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Part 6: A sunlit planet
Where the sunlight goes, and how the planet sheds the heat again.
Part 5 ended with a dry planet that made its own jet streams, trade winds and storms, kept going by a stand-in for sunlight: every layer of air nudged toward a target temperature, warm at the equator and cold at the poles. This part computes what that stand-in stood for. Some of the sunlight that arrives is reflected straight back to space, and most of the rest warms the ground. The planet sheds that energy again as infrared, from the ground and from every level of the air. Part 1 set the balance between that heating and cooling aside for a part of its own, and radiation comes first among the pieces of the real climate because it works without water.
The air in this part is dry, but its radiation is not simplified. Every figure that computes sunlight or infrared runs the climate model's own radiation code, live in your browser, with the water vapor set to zero. That leaves the absorbing gases of dry air: carbon dioxide, ozone, oxygen, methane and nitrous oxide. The first two sections place the sunlight on the sphere and through the year, and the next two follow it through the air and back out. The two after them let a column of air, and then a column at every latitude, settle into balance. The last two shine the sun on Part 5's dry sphere for a year and measure the heat its winds carry to the poles.
The sunlight that arrives
At the Earth's distance from the sun, a surface held square to the sun's rays catches about 1,361 watts on every square meter, the solar constant. The model uses 1,362 W/m² and a circular orbit. The ground, though, is rarely square to the rays. A beam that comes in at a slant spreads over more ground, so each square meter gets less, in proportion to the cosine of the angle between the sun and straight overhead, the solar zenith angle. That cosine is written μ, and the rule is Lambert's cosine law. Each cell's center is a unit vector \hat{\mathbf{x}} from the planet's center, and the direction to the sun is another, \hat{\mathbf{s}}. Their dot product is that cosine, \mu = \hat{\mathbf{x}} \cdot \hat{\mathbf{s}}, so the sunlight arriving on each square meter at the top of the atmosphere over a cell is
with S_0 = 1,362 W/m². Where the dot product is negative the sun is below the horizon, and the max sets the sunlight to zero. The figure computes exactly that, with the model's own code for \hat{\mathbf{s}}, on a coarser copy of the model's grid: N = 32, 10,242 cells about 220 km across.
The figure opens at the moment the model's clock starts, t = 0. That is the March equinox, with the sun at noon over Greenwich, on the prime meridian. The dot is the subsolar point, where the sun is straight overhead and each square meter gets the full 1,362 W/m². Away from it the sun stands lower and the color fades. At 60° north on the same meridian the sun is 30° up, and the sunlight is half the full beam, 681. The pale yellow line is the terminator, where the sun sits on the horizon, and the half of the planet beyond it gets nothing at all.
Press “Run” and the clock turns at three hours a second, bringing the subsolar point back over Greenwich every 24 hours. That solar day is four minutes longer than the turn against the stars that sets the Coriolis force in Part 2, because each day the planet also moves about a degree along its orbit.
The day slider moves the subsolar point north and south. The planet's axis is tilted 23.44° from upright to the plane of its orbit, its axial tilt, and it keeps pointing the same way in space, so for half the year the northern hemisphere leans toward the sun. The model moves the subsolar latitude δ through the year as
Starting on the equator, the point reaches 23.44° N at the June solstice, day 91, and 23.44° S at the December solstice, day 274. The dashed lines mark those two turning latitudes, the Tropic of Cancer and the Tropic of Capricorn. In the real geometry it is the sine of δ that follows a sine curve, but the two rules never differ by more than about a quarter of a degree. Set the day to 91, and the terminator passes 23.44° beyond the North Pole, which stays on the lit side all day while the Antarctic stays dark.
Whatever the day and the hour, one readout never moves: the mean over the globe, 340.5 W/m², exactly a quarter of 1,362. Press “Look from the sun” to see why. From the sun the planet is a disk that catches the full 1,362 W/m² on every square meter. The planet's surface, 4πR², is four times the disk's πR², so each square meter of it gets a quarter as much:
One factor of two is the night. The other is the slant, which fades the disk's color toward the rim. A hemisphere's area is spread evenly by height above its rim, as Archimedes showed, and the cosine is that height, so over the lit half the sunlight averages exactly half the full beam. Weighted by the sunlight each square meter receives, as the model weights it, the average μ there is two-thirds, the cosine of a sun 41.8° above the horizon. When a single column stands in for the whole planet later in this part, it gets 340.5 W/m² with the sun at that height.
