2.1. The mathematical model

We begin by considering a stably stratified parallel shear flow,

(2.1)\begin{equation} U^{*}(z)=U^{*}_{0}\left[\tanh \left( \frac{z^*-D^*}{h^*}\right)+\tanh \left(\frac{z^*+D^*}{h^*}\right) \right] \end{equation}

and

(2.2)\begin{equation} B^{*}(z)=B^{*}_{0}\left[\tanh \left( \frac{z^*-D^*}{h^*}\right)+\tanh \left(\frac{z^*+D^*}{h^*}\right) \right],\end{equation}

in which $2U^{*}_{0}$ and $2B^{*}_{0}$ are, respectively, velocity and buoyancy differences across the individual shear layer, and $2h^{*}$ is its thickness (figure 1). Both stratified shear layers are at a distance $D^{*}$ from the centre of the domain (so that the distance between the centres is $2D^*$). The domain has a vertical extent $L_z^{*}$ with upper and lower boundaries at $\pm L_z^*/2$. Asterisks indicate dimensional quantities. The Cartesian coordinates are $x^*$ (streamwise), $y^*$ (spanwise) and $z^*$ (vertical, positive upwards), and the corresponding velocity components are $u^*$, $v^*$ and $w^*$. After non-dimensionalizing velocities by $U^{*}_{0}$, buoyancy by $B^{*}_{0}$, lengths by $h^{*}$, and times by the advective time scale $h^{*}/U^{*}_{0}$, (2.1) and (2.2) become

(2.3)\begin{equation} U(z)=B(z)=\tanh \left(z-D\right)+\tanh \left(z+D\right). \end{equation}

Figure 1.

Initial mean profile for velocity and buoyancy showing dimensional parameters as defined in (2.1) and (2.2).

The evolution of the flow is governed by the Boussinesq Navier–Stokes equations, as well as the equations of buoyancy conservation and mass continuity. Non-dimensionalized, these are

(2.4)$$\begin{gather} \frac{\partial\boldsymbol{u}}{\partial t}+\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{u}={-} \boldsymbol{\nabla} p+Ri_{0}\,b\hat{\boldsymbol{z}}+\frac{1}{Re_{0}}\,\nabla^{2}\boldsymbol{u}, \end{gather}$$

(2.5)$$\begin{gather}\frac{\partial b}{\partial t}+\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla} b =\frac{1}{Re_{0}\,Pr}\,\nabla^{2}b, \end{gather}$$

(2.6)$$\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u}=0, \end{gather}$$

where $p$ is the pressure, and $\hat {\boldsymbol {z}}$ is the vertical unit vector. The equations involve three dimensionless parameters: the initial Reynolds number $Re_0=U^{*}_{0}h^{*}/\nu ^{*}$, where $\nu ^{*}$ is the kinematic viscosity, the Prandtl number $Pr=\nu ^{*}/\kappa ^{*}$, where $\kappa ^{*}$ is the diffusivity, and the initial bulk Richardson number $Ri_{0}=B^{*}_{0}h^{*}/U^{*2}_{0}$.

In general, we define the gradient Richardson number as

(2.7)\begin{equation} Ri_g(z,t)=\frac{\partial \langle b^*\rangle_{xy}/\partial z^*}{(\partial \langle u^*\rangle_{xy}/\partial z^*)^{2}}=Ri_0\,\frac{\partial \langle b\rangle_{xy}/\partial z}{(\partial \langle u\rangle_{xy}/\partial z)^2}=\frac{N^2}{S^2}. \end{equation}

Here, the notation $\langle \cdot \rangle _{r}$ represents an average over $r$, where $r$ can encompass any combination of $x$, $y$, $z$ and $t$. Also, $N^2$ is the squared buoyancy frequency, and $S$ is the mean shear. The minimum of $Ri_g$ with respect to $z$ is denoted as $Ri_{min}(t)$. In the inviscid limit, a necessary condition for instability is that $Ri_{min}$ be less than $1/4$ (Howard Reference Howard1961; Miles Reference Miles1961). For the flow described by (2.3), the initial $Ri_{min}$ increases from $Ri_0/2$ to $Ri_0$ when $D$ increases from 0 to infinity (figure 2).

Figure 2.

The dependence of initial minimum Richardson number $Ri_{min}$ on $D$. Here, $Ri_0$ is 0.16.

Boundary conditions are periodic in both horizontal directions, with periodicity intervals $L_x$ and $L_y$. The upper and lower boundaries are free-slip ($\partial u/\partial z = \partial v/\partial z = 0$), insulating ($\partial b/\partial z = 0$) and impermeable ($w=0$).

A small, random velocity perturbation is incorporated into the initial state (2.3). This initial perturbation field is purely stochastic and is applied uniformly to all three velocity components across the computational domain. The maximum amplitude of any single component is limited to 0.05, equivalent to 2.5 % of the velocity change across each shear layer. This magnitude is kept small to ensure that the initial growth phase is consistent with linear perturbation theory. For each value of $D$, an ensemble of three cases is simulated, each using a distinct seed to generate the random velocities (L22).

