Part I: Atmospheric Thermodynamics

Fundamental thermodynamic principles governing atmospheric behavior, from the equation of state through adiabatic processes, moisture physics, stability analysis, and thermodynamic diagrams.

Video Lecture: Atmospheric Thermodynamics

A comprehensive overview of the core thermodynamic concepts covered in this part, including the equation of state, hydrostatic balance, adiabatic processes, and moisture thermodynamics.

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 (the homosphere). Above this level, in the heterosphere, molecular diffusion dominates and heavier species settle relative to lighter ones. 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 (most potent natural greenhouse gas)
  • Latent heat release/absorption during phase changes
  • Cloud and precipitation formation
  • Atmospheric chemistry and hydroxyl radical production
  • Modifying air density through virtual temperature effects

The total mass of the atmosphere is approximately \(5.15 \times 10^{18}\) kg, with a mean surface pressure of \(p_0 = 1013.25\) hPa. This corresponds to a force of roughly 10 tonnes per square metre acting on every surface on Earth.

1.2 Vertical Structure

The atmosphere is divided into layers based on the temperature profile. The transition between layers is governed by the balance between radiative heating, convective transport, and photochemical processes.

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. The tropopause height varies from ~8 km at the poles to ~16 km at the equator, driven by the intensity of tropical convection. Mean tropopause temperature is approximately 220 K at the equator and 210 K at the poles.

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-B and UV-C radiation. The stratosphere is dynamically very stable owing to the positive temperature gradient, which strongly suppresses vertical mixing. The stratopause temperature reaches approximately 270 K.

Mesosphere (50 -- 85 km)

Temperature decreases with height, reaching the coldest atmospheric temperatures (~130 K in summer polar mesopause) at the mesopause. Noctilucent clouds (polar mesospheric clouds) can form here. Meteors burn up in this layer. Gravity wave breaking in the mesosphere drives a pole-to-pole meridional circulation.

Thermosphere (85 -- 600 km)

Temperature increases dramatically (to 1000--2000 K) due to absorption of solar extreme UV and X-rays by O₂ and N₂. However, the air density is so low that these high kinetic temperatures would not feel "hot." Aurora occur in this layer. The International Space Station orbits at ~400 km. Beyond the thermosphere lies the exosphere, where atoms can escape to space.

1.3 Mean Molecular Weight

For dry air, the mean molecular weight is calculated from the mixture of constituent gases. If \(\chi_i\) is the mole fraction and \(M_i\) the molecular weight of each species:

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

The specific gas constant for dry air is then:

$$R_d = \frac{R^*}{M_{\text{dry}}} = \frac{8.3145}{0.02897} = 287.05 \text{ J/(kg·K)}$$

For moist air, the effective molecular weight is lower because water vapor (\(M_v = 18.015\) g/mol) is lighter. We handle this through the virtual temperature concept (discussed in Chapter 5) rather than recomputing \(R\) for each moisture content.

1.4 Hydrostatic Equation & Barometric Formula

In the vertical, the atmosphere is nearly in hydrostatic balance: the upward pressure gradient force balances the downward gravitational force. Consider a thin slab of air with thickness \(dz\)and density \(\rho\). The weight per unit area of this slab is \(\rho g \, dz\). In equilibrium:

$$\frac{dp}{dz} = -\rho g$$
where:
\(z\) = geometric height above mean sea level [m]
\(g\) = gravitational acceleration = 9.80665 m/s² (standard value)
\(\rho\) = air density [kg/m³]
The negative sign indicates pressure decreases with increasing altitude.

For an isothermal atmosphere (\(T = \text{const}\)), substituting\(\rho = p/(R_d T)\) from the ideal gas law and integrating:

Derivation of the barometric formula:
\(\frac{dp}{dz} = -\frac{pg}{R_d T}\)(substituting \(\rho = p/(R_d T)\))
\(\frac{dp}{p} = -\frac{g}{R_d T} dz = -\frac{dz}{H}\)(defining scale height \(H = R_d T / g\))
\(\int_{p_0}^{p} \frac{dp'}{p'} = -\int_{0}^{z} \frac{dz'}{H}\)
$$p(z) = p_0 \exp\!\left(-\frac{z}{H}\right)$$
This is the barometric formula for an isothermal atmosphere.

1.5 Scale Height

The scale height \(H\) is the e-folding distance over which pressure decreases by a factor of \(e \approx 2.718\):

$$H = \frac{R_d T}{g} \approx \frac{287 \times 255}{9.81} \approx 7.5 \text{ km (global mean)}$$

For the surface temperature \(T = 288\) K, the scale height is \(H \approx 8.4\) km. The density also follows a similar exponential: \(\rho(z) = \rho_0 \exp(-z/H)\). In reality, temperature varies with altitude so the actual pressure profile deviates from a pure exponential, but the concept of scale height provides an invaluable first-order description of atmospheric structure.

Some useful altitude benchmarks from the barometric formula:

\(z = 0\) km (sea level):
\(p \approx 1013\) hPa
\(z = 5.5\) km:
\(p \approx 500\) hPa (half the atmosphere below)
\(z = H \approx 8.4\) km:
\(p \approx 373\) hPa
\(z = 2H \approx 16.8\) km:
\(p \approx 137\) hPa
\(z = 3H \approx 25.2\) km:
\(p \approx 50\) hPa

1.6 Geopotential Height

Because \(g\) varies slightly with latitude and altitude, meteorologists use geopotential height \(Z\) instead of geometric height \(z\). The geopotential \(\Phi\) at height \(z\) is the work done against gravity to raise a unit mass from sea level:

$$\Phi = \int_0^z g(z')\,dz' \qquad \text{and} \qquad Z = \frac{\Phi}{g_0}$$
where \(g_0 = 9.80665\) m/s² is the standard gravitational acceleration

In the troposphere the difference between \(Z\) and \(z\) is small (a few tens of metres). The hydrostatic equation in geopotential coordinates becomes \(dp = -\rho g_0 \, dZ\), which is exact regardless of variations in \(g\).

