← Back to Part 2: Atmospheric Thermodynamics

2.4 Adiabatic Processes

Adiabatic processes involve no heat exchange with the environment. As air parcels rise and expand (or sink and compress), their temperature changes predictably.

Dry Adiabatic Lapse Rate

$$\Gamma_d = -\frac{dT}{dz} = \frac{g}{c_p} = 9.8 \text{ K/km}$$

Rate of temperature decrease for unsaturated rising air

Derived from the first law of thermodynamics with δQ = 0 and hydrostatic balance. Approximately 10°C per 1000m of ascent.

Moist Adiabatic Lapse Rate

$$\Gamma_m = \Gamma_d \frac{1 + \frac{L_v r_s}{R_d T}}{1 + \frac{L_v^2 r_s}{c_p R_v T^2}}$$

Varies from ~4-9 K/km depending on temperature and moisture

Warm, Moist Air

Γm ≈ 4-5 K/km

More latent heat release

Cold, Dry Air

Γm ≈ 8-9 K/km

Approaches dry adiabatic

Potential Temperature

$$\theta = T \left(\frac{p_0}{p}\right)^{R_d/c_p} = T \left(\frac{p_0}{p}\right)^{0.286}$$

Temperature a parcel would have if brought adiabatically to 1000 hPa

Key Properties

  • • Conserved during dry adiabatic processes
  • • Constant on isentropic surfaces
  • • ∂θ/∂z > 0 indicates stable stratification
  • • Used to compare air masses at different altitudes

Python: Adiabatic Processes

#!/usr/bin/env python3
"""
adiabatic_processes.py - Adiabatic lapse rates and potential temperature

Run: python3 adiabatic_processes.py
Requires: pip install numpy matplotlib
"""
import numpy as np
import matplotlib.pyplot as plt

# Constants
g = 9.81          # m/s²
R_d = 287.0       # J/(kg·K)
R_v = 461.5       # J/(kg·K)
c_p = 1005.0      # J/(kg·K)
c_v = 718.0       # J/(kg·K)
L_v = 2.5e6       # J/kg
p0 = 100000       # Pa (reference pressure for θ)
epsilon = R_d / R_v

# Dry adiabatic lapse rate
Gamma_d = g / c_p * 1000  # K/km

def saturation_mixing_ratio(T, p):
    """Saturation mixing ratio (kg/kg)"""
    e_s = 6.11 * np.exp(17.27 * (T - 273.15) / (T - 35.85)) * 100  # Pa
    return epsilon * e_s / (p - e_s)

def moist_adiabatic_lapse_rate(T, p):
    """Moist adiabatic lapse rate (K/km)"""
    r_s = saturation_mixing_ratio(T, p)
    numerator = 1 + L_v * r_s / (R_d * T)
    denominator = 1 + L_v**2 * r_s / (c_p * R_v * T**2)
    return Gamma_d * numerator / denominator

def potential_temperature(T, p):
    """Potential temperature (K)"""
    return T * (p0 / p) ** (R_d / c_p)

# Altitude and pressure profile
z = np.linspace(0, 15000, 100)  # m
p = p0 * np.exp(-z / 8500)       # hydrostatic approximation

# Parcel ascent: dry then moist
T_sfc = 300  # K
T_dry = np.zeros_like(z)
T_moist = np.zeros_like(z)
theta = np.zeros_like(z)

T_dry[0] = T_sfc
T_moist[0] = T_sfc
theta[0] = potential_temperature(T_sfc, p[0])

# Find LCL (assume initial RH = 80%)
RH_initial = 0.80
r_initial = RH_initial * saturation_mixing_ratio(T_sfc, p[0])
# Rough LCL calculation
lcl_idx = 20  # approximately

for i in range(1, len(z)):
    dz = z[i] - z[i-1]

    # Dry adiabatic
    T_dry[i] = T_dry[i-1] - Gamma_d * dz / 1000
    theta[i] = potential_temperature(T_dry[i], p[i])

    # Moist adiabatic (after LCL)
    if i < lcl_idx:
        T_moist[i] = T_moist[i-1] - Gamma_d * dz / 1000
    else:
        Gamma_m = moist_adiabatic_lapse_rate(T_moist[i-1], p[i-1])
        T_moist[i] = T_moist[i-1] - Gamma_m * dz / 1000

# Plot
fig, axes = plt.subplots(1, 3, figsize=(15, 8))

# Temperature profiles
ax1 = axes[0]
ax1.plot(T_dry - 273.15, z/1000, 'r-', linewidth=2, label='Dry Adiabatic')
ax1.plot(T_moist - 273.15, z/1000, 'b-', linewidth=2, label='Moist Adiabatic')
ax1.axhline(y=z[lcl_idx]/1000, color='green', linestyle='--', label=f'LCL ≈ {z[lcl_idx]/1000:.1f} km')
ax1.set_xlabel('Temperature (°C)', fontsize=12)
ax1.set_ylabel('Altitude (km)', fontsize=12)
ax1.set_title('Adiabatic Temperature Profiles', fontsize=14)
ax1.legend()
ax1.grid(True, alpha=0.3)
ax1.set_xlim(-80, 30)

# Potential temperature
ax2 = axes[1]
ax2.plot(theta, z/1000, 'purple', linewidth=2)
ax2.axhline(y=z[lcl_idx]/1000, color='green', linestyle='--')
ax2.set_xlabel('Potential Temperature θ (K)', fontsize=12)
ax2.set_ylabel('Altitude (km)', fontsize=12)
ax2.set_title('Potential Temperature Profile', fontsize=14)
ax2.grid(True, alpha=0.3)
# θ is constant for dry adiabatic
ax2.annotate('θ = const (dry adiabatic)', xy=(theta[10], z[10]/1000),
             xytext=(320, 5), fontsize=10, color='purple',
             arrowprops=dict(arrowstyle='->', color='purple'))

# Moist adiabatic lapse rate vs temperature
ax3 = axes[2]
T_range = np.linspace(250, 310, 100)
Gamma_m_values = [moist_adiabatic_lapse_rate(T, 85000) for T in T_range]
ax3.plot(T_range - 273.15, Gamma_m_values, 'b-', linewidth=2, label='Γ_m at 850 hPa')
ax3.axhline(y=Gamma_d, color='r', linestyle='--', linewidth=2, label=f'Γ_d = {Gamma_d:.1f} K/km')
ax3.set_xlabel('Temperature (°C)', fontsize=12)
ax3.set_ylabel('Lapse Rate (K/km)', fontsize=12)
ax3.set_title('Moist Adiabatic Lapse Rate', fontsize=14)
ax3.legend()
ax3.grid(True, alpha=0.3)
ax3.set_ylim(3, 11)

plt.tight_layout()
plt.savefig('adiabatic_processes.png', dpi=150, bbox_inches='tight')
plt.show()

print("Adiabatic Lapse Rates:")
print(f"  Dry adiabatic: Γ_d = {Gamma_d:.2f} K/km")
print(f"  Moist adiabatic at 0°C: {moist_adiabatic_lapse_rate(273.15, 85000):.2f} K/km")
print(f"  Moist adiabatic at 20°C: {moist_adiabatic_lapse_rate(293.15, 85000):.2f} K/km")
print(f"\nPotential Temperature:")
print(f"  θ(300 K, 1000 hPa) = {potential_temperature(300, 100000):.1f} K")
print(f"  θ(280 K, 700 hPa) = {potential_temperature(280, 70000):.1f} K")
← 2.3 Stability & ConvectionPart 3: Atmospheric Dynamics →