← Back to Part 2: Atmospheric Thermodynamics

2.1 Thermodynamic Principles

Atmospheric thermodynamics applies the fundamental laws of thermodynamics to understand heat transfer, work, and energy transformations in the atmosphere.

First Law of Thermodynamics

$$dU = \delta Q - \delta W = \delta Q - p \, dV$$

Change in internal energy = heat added - work done

For an ideal gas:

Internal Energy

$$U = n c_v T$$

Enthalpy

$$H = U + pV = n c_p T$$

Heat Capacities

Specific Heat at Constant Pressure

c_p = 1005 J/(kg·K) for dry air

Specific Heat at Constant Volume

c_v = 718 J/(kg·K) for dry air

Ratio of Heat Capacities

$\gamma = c_p/c_v = 1.4$ for diatomic gases

Mayer's relation: $c_p - c_v = R_d$ (287 J/(kg·K) for dry air)

Specific Heat Relationships

$$c_p = \frac{\gamma}{\gamma - 1} R_d, \qquad c_v = \frac{1}{\gamma - 1} R_d$$

For dry air ($\gamma = 1.4$): $c_p = 3.5\,R_d = 1005$ J/(kg·K), $c_v = 2.5\,R_d = 718$ J/(kg·K)

Potential Temperature & Entropy

Poisson's Equation

$$\theta = T \left(\frac{p_0}{p}\right)^{R_d / c_p}$$

Potential temperature $\theta$ is conserved during dry adiabatic processes. Here $p_0 = 1000$ hPa is the reference pressure and $R_d/c_p \approx 0.286$.

Entropy of Dry Air

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

Entropy is directly related to the logarithm of potential temperature — surfaces of constant $\theta$ are isentropic surfaces.

Virtual Temperature

$$T_v = T\,(1 + 0.61\, w) \approx T\,(1 + 0.61\, q)$$

Accounts for water vapor making moist air less dense than dry air at the same T and p. Here $w$ is the mixing ratio (kg/kg).

Carnot Efficiency

$$\eta_{\max} = \frac{T_H - T_C}{T_H} = 1 - \frac{T_C}{T_H}$$

Upper bound on atmospheric heat engine efficiency. For $T_H = 300$ K (surface) and $T_C = 200$ K (tropopause): $\eta \approx 33\%$.

Interactive Simulation: Potential Temperature & Adiabatic Processes

Python

Compares temperature and potential temperature profiles for different lapse rates (superadiabatic, dry adiabatic, standard, inversion, moist adiabatic). Shows how potential temperature reveals atmospheric stability.

potential_temperature.py78 lines

import numpy as np
import matplotlib.pyplot as plt

R_d = 287.0; c_p = 1005.0; g = 9.81; L_v = 2.5e6; R_v = 461.5; eps = R_d / R_v
p0 = 100000.0  # Pa reference
kappa = R_d / c_p  # ~0.286
Gamma_d = g / c_p * 1000  # K/km

def sat_mix_ratio(T, p):
    e_s = 611.2 * np.exp(17.67 * (T - 273.15) / (T - 29.65))
    return eps * e_s / (p - e_s)

def moist_lapse(T, p):
    rs = sat_mix_ratio(T, p)
    num = 1 + L_v * rs / (R_d * T)
    den = 1 + L_v**2 * rs / (c_p * R_v * T**2)
    return Gamma_d * num / den

z = np.linspace(0, 12, 200)  # km
p_profile = p0 * np.exp(-z * 1000 / 8500)

# Environmental lapse rates to compare
lapse_rates = {'Superadiabatic (11 K/km)': 11.0, 'Dry adiabatic (9.8 K/km)': Gamma_d,
               'Standard (6.5 K/km)': 6.5, 'Inversion (stable, 3 K/km)': 3.0}
colors_lr = ['#ef4444', '#f97316', '#fbbf24', '#38bdf8']

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(13, 7))
fig.patch.set_facecolor('#0f172a')
for ax in (ax1, ax2):
    ax.set_facecolor('#1e293b'); ax.tick_params(colors='#cbd5e1')
    ax.xaxis.label.set_color('#cbd5e1'); ax.yaxis.label.set_color('#cbd5e1')
    ax.title.set_color('#cbd5e1'); ax.grid(True, color='#334155', alpha=0.5)
    for sp in ax.spines.values(): sp.set_color('#334155')

# Left: Temperature profiles and potential temperature
T_sfc = 300.0
for (name, lr), col in zip(lapse_rates.items(), colors_lr):
    T_profile = T_sfc - lr * z
    T_profile = np.maximum(T_profile, 180)
    theta = T_profile * (p0 / p_profile) ** kappa
    ax1.plot(T_profile - 273.15, z, color=col, lw=2, label=name)

# Add moist adiabatic profile
T_moist = np.zeros_like(z)
T_moist[0] = T_sfc
for i in range(1, len(z)):
    dz = (z[i] - z[i-1])
    Gm = moist_lapse(T_moist[i-1], p_profile[i-1])
    T_moist[i] = T_moist[i-1] - Gm * dz
ax1.plot(T_moist - 273.15, z, color='#a78bfa', lw=2.5, ls='--', label='Moist adiabatic (~5 K/km)')

ax1.set_xlabel('Temperature (\u00b0C)', fontsize=11); ax1.set_ylabel('Altitude (km)', fontsize=11)
ax1.set_title('Temperature Profiles by Lapse Rate', fontsize=13, fontweight='bold')
ax1.legend(loc='upper right', facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=8)

# Right: Potential temperature for each profile
for (name, lr), col in zip(lapse_rates.items(), colors_lr):
    T_profile = np.maximum(T_sfc - lr * z, 180)
    theta = T_profile * (p0 / p_profile) ** kappa
    ax2.plot(theta, z, color=col, lw=2, label=name)

ax2.axvline(T_sfc, color='#475569', ls=':', lw=1)
ax2.text(T_sfc + 1, 11, '\u03b8 = T_sfc', color='#94a3b8', fontsize=9)
ax2.set_xlabel('Potential Temperature \u03b8 (K)', fontsize=11); ax2.set_ylabel('Altitude (km)', fontsize=11)
ax2.set_title('Potential Temperature \u03b8 = T(p\u2080/p)^\u03ba', fontsize=13, fontweight='bold')
ax2.legend(loc='upper left', facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=8)

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor=fig.get_facecolor())
print("=== Potential Temperature & Lapse Rate Analysis ===")
print(f"  R_d/c_p (kappa) = {kappa:.4f}")
print(f"  Dry adiabatic lapse rate = {Gamma_d:.2f} K/km")
print(f"\nPotential temperature at z=5 km for each profile:")
i5 = np.argmin(np.abs(z - 5))
for (name, lr), col in zip(lapse_rates.items(), colors_lr):
    T5 = max(T_sfc - lr * 5, 180)
    th5 = T5 * (p0 / p_profile[i5]) ** kappa
    print(f"  {name}: T={T5-273.15:.1f} \u00b0C, \u03b8={th5:.1f} K")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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