Part III: Cloud Physics & Precipitation

Microphysics of cloud formation, droplet growth, and precipitation processes

1. Cloud Classification & Formation

1.1 Cloud Types

Clouds are classified based on their altitude and morphology according to the international system:

High Clouds (6-18 km)

Composed primarily of ice crystals due to cold temperatures (\(T < -40°C\))

  • • Cirrus (Ci): Wispy, fibrous "mare's tail" clouds composed entirely of ice
  • • Cirrostratus (Cs): Thin ice crystal sheets causing halos around sun/moon
  • • Cirrocumulus (Cc): Small white patches, "mackerel sky"

Middle Clouds (2-6 km)

Mixed-phase clouds containing supercooled water droplets and ice crystals

  • • Altostratus (As): Gray/blue-gray sheets, sun visible as through frosted glass
  • • Altocumulus (Ac): Layered patches or rolls, precipitation rarely reaches ground

Low Clouds (0-2 km)

Primarily liquid water droplets, except in winter

  • • Stratus (St): Uniform gray layer, drizzle possible
  • • Stratocumulus (Sc): Low-level patches/rolls, breaks between clouds
  • • Nimbostratus (Ns): Dark gray rain-bearing layer

Vertical Development Clouds

Convective clouds extending through multiple levels

  • • Cumulus (Cu): Puffy, cauliflower-like fair-weather clouds
  • • Cumulonimbus (Cb): Towering thunderstorm clouds with anvil tops, severe weather

1.2 Lifting Mechanisms

Clouds form when air is lifted to its saturation level. Four primary mechanisms cause lifting:

1. Orographic Lifting

Air forced over mountain barriers. Produces orographic clouds and precipitation on windward slopes, rain shadows on leeward sides. Example: Sierra Nevada range blocking Pacific moisture.

2. Frontal Lifting

Warm air forced over cold air at frontal boundaries. Produces widespread stratiform clouds along warm fronts, narrow bands along cold fronts.

3. Convective Lifting

Surface heating creates thermal updrafts. Produces cumulus and cumulonimbus clouds with strong vertical development. Dominant mechanism in tropics and summer afternoons.

4. Convergence

Air flows together horizontally, forcing upward motion. Occurs in low-pressure systems, sea-breeze fronts, and the Intertropical Convergence Zone (ITCZ).

1.3 Saturation and Supersaturation

Air becomes saturated when the relative humidity reaches 100%. The saturation vapor pressure \(e_s\) is given by the Clausius-Clapeyron equation (from Part I):

\(e_s(T) = 6.112 \exp\left[\frac{17.67(T-273.15)}{T-29.65}\right]\)[hPa]

Supersaturation is defined as:

\(S = \frac{e - e_s}{e_s} \times 100\%\)
• \(S = 0\)%: Saturated
• \(S > 0\)%: Supersaturated (condensation favored)
• \(S < 0\)%: Subsaturated (evaporation favored)

In clouds, supersaturations are typically small (\(S < 1\)% for water, \(S_{\text{ice}} < 10\)% for ice) because condensation/deposition onto existing droplets/crystals quickly reduces \(S\).

1.4 Adiabatic Cooling and Cloud Base Height

As an air parcel rises adiabatically, it cools at the dry adiabatic lapse rate \(\Gamma_d = 9.8\) K/km until it reaches saturation. The lifting condensation level (LCL)is where the cloud base forms.

Calculation of LCL height:
Starting from surface with temperature \(T_0\) and dewpoint \(T_d\):
\(T(z) = T_0 - \Gamma_d z\) (temperature of rising parcel)
\(T_d(z) \approx T_d + \Gamma_{d_w} z\) (dewpoint changes ~2 K/km)
At the LCL, \(T = T_d\), so:
\(z_{\text{LCL}} = \frac{T_0 - T_d}{\Gamma_d - \Gamma_{d_w}} \approx \frac{T_0 - T_d}{\Gamma_d} \times 125 \text{ m/K}\)

📊 Practical Example:

If surface temperature is 25°C and dewpoint is 15°C (typical summer afternoon):

\(z_{\text{LCL}} = (25-15) \times 125 = 1250\) m above ground level

This is where cumulus cloud bases form. Lower dewpoint depression (\(T_0 - T_d\)) means lower cloud bases.

2. Cloud Condensation Nuclei (CCN)

2.1 Role of Aerosols

Homogeneous nucleation (pure water condensing without a surface) requires enormous supersaturations (\(S \sim 300\)%) that never occur in the atmosphere. Instead, water vapor condenses on cloud condensation nuclei (CCN) - tiny aerosol particles suspended in the air.

Why CCN are necessary:

  • • Heterogeneous nucleation on CCN occurs at \(S \sim 0.1\text{-}1\)% (achievable in atmosphere)
  • • CCN reduce surface energy barrier for droplet formation
  • • Typical CCN concentrations: 100-1000 cm⁻³ (continental), 10-100 cm⁻³ (maritime)

2.2 Types of Aerosols

Hygroscopic Aerosols

Water-attracting particles that make excellent CCN:

  • • Sea salt: NaCl from ocean spray (maritime air masses)
  • • Sulfates: (NH₄)₂SO₄ from industrial emissions, volcanic SO₂
  • • Nitrates: NH₄NO₃ from agricultural/combustion sources
  • • Organics: Water-soluble organic compounds from biogenic sources

Hydrophobic Aerosols

Water-repelling particles, poor CCN:

  • • Black carbon (soot): From combustion
  • • Mineral dust: Desert and soil particles
  • • Some organics: Hydrophobic organic compounds

Note: Coating with soluble material can make hydrophobic particles act as CCN

2.3 CCN Activation

A CCN "activates" (becomes a cloud droplet) when the ambient supersaturation exceeds the particle's critical supersaturation (\(S_c\)). The critical supersaturation depends on:

