Immersion freezing in particle-based aerosol-cloud microphysics: a probabilistic perspective on singular and time-dependent models

This paper evaluates singular and time-dependent parameterizations for immersion freezing within probabilistic particle-based aerosol-cloud microphysics models, demonstrating that while the singular approach is limited to specific cooling rates, the time-dependent method offers a more robust framework for simulating heterogeneous ice nucleation under realistic atmospheric flow conditions.

The Gist

The Big Picture: How Clouds Turn to Ice

Imagine a cloud as a giant, floating swimming pool filled with tiny water droplets. Usually, water stays liquid even when it gets very cold (below freezing). To turn into ice, these droplets need a "kick-start" from a tiny speck of dust, pollen, or pollution floating inside them. These specks are called Ice Nucleating Particles (INPs).

The scientists in this paper are trying to figure out the best way to teach computer models how to predict when these droplets will freeze. They are comparing two different "rulebooks" (mathematical models) that scientists use to simulate this process.

The Two Rulebooks: The "Crystal Ball" vs. The "Coin Flip"

The paper compares two ways of modeling how a droplet freezes:

1. The Singular Model (The "Crystal Ball" Approach)

• How it works: Imagine you have a crystal ball that tells you the exact moment a specific droplet will freeze. In this model, every single dust particle is assigned a specific "Freezing Temperature" right at the start of the simulation.
• The Logic: If the air gets cold enough to reach that specific temperature, the droplet instantly freezes. It's deterministic (predictable).
• The Catch: This rulebook was written based on experiments done in a lab where the temperature dropped at a very specific, steady speed. The paper argues that this "Crystal Ball" only works if the weather outside behaves exactly like the lab experiment. If the air cools down too fast, too slow, or starts warming up again, the Crystal Ball gives the wrong answer. It's like using a map of a city that only works if you drive at exactly 30 mph; if you speed up or slow down, you get lost.

2. The Time-Dependent Model (The "Coin Flip" Approach)

• How it works: Instead of a fixed freezing temperature, this model treats freezing as a game of chance that happens every second. Imagine a coin flip. Every second, the computer asks: "Is the air cold enough and wet enough right now to make this droplet freeze?" It flips a coin based on the current conditions.
• The Logic: Even if the air is cold, the droplet might not freeze this second. But if it stays cold for a long time, the odds of the coin landing on "Freeze" eventually add up.
• The Benefit: This model is flexible. It works whether the air is cooling fast, cooling slow, or even warming up. It understands that freezing is a process that takes time, not just a switch that flips at a specific temperature.

The Experiment: The "Box" and the "Flow"

The researchers tested these two rulebooks in two different scenarios:

1. The Zero-Dimensional Box (The Simple Test): They put the models in a virtual "box" where they could control the temperature perfectly. They changed the cooling speed to see how the models reacted.

   • Result: When the cooling speed matched the lab experiments, both models agreed. But when they changed the cooling speed (making it faster or slower), the "Crystal Ball" model (Singular) started making huge mistakes. It either froze too many droplets or too few, depending on how fast the temperature changed. The "Coin Flip" model (Time-Dependent) stayed accurate.

2. The 2D Flow Simulation (The Real World Test): They moved to a more complex simulation that mimics a real cloud with wind and air currents moving up and down.

   • Result: In the real world, air parcels move up (cooling) and down (warming) chaotically. The "Crystal Ball" model failed miserably here. Because it couldn't handle the warming phases or the variable speeds, it predicted almost no ice would form. The "Coin Flip" model, however, correctly predicted that ice would form because it kept checking the conditions every second.

The "Rare Particle" Problem

A major challenge the paper highlights is that Ice Nucleating Particles are very rare compared to regular dust.

• The Analogy: Imagine trying to find a specific needle in a haystack, but your computer simulation only has a limited number of "haystacks" to look at. If you don't set up your simulation carefully, you might miss the needles entirely, or you might accidentally put too many needles in one spot.
• The paper shows that how you distribute these rare particles in the computer model matters just as much as the freezing rulebook you choose.

