[27] Energy Balance

What is the solar constant (total solar irradiance) at Earth's distance from the Sun?

About 1361 W/m² (sunlight on a surface facing the Sun at the top of the atmosphere).

What is the global, annual average incoming solar flux at the top of Earth's atmosphere (TOA)?

About 340 W/m², which is S/4: the solar constant (1361 W/m²) divided by 4.

What shape is used to model the area of Earth that intercepts incoming sunlight?

A disk of area πR² (Earth's cross-section), not the sphere.

Why is Earth's average incoming sunlight S/4 instead of S?

Earth intercepts sunlight over a disk of area πR² but spreads it over its whole spherical surface, 4πR².

The factor of 4 in S/4 combines which two factors of 2?

Half of Earth is in night at any moment (factor 2), and the day side receives sunlight at slanted angles (another factor 2).

In Earth's energy budget, what is the difference between 1361 W/m² and 340 W/m²?

1361 W/m² is sunlight on a surface facing the Sun (solar constant), and 340 W/m² is that spread over the whole globe (S/4).

What makes an object a blackbody, the ideal case behind the Stefan-Boltzmann and Wien laws?

It absorbs all radiation that hits it and emits the most radiation possible for its temperature (emissivity 1).

What is the Stefan-Boltzmann constant σ?

5.67 × 10⁻⁸ W m⁻² K⁻⁴.

What does the Stefan-Boltzmann law F = εσT⁴ give, and in what unit must T be?

The power emitted per square meter of surface (W/m²), with T in kelvin (ε is emissivity, 1 for a blackbody).

What is emissivity (ε) of a surface?

How efficiently it emits compared with a blackbody at the same temperature (0 to 1), so emission = εσT⁴.

By the Stefan-Boltzmann law, how much more power does a blackbody emit if its absolute temperature doubles?

16 times as much (2⁴), because emission grows with the fourth power of T.

What is Wien's displacement constant, used to find a blackbody's peak wavelength?

2898 µm·K (peak wavelength = 2898 µm·K / T).

In Earth's energy budget, which radiation comes mainly from the Sun and which from Earth: shortwave or longwave?

Shortwave comes from the Sun, and longwave (thermal infrared) is emitted by Earth's surface, atmosphere and clouds.

Why is incoming solar radiation called shortwave and Earth's outgoing radiation longwave?

By Wien's law the hot Sun (~5800 K) peaks near 0.5 µm (visible), while cool Earth (~255-288 K) peaks near 10 µm (thermal IR).

Roughly what wavelength separates solar shortwave from terrestrial longwave radiation?

About 4 µm (shortwave ~0.2-4 µm, longwave ~4-100 µm), and the two spectra barely overlap.

Which wavelength belongs to Earth's emitted longwave radiation: 0.5, 2, 30 or 200 µm?

30 µm (0.5 and 2 µm are solar shortwave, and 200 µm is far out in the tail with little energy).

True or false: at radiative equilibrium (energy in equals energy out), Earth emits at the same wavelengths at which it absorbs sunlight.

False. The energy totals match, but Earth absorbs shortwave and emits longwave.

How does the solar flux a planet receives depend on its distance d from its star?

It falls as 1/d² (inverse-square law), so doubling the distance cuts the flux to one quarter.

How do you get the flux at a planet's orbit from the flux at its star's surface?

Multiply by (R_star / d)², where R_star is the star's radius and d the orbital distance.

Which instrument measures Earth's broadband radiation budget (reflected sunlight and emitted infrared) at the top of the atmosphere?

CERES (on Aqua, Terra, Suomi NPP and NOAA-20).

What does CERES measure?

Top-of-atmosphere reflected shortwave flux, outgoing longwave flux and their net, i.e., Earth's radiation budget.

Albedo is the fraction of what?

Of incoming solar (shortwave) radiation that a surface or planet reflects, from 0 (absorbs all) to 1 (reflects all).

What is Earth's average planetary albedo (the whole planet's reflectivity, also called Bond albedo)?

About 0.30 (measured values are about 0.29-0.30).

What does Earth's albedo of 0.30 say about its shortwave budget?

