Climate Sensitivity

Stephen Schwartz of Brookhaven National Laboratory recently wrote a paper entitled Heat Capacity, Time Constant and Sensitivity of Earth's Climate System, that describes a simple "single-compartment energy balance model" to estimate “climate sensitivity": the equilibrium change in the global mean surface temperature in response to a doubling of the atmospheric carbon dioxide concentration.

Schwartz’ Method
"The equilibrium sensitivity of Earth's climate is determined as the quotient of the relaxation time constant of the system and the pertinent global heat capacity" (Schwartz, 2007)

S = τ / C

Schwartz arrives at a value of 5 ± 1 years for the time constant τ and a value of 1.1 ± 0.5 °C for the sensitivity S.

Difference Between Schwartz' result and that of IPCC
Schwartz' central value for sensitivity (1.1°C) is significantly less than the central value given by the IPCC: 3°C . In addition, even Schwartz "upper" value of 1.6°C is outside the range of "likely" values specified by the IPCC in their recent Fourth Assessment Report: "2 to 4.5°C with a best estimate of about 3°C, and ... very unlikely to be less than 1.5°C".

In addition, Schwartz’ value for τ is small compared to that accepted by many climatologists*. Schwartz defines τ as "the time-dependent response of Earth's average surface temperature to an imposed radiative forcing." It basically determines how quickly the climate system responds (temperature-wise) to an energy imbalance or “forcing" (which, in this case, is caused by an increase in atmospheric CO2).

*Some climatologists do not even accept the idea of a single "system" time constant (see below) and believe that certain components of the climate system (eg, deep ocean) equilibrate on a decade time scale.

Implications of Schwartz' results
Schwartz’ relatively short time constant (years rather than decades) means the earth reaches a new energy balance (on account of increased temperature) relatively quickly in response to an increase in the atmospheric CO2 concentration (an imposed “radiative forcing”).

If Schwartz is correct, the "global warming" (increase in the global mean surface temperature anomaly) that we have seen so far accounts for the vast majority of what we will see from the total greenhouse gas concentration increases to date. If that is the case, there is little extra “unrealized” warming still “in the “pipeline” (as is normally assumed by climate scientists).

Schwartz’ relatively small (in comparison to that of IPCC) climate sensitivity value means that the mean global surface temperature anomaly goes up a relatively small amount in response to a doubling of CO2 -- by about 1/3 the amount specified by the IPCC.

Perhaps not unexpectedly, some have already taken Schwartz results to mean that “Global warming is nothing to worry about”. Others -- including many climate scientists --are not so quick to dismiss the consensus of the IPCC.

Questions about Schwartz' results and methodology
Some have questioned the applicability of the methods Schwartz used to obtain his result and the validity of the model itself. In particular, they have noted the fact that the earth's climate system has several different components (eg, atmosphere and oceans), which have very different characteristic time constants and heat capacities. For example, the deep ocean takes a long time to heat up as compared to the atmosphere, which heats up fairly quickly in response to a radiative forcing increase. Because of this, some have questioned whether it is valid to use a single heat capacity C and a single time constant τ to estimate climate sensitivity.

Objective
My intention here is not to challenge the validity of Schwartz' basic model or even to critique his methods for estimating C and τ. Instead, I would like to take Schwartz' assumptions about the relationship between heat capacity, time constant and climate sensitivity at face value and to explore the implications in terms of warming due to CO2 over the past 3 decades (from 1975 - 2006 inclusive).

In particular, given Schwartz' values for sensitivity and time constant, how does the theoretical "realized temperature change" (for CO2 concentration increase) obtained with his method compare to the part of the "observed temperature change" that is attributable to CO2 over the last 3 decades?

It should be noted that while Schwartz' analysis was based on the entire twentieth century, his sensitivity value and time constant must nevertheless account for the observed warming attributable to CO2 over the period considered here (1975-2006) as well.

I would caution the reader that much of the analysis below is predicated on the basic assumption that Schwartz' simple "single-compartment energy balance model" is valid for estimating “climate sensitivity". If that is incorrect, all bets are off. But, as I indicated above, that is not something I wish to consider here. I will leave that for others.

Attribution of warming to CO2
In order to compare the theoretical "CO2-induced warming" over a given period to the "observed" (ie, measured) warming, it is important to know just how much of the observed warming was due to CO2. For this, it is important to know both the fraction of the observed warming that is attributable to all greenhouse gases and the fraction of the greenhouse gas-induced warming that is attributable to CO2 alone.

