Two pathways to diapycnal mixing in strongly stratified flows with no initial vertical shear

This study combines linear theory and direct numerical simulations to reveal that in strongly stratified flows with no initial vertical shear, horizontal shear instabilities inevitably drive diapycnal mixing through two distinct pathways—either via direct emergence of vertical shear or through a nonlinear evolution into columnar vortices—both of which ultimately trigger small-scale Kelvin-Helmholtz instabilities but yield different mixing efficiencies due to their excitation of distinct vertical scales.

The Gist

Imagine the ocean or the atmosphere as a giant, layered cake. The layers are made of fluids with different densities (like different flavors of cake), and they don't like to mix easily. Usually, scientists study what happens when you push these layers sideways and up or down at the same time (vertical shear). But this paper asks a different question: What happens if you only push the layers sideways, with no initial up-and-down movement, in a very strongly layered environment?

The researchers found that even if you start with a perfectly flat, horizontal flow, nature has two distinct "recipes" or pathways to create chaos (turbulence) and mix the layers. Which recipe nature chooses depends entirely on how you "seed" the experiment—essentially, what tiny little nudge you give the fluid at the very beginning.

Here is the breakdown of the two pathways using simple analogies:

The Setup: The Calm River

Imagine a wide, calm river flowing horizontally. The water is layered like a stack of pancakes (strong stratification). At first, the flow is smooth and two-dimensional (it only moves left and right, not up and down).

Pathway 1: The "Crowded Room" Effect (The Direct Route)

How it starts: You give the river a tiny, random nudge everywhere at once (like throwing a handful of confetti into the air).
What happens:

1. The Ripple: Because of the layers, the fluid doesn't just ripple left and right; it immediately starts rippling up and down in many different sizes at the same time. Think of this as a crowd of people in a room all trying to move at once, creating a chaotic, multi-directional jostle.
2. The Shear: These ripples create strong vertical currents (shear) very quickly.
3. The Breakup: These vertical currents get so strong that they snap, creating small, violent eddies (like tiny whirlpools). This is the "Kelvin-Helmholtz" instability, which looks like the breaking waves you see when wind blows over water.
   The Result: The mixing happens efficiently. Because the energy is spread out across many different sizes of ripples, the "friction" (viscous dissipation) is lower, making the mixing process relatively efficient.

Pathway 2: The "Synchronized Dance" Effect (The Indirect Route)

How it starts: You give the river a very specific, organized nudge (like a conductor waving a baton to get everyone to move in a specific pattern).
What happens:

1. The Vortex: Instead of chaotic ripples, the fluid organizes itself into long, vertical columns of swirling water (like giant tornadoes standing up in the river). For a long time, the flow stays perfectly two-dimensional, just these big swirling columns.
2. The Wobble: Eventually, these giant columns become unstable. They start to wobble in a very specific, high-frequency way. The researchers call this a "hyperbolic instability." Imagine a spinning top that starts to wobble violently right before it falls over.
3. The Breakup: This violent wobble creates very thin, sharp layers of vertical shear. These thin layers then snap into small, violent eddies, just like in Pathway 1.
   The Result: The mixing happens, but it is less efficient. Why? Because this pathway creates extremely thin, sharp layers. It takes a lot of energy (friction) to create and break these tiny, sharp layers. It's like trying to cut a thick block of cheese with a dull knife (Pathway 1) versus a razor blade (Pathway 2); the razor blade creates a much sharper, more energy-intensive cut.

The Big Takeaway

The paper proves that vertical shear (up-and-down movement) doesn't need to be there to start with. It is an inevitable by-product of horizontal shear in strongly layered fluids, provided the fluid is thick enough (high Reynolds number).

• If you start with random noise: You get Pathway 1 (Direct, efficient mixing).
• If you start with a specific pattern: You get Pathway 2 (Indirect, less efficient mixing).

The researchers used powerful computer simulations to show that these two paths are real, distinct, and that the "recipe" you choose at the start determines how much energy is wasted as heat versus how much is used to actually mix the layers.

In short: Even in a perfectly calm, layered fluid, a horizontal push will eventually create vertical chaos. But depending on how you start the push, that chaos will look different and mix the layers with different levels of efficiency.

Technical Summary

Problem Statement
The stability of large-scale, strongly stratified horizontal flows to three-dimensional (3D) perturbations is a critical mechanism for the downscale transfer of energy and diapycnal transport in geophysical and astrophysical systems. While vertically-sheared stratified flows are well-studied, their horizontally-sheared counterparts (with no initial vertical shear) have received less attention. A central question remains: how do initially two-dimensional (2D), strongly stratified ($Fr \ll 1$) horizontal shear flows transition to turbulence and generate vertical shear? Previous literature suggests two distinct pathways: one involving the direct growth of 3D primary modes leading to Kelvin-Helmholtz (KH) instabilities, and another involving the nonlinear evolution into columnar vortices followed by secondary instabilities. However, it has been unclear whether these pathways are mutually exclusive due to differences in initial conditions (e.g., hyperbolic tangent vs. sinusoidal profiles) or problem formulation (Initial Value Problem vs. body-forced), or if they represent alternative routes available within the same physical system. Furthermore, the impact of the selected pathway on diapycnal mixing efficiency has not been systematically quantified.