Computational Example:

See for a program that calculates pressure as a function of altitude using both the isothermal barometric formula and the piecewise-linear US Standard Atmosphere temperature profile.

2. Ideal Gas Law & Equation of State

2.1 Equation of State for Dry Air

The atmosphere behaves as an ideal gas to excellent approximation. The conditions for ideal gas behavior -- molecules far apart relative to their size, and interactions dominated by elastic collisions -- are well satisfied throughout the troposphere and stratosphere. The equation of state relates pressure \(p\), density \(\rho\), and temperature \(T\):

$$p = \rho R_d T$$
where:
\(p\) = pressure [Pa] (1 hPa = 100 Pa = 1 mb)
\(\rho\) = density [kg/m³] (typical sea-level value: 1.225 kg/m³)
\(T\) = absolute temperature [K]
\(R_d = R^*/M_d = 8.3145/0.02897 = 287.05\) J/(kg K) is the specific gas constant for dry air

Alternative forms of the ideal gas law:

\(p V = n R^* T\)(molar form: \(n\) = number of moles, \(R^* = 8.3145\) J/(mol K))
\(p = n_d k_B T\)(kinetic theory form: \(n_d\) = number density, \(k_B = 1.381 \times 10^{-23}\) J/K)
\(p \alpha = R_d T\)(specific volume form: \(\alpha = 1/\rho\) [m³/kg])

2.2 Virtual Temperature

Moist air is a mixture of dry air and water vapor. Since water vapor (\(M_v = 18.015\) g/mol) is lighter than dry air (\(M_d = 28.97\) g/mol), moist air is less dense than dry air at the same temperature and pressure. Rather than modifying \(R\), we define the virtual temperature \(T_v\) -- the temperature that dry air would need to have the same density as the moist air:

$$p = \rho R_d T_v$$
where the virtual temperature is:
$$T_v = T \frac{1 + r/\varepsilon}{1 + r} \approx T(1 + 0.608\,r)$$
\(r\) = mixing ratio (mass of water vapor per mass of dry air) [kg/kg]
\(\varepsilon = M_v/M_d = R_d/R_v = 0.622\)
The approximation \(T_v \approx T(1 + 0.608\,r)\) is valid for \(r \ll 1\) (always true in Earth's atmosphere)

Since \(T_v > T\) always (moisture makes air effectively warmer and lighter), moist air is more buoyant than dry air. For a typical tropical mixing ratio of \(r \approx 0.020\) kg/kg, the virtual temperature correction is about \(\Delta T_v \approx 3.6\) K -- a small but meteorologically significant difference for accurate density and buoyancy calculations.

2.3 Moisture Variables

There are several ways to quantify moisture content in the atmosphere:

Mixing Ratio \(r\)

$$r = \frac{m_v}{m_d} = \frac{\rho_v}{\rho_d} = \varepsilon \frac{e}{p - e} \approx \varepsilon \frac{e}{p}$$

Mass of water vapor per unit mass of dry air. Conserved for unsaturated air parcel motion. Typical surface values: 1--20 g/kg.

Specific Humidity \(q\)

$$q = \frac{m_v}{m_v + m_d} = \frac{r}{1+r} = \frac{\varepsilon e}{p - (1 - \varepsilon)e} \approx \varepsilon \frac{e}{p}$$

Mass of water vapor per unit mass of moist air. Numerically very close to \(r\) since \(r \ll 1\). Preferred in the conservation equations of numerical weather models.

Relative Humidity (RH)

$$\text{RH} = \frac{e}{e_s(T)} \times 100\% \approx \frac{r}{r_s(T,p)} \times 100\%$$

Ratio of actual to saturation vapor pressure. Note that RH changes when either moisture content or temperature changes. A parcel can reach saturation (RH = 100%) without adding moisture, simply by cooling.

2.4 Hypsometric Equation

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

Derivation:
\(\frac{dp}{dz} = -\rho g = -\frac{pg}{R_d T_v}\)(substituting \(\rho = p/(R_d T_v)\))
\(\frac{dp}{p} = -\frac{g}{R_d T_v}dz\)
\(\int_{p_1}^{p_2} \frac{dp}{p} = -\frac{g}{R_d} \int_{z_1}^{z_2} \frac{dz}{T_v}\)
$$\Delta Z = Z_2 - Z_1 = \frac{R_d \bar{T}_v}{g} \ln\!\left(\frac{p_1}{p_2}\right)$$
where \(\bar{T}_v\) is the mean virtual temperature of the layer between pressures \(p_1\) and \(p_2\)

Practical Application:

The hypsometric equation is fundamental to weather analysis. Warm air columns are thicker (greater distance between pressure surfaces), while cold columns are thinner. The 1000--500 hPa thickness is a standard synoptic tool: values below ~540 dam indicate cold air (snow potential), while values above ~576 dam indicate warm tropical air. Thickness gradients create the thermal wind (Part II).

