Turbulence-induced anti-Stokes flow: experiments and theory

This paper presents experimental evidence and a supporting theoretical model demonstrating that the interaction between surface waves and ambient sub-surface turbulence generates a near-surface Eulerian-mean flow that opposes and partially cancels the Stokes drift, thereby significantly altering the vertical redistribution of momentum and the transport of water-borne materials in the ocean.

The Gist

Imagine the ocean as a giant, busy highway. Usually, we think of water moving in two ways: the big, slow currents (like the traffic flow) and the waves (like the cars zooming up and down).

For a long time, scientists believed that if you dropped a piece of trash or a tiny plankton into the ocean, it would just ride the wave forward. This "forward push" is called Stokes Drift. It's like a surfer who, even though they are just bobbing up and down, slowly drifts toward the shore with every wave.

The Mystery
However, when scientists went out to measure this in the real ocean, something weird happened. They found that the trash and plankton weren't drifting forward as much as the math predicted. In fact, sometimes they weren't drifting forward at all! It was as if the ocean had a secret "brake" that canceled out the wave's push.

Previous theories couldn't explain this. Some thought it was the Earth's rotation, but that was too slow to matter. Others thought it was just the shape of the waves. But the real culprit was hiding in plain sight: Turbulence.

The New Discovery: The "Anti-Stokes" Brake
This paper reports a breakthrough experiment that finally explains the mystery. The researchers built a giant water channel in a lab (think of it as a super-controlled swimming pool) and created two things:

1. Waves: Regular, rolling waves.
2. Turbulence: They used a giant, motorized "active grid" (like a massive, chaotic whisk) to stir the water and create a chaotic, swirling mess of currents before the waves arrived.

The Analogy: The Tangled Rope
Imagine the water is a long, tangled rope.

• Without Turbulence: If you shake one end of a straight rope, the wave travels smoothly, and the rope moves forward a bit.
• With Turbulence: Now, imagine the rope is already being violently shaken and twisted by a chaotic hand (the turbulence) before you send the wave.

When the wave hits this chaotic, swirling water, it doesn't just push the water forward. Instead, the wave interacts with the swirling eddies (the little whirlpools) and tilts and stretches them.

Think of it like a dancer (the wave) trying to walk through a crowd of people spinning wildly (the turbulence). The dancer's movement forces the spinning people to change their spin. In doing so, the crowd pushes back against the dancer.

The Result: The "Anti-Stokes" Current
This interaction creates a new, hidden current.

• The wave tries to push the water forward (Stokes Drift).
• The turbulence, reacting to the wave, creates a counter-current that pushes backward.
• This backward push is the "Anti-Stokes" flow.

In the experiment, the researchers saw that near the surface, this backward current was strong enough to almost completely cancel out the forward push of the waves. It's like a car trying to drive forward on a treadmill that is suddenly speeding up in the opposite direction.

Why Does This Matter?
This is a big deal for the real world.

• Oil Spills: If an oil spill happens, we need to know exactly where it will go. If we only calculate the wave push, we might think the oil is moving fast toward the shore, when in reality, the turbulence is holding it back or pushing it sideways.
• Plastic Pollution: Microplastics are everywhere. Understanding this "brake" helps us predict where they will end up.
• Nutrients and Life: Tiny marine life and nutrients rely on these currents to travel. If we get the math wrong, our models of the ocean ecosystem are wrong.

The Bottom Line
The ocean isn't just waves and currents; it's a complex dance between order (waves) and chaos (turbulence). When they meet, they create a hidden "anti-current" that slows down the drift of everything floating on the surface. This paper proves that turbulence is the secret ingredient that changes how things move in the ocean, and we need to account for it to predict the future of our oceans accurately.

Technical Summary

Here is a detailed technical summary of the paper "Turbulence-induced anti-Stokes flow: experiments and theory."

1. Problem Statement

The transport of water-borne materials (e.g., microplastics, oil spills, plankton) in the ocean is governed by the Lagrangian-mean current ($\bar{u}_L$), which is the sum of the Eulerian-mean current ($\bar{u}$) and the Stokes drift ($u_s$).

• The Paradox: Classical theory suggests $\bar{u}_L = \bar{u} + u_s$. However, numerous field and laboratory studies have observed that waves often appear to have no discernible effect on the Lagrangian current, or that the Stokes drift is entirely cancelled by an opposing Eulerian current.
• The Gap: Existing theories could not explain why an opposing Eulerian current arises to cancel the Stokes drift, particularly in the presence of ambient turbulence. Previous explanations (e.g., Earth's rotation, viscous boundary layers) were insufficient for the observed timescales and conditions.
• The Hypothesis: The authors propose that the interaction between surface waves and pre-existing ambient turbulence generates a new Eulerian-mean flow directed opposite to the wave propagation, termed an "anti-Stokes flow."

2. Methodology

The study combines controlled laboratory experiments with statistical theory and Rapid Distortion Theory (RDT).

Experimental Setup

Experiments were conducted in a water channel at NTNU, Trondheim, using an active grid to generate tunable ambient turbulence and a wavemaker to generate waves propagating upstream against the current.

