Energy in, energy out

The conservation of energy says that energy cannot be created or destroyed; it can only be transferred between stores or carried from one place to another. In climate physics, we often draw the boundary around the whole Earth–atmosphere system. Solar radiation passes in through that boundary. Reflected solar radiation and infrared radiation pass out.

A planet’s long-term average temperature depends on this energy balance. In the simplest model,

[I{ ext{in}}=I{ ext{out}}]

where I{ ext{in}} is the mean incoming radiation intensity crossing into the system (W m⁻²), and I{ ext{out}} is the mean outgoing radiation intensity crossing out of the system (W m⁻²). If more energy enters than leaves, the system warms. If more leaves than enters, it cools. “Mean” is carrying a lot here: day/night, seasons, clouds, oceans and winds all get averaged over.

Dynamic equilibrium, not “nothing happening”

A dynamic equilibrium is a state of a system in which opposing energy transfers continue, but their average rates are equal, so there is no net long-term change in the system’s energy. Earth’s climate only fits this idea approximately. Even a small imbalance, less than a few watts per square metre, matters because it acts across the entire surface of the planet for many years.

For IB calculations, be honest about the model: write down what radiation is absorbed, what is reflected, what is emitted, then equate the relevant intensities. The physics is energy conservation. The tricky part is choosing the correct area and the correct fraction.

Grey bodies

A black body is an idealized object that absorbs all incident electromagnetic radiation and emits the maximum possible thermal radiation at a given temperature. A grey body is a model surface that emits a fixed fraction of the radiation emitted by a black body with the same area and temperature.

Emissivity is a dimensionless ratio. It compares the power radiated per unit area by a surface with the power radiated per unit area by an ideal black surface at the same temperature:

[\varepsilon=\frac{P/A}{\sigmaT^4}]

where \varepsilon is emissivity (no unit), P is the radiated power (W), A is the radiating surface area (m²), \sigma is the Stefan–Boltzmann constant (5.67 × 10⁻⁸ W m⁻² K⁻⁴) and T is thermodynamic temperature (K). Rearranging gives the grey-body form of the Stefan–Boltzmann law:

[P=\varepsilon A\sigma T^4]

Don’t treat emissivity as the same thing as “how dark it looks” in visible light. It depends on wavelength. Snow is bright in visible light but can be a good emitter and absorber in the infrared; polished metal is often a poor infrared emitter.

Net radiative exchange

A body in warm surroundings emits thermal radiation and absorbs it too. If the same emissivity is used for both directions, the net radiative power transfer can be modelled as

[P{ ext{net}}=\varepsilon A\sigma(T{ ext{h}}^4-T_{ ext{c}}^4)]

where P{ ext{net}} is the net power transferred from the hotter body to the cooler body by radiation (W), T{ ext{h}} is the temperature of the hotter surface (K) and T_{ ext{c}} is the temperature of the cooler surroundings or surface (K). Because the temperatures are raised to the fourth power, radiative balance is very sensitive to temperature.

Reflection from a macroscopic system

Albedo is a dimensionless ratio: it tells us what fraction of the incident radiation a macroscopic system scatters or reflects.

[a=\frac{P{ ext{scat}}}{P{ ext{inc}}}]

Here, a is albedo (no unit), P{ ext{scat}} is the total scattered or reflected power (W), and P{ ext{inc}} is the total incident power (W). If the albedo is 0, there is no scattering; if it is 1, all incident radiation is scattered.

So the absorbed fraction is (1-a). For radiation with incident intensity I_{ ext{inc}}, the absorbed intensity is

[I{ ext{abs}}=(1-a)I{ ext{inc}}]

where I{ ext{abs}} is the absorbed radiation intensity (W m⁻²), and I{ ext{inc}} is the incident radiation intensity (W m⁻²).

Typical albedo values show dark surfaces absorb most incoming radiation, while snow and ice reflect much more.

