# Open Mind

## Spencers Folly 2

#### July 30, 2008 · 2 Comments

Part 2: Climate Forcings and Climate Sensitivities

Solar energy enters earths climate system in the form of radiation from the sun; the vast majority of it is short-wave (SW) radiation. Some of that solar energy is simply reflected back to space (the fraction reflected back is called earths albedo). Energy is also radiated away as infrared, or long-wave (LW) radiation. If we imagine an envelope surrounding the earth, at the top of the atmosphere, we can also imagine measuring the amount of energy coming in through the envelope and the amount of energy going out. If these quantities are equal, then the earth is in energy balance. If not, the difference between the amount coming in and the amount going out is the net radiation imbalance at the top-of-the-atmosphere (TOA).

If incoming and outgoing radiation are equal (so we have equilibrium), we can use those values as reference values. Then we can define anomalies as the difference between actual values and these reference values. Lets consider the difference between TOA radiation imbalance and its reference value (the reference value will be zero because we chose an equilibrium state to define our reference values), and separate that into two parts. The part due to a change in surface temperature, well call the climate feedback (note: this uses the word feedback in the sense used by Spencer, not the more usual sense see the previous post for an explanation). The part due to other factors (e.g., more energy coming in due to an increase in solar energy, or less energy getting out because of an increase in greenhouse gases) well call the radiative forcing.

If the change is surface temperature is small compared to the surface temperature (in Kelvins) itself, then the climate feedback should be approximately proportional to the temperature change. Letting $T$ be the surface temperature anomaly, the climate feedback will then be $\lambda T$, where $\lambda$ is the feedback parameter (the same one introduced in the previous post). Let $Q$ be the radiative forcing. Then the net radiation imbalance at TOA is

$N = Q - \lambda T$.

What will affect the feedback parameter $\lambda$? The most obvious and inescapable factor is that warmer objects radiate more energy thats simple radiation physics. We can compute this factor using the Stephan-Boltzmann radiation law (just as in the previous post), giving the default or no-feedback (using the word feedback in its more common meaning, see the previous post for explanation) value $\lambda_0 = 3.3$ W/m^2/K. However, temperature change can affect climate feedback (in its Spencerian sense) in other ways. For example, higher temperature can lead to more water vapor in the atmosphere, and since water vapor is a greenhouse gas, this reduces the outgoing radiation, so therell be a water-vapor feedback term (feedback in its more common sense) $\lambda_w$ which, since it inhibits outgoing radiation, will be negative. Higher temperatures mean less ice and snow, which affects earths albedo, causing less incoming solar energy to be reflected back to space, so theres an albedo feedback $\lambda_\alpha$ which, again, will be negative. There are other feedbacks (in the more common sense) as well, and the total feedback parameter (in its Spencerian sense) will be the sum of all these factors:

$\lambda = \lambda_0 + \lambda_w + \lambda_\alpha + ...$

Now we must note an extremely important difference between the default feedback parameter and the other terms. The default parameter has its effect instantaneously. This too is straightforward physics; if an objects temperature increases, it takes no time at all for it to radiate away energy at a higher rate, according to the Stephan-Boltzmann radiation equation. But all the other terms do take time for their impact to be felt. If we raise earths temperature, it takes time (weeks to months) for more water vapor to accumulate in the atmosphere due to extra evaporation from more heat. If we raise earths temperature, it takes time for snow and ice to decrease due to increased melting from more heat. Water vapor feedback, albedo feedback, in fact all the feedbacks except the default value, take time to show themselves. For some of them, like water vapor feedback, it doesnt take very much time, only weeks to months. For others, like albedo feedback, it takes longer; extra heat can take decades or even longer to melt large ice masses and bring about albedo feedback. It can even take centuries for increased temperature to cause the release of CO2 from the warming oceans, so carbon-cycle feedback can be even more slow-moving. But the default feedback, due to the straightforward fact that warmer objects radiate more, is truly instantaneous. Its the only one.

Gregory et al. (2004, Geophysical Research Letters, 31, L03205) used this formulation of net TOA radiation imbalance to estimate climate sensitivity from computer models. They perturbed the climate system with an intial climate forcing, then studied the relationship between evolving TOA radiation imbalance $N$ and surface temperature anomaly $T$. In fact they did a linear regression to determine the best-fit coefficients $Q$ and $\lambda$ for the equation

$N = Q - \lambda T$

to estimate the radiative forcing $Q$ and the climate feedback parameter $\lambda$ (and therefore the climate sensitivity $1/\lambda$); the results agreed with values estimated by other methods. Forster and Gregory (2006, Journal of Climate, 19, 39-52) specified a time series for $Q$ and then used the radiation imbalance equation to diagnose $\lambda$ from transient observations of $N$ and $T$ during 1985-1996, when $N$ was measured by the Earth Radiation Budget Satellite.

One of the caveats expressed by Forster and Gregory is that since it takes time for climate feedbacks (in the usual sense) to express themselves, the brief time span 1985-1996 may not be long enough effectively to diagnose climate sensitivity. Forster and Taylor (2006, Journal of Climate, 19, 6181) applied the methodology to the output of coupled climate model integrations, which enabled them to examine centuries of detailed data from computer model output.

Why is this relevant to discussion of Spencers folly? Because Spencer uses this method to examine climate sensitivity, both in the simple zero-dimensional one-component model discussed in the previous post, and for data from satellite observations. In the next installment, well examine what he does with this methodology.

Categories: Global Warming

### 2 responses so far ↓

• In the next installment

Too many cliffhangers! When are we going to get to the good stuff?

Just kidding. Thanks for the posts.

[Response: Soon, I promise. Stay tuned for the exciting conclusion...]

• Ken

LOL - Michael Crichton doesnt have anything on your, when it comes to suspenseful writing. And you clearly have a significant advantage on him when it comes to understanding math and science.