Climate Sensitivity and the Linearized Response

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Climate Sensitivity and the Linearized Response

June 28, 2012 by climatephys

In discussions of climate change, it is often useful to think about the transition of the climate from one state to another, and ask how the magnitude of the response is related to changes in a control parameter (such as the solar constant, or CO2 concentration). This is the classical problem of climate sensitivity, which is intimately connected with assessing the degree to which Earth has the capacity to change. In such an analysis, we typically begin by reducing the “climate” to a single variable, commonly global mean temperature ( $T$ ), and gauge its evolution as a function of the control parameter. Of general interest to society, for example, is how the global mean temperature responds to changes in CO2 concentration. I will build up some theoretical background into climate sensitivity in this article, and move on to observations and modeling results in subsequent posts.

I write the energy balance equation as before, which is typically a close balance between the incoming solar radiation and the outgoing longwave radiation (OLR). The latter depends on the atmospheric temperature profile and opacity, as discussed in previous posts.

$\displaystyle OLR - \frac{Q(1-\alpha)}{4}= R$

where R is the net top-of-atmosphere radiation budget, which is zero in steady state, defined so that R > 0 means the planet cools down, and R < 0, it warms up. We will assume the planet radiates according to $\epsilon \sigma T_{s}^4$ where $T_{s}$ is the surface temperature, and $\epsilon$ is now introduced to represent the bulk planetary emissivity (dominated by the atmosphere's ability to trap exiting surface radiation…the surface itself has an emissivity very close to one). If the radiation to space emanated exclusively from the ground, then $OLR = \sigma T_{s}^4$ . In reality, the ground radiates about 390 W/m², and because of the greenhouse effect, about 240 W/m² exits out of the top of the atmosphere. This establishes a temperature gradient between the surface (~288 K) and the effective emission temperature of the planet many kilometers above the surface (~255 K) corresponding to a bulk emissivity of $\epsilon \approx 0.6$ . We can write,

$\displaystyle Q(1-\alpha) = 4\epsilon T_{s}^4$

Suppose we are interested in the case in which we alter the atmospheric greenhouse concentration (say, doubling CO2), by changing $\epsilon$ , but leave the incoming solar radiation unchanged. Then taking the derivative,

$\displaystyle 0 = 4\epsilon(4\sigma T_{s}^3)dT_{s} + (4\sigma T_{s}^4)d\epsilon$

For the present Earth case, we have noted that $\epsilon \sigma T_{s}^4 = 240 W/m^2$ . The radiative forcing for doubling CO2 is $\sigma T_{s}^4d\epsilon = -4 W/m^2$ (i.e., every doubling of CO2 reduces the outgoing radiation (or equivalently, the planetary emissivity) by roughly 4 W/m² at fixed temperature). Solving for the temperature change above, we have:

$\displaystyle dT_{s} = -\frac{T_{s}}{4} \; \{\frac{\sigma T_{s}^4 d\epsilon}{\epsilon \sigma T_{s}^4}\}$

which reduces to dT_s = -(288/4)(-4/240) = 1.2 K, so we expect every doubling of CO2 to change the global temperature by just over a degree.

The analysis above, in the language of climate scientists, is one that ignores feedback effects. In other words, only the temperature adjusts to the change in CO2 concentration (and correspondingly, the outgoing radiation of the planet, which is a pre-requisite to reach a new steady state). The increased or decreased radiation loss to space, as the planet warms or cools (respectively) is an extremely strong stabilizer of climate. We call this the Planck feedback. This is the default reference system used by the climate community by which we can evaluate other feedback effects (and as a reference system, is not often called a “feedback”). Feedbacks do not exert a direct forcing themselves (like external knobs such as CO2) but modify the net TOA energy balance indirectly because of their response to changes in temperature. This can further amplify or dampen the final temperature response. One such feedback already encountered was the ice-albedo feedback. Another is the water vapor feedback, which is likely the most important feedback for modern global warming.

