8.2 Aerosols

Atmospheric aerosols -- suspended solid or liquid particles ranging from nanometers to tens of micrometers -- exert a profound influence on Earth's radiation budget, cloud formation, precipitation, and air quality. They constitute the largest source of uncertainty in anthropogenic radiative forcing estimates. Understanding aerosol microphysics, optical properties, and aerosol-cloud interactions is central to climate science and remains an active frontier of research.

Aerosol Types and Sources

Aerosols are classified by composition and origin. Natural sources dominate global mass emissions, but anthropogenic aerosols have stronger per-mass radiative effects due to their smaller size and optical properties.

Sea Salt

Produced by bubble bursting and wave breaking over the ocean. Predominantly coarse mode (1--10 um). Highly hygroscopic, efficient CCN. Global emission: ~3,000--10,000 Tg/yr. Composition: NaCl with traces of Mg, Ca, K sulfates.

Mineral Dust

Wind-eroded soil from arid regions (Sahara, Gobi, Arabian deserts). Broad size range (0.1--100 um). Can serve as ice nuclei (IN). Global emission: ~1,000--3,000 Tg/yr. Absorbs and scatters radiation; iron content fertilizes remote oceans.

Sulfate Aerosol

Formed by oxidation of SO₂ (from fossil fuel burning and volcanic emissions) via gas-phase OH reaction and aqueous-phase H₂O₂/O₃ oxidation. Accumulation mode (0.1--1 um). Very efficient light scatterer. Dominant anthropogenic cooling agent.

Black Carbon (BC)

Soot from incomplete combustion of fossil fuels and biomass. Strong absorber across all wavelengths. Warms the atmosphere, reduces surface albedo when deposited on snow/ice. Radiative forcing: +0.4 to +1.2 W/m². Short atmospheric lifetime (~1 week).

Organic Carbon (OC)

Primary OC from combustion and biological debris. Predominantly scattering. Brown carbon (BrC) absorbs at UV/blue wavelengths. Mixed with BC in combustion plumes. Weakly hygroscopic.

Secondary Organic Aerosol (SOA)

Formed by oxidation of biogenic (isoprene, terpenes) and anthropogenic VOCs. Nucleation and condensation onto existing particles. Major contributor to submicron aerosol mass. Highly complex and uncertain chemistry.

Aerosol Size Distribution

The aerosol size distribution is commonly represented as a log-normal function. For a single mode, the number distribution with respect to the natural logarithm of radius is:

$$\frac{dN}{d(\ln r)} = \frac{N_0}{\sqrt{2\pi}\,\ln\sigma_g} \exp\!\left(-\frac{\ln^2(r/r_g)}{2\ln^2\sigma_g}\right)$$

where $N_0$ is the total number concentration, $r_g$ is the geometric mean radius, and $\sigma_g$ is the geometric standard deviation. Real atmospheric distributions are multimodal, typically represented as a sum of 3--4 log-normal modes:

Mode	Size Range	Typical r_g	Sources / Processes	Lifetime
Nucleation	< 0.01 um	~5 nm	Gas-to-particle conversion, new particle formation	Hours (coagulation)
Aitken	0.01--0.1 um	~30 nm	Coagulation of nucleation mode, combustion	Days
Accumulation	0.1--1 um	~150 nm	Condensation, coagulation, cloud processing	~1 week
Coarse	> 1 um	~3 um	Mechanical generation (wind, sea spray)	Hours--days (settling)

The accumulation mode has the longest atmospheric lifetime because particles in this size range are too large for efficient Brownian coagulation and too small for efficient gravitational settling or impaction scavenging. This is sometimes called the "accumulation gap" in removal efficiency.

Aerosol Optical Properties

Aerosol interactions with radiation are described by several optical parameters derived from Mie scattering theory.

Aerosol Optical Depth (AOD)

The column-integrated extinction, measuring total attenuation of a beam through the atmosphere:

$$\tau(\lambda) = \int_0^{\infty} \sigma_{\text{ext}}(\lambda, z)\,dz = \int_0^{\infty} \int_0^{\infty} Q_{\text{ext}}(x, m)\,\pi r^2\,n(r,z)\,dr\,dz$$

where $\sigma_{\text{ext}}$ is the volume extinction coefficient (m⁻¹), $Q_{\text{ext}}$ is the Mie extinction efficiency, $x = 2\pi r/\lambda$ is the size parameter, and $m = n_r + i\,n_i$ is the complex refractive index.