The Takeaway: Why This Matters

Clouds affect our climate. If a cloud freezes, it changes how much sunlight it reflects and how much rain it produces.

• The Problem: Current climate models often use the "Crystal Ball" (Singular) method because it's faster and easier for computers to run.
• The Discovery: This paper proves that the "Crystal Ball" is too rigid. It breaks when the weather gets complicated (which it always does in the real world).
• The Solution: We need to switch to the "Coin Flip" (Time-Dependent) method. It is more computationally expensive (takes more computer power), but it is robust. It gives accurate answers whether the air is cooling fast, slow, or changing direction.

In short: The paper argues that to predict our future climate accurately, we need to stop using a rigid, one-size-fits-all rule for how clouds freeze and start using a flexible, time-aware approach that understands the chaotic nature of the atmosphere.

Technical Summary

1. Problem Statement

Mixed-phase clouds, containing both supercooled liquid droplets and ice crystals, play a critical role in Earth's energy budget and climate. A key process in these clouds is immersion freezing, where ice-nucleating particles (INPs) embedded within supercooled droplets trigger freezing at temperatures above the homogeneous freezing threshold.

Current atmospheric models typically describe this process using one of two parameterizations:

• Singular (Time-Independent) Models: Assume that each INP has a specific, characteristic freezing temperature ($T_f$). Once the ambient temperature drops below $T_f$, freezing occurs deterministically. These models are computationally efficient but often derived from laboratory data with specific cooling rates.
• Time-Dependent (Stochastic) Models: Assume freezing is a probabilistic process dependent on time, temperature, and the surface area of the INP. Freezing can occur at any time a particle is in a supercooled state, with the probability increasing over time.

The Core Conflict:
While singular models are popular for their simplicity, they fail to capture the stochastic nature of nucleation and are highly sensitive to the cooling rate used during their derivation. Time-dependent models are physically more robust but computationally expensive. There is a lack of rigorous comparison between these two approaches within particle-based (super-particle) cloud microphysics models, which are essential for resolving aerosol diversity and complex flow regimes (e.g., Large Eddy Simulations). This paper addresses the trade-offs, limitations, and applicability of both approaches in such high-resolution frameworks.

2. Methodology

The authors employ a probabilistic, particle-based framework using super-particles (computational particles representing a large multiplicity of real aerosols/droplets). They implement and compare two specific schemes:

• Time-Dependent Scheme (ABIFM): Based on the water-activity-based immersion freezing model (Knopf & Alpert, 2013).
  • Mechanism: At every time step, the probability of freezing is calculated based on the current temperature, water activity, and the immersed insoluble surface area ($S$) of the super-particle. A random number determines if freezing occurs.
  • Attributes: Tracks the immersed surface area ($S$) as a dynamic attribute.
• Singular Scheme (INAS): Based on the Ice-Active Surface Site Density parameterization (Shima et al., 2020).
  • Mechanism: At initialization, each super-particle is assigned a specific freezing temperature ($T_f$) sampled from a distribution derived from the INAS density function. Freezing occurs deterministically when the ambient temperature drops below this pre-assigned $T_f$.
  • Attributes: Tracks $T_f$ as a static attribute; the surface area is used only for initialization and then discarded.

Simulation Frameworks:

1. Zero-Dimensional (Box) Model: Used to isolate the effects of cooling rates and surface area polydispersity. Simulations were run with various idealized temperature profiles (linear cooling, constant temperature, impulse drops, warming cycles).
2. Two-Dimensional (2D) Prescribed-Flow Model: A kinematic simulation mimicking an Arctic stratiform cloud. It couples an Eulerian fluid solver (transport, heat, moisture) with a Lagrangian super-particle solver. This setup tests the models under realistic, chaotic flow regimes with varying cooling rates and aerosol activation dynamics.

