Part I: Atmospheric Thermodynamics

Fundamental thermodynamic principles governing atmospheric behavior

1. Atmospheric Composition & Structure

1.1 Composition of Dry Air

The Earth's atmosphere is a mixture of gases with remarkably constant composition up to ~80 km altitude (homosphere). The major constituents by volume are:

N₂ (Nitrogen):
78.08%
O₂ (Oxygen):
20.95%
Ar (Argon):
0.93%
CO₂ (Carbon dioxide):
~0.042% (420 ppm, increasing)
Ne, He, Kr, Xe, CH₄:
trace amounts

Water vapor (H₂O) is a variable constituent, ranging from nearly 0% in polar regions to ~4% in the tropics. Despite its small concentration, water vapor plays a crucial role in:

  • Radiative transfer (greenhouse gas)
  • Latent heat release/absorption during phase changes
  • Cloud and precipitation formation
  • Atmospheric chemistry and transport

1.2 Vertical Structure

The atmosphere is divided into layers based on temperature profile:

Troposphere (0-10/16 km)

Temperature decreases with height at ~6.5 K/km (environmental lapse rate). Contains ~80% of atmospheric mass and virtually all weather phenomena. Top boundary is the tropopause.

Stratosphere (10/16-50 km)

Temperature increases with height due to ozone (O₃) absorption of UV radiation. The ozone layer (15-35 km) shields Earth's surface from harmful UV. Top boundary is the stratopause.

Mesosphere (50-85 km)

Temperature decreases with height, reaching the coldest atmospheric temperatures (~180 K) at themesopause. Noctilucent clouds can form here.

Thermosphere (85-600 km)

Temperature increases dramatically with height due to absorption of solar X-rays and UV by O₂ and N₂. Aurora occur in this layer. The International Space Station orbits here.

1.3 Mean Molecular Weight

For dry air, the mean molecular weight is calculated from the mixture:

\(M_{\text{dry}} = \Sigma \chi_i M_i = 0.7808 \times 28.014 + 0.2095 \times 31.998 + 0.0093 \times 39.948 + \ldots\)
\(M_{\text{dry}} \approx 28.97 \text{ g/mol}\)

For moist air, we use the virtual temperature concept (discussed in Chapter 6) to account for water vapor's lower molecular weight (18.015 g/mol).

2. Ideal Gas Law & Hydrostatic Balance

2.1 Equation of State for Ideal Gases

The atmosphere behaves as an ideal gas to excellent approximation. The equation of state relates pressure (p), density (ρ), and temperature (T):

\(p = \rho R T\)
where:
• \(p\) = pressure [Pa or hPa]
• \(\rho\) = density [kg/m³]
• \(T\) = absolute temperature [K]
• \(R\) = specific gas constant for dry air = \(R^*/M\) = 287 J/(kg·K)
\(R^*\) = 8.314 J/(mol·K) is the universal gas constant
\(M\) = 0.02897 kg/mol is the mean molecular weight of dry air

Alternative forms of the ideal gas law:

\(p V = n R^* T\)(using molar quantities)
\(p = n k_B T\)(using number density n, Boltzmann constant \(k_B\))
\(p \alpha = R T\)(using specific volume \(\alpha = 1/\rho\))

2.2 Hydrostatic Equation

In the vertical direction, the atmosphere is nearly in hydrostatic balance: the upward pressure gradient force balances the downward gravitational force. Consider a thin horizontal slab of air:

\(\frac{dp}{dz} = -\rho g\)
where:
• \(z\) = height above surface [m]
• \(g\) = gravitational acceleration ≈ 9.81 m/s² (varies slightly with latitude/altitude)
The negative sign indicates pressure decreases with increasing altitude.

Physical Interpretation: The pressure at any level is simply the weight of the air column above that level per unit area.

