Ignoring large-scale motions $($waves, hurricanes, etc.$)$, the tropical atmosphere can be pictured as being made up of narrow columns of rapidly rising moist air, with the rest of the atmosphere slowly subisiding. Water vapor condenses out as the air rises, forming deep clouds. Here's an animation of two simulations of this.

In the updrafts the air follows a *moist adiabat*, with its temperature decreasing by $\sim$6.5Kkm$^{-1}$. This temperature profile is quickly communicated to the rest of the atmosphere so that, to zeroth order, the whole tropical atmosphere follows a moist adiabat on average. However, when the air descends in the clear-sky regions it tries to follow a *dry adiabat*, i.e., its temperature increases by $\sim$10Kkm$^{-1}$. This would make the air too warm to be in equilibrium with the environment and so the subsiding air cools by emitting long-wave radiation.

Assuming a steady-state, this balance can be written using the thermodynamic energy equation as
$$
c_p\frac{T}{\theta}\frac{\partial\theta}{\partial z} M = -Q_c,
$$
where $c_p$ is the air's heat capacity, $T$ is temperature, $\theta$ is potential temperature, $M$ is the net upward mass flux in the clouds $($ignoring precipitation, entrainment, etc. what comes up must come down$)$ and $Q_c$ is the radiative cooling rate in the clear air. This is a way of defining *radiative-convective equilibrium*, which is often taken as a starting point for studying the dynamics of the tropical atmosphere.

Understanding the terms in this equation tells us about how the tropical atmosphere works and also about how it will respond to future climate changes. There is still much we don't know, but there are a few basic results which give us some intuition for the behavior of the tropical atmosphere.

### A Simple Explanation for the Area of Convective Updrafts

The mass flux $M$ is determined by the area of the updrafts $(\sigma_{cl})$, the speed of the updrafts $(w)$ and the density of the rising air $(\rho)$: $$ M = \sigma_{cl} w \rho. $$

Focusing on $\sigma_{cl}$, we know that the updrafts take place in a small fraction of the domain. This asymmetry is an interesting feature of moist convection: in dry convection the area of ascending motion is the same as the area of descending motion, so the presence of water vapor adds an asymmetry to the system. Bjerknes actually gave an explanation for this in a 1938 paper, which is still a useful starting point for getting some intuition (this is a summary).

For air to rise it has to be warmer than its surroundings $($it has to have positive buoyancy$)$. Letting $T_{cl}$ be the temperature in the cloudy updraft and $T_c$ be the temperature in the clear, descending region, then for the updraft to gain buoyancy over time $\frac{\partial (T_{cl} - T_c)}{\partial t}$ must be positive. In a cloudy updraft the temperature equation gives $($now assuming that the system is not in a steady-state$)$ $$ \frac{\partial T_{cl}}{\partial t} = w_{cl}(\Gamma_e - \Gamma_m) $$ where $\Gamma_m$ is the moist adiabatic lapse rate and $\Gamma_e$ is the environmental lapse rate. In the clear region $$ \frac{\partial T_c}{\partial t} = w_{c}(\Gamma_e - \Gamma_d) $$ where $\Gamma_d$ is the dry adiabatic lapse rate. Since $\Gamma_e - \Gamma_m$ is greater than zero, and $w_{cl}$ is positive we want $w_{cl}$ to be large so that $T_{cl}$ increases rapidly. $\Gamma_e - \Gamma_d$ is less than zero, but so is $w_c$ and hence $w_c$ must be small for $T_c$ to increase slowly.

If mass is approximately conserved $M_{cl} \sim M_c$ and if the density of the air is roughly the same in the two regions then since $w_{cl}$ is larger than $w_c$, $\sigma_{cl}$ must be small compared to $\sigma_c$.

### FAT

An important consequence of the radiative-convective equilibrium balance is the Fixed-Anvil-Temperature $($FAT$)$ hypothesis, first put forward by Hartmann and Larson $($2002$)$ (though inspired by some results from Hartmann et al. $($2001$)$). The FAT hypothesis is one of our best constraints on the behavior of clouds in the tropics and does well in both models and in observations.

The FAT hypothesis says that the "spreading out" of large, deep convective clouds (i.e., the anvil) always happens at the same temperature. This tells us that if the atmosphere as a whole warms the tops of these clouds will form higher, which makes the clouds a positive feedback to climate change. This is because the anvil clouds warm the troposphere by emitting radiation to space at low temperatures. If the anvils stayed at a fixed height instead of forming higher then they would emit at a warmer temperature, acting as a negative feedback on global warming by emitting more temperature to space. Instead, by moving higher they continue emitting at cold temperatures, warming the planet.

To understand the FAT hypothesis, start by assuming the radiative cooling profile $(Q_c)$ is roughly constant up to some height just below the tropopause, after which it quickly goes to zero. At this height the subsidence must also be small, from the radiative-convective equilibrium balance, and so the rising air spreads out, forming the anvil $($the air that has just risen has to go somewhere$)$. Another way of stating the FAT hypothesis then is that $dQ_c/dz$ is largest at a fixed temperature.

The question is why the radiative cooling drops off at a fixed temperature. To answer this we have to get into radiation; Ingram $($2010$)$ is a great reference for understanding this. The first assumption is that water vapor is responsible for essentially all of the radiative cooling in the clear regions. The cooling is then roughly constant up to the height at which water vapor can no longer absorb or emit energy to space efficiently, i.e the cooling is proportional to water vapor's optical depth. Ingram gives a detailed explanation for why water vapor's optical depth is proportional to temperature as part of his discussion of "Simpson's Paradox". As is explained below his equation 1, if the relative humidity and the lapse rate are assumed to be constant then water vapor's optical depth is determined by temperature and the water vapor saturation vapor pressure: $$ \tau \sim \frac{\epsilon RH}{\Gamma}\int_{T_t}^T \frac{e_{sat}}{T'}dT', $$ where $\tau$ is the optical depth, $\epsilon$ is the ratio of the molecular masses or water and air, $RH$ is the relative humidity, $e_{sat}$ is the water vapor saturation pressure and $T_t$ is the temperature at the tropopause $($or some other height at which the water vapor emission is negligible$)$.

But from Clausius-Clapeyron we know that the water vapor saturation vapor pressure decreases exponentally with temperature and so in the end everything comes down to temperature. Note: we actually need another assumption, which is that the optical depth is determined entirely by the path length $($the mass of water vapor per unit area$)$ -- I have to assume this is a reasonable approximation.

Again, the FAT hypothesis has performed well in every test so far and the physics seem fairly straight-forward, so it seems like a fair expectation to have for future climates. The one caveat would be if relative humidity changes too much, though the standard assumption is that relative humidity stays roughly fixed under warming. However, the FAT hypothesis just addresses one type of cloud, though an important one, and it also doesn't tell us whether there will be more or less of these clouds in the future. The "IRIS" hypothesis is one way of answering this question, but this is much more controversial.