The blue plot in the graph below shows the temperature of a single cell performing a complete circulation within our Planetary Greenhouse simulation. The cell exchanges no heat with its neighbors. We have mixing fraction zero, Q_heating at 0.01 K, and T_balance at 250 K. The cell warms up in the bottom row (the Sun heats the surface) and cools down in the top row (the tropopause radiates into space).
In any given row of the atmosphere, the cell is 20 K warmer on the way up than on the way down. Even though a rising cell may pass by a falling cell, they exchange no heat. The convection is adiabatic. The average temperature of the bottom row is roughly 50 K warmer than the average temperature of the top row.
Now suppose we allow the passing cells to exchange heat. The warm, rising cells cool down because they mix with the cool, falling cells. In order to reach the top, a cell will have to start off hotter than it would without mixing. Cells rise part-way, mix with their neighbors, fall, and mix with their neighbors again. The pink plot shows the variation in temperature, with mixing fraction 0.4, as a cell rises and falls during a hundred thousand iterations of the simulation. The average temperature of the bottom row is roughly 70 K warmer than the top row.
The following plot shows how bottom-row temperature varies with mixing fraction for Q_heating at 0.01 K. (Thanks to Peter Newnam for the data.) As we increase the mixing, circulation is further impeded, but but the mixing itself begins to transport sufficient heat to bring down the surface temperature.
Circulation alone is an efficient transporter of heat. The following plot shows that we see little change in the atmospheric temperature profile as we increase Q_heating by a factor of ten from 0.01 K to 0.1 K. The profile for 0.1 K is erratic, but on average is almost the same as that for 0.01 K.
When we allow mixing, we break up the continuous circulation that would otherwise transport heat from the surface to the tropopause. Instead of one continuous circulation we have a series of circulations carrying heat up in stages. Heat transport between stages takes place by mixing. The average temperature of the surface must increase because heat exchange by mixing requires an additional temperature drop over and above the adiabatic temperature drop produced by circulation.
To round off our explanation of the effect of mixing, imagine a continuous circulation from bottom to top consisting of two parallel columns of cells. One column is rising as the other falls. Now we suppose that each rising cell must be 10 K warmer than its side-by-side neighbor in the falling column, or else circulation will stop. Furthermore, we suppose that each time the circuit rotates by one cell, we allow some mixing between the side-by-side neighbors. The rising cell cools by 1 K and the falling cell warms by 1 K. Because of this mixing, a cell that has just descended will be 1 K warmer than without mixing. If it is to be 10 K cooler than its new side-by-side neighbor, this neighbor must be 1 K warmer than in circulation without mixing. The 10-K difference between the side-by-side neighbors in the column will be sustained only if the total temperature drop from bottom to top increases by 1 K multiplied by the number of cells from bottom to top. In our case, that would be 15 K.
Mixing raises the temperature difference necessary to transport heat through the atmosphere. The temperature difference is therefore greater than that dictated by adiabatic convection alone. In future posts we will enhance our simulation to allow cells within the array to radiate heat directly into space. This radiation, taking place before the cells reach the tropopause, will decrease the temperature drop between the bottom and the top. For now, however, we are satisfied that our program is effective in its simulation of mixing and circulation.