Asymmetric Kelvin-Helmholtz instabilities in stratified shear flows

In this study we explore the effect of an offset between the velocity and density interfaces on the dynamics and mixing of the Kelvin-Helmholtz (KH) instability in stratified shear flows. Most prior studies have assumed a coincident interface-symmetric KH instability. To investigate the asymmetric KH instability that emerges in the presence of offset interfaces, we conduct a linear stability analysis and direct numerical simulations, comparing results with the well-known symmetric KH instability. We find that the asymmetric KH instability is a hybrid mode of symmetric KH and Holmboe instabilities, with features of both overturning and scouring flows and a nonzero propagation speed. In contrast to the symmetric KH instability, the asymmetric KH instability does not generate a large-scale overturning of the central isopycnal but scours fluid of intermediate density from the upper portions of the interface, resulting in a significant interface deepening and sharpening of the density interface during mixing events. We observe that the dynamics and amount of mixing are strongly influenced by the degree of asymmetry (i.e., the offset distance between density and velocity interfaces) in the flow. With a larger asymmetry, the kinetic energy of the instability is larger but the cumulative mixing and mixing efficiency increase to a maximum then decrease. We find a similar transition of the gradient Richardson number distribution after the instabilities become turbulent, which has important implications for interpreting oceanographic data. Our study suggests that asymmetry should be taken into account in future studies of the KH instability.

Figure 1

Schematic illustrating the background fields of velocity (black) and density (red) of an asymmetric stratified shear layer. In the present study, the thickness of the velocity interface is the same as that of the density, $R=1$.

Figure 2

(a) Profiles of Richardson number ${\text{Ri}}_g\left(z\right)$ for the hyperbolic tangent profiles shown in Fig. 1 with ${\text{Ri}}_b=0.15$ and a variation of the degree of asymmetry $a$; (b) the vertical level $z_{min}$ of (c) the minimum value of ${\text{Ri}}_g$ as function of $a$. The dotted vertical line in (a) indicates ${\text{Ri}}_g=1/4$, and the dashed line in (b) indicates the 1:1 reference line.

Figure 3

Linear stability properties as a function of $a$ of the unstable mode: (a) growth rate contours with the most unstable wave number (red dashed line); (b) maximum growth rate (thick line) and the fastest growth rate of the Rayleigh instability of an unstratified fluid (horizontal dashed line); (c) most unstable wavelength (used in setting the initial horizontal domain for the nonlinear numerical simulations); and (d) phase speed of the maximum growth rate. The values were obtained for the initial profiles in Fig. 1 with ${\text{Ri}}_b=0.15, \text{Re}=1200$, and $\text{Sc}=9$.

Figure 4

Evolution of the three-dimensional density field at representative times (a)–(c) $a=0$: $t=\{65,110,155\}$, (d)–(f) $a=1$: $t=\{55,110,165\}$, and (g)–(i) $a=2$: $t=\{50,105,160\}$.

Figure 5

Evolution of the horizontally averaged fields for $a=0$ (left column), $a=1$ (middle column), and $a=2$ (right column). (a), (f), (k) density field, (b), (g), (l) $\overline{N^2}$, (c), (h), (m) log $\overline{\varepsilon }$, (d), (i), (n) log $\overline{\chi }$, and (e), (j), (o) $\overline{{\text{Re}}_b}$. The base of the logarithms is $e$ (natural logarithm). Superimposed on each plot are the ($-0.9$, 0, 0.9) contours of the density field.

Figure 6

Mixing due to the asymmetric KH instability for $a=1$: (a) initial (dashed line) and final (solid line) density profiles with horizontal grey lines indicating the center of the density interface; (b) evolution of $\overline{N^2}$, where the location of the maximum $\overline{N^2}$ is indicated by the red line with its magnitude given in (c).

Figure 7

Evolution of energy for $a=0$, 1, and 2: (a) two-dimensional kinetic energy ($K_{2d}$), and (b) three-dimensional kinetic energy ($K_{3d}$). In (a), the solid symbols indicate the time $t_{2d}$ when $K_{2d}$ is maximal for each case, and the dashed lines indicate the predicted growth rates from the linear stability analysis (see Table 1). In (c), the solid symbols indicate $t_{3d}$ when $K_{3d}$ is maximal.

Figure 8

Evolution of mixing parameters for $a=0$, 1, and 2: (a) turbulent dissipation rate, (b) mixing rate, (c) cumulative mixing, and (d) instantaneous mixing efficiency.

Figure 9

Summary plots for each simulation showing the influence of the asymmetry on (a) cumulative mixing and (b) cumulative mixing efficiency for the entire event from $t=0$ to $t_f$. The horizontal dashed line denotes the canonical $E_c=1/6$ suggested by Osborn [54] for the average mixing efficiency of steady stratified shear flows. This value corresponds to a flux coefficient $\mathrm{\Gamma }=E_c/(1-E_c)=0.2$.

Figure 10

Evolution of the horizontally averaged ${\mathrm{Ri}}_g$ for (a) $a=0$, (b) $a=1$, and (c) $a=2$. The vertical arrow indicates $t=t_{3d}$; the white dashed and solid lines indicate the position of the density and velocity interfaces, respectively.

Figure 11

Probability density function (PDF) of ${\mathrm{Ri}}_g(z,t)$ for a range of asymmetries. The PDF is computed after the flow becomes turbulent ($t\ge t_{3d}$) and including only turbulent regions for which ${\varepsilon }^{\prime }>\frac{2}{L_z}\frac{{\text{Ri}}_b}{\text{Re}}$ (see [25, 60]). The vertical dotted line denotes ${\mathrm{Ri}}_g=1/4$.

Figure 12

Domain averaged kinetic energy as a function of the asymmetry on (a) the maximum 2D kinetic energy ($K_{2d}$) and (b) the maximum three-dimensional kinetic energy and its associated turbulent dissipation rate.

Figure 13

Summary plots for the numerical simulations (cases 7–14). (a) Cumulative mixing and (b) cumulative mixing efficiency for the entire event from $t=0$ to $t_f$. The insignificant error bar on the solid circle (case 14) indicates the negligible difference on mixing due to different phase differences between initial perturbations (see Appendix pp2-s2). The horizontal dashed line denotes the canonical $E_c=1/6$ suggested by Osborn [54] for the average mixing efficiency of steady stratified shear flows.

Figure 14

Definition of the relative phase of the KH and subharmonic components based on $w^{\prime }(z=0$): (a) ${\theta }_{\mathrm{sub}}=0$ (case 14a) and (b) ${\theta }_{\mathrm{sub}}=-\pi /2$ (case 14b).

Figure 15

Evolution of the density field ($x-z$) at representative times: (a)–(c) case 14a: ${\theta }_{\mathrm{sub}}=0$ and (d)–(f) case 14b: ${\theta }_{\mathrm{sub}}=-\pi /2$.

Figure 16

Evolution of (a) cumulative mixing and (b) instantaneous mixing efficiency for case 14a with ${\theta }_{\mathrm{sub}}=0$ and case 14b with ${\theta }_{\mathrm{sub}}=-\pi /2$.