  • • Particle size: Larger particles have lower \(S_c\) (activate more easily)
  • • Chemical composition: More hygroscopic → lower \(S_c\)
  • • Solubility: Highly soluble salts reduce \(S_c\)
Typical critical supersaturations:
Large hygroscopic (\(r \sim 0.1\) μm):
\(S_c \sim 0.1\)%
Medium hygroscopic (\(r \sim 0.05\) μm):
\(S_c \sim 0.3\)%
Small/less hygroscopic (\(r \sim 0.01\) μm):
\(S_c \sim 1\)%

2.4 Maritime vs Continental Air Masses

CCN concentrations profoundly affect cloud properties:

Maritime Clouds

  • • Low CCN: 10-100 cm⁻³
  • • Fewer, larger droplets (\(r \sim 15\) μm)
  • • Efficient precipitation (collision-coalescence)
  • • Shorter cloud lifetime
  • • Higher albedo per unit LWP

Continental Clouds

  • • High CCN: 100-1000 cm⁻³
  • • Many smaller droplets (\(r \sim 5\) μm)
  • • Suppressed precipitation
  • • Longer cloud lifetime
  • • Brighter clouds (more reflective)

🌍 Climate Implications:

The "aerosol indirect effect" is a major source of uncertainty in climate models. Increased anthropogenic aerosols → more CCN → brighter, longer-lived clouds → cooling effect. This partially offsets greenhouse gas warming but the magnitude is poorly constrained.

2.5 CCN Activation Spectrum & Köhler Curves

The CCN activation spectrum describes the number of activated CCN as a function of supersaturation. It is often approximated by a power law:

$$N_{\text{CCN}}(S) = C \cdot S^k$$
where:
• \(C\) = CCN concentration parameter [cm⁻³]
• \(S\) = supersaturation [%]
• \(k\) = spectral shape parameter (typically 0.3-1.5)
Typical values:
• Maritime: \(C \approx 100\), \(k \approx 0.5\)
• Continental: \(C \approx 600\), \(k \approx 0.9\)

Each CCN has a unique Köhler curve that plots the equilibrium supersaturation as a function of droplet radius. The Köhler equation (detailed in Section 3) gives:

$$S = \frac{a}{r} - \frac{b}{r^3}$$
where \(a\) is the curvature (Kelvin) term and \(b\) is the solute (Raoult) term.
Key features of the Köhler curve:
  • • For very small \(r\): solute term dominates, \(S < 0\) (subsaturated equilibrium)
  • • At critical radius \(r_c\): maximum \(S_c\) (critical supersaturation)
  • • For \(r > r_c\): droplet is activated and grows freely if ambient \(S > S_c\)

🔬 Practical Interpretation of Köhler Curves:

A cloud updraft produces supersaturation that increases with height. As \(S\) increases, more CCN activate (those with \(S_c < S_{\text{ambient}}\)). The peak supersaturation determines the total droplet number concentration. Stronger updrafts → higher peak \(S\) → more droplets activated.

The competition between supersaturation production (adiabatic cooling) and depletion (condensation onto existing droplets) sets the maximum supersaturation, typically \(S_{\text{max}} \sim 0.1\text{-}1\)%.

3. Köhler Theory

Köhler theory describes the equilibrium between a solution droplet and the surrounding water vapor. It combines two competing effects:

3.1 Curvature (Kelvin) Effect

Small droplets have curved surfaces, which increases the vapor pressure above a pure water droplet compared to a flat surface:

\(e_s(r) = e_\infty \exp\left(\frac{2\sigma}{\rho_w R_v T r}\right)\)
where:
• \(e_s(r)\) = saturation vapor pressure over droplet of radius \(r\)
• \(e_\infty\) = saturation vapor pressure over flat surface
• \(\sigma\) = surface tension of water = 0.073 N/m at 20°C
• \(\rho_w\) = density of liquid water = 1000 kg/m³
• \(R_v\) = gas constant for water vapor = 461 J/(kg·K)
• \(T\) = temperature [K]

Physical interpretation: The curvature effect makes small droplets "want to evaporate" because molecules at a curved surface are less tightly bound. This creates a barrier to nucleation.

📐 Mathematical Deep Dive: Kelvin Equation Derivation

Consider the equilibrium of a pure water droplet. The chemical potential must be equal in liquid and vapor phases:
\(\mu_l(p_l) = \mu_v(e_s)\)
For a curved droplet, the pressure inside exceeds atmospheric pressure by the Laplace pressure:
\(p_l - p_\infty = \frac{2\sigma}{r}\)
The chemical potential change with pressure is:
\(d\mu_l = V_m dp_l = \frac{M_w}{\rho_w} dp_l\)
For the vapor phase (ideal gas):
\(d\mu_v = R_v T d(\ln e_s)\)
Equating the two and integrating from flat surface (\(r \to \infty\)) to curved droplet:
\(\frac{M_w}{\rho_w} \times \frac{2\sigma}{r} = R_v T \ln\left(\frac{e_s(r)}{e_\infty}\right)\)
\(e_s(r) = e_\infty \exp\left(\frac{2\sigma M_w}{\rho_w R_v T r}\right) = e_\infty \exp\left(\frac{2\sigma}{\rho_w R_v T r}\right)\)
(using \(M_w/\rho_w = 1/\rho_w\) in specific units where \(R_v\) is per kg)

3.2 Solute (Raoult) Effect

When the droplet contains dissolved salt/solute, it lowers the vapor pressure (Raoult's law). This makes the droplet stable at subsaturated conditions:

\(e_s(\text{solution}) = e_s(\text{pure}) \times \chi_w\)
where \(\chi_w\) is the mole fraction of water in the solution.
For dilute solutions with solute mass \(m_s\) and molecular weight \(M_s\):
\(\chi_w \approx 1 - \frac{m_s/M_s}{m_w/M_w} = 1 - \frac{i m_s M_w}{M_s m_w}\)
where \(i\) is the van't Hoff factor (\(i=2\) for NaCl, \(i=3\) for (NH₄)₂SO₄)

For a spherical droplet of radius \(r\) containing solute mass \(m_s\):

\(m_w = \frac{4}{3}\pi r^3 \rho_w - m_s\)
For large droplets (\(r \gg r_s\)), \(m_w \approx \frac{4}{3}\pi r^3 \rho_w\)

