Lapse Rate

Our Radiating Clouds simulation uses 350 W/m² for incoming solar power. We know from our Solar Heat calculation that this is the average power arriving at the Earth from the Sun. Our hope is that the lapse rate and surface temperature of our simulation will agree well with the actual lapse rate and surface temperature of the Earth's atmosphere. In the design of our simulation, however, we have made no effort to adjust its parameters to bring about such an agreement.

In the figure below, the blue graph shows temperature versus altitude for thermal equilibrium in the wet atmosphere of our Radiating Clouds simulation. We loaded the equilibrium state, RC_14000hr, into CC11 and instructed the simulation to print temperatures and altitudes.



For comparison, we went back to Simulated Planet Surface and loaded the equilibrium state, Day_4, which arises with the same solar heat, but with the surface entirely made of sand. Thus the pink graph shows temperature versus altitude in a dry atmosphere.

The equilibrium state of the dry atmosphere, which looks like this, shows a linear drop in temperature with altitude. According to our calculations, this slope should be −g/Cp, where g is gravity and Cp is the specific heat capacity of the dry gas. Our simulation uses 10 N/kg for gravity and 1003 J/K for Cp, so the slope of the pink graph should be close to −0.010 K/m. Its actual slope is −0.011 K/m.

The equilibrium state of the wet atmosphere, which looks like this, shows a linear drop in temperature only between altitudes 2 km and 4 km. Near the surface of the planet, the heat liberated by condensing water vapor into rising air reduces the amount by which air cools as it rises. In the tropopause, radiation from the planet surface is absorbed by the tropopause clouds, causing them to be warmer than the air immediately below. In the linear region of the graph, the slope is −0.011 K/m, but if we simply divide the total drop in temperature by the altitude of the tropopause, we obtain a net slope of −0.008 K/m.

Looking at graphs such as these, it appears that the Earth's atmosphere has a lapse rate of around −0.0065 K/m. Our simulation does not come up with this lapse rate exactly, but we see that the introduction of water does cause a substantial reduction in the net lapse rate, and with this we are well-satisfied.

The average temperature of the surface of the Earth is around 14°C, or 287 K. This is the average temperature of the air just above the planet surface, not the temperature of the surface itself. The thermometers we use to measure the surface temperature are several meters above the ground. When we use dry air and a sandy surface in our simulation, the average temperature of the surface air is 298 K. But when we add radiating clouds, rain, and snow, the average temperature drops to 288 K, or 15°C. We are well-satisfied with this agreement also.

Thermal Equillibrium

An object is in thermal equilibrium when the amount of heat entering the object is equal to the amount of heat leaving it. In the case of a planetary system, consisting of a surface and atmosphere, the system will be in thermal equilibrium when the heat arriving from the Sun is, on average, equal to the heat the system radiates into space. The planetary system of our Circulating Cells program will be in thermal equilibrium when the short-wave radiation penetrating to the planet surface from the Sun is balanced by the long-wave radiation escaping into space from its surface, atmospheric gas, clouds, rain, and snow.

When our simulated system converges upon a state where the temperature of its surface and of its atmospheric layers fluctuates around some average value, our hope is that the heat penetrating to the surface from the Sun will be equal, on average, to the heat the system radiates into space. The heat penetrating from the Sun is, of course, the incoming short-wave radiation that is not reflected back into space by our simulated clouds. The heat radiated into space is the upwelling radiation at the top of our simulated atmosphere. We refer to the top of the atmosphere at its tropopause, and to the heat radiated into space as the total escaping power.

We instructed CC11 to print out the average penetrating power from the Sun and the total escaping power from the tropopause every ten hours, and plotted these values for the fourteen thousand hours of the experiment we performed in our previous post. For the final ten thousand hours, the temperature of the surface and the atmospheric layers fluctuate by ±1°C around their average values, as you can see here.



Over the final ten thousand hours, the average penetrating power is 290.0 W/m² and the average escaping power is 289.6 W/m². Given the size of the fluctuations in both quantities, and the errors introduced by certain simplifications in our simulation's calculations, we are well-satisfied with this agreement. Our simulation converges upon an equilibrium state that is also a state of thermal equilibrium.

Radiating Clouds

The latest version of Circulating Cells implements the upwelling and downwelling radiation calculations we described in Up and Down Radiation. To run the program, download CC11 and follow the instructions at the top of the code. Clouds absorb and emit long-wave radiation as if they were black bodies. We now set the transparency fraction of our atmospheric gas to 0.5, so that it will be transparent to half the wavelengths in the long-wave spectrum and opaque otherwise. The planet surface can radiate heat directly into space at these transparent wavelengths, as it did in Simulated Planet Surface. But now we have clouds doing the same thing, while at the same time reflecting sunlight back into space.

We begin our simulation with the final state of Simulated Rain, which you will find in SR_1200hr. The initial surface air temperature is 292 K, and cloud depth is 1.5 mm. The following graph shows how air temperature and cloud depth vary in the first two thousand hours.



The following graph shows the first fourteen thousand hours. You will find the final state of the array in RC_14000hr. The average surface air temperature over the final ten thousand hours is 288 K, and the average cloud depth is 0.8 mm.



During the course of these fourteen thousand hours, the distribution of clouds in the atmosphere varies greatly. Sometimes there is a layer of clouds just above the surface of the sea. At other times there are clouds along much of the tropopause. For a view of the final state of the simulation, see here.