Mie Scattering Theory

For spherical particles, Mie theory provides exact solutions for the efficiency factors as functions of size parameter x and refractive index m:

Extinction Efficiency

$$Q_{\text{ext}} = Q_{\text{sca}} + Q_{\text{abs}}$$

Total removal of light from forward beam

Scattering Efficiency

$$Q_{\text{sca}} = \frac{2}{x^2}\sum_{n=1}^{\infty}(2n+1)(|a_n|^2 + |b_n|^2)$$

Mie coefficients a_n, b_n from boundary conditions

Absorption Efficiency

$$Q_{\text{abs}} = Q_{\text{ext}} - Q_{\text{sca}}$$

Depends on imaginary part of refractive index

Rayleigh vs Mie Regimes

When the particle is much smaller than the wavelength ($x \ll 1$), the Rayleigh approximation applies:

$$\sigma_{\text{sca}} \propto \frac{r^6}{\lambda^4} \qquad \Rightarrow \qquad \text{Blue light scattered } \sim 10\times \text{ more than red}$$

This $\lambda^{-4}$ dependence explains why the sky is blue and why sunsets are red (short wavelengths are preferentially scattered out of the direct beam). For larger particles (x ~ 1, Mie regime), the wavelength dependence weakens, producing the white appearance of clouds and haze.

Angstrom Exponent

The wavelength dependence of AOD is parameterized by the Angstrom exponent:

$$\tau(\lambda) = \tau_0 \left(\frac{\lambda}{\lambda_0}\right)^{-\alpha} \qquad \alpha = -\frac{\ln[\tau(\lambda_1)/\tau(\lambda_2)]}{\ln(\lambda_1/\lambda_2)}$$

Large $\alpha$ (> 1.5) indicates small particles (smoke, pollution). Small $\alpha$ (near 0) indicates large particles (dust, sea salt). AERONET sun photometers routinely measure AOD and alpha at multiple wavelengths worldwide.

Single Scattering Albedo and Asymmetry Parameter

Single Scattering Albedo (SSA)

$$\omega_0 = \frac{\sigma_{\text{sca}}}{\sigma_{\text{ext}}} = \frac{Q_{\text{sca}}}{Q_{\text{ext}}}$$

$\omega_0 = 1$: purely scattering (sulfate). $\omega_0 \approx 0.2$: strongly absorbing (BC). $\omega_0 \approx 0.85{-}0.95$: mixed aerosol. Critical threshold: aerosols with $\omega_0$ below ~0.85 can cause net warming.

Asymmetry Parameter

$$g = \frac{1}{2}\int_{-1}^{1} P(\cos\theta)\,\cos\theta\,d(\cos\theta)$$

$g = 0$: isotropic (Rayleigh). $g \rightarrow 1$: all forward scattering (large particles). Typical aerosol values: 0.5--0.7. Used in two-stream radiative transfer with upscatter fraction $\beta \approx (1-g)/2$.

Direct Aerosol Radiative Effect

The change in top-of-atmosphere radiative flux due to an optically thin aerosol layer over a surface with albedo $a_s$ is approximated by the Charlson-Pilat-Heintzenberg formula:

$$\Delta F \approx -\frac{S_0}{4}\,T_{\text{atm}}^2\,\left[2(1-a_s)^2\,\beta\,\omega_0\,\tau - 4\,a_s\,(1-\omega_0)\,\tau\right]$$

where $S_0 \approx 1361$ W/m² is the solar constant, $T_{\text{atm}}$ is atmospheric transmittance above the aerosol layer, $\beta$ is the upscatter fraction, and $\tau$ is AOD. The first term (scattering) always cools; the second term (absorption) always warms.

Critical Single Scattering Albedo

Setting $\Delta F = 0$ yields the critical SSA below which the aerosol warms rather than cools:

$$\omega_{0,\text{crit}} = \frac{2a_s}{2a_s + (1-a_s)^2\beta}$$

Over dark surfaces (ocean, forest) with low $a_s$, even moderately absorbing aerosols cause cooling. Over bright surfaces (desert, snow), the critical SSA is higher, so absorbing aerosols more easily cause warming -- especially relevant for BC deposited on Arctic snow.

Aerosol Indirect Effects and Cloud Interactions

Aerosols influence climate indirectly by modifying cloud properties. These effects are collectively the largest uncertainty in anthropogenic forcing estimates.