Brief Water Vapor Feedback detour

On Earth, there is a substantial reservoir of water at the surface in the form of ocean or ice. If left undisturbed for a long enough time, water vapor would continue to enter the atmosphere until it reached a point in which its partial pressure equaled the saturation vapor pressure, which is an exponentially increasing function of temperature. Therefore, the content of water vapor in our atmosphere will increase or decrease as temperature goes up or down, respectively; since water vapor is a greenhouse gas, the additional water vapor will amplify whatever caused the initial warming (and same for cooling influences). This idea can be generalized to any greenhouse substance that exists in a temperature regime allowing it to be condense on a planet or moon in consideration (such as methane on Titan). As it happens, the atmosphere does not usually reach saturation, but it turns out that we can think of the fractional change in water vapor (i.e., the slope) as a great approximation to the slope of the saturation vapor curve (i.e., the fixed relative humidity assumption, which is borne out by observation and comprehensive models). This means the actual water vapor content increases quasi-exponentially with temperature, producing a positive (or destabilizing) feedback. A negative (or stabilizing) feedback suppresses the impact of the initial perturbation. I will explore more of the nuances of the water vapor feedback in a later post.

Feedback Physics

Let ⌀ be a control parameter (say CO2 concentration), and r a feedback factor that depends on temperature, i.e., r = r(T), both of which modify the net TOA energy budget, R. Then,

$\displaystyle \frac{dR}{d\phi} = \frac{\partial R}{\partial T} \frac{\partial T}{\partial \phi} + \left (\frac{\partial R}{\partial r} \frac{\partial r}{\partial T}\right) \frac{\partial T}{\partial \phi} + \frac{\partial R}{\partial \phi}$

The new temperature response (now including feedbacks) is therefore:

$\displaystyle \frac{\partial T}{\partial \phi} = - \frac {\frac{\partial R}{\partial \phi}}{(1-f)[\frac{\partial R}{\partial T}]}$

where,

$\displaystyle f = - \left (\frac{\partial R}{\partial r}\frac {\partial r}{\partial T} \right) / \frac{\partial R}{\partial T}$

These expressions may look complicated, but are rather simple when you examine the terms individually. The numerator in equation A is a measure of the radiative forcing (i.e., how the net TOA energy budget is altered with changes in the control parameter, like CO2 concentration). The bracketed term in the denominator is a measure of how the energy budget is altered with changes in temperature, since increase in temperature in almost all circumstances lead to an increase in the outgoing radiation. With no other feedbacks, this slope is $4\sigma T^{3}$ .

Climate sensitivity is proportional to the 1/1-f term. As can see in equation B, this is a function of how a feedback variable responds to temperature, and how that alteration actually influences the planetary energy budget. For example, we can replace r with the water vapor concentration of the atmosphere. Water vapor increases with temperature (i.e., $\partial r/\partial T$ is positive) while the outgoing radiation is reduced as water vapor increases ( $\partial R/\partial r$ is negative). This makes the entire expression for f positive, implying a positive feedback.

For f greater than zero, but less than one, the feedback is positive but still smaller than the stabilizing influence of the denominator (the increased radiation loss to space as the planet warms). This is to say that OLR still increases with temperature, even if the atmospheric opacity is increasing with temperature too. This increases climate sensitivity but does not result in a climate system that “runs away.” See the figure below. This diagram shows the outgoing radiation as a function of surface temperature, and two different values for the absorbed incoming radiation. CO2 is fixed in this diagram. You can think of the applied forcing as the change in sunlight between the red and green horizontal lines. The black curve corresponds to a planet that radiates according to $\sigma T^{4}$ (i.e., with only the Planck radiation “feedback” operating). The blue curve corresponds to the same case except with a water vapor feedback, which allows the atmospheric opacity to increase with temperature. This reduces some of the curvature you’d expect with a pure T⁴ dependence. In steady state, the outgoing radiation curves intersect at the incoming solar radiation lines. Therefore, the climate sensitivity for the black OLR curve case is the difference in the distance between the two blue squares as the solar radiation is increased. The climate sensitivity for the blue OLR curve is the difference in the distance between the red circles as the solar radiation is increased. One can see that you need to increase the temperature more in the latter case to reach the same radiative equilibrium. Put another way, the more sensitive system is less efficient at shedding its infrared emission to space, and so the temperature must increase by that much more in order to come back to balance. One can include shortwave feedbacks into this picture by plotting R on the vertical axis instead of OLR, or adjusting the position of the horizontal lines (as in the ice-albedo post).