2.2. Linear stability analysis

To evaluate the linear instabilities, (2.4)–(2.6) are linearized about the initial base flow (2.3). These equations are then subjected to perturbations induced by small-amplitude, normal mode disturbances proportional to the real part of $a(z)\exp {(\sigma t+\textrm{i}kx)}$. In this context, $a(z)$ denotes the vertically varying, complex amplitude of any perturbation quantity, $\sigma$ stands for the complex exponential growth rate, and $k$ is the wavenumber in the streamwise direction. The streamwise phase speed is $c=\textrm{i}\sigma /k$. The normal mode equations are discretized using a Fourier–Galerkin method, yielding a generalized matrix eigenvalue problem that is solved using standard methods. Details may be found in § 13.3 of Smyth & Carpenter (Reference Smyth and Carpenter2019) or in Lian, Smyth & Liu (Reference Lian, Smyth and Liu2020).

2.3. Direct numerical simulations

Simulations are conducted using DIABLO (Taylor Reference Taylor2008), which utilizes a hybrid implicit–explicit time-stepping scheme with pressure projection. The viscous and diffusive components are addressed implicitly using a second-order Crank–Nicolson method, while other terms are explicitly resolved employing a third-order Runge–Kutta–Wray method. The vertical $z$ direction dependence is discretized using second-order finite differences, whereas the periodic streamwise and spanwise directions ($x, y$) are managed using the Fourier pseudo-spectral method.

To allow subharmonic mode growth, we set the streamwise periodicity interval $L_x$ to two wavelengths of the fastest-growing KH mode, as determined through linear stability analysis (§ 2.2). For the development of 3-D secondary instabilities, a spanwise periodicity interval of $L_y=L_x/4$ is adequate (e.g. Klaassen & Peltier Reference Klaassen and Peltier1985; Mashayek & Peltier Reference Mashayek and Peltier2013). The domain height is $L_z=30$ to minimize boundary effects. The computational grid is uniform and isotropic, and resolves $\sim$2.5 times the Kolmogorov length scale $L_k=(Re^{-3}/\epsilon )^{1/4}$, with $\epsilon$ representing a characteristic viscous dissipation rate after turbulence onset (e.g. Smyth & Moum Reference Smyth and Moum2000). Grid sizes are given in table 1.

Table 1.

Parameter values for five 3-member direct numerical simulations ensembles. In all cases, $Re_0=1000$, $Pr=1$, $Ri_0=0.16$. Data for the case $D=\infty$ are sourced from L22 and include only a single, isolated shear layer. The maximum initial random velocity component is 0.05.

Given the sensitivity of turbulent flows to initial conditions, we work with ensemble mean statistics where appropriate. Following L22, we use an ensemble of three cases at each separation distance $D$. Five values of $D$ are considered, for a total of 15 simulations (listed in table 1). We also employ a three-member ensemble of simulations of a single shear layer described in L22 to represent the limiting case $D\rightarrow \infty$.

To maintain our primary focus on the influence of the adjacent shear layer, we keep the initial state parameters, specifically the Richardson, Reynolds and Prandtl numbers, fixed. The choice $Ri_0=0.16$ is large enough for the pairing instability (e.g. Klaassen & Peltier Reference Klaassen and Peltier1989) to be damped by stratification when $Ri_{min}=Ri_0$ (i.e. for large $D$). In all cases, we set $Re_0 = 1000$ and $Pr=1$. While smaller than would be typical in nature, these values reflect a necessary compromise dictated by computational resource constraints.

2.4. Diagnostics

The total velocity field can be decomposed into a horizontally averaged component, referred to as the mean flow, and a perturbation

(2.8)\begin{equation} \boldsymbol{u}(x,y,z,t)=\bar{U}\,\hat{\boldsymbol{e}}^{(x)}+\boldsymbol{u}'(x,y,z,t), \quad \mathrm{where}\ \bar{U}(z,t)=\left\langle u\right\rangle_{xy}, \end{equation}

with $\hat {\boldsymbol {e}}^{(x)}$ the unit vector in the streamwise direction. Following Caulfield & Peltier (Reference Caulfield and Peltier2000), the perturbation velocity is further subdivided into two-dimensional (2-D) and 3-D components:

(2.9)\begin{equation} \boldsymbol{u}'(x,y,z,t)= \boldsymbol{u}_{2d}+\boldsymbol{u}_{3d}, \end{equation}

where

(2.10a,b)\begin{equation} \boldsymbol{u}_{2d}(x,z,t)=\left\langle\boldsymbol{u}\right\rangle_y - \bar{U}\,\hat{\boldsymbol{e}}^{(x)}\quad \mathrm{and}\quad \boldsymbol{u}_{3d}(x,y,z,t)=\boldsymbol{u}-\boldsymbol{u}_{2d}-\bar{U}\,\hat{\boldsymbol{e}}^{(x)}=\boldsymbol{u} - \left\langle\boldsymbol{u}\right\rangle_y. \end{equation}

Similarly, the buoyancy field can be decomposed as

(2.11)$$\begin{gather} b(x,y,z,t)=\bar{B}+b'(x,y,z,t), \quad\mathrm{where}\ \bar{B}(z,t)=\langle b\rangle_{xy}, \end{gather}$$

(2.12)$$\begin{gather}b_{3d}(x,y,z,t)=b-\langle b\rangle_y. \end{gather}$$

The total kinetic energy can now be partitioned as

(2.13a,b) \begin{equation} \mathscr{K}=\bar{\mathscr{K}}+\mathscr{K}',\quad\mathscr{K}'=\mathscr{K}_{2d}+\mathscr{K}_{3d}, \end{equation}

where

(2.14a–c)\begin{align} \bar{\mathscr{K}}=\tfrac{1}{2}\langle\bar{U}^2\rangle_{z},\quad \mathscr{K}_{2d}=\tfrac{1}{2}\langle u_{2d}^2+v_{2d}^2+w_{2d}^2\rangle_{xz},\quad \mathscr{K}_{3d}=\tfrac{1}{2}\langle u_{3d}^2+v_{3d}^2+w_{3d}^2\rangle_{xyz}. \end{align}

These constituent kinetic energies $\bar {\mathscr {K}}$, $\mathscr {K}'$, $\mathscr {K}_{2d}$ and $\mathscr {K}_{3d}$ can be identified as the horizontally averaged kinetic energy associated with the mean flow, the turbulent kinetic energy, and the kinetic energy related to 2-D and 3-D motions. We denote the instances in time when $\mathscr {K}_{2d}$ and $\mathscr {K}_{3d}$ reach their maximum values as $t_{2d}$ and $t_{3d}$, respectively.

