Instability and breaking of internal waves in a horizontal shear layer

This study investigates the instability and breaking of internal waves in a horizontal shear layer by combining ray-tracing theory with direct numerical simulations to identify two distinct mechanisms—refraction-induced steepening and advection-driven momentum transfer—that govern wave-breaking dynamics, energy distribution, and turbulent dissipation.

The Gist

Imagine the ocean (or the atmosphere) as a giant, multi-layered cake. The layers aren't made of sponge and cream, but of water at different temperatures and salinities. Because of this, the water is "stratified"—it wants to stay in its own layer, like oil floating on water.

Now, imagine sending a ripple through this cake. These aren't surface waves you see at the beach; they are internal waves, huge, slow-moving ripples that travel inside the ocean, between the layers.

This paper is about what happens when these internal waves run into a "wind tunnel" of moving water. Specifically, they look at what happens when a wave hits a current that is moving sideways (horizontally) relative to gravity.

Here is the story of the paper, broken down into simple concepts:

1. The Setup: The Wave and the Sideways Wind

Think of the internal wave as a surfer riding a long, invisible board. Usually, the surfer just rides the wave. But in this experiment, the surfer is also trying to ride through a strong, sideways wind (the horizontal shear flow).

The scientists wanted to know: Does the sideways wind help the wave crash, or does it just push it along?

2. The Two Ways a Wave Can "Crash"

The researchers discovered that the sideways wind can break the wave in two very different ways, depending on how the wave is set up. They created a "scorecard" (a number they call F) to predict which way it would go.

Scenario A: The "Steepening" Break (Low Score)

• The Analogy: Imagine a car driving up a hill that gets steeper and steeper. The car doesn't speed up, but the road gets so steep that the car flips over.
• What happens: The sideways wind acts like a lens, bending the wave. This bending makes the wave get taller and steeper in one specific spot. Eventually, the wave gets so steep that the top of it curls over and crashes, just like a normal ocean wave.
• The Result: The energy is split evenly between the movement of the water and the height of the wave. It's a "messy" crash involving both height and speed.

Scenario B: The "Shear" Break (High Score)

• The Analogy: Imagine a deck of cards. If you push the top of the deck sideways while holding the bottom still, the cards slide past each other. If you push hard enough, the stack gets unstable and the cards fly apart.
• What happens: Here, the wave doesn't get taller. Instead, the sideways wind "drags" the water along with it. This creates a massive difference in speed between the top and bottom of the wave (like the deck of cards). This speed difference creates a "shear" that rips the wave apart.
• The Result: The energy is mostly in the speed of the water, not the height. It's a violent, fast crash driven by friction between layers.

3. The Prediction Tool

The authors built a mathematical model (using something called "ray tracing," which is like tracking a beam of light) to predict which of these two scenarios would happen. They found that even though their math was based on simple, small waves, it actually did a great job predicting what happened in their complex, computer-simulated "turbulent" oceans.

4. The Big Surprise: The Energy Explosion

This is the most exciting part. When the wave breaks, it turns into turbulence (chaotic, swirling water).

• The Expectation: You might think the turbulence uses up the energy of the wave.
• The Reality: In the "Shear" scenario (Scenario B), the turbulence didn't just use the wave's energy. It stole energy from the background wind (the sideways current).
• The Metaphor: It's like a small spark (the wave) hitting a gas tank (the background wind). The explosion (turbulence) is way bigger than the spark. In some cases, the turbulence created 7 to 10 times more energy than the original wave had!

5. Why Should We Care?

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

This matters because the ocean is a giant heat engine. It moves heat from the equator to the poles.

• Mixing: When these waves break and create turbulence, they mix the layers of the ocean. This is how nutrients from the deep ocean get to the surface (feeding fish) and how heat gets trapped or released.
• Efficiency: The paper shows that how the wave breaks changes how well it mixes.
  • If it breaks like a steep wave (Scenario A), it mixes very efficiently.
  • If it breaks like a shear (Scenario B), it mixes very poorly, even though it's more violent.