OLR vs. surface temperature for a blackbody (black curve) and an atmosphere with CO2 and a water vapor feedback (blue curve). The horizontal lines give two values for the absorbed incoming solar radiation, and the colored shapes give possible equilibrium points. On the trajectory where water vapor exists, sensitivity is enhanced because the temperature difference between the two red circles (as sunlight goes up) is greater than the difference between the two blue circles.

Emergent Properties of the Feedback Theory

There are important consequences to the dependence of climate sensitivity on 1/1-f. One is that the effects of feedbacks on climate sensitivity are not simply additive. The second is that how uncertainties in the feedback strengths themselves propagate onto the system response, depends itself on the “true” value of the feedback strength. Some illustrative examples will clear this up.

We can re-write the above expressions as:

$\displaystyle \Delta T = \mu \Delta T_{0} = \frac {\Delta T_{0}}{1-f}$

where $\Delta T_{0}$ is the “no-feedback” temperature response (i.e., the response to a given forcing with only the reference system operating, such as the Planck response, and no other feedbacks). $\mu$ is called the system multiplier, which amplifies or dampens the sensitivity. If its value is greater than one, feedbacks are net positive; if its value is less than one, feedbacks are net negative. We can now shed light on the two properties outlined above.

1) Suppose two feedbacks are operating, both of which on their own would amplify sensitivity by 20% ( $\mu_{1} = \mu_{2} = 1.2$ ) relative to the case where only the Planck radiative response was working. This corresponds to both feedbacks factors, f₁ and f₂, both of which have values of 0.17. The total system response with both feedbacks operating is:

$\displaystyle \Delta T = \frac {\Delta T_{0}}{1-(0.17+0.17)}$

If the feedbacks impacted the system response in an additive fashion, you’d expect a 40% amplification of sensitivity ( $\mu_{1} + \mu_{2} = 1.4$ ), but in this case the response is actually 1.52 times the no-feedback reference system. This is because the feedbacks must also act upon each other, and not just on the reference system. The deviation from the additive property would be even more striking if the individual feedback multipliers were larger than 20%.

2) Suppose we don’t know the true value of f for a feedback of interest, but want to know how that uncertainty might correspond to uncertainty in the system response. Then we can differentiate the temperature with respect to a feedback factor,

$\displaystyle \frac{\delta(\Delta T)}{\delta f} = \frac {\Delta T_{0}}{(1-f)^2} = (\Delta T_{0}) \mu^2$

The dependence on $\mu^2$ means that for some uncertainty in the feedback, $\delta f$ , the uncertainty in the actual temperature response is larger for higher climate sensitivities. This property has been used by Roe and Baker, 2007 (for example) to suggest that narrowing the range of climate sensitivity may be quite difficult (however, see Hannart et al., 2009). A symmetric distribution of the uncertainty in the strength of the feedbacks results in a skewed distribution in the climate sensitivity itself, with a high probability of large values, though it should be noted that there’s no good a priori reason for uncertainties in feedbacks to be symmetric given information from observational and paleoclimate data. As a more practical matter, when using observations, sensitivity distributions depend on the prior assumptions made for the feedbacks. It took the community a while to figure out why some studies showed so dramatically different results from others. It depends largely on whether symmetric uncertainties were assumed for the feedback or the inverse of it. A linear relationship between observational data and the radiative restoring efficiency of the planet, implies that the derivative of the sensitivity with respect to the data becomes zero in the limit of high sensitivity (i.e., it would be harder to distinguish the difference between a 5 and 10 C sensitivity based on looking at the transient response to a volcanic eruption, than it would the difference between a 1.5 and 2 C sensitivity; see e.g., Frame et al., 2005 ).

The f=1 regime

One might expect based on the above equations that as f becomes close to one, the climate sensitivity blows up to infinity and a runaway effect develops (physically, the total feedback effect begins to overwhelm the stabilizing influence of the Planck radiation response). We encountered this with the Snowball Earth discussion, whereby a runaway ice-albedo feedback became sufficiently strong at low enough CO2 concentrations (or sunlight) that it forced the entire planet into an ice-bound state. This runaway feedback then ceases once the planet is covered in ice.

