Deriving the Dry Adiabatic Lapse Rate

Recommended resources

The freely available resources this lesson will direct you to.

Hello! Welcome to your next lesson in our study of the Earth's atmosphere.

Introduction

In our last two lessons, we built a foundational model of atmospheric structure. We started with the principle of hydrostatic equilibrium ( $\frac{d P}{d z} = - ρ g$ ), the balance between pressure and gravity. We then solved this equation for a simplified isothermal atmosphere, leading us to the barometric formula, which showed pressure decreasing exponentially with height.

However, we concluded by noting the major limitation of that model: the real troposphere is not isothermal. Temperature decreases with height. Today, we will replace that simplification with a more physically robust principle.

Our learning outcome is to derive the dry adiabatic lapse rate from the first law of thermodynamics and the hydrostatic equilibrium equation. We will determine the theoretical rate at which a parcel of dry air cools as it rises and expands, without any exchange of heat with its surroundings. This is a cornerstone of atmospheric science, crucial for understanding weather, cloud formation, and atmospheric stability.

1. The First Law of Thermodynamics

To understand why temperature changes with pressure, we need to consider the conservation of energy. In this context, the governing principle is the First Law of Thermodynamics. For a parcel of gas, it states that any heat added to it can do two things: increase its internal energy (i.e., raise its temperature) or be used to do work on its surroundings (i.e., expand).

Your background in mathematical economics will have given you a strong foundation in optimization and the behaviour of systems described by differential equations. Thermodynamics applies a similar formal rigour to the study of energy.

To formalize these concepts for our atmospheric parcel, we'll use a specific set of resources.

The First Law of Thermodynamics and Related Definitions

To begin, we need to establish the mathematical form of the First Law of Thermodynamics as it applies to a parcel of gas. This resource introduces the law and defines the key terms we will use in our derivation.

Please read the following sections from these lecture notes: Start at Section 3.3, 'Internal energy and mathematical statement of First Law'. Focus on understanding the equation presented as (3.4): Dq = du + Dw = du + pDα. This is the First Law expressed in terms of specific quantities (per unit mass), where q is heat, u is internal energy, and pDα is the work of expansion. Continue with Section 3.4, 'Specific heats'. The crucial part here is understanding the definitions of specific heat at constant volume (cv) and constant pressure (cp), and the two resulting forms of the First Law, labeled as (3.6) and (3.8). The second form, dq = cpdT - αdp, is the one we will use.

Let's consolidate the key takeaways from that reading. We now have two useful forms of the First Law of Thermodynamics:

$d q = c_{v} d T + p d α$
$d q = c_{p} d T - α d p$

Here, $d q$ is the heat added per unit mass, $d T$ is the change in temperature, $d α$ is the change in specific volume ( $α = 1/ ρ$ ), and $d p$ is the change in pressure. The second form, involving temperature and pressure, is the most convenient for our purpose.

2. The Adiabatic Process

We are interested in what happens to a parcel of air as it rises or sinks in the atmosphere. Air is a very poor conductor of heat, and large parcels can move vertically quite quickly (minutes to hours). Therefore, there is often negligible time for the parcel to exchange heat with the surrounding air.

A process that occurs without any heat exchange with the environment is called an adiabatic process.

For such a process, we can set $d q = 0$ . Applying this to our preferred form of the First Law gives us the governing equation for an adiabatic change:

$0 = c_{p} d T - α d p$
or
$c_{p} d T = α d p$

This simple but powerful equation tells us that for a parcel of air moving adiabatically, any change in pressure ( $d p$ ) must be accompanied by a change in temperature ( $d T$ ). As a parcel rises to a region of lower ambient pressure, it expands. This expansion does work on its surroundings, and since no heat is being added from the outside, the energy for this work must come from the parcel's internal energy, causing its temperature to drop.

3. Deriving the Dry Adiabatic Lapse Rate ( $Γ_{d}$ )

We can now combine our understanding of adiabatic processes with the hydrostatic equilibrium equation to find the rate of temperature change with altitude. The following reading performs this exact derivation.

The First Law of Thermodynamics and Related Definitions

We are now ready to derive the lapse rate. This involves combining the adiabatic energy equation (cpdT = αdp) with the hydrostatic equation (dp/dz = -ρg). The following text walks through this core derivation.

