Early onset of secondary shear instability in Kelvin-Helmholtz braids at high Reynolds number

This study demonstrates through an inviscid model and high-Reynolds-number simulations that secondary shear instability can onset early in Kelvin-Helmholtz braid regions at high stratification, potentially controlling turbulent transition and diapycnal mixing before primary billows saturate or pairing instabilities occur.

The Gist

Imagine the ocean as a giant, layered cake. Sometimes, these layers slide past each other at different speeds, creating a friction zone in the middle. In fluid dynamics, this friction creates a famous phenomenon called Kelvin-Helmholtz (KH) instability.

If you've ever seen clouds that look like rolling waves or a surfer's crest, you've seen KH instability. In the ocean, these "waves" curl up into giant, swirling tubes of water called billows.

For a long time, scientists thought the story ended there: the big billows would grow, crash, and then break down into chaotic turbulence, mixing the ocean layers together. But this new paper tells us that the story is actually much more dramatic and happens much faster than we thought.

Here is the simple breakdown of what the researchers found, using some everyday analogies.

1. The "Braids" Between the Waves

Between two giant rolling billows, there is a thin, stretched-out strip of water connecting them. The researchers call this the braid.

Think of the billows as two giant, rolling doughnuts. The braid is the thin, stretched piece of dough connecting them.

• The Old View: Scientists used to think these braids were just passive connectors, waiting for the big doughnuts to finish rolling before anything interesting happened.
• The New Discovery: The researchers found that these braids are actually the hotspots of action. They are where the real mixing happens, often before the big billows even finish growing.

2. The Tug-of-War: Stretching vs. Squeezing

Inside these braids, two opposing forces are fighting a tug-of-war:

• Force A (The Stretch): The big billows are rolling, which stretches the braid out like taffy. This stretching tends to stabilize the braid, making it calm and orderly.
• Force B (The Squeeze): Because the ocean has layers of different densities (like saltier water under fresher water), the stretching also squeezes the layers together. This creates a "baroclinic shear"—a fancy way of saying the layers get so close and steep that they become unstable.

The Analogy: Imagine you are stretching a rubber band (Force A). Usually, stretching makes it stable. But if you stretch it so fast that the material inside gets compressed and heated (Force B), it might suddenly snap or tear.

The paper shows that at high speeds (high Reynolds numbers, which is like the ocean's "speed limit"), Force B wins. The compression happens so fast that the braid becomes unstable and starts churning into turbulence while the big billows are still growing.

3. The "Early Onset" Surprise

The most exciting part of this paper is the timing.

• The Old Expectation: We thought the big billows would grow to their full size, stop, and then the braids would break.
• The Reality: The braids break early.

The Metaphor: Imagine a construction site.

• Old Theory: The crane (the billow) finishes lifting the beam, sets it down, and then the workers start welding the joints (the braid).
• New Theory: The crane is still lifting the beam, but the joints are already sparking and welding themselves because the tension is so high. The "welding" (turbulence) starts before the "lifting" is even done.

This "Early Onset" happens because the ocean is so stratified (layered) and moving so fast that the instability in the braid grows faster than the billow itself can finish forming.

4. Why Does This Matter?

You might ask, "So what? It's just water moving."

This is crucial for understanding how the Earth works:

• Ocean Mixing: The ocean needs to mix heat, salt, and nutrients. If the mixing happens in the braids (as this paper suggests) rather than in the big billows, it changes how we calculate ocean currents and climate models.
• Predicting Turbulence: For a long time, scientists thought they had to wait for the big billows to collapse to predict turbulence. This paper says, "No, look at the thin strips connecting them!" If you want to know when the ocean will get turbulent, watch the braids.

5. The "Perfect Storm" Conditions

The researchers used super-computers to simulate this. They found that this "early explosion" of turbulence only happens under specific conditions:

1. High Speed: The water must be moving fast enough (High Reynolds Number).
2. Strong Layers: The density differences must be significant (High Richardson Number).