3. Key Contributions

• Probabilistic Unification: The authors demonstrate that both singular and time-dependent models are numerical realizations of the same underlying Poissonian counting process. The singular model is essentially a time-integrated version of the time-dependent model.
• Cooling Rate Signature: They mathematically prove and numerically verify that singular parameterizations (INAS) embed the cooling rate signature of the laboratory experiments used to derive their coefficients. Consequently, they are only valid for flow regimes matching that specific cooling rate.
• Polydispersity Impact: The study highlights that neglecting the polydispersity (variability) of immersed surface areas leads to significant biases in frozen fraction predictions, comparable to multi-fold changes in cooling rates.
• Attribute Sampling Strategy: The paper details a critical sampling strategy for particle-based models to resolve rare INPs (which are orders of magnitude less abundant than CCN) without requiring prohibitive computational resources. This involves assigning lower multiplicities to INP-rich super-particles.

4. Key Results

A. Box Model Findings (Idealized Conditions)

• Cooling Rate Dependence:
  • When the ambient cooling rate matches the laboratory rate used to fit the INAS coefficients (approx. -0.75 K/min), the singular and time-dependent models produce similar results.
  • Slow Cooling: The singular model underpredicts ice formation compared to the time-dependent model because it cannot trigger freezing during periods of constant temperature or slow cooling where the time-dependent model accumulates probability.
  • Fast Cooling: The singular model overpredicts ice formation compared to the time-dependent model because it assumes instantaneous freezing once $T < T_f$, ignoring the kinetic time lag required for nucleation.
• Non-Monotonic Flows: In scenarios involving warming (downdrafts) or constant temperature, the singular model yields zero new freezing events (as $T$ is not decreasing), whereas the time-dependent model correctly continues to trigger freezing events as long as the droplet remains supercooled.
• Polydispersity: Simulations with broad surface area distributions (polydisperse) showed flattened frozen fraction curves compared to monodisperse assumptions, significantly altering the temperature range over which freezing occurs.

B. 2D Flow-Coupled Findings (Realistic Conditions)

• Ice Concentration Discrepancy: In the 2D Arctic cloud simulation, the singular model predicted ice concentrations an order of magnitude lower than the time-dependent model.
• Flow Regime Robustness: The singular model's performance was highly sensitive to the eddy frequency (flow speed). It failed to capture ice formation in downdrafts or weak updrafts where the cooling rate was insufficient to trigger the pre-assigned $T_f$ thresholds. The time-dependent model showed robust behavior across varying flow regimes.
• Limitations of Singular Schemes: The singular scheme effectively limits ice concentrations to the number of particles with $T_f$ above the minimum temperature reached in the domain. The time-dependent scheme allows ice concentrations to grow monotonically over time, consistent with the stochastic nature of nucleation.

5. Significance and Implications

• Model Selection: The study concludes that while singular models offer computational speed, they are not robust for general atmospheric modeling, particularly in Large Eddy Simulations (LES) or Global Climate Models (GCMs) where cooling rates vary chaotically and often deviate from laboratory conditions.
• Future Development: The time-dependent approach is recommended for high-resolution models. Although computationally more expensive, it:
  1. Is robust to varying flow regimes and cooling rates.
  2. Allows for online coupling with aerosol physico-chemical dynamics (e.g., changes in surface area due to coalescence or chemical processing).
  3. Correctly handles non-monotonic temperature trajectories (warming/downdrafts).
• Implementation Guidance: The authors provide a roadmap for implementing these schemes in particle-based models, emphasizing the need for careful attribute sampling to resolve rare INPs and the importance of maintaining the "identity" of particles to track their history and properties accurately.

In summary, this paper provides a rigorous probabilistic justification for moving away from singular, time-independent parameterizations toward time-dependent, stochastic approaches in particle-based cloud microphysics, ensuring more accurate representation of ice formation in complex atmospheric flows.