Please read the second half of Section 3.8, starting from the paragraph that begins 'Next, we will derive an associated quantity, the dry adiabatic lapse rate...'. Follow the derivation to see how the two fundamental equations are combined to produce the final expression for the dry adiabatic lapse rate, Γd.

Let's walk through the steps of that derivation to ensure they are perfectly clear.

Start with the First Law for an adiabatic process:
$c_{p} d T = α d p$
Recall the hydrostatic equilibrium equation from our first lesson in this module:
$\frac{d p}{d z} = - ρ g$
This can be rewritten as $d p = - ρ g d z$ .
Substitute the expression for $d p$ from the hydrostatic equation into the First Law:
$c_{p} d T = α (- ρ g d z)$
Simplify. Remember that specific volume $α$ is the inverse of density $ρ$ , so $α = 1/ ρ$ . Their product is 1:
$c_{p} d T = (\frac{1}{ρ}) (- ρ g d z) = - g d z$
Isolate the temperature gradient $\frac{d T}{d z}$ :
$\frac{d T}{d z} = - \frac{g}{c _{p}}$

This is our result. It gives the rate of change of temperature with altitude for a parcel of dry air moving adiabatically.

By convention, a "lapse rate" is defined as the rate of decrease in temperature with height. Therefore, the dry adiabatic lapse rate, denoted $Γ_{d}$ , is defined with a negative sign to make it a positive value:

$Γ_{d} \equiv - \frac{d T}{d z} = \frac{g}{c _{p}}$

4. Calculation and Physical Interpretation

Let's calculate the value of $Γ_{d}$ . Using standard values for Earth's gravity and the specific heat of dry air at constant pressure:

$g \approx 9.81 m s^{- 2}$
$c_{p} \approx 1005 J kg^{- 1} K^{- 1}$

$Γ_{d} = \frac{9.81 m s ^{- 2}}{1005 J kg ^{- 1} K ^{- 1}} \approx 0.00976 K m^{- 1}$

This is almost always expressed in kilometers:
$Γ_{d} \approx 9.8 K/km$

This means that for every kilometer a parcel of dry air rises, its temperature drops by nearly 10°C. This is a much more realistic description of the troposphere's thermal behavior than the isothermal model ( $Γ = 0$ ).

This derived rate is fundamental to atmospheric stability. The following reading briefly explains why.

Adiabatic Lapse Rate

The value we just calculated, ~9.8 K/km, is a critical benchmark. How the actual temperature profile of the atmosphere compares to this value determines whether the air will be calm or stormy. This resource explains this crucial concept.

Please read the text following the derivation in the 'Adiabatic Lapse Rate' article. Focus on the calculated value for dry air and the explanation of what happens if the actual lapse rate is faster than the adiabatic lapse rate—this leads to convective instability.

As the reading highlights, if the actual environmental lapse rate is greater than $Γ_{d}$ , a rising parcel will find itself warmer and less dense than its surroundings, causing it to accelerate upward. This is the recipe for thunderstorms. If the environmental lapse rate is less than $Γ_{d}$ , the atmosphere is stable.

You'll also note that the observed average lapse rate in the troposphere is closer to 6.5 K/km. The primary reason for this difference is the presence of water vapor and the release of latent heat during condensation, a topic we will explore in a future module on climate feedbacks.

Conclusion

In this lesson, we have moved beyond the simple isothermal model to a more dynamic and physically grounded description of the atmosphere's vertical temperature structure.

Key Takeaways:

The First Law of Thermodynamics ( $d q = c_{p} d T - α d p$ ) governs the energy balance of an air parcel.
Assuming an adiabatic process ( $d q = 0$ ) is a reasonable approximation for vertical air motion.
By combining the First Law with the hydrostatic equilibrium equation, we derived the change in temperature with height.
The dry adiabatic lapse rate is $Γ_{d} = g / c_{p} \approx 9.8 K/km$ . This is the rate at which a rising parcel of dry air cools with altitude.
This rate is a critical benchmark for determining atmospheric stability.

Preview of the Next Lesson:

We have now established the fundamental pressure and temperature structure of the atmosphere. The next step is to understand how radiation travels through this medium. In the next lesson, we will introduce the concepts of optical depth and the Beer-Lambert law. This will allow us to quantify how effectively the atmosphere absorbs and attenuates incoming solar radiation and outgoing thermal radiation, which is the physical basis of the greenhouse effect.