Scaling laws and local enhancements of buoyancy flux in stratified turbulent flows

Through extensive direct numerical simulations of stratified turbulent flows, this study reveals that buoyancy flux exhibits strong intermittency and non-Gaussian statistics driven by large-scale long-time fluctuations, with its domain-averaged behavior scaling logarithmically with the buoyancy Reynolds number and being fundamentally linked to convective instabilities that trigger burst-like energy dissipation cycles.

The Gist

Imagine the atmosphere and the ocean not as smooth, flowing rivers, but as chaotic, churning soups. Sometimes, these soups are layered like a parfait—warm on top, cold on the bottom. This layering is called stratification. Usually, you'd think this makes the soup calm and orderly, like a calm lake. But this paper reveals a surprising secret: even in these calm, layered fluids, there are sudden, violent "bursts" of chaos that happen at huge scales, not just tiny ones.

Here is a breakdown of what the researchers found, using simple analogies:

1. The "Surprise Party" in a Calm Room

In normal, unlayered turbulence (like a blender mixing a smoothie), chaos happens mostly at the very small scale—tiny little swirls. But in stratified fluids (like the ocean or upper atmosphere), the researchers found that chaos can happen big.

Think of it like a quiet library. Usually, people whisper. But suddenly, a massive, unexpected shout erupts from a specific corner. This paper shows that in the ocean and sky, these "shouts" (sudden bursts of vertical movement and temperature change) happen frequently. They aren't just tiny ripples; they are huge, localized events that can be as big as the flow itself.

2. The "Traffic Jam" of Heat and Motion

The scientists were specifically looking at buoyancy flux. Let's call this the "heat-motion handshake." It measures how much heat is moving up or down at the exact same time as air or water is moving up or down.

• The Discovery: They found that this "handshake" is incredibly erratic. Sometimes, hot air rushes up violently, and sometimes cold air rushes down violently.
• The Analogy: Imagine a busy highway. Most of the time, cars move at a steady speed. But in this study, they found that occasionally, a massive pile-up happens where cars (heat and motion) suddenly accelerate or stop in a way that is completely unpredictable and extreme. These events are so extreme that the statistical "tails" of the data are huge—meaning the "weird" events are far more common than math usually predicts.

3. The "Goldilocks" Zone of Chaos

The researchers tested many different conditions, changing how "thick" the fluid is (viscosity) and how strong the layering is (stratification). They found a specific "sweet spot" or Goldilocks zone where these extreme bursts happen most often.

• Too much layering (Strong Stratification): The fluid is like a stiff gel. It just vibrates like a guitar string (waves) and doesn't mix much.
• Too little layering (Weak Stratification): The fluid is like a chaotic blender. It mixes everything, but the bursts aren't as distinct.
• Just right (The Sweet Spot): When the layering is moderate, the fluid gets unstable. It's like a stack of Jenga blocks that is almost stable but ready to collapse. In this zone, the "shouts" (the extreme bursts) are loudest.

4. The "Energy Debt" Mechanism

Why do these bursts happen? The paper proposes a simple mechanism: Energy Debt.

Imagine you have two bank accounts: one for "Up-and-Down Motion" (Kinetic Energy) and one for "Heat Potential" (Potential Energy).

• In a perfect world, these accounts would be balanced.
• In these flows, they get out of balance. The "Up-and-Down" account gets too high compared to the "Heat" account.
• Nature hates debt. To fix this imbalance, the system suddenly dumps all that extra energy in a massive, violent burst. This creates a whirlpool or a draft that mixes the layers, pays off the debt, and then the cycle starts all over again.

5. What This Means for the "Big Picture"

The paper doesn't claim to solve climate change or predict hurricanes directly. Instead, it provides a new rulebook for understanding how these fluids behave.

• The Rule: The intensity of these chaotic bursts follows a specific mathematical pattern (a power law) based on how strong the layering is.
• The Takeaway: Even in a stable, layered ocean or sky, you cannot assume the flow is smooth. There are hidden, massive "hot spots" of mixing that happen in bursts. If you are trying to model how heat or pollution moves through the ocean or atmosphere, you have to account for these sudden, violent spikes, not just the average flow.

Summary

This paper is like discovering that a calm lake isn't actually calm. It's full of hidden, massive underwater explosions that happen in a specific "Goldilocks" zone of stability. These explosions are driven by an imbalance between motion and heat, and they follow a predictable mathematical rhythm. Understanding this rhythm helps us realize that the atmosphere and oceans are far more "spiky" and unpredictable than we previously thought.

Technical Summary

Title: Scaling laws and local enhancements of buoyancy flux in stratified turbulent flows

Problem Statement
Stratified turbulent flows, prevalent in the atmosphere and oceans, exhibit intermittency not only at small scales but also at large scales comparable to the mean flow. While fully developed homogeneous isotropic turbulence (HIT) is known to be intermittent at small scales, evidence suggests that in stratified flows, the velocity field itself can display non-Gaussian statistics and "large-scale" intermittency. This phenomenon involves the emergence of powerful vertical velocity drafts that act as local energy injection mechanisms, leading to highly variable transport properties along the direction of gravity. The specific dynamics governing these large-scale events, particularly regarding the buoyancy flux ($B_f$), and their dependence on the buoyancy Reynolds number ($Ri_B$) and Prandtl number ($Pr$), require further elucidation to understand mixing and dissipation in geophysical contexts.