Albedo is not emissivity

Albedo deals with incoming radiation. Emissivity deals with how effectively a surface emits thermal radiation compared with a black body. The two are related only in special cases, and both depend on wavelength. This is why Earth can reflect some incoming visible sunlight and still emit infrared radiation from its surface.

Why Earth does not have one fixed albedo

Earth’s average albedo is about 0.30, but that value is a global, annual mean. It shifts from day to day as cloud cover changes. It also depends on latitude, since the Sun’s angle changes and polar regions have more ice and snow. The surface matters as well: ocean, forest, desert, city surfaces, sea ice and fresh snow all scatter different fractions of incoming sunlight.

Clouds play a big part. Thick bright clouds can reflect a large fraction of sunlight back to space. Thinner clouds have a smaller effect, and they may also trap outgoing infrared radiation. So “more cloud” doesn’t give the climate system one simple instruction.

There is a useful feedback here. If warming melts snow or ice, a high-albedo surface is replaced by a lower-albedo surface such as ocean or rock. More solar radiation is then absorbed, which can drive further warming. This is one reason polar regions are so sensitive in climate models.

Radiation from the Sun at Earth’s orbit

Intensity is the incident power per unit area, measured perpendicular to the direction of the radiation:

[I=\frac{P_{ ext{inc}}}{A}]

where I is radiation intensity (W m⁻²). The solar constant is the intensity of solar radiation, across all wavelengths, arriving at Earth’s mean distance from the Sun on a plane perpendicular to the Sun’s rays. It has the symbol S. Near Earth, its accepted value is about 1.36 × 10³ W m⁻².

You can estimate the solar constant by imagining the Sun’s power output spread over a sphere centred on the Sun:

[S=\frac{L_{\odot}}{4\pir^2}]

where S is the solar constant (W m⁻²), L_{\odot} is the luminosity of the Sun, meaning its total radiated power (W), and r is the Earth–Sun distance (m). This gives an inverse-square relationship: if the distance doubles, the intensity drops to one quarter.

The solar constant isn’t exactly constant. It varies slightly with the Sun’s activity cycle, and it changes more through the year because Earth’s orbit is elliptical. Earth is a little closer to the Sun in January than in July; this changes the solar intensity received, but it does not cause the seasons. Seasons mainly come from the tilt of Earth’s rotation axis.

Why the average is S/4, not S

A planet catches sunlight as a flat disc would. It does not receive sunlight across its entire spherical surface at the same time. The solar power intercepted by a spherical planet is

[P_{ ext{in}}=S\piR^2]

where P_{ ext{in}} is the solar power intercepted by the planet before reflection (W) and R is the planet’s radius (m). Spread that power over the full spherical surface area (4\piR^2), and the mean incoming solar intensity becomes

[\bar{I}_{ ext{in}}=\frac{S\piR^2}{4\piR^2}=\frac{S}{4}]

where (\bar{I}_{ ext{in}}) is the mean incoming solar intensity over the planet’s whole surface (W m⁻²). For Earth, this gives about (1360/4), or 340 W m⁻², before reflection is included.

Including albedo, the mean absorbed solar intensity is

[I_{ ext{solar,abs}}=(1-a)\frac{S}{4}]

where I_{ ext{solar,abs}} is the mean absorbed solar intensity (W m⁻²). Using Earth’s average albedo of about 0.30 gives roughly 238 W m⁻².

Estimating equilibrium temperature

For a body with no greenhouse atmosphere, find equilibrium by setting absorbed solar intensity equal to emitted thermal intensity:

[\varepsilon\sigma T_{ ext{eq}}^4=(1-a)\frac{S}{4}]

where T_{ ext{eq}} is the equilibrium temperature (K). This is the basic IB calculation: identify the incoming radiation, subtract the reflected fraction, and equate what is left to the outgoing thermal radiation. The same method works for planets, moons, asteroids, or any body heated by a radiation source if the appropriate solar or stellar constant is supplied.

