Dry Adiabatic Lapse Rate Calculator

Lapse rate equals gravitational acceleration divided by specific heat at constant pressure

Solution

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Dry Adiabatic Lapse Rate

About 9.76 °C per km on Earth.

Γ = g ÷ cₚ

Temperature at Altitude

Estimates air temperature at any altitude.

T = T₀ − Γ × h

How It Works

As dry air rises, it expands and cools at the dry adiabatic lapse rate. Γ = g / cₚ gives about 9.76 °C/km. T = T₀ − Γh estimates temperature at altitude.

Example Problem

Surface temperature 25 °C, find temperature at 3,000 m (3 km).

  1. Identify the knowns. Surface temperature T₀ = 25 °C, altitude h = 3 km, and the dry adiabatic lapse rate Γ = 9.76 °C/km (from g/cₚ = 9.81 / 1005).
  2. Identify what we're solving for. We want the air temperature T at the target altitude assuming a dry adiabatic ascent.
  3. Write the formula in symbols: T = T₀ − Γ × h.
  4. Substitute the known values: T = 25 − 9.76 × 3.
  5. Simplify the arithmetic: T = 25 − 29.28.
  6. State the result: **T = −4.28 °C** — below freezing, so icing is possible for aircraft at this altitude.

Below freezing — icing possible.

When to Use Each Variable

  • Solve for Lapse Rate — when you know gravity and specific heat and need the temperature drop per unit altitude — e.g., comparing lapse rates on different planets.
  • Solve for Gravity — when you have a measured lapse rate and specific heat — useful for back-calculating effective gravity from atmospheric sounding data.
  • Solve for Specific Heat — when you know the lapse rate and gravity — e.g., estimating the heat capacity of an alien atmosphere from descent probe data.
  • Solve for Temperature at Altitude — when you know the surface temperature and lapse rate and need the air temperature at a given height — e.g., predicting icing conditions for aircraft.
  • Solve for Surface Temperature — when you have a temperature reading at altitude and want to extrapolate back to the ground — e.g., correcting weather station data.
  • Solve for Altitude — when you know the surface and upper temperatures and need the elevation of that temperature — e.g., estimating the freezing level.

Key Concepts

The dry adiabatic lapse rate (DALR) describes how unsaturated air cools as it rises and expands without exchanging heat with its surroundings. It equals g/cp, approximately 9.76 degrees C per kilometer on Earth. Once air becomes saturated, latent heat release slows the cooling, and the moist adiabatic lapse rate (about 5-6 degrees C/km) applies instead. The DALR is a key factor in assessing atmospheric stability.

Applications

  • Aviation: predicting temperature at cruising altitude and assessing icing risk
  • Meteorology: determining atmospheric stability by comparing the DALR to the actual environmental lapse rate
  • Wildfire management: estimating how quickly rising smoke plumes cool, which affects plume height and smoke dispersion
  • Mountain weather forecasting: predicting summit temperatures from valley observations

Common Mistakes

  • Applying the dry rate to saturated (cloudy) air — once condensation begins, the moist adiabatic rate is lower
  • Confusing lapse rate with temperature inversion — an inversion means temperature increases with altitude, not decreases
  • Using inconsistent units — mixing degrees C/km with degrees F/ft without converting

Frequently Asked Questions

What is the dry adiabatic lapse rate value?

About 9.76 °C/km (5.4 °F per 1,000 ft) on Earth, derived from Γ = g / cₚ ≈ 9.81 / 1004. The exact value depends on the assumed specific heat of dry air and local gravity, both of which are nearly constant in the troposphere.

How is dry lapse rate different from moist lapse rate?

The dry rate (~9.76 °C/km) applies to unsaturated air parcels. Once the parcel cools to the dew point and water vapor condenses, latent heat slows the cooling — the moist (saturated) adiabatic lapse rate is typically 5–6 °C/km, varying with temperature and humidity.

How do meteorologists use the lapse rate to assess stability?

Compare Γ to the measured environmental lapse rate from a sounding. If the environment cools faster than 9.76 °C/km, the air is absolutely unstable for dry parcels — rising parcels stay warmer than their surroundings and accelerate upward, fueling convection.

Why does rising air cool even though it adds no heat?

Atmospheric pressure decreases with altitude, so a rising parcel expands. Expansion does work against the surrounding pressure, and because the parcel is adiabatic (no heat exchange), that work comes from the parcel's internal energy — temperature drops.

Does the dry adiabatic lapse rate change on other planets?

Yes — Γ scales with the ratio of surface gravity to atmospheric specific heat. Mars has a similar Γ (~4.5 °C/km because cₚ for CO₂ is higher), Venus is steeper near the surface (~7.7 °C/km), and Jupiter's hydrogen-rich atmosphere has a much smaller Γ (~2 °C/km).