The seasons
The swing of the subsolar point between the tropics makes the seasons. It changes how high the sun climbs, which sets how much of its 1,362 W/m² falls on each level square meter, and how long it stays up. The map below follows the model's sun through a whole year and averages its sunlight over each day, at every latitude.
The probe starts at 45° north on the June solstice. The sun is up for 15.4 hours and climbs to 68.4° at noon, and the day's sunlight averages 499.7 W/m². Drag the probe along its row to the December solstice and everything shrinks together: 8.6 hours of daylight, a noon sun 21.6° up, and 116.6 W/m², less than a quarter as much. At the equator the day changes little, and the daily mean only moves between 433.5 and 397.8 W/m², about a third of the noon sun's 1,362.
Near the poles the length of the day wins. On the June solstice the sun circles the North Pole for all 24 hours, 23.4° above the horizon. A sun that low puts only 40 percent of its light on each level square meter, 541.8 W/m², but it does so all day. That is the most sunlight any place gets on any day of the model's year, a quarter more than the equator ever gets. The orange curve beside the map is the probe's day at every latitude, and on the June solstice every latitude from 9° north to the pole gets more than the equator does on any day.
Half a year later the same pole is dark. Poleward of 66.56°, the Arctic and Antarctic Circles, the sun stays below the horizon all day at the winter solstice. That is the polar night, and it lasts 64 days at 70° and half the year at the pole. In summer the same latitudes have the midnight sun.
Over a whole year the long summer days do not make up for the dark winters. The white curve beside the map is each latitude's average over the year: 415.6 W/m² at the equator, 307.1 at 45° and 174.1 at the poles, less than half the equator's. That difference is what drives the winds. Part 5's Held–Suarez target stood in for it, and the rest of this part follows the sunlight into the air.
The map's southern half is its northern half flipped and shifted by six months, because the model's orbit is a circle. The real Earth's orbit is slightly stretched, which makes the real sun about 3 percent brighter than average in early January and 3 percent dimmer in early July.
What the air does with sunlight
Before sunlight can warm the ground it has to get through the air, and dry air lets most of it pass. The nitrogen and oxygen that make up 99 percent of the air hardly touch visible light, which carries nearly half of the sun's energy. Ozone absorbs ultraviolet light high in the stratosphere. Oxygen and carbon dioxide absorb a few narrow bands at the red end of the spectrum and in the near infrared beyond it. Air molecules also scatter short wavelengths far more than long ones, which is called Rayleigh scattering and sends some light back to space, and the fine dust, salt and smoke in the air, the aerosol, scatter a little more and absorb a little. Of the light that reaches the ground, the share the ground reflects is its albedo.
The figure follows a beam through the midlatitude summer, a July at 45 degrees north and one of the standard atmospheres that radiation codes are tested against. It is laid on the model's 36 layers with its ozone and without its water vapor. The figure runs the model's own code twice, over the chosen ground and over a black one, whose reflection is the air's own. The top bar splits the arriving sunlight four ways, the air's share is drawn again below it ten times larger, and the profile shows how fast the sunlight warms each layer.
With the sun straight overhead, the ground absorbs 89.3 percent of the 1,362 W/m². The air sends 4.9 percent back to space. The sea reflects only 2.4 percent of a beam that strikes it head on, so the column as a whole reflects 7.1 percent. The air absorbs just 49.1 W/m², 3.6 percent, most of it in ozone. To sunlight, dry air is very nearly transparent.
Small as it is, the ozone's share heats the air directly, where there is very little air to heat. The ozone is densest near 23 km, but the heating peaks higher, at 26.5 K per day around 1.6 hPa, 45 km up, where a layer holds about a thousandth of the column's air. That heating is why the stratosphere warms with height, the inversion of Part 1 that caps the weather, up to the stratopause. Below 10 km the dry air warms by only 0.12 to 0.16 K per day, even with the sun overhead.
Now lower the sun. At 30° above the horizon the beam spreads over twice the ground and crosses twice as much air, a slant path called the air mass. The air gets more chances at it, and the share reflected nearly doubles, to 13.8 percent. A long path also uses up the ultraviolet sooner, so the heating moves up as the sun goes down. With the sun 15° up, the peak is at 0.65 hPa, 53 km up.