Of course, one cannot use this linear analysis anymore as f becomes too close to one. Feedbacks can be thought of in terms of a Taylor expansion series, and linearization corresponds to just the first term in such an expansion. The next term involve those that depend on df/dT (i.e. the dependence of the feedbacks on the climate state), then d²f/dT², etc. One can imagine that which feedbacks operate, or the relative importance of those feedbacks, depends on the climate state, and so the linear analysis is no longer valid for large enough perturbations. In the next figure, I plot the net TOA energy budget (this time, R) on the vertical axis vs temperature on the horizontal axis. We apply a radiative forcing, $F$ , that brings the climate state from the blue curve to the red curve (for instance, by adding CO2 and lowering the OLR). If there are a set of processes (like an ice-albedo feedback) that happens to give that climate the feedback structure shown in the diagram, then f = 1 actually doesn’t mean a runaway, it means a bifurcation from the first blue circle to the next. f=1 could mean a runaway if there were no equilibrium point on the other side of the bifurcation, but no information can be obtained about where such an equilibrium point might be (or if it exists) without knowledge of the non-linear Taylor expansion terms.

_{See discussion in the Pierrehumbert lecture linked below}

That is a bit of introduction into thinking about feedbacks from a theoretical perspective. Later posts will be sure to elaborate on this, and aim at more practical diagnostics of sensitivity for the modern case, in additions to discussions of other extreme bifurcations aside from Snowball Earth (e.g., the runaway greenhouse).

Some recommended reading/viewing on this is Hansen et al., 1984, Roe, 2009, and this lecture by Ray Pierrehumbert, the last one focusing on bifurcations specifically.

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Posted in Uncategorized | 17 Comments

17 Responses

on February 23, 2013 at 4:17 pm | Reply RW

Chris,

What convinces you the cloud feedback is positive?

This plot below is alone enough to show that clouds acting to further warm on incremental warming is almost certainly wrong.

http://www.palisad.com/co2/gf/st_ca.png

It’s a scatter plot of 25 years of satellite measurements from ISCCP (1983-2008). The green and blue dots are the averages for each 2.5 degree slice of latitude in each hemisphere. Each individual small orange dot represents a monthly average for one sampled grid area in a 2.5 degree slice of latitude. There are more individual grid areas in the tropics compared to the areas closer to the poles, because the total area from east to west decreases with latitude.

What makes this plot and data unique is it’s just the total cloud amount independent of cloud type or combination of cloud types. The inflection point around 0C is where the net effect of increasing/decreasing clouds switches from warming to cooling. The net effect at and above the current global average temperature is unambiguously to cool.

Do you see how at temperatures above about 0C the net effect of clouds on average is to cool, and below 0C the net effect of clouds is to warm? That is above 0C, the more clouds there are the cooler it is on average, and below 0C, the more clouds there are the warmer it is on average?

Do you agree that ice and snow are roughly as reflective to solar energy as clouds are? Do you agree that ice and snow generally only persists at temperatures at or below 0C? Do you agree that on global average, most of the Earth is not snow and/or ice covered? Is it just a coincidence that the net effect of clouds switches from warming to cooling at about the same point that the surface becomes less reflective than the clouds above?

Can you see the fundamental physical mechanism behind this? Above 0C, clouds are more reflective than the surface, so the net effect of clouds is to cool by reflecting more incoming solar energy away than is delayed beneath them (i.e. re-directed back toward the surface). Below 0C, clouds are about equally reflective to solar energy as the surface is (due to snow and ice), so the net effect of clouds is to warm by delaying more energy beneath them than is reflected away in total. Clouds on average are much more opaque to upwelling infrared radiation emitted from the surface and lower atmosphere than the clear sky is.

