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Part 7: A wet planet

Vapor, cloud and rain: the water the air carries.

Part 6 ended on a dry planet that absorbed nearly all its sunlight, far more than the full climate model does, yet the climate model's ground is the warmer, because its greenhouse is far stronger. The difference is water, as the vapor in the air, as the droplets and ice of clouds, and in the heat released and taken back as one turns into the other. This part adds it.

Water is the one part of the air that changes form at the temperatures the atmosphere reaches. It evaporates from the sea, rides the winds as an invisible gas, condenses into cloud where the air rises and cools, and falls back as rain, and at every step it moves heat. The sea pays for each kilogram that evaporates, and the air is paid back only where the vapor condenses, often kilometers higher and thousands of kilometers away.

The first section lifts a damp parcel, as Part 1 lifted a dry one. The second puts the vapor back into Part 6's radiation, and the third evaporates water from the sea. The fourth lets Part 6's column settle with vapor that follows its temperature. The fifth runs the climate model's own convection, and the sixth adds clouds. The seventh puts all of it on Part 6's sunlit sphere for a year, over a sea with no land, and the eighth follows the heat that its water carries toward the poles. Most of the figures run the climate model's own code live in your browser; the year on the sphere was computed in advance.

A damp parcel

Air is never quite dry. Mixed in with its nitrogen and oxygen is water vapor, water as an invisible gas. The model measures it as specific humidity, written q: the mass of vapor in each kilogram of air, given here in grams per kilogram. The most vapor that air can hold rises steeply with its temperature. At 1000 hPa it is 3.81 g/kg at 0 °C, 10.67 at 15 °C and 26.84 at 30 °C, about 7 percent more for each degree near freezing and 6 percent near 30 °C, by the Clausius–Clapeyron relation. The vapor the air holds, as a share of the most it could hold, is its relative humidity.

The figure is Part 1's lifted parcel again, now damp. It starts on the ground at 15 °C and 70 percent relative humidity, holding 7.47 g/kg of vapor out of the 10.67 it could hold. The gauge on the right holds its water, with a dashed line for the most vapor the parcel can hold where it is. Lift the parcel.

At first nothing is new. The parcel cools at the dry adiabatic lapse rate, 9.8 °C per kilometer, and keeps its vapor, but the cooler it gets the less it can hold, and the dashed line sinks toward the vapor. Its dew point, the temperature at which its vapor would just saturate it, starts at 9.6 °C and falls by only 1.8 °C per kilometer. The two meet at 678 m and 8.4 °C, where the relative humidity reaches 100 percent: the parcel's lifting condensation level, the base of its cloud. Moister air saturates sooner. At 90 percent the cloud base comes down to 204 m, and at 100 percent it is the ground itself, which is fog.

Above its cloud base the parcel cannot keep all of its vapor. What it holds beyond the limit condenses into droplets, a cloud, and condensing gives back the latent heat it took to evaporate the water: each gram that condenses in a kilogram of air warms it by 2.49 K. So the cloudy parcel cools more slowly, at the moist adiabatic lapse rate, 5.4 °C per kilometer at 1 km. Lift it to 5.5 km, near 500 hPa. It has made 6.26 g/kg of cloud water, and at −23.0 °C it is 14.0 K warmer than a dry parcel lifted as far. From its cloud base it cooled by 6.5 °C per kilometer on average, the “about 6 °C per kilometer” that Part 1 gave a cloudy parcel. The extra warmth is heat released inside the parcel, so its θ has risen, from 288.1 K on the ground to 305.2 K.

By 12 km the parcel has condensed 7.46 of its 7.47 g/kg, it cools at nearly the dry rate again, and its θ has stopped rising, at 308.9 K. That θ, the one a parcel ends with once all of its vapor has condensed, is its equivalent potential temperature: it counts the parcel's heat and its water together. Set the ground to 27 °C and 80 percent, like air over a tropical sea, and the parcel holds 17.99 g/kg and arrives at 12 km 32.9 K warmer than a dry one, with a θ of 353.0 K. Air that keeps that much warmth can stay warmer than the air around it and keep rising, as the air in a thunderstorm does.

The figure simplifies two things: its parcel drops its cloud water the moment it forms, a path called a pseudo-adiabat, and its droplets never freeze. The model's plume, in a later section, carries some of its water upward and freezes some of it below 0 °C.

What the model computes

The most vapor air can hold is set by the saturation vapor pressure e_s, the pressure of the vapor in air that holds all it can. Over liquid water the model takes the formula of David Bolton's 1980 paper, with T in kelvin, and from it the saturation specific humidity at a pressure p, the most vapor each kilogram of air can hold there:

Here 0.622 is R/R_v, the gas constant of dry air over that of water vapor, R_v = 461.5 J/kg/K. q_s counts the vapor against the whole kilogram of damp air, so it is a specific humidity, not a mixing ratio, which counts it against the dry air alone. The relative humidity in this part is q/q_s. For the cloud base the model inverts Bolton's formula at the parcel's own vapor pressure e to get its dew point T_d, takes a second formula of Bolton's for the temperature T_L at which the parcel, lifted dry from temperature T, saturates, and gets the pressure p_L there from the relation between θ and pressure of Part 1, Poisson's equation:

Above that level a saturated parcel cools by g/c_p for every meter it rises and is warmed by L/c_p times the vapor it condenses, with L = 2.5 × 10⁶ J/kg held fixed and c_p dry air's. With Clausius–Clapeyron for how fast q_s falls as the parcel cools, these give the moist adiabatic lapse rate:

It is 4.74 K per km at 288 K and 1000 hPa and 3.71 at 300 K, but 9.21 at 230 K and 300 hPa, against the dry 9.77. The figure follows it upward a tenth of a hectopascal at a time. The climate model uses the formula only to estimate how strong the inversion is above the flat cloud decks of the subtropical oceans; its plume finds its own temperature layer by layer. Below 0 °C the model blends Bolton's formula with one over ice by the cloud's liquid share \alpha, which falls linearly from 1 at 0 °C to 0 at −38 °C, and in the plume the heat released per kilogram becomes L + (1 - \alpha)\,L_f, with L_f = 3.34 × 10⁵ J/kg the heat of freezing. The large-scale condensation of the section on convection releases L alone, and the heat of freezing of snow goes to the lowest layer when the snow reaches the ground.