The ground decides more than the air does. With the sun overhead the column reflects 7.1 percent over the sea, 21.8 over grassland and 80.1 over fresh snow. Snow also makes the air absorb more, 69.8 W/m² against 49.1, because the light it throws back crosses the ozone and the other gases a second time on its way out. Switch the ozone off instead, and the air absorbs only 13.4 W/m² and no layer warms faster than 0.18 K per day. It is the ozone's sunlight, and almost nothing else, that warms the stratosphere.
The model does all of this in a few steps for each column. The gases and then the aerosol take their shares of the direct beam along its slanted path. The model gathers the scattering into a single layer with no height and treats it the two-stream way, lumping all the light going down into one stream and all the light coming up into another, and it adds up every bounce between that layer and the ground. So the figure's reflection by the air has no height, and the heating comes from absorption alone. With the midlatitude summer's vapor put back and the sun overhead, the same code has the air absorb 256.7 W/m² instead of 49.1, five times as much, most of it vapor's in the near infrared.
Heat that leaves as infrared
Sunlight is only half of the budget. Everything warm glows, and the ground glows in infrared, at wavelengths between about 5 and 50 micrometers, twenty times longer than the sunlight's. A perfect emitter, a black body, sends out σT⁴ watts per square meter, where σ = 5.67 × 10⁻⁸ W/m²/K⁴; that is the Stefan–Boltzmann law. The model treats the ground as a black body, so ground at 294.2 K sends up 424.8 W/m², night and day.
Nitrogen and oxygen hardly touch the infrared either, but a few rarer gases absorb it strongly, each in its own bands: carbon dioxide around 15 micrometers, ozone at 9.6, and methane and nitrous oxide near 8. These are the greenhouse gases. A layer that absorbs some share of the infrared at one wavelength also emits that same share of what a black body at its own temperature would emit there; that is Kirchhoff's law. The air is colder than the ground, so in the gases' bands the ground's glow is swallowed on the way up and replaced by the dimmer glow of colder air.
Following every wavelength is far too slow for a climate model, so the model sorts the spectrum into 34 slices, called g-points, in a method known as correlated k. A slice gathers wavelengths, wherever they lie, that the gases absorb about equally strongly, so one number per gas describes it. Those numbers were fitted to match RRTMG, a scheme many weather and climate models use. Each slice holds the share of σT⁴ that Planck's law puts at its wavelengths, and the model passes that share up the column one layer at a time. Each layer absorbs a fraction 1 − e−τ of what reaches it, where τ is the layer's optical depth in that slice, and adds the same fraction of its own share of σT⁴.
The figure runs those slices on the midlatitude summer, again without its vapor. Each bar is as wide as its slice's share of the ground's emission, the clearest on the left. Its height is where that slice's infrared leaves for space: the level where the gases above reach an optical depth of 1, from which about a third of what the air emits gets out. Where the whole column stays under 1, the bar has no height and space sees the ground.
In dry air most of the bars are missing. Slices that see the ground hold 76.4 percent of its emission. A fifth of the emission, 21.6 percent, lies in slices the dry gases barely touch, under a tenth of a percent; those belong to water vapor. Besides those, the widest slice, 21.4 percent of the emission on its own, is most of the infrared window between about 9 and 13 micrometers, and the dry gases take only 6 percent of it, most of that ozone and carbon dioxide.
The tall bars on the right are mostly carbon dioxide's, in its band around 15 micrometers. Carbon dioxide's bars that leave for space from above 500 hPa hold 16.7 percent of the ground's emission. The slice the readout starts on leaves from 236 hPa, 11.1 km up, where the air is at 227.6 K, and it sends to space only 6.1 W/m² of the 16.2 the ground emits in it. Two short bars near the ground, carbon dioxide at 986 hPa and methane, are just opaque enough that space sees air nearly as warm as the ground.
Add the slices up and 356.2 W/m² leaves the top. That is the outgoing longwave radiation, OLR: 277.6 W/m² straight from the ground and 78.6 of the gases' own glow. The shortfall, G = \sigma T_s^4 - \text{OLR}, is the greenhouse effect as the model computes it, 68.6 W/m², 16 percent of what the ground gives off. Of the ground's 424.8 W/m², the gases absorb 147.2 and send only 78.6 of their own to space in its place. They glow downward too, and 127.6 W/m² comes back to the ground.