Summary

The paper is a detective story about how invisible waves in the ocean interact with sideways currents. They found that:

1. Sideways currents can break waves in two distinct ways: by making them too steep or by shearing them apart.
2. We can predict which way it will happen using a simple number.
3. When the "shearing" happens, the wave acts like a trigger that unlocks a massive amount of energy from the background current, creating a huge burst of turbulence.

This helps scientists understand how the ocean mixes, how heat is stored, and how marine life gets fed, all by watching how a single wave crashes in a current.

Technical Summary

1. Problem Statement

The study investigates the dynamics of internal gravity waves propagating through a background horizontal shear flow, where the shear direction is orthogonal to gravity. Unlike the classical problem of waves interacting with vertical shear (which leads to critical levels where intrinsic frequency vanishes), this study focuses on horizontal shear.

Key questions addressed include:

• How do internal waves extract energy from a horizontal shear flow?
• What are the distinct mechanisms leading to wave steepening and eventual breaking?
• How do the routes to instability (convective vs. shear-driven) influence the resulting turbulence, momentum transfer, and mixing efficiency?

The authors aim to bridge the gap between linear ray-tracing theory and fully nonlinear turbulent breakdown, specifically exploring the role of the lift-up mechanism (wave advection of mean momentum) versus wave refraction (steepening due to shear).

2. Methodology

Theoretical Framework (Ray-Tracing & WKBJ)

The authors employ Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) ray-tracing theory to predict the local state of the wave prior to breaking.

• Governing Equations: They derive local polarization relations for a wave packet in a horizontal shear flow $U(y) = \Delta u \tanh(y/h)$.
• Two Mechanisms for Energy Growth:
  1. Refraction: As the wave propagates into the shear, its vertical wavenumber changes, increasing wave steepness ($s$). This can lead to convective instability (overturning).
  2. Wave Advection (Lift-up): The wave advects the background mean momentum, generating large streamwise velocity perturbations ($u'$). This creates enhanced vertical shear without necessarily increasing wave steepness.
• Dimensionless Parameter $F$: A key contribution is the construction of a dimensionless perturbation energy ratio, $F$, defined as:
  $$F^2 = \frac{k_y^2 k_z^2}{\Omega^4 |k|^4} \left(\frac{\partial U}{\partial y}\right)^2$$
  • Small $F$: Energy growth is dominated by refraction/steepening. Instability is expected to be a mix of shear and convection (convective breaking).
  • Large $F$: Energy growth is dominated by wave advection of momentum. Instability is driven by enhanced vertical shear (shear breaking).
• Stability Criterion: The authors relate $F$ to the local gradient Richardson number ($Ri_g$), showing that $Ri_g < 1/4$ (the threshold for shear instability) is achievable when $F$ is sufficiently large, even if the wave steepness $s < 1$.

Numerical Simulations (DNS)

To test theoretical predictions, the authors perform Direct Numerical Simulations (DNS) using the pseudo-spectral solver Diablo.

• Setup: A monochromatic internal wave packet is initialized in a stratified horizontal shear layer.
• Parameter Space: Simulations span a range of initial wave steepness ($s_0$), shear strength ($\Delta u$), and wavenumbers ($k_x, k_y$).
• Cases:
  • Trapping ($k_x < 0$): Waves are refracted toward a "trapping level" where group velocity vanishes.
  • Reflection/Transmission ($k_x \geq 0$): Waves reflect off or pass through the shear layer.
• Inclined Domain: To handle small $k_x$ values efficiently, simulations are performed in an inclined reference frame, allowing the wave to be independent of the streamwise coordinate.