First Indirect Effect (Twomey / Cloud Albedo Effect)

For fixed liquid water content (LWC), increasing aerosol number concentration $N_a$ increases cloud droplet number $N_d$, decreasing mean droplet effective radius $r_e$: $r_e \propto (LWC/N_d)^{1/3}$. Smaller droplets increase cloud optical depth and albedo: $\tau_c = \frac{3\,LWP}{2\,\rho_w\,r_e}$. Estimated forcing: -0.3 to -1.8 W/m².

Second Indirect Effect (Cloud Lifetime / Albrecht Effect)

Smaller cloud droplets suppress collision-coalescence, delaying or inhibiting precipitation. Clouds persist longer and have greater LWP, further increasing reflectivity. This effect is difficult to quantify because it involves complex feedbacks with dynamics and meteorology.

Semi-Direct Effect

Absorbing aerosols (BC) heat the atmosphere, stabilizing the layer and reducing relative humidity. This can evaporate existing clouds ("cloud burn-off") or suppress convective cloud formation. The net effect is typically warming, partially offsetting the indirect cooling.

Kohler Theory and CCN Activation

The critical supersaturation $S_c$ required to activate an aerosol particle as a cloud condensation nucleus (CCN) is derived from Kohler theory, balancing the Kelvin (curvature) and Raoult (solute) effects:

$$S = \frac{e_s(r)}{e_s(\infty)} = \exp\!\left(\frac{2\sigma_{w}}{n_w k_B T r}\right) \cdot \left(1 + \frac{i\,n_s}{n_w}\right)^{-1} \approx 1 + \frac{A}{r} - \frac{B}{r^3}$$

The maximum of this curve gives the critical supersaturation:

$$S_c = \left(\frac{4A^3}{27B}\right)^{1/2} \propto r_{\text{dry}}^{-3/2}$$

Larger, more soluble particles activate at lower supersaturations. Typical cloud supersaturations are 0.1--0.5%, so particles above ~50--100 nm dry diameter typically activate as CCN. The CCN spectrum $N_{\text{CCN}}(S)$ is often parameterized as $N_{\text{CCN}} = C\,S^k$ where k ranges from 0.3 (maritime) to 1.5 (continental).

Fortran: Log-Normal Integration for Bulk Optical Properties

This Fortran program integrates over a log-normal size distribution to compute bulk extinction, scattering, and absorption coefficients using a simplified Mie parameterization, and derives the single scattering albedo and asymmetry parameter.

program aerosol_bulk_optics
  ! Compute bulk optical properties from log-normal size distribution
  ! using simplified Mie efficiency parameterization
  implicit none

  integer, parameter :: dp = selected_real_kind(15, 307)
  integer, parameter :: nr = 500       ! Number of radius bins
  real(dp), parameter :: pi = 3.141592653589793_dp

  ! Distribution parameters (accumulation mode)
  real(dp) :: N0, rg, sigma_g, lambda_nm, lambda
  real(dp) :: r_min, r_max, r, dlnr, lnr, dNdlnr
  real(dp) :: x, rho, Qext, Qsca, Qabs, g_asym
  real(dp) :: sigma_ext_total, sigma_sca_total, sigma_abs_total
  real(dp) :: g_total, area_sca, omega0
  real(dp) :: m_real, m_imag, layer_H, aod
  integer  :: i, j, nwav
  real(dp) :: wavs(8)

  ! --- Parameters ---
  N0      = 500.0e6_dp       ! Number concentration (#/m^3)
  rg      = 0.15e-6_dp       ! Geometric mean radius (m)
  sigma_g = 1.6_dp           ! Geometric standard deviation
  m_real  = 1.53_dp          ! Real refractive index
  m_imag  = 0.005_dp         ! Imaginary refractive index
  layer_H = 2000.0_dp        ! Layer thickness (m)
  r_min   = 1.0e-8_dp        ! 10 nm
  r_max   = 1.0e-5_dp        ! 10 um