3.3 Complete Köhler Equation

Combining both effects, the saturation ratio \(S = e/e_\infty - 1\) over a solution droplet is:

\(S = \frac{a}{r} - \frac{b}{r^3}\)
where the curvature term is:
\(a = \frac{2\sigma}{\rho_w R_v T}\)
and the solute term is:
\(b = \frac{3 i m_s M_w}{4\pi \rho_w M_s}\)

📐 Mathematical Deep Dive: Köhler Curve Analysis

The Köhler equation \(S(r) = a/r - b/r^3\) has interesting behavior:
1. Small droplet limit (\(r \to 0\)):
The \(b/r^3\) term dominates (negative, solute effect). Droplet is in equilibrium at \(S < 0\) (subsaturated).
2. Large droplet limit (\(r \to \infty\)):
Both terms \(\to 0\), so \(S \to 0\) (equilibrium at saturation).
3. Critical point (maximum of \(S(r)\)):
Taking \(dS/dr = 0\):
\(-a/r^2 + 3b/r^4 = 0\)
\(r_c = \sqrt{\frac{3b}{a}}\)
\(S_c = \frac{a}{r_c} - \frac{b}{r_c^3} = \frac{2a}{3r_c} = \frac{2}{3}\sqrt{\frac{4a^3}{27b}}\)
For typical CCN with \(m_s = 10^{-18}\) kg (NaCl):
\(r_c \approx 0.05\) μm
\(S_c \approx 0.3\)%

3.4 Critical Radius and Activation

The critical point (\(r_c, S_c\)) represents a stability threshold:

Stable equilibrium (\(r < r_c\))

If \(r\) increases slightly, \(S(r)\) decreases → droplet is too large for ambient \(S\) → evaporates back to equilibrium. These are haze droplets in subsaturated air.

Unstable equilibrium (\(r = r_c\))

Critical radius. If ambient \(S > S_c\), the droplet "activates" and grows without bound (becomes a cloud droplet).

Activated droplet (\(r > r_c\))

For large \(r\), curvature effect becomes negligible, \(S(r) \to 0\). Droplet grows as long as ambient air is supersaturated. Growth limited by vapor diffusion.

🔬 Key Insight:

Köhler theory explains why clouds form at small supersaturations (\(S \sim 0.1\text{-}1\)%). The solute effect from hygroscopic CCN reduces the barrier to droplet formation. Larger and more hygroscopic CCN have smaller \(S_c\) and activate first as air becomes supersaturated in rising clouds.

4. Droplet Growth Mechanisms

Once activated, cloud droplets grow by two distinct mechanisms: condensation(vapor diffusion) and collision-coalescence (collision between droplets).

4.1 Condensational Growth

Diffusion of water vapor molecules to the droplet surface drives condensational growth. The growth rate is determined by mass conservation and vapor diffusion:

\(\frac{dm}{dt} = 4\pi r D \rho_v (S_\infty - S_r)\)
where:
• \(m = \frac{4}{3}\pi r^3 \rho_w\) is the droplet mass
• \(D \approx 2.5 \times 10^{-5}\) m²/s is the diffusivity of water vapor in air
• \(\rho_v\) is the saturation vapor density far from droplet
• \(S_\infty\) is the ambient supersaturation
• \(S_r\) is the supersaturation at the droplet surface (from Köhler equation)

Converting to radius growth rate and accounting for latent heat release:

\(\frac{dr}{dt} = \frac{S}{r} \frac{1}{F_k + F_d}\)
This is equivalently written in the compact form used in cloud microphysics:
$$r\frac{dr}{dt} = \frac{S-1}{F_k + F_d}$$
where \(S\) is the supersaturation and:
\(F_k = \left(\frac{L_v}{R_v T}\right)^2 \frac{\rho_w R_v T}{K_{\text{air}} e_s}\) (latent heat term)
\(F_d = \frac{\rho_w R_v T}{D e_s}\) (vapor diffusion term)
Typical values at 10°C, 850 hPa:
\(F_k \approx 1.4\), \(F_d \approx 2.0\), so \(F_k + F_d \approx 3.4\)

📐 Mathematical Deep Dive: Diffusion Growth Equation

Starting from Fick's law of diffusion:
\(J = -D \nabla \rho_v\) (vapor flux)
For spherical symmetry:
\(\frac{dm}{dt} = 4\pi r^2 J = -4\pi r^2 D \frac{d\rho_v}{dr}\)
Solving the steady-state diffusion equation \(\nabla^2 \rho_v = 0\) in spherical coords:
\(\rho_v(r) = \rho_{v,\infty} + \frac{C}{r}\)
Boundary condition at droplet surface (\(r = r_d\)):
\(\rho_v(r_d) = \rho_{v,s}\) (saturation at droplet surface)
This gives \(C = r_d(\rho_{v,s} - \rho_{v,\infty})\)
Therefore:
\(\frac{dm}{dt} = 4\pi r_d D (\rho_{v,\infty} - \rho_{v,s})\)
Including latent heat effects:
Condensation releases latent heat, warming the droplet. Heat must conduct away:
\(\frac{dQ}{dt} = 4\pi r K_{\text{air}} (T_d - T_\infty)\)
Energy balance: \(L_v \frac{dm}{dt} = 4\pi r K_{\text{air}} (T_d - T_\infty)\)
Combining diffusion and heat conduction gives the full equation with \(F_k\) and \(F_d\) terms.

4.2 Size-Dependent Growth Rates

The growth equation \(dr/dt \propto S/r\) has important implications:

  • • Small droplets (\(r < 1\) μm): Grow very rapidly, \(dr/dt \sim 1/r\) is large
  • • Large droplets (\(r > 10\) μm): Grow slowly, \(dr/dt \sim 1/r\) is small
  • • Narrow size distribution: Condensation produces nearly monodisperse droplets

📊 Practical Example:

In a cloud with \(S = 0.5\)% supersaturation at \(T = 10°C\):

• A \(r = 1\) μm droplet grows to 10 μm in ~10 minutes
• A \(r = 10\) μm droplet grows to 11 μm in ~10 minutes
• To reach raindrop size (\(r \sim 1000\) μm) by condensation alone takes ~1 day!