In a warming world, if anything, less of the surface would be snow and ice covered (not more). Hardly a case for the net effect of clouds acting to further warm instead of cool on incremental global warming.
on February 23, 2013 at 8:05 pm | Reply Chris Colose

Plotting the cloud cover distribution as a function of temperature in the current climate is just a snapshot of climatology and doesn’t have change information…the former has more to do with the atmospheric dynamics (location of subtropical regions, etc) than anything to do with feedbacks, and ISCCP is not suitable for trends. And you can’t actually diagnose the net effect of clouds as a function of latitude of the ice line…in the current climate, the shortwave and longwave effects basically cancel in the tropics for example. It’s true that clouds can warm over reflective surfaces (clouds almost certainly warm in snowball Earth for example) but that doesn’t tell you about the derivative of clouds for perturbations about the modern climate
on February 23, 2013 at 8:15 pm | Reply RW

Chris,

Relative to climate change, it’s the change in the average net behavior that counts. That’s why that particular plot is so significant.

Another way of explaining this which may be easier to see and understand is on average as clouds increase at temperatures above about 0C, the surface temperature cools and as clouds decrease the surface temperature warms. At temperatures below about 0C on the other hand, as clouds increase, the surface temperature warms and as clouds decrease the surface temperature cools. The signature of this in the data itself is independent of why the clouds increase or decrease, though if you can grasp the somewhat counterintuitive nature of how feedback works, the direction of causation is that above 0C increasing cloud coverage on average causes cooling and decreasing cloud on average coverage causes warming.

If this is still hard to see or grasp, take a look at these gain plots from the same ISCCP data set:

http://www.palisad.com/co2/plots/wbg/nh/gain.png
http://www.palisad.com/co2/plots/wbg/sh/gain.png

As the surface temperature increases, the cloud coverage increases, and as the surface temperature decreases, the cloud coverage decreases. Notice how in both the northern and southern hemispheres, the surface temperature (i.e. the ’surface out’) stays well above 0C (273K = 315 W/m^2) throughout the entire year. The ‘gain’ in the plots is just the dimentionless ratio between the surface power (i.e. net power gained at the surface) and the post albedo incident solar power. Notice also as the temperature increases and the cloud coverage increases, the ‘gain’ decreases, and as the temperature decreases and the cloud coverage decreases, the ‘gain’ increases. As the cloud coverage increases, the decreased gain reduces or attenuates the surface temperature increase, and as the cloud coverage decreases, the increased gain reduces or attenuates the surface temperature decrease. This is negative feedback in response to a surface temperature change. If the feedback was positive, as the surface temperature and cloud coveraged increased, the gain would increase – forcing or amplifying temperatures even higher.

Can you see how this is working? That cloud coverage appears to be modulating the surface temperature changes? That is, overall, when clouds are increasing the surface is too warm and trying to cool, and when clouds are decreasing the surface is too cool and trying to warm. In the end, it’s probably just basic thermodynamics and cloud physics.

Can you provide a better explanation for why the net effect of clouds is to cool by about 20 W/m^2?
on February 23, 2013 at 8:41 pm | Reply RW

Chris,

There must be a physical reason why the net effect of clouds in the current climate is to cool (by about 20 W/m^2 on global average), right? Generally, understanding why the net effect currently is to cool would be important if one is trying to figure out what the net effect will be on incremental warming, would it not? Can you at least agree with this?

Also, the data itself (in the first plot) doesn’t directly imply anything about causation in either direction, or even why the percentage of cloud coverage is what it is at a particular latitude or in a specific hemisphere. It shows that above about 0C, the more clouds there happen to be in a particular area the cooler it is on average, and below about 0C, the more clouds there happen to be the warmer it is on average. Can we agree on this point?
on February 23, 2013 at 11:18 pm | Reply Chris Colose