3. First Law of Thermodynamics

3.1 Energy Conservation

The first law of thermodynamics expresses conservation of energy for a thermodynamic system. For a unit mass of air (an air parcel), the first law states that heat added equals the increase in internal energy plus the work done by expansion:

$$\delta q = du + p\,d\alpha$$
where (all per unit mass):
\(\delta q\) = heat added to the parcel [J/kg] (not a state variable, hence \(\delta\))
\(du\) = change in specific internal energy [J/kg]
\(p \, d\alpha\) = work done by the parcel through expansion (\(\alpha = 1/\rho\) is specific volume)

3.2 Specific Heats and Mayer's Relation

For an ideal gas, internal energy depends only on temperature: \(u = u(T)\). The specific heats at constant volume and constant pressure are:

$$c_v = \left(\frac{\partial u}{\partial T}\right)_v = \frac{f}{2}\frac{R_d}{1} \quad \Rightarrow \quad du = c_v\,dT$$
$$c_p = \left(\frac{\partial h}{\partial T}\right)_p \quad \Rightarrow \quad dh = c_p\,dT$$
where \(h = u + p\alpha\) is the specific enthalpy and \(f\) is the number of degrees of freedom

For diatomic molecules (N₂, O₂) at atmospheric temperatures, there are \(f = 5\) degrees of freedom (3 translational + 2 rotational), giving \(c_v = (5/2)R_d\) and\(c_p = (7/2)R_d\). The key thermodynamic constants 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
\(\gamma = c_p/c_v = 7/5 = 1.4\) -- heat capacity ratio (adiabatic index)
\(\kappa = R_d/c_p = 2/7 \approx 0.286\) -- Poisson constant
$$c_p - c_v = R_d \qquad \text{(Mayer's relation)}$$
This follows directly from differentiating \(h = u + R_d T\) for an ideal gas

3.3 Enthalpy Form of the First Law

In meteorology, we work with pressure as the vertical coordinate rather than volume. Introducing enthalpy \(h = u + p\alpha\), we can rewrite the first law:

\(\delta q = du + p\,d\alpha\)
\(= du + d(p\alpha) - \alpha\,dp\)
\(= dh - \alpha\,dp\)
\(= c_p\,dT - \alpha\,dp\)
$$\delta q = c_p\,dT - \frac{1}{\rho}\,dp$$
This is the most useful form of the first law for atmospheric applications.

Physical interpretation: The two terms on the right side represent: (1) change in enthalpy due to temperature change, and (2) work done by expansion as the parcel moves to lower pressure. An alternative rate form useful in numerical models:

$$\frac{\delta q}{dt} = c_p \frac{dT}{dt} - \frac{1}{\rho} \frac{dp}{dt}$$

3.4 Diabatic Heating Processes

In the atmosphere, \(\delta q \neq 0\) (diabatic processes) can arise from several mechanisms:

  • Radiation: Absorption/emission of shortwave and longwave radiation. The net radiative heating rate is \(Q_{\text{rad}} = -\frac{1}{\rho c_p} \frac{\partial F_{\text{net}}}{\partial z}\) where \(F_{\text{net}}\) is the net radiative flux.
  • Latent heat release: Condensation of water vapor releases \(L_v \approx 2.5 \times 10^6\) J/kg, the dominant heating source in tropical convection.
  • Sensible heat flux: Molecular and turbulent conduction from Earth's surface into the atmospheric boundary layer.
  • Frictional dissipation: Conversion of kinetic energy to thermal energy through viscous forces (small except in strong wind shear layers).

3.5 Entropy and the Second Law

Dividing the first law by temperature defines the specific entropy \(s\):

$$ds = \frac{\delta q}{T} = c_p \frac{dT}{T} - R_d \frac{dp}{p} = c_p\,d(\ln\theta)$$

This beautiful result shows that entropy is directly related to potential temperature: \(s = c_p \ln\theta + \text{const}\). Surfaces of constant \(\theta\) are isentropic surfaces. Adiabatic motion (\(\delta q = 0\)) conserves both entropy and potential temperature, so air parcels move along isentropic surfaces.

4. Adiabatic Processes

4.1 Dry Adiabatic Process

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

  • Air is a poor thermal conductor (thermal diffusivity \(\sim 2 \times 10^{-5}\) m²/s)
  • Vertical motions occur on timescales of minutes to hours
  • Radiative heating/cooling rates are typically \(\sim 1\) K/day -- negligible over dynamical timescales
  • Turbulent mixing primarily occurs at the parcel boundary, not the interior

Setting \(\delta q = 0\) in the enthalpy form of the first law:

\(c_p dT - \alpha \, dp = 0\)
\(c_p dT = \frac{R_d T}{p} dp\)
\(c_p \frac{dT}{T} = R_d \frac{dp}{p}\)

Integrating both sides from state \((T_1, p_1)\) to \((T_2, p_2)\):

$$c_p \ln\!\left(\frac{T_2}{T_1}\right) = R_d \ln\!\left(\frac{p_2}{p_1}\right)$$
$$\frac{T_2}{T_1} = \left(\frac{p_2}{p_1}\right)^{R_d/c_p} = \left(\frac{p_2}{p_1}\right)^{\kappa}$$

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. Setting \(T_2 = \theta\) and \(p_2 = p_0\) in the adiabatic relation above:

$$\theta = T \left(\frac{p_0}{p}\right)^{R_d/c_p} = T \left(\frac{1000}{p}\right)^{0.286}$$
For dry adiabatic processes, \(\theta\) is exactly conserved: \(d\theta/dt = 0\)

Why is \(\theta\) so important? Potential temperature removes the adiabatic temperature changes caused by pressure variations. Two parcels at different pressures but with the same \(\theta\) are thermodynamically equivalent -- they have the same entropy. This makes\(\theta\) the fundamental conservative tracer for adiabatic flow:

  • A well-mixed boundary layer has constant \(\theta\) with height
  • Isentropic surfaces (\(\theta = \text{const}\)) are material surfaces for adiabatic flow
  • Potential vorticity (\(\text{PV} = -g\,\eta \cdot \nabla\theta\)) is conserved on \(\theta\)-surfaces
  • Stratospheric transport is best analysed on isentropic coordinates

Example: An air parcel at \(T = 250\) K and \(p = 500\) hPa has:

\(\theta = 250 \times (1000/500)^{0.286} = 250 \times 1.219 = 304.8\) K

4.3 Dry Adiabatic Lapse Rate

How does temperature change with altitude for a dry adiabatically rising parcel? We combine the first law (\(\delta q = 0\)) with the hydrostatic equation:

Derivation:
\(c_p dT = \alpha \, dp = \frac{1}{\rho} dp\) (first law with \(\delta q=0\))
\(c_p dT = \frac{1}{\rho}(-\rho g\,dz) = -g\,dz\) (using hydrostatic equation \(dp = -\rho g\,dz\))
$$\frac{dT}{dz}\bigg|_{\text{dry adiabatic}} = -\frac{g}{c_p} \equiv -\Gamma_d$$
$$\Gamma_d = \frac{g}{c_p} = \frac{9.81}{1004} \approx 9.8 \text{ K/km}$$
This is the dry adiabatic lapse rate (DALR) -- the rate at which unsaturated rising air cools with altitude.

Physical Interpretation:

As an air parcel rises, it enters regions of lower ambient pressure and expands. This expansion does work on the surrounding atmosphere (\(p \, d\alpha > 0\)), drawing energy from the parcel's internal energy. Since the process is adiabatic (no external heat source), the internal energy (and thus temperature) must decrease. The remarkable feature is that \(\Gamma_d\) depends only on\(g\) and \(c_p\) -- not on the parcel's initial temperature, pressure, or humidity.

4.4 Poisson's Equations

For an adiabatic process in an ideal gas, several relationships between state variables remain constant. These are collectively known as Poisson's equations:

$$T\,p^{-\kappa} = \text{constant} \qquad (\kappa = R_d/c_p \approx 0.286)$$
$$T\,\alpha^{\gamma - 1} = \text{constant} \qquad (\text{or } T\,\rho^{1-\gamma} = \text{const})$$
$$p\,\alpha^{\gamma} = \text{constant} \qquad (\text{or } p/\rho^{\gamma} = \text{const})$$

These relations are the atmospheric analogues of the well-known adiabatic relations for ideal gases. The first equation is equivalent to the definition of potential temperature. The third equation is the form most often encountered in sound wave theory, where the speed of sound is:

$$c_s = \sqrt{\gamma R_d T} \approx 340 \text{ m/s at } T = 288 \text{ K}$$

5. Moisture Thermodynamics

5.1 Clausius-Clapeyron Equation

The saturation vapor pressure \(e_s(T)\) is the equilibrium vapor pressure over a flat surface of pure water (or ice) at temperature \(T\). It is governed by the Clausius-Clapeyron equation, which follows from equating the Gibbs free energy of the two phases along the coexistence curve:

Derivation from the Clapeyron equation:
\(\frac{de_s}{dT} = \frac{L_v}{T(\alpha_v - \alpha_l)}\)(exact Clapeyron equation)
Since \(\alpha_v = R_v T / e_s \gg \alpha_l\) (vapor specific volume vastly exceeds liquid):
\(\frac{de_s}{dT} \approx \frac{L_v}{T \cdot R_v T / e_s} = \frac{L_v e_s}{R_v T^2}\)
$$\frac{de_s}{dT} = \frac{L_v\,e_s}{R_v\,T^2}$$
The Clausius-Clapeyron equation (\(R_v = R^*/M_v = 461.5\) J/(kg K))

This can be rewritten in a revealing logarithmic form:

$$\frac{d\ln e_s}{dT} = \frac{L_v}{R_v T^2} \qquad \Longleftrightarrow \qquad \frac{d\ln e_s}{d(1/T)} = -\frac{L_v}{R_v}$$

Integrating from a reference state \((T_0, e_{s0})\) with \(L_v\) assumed constant:

$$e_s(T) = e_{s0}\,\exp\!\left[\frac{L_v}{R_v}\left(\frac{1}{T_0} - \frac{1}{T}\right)\right]$$
Using \(T_0 = 273.15\) K and \(e_{s0} = 6.112\) hPa

In practice, the empirical Magnus (or Tetens) formula is preferred because\(L_v\) varies with temperature:

$$e_s(T) = 6.112\,\exp\!\left(\frac{17.67\,(T - 273.15)}{T - 29.65}\right) \quad [\text{hPa}]$$
Accurate to within 0.3% for \(-40^\circ\text{C} < T < 50^\circ\text{C}\)

Key Result:

Saturation vapor pressure increases approximately exponentially with temperature. A useful rule of thumb: \(e_s\) roughly doubles for every 10 K increase in temperature. At 0\(^\circ\)C, \(e_s \approx 6.11\) hPa; at 20\(^\circ\)C, \(e_s \approx 23.4\) hPa; at 35\(^\circ\)C, \(e_s \approx 56.2\) hPa. This exponential dependence is the fundamental reason why warm air can hold far more moisture -- and why the water cycle intensifies with warming.

