Evaporation Cycle

The following diagram presents the simplified cycle of evaporation and precipitation we propose to implement in Circulating Cells Version 10.1.



Evaporation takes place from the sea, as before. When a body of moist air rises, it cools, and microscopic droplets form by condensation. Clouds of such droplets that happen to descend from above will warm up, and some or all of their droplets will evaporate. A cloud of droplets whose temperature drops below some threshold T_f will be transformed into snowflakes by the Bergeron Process, warming the surrounding gas with latent heat of fusion. We choose T_f several degrees below the freezing point of water, so we can assume the freezing takes place rapidly.

In our simulation, snow will fall at an average of 1 m/s, which we base upon our own observations. We will implement snow fall in the same way we implemented sinking clouds. Snow that reaches the surface will melt and thus take its latent heat of fusion from the surface block. This melting at the surface is the simplest way we can think of to conserve the latent heat of fusion of the water involved in our evaporation cycle. We assume that our surface water itself never freezes, no matter how cold it gets.

Our clouds, meanwhile, will no longer sink. Real cloud droplets are of order ten microns in diameter and sink at a few millimeters per second. The contribution of such sinking to our new cycle would be negligible.

Snow that enters a gas cell at a temperature greater than T_m will melt, cooling the surrounding gas by absorbing its latent heat of fusion. The melted snowflakes become raindrops a few millimeters in diameter, and these fall at 5 m/s. With the simulation set up as we have it now, the cells are around 400 m high, so rain will take a minute or two to fall out of one cell into the next. We will choose Tm several degrees above the melting point of water so we can assume the melting takes place rapidly.

Our simplified evaporation cycle omits many interesting evaporation-related phenomena. When rain drops are carried up into cold air, for example, they form hail, which later falls to Earth. When liquid rain falls into sufficiently dry air, it evaporates and disappears altogether, giving rise to virga. When air rises at just the right speed along a mountain slope, water droplets join together to form rain drops, as in orographic precipitation. Our simulation will contain none of these interesting phenomena. But we believe it will capture the fundamental features of the Earth's evaporation cycle, and so allow us to investigate how this cycle influences the global surface temperature.

Rain

Our simulated sky never clears. Clouds fill the atmosphere almost entirely. They are forever forming in air that rises from the sea, and forever sinking to the ground, but they never come falling out of the sky all at once in the big drops we know as rain.

The droplets in our clouds are tiny. Those in our slow-sinking clouds are only 10 μm in diameter and descend at 3 mm/s. Those in our fast-sinking clouds descend at 300 mm/s. The graph we present in Falling Droplets implies that these fast-sinking droplets are 100 μm in diameter. Rain falls to Earth at several meters per second, so the same graph tells us that rain drops are at least 500 μm in diameter. A drop 500 μm in diameter contains a hundred times as much water as a droplet of 100 μm and a hundred thousand times as much water as a droplet of 10 μm. Could it be that cloud droplets collide and coalesce in order to form rain drops? If so, how long does this take, and under what circumstances does it occur?

The Wikipedia page on rain describes convective precipitation and orographic precipitation. In both these forms of rain, a cloud moves up, and encounters rain drops descending from above. If the descending drops are 500 μm in diameter, and the air is moving up at 1 m/s, the drops will remain at the same altitude. The cloud moving up and past them carries microscopic droplets that can collide with the stationary drops, coalesce with them, and so enlarge them until they are heavy enough to fall out of the rising cloud and descend to the Earth as rain.

But further reading suggests that rain formed of coalescing droplets is rare. A far more potent source of rain drops are ice crystals. In Cloud Physics, we learn of the Bergeron Process, whereby ice crystals grow, sink, melt, and become rain drops. Large rain-drops are melted hail-stones. Small rain-drops are melted snow-flakes.