  ! Wavelengths to evaluate (nm)
  wavs = (/ 340.0_dp, 380.0_dp, 440.0_dp, 500.0_dp, &
            550.0_dp, 675.0_dp, 870.0_dp, 1020.0_dp /)
  nwav = 8

  dlnr = (log(r_max) - log(r_min)) / real(nr, dp)

  print '(A)', '  Lambda(nm)  sigma_ext(1/m)  AOD      omega_0   g'
  print '(A)', '  ---------   --------------  -------  --------  -----'

  do j = 1, nwav
    lambda = wavs(j) * 1.0e-9_dp   ! Convert nm to m

    sigma_ext_total = 0.0_dp
    sigma_sca_total = 0.0_dp
    sigma_abs_total = 0.0_dp
    g_total         = 0.0_dp

    do i = 1, nr
      lnr = log(r_min) + (real(i, dp) - 0.5_dp) * dlnr
      r   = exp(lnr)

      ! Log-normal distribution: dN/d(ln r)
      dNdlnr = (N0 / (sqrt(2.0_dp * pi) * log(sigma_g))) * &
                exp(-lnr**2 / (2.0_dp * log(sigma_g)**2) + &
                    lnr * log(rg) / log(sigma_g)**2 - &
                    log(rg)**2 / (2.0_dp * log(sigma_g)**2))

      ! Size parameter
      x = 2.0_dp * pi * r / lambda

      ! Anomalous diffraction approximation for Qext
      rho = 2.0_dp * x * abs(m_real - 1.0_dp)
      if (rho > 0.01_dp) then
        Qext = 2.0_dp - 4.0_dp / rho * sin(rho) + &
               4.0_dp / rho**2 * (1.0_dp - cos(rho))
      else
        Qext = 0.0_dp
      end if
      if (Qext < 0.0_dp) Qext = 0.0_dp

      ! Simple absorption estimate
      Qabs = 4.0_dp * x * m_imag / m_real
      if (Qabs > Qext) Qabs = Qext * 0.5_dp
      Qsca = Qext - Qabs

      ! Asymmetry parameter approximation
      if (x < 0.5_dp) then
        g_asym = 0.0_dp   ! Rayleigh limit
      else
        g_asym = 0.65_dp * (1.0_dp - exp(-x / 5.0_dp))
      end if

      ! Accumulate bulk properties
      sigma_ext_total = sigma_ext_total + Qext * pi * r**2 * dNdlnr * dlnr
      sigma_sca_total = sigma_sca_total + Qsca * pi * r**2 * dNdlnr * dlnr
      sigma_abs_total = sigma_abs_total + Qabs * pi * r**2 * dNdlnr * dlnr
      g_total = g_total + g_asym * Qsca * pi * r**2 * dNdlnr * dlnr
    end do

    ! Derived quantities
    omega0 = sigma_sca_total / sigma_ext_total
    aod    = sigma_ext_total * layer_H
    if (sigma_sca_total > 0.0_dp) then
      g_total = g_total / sigma_sca_total
    end if

    print '(F8.0, 4X, ES14.4, 2X, F8.5, 2X, F7.4, 2X, F6.4)', &
      wavs(j), sigma_ext_total, aod, omega0, g_total
  end do

end program aerosol_bulk_optics

Interactive Simulation: Mie Scattering & Aerosol Optical Depth

Python

Calculate scattering efficiency vs size parameter showing Rayleigh to geometric optics regimes, and plot aerosol optical depth vs wavelength for different aerosol types (urban pollution, biomass smoke, sea salt, desert dust) with Angstrom exponent calculation.

mie_scattering_aod.py92 lines

import numpy as np
import matplotlib.pyplot as plt

# Mie Scattering Efficiency vs Size Parameter
# Using anomalous diffraction approximation (van de Hulst)
def q_ext_approx(x, m_real):
    """Approximate Mie extinction efficiency."""
    rho = 2.0 * x * abs(m_real - 1.0)
    if rho < 0.01:
        return 4.0 * rho**2 / 3.0 if rho > 0 else 0.0  # Rayleigh limit
    Q = 2.0 - (4.0/rho)*np.sin(rho) + (4.0/rho**2)*(1.0 - np.cos(rho))
    return max(Q, 0.0)

# Size parameter range
x_arr = np.logspace(-2, 2.5, 500)
m_vals = [1.33, 1.45, 1.53]  # water, organic, sulfate
labels_m = ['Water (m=1.33)', 'Organic (m=1.45)', 'Sulfate (m=1.53)']
colors_m = ['#06b6d4', '#22c55e', '#a78bfa']

fig, axes = plt.subplots(1, 2, figsize=(14, 6))
fig.patch.set_facecolor('#0f172a')
for ax in axes:
    ax.set_facecolor('#1e293b')
    ax.tick_params(colors='#cbd5e1')
    for spine in ax.spines.values():
        spine.set_color('#334155')
    ax.grid(True, color='#334155', alpha=0.4)

