Benchmarking Turbulence Models to Represent Cloud-Edge Mixing

This study benchmarks several statistical turbulence models against direct numerical simulations for cloud-edge mixing, revealing that while all models successfully capture thermodynamic evolution, they vary in their ability to accurately represent changes in cloud microphysics.

The Gist

The Big Picture: Why Clouds Are Hard to Predict

Imagine trying to predict the weather. Clouds are a huge part of that, but they are tricky. They are made of tiny water droplets mixed with air. To understand how clouds form, grow, or disappear, we need to understand how they mix with the dry air around them.

The problem is that this mixing happens on two very different scales:

1. The Big Picture: Clouds stretch for kilometers.
2. The Tiny Details: The mixing of air and water droplets happens in millimeters.

Computer models used for weather forecasting are like low-resolution cameras. They can see the big clouds, but they are too "blurry" to see the tiny, chaotic swirls of air (turbulence) that happen at the edges of the clouds. Because they can't see these tiny swirls, scientists have to use "shortcuts" (simplified models) to guess what is happening at the edges.

This paper asks a simple question: Which of these shortcuts actually works?

The Experiment: The "Cloud Filament"

To test this, the researchers created a digital experiment. Imagine a long, thin ribbon of wet, cloudy air floating in a room full of dry air. This is called a "cloud filament."

They wanted to see what happens when this wet ribbon gets mixed with the dry air. Does the water evaporate evenly? Do some droplets disappear while others stay?

They used five different methods to simulate this mixing:

1. The "Gold Standard" (DNS): This is a super-detailed simulation that solves every single physics equation for every tiny swirl of air. It's like filming the mixing process with a 4K camera. It is incredibly accurate but requires a supercomputer and takes a long time.
2. The Four "Shortcuts" (Statistical Models): These are the simpler models scientists actually use in weather forecasts. They try to guess the result without doing all the heavy math. The paper tested four specific ones:
   • LEM (Linear Eddy Model): Uses a one-dimensional map to stretch and fold the air.
   • EHM (Eddy-Hopping Model): Assumes the air jumps around randomly but treats the whole area as if it's all the same.
   • RMM (Relaxation-to-Mean Model): Assumes the air tries to return to an average state.
   • MCM (Mapping-Closure Model): Uses a complex mathematical trick to predict how the air mixes based on probability.

The Results: What Worked and What Didn't?

The researchers compared the four "shortcuts" against the "Gold Standard" (DNS) to see which one told the truth.

1. The Temperature Story (Thermodynamics)

The Verdict: All four shortcuts were good.
The Analogy: Imagine you are mixing hot coffee with cold milk. If you just want to know the average temperature of the cup, all four models got it right. They could predict how the heat and moisture changed over time just as well as the super-detailed simulation.

2. The Droplet Story (Cloud Microphysics)

The Verdict: Only some shortcuts were good.
The Analogy: Now, imagine you want to know what happens to the individual sugar crystals in that coffee.

• The Problem: When cloudy air mixes with dry air, it's not a smooth blend. Some parts of the cloud get hit by dry air and the droplets evaporate completely (disappear). Other parts stay wet, and the droplets stay the same size. This is called inhomogeneous mixing.
• The Winner (LEM, MCM, and partially RMM): These models understood that the air is messy. They realized that some droplets are in "dry pockets" and some are in "wet pockets." They correctly predicted that some droplets would vanish while others survived.
• The Loser (EHM): This model assumed everything was smooth and even. It thought all the droplets were in the same environment. So, it predicted that all droplets would shrink a little bit at the same time, but none would disappear. This is called homogeneous mixing, and the paper found this model was wrong for this specific situation.

The Key Takeaway: It's All About "Space"

The main reason the models failed or succeeded came down to one thing: Spatial Variability.

• The Failure: The Eddy-Hopping Model (EHM) treated the whole cloud as a single, uniform blob. It didn't account for the fact that dry air might be touching one side of a droplet but not the other.
• The Success: The models that worked (like LEM and MCM) kept track of where the droplets were and how the humidity varied from place to place.

The paper concludes that if you want to know how many cloud droplets survive a mixing event (which changes how clouds reflect sunlight), you must use a model that understands that humidity isn't the same everywhere. You can't just use an "average."

Summary

• Goal: Find the best simple model to represent how clouds mix with dry air.
• Method: Compare four simple models against a super-detailed "truth" simulation.
• Result: All models are good at predicting temperature and moisture averages. However, only the models that account for local differences (spatial variability) can correctly predict how cloud droplets grow or shrink.
• Implication: To improve weather and climate models, we need to use the "smart" shortcuts that remember that air isn't perfectly mixed, rather than the "dumb" shortcuts that assume everything is the same.