About 30% of incoming sunlight is reflected to space and about 70% is absorbed.

About what percent of incoming sunlight do clouds reflect back to space globally?

About 20%, roughly two-thirds of Earth's total 30% reflection.

What is the typical albedo of fresh snow?

About 0.8-0.9, the highest of common natural surfaces.

What is the typical albedo of old or melting snow?

About 0.4-0.7, lower than fresh snow.

What is the typical albedo of sea ice?

About 0.5-0.7 (higher, ~0.8, when snow-covered).

What is the typical albedo of open ocean when the Sun is high?

About 0.06, the darkest common natural surface.

What is the typical albedo of a dense forest?

About 0.08-0.15 (conifers are darkest).

What is the typical albedo of grassland?

About 0.2-0.25.

What is the typical albedo of desert sand?

About 0.3-0.4.

What is the typical albedo of thick clouds?

About 0.6-0.9, depending on their thickness and droplet number.

What is the typical albedo of fresh asphalt?

About 0.05-0.10, nearly as dark as the ocean.

Which common surface has an albedo of about 0.06?

Open ocean (with the Sun high).

Which common surface has an albedo of about 0.8-0.9?

Fresh snow.

Which common land surface has an albedo of about 0.3-0.4?

Desert sand.

Which has the highest albedo: desert sand, water, ice, or fresh snow?

Fresh snow.

Rank by albedo, lowest to highest: fresh snow, desert sand, open ocean, forest canopy.

Open ocean < forest < desert sand < fresh snow.

Rank by albedo, lowest to highest: fresh snow, asphalt, grassland, desert sand, dense forest.

Asphalt < dense forest < grassland < desert sand < fresh snow.

Rank by maximum albedo, lowest to highest: sand, asphalt, grass, ice, bare soil, snow.

Asphalt < bare soil < grass < sand < ice < snow.

Why is the albedo of fresh snow so much higher than that of open ocean?

Snow's many tiny ice grains scatter visible light back out with little absorption, while water absorbs most light that enters it.

Why does the ocean's albedo rise when the Sun is low in the sky?

Reflection off the water surface (glint) increases sharply at grazing angles.

Name three things that lower snow's albedo over time.

Grain growth as snow ages, melting (wet snow, melt ponds), and dust or soot (black carbon) on the surface.

Which has the higher albedo, asphalt or concrete?

Concrete (~0.3), versus about 0.05-0.10 for fresh asphalt.

In daytime thermal infrared imagery, which has the higher brightness temperature (the temperature a sensor infers from emitted IR), asphalt or concrete, and why?

Asphalt, because its lower albedo means it absorbs more sunlight and gets hotter.

Why might a city have a higher albedo than a nearby forest?

Light materials (concrete ~0.3, light roofs) can be brighter than dark forest (~0.1), even though asphalt is darker.

Which has the highest albedo: clouds, concrete, water or grass?

Clouds (thick clouds ~0.6-0.9), then concrete (~0.3), grass (~0.25), water (~0.06).

Rank by albedo, lowest to highest: glacial ice, powder snow, open water.

Open water (~0.06) < glacial ice (~0.3-0.6) < powder snow (~0.8-0.9).

How is the reflectance of a satellite pixel that is part snow and part soil (a mixed pixel) estimated?

Linear mixing: the area-weighted average of the pure surfaces' reflectances.

Which surface change gives the largest increase in absorbed sunlight: sea ice melting to open ocean, desert turning to grassland, or forest turning to cropland?

Sea ice melting to open ocean (albedo ~0.6 drops to ~0.06).

Which land-cover changes raise surface albedo: forest to grassland, ice-covered ocean to open water, or vegetated land to desert?

Forest to grassland and desertification both raise albedo, while melting sea ice lowers it.

Replacing dark ocean with bright sea ice changes how much sunlight Earth keeps. Is the shortwave radiative effect at the top of the atmosphere positive (energy gained) or negative (energy lost)?

Negative: more sunlight is reflected to space, a cooling effect.

What is the main climate difference between losing sea ice and losing land ice?