Water also changes how heavy the air is. A molecule of water vapor is lighter than the nitrogen and oxygen it replaces, so damp air is lighter than dry air at the same temperature and pressure, while cloud droplets add weight but take up almost no room. The model counts both through the virtual potential temperature \theta_v, which takes the place of θ in Part 4's second rule, the hydrostatic balance that turns temperature into height:

Here q_c is the cloud water, and 0.608 is R_v/R - 1. The figure's parcel on the ground, with 7.47 g/kg of vapor, is as light as dry air 1.3 K warmer.

Vapor's greenhouse

Part 6 took the vapor out of its air, and the greenhouse it found was weak: ground at 294.2 K sent up 424.8 W/m², and the dry gases held back only 68.6 W/m² of it. Its midlatitude summer came with vapor, though, 28.8 kg of it over each square meter, which would make a layer of water 28.8 mm deep if all of it fell as rain. That is the column's precipitable water, and more than half of it lies in the lowest 2 km.

Water vapor absorbs infrared in two ways. A water molecule absorbs at thousands of sharp wavelengths, its spectral lines, which crowd the infrared on both sides of the window. Between the lines it absorbs a little everywhere, a smooth background called the continuum, which reaches into the window itself and grows with the square of the humidity. The figure is Part 6's slices again, with the vapor put back, and a new row under the slices for what the air sends back down to the ground.

The column now sends 281.2 W/m² to space instead of 356.2, so G is 143.6 W/m², twice dry air's 68.6. Only 15.4 percent of the ground's infrared now goes straight to space, against 65.3 percent in dry air, and 348.1 W/m² comes back down to the ground instead of 127.6. Vapor now leads in slices that hold 78.2 percent of the emission. Most of the fifth that Part 6 left to water vapor now leaves from high up: the widest of those slices from 9.6 km, where the air is at 237.7 K. Slices that hold two-fifths of the ground's emission leave for space from between 500 and 200 hPa, from air that is colder and glows less.

The readout starts on the widest slice, most of the infrared window. Its optical depth was 0.06 in dry air; with the vapor it is 0.61, 0.48 of it the continuum. More than half of the ground's infrared in it still goes straight through, but the air now sends 37.1 W/m² of it back down to the ground, where dry air sent 3.3. Switch the continuum off, and the infrared leaving the top rises by only 6.2 W/m², while the infrared coming down falls by 44.9. Three-quarters of the window's continuum lies in the lowest 2 km, in air within 7 K of the ground's temperature, which glows to space nearly as brightly as the ground it hides but fills a part of the ground's sky that dry air left dark.

Switch off the four gases of dry air, and vapor alone holds back 95.6 W/m², more than carbon dioxide, ozone, methane and nitrous oxide together. Each doubling of the vapor holds back more than the one before, 9.8, then 12.4, then 17.1 W/m², because the continuum's depth grows fourfold with each. Doubling carbon dioxide in this moist air lowers the infrared leaving the top by only 2.5 W/m², less than the 3.3 of Part 6's dry air, because the vapor already absorbs in some of its slices. Vapor is the strongest greenhouse gas in the column, but the air holds only what evaporation supplies and its temperature allows, so its greenhouse grows as the planet warms, which a wet column, two sections on, lets it do.

Vapor also absorbs sunlight. Switch the figure to sunlight, Part 6's beam through the model's own code. With the sun straight overhead over the sea, the air absorbs 256.7 W/m² instead of 49.1, and the vapor takes 206.4 of it, most of that in the near infrared. Below 10 km the sunlight now warms the air by 2.02 to 2.70 K per day instead of 0.12 to 0.16. Under the global mean of 340.5 W/m², the vapor moves about 53 W/m² of sunlight from the ground into the air, heat the ground no longer has to pass up to the air itself.

What the model computes

In the formula of Part 6, a layer's absorption in each slice comes from the paths of its gases, and vapor adds two. The lines' path grows with the layer's vapor, scaled with pressure as the other gases' paths are. The continuum's grows with the vapor times the vapor's own pressure, and faster in cold air:

Here q_k is the layer's specific humidity, \Delta m_k its mass of air per square meter, p_k its pressure and e_k = q_k\,p_k/(0.622 + 0.378\,q_k) the pressure of its vapor. p_{\text{ref}} is 500 hPa and t_{\text{self}} is 2,521.5 K. Above about 4 hPa, the pressure p_D, the lines stop narrowing as the air thins, which is all the square root does.

Evaporation

The vapor in the air comes from the ground, almost all of it from the sea. Right at the sea's surface the air is saturated at the sea's own temperature. A little higher it holds less, and the turbulence that the wind and the warm sea stir up keeps mixing the two, so vapor moves up and the sea loses water. That is evaporation. The model computes it in every cell from the air of its lowest layer, about 20 m up, in kilograms of water per square meter each second:

Here ρ is the density of that air, Part 6's layer K, U its wind speed and q_K its specific humidity, and q_s(T_s, p_s) is that of air saturated at the sea's temperature T_s and the surface pressure p_s. C_H, the exchange coefficient, is a pure number near 10⁻³ that sets how fast the turbulence mixes. Part 6's column held it at 1.2 × 10⁻³; the climate model computes it from how rough the sea is and how stable the air above it is. A kilogram of water on a square meter is 1 mm deep, so evaporation is given in millimeters a day.