Quantification of irreversible mixing involves decomposing the total potential energy $\mathscr {P}=-Ri_0\,\langle bz\rangle _{xyz}$ into available and background components, $\mathscr {P}=\mathscr {P}_a+\mathscr {P}_b$. The background potential energy $\mathscr {P}_b$ is the minimum potential energy achievable by adiabatically rearranging the buoyancy field into a statically stable state $b^*$ (Winters et al. Reference Winters, Lombard, Riley and D'Asaro1995; Tseng & Ferziger Reference Tseng and Ferziger2001). After computing the total and background potential energies, the available potential energy is determined from the residual, $\mathscr {P}_a=\mathscr {P}-\mathscr {P}_b$. Here, $\mathscr {P}_a$ represents the potential energy available for conversion to kinetic energy, arising from lateral variations in buoyancy or statically unstable regions.

The irreversible mixing rate due to fluid motions is defined as

(2.15)\begin{equation} \mathscr{M}=\frac{\textrm{d}\mathscr{P}_b}{\textrm{d}t}-\mathscr{D}_p, \end{equation}

where

(2.16)\begin{equation} \mathscr{D}_p=\frac{Ri_0\,(b_{top}-b_{bottom})}{Re_0\,Pr\,L_z} \end{equation}

refers to the rate at which the potential energy of a statically stable buoyancy distribution would increase solely due to diffusion of the mean buoyancy profile in the absence of any fluid motion.

There exists a variety of definitions for mixing efficiency in the literature (e.g. Gregg et al. Reference Gregg, D'Asaro, Riley and Kunze2018). Here, we define the instantaneous mixing efficiency as

(2.17)\begin{equation} \eta_i = \frac{\mathscr{M}}{\mathscr{M}+\epsilon}, \end{equation}

where $\epsilon =({2}/{Re})\langle s_{ij}s_{ij}\rangle _{xyz}$ is the total dissipation rate, and $s_{ij}=(\partial u_i/\partial x_j+\partial u_j/\partial x_i)/2$ is the strain rate tensor. The mixing efficiency quantifies the fraction of energy directed towards irreversible mixing to the total kinetic energy loss that is irreversibly lost to friction (Peltier & Caulfield Reference Peltier and Caulfield2003). The cumulative mixing efficiency serves as a valuable measure for quantifying the overall efficiency of the entire mixing process, and is defined as

(2.18)\begin{equation} \eta_c = \frac{\displaystyle\int\nolimits^{t_f}_{t_i} \mathscr{M} \,\textrm{d}t}{\displaystyle\int\nolimits^{t_f}_{t_i} \mathscr{M} \,\textrm{d}t + \displaystyle\int\nolimits^{t_f}_{t_i} \epsilon \,\textrm{d}t}, \end{equation}

where $t_i\sim 2$ is the initial time after the model adjustment period, and $t_f$ is the final time of the integral at which $\mathscr {M}=\mathscr {D}_p$.

An alternative quantifier of mixing that readily shows the spatial structure is the perturbation buoyancy variance dissipation rate, defined as

(2.19)\begin{equation} \chi'(x,y,z,t)=\frac{2\,Ri_0}{Re\,Pr}\,|\boldsymbol{\nabla} b'|^2, \end{equation}

where $b'$ is the buoyancy perturbation, representing the deviation from the horizontal mean buoyancy.

The evolution equation for the kinetic energy of 3-D perturbations can be expressed in the form (Caulfield & Peltier Reference Caulfield and Peltier2000)

(2.20)\begin{align} \sigma_{3d}&=\frac{1}{2\mathscr{K}_{3d}}\,\frac{\textrm{d}}{\textrm{d}t}\mathscr{K}_{3d} \end{align}

(2.21)\begin{align} &=\mathscr{R}_{3d}+\mathscr{S\!h}_{3d}+\mathscr{A}_{3d}+\mathscr{H}_{3d}+\mathscr{D}_{3d}, \end{align}

where the first two terms represent the 3-D perturbation kinetic energy extraction from the background mean shear and the background 2-D KH billow by means of Reynolds stresses, respectively defined as

(2.22)$$\begin{gather} \mathscr{R}_{3d}={-}\frac{1}{2\mathscr{K}_{3d}}\left\langle u_{3d}w_{3d}\,\frac{\partial\bar{U}}{\partial z}\right\rangle_{xyz}, \end{gather}$$

(2.23)$$\begin{gather}\mathscr{S\!h}_{3d}={-}\frac{1}{2\mathscr{K}_{3d}}\left\langle u_{3d}w_{3d} \left(\frac{\partial u_{2d}}{\partial z}+\frac{\partial w_{2d}}{\partial x}\right) \right\rangle_{xyz}. \end{gather}$$