• Measurement Technique: Particle Image Velocimetry (PIV) and Stereo-PIV (SPIV) were used to measure 3D velocity fields.
• Three Experimental Campaigns:
  1. Experiment 1 (Wave Groups): Turbulence interacted with transient wave groups. Measurements were taken before, during, and after the group passage to capture the "spin-up" (transient) phase.
  2. Experiment 2 (Regular Waves, Low Frequency): Turbulence interacted with continuous regular waves over long durations (approx. 40 mins) to reach a quasi-steady state.
  3. Experiment 3 (Regular Waves, High Frequency): Similar to Exp 2 but with higher acquisition rates (15 Hz) to better resolve the onset of the interaction.
• Variables: The study varied turbulence intensity (via active grid actuation), wave steepness, and wavelength.

Theoretical Framework

• Statistical Theory: The authors adapt arguments from Pearson (2018) and Harcourt (2013) based on the Craik–Leibovich equations. They derive a relationship for the quasi-steady state where the vertical gradient of the Eulerian current is balanced by the Reynolds stress induced by the interaction of Stokes drift and turbulence.
• Rapid Distortion Theory (RDT): To model the transient "spin-up" phase (Experiment 1), the authors use RDT. This linearized theory assumes that the distortion of turbulence by the wave-induced Stokes drift dominates over turbulent self-interaction. It predicts the generation of Reynolds shear stress ($u'w'$) which drives the mean current.
• Data Decomposition: A critical methodological step was separating wave motion from turbulence. The authors compared Phase-Conditioned Averaging (PhCA) with Proper Orthogonal Decomposition (POD), finding POD superior for their specific data conditions, particularly in minimizing "wave residue" in the turbulence signal.

3. Key Contributions

1. Experimental Evidence: Provided the first clear, controlled experimental evidence that ambient turbulence, when interacting with waves, generates a significant Eulerian-mean current opposing the Stokes drift.
2. Mechanism Identification: Identified the physical mechanism as the reorientation and stretching of turbulent eddies by the Stokes drift, leading to a redistribution of vertical momentum (Reynolds stress) without changing the depth-integrated mass transport.
3. Theoretical Unification:
   • Derived a quasi-steady state relation: $\frac{d\bar{u}}{dz} \approx -\frac{\overline{u'u'}}{\overline{w'w'}} \frac{du_s}{dz}$. This predicts that the slope of the induced current depends on the anisotropy of turbulence and the Stokes drift gradient.
   • Developed an RDT model to predict the transient growth of the anti-Stokes current during the "spin-up" phase.
4. Resolution of the "Cancellation" Puzzle: The paper explains why previous studies observed a cancellation of Stokes drift: the wave-turbulence interaction creates an opposing Eulerian current that partially or fully negates the Stokes drift near the surface.

4. Key Results

• Observation of Anti-Stokes Flow: In all experiments, the passage of waves caused a significant change in the Eulerian-mean current ($\Delta U$). This change was:
  • Opposite to the direction of wave propagation.
  • Strongly focused near the surface (decaying exponentially with depth).
  • Correlated with turbulence intensity and wave steepness (higher turbulence/steepness $\rightarrow$ larger $\Delta U$).
• Transient vs. Steady State:
  • Experiment 1 (Groups): The flow was in a transient "spin-up" state. The ratio of the current slope to the Stokes drift slope was less than 1, indicating the system had not yet reached equilibrium.
  • Experiments 2 & 3 (Regular Waves): The flow reached a quasi-steady state. The measured ratio of the current slope to the Stokes drift slope approached the theoretical prediction based on the turbulence anisotropy ratio ($\overline{u'u'}/\overline{w'w'}$).
• Validation of Theory:
  • The RDT model successfully predicted the qualitative shape and reasonable quantitative magnitude of the transient current in Experiment 1.
  • The statistical theory (Eq. 4.7) held well for the steady-state cases, confirming that the induced current is driven by the vertical gradient of the Reynolds stress generated by the Stokes drift acting on turbulence.
• Scaling: The magnitude of the induced current scales with the square of the wave steepness ($(ak_0)^2$) in the linear regime, consistent with the Stokes drift magnitude.

5. Significance and Implications

• Ocean Transport Modeling: The findings challenge the standard practice of simply adding Stokes drift to Eulerian currents in ocean models. If ambient turbulence is present (which is nearly always the case), the net Lagrangian transport of surface materials (oil, plastics, larvae) will be significantly different from predictions that ignore this interaction.
• Turbulence-Wave Interaction: The study provides a fundamental understanding of how waves modify turbulence structure (tilting and stretching vortices) and how turbulence, in turn, modifies the mean flow.
• Stability: The induced anti-Stokes current has a velocity profile opposite to the Stokes drift, which implies a stabilizing effect on the flow regarding the Craik–Leibovich (CL2) instability mechanism responsible for Langmuir circulation.
• Methodological Advance: The successful application of POD for separating wave and turbulence components in high-turbulence, low-frequency sampling scenarios offers a robust tool for future fluid dynamics research.

In conclusion, the paper resolves a long-standing discrepancy in wave-current interaction theory by demonstrating that turbulence is the catalyst for the generation of an anti-Stokes Eulerian current, fundamentally altering our understanding of surface transport in the ocean.