Every kilogram that evaporates takes the latent heat L = 2.5\times 10^{6} J from the sea, so 1 mm a day costs the sea 28.9 W/m², paid as \class{term-d}{L E}. The air gets that heat only where the vapor condenses, and this page colors it, and the \class{term-d}{L E} that pays for it, as convection. The same turbulence carries the sensible heat of Part 6, which the climate model computes with one more term:

The g z allows for the 0.195 K that air cools by when it is lifted 20 m along Part 1's dry adiabat: air 20 m up that is 0.195 K colder than the sea takes no heat from it. The figure runs the model's own code for both fluxes over open sea.

The figure opens on a tropical sea at 27 °C, under air 1 K colder at 80 percent relative humidity and a wind of 7 m/s. The sea evaporates 4.43 mm a day, at a cost of 128.1 W/m², while the sensible heat is only 7.6 W/m². The ratio of the two, the Bowen ratio, is 0.059: over a warm sea nearly all of the heat the turbulence carries away goes as vapor. Drag the sea colder. At 15 °C it evaporates 2.16 mm a day and at 0 °C only 0.82, each degree adding 6 to 7 percent, the Clausius–Clapeyron rate of the first section. The sensible heat hardly changes, so the Bowen ratio climbs to 0.34 at 0 °C. Back at 27 °C, a wind of 12 m/s takes 7.00 mm a day. Make the air as warm as the sea, and the sensible heat turns around, to −1.7 W/m², while the sea still evaporates 3.32 mm a day.

The bars are the climate model's own ground over one day at the March equinox, run from the N = 64 state that the climate model opens with. The ground absorbs 146.9 W/m² of sunlight and sends 51.1 back up as net infrared, its own glow less what the air sends down. Evaporation takes 81.3 W/m², 2.81 mm a day, more than the infrared and five times the sensible heat: it is the largest way the ground passes the sun's energy on, where Part 6's dry ground had only the infrared and the sensible heat. Switch the bars to the sea, which covers 71 percent of the planet and supplies 89 percent of its vapor: it evaporates 3.51 mm a day, 101.6 W/m². Switch to land, and the ground evaporates only 1.09 mm a day, because it has half the sea's energy to spend and the water in its soil runs short, so its sensible heat comes close to the latent heat of its evaporation. Yet 2.28 mm of rain a day falls on the land, vapor that rose from the sea and was carried over it by the winds.

The model leaves two things out here. It takes the saturation humidity over liquid water even over sea ice, where the air at the surface is saturated over ice and holds less, and the ice pays only the heat of evaporation, not the extra heat of melting. And it has no dew: where the air holds more vapor than saturated air at the surface would, the evaporation is set to zero instead of turning around.

What the model computes

The exchange coefficient comes from Monin–Obukhov similarity theory, which describes how the wind, temperature and humidity change with height in the lowest few tens of meters of air. Near the surface each changes with the logarithm of the height, and between the surface and the air at height z the coefficient is

where 0.4 is von Kármán's constant, and z_{0m} and z_{0h} are the roughness lengths for the wind and for heat and vapor, the heights at which the logarithmic profiles reach the surface's own values. Over the sea the model takes them from the COARE 3.5 algorithm, which was fitted to measurements over the ocean:

Here u_* is the friction velocity, the square root of the wind's drag on the sea per unit of air density, ν is the air's viscosity, and α = 0.0017 U_{10} − 0.005 grows with the wind 10 m up, because a stronger wind raises steeper waves. The roughness for heat and vapor shrinks as the waves grow, so from 7 to 15 m/s C_H in neutral air stays between 0.99 × 10⁻³ and 1.03 × 10⁻³. The functions \psi_m and \psi_h correct the profiles for the air's stability, which the model reads from the bulk Richardson number,

with \theta_{v,s} the virtual potential temperature of saturated air at the surface. A negative number means air lighter at the bottom, which overturns and mixes faster. Over the figure's first sea it is −0.024, and C_H is 1.15 × 10⁻³, a sixth more than in neutral air. Slow the wind to 1 m/s and it is −1.17: C_H almost doubles, and the sea still evaporates 1.19 mm a day, a quarter as much with a seventh of the wind. Where the surface warms or moistens the air from below, rising plumes stir it even when no wind blows, and the model adds a gust for them as COARE does, U^2 = |\mathbf{v}|^2 + \max(0.2\ \text{m/s},\ 1.2\,w_*)^2, with w_* the speed of those plumes. The figure leaves the gust out; under a boundary layer 600 m deep it would add 0.08 mm a day at 1 m/s.

A wet column

Part 6's column ended with dry air over ground that reflects 30 percent of the sunlight, at 265.3 K, its convection stopping 2 km up. When Manabe and Richard Wetherald ran a column like it in 1967, they let its vapor follow its temperature by a rule that has lasted: hold the relative humidity fixed. Air that warms can hold more vapor, so under that rule it does, and since vapor is the strongest greenhouse gas in the column, anything that warms the air is amplified. That amplification is the water vapor feedback.

The figure is Part 6's column with that rule. At the ground its air holds 77 percent of the most vapor it could hold, a share that falls in a straight line with pressure to 38 percent at 500 hPa, and its ground evaporates water by the formula of the last section. Its convection lifts the lowest layer's air along its moist adiabat, and every layer above it that is colder than that path joins it, warmed onto the path by the lowest layer's sensible heat and the latent heat of the evaporated water, which rains out where it condenses. This convection is written for the figure. The climate model does not use it: it convects with the plume of the next section.

