11 August 2008

Analyzing the simplest climate model

Before we try to solve the problem of why the temperature around us averages about 15 C (288 K) instead of the -18 C (255 K) we found in the simplest model, let's look some more at the model. It depends on 3 things -- the sun's energy input to the earth, the albedo of the earth, and the average distance between the earth and sun. How much does it depend on those things? By the end, we'll also have another site to ignore for its unreliability.

We know that the sun's output can vary by about 1 Watt per square meter during a solar cycle. So try computing the temperature with 1368 instead of 1367. Try again with 1366. How much difference does that make? I get about 0.05 K (0.09 F). That's a pretty small number. We'd be hard-pressed to get a thermometer to record it.

How about the albedo? I find about a 1 K change (0.9) for a 0.01 change in albedo. Such a change in albedo (or to increase it by this much) is plausible for the earth, though 0.10 would be a wild value, not expected by anybody I know of. Quick sanity check ... why is the albedo so much more important than solar term? Albedo multiplies the solar constant. The natural variation is 1 Watt per square meter. The 0.01 change in albedo means a 13.67 Watt per square meter change in energy entering the climate system, so we expect it to be much larger. (project: why is it 19 times more important rather than just 13.67?)

The mean distance from the earth to the sun does change -- on a long enough time scale. Changes in the earth's orbital eccentricity (how circular vs. how oval-shaped) can give changes in average distance of something like 0.001 AU. This translates to about 0.18 K (0.32 F) variations in earth's temperature to space. (These are Milankovitch variations in eccentricity, with the fastest time scale of change being 100,000 years; some are over 2 million years).

If we take those sizes of temperature change and divide by how fast they occur, we get a sense of which is most important for thinking about climate change on our time scale of interest. The sun's output changes along a solar cycle of about 11 years. The earth's albedo changes with the seasonal cycle, so 1 year. And the eccentricity is 100,000 years. Looking at degrees per year, we then have:
  1. Albedo 1 degree per year
  2. Solar cycle 0.005 degrees per year
  3. Eccentricity 0.000002 degrees per year
So, if we're thinking about weather and climate over the next few years to centuries, the obvious candidate to be paying attention to is the albedo. Eccentricity changes too little and too slowly, and the solar cycle returns to its start before the century is up (net of zero change), and is much smaller than the albedo effect can be on a few year scale (albedo doesn't go in the same direction for a long time, but just how long and how much needs some thinking).

Variations in solar output and the earth's orbit are not part of the climate system itself, and the orbit can be predicted to very high accuracy for a very long time into the future. In terms of understanding the climate system, both the solar and orbit factors are good -- given these non-climate terms, we can compute a climate. If we know the albedo.

But what is albedo? It's the bouncing of energy from the sun back to space. Now what does that bouncing? Well, everything in the climate system -- clouds, gas molecules, ocean surface, trees, grass, desert, dirt, glaciers, snow cover, ice packs, ... The gas molecule term depends little on the climate, and, if I remember correctly, would bounce about 15% of the solar input even if the earth's surface were a perfect solar absorber (perfectly black). But it isn't; even the ocean, which is about the darkest part of the earth's surface, reflects at least 6% of the sun's energy (that reaches it), and that figure increases as the sun gets low in the sky.

If none of these things changed their albedo as the climate got colder or warmer, we'd only have an annoyance -- can't give a climate figure without looking at the system. But the extent of deserts, ice sheets, sea ice, ... does change depending on climate temperatures. That 'about 0.30' albedo is correct for recent times. It may not be correct in an ice age earth or an even warming than present earth. Consequently, while we can use this model to understand some points about the climate system, we can't use it to predict full climate responses. We're not surprised, since we're much warmer at the surface than the blackbody temperature. But this is an additional reason we're going to want a more complex model down the road.

On the other hand, it does help us understand the system -- we know now that albedo is very important, and how much it and solar input could be expected, on a simple basis, to affect climate.

We also, it turns out, have a tool for identifying unreliable sources. I was surprised to see this be the case, but Steve Milloy at junkscience(dot-com)/Greenhouse assumes that if there were no clouds, the earth's albedo would be zero. Even knowing nothing about the details of albedo, you know this has to be wrong. If the earth's albedo, aside from clouds, were zero, you couldn't see the earth except for the clouds. As this image reminds us, you can indeed see the earth from space. His writing where he makes the error says:
We should note that devoid of atmosphere Earth would actually be a less-cold -1 °C (272 K) because the first calculation strangely includes 31% reflection of solar radiation by clouds
He also seems to be calculating the solar constant rather than taking an observed value, and using odd values for the his calculation.

Project: what would be better values for each figure, and how would using them affect his results?


Anonymous said...

I like a solar constant of 1370 W/m**2, and albedo of 0.31 for present conditions.

Keep in mind that the "skeleton" for which the greenhouse effect operates comes through the non-volatile gases which do not precipitate from the atmosphere under Earth-like conditions (CO2, CH4, etc) and the effects of water vapor and clouds are essentially there because of those other gases, as they are temperature dependent and will condense when it becomes too cold.

The 255 K baseline temperature with no atmosphere is a measure of the incoming solar radiation with an albedo factor of 30%. If you were to remove the trace gases like CO2, CH4, etc from the atmosphere, you'd also lose your water vapor and cloud effects as it becomes so cold that they precipitate out. You'd dramatically change the albedo (losing clouds), however, you'd also expect a much higher surface albedo as you threw the planet into a snowball state. In fact, that would probably make the planetary albedo much higher than the current 0.31, which would make a no atmosphere case much colder than the 255 K baseline.

Robert Grumbine said...

In terms of reasoning forward with the simple model, I agree about the likely course of how the albedo would change.

But that kind of thing can't actually be addressed in this model. It doesn't know about time; it's a zero dimensional model. We're stuck with whatever albedo we start with.

To address the question of just what would happen, we need at least the 1 dimensional, time only, model. For a snowball earth, we'd be looking at an albedo of 0.5 to 0.8, and quite a bit cooler. (Project: compute just how much.)

You've also reminded me of an article I mean to write up -- about the solar constant.