Conclusion: Condensation is efficient for small droplets but cannot produce precipitation-sized particles in the short lifetime of clouds. This is the condensation-coalescence bottleneck.

4.3 Collision-Coalescence

Larger droplets fall faster than smaller ones due to gravity. When a large droplet overtakes a smaller one, they may collide and coalesce, forming an even larger droplet. This process accelerates precipitation formation.

Terminal Velocity

A falling droplet reaches terminal velocity when gravitational force balances drag force. For small droplets (\(r < 40\) μm), Stokes drag applies:

Stokes Law (small droplets, \(r < 40\) μm):
\(v_t = \frac{2}{9} \frac{(\rho_w - \rho_a) g r^2}{\eta}\)
Or equivalently using dynamic viscosity \(\mu\):
$$v_t = \frac{2\rho_w g r^2}{9\mu}$$
where:
• \(\rho_w = 1000\) kg/m³ (liquid water density)
• \(\rho_a \approx 1\) kg/m³ (air density, negligible compared to \(\rho_w\))
• \(g = 9.81\) m/s²
• \(\eta = \mu \approx 1.8 \times 10^{-5}\) Pa·s (dynamic viscosity of air)

For larger drops (\(r > 40\) μm) where turbulent drag applies, the general terminal velocity expression balances gravity against the drag force with drag coefficient \(C_D\):

$$v_t = \left(\frac{4 \rho_w g D}{3 C_D \rho_a}\right)^{1/2}$$
where:
• \(D\) = drop diameter
• \(C_D\) = drag coefficient (depends on Reynolds number \(Re\))
• For \(Re < 1\) (Stokes): \(C_D = 24/Re\), recovering the Stokes formula
• For \(Re \sim 10^3\) (turbulent): \(C_D \approx 0.45\), giving \(v_t \propto D^{1/2}\)
Terminal velocities:
\(r = 1\) μm (cloud droplet):
\(v_t \approx 0.003\) m/s = 0.3 cm/s
\(r = 10\) μm (large cloud droplet):
\(v_t \approx 0.3\) m/s = 30 cm/s
\(r = 100\) μm (drizzle):
\(v_t \approx 0.7\) m/s = 70 cm/s
\(r = 1000\) μm (raindrop):
\(v_t \approx 6\) m/s

Note: For \(r > 40\) μm, turbulent drag applies and \(v_t\) grows more slowly (\(v_t \propto r^{1/2}\))

Collection Efficiency

Not all collisions result in coalescence. The collection efficiency \(E\)accounts for:

\(E = E_{\text{collision}} \times E_{\text{coalescence}}\)
• \(E_{\text{collision}}\): Fraction of droplets in the path that actually collide
- Small droplets follow air streamlines around the large droplet (low \(E_{\text{collision}}\))
- Large droplets have inertia and collide (high \(E_{\text{collision}}\))
• \(E_{\text{coalescence}}\): Fraction of collisions that result in permanent coalescence
- Very small droplets may bounce off due to air cushion
- Very large droplets may break apart after collision
- Optimal coalescence for moderate size ratios
Typical collection efficiencies:
• Collector \(r = 50\) μm, collected \(r = 5\) μm: \(E \approx 0.01\) (very inefficient)
• Collector \(r = 50\) μm, collected \(r = 10\) μm: \(E \approx 0.3\)
• Collector \(r = 100\) μm, collected \(r = 10\) μm: \(E \approx 0.8\) (efficient)

Gravitational Collection Equation

The rate of mass growth for a collector drop of radius \(R\) falling through a cloud of smaller droplets:

\(\frac{dM}{dt} = E \pi R^2 |v_t(R) - v_t(r)| \times LWC\)
where:
• \(M\) = mass of collector drop
• \(E\) = collection efficiency
• \(\pi R^2\) = collection cross-section
• \(|v_t(R) - v_t(r)|\) = relative velocity between collector and collected droplets
• \(LWC\) = liquid water content (mass of cloud droplets per unit volume of air)

A simplified form of the collision-coalescence growth equation expresses the mass growth rate of a collector drop falling through a cloud of liquid water content \(\rho_L\):

$$\frac{dm}{dt} = \pi r^2 E w_r \rho_L$$
where:
• \(r\) = radius of the collector drop
• \(E\) = collection efficiency
• \(w_r\) = relative fall speed between collector and collected droplets
• \(\rho_L\) = liquid water content of the cloud [kg/m³]

Since larger drops fall faster (\(v_t \propto r^2\) for Stokes regime), the growth rate accelerates:\(dR/dt \propto R^2\). This leads to runaway growth - a few lucky droplets that get slightly larger quickly dominate and become raindrops.

⏱️ Growth Timescales:

In a maritime cloud with large droplets (\(r \sim 15\) μm) and high LWC (1 g/m³):

• A 20 μm droplet can grow to 1 mm (rain) in ~20 minutes via collision-coalescence

This explains why warm-rain showers in the tropics can form quickly without ice processes!

5. Ice Processes in Clouds

Most precipitation in mid-latitudes forms via the ice phase, even when it reaches the ground as rain. Ice processes are crucial for precipitation formation.

5.1 Ice Nucleation

Unlike water droplets, ice crystals require even more favorable conditions to form:

Homogeneous Nucleation

Pure water freezes without a nucleation site only at very cold temperatures:

  • • Small droplets (\(r < 10\) μm): Freeze at \(T < -38°C\)
  • • Larger droplets: May freeze at slightly warmer temperatures
  • • Between 0°C and -38°C: Water remains supercooled liquid

Heterogeneous Nucleation

Ice nuclei (IN) catalyze freezing at warmer temperatures. Much rarer than CCN! Four distinct modes of heterogeneous ice nucleation operate in the atmosphere:

Ice Nucleation Modes (Detailed)

1. Deposition Nucleation

Water vapor deposits directly as ice onto the surface of an IN particle, bypassing the liquid phase entirely. Requires supersaturation with respect to ice (\(S_i > 0\)%) but subsaturation with respect to liquid water. Most effective at temperatures below -15°C to -20°C.