The third term represents the stretching deformation of the 3-D motions and is defined as

(2.24)\begin{equation} \mathscr{A}_{3d}={-}\frac{1}{2\mathscr{K}_{3d}}\left\langle\frac{1}{2}\left( u_{3d}^2-w_{3d}^2\right) \left(\frac{\partial u_{2d}}{\partial x}-\frac{\partial w_{2d}}{\partial z}\right) \right\rangle_{xyz}. \end{equation}

The final two terms are the buoyancy production term and the negative-definite viscous dissipation term associated with 3-D perturbations, and are defined respectively as

(2.25)$$\begin{gather} \mathscr{H}_{3d}=\frac{Ri_0}{2\mathscr{K}_{3d}}\,\langle b_{3d}w_{3d} \rangle_{xyz}, \end{gather}$$

(2.26)$$\begin{gather}\mathscr{D}_{3d}={-}\frac{1}{\mathscr{K}_{3d}\,Re}\,\langle s_{ij}s_{ij}\rangle_{xyz}, \end{gather}$$

where $s_{ij}$ is the strain rate tensor of the 3-D motions. The time at which $\sigma _{3d}$ is a maximum is defined as $t_{\sigma _{3d}}$. The enstrophy in the three vorticity components is defined as

(2.27a–c)\begin{equation} \mathcal{Z}_x=\tfrac{1}{2}(\omega_{x}^2), \quad \mathcal{Z}_y=\tfrac{1}{2}(\omega_{y}^2), \quad \mathcal{Z}_z=\tfrac{1}{2}(\omega_{z}^2), \end{equation}

where

(2.28a–c)\begin{equation} \omega_{x}=\frac{\partial w}{\partial y}-\frac{\partial v}{\partial z},\quad \omega_{y}=\frac{\partial u}{\partial z}-\frac{\partial w}{\partial x}\quad \textrm{and}\quad \omega_{z}=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}. \end{equation}

4.1. Overview of the nonlinear development

In this subsection, we look beyond the linear regime to examine the various secondary instabilities that emerge at different separation distances $D$ and trigger the transition to turbulence (see examples in figure 9). In all cases, the initial condition consists of an unstable parallel shear flow whose primary instability grows to form 2-D periodic laminar vortices. These vortices attain maximum kinetic energy at $t=t_{2d}$ (figures 9a,e,i). As expected, the wavelength is largest (among these three examples) for $D=1$, and smallest for $D=2$, where two trains of billows combine to form the oscillatory instability (figure 9i). In the oscillatory case, $D=2$, the growth rate and the time of turbulence onset are sensitive to the details of the initial perturbations, as is evident in the contrast between the upper and lower billow trains (figures 9i,j). The evolution progresses at a comparatively slower rate for $D=1$, consistent with its relatively small linear growth rate, while growth is faster for $D=0.5$ (compare the values of $t_{2d}$ between cases). During this progression, various secondary instabilities emerge, facilitating the breakdown of the primary KH billows (e.g. figures 9b,f,j). This breakdown leads to the generation of turbulence (e.g. at $t=t_{3d}$, figures 9c,g,k). Following the turbulent mixing phase, the flow relaminarizes (figures 9d,h,l).

Figure 9.

Cross-sections through the 3-D buoyancy field for example cases with (a–d) $D=0.5$, (e–h) $D=1$, and (i–l) $D=2$, at successive times as indicated. The buoyancy values plotted range from $-1.5$ (blue) to 1.5 (red). Snapshots in (a,e,i) correspond to $t=t_{2d}$, in (c,g,k) to $t=t_{3d}$, and in (d,h,l) to $t=t_f$, the time when $\mathscr {M}=\mathscr {D}_p$.

Secondary instabilities that govern the evolution of isolated KH billows at different values of $Ri_{min}$ have been explored in previous research (e.g. Davis & Peltier Reference Davis and Peltier1979; Klaassen & Peltier Reference Klaassen and Peltier1985, Reference Klaassen and Peltier1991; Mashayek & Peltier Reference Mashayek and Peltier2012a,Reference Mashayek and Peltierb, Reference Mashayek and Peltier2013, L22). In §§ 4.2 and 4.3, we focus on secondary instabilities that contribute to 3-D perturbation kinetic energy in the regimes $D>D_c$ and $D< D_c$, wherein the linear development is dominated by the oscillatory and stationary modes, respectively. Pertinent examples include the central core instability (CCI, e.g. Klaassen & Peltier Reference Klaassen and Peltier1991, L23), which is catalysed by the initial growth of the KH instability, and the shear-aligned convective instability (SCI, e.g. Davis & Peltier Reference Davis and Peltier1979; Klaassen & Peltier Reference Klaassen and Peltier1985), which manifests when KH billows reach a sufficient size to overturn the buoyancy structure. In § 4.4, we discuss 2-D secondary instabilities: the secondary shear instability (SSI) of the braids and pairing of adjacent billows (visible in figures 9(f) and 9(b), respectively).

