Evolution of passive scalar mixing layers in stratified and unstratified homogeneous turbulence

High-resolution large-eddy simulations reveal that while passive scalar mixing in stratified homogeneous turbulence behaves similarly to unstratified cases in the transverse direction, stratification severely inhibits vertical mixing by suppressing large-scale stirring, leading to distinct growth limits and fluctuation intensities that inform specific modeling approaches for scalar flux.

The Gist

Imagine you are watching a drop of ink spread through a glass of water. If you stir the water gently, the ink spreads out evenly. This is how scientists usually think about mixing in fluids: turbulence acts like a giant spoon, stirring everything together until it's uniform.

But what happens if that water is layered? Imagine the water at the bottom is heavy and salty, while the water at the top is light and fresh. This is called stratification. In the real world, this happens in the ocean (where deep water is denser) and in the atmosphere (where air gets thinner and lighter as you go up).

This paper is a high-tech computer experiment that asks: How does this layering change the way a "stain" (like pollution or smoke) spreads through a turbulent fluid?

Here is the story of what they found, broken down into simple concepts.

The Setup: Two Types of "Stains"

The researchers created a virtual fluid that was initially swirling chaotically (turbulent). Then, they introduced two different "stains" (passive scalars) to see how they behaved:

1. The Horizontal Stain: A layer of ink spread out sideways, like a flat sheet of paper floating in the water.
2. The Vertical Stain: A layer of ink spread out up and down, like a vertical wall of color.

They ran two simulations: one in normal water (unstratified) and one in "layered" water (stratified).

The Big Discovery: The "Up-Down" vs. "Side-to-Side" Difference

1. The Side-to-Side Stain (Transverse Layer)

What happened: When the stain was spread sideways, the layering didn't stop it from spreading. In fact, it spread slightly faster in the layered water than in the normal water.
The Analogy: Imagine a crowd of people running in a hallway. If the floor is perfectly flat (unstratified), they run in all directions. If the floor has a slight, invisible slope (stratified), they still run sideways just fine, maybe even a bit more energetically. The "ink" spreads out broadly in both cases.
The Catch: While the overall spread was similar, the "ink" in the layered water was more "spiky." Instead of a smooth gradient, it had sharper, more jagged edges. It was more "intermittent," meaning there were pockets of pure ink and pockets of pure water with less of a smooth middle ground.

2. The Up-Down Stain (Vertical Layer)

What happened: This is where the magic (and the restriction) happened. In the normal water, the vertical stain spread up and down easily, just like the sideways one. But in the layered water, the spreading almost stopped completely.
The Analogy: Imagine trying to stir a thick milkshake with a spoon. If you try to move the spoon up and down, the layers of the shake resist you. The heavy stuff wants to stay at the bottom, and the light stuff wants to stay at the top. The "stirring" motion gets squashed.
The Result: The vertical stain grew a tiny bit at the very beginning, but then it hit a "ceiling." It couldn't get any wider because the stable layers of the fluid acted like a lid, preventing the turbulence from mixing things vertically. The fluid could still swirl sideways, but it couldn't mix up and down.

Why Does This Matter? (The "Why" Behind the "What")

The researchers found that in the vertical direction, the fluid behaves like a spring. Once the turbulence tries to push a heavy layer up or a light layer down, gravity pulls it back. This stops the "stirring" motion.

However, the fluid can still swirl sideways. So, the "vertical length" of the turbulence gets locked into a specific size (determined by how strong the gravity and the fluid layers are), and the stain can't grow beyond that size.

The "Recipe" for Prediction

The paper also tried to create a simple mathematical "recipe" to predict how these stains would spread without needing a supercomputer.

• If you know the shape of the stain: You can use a simple one-number formula to predict how fast it spreads sideways. It works very well.
• If you don't know the shape: You have to guess the shape (assuming it looks like a smooth curve). If you do this, you need a two-number formula. This works great after the stain has had time to settle into a rhythm with the swirling fluid.

The Bottom Line

• Sideways mixing: Stable layers (like in the deep ocean or upper atmosphere) don't stop sideways mixing; they might even make it a bit more intense and jagged.
• Vertical mixing: Stable layers act like a brake. They stop the fluid from mixing up and down almost entirely.
• The "Stirring" vs. "Mixing" distinction: The fluid can still "stir" (move around) sideways, but it cannot "mix" (blend) vertically because the layers resist being swapped.

The authors note that their experiment used a specific type of fluid property (Prandtl number of 0.7). They warn that if the fluid were "thicker" or had different properties (Prandtl number > 1), the results might change because of a "reverse" effect where the mixing creates its own buoyancy. But for the conditions they tested, the "sideways is free, up-down is blocked" rule holds true.

Technical Summary

Technical Summary: Evolution of Passive Scalar Mixing Layers in Stratified and Unstratified Homogeneous Turbulence

Problem Statement
The dispersion of particles and scalars in stably stratified environments, such as the upper troposphere, stratosphere, and oceanic mixed layers, is a critical challenge in atmospheric and oceanic science. While the effects of stable stratification on the density field and turbulence decay are well-documented, the dispersion of passive scalars (e.g., aerosols, pollutants, microplastics) remains less understood. Specifically, it is unclear how a passive scalar mixing layer (PSML) evolves when introduced into homogeneous isotropic turbulence (HIT) that subsequently decays under the influence of a stabilizing density gradient. The central question is how the introduction of buoyancy scales and anisotropy alters the self-similar evolution of scalar statistics observed in unstratified HIT.