I understand the plot (which I am assuming to be plotted correctly and meaningfully, I have not used ISCCP data before), but I don’t agree with your interpretation. Simple statements like “increased cloud amount = negative feedback” are simply not valid because you have to consider both sides of the energy budget (I actually don’t think plotting total cloud cover is meaningful at all for this question, whether or not it’s easier to understand). If cloud feedback were a simple linear function in T than the problem would have been solved a long time ago, but this isn’t what you’re looking at. Obviously regions like the Sahara desert are direct evidence that warmer doesn’t mean cloudier. In this case, it’s the General Circulation controlling the low relative humidity zone through the descending branch of the Hadley cell and cross-isentropic flow from eddies in the mid-latitudes. In the deep tropics, you have converging zones of air just from elementary considerations on our rotating planet, and wherever you have convergence you need to have some lift and outflow to conserve the column mass, and this will be associated with adiabatic cooling and condensation during ascent. Moreover, the tropical free troposphere is constrained to have weak temperature gradients in the climatology, so obviously once you have too high SSTs you will organize clumped convection and get clouds…this is just because the surface air is buoyant with respect to the free troposphere. But in a warming climate, the free troposphere is also warmer, so you have a new threshold SST for convection. This is all very elementary stuff with no straightforward application to feedback strength. The seasonal cycle also shows that simple temperature scalings are inappropriate: in areas of the NH mid-latitudes, clouds become more abundant in winter, and in the tropics, there are huge variations in the seasonal hydrologic cycle and local cloud cover despite a weak seasonal temperature change.

For these and other reasons, the spatial slope of cloud-temperature relationships with latitude may have nothing to do with the temporal slope at a given location.

I also don’t think that “why do clouds cool by 20 W/m2″ is the right question…it’s like asking “why is the Earth ~150 million km from the sun?” as if that were some magic solution that emerges from physics, when in fact you can have myriad number of orbits. Regarding clouds, that’s the climate regime the modern Earth sits in, and past climates or other planetary atmospheres will have a different cloud forcing magnitudes. I’m also not sure knowing this net effect is critical to understanding the incremental changes, just as one doesn’t need to know the mean depth of the sea in order to make sea level rise projections.

P.S. Normally I don’t mind questions in good faith…but since you came over from Roy’s blog, I’m really not interested in entertaining everyone’s crackpot theory about why there’s no greenhouse effect, why clouds have to be a negative feedback because the last decade hasn’t done much warming (when in fact I never even brought up cloud feedbacks) etc. There are a lot of elementary issues in your analysis, Roy’s, and virtually everyone commenting over there and I don’t have much time to play gotcha games with everyone’s worldview about climate dynamics that could be corrected with a little intellectual self-respect by picking up a textbook. I will respond if others want to be serious, but this is not the place to play anti-AGW for the sake of nothing but being rebellious. I’ve seen the blogs too long now and I know the games and endless cycles one can get into without anyone ever saying anything. If those people only want to have their pet theories validated by other people who don’t know what they are talking about, there’s a number of other blogs that play the “this is all interesting” free-for-all game.
- on February 24, 2013 at 6:34 am | Reply RW
  
  Chris,
  
  The ‘gain’ in those plots quantifies the same thing and has the exact same physical meaning as how they define feedback in climate science. The ‘gain’ is the ratio between the surface power and the post albedo solar power entering the system. The global average gain with a surface temperature of 287K is about 1.6 (385/239 = 1.61), where 385 W/m^2 is the net power gained at the surface to sustain 287K and 239 W/m^2 is the post albedo solar power entering the system. The global average gain is also the ultimate origin of the so-called ‘zero-feedback’ response from a doubling of CO2. For warming, a response less than the global gain indicates negative feedback and a response greater than the global gain indicates positive feedback. That is, the dimentionless ‘gain’ going down as the forcing and temperature goes up physically means and quantifies the exact same thing in regards to the net direction of the feedback and resulting temperature increase at the surface – be it net negative (less than 1.6) or net positive (greater than 1.6). I emphasize this point because it seems most everyone in the atmospheric science community doesn’t even seem to realize or recognize this.
  
  If this is still not clear, consider that the so-called ‘zero-feedback’ Planck response of about 3.3 W/m^2 per 1C of warming is itself directly derived from the global dimensionless gain ratio of 1.6 (i.e. +1C = +5.3 W/m^2 from S-B; and 5.3 W/m^2/1.6 = 3.3 W/m^2).
- on February 24, 2013 at 7:00 am | Reply RW
  
  Also, take a look at these flux plots for reference, which show the post albedo incident energy (‘power in’) and LW flux out at the TOA (‘power out’) for each hemisphere:
  
  http://www.palisad.com/co2/plots/wbg/nh/flux.png
  http://www.palisad.com/co2/plots/wbg/sh/flux.png
  