There is general agreement among climate scientists that about 60% of the total greenhouse gas "forcing" (and hence of the resultant warming) is attributable to CO2 (see Greenhouse Gas Forcing graph in this presentation by Schwartz. ) This is also shown in Fig 2 of Section 2 (Changes in Atmospheric Constituents and in Radiative Forcing) of the latest IPCC report.


Over the past decade, CO2 has actually accounted for a higher fraction of the greenhouse forcing than over previous decades, since CO2 concentration has continued upward (and its yearly change has even increased over the past 5 years) while CFC and methane concentrations have plateaued.

Most climatologists (eg IPCC) also take the view, based on an attribution analysis, that the vast majority of the warming (increase in the global mean temperature anomaly) that occurred over recent decades is attributable to greenhouse gases ( including CO2) .

Figure 9.9c (above) from the IPCC’s latest report (Section 9: Understanding and Attributing Climate Change) shows “estimated contribution for greenhouse gas (red), other anthropogenic (green) and natural (blue) components to observed global mean surface temperature changes" ( deg C per 50 years, 1950-1999).

It should be noted that the period covered by the graphs (1950-1999) is not the same as the one under consideration here (1975-2006), but most of the latter period is contained within the former, so it is plausible that the relative contribution from the 3 components ("greenhouse", "other anthropogenic" and "natural") will be roughly similar over 1975-2006 to that over the longer period.

For our purposes here, we are not concerned as much with the details as with the basic “features” of the attribution graph. The most prominent “feature” of figure 9.9 is the fact that virtually all of the warming (positive temperature change) over the latter half of the 20th century can be attributed to greenhouse gases.

The other components -- both “natural” and “other anthropogenic” -- taken as a whole, most probably produced net cooling (negative temperature change), according to Fig 9.9c.

T_t = T_g + T_oa + T_n

(where T_t= total observed temperature change, T_g="greenhouse- caused temp change", T_oa="other anthropogenic...", T_n="natural...")

If T_oa and T_n are both less than or equal to zero (as indicated in the graphic)

T_t <= T_g

This means that the warming that was observed (measured) over that period was less than or equal to what would have occurred had greenhouse gases been acting alone (ie, if there had been no “natural” and “other anthropogenic” components acting).

So the theoretical warming that is calculated for CO2 based on a particular sensitivity value must account for at least 0.6T_t (the minimum temperature change due to CO2)

The contributions shown in Fig 9.9c by the bar graphs represent what were determined to be the most probable case. There is uncertainty associated with the contribution by each component, but it should nonetheless be noted that it even if one assumes the upper limit of the uncertainty for each component in Fig 9.9c (ie, the bottom of the error bar for greenhouse gases and the top of the error bar for the other two components) all of the positive temperature change over the period (warming) is still attributable to greenhouse gases.

Note: Black carbon is one radiative forcing source that is included within the "other anthropogenic" group. It probably contributed to warming by itself (eg, through it's decrease of snow and ice albedo) , but the "other anthropogenic" group as a whole most probably had either an overall negative (or possibly zero) net contribution to the temperature change.

Other negative forcings within the group likely counter-balance the positive: eg, according to the IPCC report “Human-induced changes in land cover have likely increased the global surface albedo, leading to a negative radiative forcing of -0.2 ± 0.2 W m-2”, which means that contribution was either slightly negative or zero.

Similarly, the solar contribution to radiative forcing taken by itself was most probably positive over the twentieth century as a whole, though “about an order of magnitude less than the total greenhouse gas contributions”(IPCC Fourth Assessment report) (see Fig 2.23) and virtually flat over the last 30 years, as shown in this graphic from the Max Planck Institute (solar output shown in blue, temperature in red). But, “Over particularly the 1950 to 2005 period, the combined natural forcing has been either negative or slightly positive (less than approximately 0.2 W m-2)”.

IPCC Fig 2.23 (below) illustrates the different contributions to radiative forcing over the past century (changes from 1850) (Changes in Atmospheric Constituents and in Radiative Forcing)

In a nutshell, what the above means is that it is likely that the measured warming that occurred over the latter half of the twentieth century (and also over the period 1975--2006) can be attributed entirely to greenhouse gas increases and that the net forcing due to the "natural" and "other anthropogenic" components is most likely less than or equal to zero (ie, either led to cooling or had no net effect on temperature) .

This is an important point, because it means that the observed warming represents a minimum amount of warming that must be accounted for under any particular assumption about climate sensitivity.