2.3 Hypsometric Equation

Combining the ideal gas law with the hydrostatic equation yields the hypsometric equation, which relates the thickness of an atmospheric layer to its mean temperature:

Derivation:
\(\frac{dp}{dz} = -\rho g = -\frac{p}{RT}g\)(substituting \(\rho\) from ideal gas law)
\(\frac{dp}{p} = -\frac{g}{RT}dz\)
\(\int_{p_1}^{p_2} \frac{dp}{p} = -\int_{z_1}^{z_2} \frac{g}{RT}dz\)
\(\ln\left(\frac{p_2}{p_1}\right) = -\frac{g}{R} \int_{z_1}^{z_2} \frac{dz}{T}\)
\(z_2 - z_1 = \frac{R}{g} \bar{T} \ln\left(\frac{p_1}{p_2}\right)\)
where \(\bar{T}\) is the mean temperature of the layer

📊 Practical Application:

The hypsometric equation is fundamental to weather analysis. On weather maps, regions of warm air correspond to greater layer thickness between pressure surfaces, while cold air regions show smaller thickness. This creates the "thermal wind" (discussed in Part II).

2.4 Scale Height

For an isothermal atmosphere (\(T = \text{constant}\)), the hypsometric equation gives:

\(p(z) = p_0 \exp(-z/H)\)
\(H = \frac{RT}{g} \approx 8.4 \text{ km}\)(for \(T = 288\) K)

H is the scale height - the altitude over which pressure decreases by a factor of \(e \approx 2.718\). In reality, temperature varies with altitude, so the actual pressure profile differs from this exponential form.

💻 Computational Example:

See for a program that calculates pressure as a function of altitude using the hypsometric equation with realistic temperature profiles (US Standard Atmosphere).

3. First Law of Thermodynamics

3.1 Energy Conservation

The first law of thermodynamics expresses conservation of energy. For a parcel of air:

\(dQ = dU + p\,d\alpha\)
where (per unit mass):
• \(dQ\) = heat added to the parcel [J/kg]
• \(dU\) = change in internal energy [J/kg]
• \(p\,d\alpha\) = work done by expansion (\(p\) = pressure, \(\alpha\) = specific volume = \(1/\rho\))

For an ideal gas, the internal energy depends only on temperature: \(dU = c_v dT\), where \(c_v\) is the specific heat at constant volume.

For dry air:
\(c_v = 717 \text{ J/(kg·K)}\) (specific heat at constant volume)
\(c_p = 1004 \text{ J/(kg·K)}\) (specific heat at constant pressure)
\(\gamma = c_p/c_v = 1.4\) (heat capacity ratio)
\(R = c_p - c_v = 287 \text{ J/(kg·K)}\) (Mayer's relation)

3.2 Alternative Forms

Using the ideal gas law and Mayer's relation, the first law can be written in several equivalent forms:

Form 1: Using temperature
\(dQ = c_p dT - \alpha dp\)
Form 2: Using pressure and volume
\(dQ = c_v dT + p\,d\alpha\)
Form 3: Rate form (per unit time)
\(\frac{dQ}{dt} = c_p \frac{dT}{dt} - \frac{1}{\rho} \frac{dp}{dt}\)

3.3 Heating Processes

Atmospheric heating/cooling (\(dQ \neq 0\)) can occur through:

  • Radiation: Absorption/emission of solar and terrestrial radiation
  • Latent heat release: Condensation of water vapor
  • Sensible heat flux: Conduction from Earth's surface
  • Turbulent mixing: Vertical heat transport

When no heat is added or removed (\(dQ = 0\)), processes are called adiabatic (see Chapter 4).