The nucleation rate depends on the contact angle \(\theta\) between the ice embryo and the IN surface:

\(J_{\text{dep}} \propto \exp\left(-\frac{\Delta G^* f(\theta)}{k_B T}\right)\)where \(f(\theta) = \frac{(2+\cos\theta)(1-\cos\theta)^2}{4}\)

2. Immersion Freezing

An IN particle is immersed within a supercooled liquid droplet and triggers freezing as the droplet cools to a threshold temperature. This is the most important heterogeneous nucleation mode in mixed-phase clouds.

The freezing probability increases with decreasing temperature and follows a stochastic or singular (deterministic) model. The singular model uses an ice nucleation active site (INAS) density:

\(n_s(T) = \text{number of active sites per unit area at temperature } T\)

3. Contact Freezing

An IN particle collides with and contacts the external surface of a supercooled droplet, initiating freezing. Often more efficient than immersion freezing at the same temperature because the surface contact lowers the energy barrier.

Scavenging mechanisms that bring IN to droplet surfaces include Brownian diffusion, thermophoresis, diffusiophoresis, and inertial impaction. Contact freezing can operate at temperatures as warm as -5°C.

4. Condensation-Freezing

Water vapor first condenses as liquid onto an IN particle (acting as a CCN), and the liquid then freezes almost immediately. This two-step process is distinct from deposition because the liquid phase forms transiently.

Condensation-freezing is important for large hygroscopic particles that also serve as IN (e.g., some mineral dust coated with soluble material).

5.2 Ice Nuclei (IN)

Effective ice nuclei have crystal structures similar to ice. Common IN include:

  • • Mineral dust: Clay minerals (kaolinite, montmorillonite) - most abundant IN
  • • Biological particles: Bacteria (e.g., Pseudomonas syringae), pollen, fungal spores
  • • Combustion particles: Soot, ash from biomass burning
  • • Marine organics: Organic matter from sea spray
IN activation temperatures (typical):
-5°C:
~1 IN per liter (very few)
-10°C:
~10 IN per liter
-15°C:
~100 IN per liter
-20°C:
~1000 IN per liter
-30°C:
~10,000 IN per liter
-38°C:
All droplets freeze (homogeneous)

🌡️ Mixed-Phase Clouds:

Between 0°C and -38°C, clouds contain both supercooled liquid droplets and ice crystals. This temperature range is where the Bergeron-Findeisen process (below) operates. The exact phase depends on availability of IN and cloud dynamics.

5.3 Bergeron-Findeisen Process

The Bergeron-Findeisen (or Wegener-Bergeron-Findeisen) processis the dominant precipitation mechanism in mid-latitude clouds. It exploits a fundamental thermodynamic difference between ice and liquid water.

Saturation Vapor Pressure: Ice vs Water

At the same temperature below 0°C, the saturation vapor pressure over ice is lower than over water:

\(e_{s,\text{ice}}(T) < e_{s,\text{water}}(T)\)for \(T < 0°C\)
The difference is largest around -12°C to -15°C:
At -5°C:
\(e_{s,w} = 4.22\) hPa, \(e_{s,i} = 4.01\) hPa (5% difference)
At -12°C:
\(e_{s,w} = 2.62\) hPa, \(e_{s,i} = 2.38\) hPa (9% difference)
At -20°C:
\(e_{s,w} = 1.25\) hPa, \(e_{s,i} = 1.03\) hPa (18% difference)

Mechanism

In a mixed-phase cloud at temperature \(T\) (e.g., -12°C):

  1. Air is saturated with respect to water: \(RH_{\text{water}} = 100\)%
    Vapor pressure \(e = e_{s,\text{water}}\)
  2. Same air is supersaturated with respect to ice: \(RH_{\text{ice}} > 100\)%
    Because \(e = e_{s,\text{water}} > e_{s,\text{ice}}\)
  3. Ice crystals grow rapidly by vapor deposition
    They experience significant supersaturation
  4. Supercooled droplets evaporate
    Vapor pressure drops below \(e_{s,\text{water}}\) as ice crystals consume vapor
  5. Net result: Vapor transfer from droplets to ice crystals
    Droplets shrink, ice crystals grow large enough to fall as snow

📐 Mathematical Deep Dive: Bergeron-Findeisen Growth Rate

The growth rate of an ice crystal by vapor deposition in a mixed-phase cloud:
\(\frac{dm_i}{dt} = 4\pi C D (\rho_{v,\infty} - \rho_{v,s,i})\)
where \(C\) is the capacitance (geometry factor, \(C \approx r\) for spheres).
In a mixed-phase cloud saturated with respect to water:
\(\rho_{v,\infty} = \rho_{v,s,w}\) (vapor density at saturation over water)
\(\rho_{v,s,i}\) (vapor density at saturation over ice)
The driving force is the difference:
\(\Delta\rho_v = \rho_{v,s,w} - \rho_{v,s,i} > 0\)
At -12°C with typical cloud conditions:
• Ice crystal of 100 μm can grow to 1 mm in ~15-20 minutes
• Much faster than collision-coalescence for same size range

5.4 Ice Crystal Habits

Ice crystals grow in different habits (shapes) depending on temperature and supersaturation:

Temperature-dependent habits:
• 0 to -3°C: Thin hexagonal plates
• -3 to -5°C: Needles (columnar)
• -5 to -8°C: Hollow columns
• -8 to -12°C: Hexagonal plates
• -12 to -16°C: Dendrites (classic snowflake shape - most rapid growth)
• -16 to -25°C: Plates
• -25 to -50°C: Hollow columns, then solid columns
• Below -50°C: Compact prisms

❄️ Dendrite Zone:

The -12°C to -16°C range is optimal for snowflake formation because: (1) largest \(e_{s,w} - e_{s,i}\) difference (fast Bergeron process), and (2) dendrite habit has high surface area for vapor deposition. Heavy snowfall often occurs when cloud tops are in this temperature range.