What is a temperature inversion and how does it differ from a normal lapse rate?

An inversion is a layer where temperature increases with altitude rather than decreasing. Lapse rates are positive (warmer below); inversions are negative. Inversions trap pollutants near the surface and shut down vertical mixing, the opposite of the convective overturning a steep lapse rate produces.

Why is g/cₚ used to derive Γ instead of being measured directly?

The dry adiabatic relationship follows from the First Law of Thermodynamics for an ideal gas under reversible adiabatic ascent. The math reduces to dT/dz = −g/cₚ for any parcel that doesn't exchange heat or condense moisture — it's a theoretical limit, not an empirical fit.

Reference:

Holton, James R. 2004. An Introduction to Dynamic Meteorology. Academic Press.

Worked Examples

ICAO Standard Atmosphere

What dry adiabatic lapse rate does the ICAO standard atmosphere predict?

The ICAO standard atmosphere uses g = 9.80665 m/s² for gravitational acceleration and cp = 1004 J/(kg·K) for the specific heat of dry air at constant pressure. Compute the theoretical dry adiabatic lapse rate Γ = g / cp from those two constants — the value pilots learn as the textbook 9.8 °C/km.

  • Knowns: g = 9.80665 m/s², cp = 1004 J/(kg·K)
  • Γ = g / cp
  • Γ = 9.80665 / 1004 (°C/m)
  • Γ ≈ 0.009768 °C/m = 9.768 °C/km

Γ ≈ 9.77 °C/km (about 5.4 °F per 1000 ft)

This is the dry adiabatic value used for stability analysis. Saturated (moist) air parcels release latent heat as water vapor condenses, so the moist adiabatic lapse rate is shallower (typically 5–6 °C/km, varying with temperature and humidity).

Aviation Weather Forecasting

What is the dry-adiabatic temperature at 2 km if the surface is 25 °C?

A general-aviation pilot files a VFR flight plan and wants to know the unsaturated parcel temperature at a 2 km cruise altitude, given a surface temperature of 25 °C and the standard dry adiabatic lapse rate of 9.77 °C/km. (Compare this to the actual sounding to decide if convection is likely.)

  • Knowns: T₀ = 25 °C, Γ = 9.77 °C/km, h = 2 km = 2000 m
  • T = T₀ − Γ × h
  • T = 25 − 9.77 × 2
  • T = 25 − 19.54

T ≈ 5.46 °C at 2 km

If the actual sounding shows the air at 2 km is warmer than 5.46 °C, the atmosphere is stable for an unsaturated rising parcel (it cools faster than its surroundings and sinks back). If colder, the parcel keeps rising — convective instability and possible thunderstorm development.

Mountaineering

How high is the summit if the trailhead is 20 °C and the summit reads 3 °C?

A hiker leaves a trailhead at 20 °C and reaches a summit ridge where the temperature has dropped to 3 °C. Assuming dry-adiabatic cooling at 9.77 °C/km along the ascent (no clouds, no precipitation), back out the elevation gain.

  • Knowns: T₀ = 20 °C, T = 3 °C, Γ = 9.77 °C/km
  • h = (T₀ − T) / Γ
  • h = (20 − 3) / 9.77
  • h = 17 / 9.77

h ≈ 1.740 km = 1,740 m of elevation gain

Real-world cooling along a hike often exceeds the dry-adiabatic rate (because the atmospheric environmental lapse rate is itself the driver, not the parcel theory). On a clear, calm day with afternoon heating, 9.77 °C/km is a reasonable rough check; in cloudy or windy conditions, expect smaller temperature drops per unit altitude.

Dry Adiabatic Lapse Rate Formulas

Two related equations describe how an unsaturated air parcel cools as it rises:

Γ = g / cpDry adiabatic lapse rate from gravity and specific heat
T = T0 − Γ × hAir temperature at altitude h above the surface

Where:

  • Γ (gamma) — dry adiabatic lapse rate (°C/km or °C/m); about 9.76 °C/km on Earth
  • g — gravitational acceleration (m/s²); 9.80665 m/s² in the ICAO standard atmosphere
  • cp — specific heat of dry air at constant pressure (J/(kg·K)); ≈ 1004 J/(kg·K)
  • T — air temperature at altitude h (°C, K, or °F)
  • T0 — surface (reference-altitude) temperature (°C, K, or °F)
  • h — altitude above the reference surface (m, km, or ft)

These formulas describe a theoretical, reversible adiabatic ascent of dry air — no heat exchange with the surroundings and no condensation. Once the parcel saturates, the moist adiabatic lapse rate (typically 5–6 °C/km) replaces Γ, and the actual environmental lapse rate measured by a radiosonde can differ from both.

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