In other words, operating on the assumption that all the observed warming over the period 1975-2006 (about 0.58 deg C -- see below) can be attributed to greenhouse gases and that CO2 accounted for about 60% of that greenhouse-gas induced temperature change, that would mean that CO2 accounted for at least 0.35C over that period (no less). This will represent our “Assumption I” below, which corresponds to the case where the net contribution of "natural" plus "other anthropogenic" sources was essentially zero.


Increase in CO2 and its relation to temperature increase
At the beginning of 1975, the atmospheric CO2 concentration stood at about 330 ppm (parts per million) . Since then, it has increased yearly by the increments given on this NOAA site. Since the CO2 increases are given in yearly increments, it is natural to divide the period up into years and to calculate the theoretical temperature increase for each year based on the change in the atmospheric CO2 concentration from one year to the next using the Arrhenius equation

Delta_T = A ln (Concentration_final/Concentration_initial)

where the constant "A" depends on how much the temperature goes up for a doubling of the concentration.

For the central sensitivity value of 1.1°C that Schwartz assumed, A=1.59. So, the equation for temperature change becomes

Delta_T_1.1 = 1.59 ln(Concentration_final/Concentration_initial)

For yearly changes

Concentration_final = atmospheric CO2 concentration at the end of one given year.

Concentration_initial = atmospheric CO2 concentration at the beginning of that same year

Temperature change is not immediate
When the CO2 concentration increases in a particular year, its effect is not felt immediately, but instead, the global mean surface temperature takes a certain time to "ramp up".

With Schwartz' model, in response to a CO2 increase (or other forcing), the climate "system" as a whole (atmosphere and oceans) takes a certain characteristic time (based on the system "time constant") to reach about 63% (1-1/e) of its "final" (equilibrium) temperature. The temperature is "realized" according to the equation T(t) = T_f (1 - e^-t/τ )

“Realized” Warming
Since not all of the potential warming from previous years' CO2 increases will have shown up in the global mean surface temperature anomaly, what we actually measure is "realized warming".

"Delta_T_realized" is the temperature change for each year that has actually been "realized" to date. It varies for each year depending on the CO2 (forcing) change during that year and the number of years before present that the increase in CO2 occurred.

Delta_T_realized is obtained by multiplying the Delta_T for each year (above) by the weighting factor

1 - e^-t/τ

where τ is the time constant (5 years for the 1.1°C sensitivity that Schwartz arrived at) and “t” is the time that has elapsed in years since the end of the particular year in question (ie, during which a particular CO2 increase occurred).

To determine the total theoretical "realized temperature increase" over the time period in question, we then sum all of the "Delta_T_realized" increments over that period.

Observed warming attributable to CO2
We wish to compare the "theoretical CO2-induced warming" to the "part of the observed warming that is attributable to CO2".

Fig 3.1 from Fourth IPCC Assessment Report

The average rate of temperature increase over the period 1975-2006 is 0.018°C/year or a total increase of 0.576°C for the 32 year period. But not all of that warming (ie, temperature increase) was due to the increase in the atmospheric CO2 concentration over that period.

We will make some reasonable assumptions based on the IPCC attribution analysis, and see what that would imply for CO2-induced warming and what fraction of the warming in each case is accounted for under Schwartz' assumptions about sensitivity and time constant.

Assumption I: All the warming over the period 1975-2006 is attributable to Greenhouse gases and the net contribution of "natural" plus "other anthropogenic" sources was essentially zero, corresponding to the upper bound of the uncertainties for those two contributions on Fig 9.9c.

First, let us assume that the analysis shown in fig 9.9c of the IPCC report is at least correct in its broad brush characteristics -- ie, that 100% of the observed warming in recent decades is attributable to greenhouse gas increases. As indicated above, there is good reason to believe that this is a sound assumption.

As we will discover below, it would take an error of 25% or more in that regard to make the temperature increase predicted under Schwartz' assumptions consistent with the minimum warming attributable to CO2 over the period in question.

If we combine the above assumption with the estimate (indicated by Schwartz and IPCC) that CO2 provided 60% of the greenhouse gas forcing over the period in question , that would mean that Co2 induced warming over the period in question that was 60% of the total observed warming. In other words, under that assumption, (0.6)(0.576C) = 0.346C of the total observed warming was due to CO2 concentration increase.