5.5 Riming and Aggregation

Ice particles grow further by mechanical processes:

Riming

Supercooled water droplets collide with ice crystals and freeze instantly upon contact, coating the crystal. Light riming produces snow pellets; heavy riming produces graupel(soft hail, 2-5 mm diameter). Extreme riming in strong updrafts produces hail.

Aggregation

Ice crystals collide and stick together, forming snowflakes. Most efficient near 0°C when crystals have sticky, partially melted surfaces. Large snowflakes (several cm) form when many crystals aggregate. Common with dendrites due to their complex shapes that interlock.

5.6 Hallett-Mossop Ice Multiplication

Observations consistently show that ice crystal concentrations in clouds exceed the number of available ice nuclei by orders of magnitude (sometimes \(10^3\text{-}10^4\) times more ice crystals than IN). This discrepancy is explained by secondary ice production (SIP) mechanisms, the most well-established being the Hallett-Mossop process.

Hallett-Mossop (Rime-Splintering) Process
Conditions required:
  • • Temperature range: -3°C to -8°C (narrow zone, peak at -5°C)
  • • Presence of riming ice particles (graupel or large frozen drops)
  • • Supercooled cloud droplets with a bimodal size distribution: some \(r < 12\) μm and some \(r > 24\) μm
Mechanism:

When supercooled droplets accrete onto a riming ice particle, mechanical stress during freezing causes tiny ice splinters to be ejected. These splinters act as new ice nuclei, each capable of growing into a full ice crystal. The rate of splinter production is approximately:

\(\frac{dN_i}{dt} \approx 350 \cdot \dot{N}_{\text{rime}}\)(splinters per large drop accreted)
Impact:

Hallett-Mossop multiplication can increase ice crystal concentrations by a factor of \(10^2\text{-}10^4\)within 15-30 minutes, rapidly glaciating the cloud. This is critical for precipitation initiation in moderately supercooled clouds where primary IN are scarce.

Other Secondary Ice Production Mechanisms:

  • • Droplet shattering: Large supercooled drops fragment upon freezing, producing ice splinters
  • • Ice-ice collisions: Fragile ice crystals (dendrites, needles) break during collisions, each fragment grows independently
  • • Sublimation fragmentation: Partially sublimating ice crystals break apart, releasing small ice fragments

💻 Computational Example:

Click to view and run which simulates:

  • • Vapor deposition growth of ice crystals in mixed-phase clouds
  • • Temperature-dependent habit selection
  • • Bergeron-Findeisen process efficiency at different temperatures
  • • Competition between ice and liquid phases

Program generates saturation vapor pressure plots, supersaturation ratios, ice growth simulations, and ice crystal habit diagrams (Nakaya).

6. Precipitation Formation

6.1 Warm Rain Process

The warm rain process produces precipitation entirely by collision-coalescence, without ice. Occurs in:

  • • Tropical clouds: Cloud tops remain above 0°C (shallow convection)
  • • Maritime clouds: Large droplets, efficient collision-coalescence
  • • Summer showers: Warm cloud bases, vigorous updrafts

Requirements for efficient warm rain:

1. Large Droplets

Initial droplet size \(r > 15\) μm (maritime conditions). Speeds up collision-coalescence by providing larger collectors.

2. High Liquid Water Content

LWC \(> 1\) g/m³. More cloud water available for collection means faster growth.

3. Cloud Depth

Deep clouds (\(> 2\) km thick) provide long collection paths for growing drops.

4. Broad Droplet Spectrum

Size diversity creates differential fall speeds needed for collisions.

6.2 Cold Rain Process

The cold rain process (ice-phase precipitation) dominates in mid-latitudes and produces most precipitation globally:

  1. Mixed-phase cloud formation
    Cloud extends above 0°C isotherm. Supercooled droplets and ice coexist.
  2. Ice nucleation
    IN activate at temperatures below -5°C to -10°C.
  3. Bergeron-Findeisen process
    Ice crystals grow rapidly at expense of supercooled droplets.
  4. Riming and aggregation
    Ice particles grow to precipitation size (1-10 mm).
  5. Fallout and melting
    Snow falls through 0°C level (melting level). If surface \(T > 0°C\), arrives as rain.

🌧️ Why Cold Rain Dominates:

Even summer rain in mid-latitudes often starts as snow aloft! Deep convective clouds extend well above the 0°C level (~4-5 km in summer). Ice processes are more efficient than collision-coalescence for producing precipitation-sized particles in the available time (cloud lifetime ~30-60 min).

6.3 Precipitation Types

The type of precipitation reaching the surface depends on the vertical temperature profile:

Rain

Liquid droplets \(> 0.5\) mm diameter. Forms from melted snow or warm rain process. Terminal velocity 4-9 m/s depending on size. Maximum raindrop size ~6 mm (larger drops break apart).

Drizzle

Small liquid droplets 0.2-0.5 mm. Falls slowly (\(< 3\) m/s). Common from stratus clouds, minimal evaporation needed. Light intensity.

Snow

Ice crystals and aggregates. Requires \(T < 0°C\) from cloud to surface. Density ~0.1 g/cm³ (10:1 snow ratio). Falls at 0.5-1.5 m/s. Variety of forms: dendrites, plates, needles, columns.

Graupel (Snow Pellets)

Heavily rimed ice particles, 2-5 mm. Opaque white, conical or spherical. Falls at 1-3 m/s. Common in convective clouds and winter storms.

Hail

Ice stones \(> 5\) mm, can exceed 10 cm. Forms in severe thunderstorms with strong updrafts (\(> 20\) m/s) that recirculate particles through supercooled regions. Alternating clear/opaque layers from wet vs dry growth.

Freezing Rain

Supercooled raindrops that freeze on contact with surface (\(T < 0°C\)). Forms when snow falls through warm layer aloft (melts to rain), then through shallow surface cold layer (insufficient time to refreeze). Creates dangerous ice accumulation.

