Energy in, energy out

The conservation of energy principle says that energy cannot be created or destroyed, only transferred between stores or carried from place to place. In climate physics, we often draw the boundary around the whole Earth–atmosphere system. Solar radiation crosses that boundary inwards; reflected solar radiation and infrared radiation cross it outwards.

A planet’s long-term average temperature depends on its energy balance. The simplest version is

If more energy enters than leaves, the system warms. If more leaves than enters, it cools. “Mean” matters here: day/night, seasons, clouds, oceans and winds are all being 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 approximately. Even a small imbalance, less than a few watts per square metre, matters because it acts over the entire surface of the planet and over many years.

For IB calculations, be clear about the model: write down what radiation is absorbed, what is reflected, what is emitted, and 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, at a given temperature, emits the maximum possible thermal radiation. 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:

Rearranging gives the grey-body form of the Stefan–Boltzmann law:

$P = \varepsilon A\sigma T^4$

Don’t treat emissivity 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 in both directions, the net radiative power transfer can be modelled as

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 measuring the fraction of incident radiation that a macroscopic system scatters or reflects:

If the albedo is 0, there is no scattering; if the albedo is 1, all incident radiation is scattered.

So the absorbed fraction is $1-a$. For radiation with incident intensity $I_{\text{inc}}$, the absorbed intensity is

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 tells us 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. That’s why Earth can reflect some incoming visible sunlight while still emitting 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 changes with latitude, partly because the Sun’s angle is different and partly because polar regions have more ice and snow. Surface type matters as well: ocean, forest, desert, city surfaces, sea ice and fresh snow all scatter different fractions of incoming sunlight.

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

A useful feedback comes from this. When warming melts snow or ice, it replaces a high-albedo surface with a lower-albedo surface such as ocean or rock. The surface then absorbs more solar radiation, which can lead to 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 received per unit area, measured perpendicular to the direction of the radiation:

The solar constant is the intensity of solar radiation, across all wavelengths, incident at Earth’s mean distance from the Sun on a plane perpendicular to the Sun’s rays. It uses the symbol $S$, and near Earth its accepted value is about $1.36 \times 10^3\ \mathrm{W\,m^{-2}}$.

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

This gives an inverse-square relationship: double the distance and the intensity falls to one quarter.

The solar constant is not perfectly constant. It shifts slightly with the Sun’s activity cycle, and more noticeably through the year because Earth’s orbit is elliptical. Earth sits a little closer to the Sun in January than in July; that changes the solar intensity received, but it does not cause the seasons. Seasons come mainly 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, rather than across its entire spherical surface at the same time. The solar power intercepted by a spherical planet is

Spread that power over the planet’s full spherical surface area, $4\pi R^2$, and the mean incoming solar intensity becomes

For Earth, before reflection is included, this is about $1360/4$, or 340 $\text{W m}^{-2}$.

Including albedo gives the mean absorbed solar intensity as

Using Earth’s average albedo of about 0.30 gives roughly 238 $\text{W m}^{-2}$.

Estimating equilibrium temperature

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

In the basic IB calculation, you identify the incoming radiation, subtract the reflected fraction, then equate what remains 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’s a 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 and then re-emit radiation, so some energy returns toward the surface. The main greenhouse gases needed here are methane $CH_4$, water vapour $H_2O$, carbon dioxide $CO_2$ and nitrous oxide $N_2O$.

Nitrogen $N_2$ and oxygen $O_2$ make up most of the atmosphere, but they are not the main greenhouse gases. Their simple symmetrical molecules do not interact strongly with infrared radiation in the way needed for the greenhouse effect. The trace gases do the important absorbing.

Water vapour needs a careful wording. Human activity does not usually add water vapour as the primary driver in the same way it adds $CO_2$ or $CH_4$. Instead, warming caused by other greenhouse gases lets more water vapour remain in the atmosphere, and that strengthens the warming.

From visible sunlight to infrared emission

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

A molecular energy level is an allowed energy state of a molecule linked to a particular pattern of molecular motion, such as rotation or vibration. An infrared photon can be absorbed if its energy matches the gap between allowed molecular energy levels. The molecule doesn’t hold on to 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 moves sideways. Some travels downward and is absorbed by the surface. That downward part is why the surface is warmer than it would be without the greenhouse gases.

A transmittance is a fraction or percentage showing 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. Infrared “windows” let some surface radiation escape directly, while absorption bands due to $H_2O$, $CO_2$, $CH_4$ and $N_2O$ reduce escape at other wavelengths.

The resonance model

Resonance is a response where 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 vibrations in the molecule.

This links simple harmonic motion to climate change. In SHM, a driver transfers energy most effectively when its frequency 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 frequencies, greenhouse-gas molecules absorb it strongly.

Carbon dioxide is the usual example. $CO_2$ 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 explains the same idea using quantum language. Molecules cannot absorb just any amount of vibrational energy; they absorb photons whose energies match allowed changes between molecular energy levels. After absorption, collisions may share the energy, or the molecule may re-emit it 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. Most importantly, not every vibration absorbs infrared radiation: the vibration must involve a changing electric dipole. That is why abundant gases such as $N_2$ and $O_2$ contribute little to the greenhouse effect even though molecules can vibrate.

Natural greenhouse effect and enhanced greenhouse effect

The greenhouse effect warms a planet when greenhouse gases absorb some of the infrared radiation emitted by its surface, then re-emit radiation in all directions. That raises the average surface temperature. Earth depends on the natural greenhouse effect; without it, the average surface temperature would be far lower and liquid water would be much less widespread.

The enhanced greenhouse effect is the human-caused strengthening of the greenhouse effect, caused by increased concentrations of greenhouse gases. Burning fossil fuels is a main cause because it moves carbon stored underground into atmospheric $CO_2$. Agriculture, land-use change and some industrial processes also raise greenhouse-gas concentrations.

Pre-industrial and recent atmospheric concentrations of key greenhouse gases.

A simple surface–atmosphere exchange model

One compact model treats the atmosphere as returning a fraction of the infrared radiation emitted by the surface. A simple balance is

In this model, a higher greenhouse-gas concentration increases $k$. With the same absorbed solar input, balance is restored only if $T_{\text{s}}$ rises enough for sufficient radiation to escape to space.

This is why energy-balance questions may include energy exchanged between the surface and the atmosphere. The surface is not just radiating into empty space; it receives solar radiation and downward infrared radiation, while losing infrared radiation upward and transferring energy by other processes such as convection and evaporation. In the exam, the diagram will usually be simplified, but the accounting still has to obey conservation of energy.

Feedbacks and human choices

Warming can set off feedbacks. When ice melts, albedo falls, so more solar radiation is absorbed. Warmer oceans hold less dissolved $CO_2$, leaving more $CO_2$ in the atmosphere. More water vapour can strengthen warming as well, because $H_2O$ is itself a greenhouse gas.

Different ways of generating electricity affect the atmospheric energy balance through their greenhouse-gas emissions. Fossil-fuel power stations release $CO_2$ during operation, which strengthens infrared absorption. Nuclear, wind, hydroelectric and solar generation do not release $CO_2$ in the same direct large-scale way during operation, although construction, mining and transport still have life-cycle impacts. The physics link is direct: methods that add greenhouse gases tend to increase atmospheric infrared absorption and therefore the value of $k$ in the simple model.

Climate change also drives science and technology. It pushes better climate models, satellite monitoring, low-carbon electricity, energy storage, electric transport, improved building insulation, carbon capture and research into alternative energy sources. This is a good example of physics feeding into public decisions: the equations do not make the policy, but they do constrain what is physically possible.