3. Key Contributions

1. Identification of Dual Breaking Mechanisms: The paper rigorously distinguishes between two pathways to wave breaking in horizontal shear:
   • Convective Breaking: Driven by wave steepening (refraction), leading to density overturning.
   • Shear Breaking: Driven by the "lift-up" mechanism, where wave advection of momentum creates strong vertical shear, leading to Kelvin-Helmholtz-like billows even without density overturning.
2. Predictive Parameter $F$: The introduction of the parameter $F$ allows for the prediction of the dominant instability mechanism based solely on initial conditions. It serves as a proxy for the minimum gradient Richardson number.
3. Validation of Linear Theory in Nonlinear Regime: Despite the breakdown of WKBJ assumptions (scale separation) near breaking, the linear ray-tracing theory successfully predicts the partition of kinetic and potential energy and the type of instability observed in fully nonlinear DNS.
4. Mixing Efficiency Variability: The study demonstrates that mixing efficiency is not constant but depends critically on the breaking mechanism.

4. Key Results

Dynamics of Wave Breaking

• Wave Trapping ($k_x < 0$):
  • Waves focus energy at a trapping level, increasing steepness ($s$).
  • If $F$ is small, breaking is convective (density overturning).
  • If $F$ is large (strong shear), the wave advection mechanism amplifies vertical shear, leading to shear instability.
  • A "mean flow jet" forms due to wave-induced acceleration, which can further destabilize the flow.
• Wave Reflection/Transmission ($k_x \geq 0$):
  • When $k_x = 0$ (no refraction), the wave does not steepen, but the lift-up mechanism generates significant vertical shear.
  • For large $F$, this leads to pure shear instability (Kelvin-Helmholtz billows) even when the wave steepness remains low ($s < 1$).
  • For $k_x > 0$, interference between incident and reflected waves can create localized static instability, but the subsequent turbulence is still driven by the enhanced vertical shear.

Energy and Momentum Budgets

• Energy Dissipation: In cases dominated by shear instability (high $F$), the total turbulent dissipation can exceed the initial wave energy by a factor of 5 to 7.5. This indicates a subcritical transition where the background shear flow itself supplies significant energy to the turbulence.
• Momentum Transfer: Wave breaking leads to significant deceleration of the background shear flow. In wave-trapping cases, a narrow jet forms before breaking; in shear-driven cases, the momentum transfer is more uniform but results in substantial drag.
• Internal Wave Emission: Turbulent breakdown generates secondary internal waves. In some cases, the energy radiated away as new waves is up to 25% of the total energy dissipated.

Mixing Efficiency

• Strong Dependence on Breaking Route:
  • Convective Breaking (Low $F$): Kinetic and potential energy are roughly equipartitioned. Mixing efficiency ($\eta$) is high, reaching up to 0.42.
  • Shear Breaking (High $F$): Kinetic energy dominates. Mixing efficiency is low, dropping to 0.08.
• This challenges the assumption of constant mixing efficiency in ocean models, suggesting that the "history" of the wave breaking mechanism dictates the mixing outcome.

5. Significance and Implications

• Oceanic Mixing: The findings suggest that internal wave breaking in the ocean (where horizontal shear is common due to currents and tides) may be a primary source of highly variable mixing efficiency. The "lift-up" mechanism provides a route to turbulence that does not require large wave amplitudes, potentially explaining mixing in regions where waves appear stable.
• Parameterization: Current ocean models often use fixed mixing efficiencies or parameterizations based on vertical shear. This study argues for the inclusion of the horizontal shear interaction and the $F$ parameter to better predict energy dissipation and mixing.
• Theoretical Robustness: The success of ray-tracing theory in predicting the type of instability (convective vs. shear) despite the highly nonlinear nature of the final breakdown validates the use of linear theory as a guide for complex turbulent transitions in stratified flows.
• Potential Vorticity: The study highlights that horizontally bounded wave packets carry potential vorticity (via Lagrangian vorticity dipoles), which may play a crucial role in the energy cascade to small scales, a feature often missing in idealized vertical shear models.

In summary, this paper provides a comprehensive framework for understanding how internal waves break in horizontal shear, identifying a new dominant mechanism (lift-up) that drives shear instability and significantly alters the energetics and mixing properties of the resulting turbulence.