The column settles on day 1,507 with its ground at 286.6 K, 21.4 K warmer than Part 6's dry column. Its air holds 12.5 kg/m² of vapor, and every day the ground evaporates 2.07 mm of water, at a cost of 59.9 W/m², and the same amount rains out. The column holds back G = 133.2 W/m², where the dry column held back 47.9.

Convection now reaches 318 hPa, 8.5 km up, instead of 1.9 km, and below that the temperature lies on the dashed moist adiabat. At the probe's 517 hPa the vapor's sunlight warms the air by 0.60 K a day and its infrared cools it by 1.67, where the dry column's layer balanced at 0.04 K a day each way. Convection makes up the difference, 1.07 K a day. In all it hands 79.3 W/m² to the layers above the lowest, three-quarters of it the latent heat the ground paid to evaporate the water, coming back where the vapor condenses.

Convection matters far more than it did in dry air. Once the column has settled, choose “frozen” to hold each layer's vapor where it is, and switch convection off. The ground climbs 21.9 K, to 308.5 K, where over the same ground the dry column warms by only 1.4 K when its convection stops. Dry convection instead leaves the ground at 294.2 K, 7.5 K above the moist column: its temperature then falls 9.7 K per km between 850 and 300 hPa against the moist one's 8.1, so the air aloft is colder and sends less infrared to space.

With water the top of convection rises to 8.5 km, and over ground as dark as the sea to 14.2 km, within the 10 to 15 km of the real troposphere. It is the top of the weather, where the thunderstorms of Part 1 spread into anvils. The coldest air, which ozone's sunlight sets and which Parts 1 and 6 called the tropopause, does not follow it: it sits 15.4 km up, against 15.8 km in the dry column, and over dark ground 23.6 km, against Part 6's 19.6 km. In the real tropics the two levels nearly meet. In this column, with no winds and no clouds, they lie about 7 km apart, and radiation alone sets the temperature between them.

Vapor also darkens the column. It absorbs 46.2 W/m² of the sunlight before the light reaches the bright ground, so the column reflects 26.7 percent of it where the dry column reflected 31.6.

Now double the carbon dioxide. The ground settles 2.36 K warmer, with 24 percent more vapor but only 11 percent more rain: at balance the latent and sensible heat together only make up the air's net loss by radiation, which grows slowly. Go back to ×1, let the column settle, and choose “frozen” before doubling: the ground warms by only 0.73 K, so the feedback has more than tripled the warming. “Frozen” holds only the vapor the radiation sees; the lifted air still follows the moist adiabat of warmer air, which warms the upper air most and lets it shed more infrared. Manabe and Wetherald also got 2.36 K, and 1.33 K with the vapor fixed, but the match is a coincidence: their convection held the temperature falling at a fixed 6.5 K per km. This warming for a doubling is the climate sensitivity. Take the carbon dioxide away instead, and the ground falls 19.3 K and G to 46.9 W/m², about what the dry column holds back. Vapor amplifies a warming, but the gases that do not condense set the level it amplifies.

Last, set carbon dioxide back to ×1 and make the ground as dark as the sea. The column settles at 306.8 K with 65.0 kg/m² of vapor, five times as much, rains 5.45 mm a day, and convects up to 14.2 km. Between the two grounds the infrared leaving the top rises by 2.7 W/m² for each kelvin of warmer ground, where the dry column's rose by 3.6. With vapor in the air, warmer ground sends its warmth to space less readily, as Part 6 said of the real tropics.

What the model computes

Each layer's vapor is a fixed share h of q_s, the most it could hold:

Above 100 hPa the vapor is capped at 3\times 10^{-6} kg/kg. After every step the vapor is set again from the new temperatures, so the column keeps no account of its water, and what evaporates and what rains out are equal. The ground's budget is Part 6's, with one more term, the latent heat it pays to evaporate water at the rate E:

with Part 6's C_H = 1.2\times 10^{-3} and a wind U of 5 m/s. The convection lifts the lowest layer's air, with the vapor h(\sigma_K)\,q_s(T_K, p_K) it holds, along the path of the first section, whose temperature T^{\text{ad}} falls with height at g/c_p below its cloud base and \Gamma_m above it. The search starts one layer below the last step's top of convection, every layer colder than the path joins the convecting layers t to K, and the lowest layer's temperature on the path is chosen so that together the layers gain exactly the latent heat of the step's evaporation:

In the climate model, evaporation goes into the vapor of the lowest layer after each step of the winds,

and from there the boundary layer mixes it upward, the winds of Part 4 carry it, and the plume of the next section lifts it, condenses it and rains it out.

Convection and rain

The rising core of a thunderstorm is a few kilometers across. At N = 128 the centers of the model's cells are 60 km apart, and each cell is about 56 km across by area, so the grid cannot draw one. To draw one, the cells would have to be about a kilometer across, three thousand times as many as N = 128 has, each stepped fifty times as often so that no wind crosses a cell in one step: well over a hundred thousand times the work of a run that already takes 48 seconds per model day on a laptop's graphics chip, with a state of hundreds of gigabytes to stream through the processor at every step. No single computer can do that. The models that resolve storms over the whole globe run on the largest supercomputers, and at that scale the hydrostatic balance that Parts 1 and 4 built on no longer holds, so the equations themselves would have to change. Instead the model stands in for all the convection in a cell with a single current of rising air, a plume, worked out from the cell's own column at every step. A rule like this, which computes the effect of what the grid cannot resolve from what it can, is called a parameterization. The model's plume follows the one in the IFS, the forecast model of the European Centre for Medium-Range Weather Forecasts.

