Moist Adiabatic Lapse Rate Calculator

The moist (or saturated) adiabatic lapse rate is the rate at which a rising, saturated air parcel cools with height, typically between 4 and 9 C per kilometer. It's always lower than the dry adiabatic lapse rate of 9.8 C/km because condensing water vapor releases latent heat that offsets some of the cooling.

Gamma_m = g(1 + Lv*ws/(Rd*T)) / (cp + Lv^2*ws*epsilon/(Rd*T^2)), where g is gravity, Lv is the latent heat of vaporization, Rd is the dry air gas constant, ws is the saturation mixing ratio, epsilon is 0.622, cp is the specific heat of dry air, and T is temperature in kelvin.

As saturated air rises and cools, water vapor condenses into liquid droplets, releasing latent heat back into the parcel. That released heat partially offsets the adiabatic cooling from expansion, so the parcel cools more slowly than dry air would, roughly 4 to 7 C/km instead of the fixed dry rate of 9.8 C/km.

Warmer air can hold far more water vapor (per the Clausius-Clapeyron relation embedded in the Tetens formula), so there's more latent heat available to release during condensation. That's why warm tropical air has a moist lapse rate near 3.5 to 4 C/km, while cold polar air has almost no vapor to condense and its moist rate climbs close to the dry rate of 9.8 C/km.

The saturation mixing ratio (ws) is the maximum mass of water vapor a parcel of air can hold per unit mass of dry air at a given temperature and pressure, expressed in grams of water vapor per kilogram of dry air. This calculator computes it from the Tetens saturation vapor pressure formula.

The Tetens formula estimates saturation vapor pressure as es = 6.112 times exp(17.67*T / (T + 243.5)) in hectopascals, with T in Celsius. It's a widely used empirical approximation, accurate to about 1 percent across typical atmospheric temperatures, and much simpler to compute than the full Clausius-Clapeyron integral.

Lower pressure (higher altitude) increases the saturation mixing ratio at a given temperature, because ws = epsilon*es/(P minus es) grows as P shrinks toward es. That means, all else equal, the moist lapse rate is slightly lower at lower pressure for the same temperature.

A dry adiabatic process describes unsaturated air, which cools at a constant 9.8 C/km as it rises with no phase change involved. A moist adiabatic process describes saturated air, where rising motion triggers condensation, and the released latent heat slows the cooling to somewhere between about 4 and 9 C/km depending on temperature and pressure.

Meteorologists compare the moist adiabatic lapse rate of a rising cloud parcel to the environmental lapse rate around it. If the environment cools with height faster than the moist rate, the rising saturated parcel stays warmer (and less dense) than its surroundings, keeping it buoyant, this is the basic mechanism behind deep convection and thunderstorm development.

The commonly cited 6.5 C/km 'average environmental lapse rate' is a global average of the actual atmosphere, not the moist adiabatic rate for any specific parcel. The true moist adiabatic lapse rate varies significantly with temperature and pressure, from about 3.5 C/km in warm tropical air near the surface to nearly 9 C/km in cold, dry air, so calculating it directly (as this tool does) gives a more accurate value for any specific scenario.

This calculator accepts temperatures from -60 to 50 degrees Celsius and pressures from 1 to 1100 hectopascals, covering the full range of conditions found from the surface up through the mid-troposphere.