Sea ice loss mainly lowers albedo (little sea level change), while land ice loss raises sea level.

Why does increasing surface albedo lower average temperature?

More sunlight is reflected and less absorbed, so less energy heats the surface.

True or false: a decrease in Earth's albedo would reduce the energy absorbed by the climate system.

False. Absorbed solar is proportional to (1 - albedo), so lower albedo means more absorbed energy.

How do you find the absorbed solar flux from the incident flux and the albedo?

Absorbed = incident × (1 - albedo).

How do you compute a planet's albedo from patches of different surfaces?

Take the area-weighted average: sum of (area fraction × albedo), with the fractions adding to 1.

In climate science, what are aerosols?

Tiny solid or liquid particles suspended in the air, such as sulfate, dust, sea salt, smoke and soot (black carbon).

What two-fold effect do phytoplankton have on albedo?

Their chlorophyll darkens the ocean surface (lower albedo), while the dimethyl sulfide (DMS) they release forms sulfate aerosols that seed brighter clouds (higher albedo).

If chlorophyll (phytoplankton) near the sea surface increases, how does water temperature just below change?

It cools, because the pigments absorb sunlight near the surface and shade deeper water.

Which instrument provides the standard global surface albedo product used to track snow and ice albedo change?

MODIS, the multispectral imager on Terra and Aqua (from multi-angle surface reflectance).

MODIS and CERES both relate to albedo. What is the difference?

MODIS maps surface albedo at about 500 m, while CERES measures top-of-atmosphere (planetary) reflected shortwave at coarse (~20 km) resolution.

On a MODIS land albedo map, which two land surface types show the highest albedo?

Snow and ice (Greenland, Antarctica, ~0.8) and bright deserts (Sahara, Arabia, ~0.3-0.45).

Why can Greenland's ice sheet have negative net radiation (it absorbs less radiation than it emits) even in summer?

Its high-albedo snow (~0.8+) reflects most sunlight, so absorbed shortwave is smaller than the longwave it emits.

Which planet has the highest albedo, about how high is it, and why?

Venus, about 0.76, from its thick sulfuric acid clouds.

What is the basic idea behind the planetary equilibrium temperature equation?

Energy absorbed from the star equals energy emitted by the planet, so the temperature stays steady.

If absorbed solar power equals emitted infrared power, what state is the planet in?

Radiative equilibrium (temperature steady).

Write a planet's energy balance in terms of total power absorbed and total power emitted.

πR²S(1 - α) = 4πR²σT⁴: absorbed over the cross-section disk, emitted from the whole sphere.

Write the planetary equilibrium (effective) temperature formula.

T = [S(1 - α) / (4σ)]^(1/4), with S the solar constant, α the albedo and σ the Stefan-Boltzmann constant.

Why does a planet's radius not affect its equilibrium temperature?

Absorbed power (∝ πR²) and emitted power (∝ 4πR²) both scale with R², so R cancels.

In T = [S(1 - α)/(4σ)]^(1/4), what does the quantity S(1 - α)/4 represent?

The global average absorbed solar flux (about 238-240 W/m² for Earth).

What is Earth's effective (equilibrium, no-greenhouse) temperature?

About 255 K (-18 °C), using S = 1361 W/m² and α = 0.30.

What is Earth's actual global mean surface temperature?

About 288 K (15 °C).

For Earth, what do the temperatures 255 K and 288 K represent?

255 K is the effective (equilibrium) temperature, and 288 K is the actual mean surface temperature.

Earth's mean surface (288 K) is about 33 K warmer than its effective temperature (255 K). What causes the difference?

The natural greenhouse effect: greenhouse gases absorb outgoing IR and re-emit part of it back down (the simple model has no atmosphere).

If Earth had no atmosphere and zero albedo (a perfect blackbody), about what would its equilibrium temperature be?

About 278-280 K, warmer than 255 K because no sunlight is reflected.

Does adding albedo to a blackbody planet model raise or lower its equilibrium temperature?

Lowers it (less sunlight is absorbed).

If a planet's albedo decreases because melting ice exposes darker surfaces, what happens to its equilibrium temperature?