If the water is moving too slowly or the layers are too weak, the "stretching" force wins, and the braid stays calm. But in the real ocean, where speeds and layers are strong, the "squeezing" force wins, and the braid goes chaotic early.

Summary

This paper is like discovering that the cracks in a breaking egg happen before the shell fully shatters.

By focusing on the thin, stretched connections (braids) between the big swirling waves (billows), the authors found that these connections become unstable and start mixing the ocean much earlier than previously thought. This changes our understanding of how the ocean mixes heat and energy, suggesting that the "braids" are the true engines of turbulence in the deep sea.

Technical Summary

1. Problem Statement

The transition from laminar to turbulent flow in stably stratified, parallel shear flows via Kelvin–Helmholtz (KH) instability is a fundamental problem in geophysical fluid dynamics. While the primary KH instability produces large-scale vortex structures ("billows"), the subsequent transition to three-dimensional turbulence is governed by secondary instabilities.

Traditionally, it was assumed that secondary instabilities (such as vortex pairing or convective instabilities in the billow core) only develop after the primary billow saturates. However, field observations in the ocean (Reynolds numbers $Re \approx 10^5 - 10^8$) frequently show intense turbulent mixing concentrated in the braids (the thin strips of vorticity connecting adjacent billows) rather than the billow cores. This suggests that a specific secondary instability, Secondary Shear Instability (SSI), may trigger turbulence before the primary billow saturates.

The paper addresses the gap in understanding when and why this early onset of SSI occurs, particularly at geophysically relevant high Reynolds numbers ($Re$) and Richardson numbers ($Ri$). Previous models (e.g., Corcos & Sherman 1976; Staquet 1995) often assumed a steady or "frozen" background flow after billow saturation, failing to capture the dynamics of the evolving braid during the growth phase.

2. Methodology

The authors employ a dual approach combining a semi-analytical inviscid model and high-resolution Direct Numerical Simulations (DNS).

A. Inviscid Theoretical Model

• Coordinate System: The authors modify the classical analysis of Corcos & Sherman (1976) by defining a braid-aligned coordinate system ($x', z'$) tilted at an angle $\phi(t)$ relative to the horizontal.
• Governing Equations: They derive simplified equations for buoyancy ($b$) and vorticity ($\Omega$) in the braid region, assuming a uniform compressive strain field ($w' = -\gamma z'$) and neglecting diffusion in the inviscid limit ($Re \to \infty$).
• Similarity Solutions: The model assumes similarity solutions for the thickness of the buoyancy and velocity layers ($\delta(t)$).
• Key Dynamics: The model tracks two competing effects:
  1. Strain: Compresses the braid, thinning it and stabilizing the flow.
  2. Baroclinic Shear: The compression of tilted isopycnals generates shear ($S(t)$), which destabilizes the flow.
• Stability Criterion: Instead of relying solely on the gradient Richardson number ($Ri_g < 1/4$), which is often insufficient in strained flows, the authors adopt a criterion based on the ratio of strain rate to shear ($\gamma/\Omega$). Instability is predicted when this ratio drops below a critical threshold $\Gamma^*$.

B. Numerical Simulations (DNS)

• Solver: 2D Direct Numerical Simulations were performed using Oceananigans.jl (GPU-accelerated).
• Parameter Space: 16 simulations covering:
  • Richardson numbers: $Ri = [0.05, 0.10, 0.15, 0.20]$
  • Reynolds numbers: $Re = [10^4, 10^5, 10^6, 10^7]$
• Setup: The domain was restricted to a single primary billow wavelength to isolate braid dynamics and exclude vortex pairing instabilities.
• Diagnostics: The simulations tracked the evolution of the braid angle ($\phi$), strain rate ($\gamma$), and the ratio $\gamma/\Omega$ to determine the onset time of SSI ($t_S$) compared to the primary billow saturation time ($t_{2D}$).