4. Adiabatic Processes

4.1 Dry Adiabatic Process

An adiabatic process occurs without heat exchange (\(dQ = 0\)). This is an excellent approximation for rising and sinking air parcels because:

  • Air is a poor conductor of heat
  • Vertical motion often occurs rapidly (minutes to hours)
  • Radiative heating/cooling is slow compared to dynamical timescales

For a dry adiabatic process, the first law gives:

\(c_p dT - \alpha dp = 0\)
\(c_p dT = \alpha dp = \frac{RT}{p} dp\)
\(\frac{dT}{T} = \frac{R}{c_p} \frac{dp}{p}\)

Integrating and using \(R/c_p = (\gamma-1)/\gamma = 2/7\):

\(T p^{-R/c_p} = \text{constant}\)
or equivalently:
\(T p^{-2/7} = \text{constant}\)

4.2 Potential Temperature

The potential temperature (\(\theta\)) is defined as the temperature an air parcel would have if brought adiabatically to a reference pressure \(p_0 = 1000\) hPa:

\(\theta = T \left(\frac{p_0}{p}\right)^{R/c_p}\)
For dry adiabatic processes, \(\theta\) is conserved: \(d\theta/dt = 0\)

🎯 Key Insight:

Potential temperature is a conserved quantity for adiabatic motion and serves as a tracer of air mass properties. It's more useful than temperature because it removes the effects of compression and expansion due to pressure changes.

4.3 Dry Adiabatic Lapse Rate

How does temperature change with altitude for a dry adiabatic process? Combining the first law with the hydrostatic equation:

Derivation:
\(c_p dT = \alpha dp\) (first law, \(dQ=0\))
\(c_p dT = -\frac{RT}{p} \rho g dz\) (using \(dp=-\rho g dz\) and \(\alpha=1/\rho\))
\(c_p dT = -g dz\)
\(\frac{dT}{dz} = -\frac{g}{c_p} = \Gamma_d\)
\(\Gamma_d = 9.8 \text{ K/km}\)
This is the dry adiabatic lapse rate - the rate at which temperature decreases with height for a rising unsaturated air parcel.

📊 Physical Interpretation:

As an air parcel rises, it expands due to decreasing pressure. This expansion does work against the surrounding atmosphere, converting internal energy to work. Since the process is adiabatic (no heat added), the internal energy (and thus temperature) decreases. The rate is independent of the initial temperature - determined only by g and c_p.

4.4 Poisson's Equations

For adiabatic processes, several quantities remain constant:

\(T p^{-\kappa} = \text{constant}\)where \(\kappa = R/c_p = 2/7\)
\(T^\gamma p^{-\gamma+1} = \text{constant}\)where \(\gamma = c_p/c_v = 7/5\)
\(\rho^\gamma p^{-1} = \text{constant}\)

5. Atmospheric Stability

5.1 Static Stability Concept

Static stability determines whether vertical displacements of air parcels are amplified (unstable) or suppressed (stable). Consider a parcel displaced vertically from its equilibrium position:

Stable Atmosphere

If displaced upward, the parcel becomes cooler and denser than its surroundings → experiences downward buoyancy force → returns to original level. Oscillations occur (buoyancy oscillations or gravity waves).

Unstable Atmosphere

If displaced upward, the parcel remains warmer and less dense than surroundings → experiences upward buoyancy force → continues to rise. Convection develops.

Neutral Atmosphere

If displaced, the parcel has the same temperature/density as surroundings → no buoyancy force → remains at the new level.

5.2 Stability Criteria

Compare the environmental lapse rate \(\Gamma = -dT/dz\) (actual temperature profile) with the dry adiabatic lapse rate \(\Gamma_d = 9.8\) K/km:

✓
\(\Gamma < \Gamma_d\)→ Absolutely stable

(\(d\theta/dz > 0\): potential temperature increases with height)

~
\(\Gamma = \Gamma_d\)→ Neutral (adiabatic atmosphere)

(\(d\theta/dz = 0\): potential temperature constant with height)

✗
\(\Gamma > \Gamma_d\)→ Absolutely unstable

(\(d\theta/dz < 0\): potential temperature decreases with height - rare!)