4.2. Regime $D>D_c$

We examine the regime $D>D_c$, using ensembles of simulations with $D\rightarrow \infty$, $D=3$ and $D=2$ as examples. When the shear layers are infinitely separated ($D\rightarrow \infty$), they are independent of each other, and each exhibits the standard KH instability (e.g. L22). The 3-D perturbation kinetic energy $\mathscr {K}_{3d}$ (figure 10a) is created mostly by shear production $\mathscr {R}_{3d}$, which draws energy from the mean flow (blue curve). The growth of $\mathscr {R}_{3d}$ can be attributed to the sinusoidal distortion of the spanwise vortex tube at the core of each nascent KH billow, which redirects spanwise ($\kern0.7pt y$) vorticity towards the $x\unicode{x2013}z$ plane. The tilt of the sinusoidal distortion is such that the Reynolds stress $\langle u_{3d}w_{3d}\rangle _{xyz}$ becomes negative (see figure 14 of Lasheras & Choi (Reference Lasheras and Choi1988), figure 9 of Smyth & Winters (Reference Smyth and Winters2003), or figure 8 of Smyth Reference Smyth2006). This negative 3-D stress field works with the positive mean shear $\textrm{d}\bar {U}/\textrm{d}z$ to generate 3-D kinetic energy. By $t=t_{\sigma _{3d}}$ (the time of maximum 3-D growth), $\textrm{d}\bar {U}/\textrm{d}z$ is no longer a maximum in the billow core, but the Reynolds stress is. Therefore, the dominant contributor to energy growth, quantified by $\mathscr {R}_{3d}$, arises in this region. We identify this mode as the CCI.

Figure 10.

Time variation of terms of the $\sigma _{3d}$ equation (2.21) when (a) $D=\infty$, (b) $D=3$, and (c) $D=2$. All curves are ensemble averaged. Vertical dashed lines show the time at which the growth $\sigma _{3d}$ is a maximum. Note that the time for each ensemble case is shifted, such that the 3-D growth rate $\sigma _{3d}$ is a maximum at $t-t_{\sigma _{3d}}=0$. Terms except for $\sigma _{3d}$ are obtained from cubic spline fits.

The buoyancy production $\mathscr {H}_{3d}$ (red curve) is positive but much smaller than $\mathscr {R}_{3d}$. In the current case with $Ri_0=0.16$, previous work suggests that the SCI (signalled by positive $\mathscr {H}_{3d}$) should be suppressed. Based on secondary stability analysis, the SCI grows only when $0.065< Ri_0<0.13$ (Klaassen & Peltier Reference Klaassen and Peltier1991). The dominance of shear production $\mathscr {R}_{3d}$ and suppression of buoyancy production $\mathscr {H}_{3d}$ when $D\rightarrow \infty$ are also consistent with the findings of Mashayek, Caulfield & Peltier (Reference Mashayek, Caulfield and Peltier2013), particularly in their case $Ri_0=0.16$, $Re=6000$.

As the separation distance between two shear layers is decreased from infinity to values approaching $D_c$ (e.g. our examples $D=3$ and 2), interactions become evident. When $D=3$, the evolution of each perturbation energy term resembles the infinite separation case (compare figures 10a,b), suggesting only a weak interaction between the upper and lower instabilities. When $D=2$, $\mathscr {R}_{3d}$ remains the dominant term (i.e. the principal secondary instability is still the CCI); however, a reduction in $\mathscr {H}_{3d}$ (figure 10c) is observed. At $t=t_{\sigma _{3d}}$, for example, the reduction is ${\sim }40\,\%$ compared to case $D\rightarrow \infty$. This reduction can be attributed to the close proximity of the shear layers, which results in additional suppression of SCI beyond the inherent effects of high $Ri_{min}$. Because the upper and lower billows co-rotate, roll-up is suppressed, reducing overturning. This is reminiscent of the effect of a nearby boundary on the SCI (L23).

4.3. Regime $D< D_c$

We now examine distinctions that arise when $D< D_c$, using examples $D=0$, $0.5$ and 1. When $D=0$, the two shear layers add to form a single shear layer with $Ri_{min}=Ri_0/2=0.08$. Thus the instability behaves similarly to a weakly stratified shear instability, and we expect to encounter the SCI. During the earliest stage of 3-D growth ($t-t_{\sigma _{3d}}\sim -18$ to $-6$), the $\mathscr {K}_{3d}$ budget is dominated by the shear production term $\mathscr {R}_{3d}$ due to the CCI. By $t \sim t_{\sigma _{3d}}$, the billow has rolled up enough to form convectively unstable layers. Consistent with the low initial $Ri_{min}$, the SCI is now the principal secondary instability that breaks down the KH billow structure. This is indicated in the $\mathscr {K}_{3d}$ budget (figure 11) by increased values of $\mathscr {H}_{3d}$ as well as $\mathscr {S\!h}_{3d}$ and $\mathscr {A}_{3d}$. One would expect the buoyancy production term $\mathscr {H}_{3d}$ to be substantial due to the prevailing influence of the SCI (Caulfield & Peltier Reference Caulfield and Peltier2000, L23). Surprisingly, both $\mathscr {S\!h}_{3d}$ and $\mathscr {A}_{3d}$ exhibit larger magnitudes than $\mathscr {H}_{3d}$ (figure 11a). This finding is distinguished from previous studies (Mashayek & Peltier Reference Mashayek and Peltier2013, L23), where buoyancy production dominated in the presence of the SCI. This may reflect a difference in the initial perturbations; the buoyancy field was perturbed in the previous studies but not in the present work.

Figure 11.

As in figure 10, but with $D=0$.

