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2.2 Moisture & Humidity

Water vapor is the most important variable constituent of the atmosphere. Its phase changes release and absorb latent heat, driving convection and weather systems.

Humidity Variables

Mixing Ratio (r)

$$r = \frac{m_v}{m_d} = \frac{\rho_v}{\rho_d}$$

Mass of water vapor per mass of dry air (g/kg)

Specific Humidity (q)

$$q = \frac{m_v}{m_v + m_d} = \frac{r}{1+r} \approx r$$

Mass of water vapor per total mass (g/kg)

Relative Humidity (RH)

$$RH = \frac{e}{e_s} \times 100\%$$

Ratio of actual to saturation vapor pressure

Clausius-Clapeyron Equation

$$\frac{de_s}{dT} = \frac{L_v e_s}{R_v T^2}$$

Rate of change of saturation vapor pressure with temperature

Integrated form (Tetens formula):

$$e_s(T) = 6.11 \exp\left(\frac{17.27 (T-273.15)}{T - 35.85}\right) \text{ hPa}$$

Saturation vapor pressure approximately doubles for every 10°C increase

Dew Point & Lifting Condensation Level

Definition

Temperature to which air must be cooled at constant pressure to become saturated

Dew Point Depression

T - T_d: larger values indicate drier air

Lifting Condensation Level (LCL)

$$z_{\text{LCL}} \approx 125\,(T - T_d) \quad \text{(metres)}$$

Quick estimate of cloud base height from surface temperature and dew point (both in °C). Based on the different lapse rates of temperature and dew point with height.

Wet-Bulb Temperature Relationship

$$T_d \leq T_w \leq T$$

The wet-bulb temperature $T_w$ is found by evaporating water into air at constant pressure until saturation:

$$c_p\,(T - T_w) = L_v\bigl(w_s(T_w) - w\bigr)$$

where $w_s(T_w)$ is the saturation mixing ratio at the wet-bulb temperature

Advanced Moisture Thermodynamics

Equivalent Potential Temperature

$$\theta_e \approx \theta \exp\!\left(\frac{L_v\, w_s}{c_p\, T}\right)$$

Conserved during both dry and moist adiabatic processes. Represents the potential temperature a parcel would reach if all its moisture were condensed out and the latent heat used to warm the parcel.

Precipitable Water

$$W = \frac{1}{\rho_w\, g} \int_0^{\infty} \rho\, w \, dz = \frac{1}{g} \int_0^{p_s} q \, dp$$

Total water vapor in a column expressed as liquid depth (typically 10-50 mm). Here $\rho_w$ is the density of liquid water, $w$ is mixing ratio, and $q$ is specific humidity.

Interactive Simulation: Clausius-Clapeyron Saturation Curve

Python

Plots saturation vapor pressure over liquid water and ice from -40 to 45 degrees C. Shows the Bergeron process region, common atmospheric conditions, dewpoint depression at various relative humidities, and prints a table of key values.

clausius_clapeyron.py75 lines

import numpy as np
import matplotlib.pyplot as plt

def es_liquid(T_c):
    """Saturation vapor pressure over liquid water (hPa) - Tetens formula"""
    return 6.112 * np.exp(17.67 * T_c / (T_c + 243.5))

def es_ice(T_c):
    """Saturation vapor pressure over ice (hPa) - Murphy-Koop approx"""
    return 6.112 * np.exp(22.46 * T_c / (T_c + 272.62))

T = np.linspace(-40, 45, 500)
es_liq = es_liquid(T)
T_ice = np.linspace(-40, 0, 250)
es_i = es_ice(T_ice)

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(13, 7))
fig.patch.set_facecolor('#0f172a')
for ax in (ax1, ax2):
    ax.set_facecolor('#1e293b'); ax.tick_params(colors='#cbd5e1')
    ax.xaxis.label.set_color('#cbd5e1'); ax.yaxis.label.set_color('#cbd5e1')
    ax.title.set_color('#cbd5e1'); ax.grid(True, color='#334155', alpha=0.5)
    for sp in ax.spines.values(): sp.set_color('#334155')

# Left plot: saturation vapor pressure curves
ax1.semilogy(T, es_liq, color='#38bdf8', lw=2.5, label='Over liquid water')
ax1.semilogy(T_ice, es_i, color='#a78bfa', lw=2.5, ls='--', label='Over ice')

# Mark common conditions
conditions = [(-20, 'Arctic winter', '#94a3b8'), (0, 'Freezing', '#e2e8f0'),
              (20, 'Room temp', '#fbbf24'), (35, 'Hot summer', '#ef4444')]
for tc, lbl, col in conditions:
    es_val = es_liquid(tc)
    ax1.plot(tc, es_val, 'o', color=col, ms=8, zorder=5)
    ax1.annotate(f'{lbl}\n{es_val:.1f} hPa', (tc, es_val), textcoords='offset points',
                 xytext=(10, 10), color=col, fontsize=8,
                 arrowprops=dict(arrowstyle='->', color=col, lw=1))

# Shade supercooled region
ax1.fill_between(T_ice, es_i, es_liquid(T_ice), alpha=0.15, color='#c084fc')
ax1.text(-25, 0.8, 'Bergeron\nprocess\nregion', color='#c084fc', fontsize=8, ha='center')

ax1.set_xlabel('Temperature (\u00b0C)', fontsize=11)
ax1.set_ylabel('Saturation Vapor Pressure (hPa)', fontsize=11)
ax1.set_title('Clausius-Clapeyron Saturation Curves', fontsize=13, fontweight='bold')
ax1.legend(loc='upper left', facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

# Right plot: dewpoint depression visualization
T_air = 30.0
RH_vals = np.array([30, 50, 70, 90, 100])
es_air = es_liquid(T_air)
e_vals = RH_vals / 100.0 * es_air
Td_vals = 243.5 * np.log(e_vals / 6.112) / (17.67 - np.log(e_vals / 6.112))
depression = T_air - Td_vals

bars = ax2.barh(range(len(RH_vals)), depression, color=['#ef4444','#f97316','#fbbf24','#4ade80','#38bdf8'], height=0.6)
ax2.set_yticks(range(len(RH_vals)))
ax2.set_yticklabels([f'RH={r}%' for r in RH_vals])
ax2.set_xlabel('Dewpoint Depression T - T_d (\u00b0C)', fontsize=11)
ax2.set_title(f'Dewpoint Depression at T={T_air:.0f}\u00b0C', fontsize=13, fontweight='bold')
for i, (dep, td) in enumerate(zip(depression, Td_vals)):
    ax2.text(dep + 0.3, i, f'T_d={td:.1f}\u00b0C', va='center', color='#cbd5e1', fontsize=9)

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor=fig.get_facecolor())
print("=== Clausius-Clapeyron Saturation Vapor Pressure ===")
print(f"{'T (\u00b0C)':>8} {'es_liq (hPa)':>13} {'es_ice (hPa)':>13}")
print("-" * 36)
for tc in [-40, -20, -10, 0, 10, 20, 30, 40]:
    el = es_liquid(tc)
    ei = es_ice(tc) if tc <= 0 else float('nan')
    ice_str = f'{ei:13.3f}' if not np.isnan(ei) else '          N/A'
    print(f"{tc:8.0f} {el:13.3f} {ice_str}")
print(f"\nAt T=20\u00b0C: es = {es_liquid(20):.2f} hPa")
print(f"At T=30\u00b0C: es = {es_liquid(30):.2f} hPa  (~7%/\u00b0C increase)")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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