As we have discussed many times before, the absorption of long-wave radiation by the atmosphere gives rise to the greenhouse effect. The more opaque the atmosphere, the more heat must be radiated into space by the tropopause instead of the planet surface. In order to radiate more heat, the tropopause must be warmer. If the tropopause is warmer, the planet surface must be warmer too, in order to motivate convection to carry heat to the tropopause. When we change our atmospheric gas from 0% to 50% transparency, we expect the surface temperature drop. And indeed it does: by 4°C.

This cooling of 4°C is, however, far less than the cooling of 31°C we observed when we increased the transparency of our gas from 0% to 50% in the absence of simulated clouds. As we have already discussed, clouds and rain greatly reduce the sensitivity of surface temperature to changes in solar power. Now we find that they also greatly reduce the sensitivity of surface temperature to changes in the transparency of the atmospheric gas.

Up and Down Radiation

We are going to add to our Circulating Cells simulation the absorption and emission of long-wave radiation by clouds. As we showed earlier, a liquid water depth of 100 μm absorbs over 99% of all long-wave radiation. Rain contains liquid water also, and ice is a good absorber of long-wave radiation too. We will add the equivalent depth of snow, rain, and cloud droplets for each cell, and so obtain the depth of water within the cell that acts to absorb long-wave radiation.

We note that the same addition of rain, snow, and cloud droplets does not apply to the transmission of short-wave radiation. Water is transparent to short-wave radiation, and clouds reflect it by refracting it through millions of microscopic droplets. But rain and snow contain thousands of times fewer drops and crystals for a given depth of water, so they are thousands of times less effective at refracting sunlight.

For simplicity, we will assume the water in a cell is either transparent or opaque to long-wave radiation, but not in-between. If the combined concentration of rain, snow, and cloud droplets in a cell is greater than wc_opaque, we will assume the entire gas cell is opaque to long-wave radiation. Otherwise, the cell will absorb long-wave radiation as if it were dry, as determined by our transparency_fraction. With our 300-kg cells, a concentration of 0.33 g/kg corresponds to 100 μm of water.

Now we are faced with the possibility of multiple layers of cloud, snow, and rain, all absorbing and emitting long-wave radiation in all directions. The first simplification we make is to assume each gas cell radiates only vertically upwards and downwards. Because our columns of cells are much the same as one another on average, the net effect of this simplification will be small. Even with this simplification, we see that a cloud can absorb radiation from a cloud below, and emit radiation back to that same cloud below, and upwards to a third cloud.

We will calculate the effect of long-wave radiation in the following way. We start at the surface and allow it to radiate as a black body. We allow this upward radiation to enter the first gas cell. We calculate how much is absorbed by the cell and how much keeps going. We calculate how much power the gas cell itself radiates upwards. We add this to the existing upward radiation. We move on to the cell above, and so on, until we get to the tropopause. At the tropopause, we assume the atmosphere above is transparent to long-wave radiation, so all upward-going radiation passes out into space.

We repeat the same process, going down. We start with the tropopause gas cell in each column and move down cell by cell until we arrive at the bottom, at which point all the downward-going radiation is absorbed by the surface. We first considered this kind of downward-going long-wave radiation in our Back Radiation post. It is distinct from the solar radiation that penetrates the atmospheric clouds because it is radiation emitted by the clouds, rain, snow, and atmospheric gas themselves.

In any cell, the long-wave radiation going up is the upwelling radiation and the long-wave radiation going down is the downwelling radiation. At the tropopause, the upwelling radiation is the heat leaving our planetary system. It is our total escaping power. When our simulation converges to equilibrium, we should find that the average solar power penetrating to the surface is equal to the average total escaping power.

Sinking Restored

When we added rain and snow to our Circulating Cells program, we removed the slow descent of microscopic water droplets, saying that their movement would be insignificant compared with that caused by convection. This is indeed the case when an equilibrium with plenty of atmospheric convection is established.

Nevertheless, we have found in our recent tests, in which we are allowing the clouds to radiate heat directly into space, that clouds can form and sit directly upon the surface, where they block the Sun's light. The surface cools beneath these clouds, and the clouds themselves cool by radiating into space, and we have seen them sit there fore hundreds of hours. This is unrealistic, because in a hundred hours, a cloud will sink by at least a few hundred meters.
So we restored the sinking of cloud droplets to our simulation, at 3 mm/s, which is realistic for cloud droplets 20 μm in diameter.

When we restored the sinking, we noticed that our previous implementation had allowed the clouds to sink only when the cells containing them took part in a convection circulation. As a result, the clouds were sinking through our simulated atmosphere a hundred times slower than they should have been. The fast-sinking implemented by CC9 were in fact sinking at 3 mm/s instead of 300 mm/s, and the slow-sinking clouds were sinking at only 0.03 mm/s instead of 3 mm/s. Thus our fast-sinking clouds were a more realistic simulation of the manner in which actual clouds would sink, while our slow-sinking clouds were unrealistically slow. We run our sinking cloud experiments with a corrected version of CC9, and the new slow-sinking result looked like the former fast-sinking result.

When clouds sink at 3 mm/s, they can sit on the surface for a few hours, but after a hundred hours, they disappear. The droplets will sink 1000 m in a hundred hours, which is three times the height of a gas cells resting upon the surface. The slow descent of the droplets removes clouds that would otherwise freeze the surface, and therefore plays an important role in our simulation, despite the fact that convection, rain, and snow cause movements that are thousands of times faster.