5.2 Latent Heats

Phase changes of water involve enormous energy transfers relative to the heat capacity of air:

\(L_v\) (vaporization, 0\(^\circ\)C):
\(2.501 \times 10^6\) J/kg
\(L_v\) (vaporization, 25\(^\circ\)C):
\(2.442 \times 10^6\) J/kg
\(L_f\) (fusion, 0\(^\circ\)C):
\(3.337 \times 10^5\) J/kg
\(L_s\) (sublimation, 0\(^\circ\)C):
\(2.834 \times 10^6\) J/kg
Note: \(L_s = L_v + L_f\). Also, \(L_v\) decreases with temperature: \(L_v(T) \approx 2.501 \times 10^6 - 2370\,(T - 273.15)\) J/kg

To put these values in perspective, condensing 1 g of water vapor releases enough energy to warm 2.5 kg of air by 1 K. The total latent energy in a tropical column with precipitable water of 50 mm is roughly \(1.25 \times 10^8\) J/m² -- comparable to the kinetic energy of a hurricane.

5.3 Wet-Bulb Temperature

The wet-bulb temperature \(T_w\) is the lowest temperature to which air can be cooled by evaporating water into it at constant pressure. It is measured by wrapping a thermometer in a wet wick and ventilating it. The relationship between \(T\), \(T_w\), and \(T_d\) is:

$$T_d \leq T_w \leq T$$
Equality holds only at saturation (RH = 100%)

The wet-bulb temperature is determined by the energy balance between sensible cooling and latent heating:

$$c_p(T - T_w) = L_v\bigl[r_s(T_w) - r\bigr]$$
where \(r_s(T_w)\) is the saturation mixing ratio at the wet-bulb temperature

The wet-bulb temperature is conserved during adiabatic, isobaric evaporation and is a critical heat stress parameter: \(T_w > 35^\circ\) C is considered the upper limit of human survivability.

5.4 Lifted Condensation Level (LCL)

When an unsaturated air parcel rises, it cools at the dry adiabatic lapse rate while its mixing ratio remains constant. The dewpoint also decreases, but more slowly (\(\sim 1.7\) K/km). The level where \(T = T_d\) (the parcel reaches saturation) is the Lifted Condensation Level (LCL):

$$z_{\text{LCL}} \approx \frac{T - T_d}{(\Gamma_d - \Gamma_{T_d})} \approx \frac{T - T_d}{8.0} \text{ km}$$
Or equivalently, \(z_{\text{LCL}} \approx 125 (T - T_d)\) metres (the Espy formula, with \(T, T_d\) in \(^\circ\)C)

The LCL represents the cloud base for convective clouds. Above the LCL, the parcel follows the moist (saturated) adiabat instead of the dry adiabat.

5.5 Moist Adiabatic (Saturated) Lapse Rate

When a saturated parcel rises, continued cooling causes condensation, which releases latent heat and partially offsets the adiabatic cooling. The first law for a saturated parcel becomes:

$$c_p\,dT + g\,dz = -L_v\,dr_s$$
The right side represents latent heating from condensation (\(dr_s < 0\) when rising)

Using the Clausius-Clapeyron equation to express \(dr_s\) in terms of \(dT\), one obtains the saturated adiabatic lapse rate:

$$\Gamma_s = \frac{g}{c_p} \cdot \frac{1 + \dfrac{L_v\,r_s}{R_d\,T}}{1 + \dfrac{\varepsilon\,L_v^2\,r_s}{c_p\,R_d\,T^2}}$$
where \(\varepsilon = R_d/R_v \approx 0.622\) and \(r_s = r_s(T, p)\)
Typical values of \(\Gamma_s\):
At \(-40^\circ\)C: \(\Gamma_s \approx 9.2\) K/km (very little moisture, nearly dry adiabatic)
At 0\(^\circ\)C: \(\Gamma_s \approx 6.5\) K/km
At 20\(^\circ\)C: \(\Gamma_s \approx 4.5\) K/km (substantial latent heating)
At 35\(^\circ\)C: \(\Gamma_s \approx 3.3\) K/km (very strong latent heating)
Note: \(\Gamma_s < \Gamma_d\) always, and \(\Gamma_s \to \Gamma_d\) for very cold temperatures (low \(r_s\))

5.6 Equivalent Potential Temperature

The equivalent potential temperature \(\theta_e\) is the potential temperature a parcel would have if all its moisture were condensed out and the latent heat released was used to warm the parcel. It is conserved for both dry and saturated adiabatic processes (as long as condensate falls out -- the pseudoadiabatic assumption):

$$\theta_e = \theta \exp\!\left(\frac{L_v\,r_s}{c_p\,T}\right) \approx \theta \exp\!\left(\frac{L_v\,r}{c_p\,T_{\text{LCL}}}\right)$$
\(\theta_e\) is conserved for reversible moist adiabatic processes (to good approximation)

A related quantity is the wet-bulb potential temperature \(\theta_w\), which is the temperature reached by saturated adiabatic descent from the LCL to 1000 hPa. Both \(\theta_e\) and \(\theta_w\) are extremely useful for air mass identification and stability analysis because they account for both temperature and moisture content in a single variable.

