Dynamics of asymmetric stratified shear instabilities

Most idealized studies of stratified shear instabilities assume that the shear interface and the buoyancy interface are coincident. We discuss the role of asymmetry on the evolution of shear instabilities. Using linear stability theory and direct numerical simulations, we show that asymmetric shear instabilities exhibit features of both Holmboe and Kelvin-Helmholtz (KH) instabilities, and develop a framework to determine whether the instabilities are more Holmboe-like or more KH-like. Further, the asymmetric instabilities produce asymmetric mixing that exhibits features of both overturning and scouring flows and that tends to realign the shear and buoyancy interfaces. In all but the symmetric KH simulations, we observe a collapse in the distribution of gradient Richardson number (${\text{Ri}}_g$), suggesting that asymmetry reduces the parameter dependence of KH-driven mixing events. The observed dependence of the turbulent dynamics on small-scale details of the shear and stratification has important implications for the interpretation of oceanographic data.

Figure 1

Representative profiles of the background horizontal velocity and buoyancy fields. The associated gradient Richardson number ${\text{Ri}}_g$ (22) for these profiles is also included. These profiles are similar to the asymmetric Holmboe case.

Figure 2

Top row: Example profiles of ${\mathcal{M}}_v$ and ${\mathcal{M}}_b$ for the fastest-growing modes of (a) a KH-type instability far from from the interface ($R_0=3, a_0=3$), (b) an asymmetric KH instability ($R_0=1, a_0=0.5$), and (c) a near-symmetric Holmboe instability ($R_0=3, a_0=0.001$). The horizontal dotted lines indicate the height of maximum shear and stratification, and the yellow and purple shaded regions represent contributions to ${\mathcal{M}}_v^{-}$ and ${\mathcal{M}}_v^{+}$, respectively. ${\text{Ri}}_b=0.15$ for all three cases. Middle and bottom rows: Growth rates (contours) and pseudomomentum ratios (colors) for the base flow Eqs. (2) for the symmetric and asymmetric KH and Holmboe configurations with $\text{Re}=1200$ and $\text{Pr}=9$. (d) $R_0=1, {\text{Ri}}_b=0.15$. (e) $R_0=3, {\text{Ri}}_b=0.15$. (f) $R_0=1, {\text{Ri}}_b=0.20$. (g) $R_0=3, {\text{Ri}}_b=0.20$. The contour interval is 0.01 and the stars denote parameters for the nonlinear simulations described in Table 1.

Figure 3

Snapshots of buoyancy field at representative times throughout the flow evolution. Vertical ($x-z$) slices are provided for (a)–(d), case 4: $R_0=1, a_0=0,t=\{55,85,115,145\}$; (e)–(h), case 5: $R_0=1, a_0=0,t=\{55,85,115,145\}$; (i)–(l), case 6 – $R_0=\frac{1}{3}, a_0=0,t=\{70,105,140,175\}$; and (m)–(p), case 6: $R_0=\frac{1}{3}, a_0=0,t=\{120,170,220,270\}$. In all included cases, $\text{Re}=1200,{\text{Ri}}_b=0.15,\text{Pr}=9.$

Figure 4

Three-dimensional visualizations of the buoyancy field as the secondary instabilities develop. The output times are $t=85$ (SKH), $t=85$ (AKH), $t=105$ (AHI), and $t=170$ (SHI), i.e., the same times as in Figs. 3, 3 3, and 3. These plots were constructed using VisIt's [43] volume plot, which sets the opacity of the buoyancy field based on its value.

Figure 5

TKE budgets for the cases listed in Table 1. Note that the time axes were trimmed [e.g., (d)] to have all the TKE budgets on a consistent and legible horizontal scale.

Figure 6

Contour plots of the horizontally averaged (a)–(d) buoyancy, (e)–(h) $N^2$, (i)–(l) $S^2$, (m)–(p) $log\varepsilon$, (q)–(t) $log\chi$, and (u)–(x) ${\mathrm{Re}}_B$ as a function of time. The base of the logarithms is $e$ (natural logarithm). Data is plotted for case 4 (left), case 5 (middle left), and case 6 (middle right), and case 7 (right): $\text{Re}=1200,{\text{Ri}}_b=0.15$. Superimposed on each plot are the ($-0.9$, 0, 0.9) contours of $\overline{\mathbf{u}}$ (green) and $\overline{b}$ (yellow). Anomalous values of large $\chi$ and ${\mathrm{Re}}_B$ associated with $N^2\to 0$ have been masked our as described in Sec. 3f.

Figure 7

Plot of the initial and final profiles of $-\frac{\partial \overline{b}}{\partial z}$ and $\frac{\partial \overline{u_1}}{\partial z}$ of the (a) SKH, (b) AKH, (c) AHI, and (d) SHI for $\text{Re}=1200,{\text{Ri}}_b=0.15$. The evolution of the (e) interface thicknesses $R$ and (f) offset $\mathbf{a}$ are included as a function of time.

Figure 8

(a)–(d) Time-series plots of the gradient Richardson number of the horizontally averaged flow, $N^2/S^2$. (e)–(h) Local mixing efficiency, $\eta$, as defined in Eq. (23). The opaque regions in (a)–(h) indicate the portions of the flow for which ${\varepsilon }^{\prime }<2{\text{Ri}}_b/\left(\text{Re}{L}_z\right)$. (i)–(l) Profiles of $R{i}_g$ and $\eta$ of the horizontally averaged flow at $t=t_{3\text{D},\text{max}}$ (dark) and averaged from $t_{3\text{D},\text{max}}$ (light) to the end of the simulation. The thin dashed lines indicate $R{i}_g=1/4$ and $\eta =1/6$. All cases shown have $\text{Re}=1200$ and ${\text{Ri}}_b=0.15$.

Figure 9

PDFs of ${\text{Ri}}_g$ and $\eta$ where the flow is turbulent for (a), (b) SKH; (c), (d) AKH; (e), (f) AHI; and (g), (h) SHI. Vertical dashed lines are included at ${\text{Ri}}_g=\frac{1}{4}$ and $\eta =\frac{1}{6}$, respectively.