When $D$ is slightly above 0 (typified here by $D=0.5$), $Ri_{min}$ is small enough that the KH billow is again susceptible to SCI (Klaassen & Peltier Reference Klaassen and Peltier1991). During the initial growth phase (dot-dashed line in figure 12a), large positive values of $\mathscr {R}_{3d}$ concentrate in the billow core, indicating the CCI. This mechanism can be discerned qualitatively in the spanwise-averaged $x\unicode{x2013}z$ representation of $\mathscr {R}_{3d}$ (figure 12(b), region 1). Simultaneously, small areas of positive $\mathscr {S\!h}_{3d}$ manifest at the upper and lower extents of the billows (figure 12(c), region 2). Moreover, positive $\mathscr {A}_{3d}$ emerges along the braids (figure 12(d), region 3). These results are associated with the mechanism illustrated in figure 12 of Lasheras & Choi (Reference Lasheras and Choi1988), which shows that vortex filaments present in the braids undergo amplification through stretching along the principal plane of positive strain. These vortex filaments eventually envelop the spanwise vortex tubes of the central core, resulting in positive $\mathscr {S\!h}_{3d}$ in the upper and lower regions of each billow, and positive $\mathscr {A}_{3d}$ at the braids. Owing to the wrapping of these vortex filaments, the spanwise vortex tubes undulate (figure 14 in Lasheras & Choi Reference Lasheras and Choi1988), creating positive $\mathscr {R}_{3d}$ in the core. Nonetheless, $\mathscr {S\!h}_{3d}$ is mostly negative in the braids and in the billow cores, leading to an overall negative volume average (dashed line in figure 12a). Positive $\mathscr {H}_{3d}$ in the eyelids (region 4 of figure 12e) indicates the SCI.

Figure 12.

Dominant stationary mode when $D=0.5$. (a) As in figure 10. Vertical dot-dashed line and dashed line show the times at $t-t_{\sigma _{3d}}=-11$ and 0, respectively. (b–e) Spatial distribution of each energy term: $\mathscr {R}_{3d}$, $\mathscr {S\!h}_{3d}$, $\mathscr {A}_{3d}$ and $\mathscr {H}_{3d}$, respectively, at $t_{\sigma _{3d}}=-11$ (dot-dashed line in a). (f–i) Same at $t-t_{\sigma _{3d}}=0$ (dashed line in a).

At $t=t_{\sigma _{3d}}$, similar to $D=0$, $\mathscr {H}_{3d}$ is smaller than both $\mathscr {S\!h}_{3d}$ and $\mathscr {A}_{3d}$ (figure 12a). The SCI induces the formation of shear-aligned convective rolls, consistent with increased buoyancy production $\mathscr {H}_{3d}$ (figure 12(i), region 7). Positive $\mathscr {S\!h}_{3d}$ coincides with these convective rolls (region 5), suggesting that the SCI could be responsible for its generation. During the early growth phase, $\mathscr {H}_{3d}$ (region 4) begins to increase on the eyelids of each billow, whereas $\mathscr {S\!h}_{3d}$ remains small or negative in that area (figure 12c). This implies that as time progresses, the increase in positive $\mathscr {S\!h}_{3d}$ on the eyelids results from the formation of shear-aligned convective rolls with circulations tilted against the 2-D shear (figure 13). Vortex tubes at the periphery of the billows also undergo stretching, as quantified by $\mathscr {A}_{3d}$. Stretching occurs when denser fluid descends on the upper right portion of the billow under the action of gravity, while lighter fluid ascends on the lower left (figure 12(h), region 6).

Figure 13.

Schematic showing shear-aligned convective rolls tilting and stretching to form positive $\mathscr {S\!h}_{3d}$ and $\mathscr {A}_{3d}$, respectively.

During this phase of maximum growth, negative $\mathscr {R}_{3d}$ emerges at the margins of the billows (figure 12f). Consequently, the volume-averaged value is negative (figure 12(a), indicated by the blue curve at $t=t_{\sigma _{3d}}$). This suggests that the background mean flow contributes little to the 3-D perturbation kinetic energy in this instance. Instead, the perturbation energy is partially created by buoyancy production, but is predominantly due to shear production and the stretching of vortex tubes as discussed above.

In the case $D=1$, although the oscillatory mode is unstable, the dynamics is governed primarily by the stationary mode. The perturbation energy terms evolve similarly to the case $D=0.5$ (compare figures 12(a) and 14). This is interesting because $Ri_{min}=0.15$, which is outside the range 0.065–0.13 where the SCI is expected based on secondary stability analysis of an isolated shear layer (Klaassen & Peltier Reference Klaassen and Peltier1991), yet the roll motions are visible, for example, on the right-hand face of figure 9(f). We conclude that as in the case $D=0.5$, the SCI gives rise to shear-aligned convection rolls, consistent with positive values of $\mathscr {H}_{3d}$. The dominant source terms are again $\mathscr {S\!h}_{3d}$ and $\mathscr {A}_{3d}$ (figure 14).

Figure 14.

Dominant stationary mode when $D=1$. As in figure 10.

4.4. The SSI and pairing

We discuss the SSI and pairing separately as they affect $\mathscr {K}_{3d}$ negligibly. The SSI grows on the braids of the primary billows where the flow is nearly parallel and the shear is intensified by the strain of the large billows (Corcos & Sherman Reference Corcos and Sherman1976; Staquet Reference Staquet1995; Smyth Reference Smyth2003; Mashayek & Peltier Reference Mashayek and Peltier2012a). Staquet (Reference Staquet1995) and Smyth (Reference Smyth2003) find that the SSI tends to occur at higher $Re_0$. When $D< D_c$, the initial mean flow resembles a single shear layer with increased thickness and velocity change, i.e. with a larger Reynolds number. Therefore, the SSI may occur, depending on the initial noise field. An example is seen in figure 9(f). This secondary instability plays a notable role in generating turbulent mixing (to be discussed in § 5). At $t=t_{\sigma _{3d}}$, when $\sigma _{3d}$ is a maximum, the enstrophy of the spanwise component $\mathcal {Z}_y$ (figure 15b) is significantly stronger than that of the other two components combined, $\mathcal {Z}_x+\mathcal {Z}_z$ (figure 15a). The same is true for later times (figures 15(c,d) in the SSI-affected region), further confirming the 2-D nature of the SSI.