Turn the gases off one at a time. Without carbon dioxide, G falls to 24.2 W/m². Carbon dioxide alone holds back 45.2, nearly the 44.4 it takes away from the four together, because in dry air the gases' bands barely overlap.
Now change the amount of carbon dioxide, from the model's 388.75 ppm for 2010. Doubling it lowers the outgoing infrared by 3.3 W/m², and doubling again by 3.6 more, while the first half of the gas, from none to 194 ppm, takes away 40.9. The middle of the band is already opaque, so more gas only lifts each bar higher into colder air: doubling moves the 236 hPa slice up to 167 hPa, and it sends 5.4 W/m² instead of 6.1. In the stratosphere, where the air warms with height, the bars rise into warmer air and send more, but the losses win. That drop of 3.3 W/m², with nothing else changed, is the radiative forcing of doubled carbon dioxide on this column. The planet now absorbs more than it sends away and has to warm until the gap closes, which the next section lets it do.
Radiation and convection
Part 1 left a question hanging. Convection erases any fall of θ with height, yet the real atmosphere is stable overall. The answer it gave was a balance: sunlight warming the ground, the air radiating heat to space from every level, and convection carrying heat up from below. This section puts the radiation to work in a single column of air and lets it run until nothing changes. That end state is called radiative–convective equilibrium.
The column stands for the whole planet, with the 340.5 W/m² of the first section and the sun 41.8° up. Its ground reflects 7 percent of the light, a little more than the model's open sea, and stores as much heat as 5 m of water. Its air is the model's dry air, with 389 ppm of carbon dioxide. Every 3 simulated hours the model's own radiation heats or cools each of the 36 layers and the ground. With convection on, the ground also warms the air touching it, through the sensible heat flux. And after every step, any layers in which θ falls with height are mixed to a single θ. This is Part 1's overturning done in one stroke, called convective adjustment. Everything starts at 288 K.
The stratosphere forms first, within a month, as the air at 54 hPa cools fast while the ozone layer near 1.6 hPa holds its warmth; the whole column takes about 970 days to settle. At the end the ground is at 283.3 K, and the column sends as much infrared to space as the 298.0 W/m² of sunlight it absorbs. The coldest air, 208.1 K, sits at 54 hPa, 19.6 km up, and marks the tropopause. That is far higher than the top of the real troposphere, 10 to 15 km up: without water, convection stops at 2 km, so above it radiation alone sets the temperature. A black body would shed that much at 269.3 K, the column's effective temperature, T_e = (\text{ASR}/\sigma)^{1/4}, where ASR is the absorbed sunlight. The 14.1 K by which the ground is warmer is the work of dry air's greenhouse effect.
The arrows show how the balance is kept. In the stratosphere every layer balances on its own, sunlight against infrared. At 1.6 hPa ozone's sunlight heats the air by 8.65 K a day, and the air radiates exactly as much away, so the stratosphere is in radiative equilibrium. So is the troposphere above 2 km, with heating tens to hundreds of times weaker, 0.04 to 0.2 K a day each way. Only the lowest 2 km are different. There every layer above the lowest radiates away 0.4 to 0.8 K a day more than it absorbs, and convection makes up the difference with heat from the ground. The figure's probe starts on the lowest layer. The ground heats it by 31.1 K a day, the 17.6 W/m² of sensible heat poured into a layer only 40 m thick. The layer also gains about half a kelvin a day from the ground's infrared, so convection carries 31.6 K a day up to the layers above, the ground's 31.1 plus that. In every layer the arrows cancel, and the one for them all together shrinks to a dot.
Switch to the lowest 3 km. Up to 2.0 km the temperature falls by 9.77 K per km, which is g/cp, the dry adiabatic lapse rate of Part 1. Now turn convection off. The column drifts onto the faint line, radiation alone, and cools by 33.7 K per km up to 530 m, so a parcel nudged upward arrives warmer than its new neighbors and keeps going, the runaway case of Part 1's bobbing parcel. Radiation alone builds such a column because the ground takes 95 percent of the sunlight the column absorbs, so the heat arrives at the bottom.
Yet the ground barely notices. Switching convection on cools it by about one and a half kelvin, from 284.9 to 283.3 K, while the air 1.4 km up warms by 13.5 K. In dry air two-thirds of the ground's infrared goes straight through to space, so the ground hardly needs convection. When Syukuro Manabe and Richard Strickler computed a column like this in 1964, convection mattered far more, because their air held water vapor. Vapor absorbs most of the ground's infrared in the lowest kilometers, and with a midlatitude summer's vapor held fixed in this column, convection lowers the ground by 15.9 K instead of 1.5.