Methodology
The authors combine linear stability theory with Direct Numerical Simulations (DNS) to investigate these transitions in a triply-periodic domain. The study focuses on a vertically-invariant, horizontal, plane-parallel Kolmogorov flow ($\bar{u}(y) = \sin(y)\mathbf{e}_x$) in a linearly stratified fluid with constant buoyancy frequency $N$. The simulations are conducted at high Reynolds ($Re = 10,000$) and Péclet ($Pe = 10,000$) numbers, with low Froude numbers ($Fr = 0.15$ and $Fr = 0.1$), resulting in buoyancy Reynolds numbers ($Re_b = Fr^2 Re$) of 225 and 100, respectively.

To isolate the effect of initial conditions on pathway selection, two sets of DNS are performed:

1. DNS1 (Pathway P1): Initialized with grid-scale white noise, allowing all unstable modes of the primary horizontal shear instability to grow.
2. DNS2 (Pathway P2): Initialized with grid-scale noise plus a specific "seed" for the fastest-growing 2D ($k_z=0$) eigenmode of the primary instability.

Additionally, the authors perform a linear stability analysis of a "frozen-in" analytical model of the columnar flow that emerges in DNS2. This model approximates the nonlinearly evolved 2D flow as a Taylor-Green vortex array superimposed with sinusoidal flows, allowing for the calculation of eigenmodes and growth rates of secondary 3D perturbations.

Key Contributions and Results

1. Confirmation of Two Pathways in a Single Setup:
   The study demonstrates that both pathways to turbulence can occur within the exact same model setup, distinguished solely by the initial perturbation spectrum:

   • Pathway P1 (Direct 3D Growth): When initialized with white noise, $k_z \neq 0$ eigenmodes of the primary horizontal shear instability grow directly. These modes generate vertical shear on a broad range of scales, which subsequently becomes unstable to secondary, small-scale KH instabilities when $Re_b$ is sufficiently large. This pathway aligns with observations in body-forced Kolmogorov flows.
   • Pathway P2 (Columnar Evolution): When the $k_z=0$ mode is seeded, it dominates the flow, driving the background into a long-lived, time-dependent 2D columnar (vortical) flow. This 2D state is subsequently unstable to secondary 3D modes. The authors identify these secondary modes as hyperbolic instabilities arising from the hyperbolic stagnation points in the vortex array. These instabilities have high vertical wavenumbers ($k_z \sim Fr^{-1}$) and drive the emergence of vertical shear, which eventually triggers tertiary KH instabilities.

2. Characterization of the Secondary Instability in Pathway P2:
   Through linear stability analysis of the frozen columnar flow, the authors show that the secondary instability is a high-$k_z$ hyperbolic instability. The growth rate of this instability scales with the local strain rate of the hyperbolic points ($\Delta_h$) and is approximately $\lambda \approx 0.46 \Delta_h$, largely independent of stratification strength for large amplitudes. The instability manifests as "ripples" in the vertical velocity field centered around hyperbolic points, distinct from the traditional "zigzag" instability observed in isolated vortex pairs.

3. Impact on Mixing Efficiency:
   The study quantifies the diapycnal mixing and mixing efficiency ($\eta$) for both pathways.

   • Buoyancy Dissipation ($\chi$): The peak rate of buoyancy variance dissipation is comparable in both pathways, as it is controlled by small-scale motions ($k_z \gtrsim Fr^{-1}$) in both cases.
   • Viscous Dissipation ($\epsilon$): Pathway P2 exhibits significantly higher viscous dissipation than Pathway P1. This is because the hyperbolic instability in P2 excites a narrow, high-$k_z$ spectrum, leading to intense dissipation in thin vertical shear layers. In contrast, P1 excites a broad spectrum of vertical scales, distributing the dissipation more widely.
   • Mixing Efficiency: Consequently, the peak mixing efficiency is significantly higher in Pathway P1 ($\eta_{max} \approx 0.25$) than in Pathway P2 ($\eta_{max} \approx 0.165$). The difference arises because P2 dissipates more kinetic energy relative to the amount of buoyancy mixed.

Significance and Claims
The paper claims that the emergence of vertical shear driving small-scale KH instabilities is an inevitable by-product of horizontal shear instabilities in strongly stratified flows at sufficiently large $Re_b$, provided the flow has an inflection point. The selection between the two pathways is determined by the initial conditions: specifically, the ratio of the initial amplitude of the $k_z=0$ mode relative to the $k_z \neq 0$ modes.

The authors suggest that the prevalence of Pathway P2 in previous literature (specifically in hyperbolic tangent shear layer studies) may be an artifact of initialization practices where a $k_z=0$ seed is often added to accelerate computations, thereby suppressing the direct growth of 3D modes (Pathway P1). Conversely, the lack of P1 in some contexts may be due to weaker stratification limits where the range of unstable $k_z \neq 0$ modes is limited.

Ultimately, the work establishes that horizontal shear instabilities are key players in generating stratified turbulence and diapycnal mixing, even in the absence of pre-existing vertical shear. It highlights that the specific route to turbulence (direct 3D growth vs. 2D-to-3D transition) fundamentally alters the energy dissipation spectrum and, consequently, the efficiency of mixing, a factor that must be considered when parameterizing turbulence in climate and geophysical models. The authors note that while their results are robust for idealized flows, the interplay with rotation and ambient wave fields in realistic geophysical settings remains an open question.