The layer model with greenhouse effect

Our simple model is too cold because it lacks the greenhouse effect. We had no atmosphere on our planet; what we calculated was the temperature of a bare rock in space (Fig. 3.3), like the Moon. Of course the surface of the moon has a very different temperature on the sunlit side than it does in the shade, but if the incoming sunlight were spread out uniformly over the Moon's surface, or if somehow the heat from one side

Fig. 3.4 An energy diagram for a planet with a single pane of glass for an atmosphere. The glass is transparent to incoming visible light, but a blackbody to IR light.

of the planet conducted quickly to the other side, or if the planet rotated real fast, our calculation would be pretty good. But to get the temperatures of Earth, Venus, and Mars, we need a greenhouse effect.

In keeping with the philosophy of the layer model, the atmosphere in the layer model is simple to the point of absurdity. The atmosphere in the layer model resembles a pane of glass suspended by magic above the ground (Fig. 3.4). Like glass, our atmosphere is transparent to visible light, so the incoming energy from the Sun passes right through the atmosphere and is deposited on the planet surface, as before. The planet radiates energy as IR light according to so Tground, as before. In the IR range of light, we will assume that the atmosphere, like a pane of glass, acts as a blackbody, capable of absorbing and emiting all frequencies of IR light. Therefore the energy flowing upward from the ground, in units of Watts per square meter of the Earth's surface, which we will call /up,ground, is entirely absorbed by the atmospheric pane of glass. The atmosphere in turn radiates energy according to so Tatmosphere. Because the pane of glass has two sides, a top side and a bottom side, it radiates energy both upward and downward,

^up, atmosphere and iiown, atmosphere.

The layer model assumes that the energy budget is in steady state; energy in = energy out. This is true of any piece of the model, such as the atmosphere, just as it is for the planet as a whole. Therefore, we can write an energy budget for the atmosphere, in units of Watts per area of the earth's surface, as

^up, atmosphere + iiown, atmosphere = ^up, ground or

2so Tatmosphere = SO Tground

The budget for the ground is different from before because we now have heat flowing down from the atmosphere. The basic balance is

We can break these down into component fluxes

Tup, ground — In , solar + Tdown, atmosphere and then further dissect them into

T4 — (1 — a) T _i_ T4 Sa 1 ground — 4 Tsolar + ea 1 atmosphere

Finally, we can also write a budget for the earth overall by drawing a boundary above the atmosphere and figuring that if energy gets across this line in, it must also be flowing across the line out at the same rate.

Tup, atmosphere — Tin, solar

The intensities are comprised of individual fluxes from the Sun and from the atmosphere

Sa 1 atmosphere — 4 Tsolar

There is a solution to the layer model for which all the budgets balance. We are looking for a pair of temperatures Tground and Tatmosphere. Solving for Tground and Tatmosphere is a somewhat more complex problem algebraically than the bare rock model with no atmosphere we solved above, but we can still do it. We have two unknowns, and we appear to have three equations, the budgets for the atmosphere, for the ground, and for the Earth overall. Three equations and two unknowns might be a recipe for an unsolvable system,but it turns out that in this problem we are free to use any two of the three budget equations to solve for the unknowns Tground and Tatmosphere. The third equation is simply a combination of information from the first two. The budget equation for the Earth overall, for example, is just the sum of the budget equations for the ground and the atmosphere (verify this for yourself).

There are laborious ways of approaching this problem, and there is also an easy way. Shall we choose the easy way? OK. The easy way is to begin with the energy budget for the Earth overall. This equation contains only one unknown, Tatmosphere. Come to think of it, this equation looks a lot like Eqn. (3.1), describing the surface temperature of the bare planet model above. If we solve for Tatmosphere here, we get the same answer as when we solved for Tbareearth. This is an important point, more than just a curiosity or an algebraic convenience. It tells us that the place in the Earth system where the temperature is most directly controlled by the rate of incoming solar energy is the temperature at the location that radiates to space. We will call this temperature the skin temperature of the Earth.

What about temperatures below the skin, in this case Tground? Now that we know that the outermost temperature, Tatmosphere, is equal to the skin temperature, we can plug that into the budget equation for the atmosphere to see that

2sa Tatmosphere = sa Tground or

Tground — V^2Tatmosphere

The temperature of the ground must be warmer than the skin temperature, by a factor of the fourth root of two, an irrational number that equals about 1.189. The ground is warmer than the atmosphere by about 19%. When we do the calculation Tground for Venus, Earth, and Mars in Table 3.1, we see that we are getting Earth about right, Mars too warm, and Venus not yet warm enough.

The blackbody atmospheric layer is not a source of energy, like some humungous heat lamp in the sky. How then does it change the temperature of the ground? I am going to share with you what is perhaps my favorite earth sciences analogy, that of the equilibrium water level in a steadily filled and continuously draining sink. Water flowing into the sink, residing in the sink for a while, and draining away is analogous to energy flowing into and out of the planet. Water drains faster as the level in the sink rises, as the pressure from the column of water pushes water down the drain. This is analogous to energy flowing away faster as the temperature of the planet increases, according to sa T4. Eventually the water in the sink reaches a level where the outflow of water balances the inflow. That's the equilibrium value and is analogous to the equilibrium temperature we calculated for the layer model. We constrict the drain somewhat by putting a penny down on the filter. For a while, the water drains out slowly, and the water level in the sink rises because of the water budget imbalance. The water level rises until the higher water level pushes water down the drain fast enough to balance the faucet again. A greenhouse gas, like the penny in the drain filter, makes it more difficult for the heat to escape the Earth. The temperature of the Earth rises until the fluxes balance again.