Now double the carbon dioxide, to 778 ppm. The infrared leaving the top drops at once by 3.4 W/m², so the column starts to gain heat, yet the upper stratosphere cools, and fast. Up there the only heating is ozone's sunlight and most of the cooling is carbon dioxide radiating to space, so the air at 0.65 hPa ends 17.1 K colder, halfway there in 3.5 days. The ground takes 54 days to get halfway and settles 1.3 K warmer. The infrared leaving the top climbs back to match the sunlight absorbed, but G is now 73.8 W/m² instead of 67.3: the same infrared leaves from behind more gas, so the ground must be warmer to push it out. A second doubling adds another 1.5 K.
Take the carbon dioxide away, and the ground falls 11.8 K and G drops to 10.2 W/m²: in dry air, most of the greenhouse effect is carbon dioxide's. Ozone, methane and nitrous oxide radiate poorly, so ozone's sunlight heats the air at 0.65 hPa to 352.5 K, and the coldest air sinks to 440 hPa, 6 km up, with the air warming all the way up to 0.65 hPa.
Last, put the carbon dioxide back to ×1 and make the ground reflect 30 percent of the sunlight, about the share the whole Earth reflects, though the Earth gets most of it from clouds. The column then absorbs 232.8 W/m² instead of 298.0, and the ground settles 18.0 K colder, at 265.3 K.
What the model computes
Each layer warms by the net flux of energy into it, in watts per square meter. That flux has three parts: \class{term-a}{Q^{\text{sun}}_k} from sunlight, \class{term-b}{Q^{\text{ir}}_k} from infrared and, for the lowest layer only, \class{term-c}{Q^{\text{ground}}_k} from the ground. Divided by the layer's heat capacity, they make the heating \class{term-e}{H} of Part 4's fourth rule:
Here π is the surface pressure of Part 4, not the 3.14159 of the first section, so \pi\,\Delta\sigma_k/g is the layer's mass per square meter, and \overline{\Pi}_k turns a change of temperature into a change of θ. The model adds \Delta t\,\class{term-e}{H_k} to θ after each RK4 step, as one of the processes that operator splitting applies in turn. \class{term-a}{Q^{\text{sun}}_k} is the sunlight that the gases and aerosol absorb in the layer, along the slanted path of the section on sunlight. The infrared is computed separately in each of the 34 slices. In slice j, a layer absorbs the fraction
of the infrared crossing it. In that formula a_{j,m} is how strongly gas m absorbs in the slice, and u_{k,m} is how much of the gas the layer holds, scaled with pressure and, for carbon dioxide and water vapor's continuum, with temperature, as the fit to RRTMG requires. The 1.66, called the diffusivity factor, stands in for infrared that crosses the layer at every slant. By Kirchhoff's law the layer emits the same fraction of B_{k,j} = s_j(T_k)\,\sigma T_k^4, both up and down. That is the slice's share s_j of what a black body at the layer's temperature would emit, which the model fits as a quartic in temperature. D and U are the downward and upward fluxes on the interfaces between layers. The model sweeps down from the top, where D_{1/2,j} = 0, and up from the ground, which emits U_{K+1/2,j} = s_j(T_s)\,\sigma T_s^4. Each layer keeps what it absorbs less what it emits:
Summed over the slices, the flux leaving the top is the OLR, and the flux reaching the ground is the downward longwave radiation, DLR:
The ground has no layers. Its temperature changes with the sunlight it absorbs, the infrared it receives and emits, and the sensible heat it gives the lowest layer, layer K:
C is the ground's heat capacity, 2.1 × 10⁷ J/m² per K in the figure, and ρ is the density of the air in the lowest layer. C_H = 1.2\times 10^{-3} is the exchange coefficient over the sea, and |u| is the wind near the ground, 5 m/s in the figure and never counted as less than 3 m/s. Then comes the convective adjustment. Working up from the ground, the model gathers each run of neighboring layers in which θ does not rise with height and gives them all one θ, chosen so that the run's total heat, c_p T times mass, does not change:
The figure's convection arrows are this change in each layer, divided by the 3-hour step. In the climate model the sunlight on each cell follows the sun through the day and the year, by the rule of the first section, S = S_0\max(0,\, \mu) with \mu = \hat{\mathbf{x}}\cdot\hat{\mathbf{s}}. The figure's column gets the global average instead: S_0/4 at μ = 2/3.