5.3 Brunt-Väisälä Frequency

For a stable atmosphere, displaced parcels oscillate with the Brunt-Väisälä frequency (or buoyancy frequency):

\(N^2 = \frac{g}{\theta} \frac{d\theta}{dz} = \frac{g(\Gamma_d - \Gamma)}{T}\)
For a stable atmosphere (\(\Gamma < \Gamma_d\)):
• \(N^2 > 0\) → \(N\) is real → oscillatory motion
• Typical values: \(N \sim 0.01\text{-}0.02 \text{ s}^{-1}\)
• Period: \(T = 2\pi/N \sim 5\text{-}10\) minutes

🌊 Gravity Waves:

The Brunt-Väisälä frequency determines the frequency of internal gravity waves in a stratified atmosphere. These waves are commonly observed as wave clouds and play important roles in momentum transport and atmospheric mixing.

5.4 Temperature Inversions

A temperature inversion occurs when temperature increases with height (Γ < 0, or dT/dz > 0). Inversions are extremely stable and act as "lids" that suppress vertical motion.

Radiation Inversion

Forms at night when the ground cools by infrared radiation. Common in clear, calm conditions. Traps pollutants near the surface.

Subsidence Inversion

Forms aloft when sinking air warms adiabatically. Common in high-pressure systems. Caps convective clouds below.

Frontal Inversion

Occurs when warm air overruns cold air at a frontal boundary. Associated with weather fronts.

Marine Inversion

Forms over cool ocean surfaces. Warm air aloft overlies cool, moist marine layer. Common along west coasts.

💻 Computational Example:

See for a program that:

  • • Analyzes atmospheric soundings for stability
  • • Calculates CAPE (Convective Available Potential Energy)
  • • Computes the lifted condensation level
  • • Plots temperature and dewpoint profiles
  • • Determines stability indices (K-index, Showalter index)

6. Water Vapor & Phase Changes

6.1 Humidity Variables

Several measures quantify atmospheric moisture content:

Mixing Ratio (\(r\))

\(r = \frac{m_v}{m_d}\)[g/kg or kg/kg]

Mass of water vapor per unit mass of dry air. Approximately conserved for air parcel motion (unless condensation/evaporation occurs).

Specific Humidity (\(q\))

\(q = \frac{m_v}{m_v + m_d} \approx \frac{r}{1+r}\)[g/kg]

Mass of water vapor per unit mass of moist air. Very close to \(r\) for typical atmospheric values.

Vapor Pressure (\(e\))

\(e = \text{partial pressure of water vapor}\)[hPa]

Related to mixing ratio by: \(r \approx 0.622 \, e/p\) (where \(p\) is total pressure, \(0.622 = M_w/M_d\))

Relative Humidity (RH)

\(\text{RH} = \frac{e}{e_s} \times 100\%\)or\(\text{RH} = \frac{r}{r_s} \times 100\%\)

Ratio of actual vapor pressure to saturation vapor pressure. Commonly reported in weather forecasts.

Dewpoint Temperature (\(T_d\))

Temperature to which air must be cooled (at constant pressure) to reach saturation. Directly related to vapor pressure via the Clausius-Clapeyron equation.

6.2 Clausius-Clapeyron Equation

The saturation vapor pressure \(e_s\) depends strongly on temperature and is governed by the Clausius-Clapeyron equation:

Derivation from thermodynamics:
\(\frac{de_s}{dT} = \frac{L_v}{T \Delta\alpha}\) (Clapeyron equation)
where \(L_v\) = latent heat of vaporization, \(\Delta\alpha = \alpha_{\text{vapor}} - \alpha_{\text{liquid}}\)
Assuming ideal gas for vapor, \(\alpha_{\text{liquid}}\) negligible:
\(\frac{de_s}{dT} = \frac{L_v e_s}{R_v T^2}\)

Integrating (assuming \(L_v\) constant):