For an unsaturated parcel with temperature \(T\), mixing ratio \(r\), and pressure \(p\), a practical approximation (Bolton 1980) for \(\theta_e\) is:

$$\theta_e = T\left(\frac{1000}{p}\right)^{0.2854(1 - 0.28\,r)} \exp\!\left[r(1 + 0.81\,r)\left(\frac{3376}{T_{\text{LCL}}} - 2.54\right)\right]$$

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, LCL, LFC, and CAPE/CIN

6. Atmospheric Stability

6.1 Static Stability Concept

Static stability determines whether a small vertical displacement of an air parcel will be amplified (unstable), suppressed (stable), or neutral. Consider displacing a parcel vertically by \(\delta z\) from its equilibrium position. The buoyancy force per unit mass acting on the displaced parcel is:

$$B = -g\,\frac{\rho' - \rho_{\text{env}}}{\rho_{\text{env}}} = g\,\frac{T'_v - T_{v,\text{env}}}{T_{v,\text{env}}}$$
where primes denote parcel quantities and env denotes environmental

Stable Atmosphere (\(B < 0\) for upward displacement)

The displaced parcel becomes cooler (denser) than its surroundings and experiences a restoring downward buoyancy force. The parcel oscillates about its equilibrium level -- these are buoyancy oscillations (gravity waves).

Unstable Atmosphere (\(B > 0\) for upward displacement)

The displaced parcel remains warmer (less dense) than surroundings and accelerates upward. This leads to convective overturning. Absolutely unstable conditions (\(\Gamma > \Gamma_d\)) are rare and quickly eliminated by convective mixing, but conditional instability is common.

Neutral Atmosphere (\(B = 0\))

The displaced parcel has exactly the same temperature/density as its environment. No net buoyancy force acts. This occurs when \(\Gamma = \Gamma_d\) (dry-neutral) or \(\Gamma = \Gamma_s\) (moist-neutral).

6.2 Stability Criteria

Let \(\Gamma = -dT/dz\) be the environmental lapse rate (the actual temperature profile). The stability classification depends on the comparison with \(\Gamma_d\) and \(\Gamma_s\):

1.
Absolutely Stable: \(\Gamma < \Gamma_s < \Gamma_d\)

Stable for both saturated and unsaturated displacements. \(d\theta/dz > 0\) and \(d\theta_e/dz > 0\).

2.
Conditionally Unstable: \(\Gamma_s < \Gamma < \Gamma_d\)

Stable for unsaturated parcels, unstable for saturated parcels. This is the most common state of the troposphere! \(d\theta/dz > 0\) but \(d\theta_e/dz < 0\) typically.

3.
Absolutely Unstable: \(\Gamma > \Gamma_d > \Gamma_s\)

Unstable for any displacement. \(d\theta/dz < 0\). Rare except in the surface superadiabatic layer on hot days (lowest ~100 m).

4.
Dry Neutral: \(\Gamma = \Gamma_d\)

\(d\theta/dz = 0\): well-mixed conditions. Common in the convective boundary layer during daytime.

5.
Saturated Neutral: \(\Gamma = \Gamma_s\)

\(d\theta_e/dz = 0\): the lapse rate of a cloud layer in which saturated vertical mixing occurs.

6.3 Brunt-Vaisala Frequency

For a stable atmosphere, the equation of motion for a small vertical displacement \(\delta z\) of an unsaturated parcel can be derived from Newton's second law with the buoyancy force:

\(\frac{d^2(\delta z)}{dt^2} = B = g \frac{T'_{\text{parcel}} - T_{\text{env}}}{T_{\text{env}}}\)
The parcel cools at \(\Gamma_d\), the environment at \(\Gamma\):\(T' - T_{\text{env}} = -(\Gamma_d - \Gamma)\delta z\)
\(\frac{d^2(\delta z)}{dt^2} = -\frac{g(\Gamma_d - \Gamma)}{T}\delta z = -N^2 \delta z\)
$$N^2 = \frac{g}{\theta}\frac{d\theta}{dz} = \frac{g(\Gamma_d - \Gamma)}{T}$$
Stable atmosphere (\(\Gamma < \Gamma_d\)): \(N^2 > 0\), \(N\) is real, parcel oscillates with period \(\tau = 2\pi/N\)
Typical tropospheric values: \(N \sim 0.01\) s\(^{-1}\), period \(\tau \sim 10\) minutes
Stratospheric values: \(N \sim 0.02\) s\(^{-1}\), period \(\tau \sim 5\) minutes (more stable)
Unstable atmosphere (\(\Gamma > \Gamma_d\)): \(N^2 < 0\), displacement grows exponentially

The Brunt-Vaisala frequency \(N\) is one of the most fundamental parameters in atmospheric dynamics. It determines the frequency of internal gravity waves, the vertical structure of mountain waves, the Froude number for flow over topography (\(\text{Fr} = U/(NH)\)), and the Richardson number for shear instability (\(\text{Ri} = N^2/(dU/dz)^2\)).

6.4 Convective Available Potential Energy (CAPE)

CAPE quantifies the total positive buoyancy available to an ascending parcel from its Level of Free Convection (LFC) to the Equilibrium Level (EL). It represents the maximum kinetic energy a parcel can acquire from buoyancy:

$$\text{CAPE} = \int_{\text{LFC}}^{\text{EL}} g\,\frac{T_{v,\text{parcel}} - T_{v,\text{env}}}{T_{v,\text{env}}}\,dz = -R_d \int_{p_{\text{EL}}}^{p_{\text{LFC}}} (T_{v,p} - T_{v,e})\,d\ln p$$
Units: J/kg (equivalent to m²/s²)
Maximum updraft speed: \(w_{\max} = \sqrt{2 \cdot \text{CAPE}}\)
CAPE classification:
0 -- 300 J/kg:
Marginal instability
300 -- 1000 J/kg:
Moderate instability
1000 -- 2500 J/kg:
Large instability
2500 -- 4000 J/kg:
Extreme instability
> 4000 J/kg:
Extreme (severe weather likely)

The related quantity CIN (Convective Inhibition) represents the negative buoyancy a parcel must overcome between the surface and the LFC:

$$\text{CIN} = -\int_{\text{SFC}}^{\text{LFC}} g\,\frac{T_{v,\text{parcel}} - T_{v,\text{env}}}{T_{v,\text{env}}}\,dz > 0$$
CIN acts as a "cap" that prevents convection. Typical values of 50--200 J/kg must be overcome by surface heating or mechanical lifting for thunderstorms to initiate.

