Moist adiabatic lapse rate

(From Deepseek, 2025/9/28)

Moist Adiabatic Lapse Rate

The moist adiabatic lapse rate (MALR), often denoted by <math>\Gamma_m</math>, is a fundamental concept in atmospheric science. The key thing to understand is that the MALR is not a constant value. It varies with temperature and pressure because the amount of moisture the air can hold is highly dependent on its temperature. Warmer air can hold more water vapor, so when condensation occurs in a warm, saturated parcel, more latent heat is released, which significantly offsets the cooling due to expansion. This makes the MALR much smaller than the dry adiabatic lapse rate (DALR). A typical value used for the MALR near the surface is ~5 °C/km, compared to the constant DALR of ~9.8 °C/km.

The Equation

The full equation for the moist adiabatic lapse rate is: <math>\Gamma_m = -\frac{dT}{dz} = g \frac{ \left(1 + \frac{L_v r_s}{R_d T} \right) } { c_{pd} + \frac{L_v^2 r_s \epsilon}{R_d T^2} }</math> Where: <math>\Gamma_m</math> = Moist Adiabatic Lapse Rate (K/m or °C/m) <math>\frac{dT}{dz}</math> = The change in temperature (T) with height (z) <math>g</math> = Acceleration due to gravity (9.81 m/s²) <math>L_v</math> = Latent heat of vaporization for water (~2.5 × 10⁶ J/kg) <math>r_s</math> = Saturation mixing ratio of water vapor (kg/kg) (This is the variable that makes the rate change) <math>R_d</math> = Gas constant for dry air (287 J/kg·K) <math>T</math> = Temperature (Kelvin) <math>c_{pd}</math> = Specific heat of dry air at constant pressure (1005 J/kg·K) <math>\epsilon</math> = Ratio of the molecular weight of water to dry air (<math>\epsilon = R_d/R_v \approx</math> 0.622)

Why is it so complex?

The complexity arises because the saturation mixing ratio (<math>r_s</math>) is a function of both temperature and pressure. As the air parcel rises and cools, its temperature drops, which decreases <math>r_s</math>. This change means the amount of latent heat released from condensation is constantly changing. Therefore, you must solve this equation step-by-step through the atmosphere; you cannot use a single value for all conditions.

Simplified Interpretation

The equation essentially states that the lapse rate is a balance between two main effects: Cooling from expansion: The parcel cools as it rises and expands (described by <math>g / c_{pd}</math> in the dry case). Heating from condensation: The release of latent heat (<math>L_v</math>) when water vapor condenses, which warms the parcel and counteracts the cooling. The denominator is larger than in the dry case due to the latent heat term (<math>\frac{L_v^2 r_s \epsilon}{R_d T^2}</math>). This larger denominator results in a smaller lapse rate (<math>\Gamma_m</math>).

Practical Use: Approximate Values

Because the full equation is cumbersome for quick calculations, meteorologists often use approximate values based on typical conditions: In the warm, moist lower troposphere: <math>\Gamma_m \approx</math> 4 - 7 °C/km A common rule of thumb: <math>\Gamma_m \approx</math> 6 °C/km (or 6.5 °C/km for some applications) In the cold, dry upper troposphere (where the air can hold very little moisture), the MALR approaches the DALR (~9.8 °C/km).

Comparison to Dry Adiabatic Lapse Rate

For context, it's always helpful to compare it to the dry adiabatic lapse rate, which is a constant.

Property	Dry Adiabatic Lapse Rate (DALR)	Moist Adiabatic Lapse Rate (MALR)
Definition	Rate a dry/unsaturated air parcel cools Rate a saturated air parcel cools Key Process Adiabatic expansion Adiabatic expansion + latent heat release - Value Constant: ~9.8 °C/km Variable: ~4-7 °C/km (typically ~6 °C/km) - Governing Eq. <math>\Gamma_d = \frac{g}{c_p} = 9.8 ^{\circ}\text{C/km}</math> <math>\Gamma_m = g \frac{ \left(1 + \frac{L_v r_s}{R_d T} \right) } { c_{pd} + \frac{L_v^2 r_s \epsilon}{R_d T^2} }</math> } Summary In summary, the moist adiabatic lapse rate is variable and complex to calculate precisely because it depends on the ever-changing moisture content (saturation mixing ratio) of the rising air parcel.