\(e_s(T) = e_s(T_0) \exp\left[\frac{L_v}{R_v}\left(\frac{1}{T_0} - \frac{1}{T}\right)\right]\)

A common approximation (Magnus formula) used in practice:

\(e_s(T) = 6.112 \exp\left[\frac{17.67(T-273.15)}{T-29.65}\right]\)[hPa]
(valid for temperatures between -40°C and +50°C)

📈 Key Result:

Saturation vapor pressure increases exponentially with temperature. At 0°C, \(e_s \approx 6.11\) hPa. At 30°C, \(e_s \approx 42.4\) hPa. This exponential dependence is why warm air can hold much more moisture than cold air - crucial for understanding precipitation and climate.

6.3 Latent Heats

Phase changes of water involve large energy transfers:

\(L_v\) (vaporization):
\(2.5 \times 10^6\) J/kg at 0°C
\(L_f\) (fusion/melting):
\(3.34 \times 10^5\) J/kg
\(L_s\) (sublimation):
\(2.83 \times 10^6\) J/kg
Note: \(L_s = L_v + L_f\)

For comparison, the specific heat of air is \(c_p = 1004\) J/(kg·K). The latent heat of vaporization can heat air by ~2500 K! This enormous energy explains:

  • Why thunderstorms are so energetic (latent heat release drives convection)
  • Why hurricanes form over warm oceans (evaporation provides energy)
  • The importance of moisture in weather systems

6.4 Virtual Temperature

Water vapor has lower molecular weight (\(M_v = 18.015\) g/mol) than dry air (\(M_d = 28.97\) g/mol), so moist air is less dense than dry air at the same temperature and pressure. We account for this using virtual temperature (\(T_v\)):

\(T_v = T(1 + 0.61r)\)
\(T_v\) is the temperature dry air would need to have the same density as moist air.
The ideal gas law for moist air becomes: \(p = \rho R T_v\)

For typical mixing ratios (\(r \sim 0.01\) kg/kg), \(T_v\) is about 0.6% higher than \(T\). This small difference is important for accurate density calculations in weather models.

6.5 Moist Adiabatic Process

When a saturated air parcel rises and cools, water vapor condenses, releasing latent heat. This reduces the cooling rate compared to the dry adiabatic lapse rate:

\(\Gamma_m = \frac{g}{c_p} \times \frac{1 + L_v r_s/(RT)}{1 + \varepsilon L_v^2 r_s/(c_p R_v T^2)}\)
where \(\varepsilon = R/R_v \approx 0.622\)
Typical values:
• At -40°C: \(\Gamma_m \approx 9.2\) K/km (nearly dry adiabatic, little moisture)
• At 0°C: \(\Gamma_m \approx 6.5\) K/km
• At 25°C: \(\Gamma_m \approx 4.3\) K/km (strong latent heating)

The moist adiabatic lapse rate is not constant - it decreases with increasing temperature because warmer air can hold more moisture (Clausius-Clapeyron), leading to more latent heat release.

💻 Computational Example:

See for a program that:

  • • Plots atmospheric soundings on a Skew-T log-p diagram
  • • Draws dry adiabats (constant \(\theta\) lines)
  • • Draws moist adiabats (constant \(\theta_e\) lines)
  • • Draws saturation mixing ratio lines
  • • Calculates lifted parcel paths and CAPE/CIN

Summary

In Part I, we've established the thermodynamic foundation for understanding atmospheric processes:

  • ✓ The atmosphere behaves as an ideal gas in hydrostatic balance
  • ✓ Adiabatic processes conserve potential temperature
  • ✓ Stability depends on the environmental lapse rate vs. adiabatic lapse rate
  • ✓ Water vapor dramatically affects atmospheric thermodynamics through latent heat
  • ✓ Phase changes of water are central to cloud formation and precipitation

These principles form the basis for understanding atmospheric dynamics (Part II), cloud physics (Part III), and climate systems (Part IV).