6.5 Temperature Inversions

A temperature inversion occurs when temperature increases with height (\(\Gamma < 0\), i.e., \(dT/dz > 0\)). Inversions represent extreme stability and act as "lids" that suppress vertical mixing and trap pollutants.

Radiation Inversion

Forms at night when the ground cools by infrared radiation faster than the air above. Common in clear, calm conditions. Strongest in winter at high latitudes. Traps fog and pollutants near the surface.

Subsidence Inversion

Forms aloft when large-scale sinking (subsidence) warms air adiabatically, creating a warm layer over cooler air below. Common in subtropical high-pressure systems. Caps the planetary boundary layer and limits cumulus cloud tops.

Frontal Inversion

Occurs when warm air overrides cold air at a frontal boundary. The warm air mass aloft creates a temperature increase with height. Can produce freezing rain when surface temperatures are below 0\(^\circ\)C.

Marine Inversion

Forms over cool ocean surfaces where warm air subsides over a cool marine boundary layer. Extremely persistent along west coasts of continents (California, Chile, Namibia). Produces persistent stratus/stratocumulus cloud decks.

Computational Example:

See for a program that:

  • Analyzes atmospheric soundings for stability classification
  • Calculates CAPE, CIN, LCL, LFC, and equilibrium level
  • Computes the Brunt-Vaisala frequency profile
  • Plots temperature/dewpoint profiles with parcel ascent curves
  • Determines stability indices (K-index, Showalter, Total Totals, SWEAT)

7. Thermodynamic Diagrams

7.1 Purpose and Construction

Thermodynamic diagrams are graphical tools that plot the state of the atmosphere (temperature, moisture, pressure) in a way that makes stability analysis and parcel lifting calculations intuitive. A good thermodynamic diagram has several desirable properties:

  • Area on the diagram is proportional to energy (work or heat)
  • Fundamental lines (isotherms, adiabats, mixing ratio lines) are nearly straight
  • Large angle between isotherms and dry adiabats (for easy reading)
  • Pressure decreases upward (mimicking the real atmosphere)

On any thermodynamic diagram, five sets of lines are plotted:

1. Isobars -- lines of constant pressure \(p\)
2. Isotherms -- lines of constant temperature \(T\)
3. Dry adiabats -- lines of constant \(\theta\): \(T = \theta (p/p_0)^{\kappa}\)
4. Saturated (moist) adiabats -- lines of constant \(\theta_e\) (or \(\theta_w\))
5. Saturation mixing ratio lines -- lines of constant \(r_s\): \(r_s = 0.622\,e_s(T)/[p - e_s(T)]\)

7.2 Skew-T Log-P Diagram

The Skew-T log-P diagram is the most widely used thermodynamic diagram in North American meteorology. Its coordinates are:

$$x = T + k\,\ln(p_0/p) \qquad \text{(horizontal axis)}$$
$$y = -\ln(p/p_0) \qquad \text{(vertical axis)}$$

The "skew" refers to the fact that isotherms slope to the upper right rather than being vertical. This increases the angle between isotherms and dry adiabats, making it easier to distinguish temperature changes from adiabatic processes. Key features:

  • Pressure axis is logarithmic (equal-\(\ln p\) spacing), giving uniform vertical resolution
  • Isotherms are straight lines tilted ~45\(^\circ\) to the right
  • Dry adiabats curve from lower-right to upper-left
  • Moist adiabats curve from lower-right to upper-left, more steeply at low altitudes
  • Area on the diagram is proportional to energy

Reading a sounding on a Skew-T: The environmental temperature profile (solid red line) and dewpoint profile (solid green line) are plotted. The area between the lifted parcel curve and the environmental temperature, where the parcel is warmer (between LFC and EL), is proportional to CAPE. Where the parcel is cooler (below LFC), the area represents CIN.

7.3 Other Common Diagrams

Tephigram (T-\(\phi\) gram)

Used primarily in the UK, Canada, and Australia. Coordinates are temperature \(T\) (x-axis) and entropy \(\phi = c_p \ln\theta\) (y-axis). Dry adiabats are exactly horizontal and isotherms are exactly vertical. The area enclosed is exactly proportional to energy, making it theoretically the most "ideal" diagram:

$$\oint T\,d\phi = \oint T\,c_p\,d(\ln\theta) = \oint \delta q$$

Emagram (Energy per unit Mass diAGRAM)

Coordinates are \(T\) (x-axis) and \(-\ln p\) (y-axis, increasing upward). Isotherms are vertical and isobars horizontal. Dry adiabats are slightly curved lines running from lower-right to upper-left. The area bounded by a thermodynamic cycle is proportional to the energy.

Stuve Diagram

Coordinates are \(T\) (x-axis) and \(p^{\kappa}\) (y-axis, decreasing upward, where \(\kappa = R_d/c_p\)). The clever choice of the vertical coordinate makes dry adiabats plot as exactly straight lines. However, area is not proportional to energy. Its simplicity makes it popular for educational purposes.

Aerological Diagram (Pseudoadiabatic Chart)

Uses \(\theta\) as the x-axis and \(p\) as the y-axis (logarithmic, decreasing upward). Dry adiabats are vertical lines, making it particularly easy to identify well-mixed layers. This diagram is sometimes used in research for boundary layer analysis.

