"Building a planetary model from the ground up." — Jan 19 & 20 Lectures
As developed in our recent lectures, we build this model by applying conservation laws to Energy ($E$). Every climate model, regardless of complexity, is built on four core pillars:
Primarily Temperature ($T$). In this 0D setup, $T$ serves as both the measurable state we input to find parameters and the output we solve for.
Conservation of Energy: The time-rate of change $\frac{dE}{dt}$ equals Energy In minus Energy Out.
Changeable planetary properties like Albedo ($\alpha$) and the Solar Constant ($L_*$).
Universal, unchanging values like the Stefan–Boltzmann constant ($\sigma$).
The 0D Balance Condition
When applying $\sigma T^4$, we make a vital conceptual jump. We don't think about emission from individual gas particles. Instead, we bridge microscopic behavior with macroscopic rules using Local Thermodynamic Equilibrium (LTE). This allows us to treat the atmosphere as a continuum—a blackbody at temperature $T$—where radiation depends only on $T$.
How do we map our abstract "isothermal dot" $T$ to a real, 3D planet? This is the Data Assimilation problem.
A. Spatial Weighting (The Grid Problem)
Satellite data is usually provided on equiangular grids ($1^\circ \times 1^\circ$). However, a $1^\circ$ "box" at the Equator covers far more surface area than a $1^\circ$ "box" near the Pole. To infer a single global $T$, we must apply Cosine Weighting to account for the true geometry of the sphere.
B. Mass & Pressure Weighting
Temperature is a proxy for Average Kinetic Energy. Since 90% of Earth's air mass is in the lower atmosphere, we must use Pressure Weighting ($dp/g$). A warming at the surface (high mass) contributes far more to the planetary energy stock than the same warming in the thin stratosphere.
Because $\sigma T^4$ is non-linear, the global average of $T^4$ is not the same as the global average of $T$ raised to the fourth power. To conserve energy, we must average fluxes first:
Modelers approximate the latitudinal temperature drop as: $T(\phi) = T_{eq} - \Delta T \sin^n(\phi)$.
Interpreting Profile Shapes
The primary goal is to validate the Top-of-Atmosphere (TOA) boundary. Since the solar and Earth spectra are distinct, we need three distinct sensors:
Sensor 1: Incoming Solar
Shortwave (< 4 μm). Measures $L_*$ directly from the Sun.
Sensor 2: Reflected Solar
Shortwave (< 4 μm). Measures reflected flux to compute Albedo ($\alpha$).
Sensor 3: Emitted Infrared
Longwave (> 4 μm). Measures OLR (Outgoing Longwave Radiation).
Imbalance detection: Once validated, these sensors reveal that Earth is currently trapping $\approx 3\text{ W/m}^2$. This "un-sustainable" energy storage is the core of global warming.
Where does the trapped 3W/m² go? Over 90% enters the oceans. Monitoring Ocean Heat Content (OHC) serves as a terrestrial, independent validation of our satellite measurements.
Our model predicts $T \approx 255\text{K}$. Using RRTMG (available in climt) allows us to locate the Effective Radiating Level in the vertical.
To find how temperature responds to a change in albedo, we differentiate our equilibrium equation with respect to $\alpha$:
$$\frac{d}{d\alpha}(\sigma T^4) = \frac{d}{d\alpha} \left[ \frac{L_*}{4}(1 - \alpha) \right]$$
Using the power rule on the left and evaluating the derivative on the right:
$$4\sigma T^3 \frac{dT}{d\alpha} = -\frac{L_*}{4}$$
Solving for $\frac{dT}{d\alpha}$ gives us the Base Planck Sensitivity ($\lambda_0$):
$$\lambda_0 = \frac{dT}{d\alpha} = - \frac{L_*}{16 \sigma T^3}$$
In nature, albedo ($\alpha$) is not an external parameter but a function of the state variable $T$. Melting ice reduces albedo, absorbing more energy—a classic positive feedback.
We start by introducing a small external radiative forcing $\Delta F$ to the equilibrium balance:
$$\sigma T^4 = \frac{L_*}{4} (1 - \alpha(T)) + \Delta F$$
Differentiating both sides with respect to $T$ to see how the system responds to this "push":
$$4\sigma T^3 \Delta T = -\frac{L_*}{4} \left( \frac{d\alpha}{dT} \right) \Delta T + \Delta F$$
Let $\gamma(T) = \frac{d\alpha}{dT}$. Grouping $\Delta T$ terms to solve for sensitivity $\lambda = \Delta T / \Delta F$:
$$\lambda = \frac{1}{4\sigma T^3 + \frac{L_*}{4} \gamma(T)}$$
To relate this to the Base Planck Sensitivity ($\lambda_0 = 1/4\sigma T^3$), we factor $4\sigma T^3$ out of the denominator:
$$\lambda = \frac{\lambda_0}{1 + \lambda_0 \left( \frac{L_*}{4} \gamma(T) \right)}$$
Finally, by defining the Feedback Gain Factor $g(T) = -\lambda_0 \frac{L_*}{4} \gamma(T)$, we obtain the universal feedback form:
$$\lambda = \frac{\lambda_0}{1 - g(T)}$$
The climate system is a web of interacting processes. To understand the total sensitivity of the planet, we expand our framework to include multiple independent feedbacks.
As established by the Clausius-Clapeyron relation, a warmer atmosphere holds more water vapor. Since water vapor is a potent greenhouse gas, it reduces the efficiency of the planet's cooling (decreases $OLR$).
When multiple processes (Ice-Albedo, Water Vapor, Clouds, Lapse Rate) all respond to temperature, their individual Gains ($g_i$) are additive in the denominator. The total climate sensitivity $\lambda_{total}$ becomes:
$$\lambda_{total} = \frac{\lambda_0}{1 - (g_{ice} + g_{wv} + g_{clouds} + ...)}$$
This formal framework allows us to study feedbacks in isolation and then sum their effects. It demonstrates why the system is so sensitive to small errors in measuring individual processes: if the sum of gains approaches 1, the planet enters a Runaway State.
A climate is stable only if $\sum g_i < 1$. On Earth, the Planck feedback ($g_{planck} = 0$) provides the fundamental restoring force, but positive gains from vapor and ice significantly amplify our sensitivity to forcing ($\Delta F$).
The feedback sum $\sum g_i$ determines if an exoplanet can maintain liquid water or if it is doomed to a runaway greenhouse (Venus) or runaway glaciation.
How did the early Earth stay warm when the Sun was 30% dimmer? Higher greenhouse forcing ($\Delta F$) and strong positive feedbacks ($g_{wv}$) were likely required.