The cloud that cools more slowly than the air

The layer a parcel cannot leave displaced a parcel of air in a column and asked whether it came back. A parcel carried upward expands into the lower pressure, does work on its surroundings and cools, at $g/c_p$ — 9.8 kelvin per kilometre for dry air — and the column is stable if the parcel ends up colder than the air beside it. The essay closed by naming what it had left out: the water. A rising parcel that saturates begins to condense, the latent heat released warms it, and it cools more slowly. A column can therefore be stable to dry displacements and unstable to saturated ones.

Later essays followed waves and mixtures in stratified fluids: the salt that sinks through a stable sea, where two substances diffusing at different rates release an instability hidden in a column that looks stable, and the mixture heavier than either water, where mixing two stable waters makes an unstable one. This essay returns to the parcel and adds the water, and finds the same shape of result as those two: a column stable by one measure and unstable by another, with the switch between them thrown by a change of phase.

The heat a cloud carries #

Water vapour is a fuel carried in the air without being burned. Evaporating a kilogram of water takes 2.5 million joules, which the heat that changes no temperature traced into the work of pulling molecules apart, and condensing it gives the same back. A kilogram of humid tropical air holds about eighteen grams of vapour, so condensing all of it would release about forty-five thousand joules, enough to warm the air by about forty-five kelvin.

How much a rising parcel can hold is set by the vapour-pressure curve, which rises by about seven per cent for each kelvin. A parcel that rises cools, the amount of vapour it can hold falls, and once it has cooled to its dew point the excess must condense. From then on, every hundred metres of rise cools the parcel, which forces some vapour out, which warms the parcel back up by part of what it lost. The cooling and the warming balance at a rate below the dry one.

The saturated, or moist adiabatic, lapse rate follows from the first law with the latent heat included:

$$\Gamma_m = g\,\frac{1 + L r_s/R_d T}{c_p + L^2 r_s \varepsilon/R_d T^2},$$

where $r_s$ is the mass of vapour per mass of dry air at saturation, L the latent heat, $R_d$ the gas constant of dry air and ε the ratio of the molecular weights of water and air. When there is little vapour, the fractions collapse to $g/c_p$, the dry rate. When there is a lot, the second term in the denominator dominates and the rate is small. Saturated air at 25 degrees near the ground cools at only 3.8 kelvin per kilometre, less than half the dry rate; at minus twenty degrees, half-way up the troposphere, it has so little vapour left that it cools at 7.7, approaching the dry rate.

The quantity a cloud keeps #

A dry parcel has a temperature that changes as it rises and a quantity that does not: its potential temperature, the temperature it would have if brought back down to a reference pressure without gaining or losing heat, which the water that warms on the way down needed for the same reason in the deep ocean. Two dry parcels can be compared by their potential temperatures wherever they happen to be, and the one with the higher value is the lighter at any common level.

A saturated parcel loses that conservation, because condensation heats it. What it keeps instead is a quantity that counts the heat its vapour would release, the equivalent potential temperature: the potential temperature the parcel would have if all its vapour were condensed and the latent heat kept. Through a cloud it stays nearly constant while the temperature, the potential temperature and the vapour content all change. The two together describe a column far better than its temperature alone. A layer whose potential temperature increases with height is stable to dry displacements; if its equivalent potential temperature decreases with height — humid air beneath drier air — the layer can become unstable when it is lifted as a whole, its lower part saturating first and cooling more slowly than its upper part. The humid ground layer of a summer day under a dry layer aloft is such a column, and a cold front lifting it as a block can release a line of storms.

Stable, unstable, and the band between #

Two lapse rates make three regimes. A layer that cools with height more slowly than the saturated rate is stable to every parcel, dry or saturated: displace any parcel upward and it ends colder than its surroundings. A layer that cools faster than the dry rate is unstable to every parcel and overturns at once; that happens in the air just above sunlit ground on a summer afternoon, and almost nowhere else, because the overturning removes it. Between the two the layer is conditionally unstable — stable to an unsaturated parcel, unstable to a saturated one.

The atmosphere’s average cooling rate, 6.5 kelvin per kilometre, lies in that middle band at every temperature above about freezing in the lower atmosphere. So the troposphere, almost everywhere, is a store of instability that dry air cannot release. Push dry air up and it sinks back; lift humid air until it condenses, and it may keep going on its own. What the column will do is not decided by its temperature profile alone but by its humidity and by whatever does the lifting — a front, a sea breeze, a mountain, the heating of the ground.

