of perturbations increases. In the wave with $L_{\mathrm{Ri}}/L_{\text{wave}}\approx 0.88$, further perturbing the flow field with sufficiently large perturbations leads to the growth of instability on the upstream side of the wave's crest, a result close to the optimal transient growth, even though the perturbations are in the form of normal-mode disturbances.

Figure 1

(a) Density and (b) horizontal velocity fields of the large wave at $t=0$ s. The velocity in (b) is measured in meters per second. The Richardson number contours (black) have values of 0.1 (inside) and 0.25 (outside). The wave's crest is indicated by the dashed lines (white).

Figure 2

Shaded density contours showing the evolution of shear instability in the base case at four different times. Detail of the billows in (b) is shown in Fig. 3.

Figure 3

Zoomed-in view of Fig. 2, showing detail of the billows at $t=20$ s.

Figure 4

Three-dimensional volume plots showing the density field in the base case at (a) $t=22$ s, (b) $t=24$ s, and (c) $t=26$ s.

Figure 5

Density contours showing the billow formation in the cases [(a)–(c)] LR5 and [(d)–(f)] LR10.

Figure 6

Same as in Fig. 5 but for the cases [(a)–(c)] MR5 and [(d)–(f)] MR10.

Figure 7

Time series of the maximum vorticity of the flow fields in the cases (a) LR5 (blue) and LR10 (red) and (b) MR5 (blue) and MR10 (red).

Figure 8

Wave profiles as functions of $z$ across the large wave's crest in LR5S1 (red), LR5 (green), and LR5S20 (blue) at $t=10$ s and for the initial wave at $t=0$ s (black).

Figure 9

Density contours showing detail of the billows in the cases (a) LR5S1, (b) LR5, and (c) LR5S20 at $t=20$ s.

Figure 10

Same as in Fig. 5 but for the cases [(a)–(c)] SR5P-4 and [(d)–(f)] SR5P-3. Note that we have varied the $x$ axis for each panel in order to show the billow formation.

Figure 11

Density contours showing the onset of instability in four different cases. Note that we have varied the $x$ axis and the plot times for each panel in order to show the same maturity billows. The dashed lines indicate the locations of the wave's crest.

Figure 12

Growth rate of waves of disturbance as a function of wavelength, in a background environment extracted from (a) the large wave and (b) the small wave.

Figure 13

Schematic diagram of the model setup for ISW-free wave interaction.

Figure 14

Density contours showing the interaction of the ISWs with normal modes.

Figure 15

Hovmöller plot of the vertically averaged, high-pass filtered density field of the base case, in a periodic domain moving with the wave. The dashed line indicates location of the wave's crest.