Figure 15.

Spatial distribution of enstrophy during and after maximum secondary instability growth. Data are from a sample simulation in the $D=1$ ensemble: (a,c) $x$ and $z$ components combined; (b,d) $y$ component. (a,b) are both at $t-t_{\sigma _{3d}}=0$, and (c,d) are at $t-t_{\sigma _{3d}}=7$. Horizontal streaks are artefacts of limited spatial resolution.

Secondary billows can be created either in pairs straddling the braid stagnation point or individually (Smyth Reference Smyth2003), as seen at $t-t_{\sigma _{3d}}=7$ in figure 15(d). Between the large billow cores, a pair of smaller billows emerges at the stagnation point. The pair eventually merges and becomes a larger single vortex, which then creates its own tertiary shear instability in its surroundings, a vivid illustration of a self-similar downscale energy cascade. Other secondary billows developed away from the stagnation point are advected outwards by the extensional strain.

Vortex pairing is also affected by a nearby shear layer. Pairing is more likely to occur when $D$ is small (e.g. $D=0$ and 0.5), due to small $Ri_{min}$ (figure 9b). L22 found that pairing is laminar (i.e. it occurs prior to the onset of turbulence) in cases with $Ri_{min}$ less than 0.14, and we expected this to remain true in the present cases where $Ri_{min}$ is considerably smaller. However, figure 9(c) indicates turbulent pairing. This is likely due to the difference in shape between the present shear layer and the single hyperbolic tangent profile assumed in L22. When $D\sim 0$, pairing precedes the onset of the SSI, leading to the disappearance of alternate braids. Subsequently, if the braids are not yet turbulent, then the SSI is likely to appear. The timing of turbulence onset, which itself depends on the choice of initial perturbation (L22), partly determines the occurrence of pairing and the SSI.

5.1. Instantaneous mixing properties

We first examine cases with an isolated shear layer, namely, $D=0$ and $D\rightarrow \infty$, to set the stage for cases with $D\sim O(1)$. When $D=0$, mixing efficiency peaks as the billows roll up ($t\sim 60$, black curve in figure 16c). At this pre-turbulent stage, the mixing rate is large (figure 16a) due to sharp scalar gradients, while the dissipation rate (figure 16b) remains small, and mixing efficiency is therefore large (Winters et al. Reference Winters, Lombard, Riley and D'Asaro1995; Caulfield & Peltier Reference Caulfield and Peltier2000; Smyth & Moum Reference Smyth and Moum2001; Smyth Reference Smyth2020). Subsequently, the billow structure collapses due to the SCI (§ 4), leading to an increase in both mixing and dissipation rates. Thus mixing efficiency is reduced at $t\sim 70$ as the flow becomes turbulent. As the billows pair and merge into a single large vortex ($t\sim 110$), the mixing and dissipation rates begin to rise.

In the case $D\rightarrow \infty$, $Ri_{min}$ doubles to 0.16. Therefore, mixing is visibly weaker than at $D=0$ (compare black solid and dashed curves in figure 16a). However, since the dissipation rate is also smaller, the peak mixing efficiency at $t\sim 208$ ($\eta _i=0.63$) for $D\rightarrow \infty$ is not very different from the peak value for $D=0$ at $t\sim 62$ ($\eta _i=0.78$). The two peaks of $\mathscr {M}$ are associated respectively with the breakdown of the billow and with mixing due to fully developed turbulence (cf. Kaminski & Smyth Reference Kaminski and Smyth2019, L23).

When $D=0.5$, the mixing characteristics resemble those at $D=0$. Mixing efficiency exhibits a peak during roll-up (as $t=76$, marked by the blue diamond in figure 16c). Strong mixing, quantified by the buoyancy variance dissipation rate $\chi '$ (2.19), begins along the braids and extends inwards through overturned layers surrounding the core (figure 17a).

When $D=1$, the time at which the billows roll up is the latest compared with other cases of $D$, consistent with its smallest growth rate (figure 4). The increase of mixing efficiency begins at $t\sim 180$ (red curve in figure 16c). Subsequently, the mixing rate increases rapidly due to the amplifying KH billow. At $t=217$, the 2-D kinetic energy reaches its maximum, while dissipation remains relatively weak, accounting for the highly efficient mixing. Before the SCI collapses the KH billow structure, the SSI emerges along the braids. The emergence of the SSI leads to a surge of highly efficient mixing ($t=215\unicode{x2013}235$ in figure 16a). Mixing is most intense in the braids, where it coincides with the secondary KH billows (figure 17b), and is most efficient at $t=235$ (red diamond) because the secondary billows have not yet become turbulent, with $\eta _i\sim 0.8$ (figure 16c).

The SSI billows travel along the braids towards the primary KH billow, and then intermingle with the shear-aligned convective rolls at the eyelids (at $x=40$, figure 17b). The primary KH billow then collapses, and the flow becomes more turbulent ($t\sim 250$). At this time, the mixing and dissipation rates approach their peak values, coinciding with a precipitous drop in the mixing efficiency (figure 16c).