A full call of the radiation is the most expensive piece of a step, so the climate model makes one every 11.25 simulated minutes and holds the result in between: every 4 steps at N = 128, every 2 at N = 64, and every step on coarser grids. The call takes the sun's height as the mean of μ over the steps it covers, with each μ weighted by itself:
Between calls the sunlight follows the sun's height. The infrared stays as it was except for the ground's emission. Of any change in that emission, each layer takes the share f_k it absorbed at the call, and the rest leaves for space:
At N = 128, calling every 4 steps instead of every step changes the infrared leaving the top by only 0.07 W/m². Against RRTMG itself, on the midlatitude summer with its water vapor and its own carbon dioxide, methane and nitrous oxide, about 355 ppm, 1.6 ppm and 296 ppb, the model's infrared leaving the top is 282.0 W/m² and the infrared reaching the ground is 347.7 W/m², where RRTMG gives 281.5 and 348.5. On that same column, doubling carbon dioxide from 287 to 574 ppm cuts the infrared leaving the top by 2.66 W/m², 6 percent less than the 2.84 W/m² of LBLRTM, the line-by-line calculation that RRTMG was itself fitted to.
Every latitude for itself
No real place gets the sunlight of the last section's column. Averaged over a year, the top of the air over the equator receives 415.6 W/m² and over the pole 174.1, as the seasons showed, and the sun, weighted by its light, stands 48.9° above the horizon at the equator and 18.3° at the pole. Part 5's Held–Suarez recipe nudged every latitude toward a target that stood in for the temperature each latitude would reach on its own. At the ground that target is 315 - 60\sin^2\varphi K, where φ is the latitude. The figure computes that temperature instead, running the column of the last section at ten latitudes, those of the N = 16 grid's cells nearest 0°, 10°, 20° and so on up to the pole, each with its own year-round sunlight, sun height and ozone. No air moves from one column to another, so no heat passes between them, and each column settles into a radiative–convective equilibrium of its own: every latitude for itself.
Each column settles with its ground at the temperature where it sheds exactly what it absorbs: 298.4 K at the equator, 256.9 K at 60° and 234.7 K at the pole. The shape is the target's. The ground falls by 63.7 K from the equator to the pole, against the 60 K Held and Suarez asked for. The level is not. Every latitude is colder than the target, by 16.6 K at the equator and 20.3 K at the pole. The target was chosen to give an Earth-like climate, and the Earth's air holds water vapor. Dry air has a weak greenhouse.
The air above follows the ground: at 500 hPa, halfway up the atmosphere by mass, the equator's column is at 252.0 K and the pole's at 203.4 K. The coldest air sits higher over the equator, 17.3 km up, than over the pole, 12.6 km, but in these dry columns it is highest of all, near 21 km, over the midlatitudes.
Switch the figure to sunlight and infrared. At every latitude the two lines lie on top of each other: 365.8 W/m² in and out at the equator, 145.3 at the pole. The shaded band above them is the sunlight reflected, 12.0 percent at the equator and 16.5 percent at the pole, although the ground reflects 7 percent everywhere. Near the pole the light slants in through more air, and the air scatters more of it back to space.
Averaged over the area of the globe, the ten columns absorb 297.3 W/m², and their ground averages 281.4 K. The single column of the last section absorbed almost the same, 298.0 W/m², yet settled 1.9 K warmer. An uneven planet sheds heat more easily than an even one: infrared emission grows with the fourth power of temperature, so the warm tropics shed far more than their share.
Press “Start every column at 288 K” to watch the columns settle at the default settings. The equator is in balance after about 560 days and the pole only after about 1,190: the pole has 53 K to fall, and a cold column sheds its heat slowly. Double the carbon dioxide and every column warms, the equator by 1.3 K and the pole by only 1.0 K. The real planet warms most near the poles, which is called polar amplification, through melting snow and ice, water vapor and winds that carry heat, none of which these columns have. Press Reset, raise the ground's reflection to 30 percent, and the average falls by 17.9 K while the contrast from the equator to the pole becomes 60.0 K, the target's own.
