from openap import contrail7 Contrails
Contrails (condensation trails) are line-shaped ice clouds produced by aircraft engine exhaust. The openap.contrail module provides functions to calculate contrail formation conditions and their climate impact through radiative forcing.
7.1 Overview
The module includes functions for:
- Saturation pressure: Water vapor saturation over liquid water and ice (Murphy & Koop 2005)
- Relative humidity: Conversions between specific humidity and relative humidity
- Critical temperature: Schmidt-Appleman criterion for contrail formation
- Radiative forcing: Shortwave (cooling) and longwave (warming) effects
7.2 Saturation Pressure
Saturation vapor pressure is the pressure at which water vapor is in equilibrium with liquid water or ice at a given temperature. These functions use the Murphy & Koop (2005) formulations.
7.2.1 Over liquid water
Valid for temperatures from 123 K to 332 K:
import numpy as np
# Temperature in Kelvin (typical cruise altitude: ~220 K)
T = np.array([210, 220, 230, 240])
# Saturation pressure over liquid water (Pa)
p_sat_water = contrail.saturation_pressure_over_water(T)
for t, p in zip(T, p_sat_water):
print(f"T = {t} K: {p:.2f} Pa")T = 210 K: 1.23 Pa
T = 220 K: 4.36 Pa
T = 230 K: 13.55 Pa
T = 240 K: 37.67 Pa7.2.2 Over ice
Valid for temperatures from 110 K to 273 K:
# Saturation pressure over ice (Pa)
p_sat_ice = contrail.saturation_pressure_over_ice(T)
for t, p in zip(T, p_sat_ice):
print(f"T = {t} K: {p:.2f} Pa")T = 210 K: 0.70 Pa
T = 220 K: 2.65 Pa
T = 230 K: 8.95 Pa
T = 240 K: 27.27 PaThe ratio of these pressures determines contrail persistence:
ratio = p_sat_water / p_sat_ice
for t, r in zip(T, ratio):
print(f"T = {t} K: p_water/p_ice = {r:.3f}")T = 210 K: p_water/p_ice = 1.757
T = 220 K: p_water/p_ice = 1.643
T = 230 K: p_water/p_ice = 1.514
T = 240 K: p_water/p_ice = 1.3817.3 Relative Humidity
Convert between specific humidity (kg water / kg air) and relative humidity.
# Typical cruise conditions
specific_humidity = 0.00005 # kg/kg (very dry at cruise altitude)
pressure = 25000 # Pa (~FL350)
temperature = 220 # K
# Relative humidity with respect to ice
rhi = contrail.relative_humidity(specific_humidity, pressure, temperature, to="ice")
print(f"Relative humidity (ice): {rhi:.1%}")
# Relative humidity with respect to water
rhw = contrail.relative_humidity(specific_humidity, pressure, temperature, to="water")
print(f"Relative humidity (water): {rhw:.1%}")Relative humidity (ice): 75.7%
Relative humidity (water): 46.1%7.3.1 Converting between water and ice reference
When atmospheric data provides relative humidity with respect to water, you can convert to ice:
# Convert RH w.r.t. water to RH w.r.t. ice
rhi_converted = contrail.rhw2rhi(rhw, temperature)
print(f"RH (water) = {rhw:.1%} -> RH (ice) = {rhi_converted:.1%}")RH (water) = 46.1% -> RH (ice) = 75.7%7.4 Schmidt-Appleman Criterion
The Schmidt-Appleman criterion determines when contrails can form. It calculates the critical temperature below which the aircraft exhaust plume becomes saturated with respect to water, allowing ice crystals to form.