In our simulation, whenever the concentration of water vapor exceeds the saturation concentration, we assume the excess water condenses. It turns out, however, that the surface tension of liquid water makes it hard for water to condense into floating, microscopic droplets. If we provide a solid surface for the water to condense against, such as a blade of grass or a glass mirror, the water will condense when it reaches the saturation concentration, but in a body of air high above the ground, the only such surfaces would be dust particles, and these may be rare. Each one will serve as a catalyst for condensation until a droplet forms around it.

But the same is not true of ice crystals. In air saturated with water vapor and below the freezing point of water, an ice crystal can form on a grain of dust, and after that it will continue to grow. Water vapor deposits directly upon the surface of the crystal, thus changing state from gas to solid in one step, and the newly-created ice surface is an ideal foundation for further growth.

Now, suppose a cloud of microscopic water droplets rises until its temperature drops to −20°C. We might assume that the droplets will freeze. But pure water droplets resist freezing until they drop to −40°C. Ice crystals form in the midst of the cloud of super-cooled water droplets. As water vapor is deposited on the crystals, the concentration of water vapor in the air drops.

And here we encounter another curious physical phenomenon. The saturation concentration of water vapor with respect to an ice crystal turns out to be lower than the saturation concentration of water vapor with respect to super-cooled liquid water. Water vapor will deposit on the ice crystals until the concentration of water vapor drops to the saturation concentration of water vapor with respect to ice crystals. Because this concentration is below the saturation concentration with respect to super-cooled liquid water, the water droplets actually start to evaporate. The droplets evaporate, and their water is deposited onto larger and larger ice crystals.

Once the ice crystals are large enough, they start to fall, and they eventually fall into air that is warm enough to melt them. They turn into drops of water and fall to Earth as rain. It is this process that we will attempt to simulate in the next version of our Circulating Cells program.

Less Reflection

With 350 W/m² arriving from the Sun, 75% of the surface covered by water, clouds sinking at 300 mm/s, and each 3 mm of cloud reflecting 63% of sunlight, our CC9 simulation converges upon a surface air temperature of −12°C. When we increase the Sun's power to 400 W/m², the temperature rises by a mere 0.5°C. Our simulated planet is kept cold by thick clouds that reflect the Sun's light back into space. Ice crystals drift down from the sky in some places, while elsewhere water evaporates from the frozen seas.

The surface of the Earth is at an average temperature well above the freezing point of water, and the Earth's sky is frequently clear of clouds. Our simulated sky never clears, and the surface is frozen. It never rains in our simulation, nor do our simulated clouds emit or absorb radiation. Perhaps these two omissions are responsible for our permanent clouds and frozen seas. Before we attempt to rectify them, however, let us consider the effect of decreasing the reflecting power of our simulated clouds.

We increased Lc_water from 3.0 mm to 6.0 mm, so that it now takes 6.0 mm of cloud water to reflect 63% of the Sun's light. With the reflecting power divided in half, we ran our simulation for eight thousand hours from the starting point CS_0hr. You will find the final state in LR_8000hr.



Compared to before, we now have more clouds in the sky. The following graph shows how cloud depth and surface air temperature vary with time.



Compared to before, we see the atmosphere reaches equilibrium in on third the time. The new temperature is higher and the cloud cover is thicker. The following table compares the state of the atmosphere for both types of clouds.



Our seas are now at −3°C. If they contain salt, they will not freeze. The air a few meters above our sandy island will be just below freezing. Our simulated world is still much colder than the Earth, and nobody standing on the island would ever see the Sun. We are, however, gratified to find that our simulation remains stable with such a large drop in cloud reflectance.

Negative Feedback

With fast-sinking clouds, our circulating cells program reaches equilibrium in eight hundred hours of simulation time. With the 350 W/m² arriving continuously from the Sun, the surface air temperature settles to 261 K.

If we increase the power arriving from the sun, it seems reasonable to suppose that the surface temperature of our planet will rise. Indeed, before we added clouds to our simulation, we could use Stefan's Law to answer this question. The planet surface absorbed all the Sun's heat and the surface and tropopause radiated it all back into space, so if we increased Solar power by 4%, the absolute temperature of the surface and the tropopause would increase by 1%. But with clouds reflecting light from the Sun, we can no longer assume that all the Sun's heat will be absorbed, nor even that a constant fraction of it will be absorbed.