The figure runs one step of the model's own moist physics on a single column 53.4 km across, one of the grid’s pentagons and a little smaller than the average cell at N = 128. The column starts with no past, set to C. L. Jordan's 1958 average of the soundings, balloon flights up through the air, made over the West Indies in the hurricane season. Near the sea its air is at 26.3 °C and 89 percent relative humidity; higher up it holds about half of what it could.

The plume's air, the average of the lowest 50 hPa, starts at 300.0 K with 17.9 g/kg of vapor and saturates 450 m up, at the dot. From there it rises as cloud to 153 hPa, 14.1 km up, and it is never much warmer than the air around it: at most 1.14 K, at 440 hPa, counting its vapor, which makes it lighter, and its cloud water, which weighs it down. The energy this buoyancy gives each kilogram of the plume on the way up is its convective available potential energy, CAPE, here 259 J/kg, and the shaded area is drawn in proportion to it. Above 10.7 km the plume is colder than the air, but it is still rising and overshoots, to stop 4.8 K colder than the air. In the layers between, it leaves its air behind, where a real storm spreads into its anvil, and that cold air chills them by up to 18 K a day.

The plume stays so close to the air's temperature because it mixes that air in as it rises, which is called entrainment. It mixes in more where the air is drier and less as it climbs: near its base it replaces half of its air within 1.2 km, and above 500 hPa it hardly mixes at all. Lifted without mixing, like the first section's parcel, the same air would end up 5.7 K warmer than the air near 400 hPa, with a CAPE of 1,940 J/kg.

How much air the plume lifts is its mass flux, 0.0224 kg/m²/s at the cloud base, and a downdraft of air cooled by evaporating rain sinks from the middle of the cloud with 30 percent of it. The model sets the base flux by a rule called the closure: strong enough to use up the plume's CAPE in a time τ, the 35 minutes its air takes to cross the cloud, stretched by a factor that grows with the width of the cell. Here τ is 3,575 seconds, an hour.

The last panel shows what one step of 168.75 seconds, the climate model's step at N = 128, does to the column. The plume's condensation warms the plume, not the air around it. That air warms by up to 19.3 K a day, at 440 hPa, mostly by sinking: the air the plume lifts must come down again around it, and at 440 hPa the readout's surrounding air sinks 0.0481 kg/m²/s, as much as the plume lifts there. Sinking air warms by compression, like the air under the piston of Part 1. It brings drier air down too, so the drying peaks lower, at 791 hPa, and near the ground the plume draws away the warmest, moistest air. Added up through the column, the heating is 1,159 W/m², exactly the latent heat of the rain that reaches the ground, 40.1 mm a day, heavy rain for an average over a whole cell: the heat that evaporation took from the sea, given back high in the column.

The heating dips at 610 hPa, the first layer warmer than 0 °C. Above that layer part of the plume's water freezes, and the heat of freezing warms the plume; its frozen rain melts again in that first warm layer and takes the heat back. Switch the ice off, and the plume, less buoyant, has a CAPE of 142 J/kg, stops at 211 hPa and rains 22.1 mm a day. Switch to cells as at N = 64, and the column is 106.8 km across; τ grows to 5,058 seconds, so everything the plume does shrinks in proportion, to 28.3 mm of rain a day.

Now dry the air above 850 hPa. The plume mixes in drier air, which evaporates some of its cloud water and cools it, and by ×0.92 its rain has fallen to about 14 mm a day. At ×0.91 it regroups: its top rises a layer, its base flux grows more than fivefold, its heating moves down to 791 hPa, and it rains 28.7 mm a day. At ×0.89 it collapses, to a top at 659 hPa and 2.4 mm a day. The response jumps both ways because the plume's top can stop only at an edge between layers, and the closure weighs every layer at once. Warm the lowest kilometer instead, keeping its relative humidity, and half a kelvin nearly doubles the CAPE, to 489 J/kg, and more than doubles the rain, to 93.9 mm a day. The plume lives on the warmth and vapor of the air near the sea.

The figure shows the rates of one step, not a storm, because on its own the column soon stops. Nothing replaces the warm, moist air the plume draws away: stepped on, the deep plume rains for about 20 minutes, 0.43 mm in all, and then gives way to the shallow plume. In the climate model the sea's evaporation, the sunlight and the winds keep resupplying that air.

What the model computes

The model's moist physics runs on every column once per step, after the winds have carried the vapor and the cloud water and the boundary layer has mixed them, in three parts. None of them makes or loses heat or water: they move both between the layers, or out of the column as rain, leaving the rain's latent heat behind in the air.

The first part condenses vapor wherever the large-scale motion has brought a layer near saturation. The model takes a layer's total water q_t, vapor and cloud together, to be spread evenly across the cell about its mean, and the part of the spread above saturation is cloud. The spread is as wide as it must be for cloud to begin once the layer's relative humidity passes a critical value,

which is 0.975 at the ground and 0.761 at 500 hPa when the surface pressure p_s is 1000 hPa. With q_s the saturation humidity, the cloud water q_c and the share f of the cell it covers are

between Q = -\beta, where the cell is clear, and Q = \beta, where it is overcast and q_c = Q. The denominators allow for the condensing, which warms the air and raises q_s. No layer of Jordan's column passes its \text{RH}_c, so in the figure this part condenses nothing.