7.4.1 Critical temperature for contrail formation
# Pressure at different flight levels
pressures = np.array([20000, 25000, 30000, 35000]) # Pa
# Calculate critical temperature (default propulsion efficiency = 0.4)
T_crit = contrail.critical_temperature_water(pressures)
for p, t in zip(pressures, T_crit):
fl = round(44330 * (1 - (p/101325)**0.19) / 0.3048 / 100)
print(f"FL{fl:03d} ({p/100:.0f} hPa): T_crit = {t:.1f} K ({t-273.15:.1f} °C)")FL386 (200 hPa): T_crit = 230.5 K (-42.7 °C)
FL340 (250 hPa): T_crit = 232.8 K (-40.3 °C)
FL300 (300 hPa): T_crit = 234.8 K (-38.4 °C)
FL266 (350 hPa): T_crit = 236.5 K (-36.7 °C)7.4.2 Effect of propulsion efficiency
Modern engines with higher propulsion efficiency produce contrails more easily (higher critical temperature):
pressure = 25000 # Pa
# Different engine efficiencies
efficiencies = [0.3, 0.35, 0.4, 0.45]
for eta in efficiencies:
T_crit = contrail.critical_temperature_water(pressure, propulsion_efficiency=eta)
print(f"η = {eta}: T_crit = {T_crit:.1f} K ({T_crit-273.15:.1f} °C)")η = 0.3: T_crit = 231.2 K (-42.0 °C)
η = 0.35: T_crit = 232.0 K (-41.2 °C)
η = 0.4: T_crit = 232.8 K (-40.3 °C)
η = 0.45: T_crit = 233.7 K (-39.4 °C)7.4.3 Persistent contrails
Contrails persist when the atmosphere is supersaturated with respect to ice. The critical_temperature_water_and_ice function returns both thresholds:
pressure = 25000 # Pa
T_crit_water, T_crit_ice = contrail.critical_temperature_water_and_ice(pressure)
print(f"Contrail formation: T < {T_crit_water:.1f} K ({T_crit_water-273.15:.1f} °C)")
print(f"Contrail persistence: T < {T_crit_ice:.1f} K ({T_crit_ice-273.15:.1f} °C)")Contrail formation: T < 232.8 K (-40.3 °C)
Contrail persistence: T < 226.3 K (-46.9 °C)7.5 Radiative Forcing
Contrails affect Earth’s radiative balance through two mechanisms:
- Shortwave (SW): Reflect incoming solar radiation (cooling effect, negative RF)
- Longwave (LW): Trap outgoing thermal radiation (warming effect, positive RF)
7.5.1 Shortwave radiative forcing
Depends on solar geometry and contrail optical properties:
# Solar zenith angle (0° = overhead, 90° = horizon)
zenith = np.array([0, 30, 60, 80, 100]) # degrees
# Contrail optical depth (typical: 0.1-0.5)
tau = 0.3
# Background cirrus optical depth
tau_c = 0.2
rf_sw = contrail.rf_shortwave(zenith, tau, tau_c)
for z, rf in zip(zenith, rf_sw):
status = "night" if z > 90 else "day"
print(f"Zenith = {z:3d}° ({status}): RF_SW = {rf:7.2f} W/m²")Zenith = 0° (day): RF_SW = -8.81 W/m²
Zenith = 30° (day): RF_SW = -9.70 W/m²
Zenith = 60° (day): RF_SW = -16.32 W/m²
Zenith = 80° (day): RF_SW = -20.77 W/m²
Zenith = 100° (night): RF_SW = 0.00 W/m²Note that shortwave forcing is zero at night (zenith > 90°) and negative (cooling) during daytime.
7.5.2 Longwave radiative forcing
Depends on outgoing longwave radiation (OLR) and temperature:
# Outgoing longwave radiation (typical: 200-300 W/m²)
olr = 250 # W/m²
# Contrail temperature
temperatures = np.array([210, 220, 230, 240]) # K
rf_lw = contrail.rf_longwave(olr, temperatures)
for t, rf in zip(temperatures, rf_lw):
print(f"T = {t} K: RF_LW = {rf:.1f} W/m²")T = 210 K: RF_LW = 137.8 W/m²
T = 220 K: RF_LW = 118.4 W/m²
T = 230 K: RF_LW = 99.1 W/m²
T = 240 K: RF_LW = 79.7 W/m²Longwave forcing is always positive (warming) and increases with colder contrails.