Methodology
The authors employed high-resolution large-eddy simulations (LES) to investigate the decay of both unstratified and stably stratified homogeneous turbulence. The simulations utilized the non-divergent Navier-Stokes equations with the non-hydrostatic Boussinesq approximation.

• Flow Configuration: Two simulations were conducted. The first was unstratified HIT. The second was identical in initial conditions but included a uniform stabilizing density gradient ($g \neq 0$), causing the flow to transition to strongly stratified turbulence after approximately one buoyancy period.
• Passive Scalars: Two passive scalar mixing layers were initialized: one with a mean gradient in the vertical direction ($\xi_z$) and one in the transverse direction ($\xi_y$). These were modeled as mixture fractions ($0 \le \xi \le 1$).
• Numerical Details: To minimize diffusive effects that cause departures from self-similarity, the simulations used hyperviscosity and hyperdiffusivity with equal coefficients. The molecular Prandtl number was set to $Pr = 0.7$. The Reynolds numbers were high ($Re_h > 15,000$ and $Re_\lambda > 450$ for the unstratified case), ensuring the existence of inertial-convective and inertial-diffusive ranges. The resolution was sufficient to resolve the buoyancy length scale ($L_b/\Delta \gg 1$).
• Analysis: The study tracked the evolution of scalar profile widths, variance, higher-order moments (skewness, kurtosis), fluxes, and mixing efficiency. It also evaluated eddy-diffusivity models for the transverse scalar flux under two conditions: when the mean scalar profile is known and when it must be assumed.

Key Results

1. Transverse Mixing ($\xi_y$): In the direction perpendicular to gravity, the evolution of the passive scalar in the stratified case is broadly similar to the unstratified case, though it spreads slightly faster.

   • The scalar fluctuations are more intense in the stratified case, with peak variance approximately 15% higher.
   • The turbulent/non-turbulent interface is more intermittent in the stratified case.
   • Once the scalar field reaches quasi-equilibrium with the velocity field, the normalized flux profiles for both stratified and unstratified cases collapse, indicating similar underlying transport mechanisms despite the stratification.

2. Vertical Mixing ($\xi_z$): Stratification drastically inhibits vertical mixing.

   • The vertical layer width ($\delta_z$) grows initially but rapidly ceases to expand after approximately one buoyancy period ($t^* \approx 3$).
   • This cessation of growth is attributed to the suppression of large-scale vertical stirring by buoyancy forces.
   • The final width of the layer is proportional to the vertical integral length scale of the horizontal velocity ($L_v$), which is constrained to maintain a vertical Froude number of order one ($Fr_v \sim O(1)$).
   • While the mean profile shows monotonic growth due to molecular diffusion, the flux analysis reveals a "reverse flux" phenomenon where stirring occasionally sweeps scalar tendrils back toward their original side, consistent with internal wave-like behavior.

3. Mixing Efficiency: The cumulative mixing (dissipation of scalar variance) in the stratified case eventually exceeds that of the unstratified case at late times. This is attributed to the fact that mixing follows stirring, and the horizontal stirring rate in the stratified flow is proportional to the horizontal length scale, which grows larger than in the unstratified case.

4. Modeling Flux:

   • Known Profile: If the mean scalar profile is known, a single-constant eddy-diffusivity model ($F_y = -c_1 \delta_y v' \partial \xi / \partial y$) effectively predicts the flux for both stratified and unstratified cases, with $c_1 \approx 0.4$.
   • Assumed Profile: If the profile shape (assumed to be an error function) and the scalar length scale must be estimated from velocity statistics, a two-constant model is required. This model performs well at late times when the scalar is in quasi-equilibrium with the velocity field, allowing the scalar length scale to be scaled from the kinetic energy dissipation rate.

Significance and Claims
The paper claims to provide a detailed characterization of passive scalar mixing in decaying stratified turbulence, bridging the gap between theoretical predictions of anisotropic diffusion and numerical observations.

• Anisotropy: The study confirms that stratification creates a stark dichotomy in mixing: horizontal mixing remains active and even enhanced, while vertical mixing is effectively suppressed once the flow becomes strongly stratified.
• Modeling Utility: The authors demonstrate that standard eddy-diffusivity approaches remain viable for stratified flows, provided the scalar is in quasi-equilibrium with the velocity field. They highlight that assuming a known profile significantly simplifies the modeling requirements compared to assuming a profile shape.
• Limitations and Future Work: The authors modestly note that their results are specific to a Prandtl number of 0.7. They explicitly state that recent literature suggests higher Prandtl numbers ($Pr > 1$) induce reverse buoyancy fluxes at small scales, which could alter flow decay rates and, consequently, passive scalar mixing rates. Therefore, the current findings do not directly apply to high-Prandtl-number scenarios without further investigation.

The study concludes that while stratification fundamentally alters the vertical dynamics of a mixing layer, the transverse mixing retains many characteristics of unstratified turbulence, albeit with increased intermittency and fluctuation intensity.