Methodology
The authors performed a large parametric exploration using Direct Numerical Simulations (DNS) of the Boussinesq equations. The study utilized the GHOST pseudo-spectral code with hybrid parallelization (MPI, OpenMP, CUDA).

• Governing Equations: The incompressible Navier-Stokes equations coupled with a temperature equation, including buoyancy terms characterized by the Brunt-Väisälä frequency ($N$).
• Parameters: The study varied the Froude number ($Fr$) over a geophysically relevant range ($0.01 \le Fr \le 1$), corresponding to buoyancy Reynolds numbers ($Ri_B$) from $0.06$ to $2300$. Two Prandtl numbers were examined: $Pr = 1$ and $Pr = 6$.
• Forcing Schemes: Two types of forcing were employed for the velocity field: a random Pouquet-Patterson forcing (PP) and a Taylor-Green forcing (TG).
• Simulation Scope: A total of 60 runs were analyzed, including series with $Pr=1$ (PP-Pr1, TG-Pr1), $Pr=6$ (PP-Pr6), and passive scalar simulations ($N=0$). Resolutions ranged from $128^3$ to $512^3$ grid points.
• Analysis: The study focused on high-order statistics (skewness and kurtosis) of vertical velocity ($w$), temperature fluctuations ($\theta$), and the buoyancy flux ($B_f = Nw\theta$). The analysis included volume and time averaging, as well as the examination of probability density functions (PDFs) and joint PDFs.

Key Contributions and Results

1. Large-Scale Intermittency in Buoyancy Flux:
   The study confirms that the buoyancy flux ($B_f$) exhibits strong non-Gaussian tails, with kurtosis values reaching as high as $\approx 10^2$. This indicates that stratified geophysical flows can be characterized by highly variable transport properties, even under stable stratification. This intermittency is associated with long-time, large-scale intermittent behavior of vertical velocity and temperature.

2. Scaling Laws and $Ri_B$ Dependence:

   • Skewness: The skewness of the buoyancy flux ($S_{B_f}$) increases with $Ri_B$ following a power-law scaling ($S_{B_f} \propto Ri_B^{0.28}$) as stratification weakens. This trend saturates in the passive-scalar limit ($N \to 0$), where temperature decouples from the velocity field.
   • Kurtosis Peaks: The maxima for the kurtosis of $w$, $\theta$, and $B_f$ occur within a narrow range of $1 < Ri_B < 2$. This range corresponds to a specific regime of wave-eddy balance.
   • Mean Buoyancy Flux: The domain-averaged buoyancy flux $\langle B_f \rangle$ exhibits two distinct trends: it scales logarithmically with $Ri_B$ ($\langle B_f \rangle \propto \ln(Ri_B)$) in the intermediate regime, and approaches a small offset as stratification strengthens ($Ri_B \to 0$).

3. Role of Prandtl Number:
   The variation of the Prandtl number ($Pr = 1$ vs. $Pr = 6$) does not appear to significantly affect the development of extreme vertical drafts or the emergence of large-scale intermittency, at least within the values considered. The shape of the scaling curves for different $Pr$ series is nearly identical, differing only in magnitude.

4. Mechanism of Extreme Events:
   A simple model for the temporal evolution of energy and buoyancy flux indicates that the "energy defect" between vertical kinetic energy and potential energy drives the strong events in buoyancy flux.

   • The model (based on Vieillefosse dynamics extended to stratified flows) suggests that a lack of equipartition between vertical and potential energy makes the buoyancy flux susceptible to local wave dynamics.
   • This energy imbalance leads to convective instabilities, the formation of 2D and 3D eddies, and rapid dissipation on the scale of a turnover time, allowing the energetic cycle to restart in bursts.
   • Joint PDFs reveal that at the peak of intermittency ($Ri_B \approx 1.78$), vertical velocity can reach up to $15\sigma_w$, while horizontal velocity remains limited, highlighting the anisotropic nature of these events.

5. Mixing Efficiency:
   The study analyzes mixing efficiency ($\langle B_f \rangle / \langle \epsilon_V \rangle$) and irreversible mixing efficiency ($\hat{\Gamma}$). Both quantities show a decrease as stratification strengthens until they saturate at low $Ri_B$. The authors note that while extreme events significantly affect local fluxes, they do not globally alter the mixing efficiency when averaged over the flow volume and time. However, local values of $B_f$ can differ by an order of magnitude between regions with extreme events and those without.

Significance
The paper claims that identifying these scaling laws and the specific conditions triggering extreme events is essential for improving the modeling and parametrization of geophysical and climatological flows.

• Predictability: The findings suggest that the rapid intensification of phenomena like hurricanes could be better predicted by tracking variations in vertical velocity, given the established link between vertical drafts and large-scale intermittency.
• Geophysical Relevance: The identified range of $Ri_B$ and $Fr$ where these extreme events occur is pertinent to both the atmosphere and the oceans.
• Modeling: Understanding the local enhancements of buoyancy flux and the energy defect mechanism provides a basis for refining sub-grid scale models and parameterizations used in climate and ocean models, particularly regarding how turbulence and waves interact to drive mixing.

The authors conclude that while the global mixing efficiency may remain relatively stable, the local, burst-like nature of dissipation and transport driven by these large-scale intermittent drafts is a critical feature of stratified turbulence that must be accounted for in geophysical applications.