7.5.3 Net radiative forcing
The net effect combines both components:
# Daytime scenario
zenith_day = 45 # degrees
tau = 0.3
tau_c = 0.2
olr = 250 # W/m²
temperature = 220 # K
rf_net_day = contrail.rf_net(zenith_day, tau, tau_c, olr, temperature)
rf_sw_day = contrail.rf_shortwave(zenith_day, tau, tau_c)
rf_lw_day = contrail.rf_longwave(olr, temperature)
print(f"Daytime (zenith = {zenith_day}°):")
print(f" SW: {rf_sw_day:.1f} W/m² (cooling)")
print(f" LW: {rf_lw_day:.1f} W/m² (warming)")
print(f" Net: {rf_net_day:.1f} W/m²")Daytime (zenith = 45°):
SW: -12.1 W/m² (cooling)
LW: 118.4 W/m² (warming)
Net: 106.4 W/m²# Nighttime scenario
zenith_night = 120 # degrees (sun below horizon)
rf_net_night = contrail.rf_net(zenith_night, tau, tau_c, olr, temperature)
rf_sw_night = contrail.rf_shortwave(zenith_night, tau, tau_c)
print(f"\nNighttime (zenith = {zenith_night}°):")
print(f" SW: {rf_sw_night:.1f} W/m² (no cooling)")
print(f" LW: {rf_lw_day:.1f} W/m² (warming)")
print(f" Net: {rf_net_night:.1f} W/m²")
Nighttime (zenith = 120°):
SW: 0.0 W/m² (no cooling)
LW: 118.4 W/m² (warming)
Net: 118.4 W/m²This explains why nighttime contrails have a stronger warming effect—there’s no shortwave cooling to offset the longwave warming.
7.6 Contrail Optical Evolution
Contrails evolve over time, spreading and changing their optical properties:
ages = [0.5, 1.5, 3, 5, 8] # hours
print("Age (h) | tau | width (m) | tau_c")
print("--------|-------|-----------|------")
for age in ages:
tau, width, tau_c = contrail.contrail_optical_properties(age)
print(f" {age:4.1f} | {tau:.2f} | {width:5.0f} | {tau_c:.3f}")Age (h) | tau | width (m) | tau_c
--------|-------|-----------|------
0.5 | 0.40 | 500 | 0.360
1.5 | 0.60 | 1500 | 0.540
3.0 | 0.68 | 3500 | 0.612
5.0 | 0.70 | 6500 | 0.630
8.0 | 0.71 | 10500 | 0.6397.7 Practical Example: Contrail Formation Check
Determine if contrails will form for a given flight condition:
import openap
# Flight conditions
altitude = 35000 # ft
temperature_isa_deviation = 0 # K
# Get atmospheric conditions
h_m = altitude * openap.aero.ft
pressure = openap.aero.pressure(h_m)
temperature = openap.aero.temperature(h_m, temperature_isa_deviation)
# Calculate critical temperature
T_crit = contrail.critical_temperature_water(pressure, propulsion_efficiency=0.4)
# Check formation
will_form = temperature < T_crit
print(f"Flight level: FL{altitude//100}")
print(f"Pressure: {pressure:.0f} Pa ({pressure/100:.0f} hPa)")
print(f"Ambient temperature: {temperature:.1f} K ({temperature-273.15:.1f} °C)")
print(f"Critical temperature: {T_crit:.1f} K ({T_crit-273.15:.1f} °C)")
print(f"Contrails will form: {will_form}")Flight level: FL350
Pressure: 23836 Pa (238 hPa)
Ambient temperature: 218.8 K (-54.3 °C)
Critical temperature: 232.3 K (-40.8 °C)
Contrails will form: True7.8 Loading OLR Data
For climate impact calculations with real atmospheric data, use load_olr to interpolate OLR from netCDF files:
# Load OLR data (requires xarray)
olr_values = contrail.load_olr(
filepath="path/to/olr_data.nc",
lat=52.3, # latitude
lon=4.8, # longitude
time="2024-01-15T12:00:00"
)7.9 Physical Constants
The module provides several physical constants used in the calculations:
| Constant | Value | Unit | Description |
|---|---|---|---|
ei_water |
1.2232 | kg/kg | Water vapor emission index |
spec_combustion_heat |
43×10⁶ | J/kg | Specific combustion heat of jet fuel |
gas_constant_water_vapor |
461.51 | J/(kg·K) | Gas constant for water vapor |
gas_constant_dry_air |
287.05 | J/(kg·K) | Gas constant for dry air |
7.10 References
- Murphy, D. M. and Koop, T. (2005). Review of the vapour pressures of ice and supercooled water for atmospheric applications. Q. J. R. Meteorol. Soc., 131, 1539-1565.
- Schumann, U. (1996). On conditions for contrail formation from aircraft exhausts. Meteorol. Z., 5, 4-23.