Mathematical models for atmospheric pressure distribution and climate sensitivity derived from first principles in Mathematical Geoenergy (GEM), Chapter 17 (Solar Energy: Thermodynamic Balance), Sections 17.1 and 17.3.
This module covers two derivations from GEM Chapter 17 that together explain the basic thermodynamic structure of Earth’s atmosphere.
The atmosphere in hydrostatic equilibrium is the canonical textbook application of the Principle of Maximum Entropy (MaxEnt). Given only the mean thermal energy $\langle E\rangle = RT$ of an air molecule, MaxEnt uniquely selects the exponential (Boltzmann) energy distribution:
\[p(E) = \frac{1}{\langle E\rangle}\cdot e^{-E/\langle E\rangle}\]Identifying the gravitational potential energy $E_\text{grav} = Mgz$ as the relevant energy variable and equating $\langle E\rangle = RT$ (ideal gas thermal energy per mole) yields the barometric (hypsometric) formula for pressure as a function of altitude:
\[P(z) = P_0\cdot\exp\left(-\frac{Mgz}{RT}\right)\]The characteristic decay length — the scale height — is $H_s = RT/(Mg) \approx 8.5$ km for Earth’s standard atmosphere.
Earth’s present mean surface temperature of $\approx 289$ K is 34°C warmer than the $\approx 255$ K it would be without greenhouse gases. This section derives the 33°C discrepancy analytically using two coupled equations:
Log-sensitivity (standard radiative forcing law):
\[T = T_0 + \alpha\ln\left(\frac{C}{C_0}\right)\]Clausius-Clapeyron activation (water-vapor outgassing as a Boltzmann factor with heat of vaporisation $H = 0.42\;\text{eV}/k_B \approx 4873$ K):
\[\frac{C}{C_0} = \beta\cdot e^{-H/T}\]Substituting (2) into (1) and using $\ln(e^x)=x$ yields a quadratic in $T$:
\[T^2 - \gamma T + \alpha H = 0, \qquad \gamma \equiv T_0 + \alpha\ln\beta\]whose two roots correspond to the cold no-GHG equilibrium ($T_\text{low} \approx 255$ K) and the warm present-day equilibrium ($T_\text{high} \approx 289$ K). Applying Vieta’s formulas ($T_\text{low}+T_\text{high}=\gamma$, $T_\text{low}\cdot T_\text{high}=\alpha H$) gives the climate sensitivity:
\[\alpha = \frac{\gamma^2 - 4\Delta T^2}{4H} \approx 15.1\\;^\circ\text{C per ln-unit}\]where $\Delta T = (T_\text{high}-T_\text{low})/2 = 17$ K.
The differential Clausius-Clapeyron relation then quantifies the water-vapor feedback triggered by a CO₂-induced warming $\Delta T_{\mathrm{CO_2}}$:
\[\frac{dC}{C} = \frac{H}{T^2}\\,dT \quad\Longrightarrow\quad \alpha\ln\left(1+\frac{H}{T^2}\Delta T_{\mathrm{CO_2}}\right) \approx 1.05\\;^\circ\text{C}\]Adding the direct CO₂ contribution (1.23°C), the water-vapor feedback (1.05°C), and other feedbacks (∼0.72°C) recovers the consensus 3°C climate sensitivity for a doubling of CO₂.
Eq. 17-1 — MaxEnt Boltzmann energy distribution
\[p(E) = \frac{1}{\langle E\rangle}\cdot e^{-E/\langle E\rangle}, \quad E \geq 0\]Applying the Principle of Maximum Entropy with only a mean-energy constraint yields the unique exponential (Gibbs–Boltzmann) distribution. The mean energy per mole at temperature $T$ is $\langle E \rangle = RT$.
Eq. 17-2 — Barometric pressure formula
\[P(z) = P_0\cdot\exp\left(-\frac{Mgz}{RT}\right)\]| Symbol | Meaning | Value (std. atm.) |
|---|---|---|
| $P_0$ | Surface pressure | 101 325 Pa |
| $M$ | Mean molar mass of air | 0.029 kg/mol |
| $g$ | Gravitational acceleration | 9.81 m/s² |
| $R$ | Ideal gas constant | 8.314 J/(mol·K) |
| $T$ | Temperature | 288 K |
| $H_s = RT/(Mg)$ | Scale height | ≈ 8417 m |
Scale height:
\[H_s = \frac{RT}{Mg} \approx 8.5\\;\text{km}\]Eq. 17-3 — Log-sensitivity to GHG concentration
\[T = T_0 + \alpha\ln\left(\frac{C}{C_0}\right)\]$\alpha$ is the climate sensitivity in °C per e-folding of concentration. For a doubling $C/C_0 = 2$: $\Delta T = \alpha\ln 2$.