# Panel 1: Q_ext vs size parameter
for m, lbl, clr in zip(m_vals, labels_m, colors_m):
    Q = np.array([q_ext_approx(xi, m) for xi in x_arr])
    axes[0].loglog(x_arr, Q, color=clr, lw=2, label=lbl)

# Rayleigh regime reference
axes[0].axvline(x=0.1, color='#475569', ls=':', alpha=0.5)
axes[0].text(0.04, 3.0, 'Rayleigh\n(x<<1)', color='#94a3b8', fontsize=8, ha='center')
axes[0].axvline(x=10, color='#475569', ls=':', alpha=0.5)
axes[0].text(40, 3.0, 'Geometric\n(x>>1)', color='#94a3b8', fontsize=8, ha='center')
axes[0].text(1.0, 3.5, 'Mie regime', color='#94a3b8', fontsize=9, ha='center')
axes[0].set_xlabel('Size Parameter x = 2*pi*r/lambda', color='#cbd5e1')
axes[0].set_ylabel('Extinction Efficiency Q_ext', color='#cbd5e1')
axes[0].set_title('Mie Scattering Efficiency', color='#e2e8f0', fontweight='bold')
axes[0].legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=9)
axes[0].set_ylim(1e-4, 5)

# Panel 2: AOD vs wavelength for different aerosol types
wavs = np.linspace(0.3, 1.1, 80)  # micrometers
aerosol_types = {
    'Urban pollution': {'rg': 0.12, 'sg': 1.6, 'm': 1.53, 'N': 5000, 'H': 1500, 'color': '#ef4444'},
    'Biomass smoke':   {'rg': 0.08, 'sg': 1.7, 'm': 1.50, 'N': 8000, 'H': 2000, 'color': '#f97316'},
    'Sea salt':        {'rg': 1.0,  'sg': 2.0, 'm': 1.38, 'N': 100,  'H': 1000, 'color': '#06b6d4'},
    'Desert dust':     {'rg': 0.5,  'sg': 2.2, 'm': 1.56, 'N': 200,  'H': 3000, 'color': '#eab308'},
}

print("=== Aerosol Optical Depth & Angstrom Exponent ===")
print(f"{'Type':<20}{'AOD@550nm':>12}{'Angstrom':>10}")

for name, props in aerosol_types.items():
    r = np.logspace(-3, 1, 200)  # um
    dr = np.gradient(r)
    dNdr = (props['N'] / (r * np.sqrt(2*np.pi) * np.log(props['sg']))) * \
           np.exp(-np.log(r/props['rg'])**2 / (2*np.log(props['sg'])**2))
    aod = np.zeros_like(wavs)
    for i, lam in enumerate(wavs):
        sigma_ext = 0.0
        for j in range(len(r)):
            xi = 2 * np.pi * r[j] / lam
            Qe = q_ext_approx(xi, props['m'])
            sigma_ext += Qe * np.pi * (r[j]*1e-6)**2 * (dNdr[j]*1e6) * (dr[j]*1e-6)
        aod[i] = sigma_ext * props['H']
    axes[1].semilogy(wavs*1000, aod, color=props['color'], lw=2, label=name)

# Angstrom exponent
    i440 = np.argmin(np.abs(wavs - 0.44))
    i870 = np.argmin(np.abs(wavs - 0.87))
    if aod[i440] > 0 and aod[i870] > 0:
        alpha = -np.log(aod[i440]/aod[i870]) / np.log(0.44/0.87)
    else:
        alpha = 0.0
    i550 = np.argmin(np.abs(wavs - 0.55))
    print(f"{name:<20}{aod[i550]:>12.4f}{alpha:>10.2f}")

axes[1].set_xlabel('Wavelength (nm)', color='#cbd5e1')
axes[1].set_ylabel('Aerosol Optical Depth', color='#cbd5e1')
axes[1].set_title('AOD Spectra by Aerosol Type', color='#e2e8f0', fontweight='bold')
axes[1].legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=9)

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor=fig.get_facecolor())
print("\nSmall particles (smoke) -> steep wavelength dependence (high Angstrom)")
print("Large particles (dust, sea salt) -> flat spectrum (low Angstrom)")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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