7.4 Practical Use: Parcel Method

The step-by-step procedure for analysing an atmospheric sounding on a thermodynamic diagram:

  1. Plot the sounding: Plot \(T(p)\) and \(T_d(p)\) from radiosonde data.
  2. Select starting parcel: Choose surface conditions (or mixed-layer averages for \(T\) and \(r\)).
  3. Lift dry-adiabatically from the surface along a constant-\(\theta\) line until the parcel reaches saturation (where the dry adiabat crosses the mixing ratio line corresponding to the parcel's \(r\)). This is the LCL.
  4. Lift moist-adiabatically from the LCL along a constant-\(\theta_e\) line. Where this curve crosses the environmental \(T\) profile going from cooler to warmer than the environment is the LFC.
  5. Continue moist-adiabatically until the parcel curve crosses the environment again going from warmer to cooler. This is the EL (Equilibrium Level).
  6. Compute CAPE and CIN from the enclosed areas between parcel and environment curves.

7.5 Important Derived Quantities from Soundings

Several stability indices and derived parameters are routinely computed from thermodynamic diagrams:

Lifted Index (LI): \(\text{LI} = T_{\text{env}}(500) - T_{\text{parcel}}(500)\) [K]. Negative values indicate instability at 500 hPa. LI \(< -6\) suggests severe thunderstorm potential.
K-Index: \(K = (T_{850} - T_{500}) + T_{d,850} - (T_{700} - T_{d,700})\). Values above 30 indicate high thunderstorm probability.
Total Totals (TT): \(\text{TT} = T_{850} + T_{d,850} - 2 T_{500}\). Values above 50 suggest severe thunderstorm potential.
Showalter Index (SI): Similar to LI but lifts a parcel from 850 hPa instead of the surface.\(\text{SI} = T_{\text{env}}(500) - T_{\text{parcel,850 lifted}}(500)\)
Precipitable Water (PW): \(\text{PW} = \frac{1}{\rho_w g} \int_0^{p_s} q\,dp\) [mm]. Total column water vapor expressed as depth of liquid water.

Summary

In Part I, we have established the complete thermodynamic foundation for understanding atmospheric processes. The key results and their interconnections are:

  • 1.The atmosphere behaves as an ideal gas (\(p = \rho R_d T_v\)) in hydrostatic balance (\(dp/dz = -\rho g\)), giving rise to the barometric formula \(p = p_0 e^{-z/H}\) and the hypsometric equation.
  • 2.The first law of thermodynamics (\(\delta q = c_p\,dT - \alpha\,dp\)) leads naturally to potential temperature \(\theta\), which is conserved for adiabatic processes and is directly related to entropy.
  • 3.Dry adiabatic ascent produces cooling at \(\Gamma_d = g/c_p \approx 9.8\) K/km, while saturated ascent produces slower cooling at \(\Gamma_s < \Gamma_d\) due to latent heat release governed by the Clausius-Clapeyron equation \(de_s/dT = L_v e_s/(R_v T^2)\).
  • 4.Atmospheric stability depends on the comparison of \(\Gamma\) with \(\Gamma_d\) and \(\Gamma_s\). The Brunt-Vaisala frequency \(N^2 = (g/\theta)(d\theta/dz)\) quantifies static stability, while CAPE quantifies the total energy available for deep convection.
  • 5.Equivalent potential temperature \(\theta_e\) is conserved for moist adiabatic processes and serves as the fundamental tracer for air masses containing moisture.
  • 6.Thermodynamic diagrams (Skew-T, tephigram, emagram, Stuve) provide powerful graphical tools for analysing soundings, computing CAPE/CIN, and identifying key atmospheric levels (LCL, LFC, EL).

These thermodynamic principles form the essential basis for understanding atmospheric dynamics (Part II), cloud physics and precipitation (Part III), climate systems (Part IV), and weather analysis and forecasting (Part V).

NPTEL: Introduction to Atmospheric Science

Lectures from the NPTEL Introduction to Atmospheric Science course, covering atmospheric structure, thermodynamic principles, and the Skew-T diagram.

Lec-01 Introduction

Lec-02 Atmosphere — Pressure, Temperature and Composition

Lec-03 Vertical Structure of the Atmosphere

Lec-10 Atmospheric Thermodynamics — Introduction

Lec-11 The Hydrostatic Equation

Lec-12 Hypsometric Equation and Sea-Level Pressure

Lec-13 Basic Thermodynamics

Lec-14 Air Parcel Concept and Dry Adiabatic Lapse Rate

Lec-15 Potential Temperature

Lec-16 Skew-T ln-P Chart

Lec-17 Problems Using Skew-T ln-P Chart (1)

Lec-18 Problems Using Skew-T ln-P Chart (2)

Atmospheric Radiation and Energy Balance

Lectures on atmospheric radiation, energy balance, greenhouse gases, and the sulfur cycle from the CLEX Biogeochemistry course.

Atmospheric Radiation, Energy Balance and GHGs

The Atmospheric Sulfur Cycle, Water, Clouds and Aerosols

Yale GG 140: Atmospheric Fundamentals

Lectures from Yale's 'The Atmosphere, the Ocean, and Environmental Change' course covering atmospheric basics, gas laws, hydrostatic balance, and horizontal transport.

01. Introduction to Atmospheres

02. Retaining an Atmosphere

03. The Perfect Gas Law

04. Vertical Structure and Residence Time

05. Earth Systems Analysis (Tank Experiment)

06. Greenhouse Effect and Habitability

07. Hydrostatic Balance

08. Horizontal Transport