The second part is the plume. A test parcel sorts the column first. It starts from the plume's source and mixes in the air around it all the way up, whatever that air's humidity, at 0.4 \times 1.75\times 10^{-3}\ \text{m}^{-1}\,\left(q_s(T)/q_s(T_K)\right)^{3}, with T_K the lowest layer's temperature, and it drops half of its cloud water at each level. If its cloud is deeper than 200 hPa the column gets the deep plume; otherwise a shallow plume runs, reaching at most 700 hPa, and it never rains. The deep plume rises unmixed to its lifting condensation level. Above that, in every layer where it was buoyant in the layer below, it mixes in the air around it at the fractional rate, per meter of height,

where RH is the relative humidity of that air, and the last factor, the saturation humidity there against that of the first cloud layer, makes the plume mix less as it climbs into colder air. Its vertical speed starts at 1 m/s at the cloud base and changes with height as

where T_v = T\,(1 + 0.608\,q - q_c) is the virtual temperature, and u marks the plume's. Only a third of the buoyancy goes into speeding the plume up, a = 1/3, because it must also push aside the air in its way, and b = 1.949 adds a drag to the mixing. Where w^2 reaches zero the plume stops. Part of its cloud water rains out on the way up. A saturated downdraft starts in the layer of least moist static energy, c_p T + g z + L q, with 30 percent of the base flux, kept saturated by the plume's rain evaporating into it. The base flux M_b comes from the closure that Peter Bechtold and his colleagues published in 2014:

PCAPE adds up the plume's buoyancy over the mass of its cloud, and F_P is the same sum for the warming a plume of unit base flux would cause, so the base flux is the one that would remove the PCAPE in the time τ. That time is the cloud's depth, from its base z_b to its top z_t, over the plume's mean speed through it, stretched by \alpha_x, which grows with the width \Delta x of the cell, and held between 12 minutes and 3 hours. In the running model the closure first takes off the part of the PCAPE that the warming below the cloud base keeps producing, which the figure's column, with no past, does not have. The base flux is also capped, and it is turned off where the plume must first push through more than 25 to 75 J/kg of convective inhibition. Each layer then changes by what the plume, the downdraft and the sinking air around them carry across its edges, and by the latent heat of the rain made in it less that of the rain that evaporates there, so the heating, added up, is the latent heat of the rain.

The third part turns cloud water into rain, which is called autoconversion, by the rule of Edwin Kessler: cloud water above 0.2 g/kg turns into rain at a rate of 10^{-3} per second, and all cloud water decays into rain with a lifetime of an hour, or three hours for flat cloud under an inversion. The rain falls through the column within the step, evaporating into the clear, unsaturated layers below, and what reaches the ground is the large-scale rain.

Clouds

Two sections back, the wet column's ground reflected 30 percent of the sunlight as a stand-in for clouds. Real clouds do two things to the radiation at once. A thick cloud sends much of the sunlight back to space, which cools the planet. And it absorbs the infrared from below and glows in its place at its own temperature, colder the higher the cloud, so less infrared escapes and the planet keeps more heat. Each effect is the change the cloud makes at the top of the air, and together they are the cloud's radiative effect, also called cloud forcing.

In the model a cloud is water held in a layer: droplets where the air is warmer than 0 °C, ice below −38 °C, and a mix of the two in between, where some droplets stay liquid below freezing, supercooled. What it does depends on its water path, the mass of its water above each square meter, and on its cover, the share of the sky it fills. The figure places one cloud in the moist midlatitude summer of the section on vapor, over ground that reflects 7 percent, under the planet's average sunlight, and runs the model's own radiation with the cloud and without it; the column without it is what the model calls clear sky. The map colors every cloud the figure can place by its two effects together, and the arrows split the chosen cloud's effect into its sunlight and its infrared.

The figure opens on a low cloud at 894 hPa, 1.1 km up, holding 100 g/m² of droplets and covering the whole sky, like the flat sheets that Part 1's inversion caps over the subtropical oceans, which are called stratocumulus. It raises the sunlight reflected to space from 37.1 to 153.2 W/m², a sunlight effect of −116.1 W/m². Its top is only 4.9 K colder than the ground, so its infrared effect is only +6.0, and together it cools the planet by 110.1 W/m². That top radiates up into the drier, colder air above, and the cloud's layer loses 21.4 K a day to infrared, against 2.4 without the cloud, a chill that in a real deck stirs the layer beneath and keeps the deck fed with moist air from the sea.

Lift the cloud and its infrared effect grows, because its top gets colder. At 608 hPa the same 100 g/m², nearly all of it supercooled droplets, keeps in 36.6 W/m² and cools the planet by 78.2. At 270 hPa, 10.2 km up, the cloud is ice. Make it a thin veil of cirrus, 20 g/m², and the two effects trade places: it adds only 38.1 W/m² to the sunlight reflected but cuts the infrared leaving the top by 79.4, and warms the planet by 41.2. Ice crystals are larger than droplets, so a gram of ice catches less light, and these 20 g/m² have an optical depth of only 1.5, where 100 g/m² of droplets has 12.7.

The dashed line on the map is where the two effects cancel. Every cloud up to 3 km cools, however little water it holds. Higher up, thin clouds warm and thick ones cool, and the higher the cloud, the more water it takes to tip it, 118 g/m² at 10.2 km. A droplet cloud is nearly black to the infrared by 20 or 30 g/m², an ice cloud by nearly three times that, and once a cloud is black, more water adds only to its reflection. So a thick anvil, 400 g/m² of ice at 10.2 km, keeps in 117 W/m² of infrared, but its 150 W/m² of reflection outweighs that, and it cools the planet by 33.3 W/m².

Choose night, and every cloud warms. The low cloud sends 66.1 W/m² more infrared down to the ground, which is why a cloudy night stays mild. Switch the air to dry instead, and the low cloud's infrared effect triples, to 18.1 W/m², and the cirrus's grows to 129.2. In dry air 65.3 percent of the ground's infrared goes straight to space, and a cloud that blocks it makes a large difference. In moist air only 15.4 percent does, and the vapor above a low cloud has already hidden most of what the cloud would.

