Cloud Parcel Modelling - Part 4: K-Kohler Droplet Activation

❓Why Include κ-Köhler Theory If the Cloud Parcel Model Starts After Activation?

Even though the cloud parcel model tracks only activated droplets, it still needs to know which CCN activate and when. That’s where κ-Köhler theory becomes essential — it’s the bridge between aerosol properties and cloud formation.

🧩 The Role of κ-Köhler in the Parcel Model

Purpose Why It Matters
🎯 Determines Critical Supersaturation (Sc) The parcel model uses κ-Köhler to determine which aerosol particles activate into droplets.
🔢 Sets Initial Number of Activated Droplets (Nd) This number directly affects cloud microphysics: supersaturation evolution, latent heat release, droplet competition, etc.
⏳ Controls Timing of Activation Activation doesn't happen instantly — the model checks if supersaturation exceeds Sc for a given κ and particle size.
🌎 Links Cloud Behavior to Aerosol Properties κ varies by composition (e.g., NaCl ≈ 1.2, organics ≈ 0.1–0.3, black carbon ≈ 0), affecting which particles activate.

🚫 What Happens Without It?

  • The model won’t know how many CCN activate at a given supersaturation.
  • You must assume activation manually (e.g. fixed Nd), limiting realism in cloud-aerosol interaction studies.
  • You lose the connection between aerosol composition and cloud properties — a key advantage of detailed modeling.

🌧️ Part 4 – Droplet Activation: From Aerosol to Cloud Droplet

In this part, we explore how cloud condensation nuclei (CCN) activate to form droplets based on the κ-Köhler framework. It provides a simplified but powerful way to relate aerosol hygroscopicity and size to critical supersaturation.

  • 📈 Köhler Curve: Illustrates how equilibrium saturation changes with droplet size.
  • 🧪 κ-Köhler Parameterization: Introduces κ (kappa) as a single-value measure of particle hygroscopicity.
  • 🔬 Activation Condition: A particle activates when environmental supersaturation S exceeds its critical Sc.
  • 🔗 Link to Cloud Microphysics: Activated droplet number Nd controls cloud properties: optical depth, albedo, and more.

💡 Summary: κ-Köhler theory is not about post-activation behavior — it defines the gateway into the cloud parcel model. It determines who gets in and how many.

🧪 Part 4 – From Aerosol to Cloud Droplet: κ-Köhler Activation Theory

To understand how aerosol particles transition into cloud droplets, we need to understand how activation happens. This process is governed by the Köhler theory, which links aerosol size and composition to the relative humidity (or more precisely, supersaturation) needed to form a droplet.

🌫️ What Is Activation?

A cloud condensation nucleus (CCN) activates when the surrounding air reaches a high enough supersaturation (S > Sc), such that the particle can spontaneously grow into a cloud droplet — usually larger than 1 µm in diameter.

Before activation, particles may absorb water and swell. But only when a particle reaches its critical diameter and exceeds its critical supersaturation (Sc) does it transition into an activated droplet.

📈 The Köhler Curve

The Köhler equation combines two effects:

  • 🔹 Raoult effect (solute): lowers vapor pressure due to dissolved salts or organics.
  • 🔹 Kelvin effect (curvature): increases vapor pressure over small droplets.

These two effects produce the Köhler curve — a plot of equilibrium saturation Seq versus droplet size. At a certain droplet radius, the saturation peaks: this is the critical point where activation occurs.

🧮 κ-Köhler Equation

The κ-Köhler formulation (Petters and Kreidenweis, 2007) simplifies the Köhler equation by introducing a single hygroscopicity parameter, κ:

Seq(r) = (1 + κ) · (Vs/Vw) - (A / r)

Where:

  • κ = hygroscopicity parameter (depends on composition: NaCl ≈ 1.2, organics ≈ 0.1–0.3, BC ≈ 0)
  • Vs = dry particle volume
  • Vw = water volume
  • A = Kelvin effect term = (2σMw) / (RTρw)

📊 Critical Supersaturation (Sc) and Activation Radius (rc)

Each particle has a critical supersaturation and radius that depends on κ and its dry size. When S > Sc, the droplet grows spontaneously.

In cloud parcel models, we use κ-Köhler theory to determine:

  • ➕ Which aerosol particles activate (based on size and composition)
  • 🔢 How many droplets are activated (Nd)

🔗 Why This Matters for the Cloud Parcel Model

While the model simulates cloud droplet growth after activation, it needs to know:

  • ✅ Which CCN activate at a given S
  • ✅ When they activate (timing affects latent heat release)
  • ✅ How many droplets form (controls supersaturation, optical depth, and precipitation potential)

📌 Summary Box

Term Definition
Köhler Curve Shows equilibrium supersaturation as a function of droplet size
κ (kappa) Single parameter that quantifies particle hygroscopicity
Sc Critical supersaturation needed for activation
rc Droplet radius at which Seq is maximum
Nd Number of activated droplets

💡 Key Insight: Activation is the gateway to the cloud parcel model. Without κ-Köhler theory, you cannot realistically simulate how aerosol properties affect cloud formation.

🔢 Activation Spectrum and Predicting Nd

While κ-Köhler theory gives us the critical supersaturation (Sc) for individual particles, cloud models must go further: they must estimate how many particles actually activate. This is where the CCN activation spectrum comes into play.