A parcel that leaves the air behind #

Follow one parcel up a humid tropical column. From the ground it rises unsaturated and cools at the dry rate, faster than the air round it, so it is colder and heavier: it must be pushed. At 890 metres it reaches its dew point and a cloud base forms — the flat base of a cumulus cloud is this height, the same for every cloud lifted from the same surface air, about 125 metres for every kelvin between the ground’s temperature and its dew point. Above it the parcel cools at the saturated rate, slower than the surroundings, and catches up with them. At 2.1 kilometres it is as warm as the air beside it; above that it is warmer, lighter, and rises on its own, accelerating, until the saturated rate has steepened with height enough to bring it back below the surroundings near the top of the troposphere, at 13.4 kilometres.

The shaded area between the parcel’s path and the environment, the buoyancy integrated through the height over which it is positive, is the convective available potential energy, CAPE: here 2,543 joules per kilogram. If all of it went into the parcel’s motion it would reach seventy-one metres a second, which is an upper bound. Real updrafts are slowed by mixing in drier surrounding air, by the weight of the water they carry, and by pressure forces from the air they push aside, and the strongest measured are about half that. They are enough to hold hailstones aloft. The smaller area below the level of free convection, a hundred joules per kilogram here, is the convective inhibition, the work that must be done to lift the parcel to where it can rise freely. A column with a large CAPE and a modest inhibition can sit through a hot morning and erupt in an afternoon, once the ground’s heating or a passing disturbance has paid the inhibition.

Flat bottoms and ragged tops #

The ascent explains a familiar shape. On a summer morning the air near the ground is well mixed, every parcel lifted from it has much the same temperature and dew point, and every one saturates at the same height. A field of cumulus clouds therefore has flat bases at a common level, which rises through the day as the ground warms faster than it moistens and the gap between temperature and dew point widens. The tops are another matter: each parcel stops where it has used up its buoyancy, which depends on how much drier air it has mixed with on the way, so tops are ragged and varied, and the tallest towers punch past the level where their buoyancy ran out, carried by their momentum, and spread into an anvil against the stable layer above the tropopause.

The same reasoning puts clouds in fixed places in a moving airstream. Air crossing a mountain is set oscillating, and where the oscillation lifts it past its condensation level a cloud forms and sits still while the air blows through it — the wave that is required to stand still — with a flat base at the condensation level and a smooth top where the air descends again.

A sounding, read twice a day #

The figures are drawn from an idealised column, and forecasters do the same thing with measured ones. Twice a day, at the same hour round the world, several hundred balloons are released carrying instruments that radio back temperature, humidity and pressure as they rise, and their profiles are plotted on charts designed so that dry and saturated adiabats appear as families of lines. A forecaster lifts a parcel from the surface by eye along those lines and reads the cloud base, the level of free convection, the equilibrium level and the area of positive buoyancy, exactly as the ascent figure does by arithmetic. Satellites now supplement the balloons by inferring temperature at many heights from the spectrum that is a thermometer at every height, though they are coarser near the ground, where the parcel starts.

What the water pays #

The buoyancy has a precise source. By ten kilometres the parcel has condensed more than nine-tenths of its water, and the latent heat released has warmed it by forty kelvin relative to dry air lifted the same distance. That forty kelvin is the margin by which it outruns the cooling of the environment. The water that condenses forms cloud droplets, some of which grow into rain and fall out, and the rain that reaches the ground is a measure of the energy the cloud has used: a centimetre of rain falling on a square kilometre released about twenty-five million million joules higher up.

That is the sense in which a thunderstorm is a heat engine, running between the warm, humid surface and the cold upper troposphere, and its fuel is vapour evaporated somewhere else, often days earlier and thousands of kilometres away. Most of the heat that the tropics move away from the surface goes this way, as latent heat carried up in convective towers, rather than by the air being warmed directly.

The engine description can be taken literally for the largest storms. A hurricane draws heat from the sea surface at about 300 kelvin, mostly as evaporated water, carries it up in the rings of cloud round its eye, and gives it up by radiation from the outflow at the top of the troposphere, near 200 kelvin. Treated as a heat engine between those two temperatures, its efficiency is bounded by the Carnot ratio, about a third — the bound the ceiling on every engine set before any engine was designed — and the work it does is spent against friction at the sea surface. Kerry Emanuel turned that bookkeeping into an estimate of the strongest wind a given sea and atmosphere can support, which matches the strongest storms observed. It also says why hurricanes need sea water warmer than about 26 °C: below that, the vapour the sea supplies is too little to make the inflowing air buoyant enough to carry the engine.