This is the planet that Part 5's target stood in for, now computed: a drop of 63.7 K from the equator to the pole, with nothing to carry heat across it. It cannot last. That much contrast makes the sloping surfaces of potential temperature that broke into storms in Part 5, and the storms carry heat toward the poles. The next section lets the columns trade heat that way, on Part 5's dry sphere.
Sunlight instead of nudging
Part 5's dry sphere could trade heat, but its sunlight was a stand-in. This section gives it the real sun. The run below is Part 5's dry sphere again: the N = 16 grid with 27 layers, steps of 450 seconds and the same friction near the ground. The Held–Suarez nudging is gone. In its place is the model's own radiation, the same code the figures above run, on every column of the grid under the sun of each hour of the year. Below the air lies the ground of the columns, with the same reflection of 7 percent and the same heat capacity. It warms the lowest layer through sensible heat, and convective adjustment mixes away any fall of θ with height. Like the columns, the planet has no water, clouds, snow or ice. The run began with 60 days of Held–Suarez from rest, so that the sun would rise on a planet already full of weather. The sun then shone for 150 days before one model year was recorded. Those 150 days brought the planet most of the way to a balance, but not all the way. It was still cooling slowly while the year was recorded, and its ground ended the year about 0.85 K colder, most of that in the south. The poles were still adjusting through the first months of the record. The whole run took about four and a half hours of computing, so the figure plays it back.
The figure opens on the March equinox at the start of the recorded year, with the sun over the equator. Its dates follow the calendar from March 20, but the model's year has four equal seasons. Its solstices and its September equinox therefore fall a few days before the calendar's: the June solstice on June 19, the September equinox on September 19 and the December solstice on December 19. Play the year and watch the ground. The tropics barely change: the ground at the equator stays between 294 and 298 K all year. The middle latitudes swing widely, because the ground stores little heat and warms and cools quickly. Around 45° north it warms to 298.8 K by mid-August and cools to 244.6 K by early February, a swing of 54 K. Both come seven to eight weeks after the solstices, the time the ground takes to catch up with the sun. Switch the color to the infrared leaving the top, and it follows the ground: brightest over the summer hemisphere's warm ground, 389 W/m² near 22° north in mid-August, and dark over the winter pole, under 100 W/m².
Near the poles the swing is nearly 90 K. Around the North Pole the ground warms to 285.7 K in early August, under the midnight sun, and cools through the polar night to 196.5 K on March 20, when the sun returns, a swing of 89 K, and the northern ground was still cooling when the record ended. The southern ground bottoms out at 197.3 K on September 26, a week after its sun returns. A year earlier, at the start of the record, the northern night had ended at 201.9 K, 5 K warmer. Through the dry air the dark ground loses heat to space faster than the air above can give it back, so the ground ends colder than the air. In January the ground near the North Pole averages 211 K, and the air 1.5 km up, the figure's third choice, is 17 K warmer. That is the inversion Part 1 found over the ground on a clear, calm night, here lasting all winter.
Over the recorded year the planet absorbs 295.9 W/m² of sunlight and sends 296.0 W/m² back to space. From the equator to the poles its ground falls by 59 K, from 295.8 to 236.7 K, 5 K less than the columns' 63.7 K. The air in the lowest layer falls by 56 K, close to the 60 K that Held and Suarez asked for. About 600 m up, the air at 60° is 32 K colder than at the equator, against 37 K at the same height on Part 5's settled planet. The winds have warmed the polar air far more than the polar ground. At 500 hPa the air over the poles averages 226 K, 23 K warmer than over the polar column, while the ground under it is only 2 K warmer. From the equator to 50° the ground, under air that the winds have warmed or stirred, is 2 to 4 K colder than its column, so the ground of the whole planet averages 279.3 K, 2.1 K colder than the ten columns.
Switch to the wind 10 km up. For most of the year the wind at 250 hPa is weaker than Part 5's 32 m/s jet: averaged over the year and along each circle of latitude, it peaks at about 19 m/s. It reaches 35 m/s near 44° north in March and April, and the southern westerlies peak half a year later, at 30 m/s in September. What the map cannot show is that these westerlies are only the lower flank of a wind that keeps strengthening with height all the way to the model's top, where it reaches 150 m/s near 1 hPa, with no core near the tropopause as Part 5's jet had. The cause is temperature. Part 5's nudging held the air above the tropopause near 200 K at every latitude, while here the winter pole cools in the dark all the way to the top, so the contrast with the tropics, and with it the wind, keeps growing with height. In the real atmosphere, gravity waves like Part 5's, launched upward by storms and mountains, break in the thin air and drag on it, which keeps the real winter stratosphere from getting that cold and that fast; the climate model adds that drag. Near the ground the westerlies are weaker than Part 5's 9 m/s and wander with the seasons, reaching about 7 m/s near 50° to 65° in spring; averaged over the year they blow at no more than about 2 m/s. The trades swing across the equator with the seasons, blowing at about 9 m/s near 12° north in January and as strongly near 12° south in July.
