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Energy Balance Models

Table of Contents

1. Overview

Energy balance models appear to balance the incoming and outgoing radiation at the Earth's surface. The simplest model is "zero-dimensional" without an atmosphere, which means we just model the Earth as a point particle. We can refine this by adding an atmosphere to the zero-dimensional model, then add several atmosphere layers. Or we could make the model one-dimensional by having the amount of radiation balanced depend on the latitude alone.

2. Zero Dimensional Model

The basic model is to use the Stefan-Boltzmann law to describe how the Earth radiates off its energy balance, while the incoming radiation is described using the solar constant. We expect to recover the effective temperature of the Earth. But these contributions give us

\begin{equation} C\frac{\D T_{s}}{\D t} = \frac{(1-\alpha)S}{4} - \varepsilon\sigma T_{s}^{4} \end{equation}

where:

  • \(T_{s}\) is the surface temperature of the Earth
  • \(\varepsilon\) is the emissivity of the Earth
  • \(C\approx 2.08\times 10^{8}J\,K^{-1}m^{-2}\) is the effective heat capacity of the Earth
  • \(\sigma = 5.67\times 10^{-8}W\,m^{-2}\,K^{-4}\) is the Stefan–Boltzmann constant
  • \(\alpha\) is the Earth's albedo
  • \(S\approx 1370 \,W\,m^{-2}\) is the solar constant

For equilibrium, the time-derivative of temperature vanishes. So now we're just trying to solve the equation:

\begin{equation} \frac{(1-\alpha)S}{4} = \varepsilon\sigma T_{s}^{4}. \end{equation}

2.1. Emissivity vs Temperature

If we keep the quartic equation, then we can create a table for various values of emissivity (a parameter between 0 and 1). 

(defun full-model (emissivity)
  (let ((solar-constant 1361.1972737230658)
        (stefan-boltzmann 5.67037441918443e-008)
        (albedo 0.32))
    (expt (/ (* 0.25 (- 1 albedo) solar-constant)
             (* emissivity stefan-boltzmann))
          0.25)))

(cons '(emissivity temperature)
(mapcar (lambda (emissivity)
          (list emissivity (full-model emissivity)))
        '(0.01 0.25 0.5 0.75 0.8 0.85 0.9 0.95 0.9655 1)))
emissivity temperature
0.01 799.2627332202043
0.25 357.4411606725312
0.5 300.5709906736585
0.75 271.5965493997138
0.8 267.249601984984
0.85 263.229665812042
0.9 259.4949655115338
0.95 256.0110215222003
0.9655 254.9772860954031
1 252.7490685867371

2.2. Linearization

Usually the quartic temperature expression is linearized. After all, the Earth's surface temperature varies at most by 30 degrees about its long-term average (of 273 K). So we make the switcheroo:

\begin{equation} \varepsilon\sigma T_{s}^{4} = A + BT_{s} \end{equation}

where \(A=315 W\,m^{-2}\) and \(B=4.6\,W\,m^{-2}K^{-1}\) are empirically measured constants. Then we can solve

\begin{equation} T_{s} = \left(\frac{S(1-\alpha)}{4}-A\right)\frac{1}{B} \approx 255\,K. \end{equation}

The exact value may be determined: 

(defun celsius->kelvin (temp)
  (+ temp 273.15))

(defun linearized-solution ()
  (let ((solar-constant 1361.1972737230658)
        (stefan-boltzmann 5.67037441918443e-008)
        (albedo 0.32)
        (a 315.0)
        (b 4.6))
    (celsius->kelvin
     (/ (- (* 0.25 (- 1 albedo) solar-constant) a)
        b))))

(linearized-solution)

254.97685576802633

We can estimate the emissivity for this value of the Earth's surface temperature: 

(defun equilibrium-emissivity (temperature)
  (let ((solar-constant 1361.1972737230658)
        (stefan-boltzmann 5.67037441918443e-008)
        (albedo 0.32))
    (/ (* 0.25 (- 1 albedo) solar-constant)
       (* (expt temperature 4) stefan-boltzmann))
       ))

(equilibrium-emissivity (linearized-solution))

0.9655065179583904

2.3. Evaluating Results

We find the linearized approximation yield results similar to the quartic equation, which is good. Further the effective temperature of the surface of the Earth \(T_{s}\approx 255\,K\) is less than the effective temperature we estimated using only the Stefan–Boltzmann law, but that should be expected because we're now also modeling the fact the Earth radiates off heat. And again we've yet to include any modeling of greenhouse gases.

3. References

Last Updated 2021-06-01 Tue 10:00.