Technical Summary

Technical Summary: Benchmarking Turbulence Models to Represent Cloud-Edge Mixing

Problem Statement
Understanding cloud development requires resolving multi-scale processes ranging from large-scale cloud fields (100 km) to the smallest scales of turbulence (1 mm) where aerosols and droplets interact. While Direct Numerical Simulations (DNS) can accurately resolve these small-scale turbulent mixing processes, they are computationally prohibitive for large domains, typically limited to scales of up to one meter. Consequently, atmospheric models operating at larger scales (e.g., Large-Eddy Simulations or Global Climate Models) must parameterize subgrid-scale mixing. A critical challenge is representing the mixing of cloudy filaments with dry, subsaturated air, which alters cloud microphysics (droplet number concentration and size distribution) and thermodynamics. The nature of this mixing—whether "homogeneous" (reducing droplet size but not number) or "inhomogeneous" (evaporating some droplets completely while leaving others intact)—significantly impacts cloud radiative properties. However, there has been a lack of systematic comparison between different statistical turbulence models regarding their ability to capture these microphysical responses.

Methodology
This study benchmarks four statistical turbulence models against a set of new Direct Numerical Simulations (DNS) that serve as the reference "ground truth." The DNS solves the incompressible, Oberbeck-Boussinesq-approximated Navier-Stokes equations coupled with thermodynamic equations for temperature and water vapor, and Lagrangian equations for cloud droplet dynamics.

The four statistical models compared are:

1. Linear-Eddy Model (LEM): A one-dimensional statistical model that mimics turbulent stretching and folding via a mapping approach to relocate scalars, while solving for molecular diffusion and condensation/evaporation.
2. Eddy-Hopping Model (EHM): A statistical model treating supersaturation fluctuations as an Ornstein-Uhlenbeck process. It assumes spatial homogeneity, where all fluid elements experience similar mixing dynamics.
3. Relaxation-to-Mean Model (RMM): A one-dimensional model that tracks Lagrangian fluid elements and relaxes their supersaturation toward an ensemble-averaged mean, allowing for some spatial variability.
4. Mapping-Closure Model (MCM): A model based on mapping-closure approximations that predicts the non-Gaussian evolution of supersaturation from Gaussian statistics, utilizing a mapping function derived from DNS data.

All models utilize a comparable Lagrangian representation of cloud microphysics. The study simulates the mixing of a cloudy filament with dry air across various turbulence intensities (energy dissipation rates $\epsilon$ from 1 to 1000 cm² s⁻³) and ambient humidities (cases n0 to n3). The models are initialized with specific profiles for water vapor mixing ratio and temperature, and results are analyzed for domain-averaged thermodynamic quantities, droplet number concentration ($N_c$), mean droplet radius ($r_m$), and the spectral width of droplet size distributions.

Key Results

• Thermodynamics: All four statistical models successfully capture the evolution of thermodynamic quantities (water vapor mixing ratio, temperature, and domain-averaged supersaturation) during the mixing process, regardless of turbulence intensity or ambient humidity. Minor discrepancies in mixing rates were observed but were negligible compared to the variations caused by changing turbulence parameters.
• Cloud Microphysics: Significant differences emerged in the representation of cloud microphysical properties.
  • Inhomogeneous Mixing: DNS, LEM, and MCM successfully captured the gradual decrease in droplet number concentration ($N_c$), indicating inhomogeneous mixing where some droplets evaporate completely while others remain.
  • Homogeneous Bias: The EHM failed to capture this behavior, predicting a constant $N_c$ until all droplets evaporated simultaneously. This suggests a bias toward homogeneous mixing.
  • Partial Success: The RMM showed partial success in representing inhomogeneous mixing but tended to evaporate the largest droplets too quickly compared to DNS.
• Supersaturation Spectra: The ability to represent inhomogeneous mixing was directly linked to the models' capacity to reproduce the broad, asymmetric distribution of Lagrangian supersaturation experienced by droplets.
  • DNS and LEM reproduced the broad, asymmetric spectra resulting from the mixing of high-supersaturation cloud air with low-supersaturation ambient air.
  • MCM agreed well with DNS for high turbulence ($\epsilon = 1000$ cm² s⁻³) but struggled with low turbulence ($\epsilon = 1$ cm² s⁻³) due to its design to relax toward a Gaussian distribution.
  • EHM and RMM failed to represent the full broadening of the supersaturation distribution, with EHM assuming a very narrow distribution.
• Resolution Sensitivity: The study highlighted that the LEM's ability to represent inhomogeneous mixing is resolution-dependent. Coarser resolutions in LEM decelerated initial condensation and subsequent evaporation, artificially moistening the ambient air and reducing cloud fragmentation.

Significance and Claims
The paper claims that while simpler statistical models can effectively represent the thermodynamic evolution of cloud-edge mixing, they do not all equally capture the resulting cloud microphysical changes. The study demonstrates that accurately representing the spatial variability of supersaturation is a necessary condition for models to correctly simulate inhomogeneous mixing and the associated changes in droplet number concentration and spectral width.

The authors conclude that the LEM and MCM (with appropriate closures) are better suited than the EHM for representing these microphysical processes as subgrid-scale schemes in larger-scale models. However, the MCM requires external data (e.g., from DNS or machine learning) to generate scenario-specific closures for the mapping function. The EHM requires tuning parameters that may vary with environmental conditions, posing a challenge for its application as a subgrid-scale model. Ultimately, these statistical models offer a practicable means to bridge the gap between small-scale turbulence and cloud physics, though DNS remains essential for resolving the intricate physical processes that these models approximate.