The regime $D>D_c$, in which the oscillatory instability dominates (§ 3), is typified here by the cases $D=2$ and $D=3$. Both the mixing rate and dissipation rate are weak compared to cases where stationary mode dominates, e.g. $D=0$, $0.5$ and 1 (figures 16a,b). This weakening is due to the stronger stratification, which tends to damp both the SCI and pairing. In addition, the mutual interference of neighbouring billows suppresses the growth of the primary KH instability, leading to reduced overturning and 3-D convection, hence smaller $\mathscr {M}$ (compare $D=2$ and $D=\infty$ in figure 18a).

Figure 18.

Dependence of (a) cumulative mixing, (b) cumulative dissipation, and (c) cumulative mixing efficiency on $D$. All ensemble cases are plotted. Red curves are the ensemble mean. The end time for the time integral is when $\mathscr {M}=\mathscr {D}_p$. The vertical dashed lines denote the critical separation distance $D_c$. The horizontal line denotes the canonical $\eta _c=1/6$ suggested by Osborn (Reference Osborn1980).

While the general pattern of mixing, dissipation, and mixing efficiency remains largely consistent across all cases when $D>D_c$, there is a reduction in mixing efficiency as $D\rightarrow D_c^+$ (compare peak values for $D=\infty$, $D=3$ and $D=2$ in figure 16c). This reduction mainly reflects the diminished mixing rate $\mathscr {M}$ observed at smaller values of $D$. As shown in figures 17(c) and 17(d), $\chi '$ is less pronounced in case $D=2$ compared to case $D=3$. This suggests that as the nonlinear interaction between the upper and lower shear layers intensifies, mixing is suppressed.

5.2. Cumulative mixing properties

We next investigate the dependence of the cumulative mixing ($\mathscr {M}_c$), dissipation ($\epsilon _c$) and mixing efficiency ($\eta _c$) on the separation distance $D$. When $D< D_c$, the net mixing and dissipation are $\sim$1 order of magnitude larger than when $D>D_c$ (figures 18a,b). There is less disparity in $\eta _c$, indicating an approximate balance between mixing and dissipation that tends to preserve mixing efficiency. Even so, mixing is typically more efficient by a factor $\sim$2 when $D< D_c$ compared to when $D>D_c$. At the extremes $D=0$ and $D\rightarrow \infty$, $\eta _c$ takes the high values (0.3–0.4) expected for an isolated shear layer (Winters et al. Reference Winters, Lombard, Riley and D'Asaro1995; Caulfield & Peltier Reference Caulfield and Peltier2000; Smyth, Moum & Caldwell Reference Smyth, Moum and Caldwell2001).

In the oscillatory regime, the overall reduction in total amount of mixing as $D$ approaches $D_c$ from above may be attributed to the suppression of both the primary KH instability (due to interference between neighbouring billows impeding the phase-locking of resonant waves, as discussed in § 3) and secondary instabilities. The SCI, which plays a major role in driving mixing, can be impacted by both the reduced overturning in the suppressed primary KH instability and the neighbouring effect (§ 4.2). This suppression of the SCI becomes more pronounced as $D\rightarrow D_c^+$, potentially leading to a complete prevention of mixing – auxiliary simulations with $D=1.5$, not shown here, failed to generated detectable instability or mixing. While $\mathscr {M}_c$ decreases as $D\rightarrow D_c^+$, there is little corresponding change in total dissipation (figure 18b), leading to an overall decrease in mixing efficiency.

In the stationary regime $D< D_c$, there is a slight tendency towards stronger mixing and dissipation (figures 18a,b) with decreasing $D$. This is likely associated with the slight reduction of $Ri_{min}$.

5.3. Emergence of marginal instability

Geophysical stratified shear flows are often in a state of marginal instability (MI), wherein the mean flow fluctuates around a stability boundary approximated by $Ri_g=1/4$ (see Smyth (Reference Smyth2020), for a recent review). In the present simulations, we find MI-like behaviour when $D=2$ (figures 19b,e,h). As turbulence decays ($t\sim 250$), a layer of near-critical $Ri_g$ (i.e. clustered around a value near $1/4$) emerges around $z=0$ (figure 19(b), symbol ${\textsf{x}}$). This near-critical $Ri_g$ corresponds to a new stratified shear layer that forms between the two original layers (figures 19e,h) as mixing brings fluid from the upper and lower turbulent layers into close contact in the middle region, leading to local amplification of the mean buoyancy and velocity gradients.

Figure 19.

Horizontally averaged time series of (a–c) $Ri_g$, (d–f) $N^2$, and (g–i) $S^2$. The symbol $\textsf{x}$ indicates the potential location for MI to occur.

The MI appears only in a restricted range of $D$, namely when the instability is in the oscillatory regime ($D>D_c$) but $D$ is not much greater than $D_c$. Conversely, for $D< D_c$, the mixing characteristics resemble those of a typical KH instability, where both stratification and shear are smoothed due to strong overturning (figures 19d,g). This leads to an increase of $Ri_g$ towards a stable state (figure 19a). When $D$ is much greater than $D_c$, e.g. $D=3$, the upper and lower shear layers remain too distant to overlap despite their expansion. Consequently, the weakly stratified and weakly sheared middle layer (at $z=0$) persists (figures 19f,i) such that $Ri_g$ is much greater than $1/4$ (figure 19c).