Numerical simulations of the shear instability and subsequent degeneration of basin scale internal standing waves

We present high-resolution simulations of the instability and subsequent breakdown of standing waves, or seiches, in a fluid continuously stratified in the vertical direction. It is well known that such waves can evolve to form nonlinear, dispersive wave trains. When the initial dimensionless amplitude is large, it is possible that a stratified shear instability develops, possibly at the same time as dispersive wave trains. While both dispersion and shear instability serve to move energy from large to small scales, they are fundamentally different. The development into wave trains is nondissipative in nature, and in the asymptotic limit of small but finite amplitude seiches may be described by variants of the Korteweg–de Vries (KdV) equation. Shear instability, on the other hand, yields Kelvin-Helmholtz billows, which in turn provide one of the basic archetypes of transition to turbulence, with greatly increased rates of mixing and viscous dissipation. We discuss how the two phenomena vary as the aspect ratio of the tank and the height of the interface between lighter and denser fluids are changed, highlighting cases where the two phenomena coexist. A quantitative accounting for the evolution of the horizontal modewise decomposition of the kinetic energy of the system in addition to the creation of a semianalytical model of the evolution of the fundamental mode is presented. Finally, the mixing is characterized during the evolution of the seiche, illustrating a fundamental transition that occurs as the aspect ratio is decreased.

Figure 1

Schematic of the initial condition indicating each of the layer depths ($h_1$ and $h_2$), the total depth ($H$), the length ($L$), amplitude (${\eta }_0$), and pycnocline mean depth ($z_0$). The density interface is drawn as a single curve. Here, $h_1$ and $h_2$ are the mean upper and lower depths of the fluid.

Figure 2

Snapshots of the density field, with the interface particularly prominent, for the L5-2m case. Density fields taken at $\tau =0.25,0.7,0.9,$ and 1.25.

Figure 3

Snapshots of the density field, with the interface particularly prominent, for the L5-8m cases. Density fields taken at $\tau =0.25,0.7,0.9,$ and 1.25.

Figure 4

The vorticity field for the L5-2m (top) and L5-8m (bottom) cases at $\tau =0.9$.

Figure 5

The KE field for the L5-2m (top) and L5-8m (bottom) cases at $\tau =0.9$.

Figure 6

[(a), (c)] Space-time plot for the vertically integrated KE normalized by the maximum for the L5-2m and L5-8m cases, respectively. [(b), (d)] Green line represents the total KE normalized by the maximum. The red line represents the energy stored in the first four harmonics, and the blue line represents the rest of the energy for each case.

Figure 7

Snapshots of the density field, with the interface particularly prominent, for the 15L5-8m case. Density fields taken at $\tau =0.25,0.7,0.9,$ and 1.25.

Figure 8

The vorticity field for the 15L5-8m (top) and 25L5-8m (bottom) cases at $\tau =0.7$.

Figure 9

The KE field for the 15L5-8m (top) and 25L5-8m (bottom) cases at $\tau =0.7$.

Figure 10

[(a), (c)] Space-time plot for the vertically integrated KE normalized by the maximum for the 15L5-8m and 25L5-8m cases, respectively. [(b), (d)] Green line represents the total KE normalized by the maximum. The red line represents the energy stored in the first four harmonics, and the blue line represents the rest of the energy for each case.

Figure 11

The vertically averaged KE in each harmonic from modes 2 to 17. The first harmonic is ignored because it composes most of the KE and thereby skews the plots. The horizontal axis is broken down by half-period groupings. The vertical axis is normalized by the total KE for time average over each half period: (a) L5-2m, (b) L5-8m, (c) 15L5-8m, and (d) 25L5-8m.

Figure 12

The logarithm of the decay parameter $\lambda$ calculated via linear regression of the maximums of the potential energy from the first harmonic, as a function of the dimensionless amplitude ${\eta }_0/h_1$ and aspect ratio $\mu =H/L$.

Figure 13

Fitting parameters for the model of $\lambda (W^{-1},\mu )$. Definitions are given in the text.

Figure 14

How $\lambda$ changes in $(W^{-1},\mu )$ parameter space. Each circle corresponds to a particular case run and where it lies in the parameter space.

Figure 15

Comparison of the model decay parameter (dashed line) and the calculated decay parameter (dotted line) against the mode 1 contribution to the $E_a$ (solid line) for two cases. The 25L5-8m case is in black and the L5-2m case is in red. The oscillating component out of the model was removed because there is no decay in mode 1 $E_a$ built into the oscillating part.

Figure 16

The $M_E$ time series for every case in Table 1. Panels (a)–(d) show the time series for the 1-m, 2-m, 4-m, and 8-m cases respectively, with the initial dimensionless amplitude denoted by different colors.

Figure 17

The change in $E_b$ as a percentage of the initial $E_b$ plotted against $\mu$. The change in $E_b$ is taken as the final value $E_b$ of the fluid at $\tau =2$ minus the initial background potential energy of the pycnocline. The points with the same marker shape are of equal $W^{-1}$. Read left to write, the aspect ratios of columns of points are $\mu =0.03125$, $\mu =0.0625$, $\mu =0.125$, and $\mu =0.25$. Filled markers are cases where Kelvin-Helmholtz (or similar) billows are seen.

Figure 18

A qualitative degeneration regime diagram for a laboratory-scale seiche. A “linear waves” bubble is included to show that at small amplitudes, linear theory is sufficient to predict wave motion. For large amplitude and reasonably short tanks, shear instabilities form (L5-1m, L5-2m). For midlength and midamplitude waves, wave trains form and for longer tanks with large amplitudes, both shear instabilities and wave trains form (15L5-4m, 15L5-8m, 25L5-4m, 25L5-8m).