  The ‘gain’ is literally 180 degrees out of phase with input power. This is the unmistakable signature of a system dominated by net negative feedback.
  - on February 24, 2013 at 10:36 pm Chris Colose
    
    Now you’re just making things up, supported by graphs which look like a Microsoft Paint job from an unreliable site. I’m really not interested. See this post for the proper description of feedback terminology
  - on February 25, 2013 at 11:18 pm RW
    
    If you’re not intersted, I guess you’re not interested. Dimentionless ‘gain’ is basic engineering concept. That the climate science community doesn’t recognize or understand that it quantifies the same exact thing in regards to the net direction and magnitude of the feedback is really poor on their part.
    
    There is more detailed information regarding a lot of this data I’ve presented here:
    
    http://www.palisad.com/co2/eb/eb.html
  - on February 25, 2013 at 11:42 pm RW
    
    That is, the ‘gain’ is just the ratio between the power supplied into the system and the power gained at the surface. When the post albedo solar power supplied in increases and the surface temperature increases, the dimentionless ratio of these power densities decreases and vice versa, which indicates negative feedback.
on February 26, 2013 at 12:01 am | Reply Chris Colose

The climate science community is well aware of the definitions, see the Roe 2009 review article for example. The problem is that you have no clue what you are looking at because you’ve ignored many terms in the energy budget equation, haven’t even applied energy balance to the planet or surface properly, have made up your own definitions, and have ignored real-world evidence that cloud feedbacks cannot simply be scaled to temperature. If you actually want to move beyond faith in your flawed analysis, try reading http://isccp.giss.nasa.gov/role.html for a basic understanding of the problem. There’s nothing worse than a fanatic who knows something that just ain’t so…thus, this is all I’m going to allow until you give a real effort to learn the basics.
on March 1, 2013 at 12:54 am | Reply RW

Chris,

“The problem is that you have no clue what you are looking at because you’ve ignored many terms in the energy budget equation, haven’t even applied energy balance to the planet or surface properly, have made up your own definitions, and have ignored real-world evidence that cloud feedbacks cannot simply be scaled to temperature.”

How about some specifics here? You seem so absolutely sure, which makes seem more than fair to ask.
- on March 9, 2013 at 2:13 am | Reply RW
  
  Chris,
  
  Let me ask you this, do you agree that for a state of energy balance, net power gained at the surface is the same as the power directly radiated from the surface?
  - on March 10, 2013 at 2:14 am Chris Colose
    
    No, there’s a lot of non-radiative fluxes and horizontal transport.
  - on March 10, 2013 at 7:57 pm RW
    
    Of course. The balance at the surface is the sum of a radiant and non-radiant flux where additive superposition applies to the effects of energy (and power) on the surface temperature. To the extent that the surface receives more direct radiative power from the atmosphere and Sun than it emits, the excess must be replacing non-radiative power leaving the surface, but not returned (effectively making it net zero energy flux entering the surface).
    
    The point is, in the steady-state (for a surface temperature of about 288K), all power in excess of 390 W/m^2 incident on the surface has to be exactly offset by power in excess of 390 W/m^2 leaving the surface, and that the surface specifically emits 390 W/m^2 of radiative power solely due to its temperature (and emissivity, which is really close to 1). Moreover, all non-radiative power leaving the surface has to be in excess of that directly radiated from the surface, otherwise the surface temperature would be higher, where as there is no such requirement for the proportions of radiant and non-radiant power incident on the surface from the atmosphere. Thus, 390 W/m^2 is the net power gained at the surface required to sustain 288K, which is then also the power directly radiated from the surface at a temperature of 288K.
  - on March 10, 2013 at 8:00 pm RW
    
    Put more succinctly, do you agree that 390 W/m^2 is the net power gained at the surface to sustain 288K, and that if more than 390W/m^2 is gained, the surface warms, and that if less than 390 W/m^2 is gained, the surface cools? And that this is entirely independent of how the joules are supplied to the surface?
    
    If not, why not?
  - on May 19, 2013 at 3:11 pm RW
    
    Still no reply, Chris?

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