So far the figure has placed its cloud by hand. Switch to how much cloud, and it makes the cloud as the model makes most of its own, from humidity that varies across each cell, about 56 km across at N = 128. The model takes the water in each layer, vapor and cloud together, to vary evenly across the cell between a driest and a moistest part, and whatever lies above saturation condenses, by the rule of the section on convection. Cloud begins at a critical relative humidity, the mean at which the moistest part just saturates, 91.9 percent at 894 hPa. At a mean of 95 percent, 19 percent of the cell is cloud. At saturation half of it is, with 62 g/m² where it covers, and it cools the planet by 43.3 W/m². The plume of the last section adds cumulus cloud, and a separate model of the boundary layer adds the subtropical decks.

Two of the model's simplifications show in this figure. Vapor takes its share of the sunlight from the whole beam before any cloud scatters it, so a cloud never shades the vapor beneath it. And a cloud absorbs sunlight alike at every wavelength, 3.9 percent of what it reflects or passes on at 100 g/m².

In a five-year run at N = 128 on the model as it stood in early October 2026, the climate model's clouds took 54.1 W/m² off the sunlight the planet absorbed and kept in 26.6 W/m² of infrared, so they cooled it by 27.5. Without them the planet would have absorbed about 292.0 W/m² of sunlight, nearly the 298.0 of Part 6's dry column; with them it absorbed 237.9. It reflected 30.1 percent of the sunlight, where its clear sky would have reflected 14.2: clouds more than doubled what the planet reflected. The satellites of CERES put the Earth's cloud effects at −47 ± 4 W/m² in sunlight and +26 ± 3 in infrared, so that run's clouds kept in about the right infrared and reflected somewhat too much sunlight, though the satellites' clear sky, measured where the sky happens to be clear, is usually drier than the model's.

A wet planet

Part 6's sunlit sphere carried heat to the poles in dry air. The run below gives it water: it is the climate model itself on the N = 16 grid, with 36 layers, steps of 450 seconds and all the physics of this part on every column. There is no land: the whole planet is a sea without currents, a slab of water 10 m deep that warms and cools where it lies and freezes into sea ice. The run started from the climate model's own opening state, a sea at 28 °C on the equator with ice near each pole, and ran for a year before it recorded the next. That year brought it most of the way to a balance: over the recorded year the north still cooled by 2.1 K and its ice grew by nearly half, while the south warmed by 1.1 K.

A first attempt, with the model's usual bright ice over a slab 5 m deep, was stopped after 242 days because its ice seemed to be running away. Bright ice reflects the sunlight that would melt it, so ice that spreads helps itself spread, a loop called the ice–albedo feedback. That run's southern ice covered 48 percent of its hemisphere at its peak, against 38 in the spin-up of the run shown here, and a month before the southern summer it still covered 39 percent against 29, with the planet 2.7 K colder. Whether it would have kept growing, the run was too short to show. For the run shown, two settings changed at once: the bare ice reflects 7 percent of the sunlight, about as little as open water, and the slab is 10 m deep. Snow on the ice keeps its brightness, 70 to 85 percent, until it melts, so the feedback is weaker here but not gone.

The figure opens on the rain at the March equinox. Most of it falls in a band of storms, centered on this day between 10° and 15°S, where the trade winds of the two hemispheres meet: the intertropical convergence zone, or ITCZ. Play the year and the band wanders far, a few degrees on the equator's side of the warmest sea, which lags the sun by about three months: it reaches 23°N in mid-September and is back at 18°S by the March that ends the record. Under the sinking air of the winter subtropics hardly any rain falls, less than 0.7 mm a day between 5°S and 30°S in September, where the air holds 10 to 25 kg/m² of vapor against 45 to 53 under the band. Show a month of weather, a January, and the rain of the northern storms sweeps east in long bands. The cloud water follows the rain.

Switch to the surface temperature. Near 45°N the sea is warmest in mid-September, at 294.4 K, three months after the June solstice, where Part 6's ground took seven to eight weeks: the slab holds twice the heat for each kelvin, so it warms more slowly. By March the sea there has partly frozen, and the surface is coldest at 274.2 K in the March that opens the record and at 268.8 K in the one that ends it, a swing of 20 to 26 K where Part 6's ground swung 54 K. Near the poles it swings by about 45 K, against 87 to 89. There the winter surface is the top of the ice, which holds little heat, but the sea's heat conducts up through the ice, and the moist, cloudy air and the winds bring more from above.

In March ice covers 18 percent of the northern hemisphere at the start of the record and 23 percent at its end, and in September it covers a third of the southern one. The hemispheres get the same sunlight half a year apart, so the difference is the run's memory of its start at a March equinox, when the south went straight into winter and the north's first summer melted its starting ice. The two were drawing together: the south's ice peaked at 38 percent of its hemisphere in the spin-up and at 33 in the recorded year.

Over the recorded year the planet absorbs 216.4 W/m² of sunlight and sends out 217.6 W/m² of infrared. Its surface averages 285.4 K, 6.1 K warmer than Part 6's dry sphere although it reflects 36.4 percent of the sunlight instead of 13.1: its greenhouse holds the surface 36.8 K above the planet's effective temperature, against the dry sphere's 10.5 K. Its clouds take 77.9 W/m² off the sunlight and keep in 31.4 W/m² of infrared, against 54.1 and 26.6 in the climate model's run of the last section and the satellites' 47 and 26, so they reflect far more than the Earth's. The planet lost 1.2 W/m² at the top, but its sea, ice and air gave up only 0.2. The other 1.0 W/m² is a gap in the run's books, heat it gained from no source it records, where Part 6's run lost 0.7 W/m² to friction and smoothing.

The second figure averages the run around each circle of latitude, with arrows for how the air moves north or south and up or down. It opens on the relative humidity over the year.