There is a clear limit to the model. It treats the body as if its surface radiates directly to space. Earth’s real atmosphere absorbs and re-emits infrared radiation, so a surface–atmosphere exchange model is needed for a better estimate.

The small atmospheric fraction that matters

A greenhouse gas is a gas in the atmosphere whose molecules absorb infrared radiation emitted by a planet’s surface, then re-emit radiation so that some energy returns toward the surface. The main greenhouse gases needed here are methane CH₄, water vapour H₂O, carbon dioxide CO₂ and nitrous oxide N₂O.

Nitrogen N₂ and oxygen O₂ make up most of the atmosphere, but they are not the main greenhouse gases. Their simple symmetrical molecules do not interact strongly enough with infrared radiation to drive the greenhouse effect. The trace gases do that absorbing.

Water vapour needs a careful distinction. Human activity does not usually add water vapour as the primary driver in the same way it adds CO₂ or CH₄. Instead, warming caused by other greenhouse gases lets more water vapour stay in the atmosphere, and that strengthens the warming.

From visible sunlight to infrared emission

Most of the solar radiation reaching the ground lies in the visible and near-infrared parts of the spectrum. Earth’s surface is far cooler than the Sun, so the radiation it gives off is thermal radiation mainly at longer infrared wavelengths. That wavelength shift sits at the centre of the greenhouse effect.

A molecular energy level is an allowed energy state of a molecule associated with a particular pattern of molecular motion, such as rotation or vibration. An infrared photon can be absorbed if its energy matches the separation between allowed molecular energy levels. The molecule doesn’t store that energy forever. It passes energy on through collisions and by re-emitting infrared radiation.

The key directional idea is simple: re-emission happens in all directions. Some emitted infrared radiation travels upward and can escape to space. Some travels sideways. Some travels downward and is absorbed by the surface. That downward part keeps the surface warmer than it would be without the greenhouse gases.

A transmittance is a fraction or percentage that describes how much radiation at a given wavelength passes through a material or layer. In some wavelength ranges, Earth’s atmosphere has high transmittance; in others, it has low transmittance. The infrared “windows” let some surface radiation escape directly, while absorption bands due to H₂O, CO₂, CH₄ and N₂O reduce escape at other wavelengths.

The resonance model

Resonance is a response in which an oscillating system absorbs energy efficiently from a periodic driver when the driving frequency matches one of the system’s natural frequencies. A greenhouse-gas molecule can be pictured, roughly, as atoms connected by spring-like bonds. Infrared electromagnetic radiation supplies an oscillating electric field, which can drive molecular vibrations.

This links simple harmonic motion to climate change. In SHM, a driver transfers energy most effectively when it matches a natural oscillation frequency. In the atmosphere, some molecular vibrations have natural frequencies in the infrared region. If outgoing infrared radiation from Earth has those matching frequencies, greenhouse-gas molecules absorb it strongly.

Carbon dioxide is the usual example. CO₂ has vibrational modes including symmetric stretching, anti-symmetric stretching and bending. The symmetric stretch does not produce strong infrared absorption, because it does not create the required changing unevenness in charge distribution. The bending and anti-symmetric stretching modes do, so they can interact strongly with infrared radiation.

The energy-level model

The molecular energy-level model gives the same idea in quantum terms. Molecules cannot absorb just any amount of vibrational energy; they absorb photons whose energies correspond to allowed changes between molecular energy levels. After absorption, the energy may be shared in collisions or re-emitted as infrared radiation in random directions.

Limits of the resonance picture

The resonance model helps, but it is not the full explanation. Real molecules have quantized energy levels rather than a continuous set of classical spring motions. Absorption bands also have widths because of molecular collisions, pressure effects, temperature and molecular rotation-vibration structure. Above all, not every vibration absorbs infrared radiation: the vibration must involve a changing electric dipole. That is why abundant gases such as N₂ and O₂ contribute little to the greenhouse effect even though molecules can vibrate.