🌫️ What is an Activation Spectrum?

The CCN activation spectrum describes the relationship between supersaturation (S) and the cumulative number of aerosol particles that activate as cloud droplets. In simple terms:

As S increases → More aerosol particles have Sc ≤ S → More particles activate → Nd increases.

This is usually plotted as Nd versus S, forming an S-shaped curve.

📏 Role of Aerosol Number-Size Distribution

Since each particle's Sc depends on its size and κ value, we must consider the full aerosol number-size distribution (e.g., dN/dlogD). The integration of all particles with Sc below the current S gives us the total activated droplet number (Nd).

Thus, for a given aerosol distribution and composition:

  • Small particles → higher Sc, less likely to activate
  • Larger particles → lower Sc, activate earlier

Shape and width of the distribution control how steep or gradual the activation curve is.

📈 Link to Supersaturation Evolution

In the parcel model, the peak supersaturation (Smax) is the key value. It's determined by a balance between:

  • Vapor supply (due to adiabatic cooling)
  • Vapor demand (due to condensation by growing droplets)

Once Smax is known, it is used to predict Nd from the activation spectrum:

Nd = ∫ n(D) dD, for all particles with Sc(D, κ) ≤ Smax

This is often calculated numerically using a lookup table or by evaluating κ-Köhler equation across the distribution.

🧠 Why This Matters

The value of Nd sets the stage for everything that follows in the cloud parcel model:

  • It affects supersaturation depletion rate
  • Determines latent heat release
  • Impacts droplet size distribution shape and optical depth

Without predicting Nd properly, you lose the crucial link between aerosols and cloud microphysics.

📌 Summary Box

🔍 What Does the Activation Spectrum Add?
It bridges the gap between individual droplet theory (κ-Köhler) and bulk cloud properties (Nd). This makes the parcel model realistic and responsive to aerosol variability.

Komentar

Posting Komentar

Postingan populer dari blog ini

Cloud Parcel Model – Part 0: Introduction and Context (A)

In this introductory part of the Cloud Parcel Modelling series, we clarify what conditions define the beginning of a cloud parcel simulation. When does the model start? What processes are included — and what’s left out? This foundational understanding is crucial before diving into growth equations and microphysical feedbacks. So, the cloud parcel model is conditioned after CCN particles activate? ✅ A Cloud Parcel Model typically starts from the point when: 💧 The rising air parcel becomes supersaturated (RH > 100%) ➕ Cloud Condensation Nuclei (CCN) begin to activate into cloud droplets 🔄 Step-by-Step View of What the Cloud Parcel Model Covers: Air parcel rises due to updraft Cooling occurs due to adiabatic expansion Relative humidity increases At RH ≈ 100% , the supersaturation (S > 0) triggers: Some CCN → activated Droplet formation begins From that point on , the parcel model: Simulates condensation growth of activated droplets Tracks lat...
Read more »

Cloud Parcel Modeling – Part 3: Supersaturation Evolution, Droplet Size Distribution, and and the Core Understanding in Cloud Parcel Modeling

☁️ Cloud Parcel Modeling – Part 3: Droplet Size Distribution and Supersaturation Evolution 📌 Core Idea: Once cloud droplets activate, the system evolves in a coupled way — supersaturation changes, droplets grow, and their size distribution shapes the cloud’s microphysical and radiative properties. 1. 🌡️ Supersaturation (S) is Dynamic, Not Static As an air parcel ascends: It cools → water vapor condenses → latent heat is released Droplets grow by condensation → reduces ambient water vapor This causes a feedback: Supersaturation increases, peaks, and then declines Equation: dS/dt = α ⋅ w - β ⋅ G(S, r) Where: α ~ adiabatic cooling rate (from updraft w ) β ⋅ G ~ condensation sink (depends on total surface area of droplets) ✅Thinking : What does dS/dt and G(S,r) relationship infer? The relationship infer that current values of S ( t ) S(t)  and r ( t ) r(t)  affect the droplet growth rate d r / d t dr/dt , which is encapsulated in G ( S , r ) G(S, ...
Read more »

Building the Cloud Parcel Model on Matlab - Part 2: Create the core ODE system (A. basic)

 Here’s how you can construct and describe the ODE system of a cloud parcel model—based on Nenes et al. (2001, 2002) and examples like Pyrcel —for full implementation in MATLAB. This clarifies the variables, equations, and overall structure you'll include in your solver ( ode15s ). 🌫️ 1. State Vector ( y ) Your ODE state vector y should stack the following: Temperature : T ( t ) T(t)  — air parcel temperature [K] Water vapor : w ( t ) w(t)  — mass mixing ratio [kg/kg] Droplet radii : r i ( t ) r_i(t) , for i = 1 , … , N i = 1,\dots,N i = 1 , … , N — each aerosol bin radius [m] So in MATLAB: y = [ T; w; r1; r2; …; rN ]; 🔢 2. ODEs Formulation You’ll define the right-hand side function parcel_ode(t, y, params) , which returns: d y d t = [   d T / d t ;    d w / d t ;    d r 1 / d t ;    …   ;    d r N / d t   ] 2.1 Droplet Growth ( dr_i/dt ) Using κ-Köhler theory (as in Nenes 2001, Pyrcel implementation   d r i d t = G ( T , p , …   )   S r i...
Read more »