Ice Pellets (Sleet)

Frozen raindrops, 2-5 mm transparent ice. Same vertical profile as freezing rain but deeper cold layer allows complete refreezing before reaching surface. Bounces on impact.

Melting Level and Rain/Snow Transition:

The 0°C isotherm altitude determines precipitation type. However, the transition is not sharp:

  • • Falling snow cools air by sublimation/melting → can lower 0°C level by ~300 m
  • • Wet-bulb temperature often better predictor than dry-bulb temperature
  • • Rain typically occurs when surface \(T > 2°C\), snow when \(T < 0°C\), mixed 0-2°C

6.4 Drop Size Distribution

Precipitation particle sizes follow statistical distributions. The Marshall-Palmer distribution describes raindrop sizes:

$$N(D) = N_0 \exp(-\Lambda D)$$
where:
• \(N(D)\) = number concentration of drops with diameter \(D\) [m⁻³ mm⁻¹]
• \(N_0 = 8000\) m⁻³ mm⁻¹ (intercept parameter, approximately constant)
• \(\Lambda = 41 R^{-0.21}\) mm⁻¹ (slope parameter)
• \(R\) = rainfall rate [mm/h]

Interpretation: Exponential decrease in number with size. Light rain (\(R = 1\) mm/h) has steep slope (mostly small drops). Heavy rain (\(R = 25\) mm/h) has shallower slope (more large drops).

6.5 Radar Reflectivity

Weather radars measure precipitation by detecting microwave energy scattered by hydrometeors. The received power from a radar pulse is given by the weather radar equation:

$$P_r = \frac{C |K|^2 Z}{r^2}$$
where:
• \(P_r\) = received power at the radar antenna
• \(C\) = radar constant (depends on wavelength, antenna gain, pulse width, beamwidth)
• \(|K|^2\) = dielectric factor (0.93 for water, 0.197 for ice)
• \(Z\) = radar reflectivity factor [mm⁶/m³]
• \(r\) = range (distance) to the target

The radar reflectivity factor (Z) is defined as:

\(Z = \int_0^\infty N(D) D^6 dD\)[mm⁶/m³]
Reflectivity is proportional to the 6th power of diameter - large drops dominate!
For Marshall-Palmer distribution:
\(Z = \frac{720 N_0}{\Lambda^7} \approx 200 R^{1.6}\)

In practice, reflectivity is expressed in dBZ (decibels of Z):

\(dBZ = 10 \log_{10}(Z)\)
Typical reflectivity values:
Light rain (\(R = 1\) mm/h):
~20 dBZ
Moderate rain (\(R = 5\) mm/h):
~35 dBZ
Heavy rain (\(R = 25\) mm/h):
~50 dBZ
Hail:
60-70 dBZ

The Z-R relationship converts radar reflectivity to rainfall rate using the general power-law form:

$$Z = aR^b$$
where \(Z\) is in mm⁶/m³, \(R\) is rain rate in mm/h, and \(a, b\) are empirical constants.
Common Z-R relationships:
• Stratiform rain: \(Z = 200 R^{1.6}\) (Marshall-Palmer, \(a=200, b=1.6\))
• Convective rain: \(Z = 300 R^{1.4}\) (larger drops, \(a=300, b=1.4\))
• Snow: \(Z = 2000 R^{2.0}\) (lower density, higher Z for given R)
• Tropical rain: \(Z = 250 R^{1.2}\) (broad drop spectrum)

6.6 Dual-Polarization Radar Variables

Modern weather radars transmit and receive electromagnetic waves in both horizontal (H) and vertical (V) polarizations. Dual-polarization (dual-pol) radar provides additional information about hydrometeor shape, size, phase, and orientation, greatly improving precipitation estimation and hydrometeor classification.

Differential Reflectivity (\(Z_{DR}\))

Ratio of reflectivity in horizontal to vertical polarization (in dB):

\(Z_{DR} = 10 \log_{10}\left(\frac{Z_H}{Z_V}\right)\)
  • • \(Z_{DR} > 0\) dB: Oblate particles (raindrops flatten as they fall; large rain: 1-4 dB)
  • • \(Z_{DR} \approx 0\) dB: Spherical or tumbling particles (drizzle, hail, graupel)
  • • \(Z_{DR} < 0\) dB: Prolate particles (vertically oriented ice crystals - rare)

Correlation Coefficient (\(\rho_{HV}\))

Correlation between H and V return signals. Indicates hydrometeor diversity within the radar volume:

  • • \(\rho_{HV} > 0.97\): Uniform hydrometeors (pure rain, dry snow)
  • • \(0.90 < \rho_{HV} < 0.97\): Mixed-phase precipitation, melting layer
  • • \(\rho_{HV} < 0.90\): Non-meteorological targets (ground clutter, debris, biological)
  • • Melting layer ("bright band"): \(\rho_{HV}\) drops to 0.85-0.95 as wet snow produces mixed signals

Specific Differential Phase (\(K_{DP}\))

Rate of change of the differential phase shift between H and V signals [°/km]:

\(K_{DP} = \frac{1}{2}\frac{d\Phi_{DP}}{dr}\)
  • • Proportional to liquid water content (not affected by hail or ice)
  • • Immune to radar calibration errors and attenuation
  • • Excellent rainfall estimator: \(R \approx 44 |K_{DP}|^{0.822}\) mm/h
  • • \(K_{DP} > 3\) °/km indicates heavy rain (\(>50\) mm/h)

Linear Depolarization Ratio (\(LDR\))

Ratio of cross-polar to co-polar return. Sensitive to mixed-phase hydrometeors and wet snow. Values: \(LDR < -30\) dB for rain, \(-30 < LDR < -20\) dB for wet snow/melting hail,\(LDR > -20\) dB for large wet hailstones.