Near the sea the air holds 74 to 92 percent of the vapor it could. Above it in the subtropics lies the dry, sinking branch of the Hadley cells: near 500 hPa the air holds 28 percent of what it could near 22°S and 37 near 22°N, against 52 to 56 in the middle latitudes, because it gave up its water as rain when it rose in the band and warms as it sinks. Switch to the water vapor: near the sea the trades carry 9 to 12.5 g/kg of it back toward the band. In September the winter hemisphere's cell reaches across the equator, its trades blow north across it at nearly 9 m/s, and the air sinking over the southern subtropics holds only 9 percent of the vapor it could.

Switch to the eastward wind. The westerlies peak between 160 and 200 hPa, at 42 m/s near 33°S and 32 m/s near 38°N over the year, and at the end of each winter, when the sea is coldest, at 61 m/s near 38°S in September and 57 m/s near 33°N in March. They are jet streams with a core, where Part 6's dry westerlies kept strengthening up to the model's top. The temperature shows why. The rain band's plumes carry the sea's warmth high, the air over the equator is coldest near 100 hPa, at 193 K, and above about 180 hPa the tropics are colder than the air near 60°N, so by Part 5's thermal wind the westerlies weaken upward. In winter, poleward of about 50°, they keep strengthening above 100 hPa, where the pole in its months of night is far colder than the middle latitudes: in September near 52°S they grow from 21 m/s at 300 hPa to 46 m/s at 37 hPa.

Water carries heat

Over the recorded year the sea evaporates 2.66 mm of water a day, averaged over the planet, and 2.66 mm falls back as rain and snow. Paying for the evaporation takes 77 W/m² from the sea, five and a half times the 14.0 W/m² of its sensible heat. A little more than half of the rain, 53 percent, comes from the plume, and the rest from the large-scale condensation. The air holds 23.0 kg/m² of vapor, enough for 8.7 days of rain, so water spends about nine days in the air between leaving the sea and falling back, its residence time, about the same as on the Earth. In those days the winds carry it far, and since the sea pays the latent heat where the water evaporates and the air is paid where it condenses, the heat travels with it. The figure follows this water cycle over the year: the water above, and below it the heat, added up across each latitude as Part 6 did.

Water leaves the sea fastest between 25°S and 25°N, 3.7 to 4.0 mm a day, and more slowly toward the poles, where the sea is cold and often frozen. The rain peaks near 16°N, 4.2 mm a day over the year, in the band the ITCZ sweeps across, and again in the storm tracks, 3.0 mm near 43°N and 2.5 near 42°S. Outside the band and the storm tracks the sea is the planet's source of vapor: from 37°S to 6°N and from 22°N to 37°N it gives up more water than falls back on it, up to 1.7 mm a day near 26°S and 1.1 near 29°N. The rain band between 6°N and 22°N and everything poleward of 37° are its sinks.

In all, the air carries 4.2 PW north across 40°N and 5.0 PW south across 40°S, four to five times the 1.1 PW of Part 6's dry sphere, the faint line. The Earth's winds and ocean currents together carry 5 to 6 PW at their peak, so this planet's air alone carries nearly as much. The gap it has to close is wider. From the equator to the poles the sunlight absorbed falls by 189 W/m², against the dry sphere's 218, but the infrared leaving the top falls by only 47 W/m², where the dry sphere's fell by 179. Switch the globe to the infrared leaving the top. On the opening day the rain band, over sea at 298 K, sends out 171 W/m², averaged around its latitude, about as little as the frozen north, while the dry air on either side sends out up to 267. Vapor and high cloud hide the warm sea's glow, as the wet column and the section on clouds showed.

More heat goes south than north, and 0.6 PW crosses the equator, but most of that difference is the drift of the last section: heat leaving the cooling north for the warming south. From the top of the air alone, leaving out the heat the sea, ice and air stored or gave up, the peaks would be 4.6 PW in the north and 4.7 in the south, and only 0.1 PW would cross the equator. The figure also spreads evenly over the globe the 1.0 W/m² that the run's books cannot place. Where that heat really belongs could move the crossing at the equator by about a quarter of a petawatt either way.

Of the heat the air carries, the vapor carries two-fifths to half in the middle latitudes, 2.1 PW across 35° in each hemisphere, close to the 2 PW that the Earth's air carries as vapor there. The storms carry warm, damp air poleward and cold, dry air back, and the vapor rains out on the way. In the southern tropics the latent part runs the other way. From 21°S the southern trades carry the vapor of the subtropical sea north into the rain band, which on the year's average lies near 16°N, and 1.3 PW of latent heat crosses the equator with it. North of the equator the vapor never turns back: it runs north at every latitude, into the band and on to the storm tracks. From 21°S to 11°N the heat of the dry air runs against it, 1.9 PW south across the equator and up to 4.2 PW south across 20°S, carried away aloft by the air that the condensing vapor warmed in the band. The Hadley cell's low branch brings the vapor in and its high branch takes the heat out, so there the two parts oppose each other, and what the tropics export in all is the difference: 3.3 PW across 15°S, where the dry air alone carries 3.9. On the Earth, whose rain band stays nearer the equator, the trades of both hemispheres carry vapor toward it.

Water turns Part 6's dry planet into something close to the climate model. Over ground that reflects 30 percent of the sunlight, where the dry column settled at 265.3 K, the wet column settles at 286.6 K. The wet planet, brighter than the Earth under its clouds and snowy ice, keeps its surface at 285.4 K, and the climate model's run of early October kept its ground at 289.0 K. Vapor and clouds hold back more than 40 percent of the infrared the sea sends up, the plume carries the sea's heat high into the air, and the vapor carries close to half of the heat that moves toward the poles. The sea of this part only warms and cools where it lies; the next part lets it move.