Now show a month of weather, a January with four frames a day, and the temperature 1.5 km up. In the northern winter the storms of Part 5 return, swinging tongues of cold air toward the equator and tongues of warm air toward the pole. Between 35° and 65° north the temperature typically strays 4.7 K from the average around its latitude, more than the 4 K the same measure gives for Part 5's settled month. In the southern summer it strays only 1.5 K. The storms come with the winter, when the contrast that feeds them is sharpest, and they carry heat toward the poles.
Carrying heat to the poles
In the columns of the section before last, each latitude sent back out exactly the sunlight it absorbed. On the sunlit sphere that no longer holds. The figure takes the year's average of both at the top of the air, in bands 5° of latitude wide. Over a year, once the heat the planet itself stored or shed is taken out, a band that absorbs more than it sends out must pass the surplus to its neighbors, carried by the air across its edges. Adding up those surpluses and losses from the South Pole northward therefore gives the heat that crosses each latitude. It is measured in petawatts, PW, and a petawatt is a million billion watts.
The tropics absorb more than they send out. At the equator they take in 363.6 W/m² of sunlight and send out 351.0 W/m² of infrared, keeping 12.6. The poles do the reverse, absorbing 145.4 W/m² and sending out 172, a loss of 26 to 27. Over the recorded year a fifth to a third of that came out of the polar ground and air as they cooled, and heat from lower latitudes made up the rest. With the stored heat taken out, the surplus turns into a loss near 40° in each hemisphere, and that is where the most heat crosses: about 1.1 PW, northward in the north and southward in the south, with almost nothing crossing the equator, and the difference between the two hemispheres is within what the correction can tell. Without that correction the southern peak would read 1.27 PW, because the heat the south's own ground and air gave up would count as heat carried there. The heat the planet shed, 0.76 W/m², and the 0.1 W/m² it lost at the top differ by 0.7 W/m², heat this bare run of the core loses on the way: its friction and smoothing take energy out of the wind without returning it as warmth, as Part 4's energy budget showed. The figure spreads that remainder evenly over the globe. On the figure's latitude scale equal widths hold equal areas, so the shaded surplus and the shaded losses hold nearly the same amount of heat.
The real Earth carries 5 to 6 PW at its peak, in its winds and ocean currents together, about five times as much. The dry planet's two lines lie close together because so much of its infrared comes straight from the ground: a latitude with warm ground sends out strong infrared and sheds most of its sunlight where it falls. From the equator to the poles the infrared leaving the top falls by 179 W/m², nearly as much as the sunlight absorbed, 218. On the real planet water vapor and clouds hide most of the ground's glow, the infrared of the tropics rises far less with their warmth, and the gap that the winds and oceans have to close is far wider.
In the middle latitudes the heat travels with the storms of the last section. In the tropics it travels with the slow overturning of the Hadley cells of Part 1, air of high θ spreading poleward aloft while air of lower θ returns near the ground. Without them every latitude would be on its own, as in the columns.
The dry planet of this part also absorbs nearly all its sunlight. With no clouds, snow or ice, its column reflects only 12.5 percent of the sunlight and the sunlit sphere 13.1 percent, where the full climate model reflects 30.1 percent. Yet the climate model's ground is the warmest, 289.0 K against the column's 283.3 K and the sphere's 279.3 K, because its greenhouse is far stronger: its ground sits 34.5 K above its effective temperature, and the dry column's only 14.1 K. The sphere's margin, 10.5 K, is smaller still, partly because an uneven planet sheds heat more easily, as the latitude section showed. Clouds reflect much of the sunlight the dry planet keeps, and with the midlatitude summer's vapor put back, the column of the section on infrared holds back 143.6 W/m² instead of 68.6. Between the climate model and the dry planet, the difference is water.