Hydrometeor Classification Algorithm (HCA):

Dual-pol variables are combined using fuzzy logic or machine learning to automatically classify radar echoes into hydrometeor types: drizzle, rain, big drops, hail, graupel, dry snow, wet snow, ice crystals, and non-meteorological echoes. This dramatically improves:

  • • Quantitative Precipitation Estimation (QPE) accuracy
  • • Hail detection and severe weather warnings
  • • Winter precipitation type discrimination (rain vs. snow vs. freezing rain)
  • • Removal of non-meteorological clutter

💻 Computational Example:

Click to view and run which simulates:

  • • Simulate Marshall-Palmer drop size distributions
  • • Calculate radar reflectivity and rainfall rates
  • • Model terminal velocities and collection efficiencies
  • • Implement bulk microphysics schemes used in NWP models

Program generates Z-R relationships, terminal velocity curves, drop size distribution plots, and Kessler bulk microphysics simulations.

7. Cloud Electrification & Lightning (Advanced Topic)

Thunderstorms generate enormous electric fields through charge separation processes. When the field exceeds the dielectric breakdown of air (~3 MV/m), lightning occurs.

7.1 Charge Separation Mechanisms

The dominant mechanism for thunderstorm electrification is the non-inductive charging mechanism (ice-ice collisions):

  1. Riming occurs in mixed-phase region
    Graupel/hail particles grow by collecting supercooled droplets (-10°C to -20°C)
  2. Collisions between ice crystals and graupel
    During collision, charge transfer occurs (still not fully understood)
  3. Temperature-dependent polarity
    • Warmer than -15°C: Graupel becomes negatively charged, ice crystals positive
    • Colder than -15°C: Polarity reverses
  4. Gravitational separation
    Heavy graupel/hail falls → negative charge at cloud base
    Light ice crystals carried upward by updraft → positive charge at cloud top

This creates the classic tripole structure in thunderstorms:

• Upper positive charge (~10-12 km, -40°C to -60°C): Small ice crystals
• Main negative charge (~6-8 km, -10°C to -20°C): Graupel/hail
• Lower positive charge (~3-5 km, 0°C to -10°C): Smaller region, screening layer

7.2 Lightning Formation

When the electric field exceeds ~3 MV/m, air breaks down and conducts:

Intracloud (IC) Lightning

Most common type (~75% of all lightning). Discharge between positive and negative charge regions within the cloud. Often seen as diffuse "sheet lightning" when cloud blocks direct view.

Cloud-to-Ground (CG) Lightning

~25% of lightning. Most dangerous and well-studied. Multi-step process:

  1. Stepped leader: Negative charge descends in 50 m steps toward ground (~1 ms between steps)
  2. Attachment: When leader nears ground (~50-100 m), upward streamers from tall objects connect
  3. Return stroke: Bright discharge propagates upward at ~1/3 speed of light, 20,000-30,000 A current
  4. Continuing current: Lower-intensity current flow (100-1000 A) for 10-100 ms
  5. Dart leader: Subsequent strokes follow same channel (2-5 strokes typical, up to 20+)

Positive CG Lightning

~10% of CG strikes. Originates from upper positive charge region. Typically single stroke but much more powerful (100,000-400,000 A peak current). Can strike 10+ km from parent storm ("bolt from the blue"). More likely to start wildfires.

7.3 Thunder

Lightning heats the air channel to ~30,000 K (5× surface of the Sun) in microseconds. This explosive heating creates a shock wave that we hear as thunder.

Thunder characteristics:

  • • Initial sharp crack: From nearest part of lightning channel
  • • Rumbling: Sound from distant parts arrives later due to finite sound speed (~343 m/s)
  • • Duration ~5-20 seconds: Indicates channel length of 2-7 km
  • • Distance estimation: Count seconds between flash and thunder, divide by 3 for distance in km (or divide by 5 for miles)

⚡ Lightning Facts:

  • • Global lightning rate: ~100 flashes per second (~8.6 million per day)
  • • Peak return stroke current: 20,000-200,000 A (positive CG can exceed 400,000 A)
  • • Channel temperature: ~30,000 K
  • • Energy per flash: ~250 kWh (mostly dissipated as heat, light, sound)
  • • Flash duration: 0.2-2 seconds (multiple strokes)
  • • Channel diameter: 2-3 cm (return stroke), expands to ~10-20 cm

Summary

Cloud physics and precipitation processes are central to atmospheric science. Key concepts covered:

  • ✓ Clouds form via four lifting mechanisms: orographic, frontal, convective, convergence
  • ✓ Cloud condensation nuclei (CCN) enable droplet formation at small supersaturations
  • ✓ Köhler theory explains CCN activation through curvature and solute effects
  • ✓ Droplets grow by condensation (fast for small \(r\)) and collision-coalescence (needed for rain)
  • ✓ Ice processes dominate mid-latitude precipitation via Bergeron-Findeisen mechanism
  • ✓ Precipitation type depends on vertical temperature profile and microphysical processes
  • ✓ Lightning results from charge separation in mixed-phase clouds

Understanding these microphysical processes is essential for weather prediction, climate modeling, and quantifying aerosol-cloud-precipitation interactions. These principles bridge the gap between thermodynamics (Part I), dynamics (Part II), and larger-scale climate systems (Part IV).

NPTEL: Introduction to Atmospheric Science

Lectures from the NPTEL Introduction to Atmospheric Science course, covering condensation, cloud formation, atmospheric stability, and convective instability.

Lec-19 Problems Using Skew-T ln-P Chart (3)

Lec-20 Lifting Condensation Level (LCL)

Lec-21 Lifting Condensation Level (continued)

Lec-22 Saturated Adiabatic and Pseudo-adiabatic Processes

Lec-23 Equivalent Potential Temperature and Wet-Bulb Potential Temperature

Lec-24 Normand's Rule — Chinook Winds

Lec-25 Chinook Wind Problems and Static Stability

Lec-26 Static Stability — Brunt-Väisälä Frequency

Lec-27 Conditional and Convective Instability

Yale GG 140: Water, Clouds and Precipitation

Lectures on water vapor in the atmosphere, cloud formation, and precipitation processes.

09. Water in the Atmosphere I

10. Water in the Atmosphere II

11. Clouds and Precipitation (Cloud Chamber Experiment)