We ran our fast-sinking clouds simulation repeatedly from the same CS_0hr starting conditions, each time with a different Solar power. Each time we stopped the simulation after a thousand simulated hours, so that we could be sure it had reached equilibrium, and recorded the surface air temperature. We obtained the following graph.



Without clouds, a doubling of Solar power would cause the surface temperature to increase by roughly 20%. Here we increase Solar power from 200 W/m² to 400 W/m² and the surface temperature increases by only 1.5%. The following graph shows how the cloud depth increases with Solar power, thus decreasing the fraction of Solar power that penetrates to the surface.



The Sun's light, arriving at the surface, causes evaporation. This evaporation leads to clouds. But these same clouds reflect the Sun's light back into space. Thus one effect of Sunlight arriving at the surface is to reduce the amount of Sunlight arriving at the surface. The effect of clouds is an example of negative feedback. This negative feedback reduces the sensitivity of surface temperature to Solar power by more than a factor of ten.

Fast-Sinking Clouds

In our previous post we allowed the clouds in our simulation to sink to the surface at 3 mm/s. We implemented this sinking by allowing an average of 0.001% of each cell's water droplets to drop down out of the cell every second (0.1% every 100 s). Today we repeat our experiment from the same starting point, but this time the gas cells lose 0.1% of their water droplets per second, which corresponds to droplets sinking at 300 mm/s. Here is the state of the atmosphere after thirteen thousand hours.



Although our screen shot is taken at thirteen thousand hours, the atmosphere converges to its equilibrium state in a mere eight hundred hours. Our previous simulation converged only after eight thousand hours. The following graph shows surface gas temperature and cloud depth versus time.



The following table compares the equilibrium state of the atmosphere at the end of our two experiments.



The faster-sinking clouds cause the surface to warm by 5.3 K. The Solar power penetrating to the surface increases by 13 W/m² because the clouds are slightly thinner. You may recall that our current simulation of clouds does not implement their absorption and emission of long-wave radiation, so we are working with transparency fraction set to 0.0, indicating an atmospheric gas that is opaque to long-wave radiation. The only place for this artificial atmosphere to radiate is at the tropopause. So we expect to see the tropopause radiating the same amount of heat that penetrates to the surface: the heat leaving the system must be equal to the heat entering. And indeed this is the case to within a couple of Watts per square meter.

We see that faster-sinking clouds cause the world to warm up, and this is in keeping with our expectation. The icy surface must warm up so that evaporation will keep up with the greater rate of return of water to the surface.

UPDATE: It turns out that our code was allowing clouds to sink only when they took part in a circulation, which resulted in them sinking roughly a hundred times slower than they should have, so our effective sinking rate here was more like 3 mm/s. When we correct our error, so that the clouds really do sink at 300 mm/s, the surface temperature warms by roughly 7 K. [07-JAN-12]

Slow-Sinking Clouds

In Falling Droplets, we concluded that 10-μm water droplets will sink at 3 mm/s in air at pressure 100 kPa. The lowest atmospheric cells in our Circulating Cells program are at 100 kPa, and they are roughly 300 m high, so we see that it will take a hundred thousand seconds for a droplet to fall the height of the cell. Cells higher up are taller, but the gas within them is thinner. A droplet must fall more quickly through thinner air before its weight is matched by air resistance. For simplicity, we will assume that the time it takes a 10-μm droplet to fall the height of a cell is the same regardless of altitude.

As we found in Simulation Time, our program checks each gas cell every one hundred iterations on average, which corresponds to every 100 s. If it takes a droplet one hundred thousand seconds to fall the height of a cell, and the droplets in the cell are evenly distributed, 0.1% of the droplets will sink out of the cell every 100 s. If the cell rests upon a surface block, these droplets will return to the surface. We now have a way for water to leave the surface, by evaporation, and a way for water to return to the surface, by sinking. If the gas cell rests upon another gas cell, the droplets enter the cell below, where they may evaporate.