3. Key Contributions

1. Time-Dependent Braid Model: The study relaxes the "frozen flow" assumption of previous literature. It explicitly models the time-evolving steepening of the braid, demonstrating that the braid dynamics are critical for predicting instability onset.
2. Identification of Early Onset Mechanism: The paper establishes that at sufficiently high $Ri$, increased stratification slows the growth of the primary billow (delaying saturation) while simultaneously accelerating baroclinic shear production in the braid. This creates a window where SSI can grow before the primary billow saturates.
3. Refined Stability Criterion: The authors validate that the ratio of strain to shear ($\gamma/\Omega$) is a robust predictor for SSI onset, identifying a critical threshold of $\Gamma^* \approx 0.02$.
4. Viscosity Effects: The study quantifies how viscosity delays braid thinning at lower $Re$, effectively preventing the early onset of SSI in regimes where the inviscid model predicts it.

4. Key Results

• Inviscid Limit ($Re \to \infty$): The theoretical model predicts that SSI onset time ($t^*$) scales as $1/\sqrt{\gamma_1}$ (where $\gamma_1$ is the strain growth rate). For $Ri \gtrsim 0.10$, $t^*$ occurs significantly earlier than the primary billow saturation time ($t_{2D}$).
• High Reynolds Numbers ($Re \ge 10^6$): DNS results confirm the inviscid model. SSI develops early in the braid while the primary billow is still growing. The braid dynamics are essentially inviscid, and the onset time matches the theoretical prediction.
• Intermediate Reynolds Numbers ($Re = 10^5$): Viscosity slightly slows braid thinning, delaying $t^*$ and narrowing the time window ($t_{2D} - t^*$) available for SSI growth. For $Ri=0.10$, SSI becomes marginal and requires additional noise to trigger.
• Low Reynolds Numbers ($Re = 10^4$): Viscous diffusion halts braid thinning before the critical shear is reached. Consequently, SSI is suppressed for $Ri = 0.10$ and $0.15$, occurring only marginally at $Ri=0.20$.
• Stratification Dependence: As $Ri$ increases, the primary billow grows slower (flatter cores), but the baroclinic shear in the braid intensifies faster. This separation in timescales is the primary driver for early SSI onset.
• Comparison with Other Instabilities: The study notes that at high $Ri$, SSI outpaces both vortex pairing (which slows with increasing $Ri$) and convective instabilities (which typically require the billow core to be fully formed).

5. Significance

• Mechanistic Explanation for Field Observations: The findings provide a physical basis for the widespread observation of "braid-dominated mixing" in the ocean. It explains why intense mixing occurs in the braids of KH billows even when the billow cores remain relatively laminar.
• Predictive Capability: The study offers a predictive framework for when 3D turbulence will transition in stratified shear flows. It suggests that at geophysically relevant $Re$ and $Ri$, SSI controls the transition, pre-empting other instability mechanisms.
• Implications for Modeling:
  • Mixing Parameterizations: Ocean models must account for braid-dominated mixing, which may arrest primary billow growth and alter diapycnal mixing rates.
  • Numerical Resolution: The study highlights a computational challenge: capturing the rapid thinning of the braid requires extremely fine grid resolution ($\delta > 10 \Delta z$) to avoid spurious instabilities. This constraint may be more restrictive than standard Kolmogorov scale requirements in 3D simulations.
  • Prandtl Number: While the current study assumes $Pr=1$, the authors note that in the ocean ($Pr \approx 10-1000$), higher Prandtl numbers may further promote SSI by reducing the interface thickness where diffusion acts, though this requires further investigation.

In conclusion, this paper fundamentally shifts the understanding of KH transition in stratified flows, demonstrating that secondary shear instability in the braids is not a late-stage phenomenon but a primary driver of turbulence that can occur well before the main vortex structures saturate.