The wind that comes down warmer #

The asymmetry between the two lapse rates shows up most plainly on either side of a mountain range. Air forced up the windward slope cools at the dry rate to its condensation level and then at the saturated rate, raining on the slope as it goes. At the crest it has lost much of its water. Descending the far side, it is compressed and warms, and with no cloud droplets left to evaporate it warms at the full dry rate the whole way down — faster than it cooled on most of the way up. It arrives at the foot of the lee slope warmer and much drier than it started: in the figure, ten kelvin warmer, with its dew point fallen from fourteen degrees to six.

Nothing has heated the air but its own vapour, condensed on one side of the range and left there as rain. The warm, dry wind on the lee side of a range is called the foehn in the Alps and the chinook east of the Rockies, where it is known as the snow-eater for how quickly it removes snow by evaporation. Chinooks have raised the temperature of a town by more than twenty degrees in minutes, when the warm air displaced a shallow layer of cold air sitting against the foot of the mountains. The lee of a range is a rain shadow for the same reason it is warm.

A tropical atmosphere set by its clouds #

Over the warm tropical oceans the arrangement goes further: the atmosphere’s temperature profile is set by convection itself. Wherever the column becomes unstable, deep convection erupts and carries heat upward until the instability is spent, and the air between the storms, warmed by subsidence, ends up with a temperature profile close to the moist adiabat of the air at the bottom. The lapse rate of the tropical troposphere is not 6.5 kelvin per kilometre by coincidence; it is the saturated rate of tropical surface air, imposed on the whole column by the clouds.

That has a consequence for a warming climate which follows from the first figure alone. Warm the surface air by a kelvin and its saturated lapse rate falls, because warmer air carries more vapour; the moist adiabat from the warmer surface is less steep, and the upper troposphere warms by more than the surface — by one and a half to two times as much at ten kilometres in the tropics. The amplified warming aloft is a prediction of every climate model and a direct consequence of the curve above, and how closely the measured warming of the tropical upper troposphere matches it has been checked repeatedly with balloons and satellites.

What the parcel leaves out #

The parcel is idealised in several ways the figures do not hide. It is a sealed bag: real cloudy air mixes with its surroundings, and the drier air it entrains evaporates some of its droplets and cools it, which is why measured updrafts fall short of $\sqrt{2\,\mathrm{CAPE}}$. All the condensate is assumed to fall out at once, the pseudo-adiabatic assumption; carrying the water instead adds its weight and a little to its heat capacity. Saturation is taken over liquid water at all temperatures, though below freezing some droplets freeze and release a further latent heat of fusion, which adds buoyancy high in a storm. The environment is a single straight-lapse-rate profile, where real soundings have inversions, dry layers and moist layers. And the density of air depends slightly on its vapour content, which the figures neglect.

Each of those is a refinement that a forecaster’s tools include. None of them changes the asymmetry: a saturated parcel cools more slowly than a dry one, and a column between the two rates is stable or unstable according to whether its air has been brought to saturation.

Still open: how clouds change as the climate warms #

The parcel argument gives the amount of energy available to convection, and a warmer, moister atmosphere has more of it. What it does not give is how often, how deeply and how widely the energy is released, and that is where the uncertainty in projections of climate is largest. Clouds reflect sunlight and trap infrared, and whether low clouds over the subtropical oceans thin or thicken as the climate warms changes the warming itself by a large factor. Their behaviour depends on the mixing of their tops with dry air above, on turbulence too small for global models to resolve, and on droplet physics; recent satellite records and high-resolution simulations have narrowed the range, and how much further it can be narrowed is an open question.

The habit worth carrying away is to ask of a stable state what it is stable to. Air cooling at 6.5 K per kilometre returns a dry parcel and releases a saturated one, which cools at 3.8 to 7.7 K/km as its condensing vapour pays it latent heat — 40 K of warming by ten kilometres for a parcel that started at 30 °C, and 2,543 J/kg of available energy. The column was never stable; it was stable to the parcels that had not yet condensed.