In Circulating Cells Version 9.1, we specify the sinking speed of droplets at 100 kPa with sinking_speed_mps in units of meter per second. We set sinking_speed_mps to 0.003 m/s and loaded CS_0hr into our array, which is the starting condition we used in Cold Start.

The planet warms quickly in the steady light of the Sun. After two hundred hours, the atmosphere is full of clouds and the planet starts to cool. The clouds sink towards the surface. After four thousand hours, they are thin enough that the sun starts to warm the surface. After eight thousand hours, this warming is stopped by the formation of new clouds. After a thirty thousand hours, the atmosphere settles to the steady state shown below, which you will find saved as a text array here.



The graph below plots temperature and cloud depth versus time for the first thirty thousand hours.



After thirty thousand hours, the sand and water surfaces are both at 260 K (−13°C), and the lower gas cells are at 255 K (−18°C). The average cloud depth is 3.2 mm and the average power reaching the surface is 120 W/m².

The combination of evaporation and sinking gives rise to an equilibrium in which the clouds allow just enough heat to reach the surface so that evaporation balances the return of water to the sea in the form of sinking droplets. This balance between evaporation and sinking controls the temperature of the planet surface. In our next post, we will increase the sinking speed by a factor of a hundred and see how this affects the surface temperature. We expect the surface temperature to go up, because only then will evaporation keep up with the increased loss by sinking.

UPDATE: It turns out that our code was allowing clouds to sink only when they took part in a circulation, which resulted in them sinking roughly a hundred times slower than they should have, so our effective sinking rate in this simulation was more like 0.03 mm/s. When we correct our error, so that the clouds really do sink at 3 mm/s, the surface temperature warms by roughly 7 K. [07-JAN-12]

Falling Droplets

In Clouds Without Rain we saw immortal clouds circulating above a frozen planet, reflecting the Sun's heat back into space. We concluded that it is rain that saves our planet from freezing. It is rain that clears the skies so that the Sun's heat can reach us.

We must implement rain in our simulation, so that water vapor has some way of returning to the surface. Let us begin by considering how fast water drops fall through air. A falling drop accelerates until air resistance matches its weight. At that point, it continues to fall but it does not accelerate. It has reached its terminal velocity. In The Terminal Velocity of Fall for Water Droplets in Stagnant Air, Gunn et al. describe their apparatus for measuring the terminal velocity of water droplets, and present their measurements in graphs and tables. (The paper, published in 1948, is an enjoyable read that you can download here.)

We can calculate the terminal velocity of rigid, spherical objects using Stokes Law. Gunn et al. show that Stokes' Law applies well to water droplets of diameter less than 100 μm (one tenth of a millimeter). You may recall that the droplets in our simulated clouds are roughly 10 μm in diameter (one hundredth of a millimeter). For the droplets are small enough, surface tension is able to maintain a spherical shape in the face of air resistance.

But for droplets larger than 100 μm, Stokes Law over-estimates the terminal velocity. Larger droplets assume flattened shapes as they fall, and they are in constant motion, so that the air resistance they encounter is far greater than it would be for a rigid sphere. When the diameter exceeds 5 mm, the motion of the drop becomes so vigorous that the drop breaks into smaller drops.

The following graph shows the Gunn et al. measurements of terminal velocity, plotted against droplet diameter. We see that the maximum terminal velocity for the largest possible water droplets is around 10 m/s. For diameters less than 0.1 mm, Gunn et al. assure us we can use the terminal velocity given by Stoke's Law, so we also plot the terminal velocity calculated from Stokes' Law.



The 10-μm droplets in our simulated clouds will fall at a mere 3 mm/s through our gas cells. Given that our cells are a few hundred meters high, it will take a day or two for a cloud to fall from one cell to the cell below. Slow as this may be, the sinking of clouds does provide a way for water to move from one cell to another, and ultimately to return to the planet surface. In our next post, we will see how sinking clouds affect the result of our Cold Start simulation.