It increases, because more sunlight is absorbed.

How does equilibrium temperature scale with (1 - albedo)?

T ∝ (1 - α)^(1/4), so T₂/T₁ = [(1 - α₂)/(1 - α₁)]^(1/4).

About how much does Earth's effective temperature rise if its albedo drops by 0.01 (0.30 to 0.29)?

About 0.9 K, since (0.71/0.70)^(1/4) ≈ 1.0036.

How does equilibrium temperature scale with the solar constant S?

T ∝ S^(1/4), so a 10% brighter Sun raises T by only about 2.4% (1.1^(1/4) ≈ 1.024).

How does a planet's equilibrium temperature scale with its distance d from its star?

T ∝ 1/√d, because flux ∝ 1/d² and T ∝ flux^(1/4).

How does a bare planet's equilibrium temperature depend on its star's temperature T_star, radius R_star and the orbital distance d?

T_planet = T_star × (1 - α)^(1/4) × √(R_star / 2d).

Why does a no-albedo, no-greenhouse model of Earth (~279 K) land closer to the real 288 K than the model with albedo (255 K)?

Coincidence: it leaves out two effects (albedo cooling and greenhouse warming) that partly cancel.

If you forget the factor of 4 in the equilibrium temperature formula, how wrong is your answer?

Too high by 4^(1/4) = √2 ≈ 1.41: about 360 K instead of 255 K for Earth.

Why is Mars's calculated equilibrium temperature (~210 K) close to its actual mean temperature?

Its thin atmosphere produces very little greenhouse warming.

Venus's effective temperature is only ~230 K, but its surface is ~737 K. What explains the difference?

A very strong greenhouse effect from its thick (~92 bar), ~96% CO2 atmosphere, left behind by a past runaway greenhouse.

Why is Venus's effective temperature lower than Earth's even though Venus is closer to the Sun?

Its very high albedo (~0.76) reflects most sunlight.

How big is the greenhouse effect (surface temperature minus effective temperature) on Earth versus Venus?

Earth about 33 K, Venus about 500 K.

Solving the no-atmosphere energy balance for the albedo that would give Venus its real 737 K surface gives about -24. What does that impossible result tell you?

No albedo can explain the heat (albedo must lie between 0 and 1), so a huge greenhouse effect must be warming Venus's surface.

What is the runaway greenhouse effect, and where did it likely occur?

Warming adds water vapor, which causes more warming, until the oceans evaporate and the planet cannot shed heat. It likely happened on Venus.

What does Kirchhoff's law say about absorptivity and emissivity?

At a given wavelength they are equal: a good absorber is a good emitter.

An object is a graybody with ε = 0.6 at all wavelengths instead of a blackbody. Does its equilibrium temperature change?

No. It absorbs 0.6 of the incoming power and emits 0.6σT⁴, so ε cancels.

When does emissivity change a planet's equilibrium temperature?

When its absorptivity for incoming shortwave differs from its emissivity for outgoing longwave, as for Earth with its atmosphere.

A thin cirrus cloud (a high, wispy ice cloud) has emissivity 0.3 in an IR band. What fraction of the upwelling radiation in that band does it absorb, and why?

30%, because by Kirchhoff's law absorptivity equals emissivity at a given wavelength.

A planet's longwave emissivity drops while its absorbed sunlight stays the same. Does its temperature rise or fall?

Rises: it emits less efficiently, so it must be hotter to radiate the same energy, T = [S(1 - α)/(4εσ)]^(1/4).

In the effective-emissivity model S(1 - α)/4 = εσT_s⁴, what ε gives Earth's 288 K, and what does it represent?

About 0.61, the ratio of outgoing longwave (~240 W/m²) to surface emission (~390 W/m²). Smaller ε means a warmer surface.

In a simple energy balance model, which change raises equilibrium temperature without changing solar output?

Less infrared escaping to space at a given temperature (more greenhouse absorption, or lower emissivity).

How do greenhouse gases warm a planet's surface?

They let most sunlight through but absorb outgoing thermal IR and re-emit it in all directions, partly back down to the surface.