Eq. 17-4 — Clausius-Clapeyron activation relation
\[\frac{C}{C_0} = \beta\cdot e^{-H/T}\] \[H = \frac{0.42\\;\text{eV}}{k_B} \approx 4873\\;\text{K}\]The water-vapor partial pressure is activated by a Boltzmann factor with the heat of vaporisation as the activation energy.
Eq. 17-5 — Quadratic reduction (substituting Eq. 17-4 into Eq. 17-3)
\[T^2 - \gamma T + \alpha H = 0, \qquad \gamma \equiv T_0 + \alpha\ln\beta\]Eq. 17-6 — Two equilibrium temperature roots
\[T = \frac{\gamma \pm \sqrt{\gamma^2 - 4\alpha H}}{2}\]The two roots correspond to the cold ($T_\text{low} \approx 255$ K) and warm ($T_\text{high} \approx 289$ K) stable equilibria. By Vieta’s formulas:
\[T_\text{low} + T_\text{high} = \gamma, \qquad T_\text{low}\cdot T_\text{high} = \alpha H\]Eq. 17-7 — Climate sensitivity from observed temperatures
\[\alpha = \frac{\gamma^2 - 4\Delta T^2}{4H} \approx 15.1\\;^\circ\text{C}\]where $\Delta T = (T_\text{high} - T_\text{low})/2 = 17$ K.
Doubling sensitivity:
\[\alpha\ln 2 \approx 10.5\\;^\circ\text{C}\]Eq. 17-8 — Differential Clausius-Clapeyron and CO₂ feedback
\[\frac{dC}{C} = \frac{H}{T^2}\\,dT\]For CO₂ doubling ($\Delta T_{\mathrm{CO_2}} = 1.23$ °C, $T = 289$ K):
\[\frac{dC}{C} = \frac{4873}{289^2}\times 1.23 \approx 7.2\%\] \[\alpha\ln\left(1 + \frac{dC}{C}\right) \approx 1.05\\;^\circ\text{C}\]| File | Purpose |
|---|---|
atmos_thermo_symbolic.py |
Symbolic SymPy derivation of all equations (Secs. 17.1 & 17.3) |
atmos_thermo_numerical.py |
Numerical implementation, validation, and composite figure |
atmos_thermo_model_output.png |
Output figure (6 panels) generated by atmos_thermo_numerical.py |
Install dependencies (from models/requirements.txt):
pip install -r ../requirements.txt
Run symbolic derivation (all assertions print ✓):
python atmos_thermo_symbolic.py
Run numerical model and generate figure:
MPLBACKEND=Agg python atmos_thermo_numerical.py
See also set_point_co2.py for a plot.
MaxEnt is all you need for the barometric formula: the exponential decrease of atmospheric pressure with altitude is not an empirical fit — it is the unique consequence of applying the Principle of Maximum Entropy to a gas in a gravitational field with a known mean thermal energy.
Earth is not an ice-box because of a quadratic feedback: coupling the log-sensitivity law for GHG forcing with the Clausius-Clapeyron activation produces a quadratic equation in temperature. Its two roots define a cold ($\approx 255$ K) and a warm ($\approx 289$ K) stable state separated by exactly 34°C — the observed discrepancy.
Climate sensitivity is fixed by two observables: given the cold and warm equilibrium temperatures and the activation energy $H$, the sensitivity $\alpha \approx 15.1$ °C per ln-unit is uniquely determined — no free parameters.
Water-vapor is the amplifier, CO₂ is the control knob: the differential Clausius-Clapeyron shows that a 1.23°C CO₂-induced warming drives a 7.2% rise in water vapor, which in turn raises the temperature by a further 1.05°C — consistent with IPCC consensus estimates.
Universal exponential structure: the same Boltzmann factor $e^{-E/\langle E\rangle}$ that governs barometric pressure also governs Clausius-Clapeyron outgassing, illustrating how a single MaxEnt principle underpins both atmospheric structure and climate feedback.