Theoretical "Realized warming” compared to “observed" warming

Assume climate sensitivity of 1.1°C (and that 60% of all the warming is attributable to CO2)

For a climate sensitivity of 1.1C, if one adds up all the weighted temperature increments (Delta_T_realized increments) over the years 1975-2006 (inclusive), one gets a temperature change of 0.20°C. This is the theoretical "realized' warming due to the CO2 concentration increase Delta_T_realized_1.1 = 0.20°C

Delta_T_realized_1.1 / Delta_T_observed = 0.20°C/ 0.346°C = 0.58

In other words, using a sensitivity of 1.1C (Schwartz' central value) yields a theoretical "realized temperature change" that is only about 60% of the observed CO2-induced warming over that period -- ie, it fails to account for about 40% of the part of the observed warming that is attributable to CO2.

But the mismatch between Delta_T_realized_1.1 and the actual warming caused by Co2 over the period in question might be worse than indicated here. Our "Assumption I" was that the net contribution of "natural" plus "other anthropogenic" sources was essentially zero over the period in question -- instead of negative, which would have made the amount of actual warming due to CO2 larger than the Delta_T_observed obtained above.

Note that with no time lag and a sensitivity of 1.1°C, one would expect that the total CO2 concentration increase over that period would have produced about 0.23°C temperature increase. So, in the above case (with time constant of 5 years), about 0.03°C ( 0.23°C - 0.20°C) more warming would be “in the pipeline”. In other words, that would be the amount of yet “un-realized” warming due to CO2 over that period. The "un-realized" warming due to all greenhouse gases (combined) over the same period would be about 0.05°C (assuming CO2 accounts for about 60%).


Assume climate sensitivity of 3.0°C (and that 60% of all the warming is attributable to CO2)
It is interesting to see what result we get if we perform the same calculations that we did above using a different assumption about sensitivity and time constant τ. Namely, assume sensitivity is instead 3.0°C (the "best" value provided by Annan et al and adopted by the IPCC). Under Schwartz theory, that would imply the time constant would be about 13.6 years (instead of 5 years), with the same heat capacity.

If we assume a sensitivity of 3.0 °C for CO2 doubling, the equation for temperature change becomes

Delta_T_3 = 4.33 ln (Concentration_final/Concentration_initial)

Multiplying each of the theoretical (equilibrium) yearly increments in temperature by the time-dependent factor as before and summing all the increments, we get a total theoretical “realized temperature change” over that period Delta_T_realized_3.0 = 0.38C, which fully accounts for the observed temperature change attributed to CO2.

Delta_T_realized_3.0 / Delta_T_observed = 0.38°C/ 0.346°C = 1.10

In other words, a sensitivity of 3.0°C and a time constant of 13.6 years yields a theoretical "realized temperature change" that is 110% of observed, so, it does account for the minimum amount of warming estimated to be due to CO2. (again, on the above assumption that CO2 caused 60% of total warming)

It should again be noted that our "Assumption I" corresponds to the "minimum" case where the net contribution of "natural" plus "other anthropogenic" sources was assumed essentially zero over the period in question -- instead of negative, which would have made the amount of actual warming due to CO2 larger than Delta_T_observed.

If one assumes there was significant counterbalancing cooling due to aerosols and other sources over the period, the match might be worse than the 1.10 ratio we obtained for Delta_T_realized_3.0 / Delta_T_observed.

Also, the time constant of 13.6 years for a sensitivity of 3.0°C that was calculated using Schwartz' method might not be correct. If the actual time constant is longer (presuming one can use a single time constant to characterize the climate system, which is by no means certain), that would mean the theoretical "realized warming" due to CO2 (Delta_T_realized_3.0 ) would be less than the value we got above. That would also make the match worse.

So, the close match above between Delta_T_realized_3.0 and Delta_T_observed should be taken with a grain of salt (or two).

Finally, note that with no time lag and a sensitivity of 3.0°C, one would expect that the total CO2 concentration increase over that period would have produced about 0.63°C temperature increase. So, in the above case (time constant of 13.6 years) , about 0.25°C (0.63°C - 0.38°C) more warming would be “in the pipeline” from that period. The "un-realized" warming due to all greenhouse gases (combined)over the same period (1975-2006) would be about 0.42°C.

Assume climate sensitivity of 1.6°C (and that 60% of all the warming is attributable to CO2)

Finally, what about a sensitivity at the top of the range given by Schwartz (1.6°C)?

If the sensitivity were 1.6°C, the corresponding time constant would be about 7.3 years and the "realized" warming over the period 1975-present would be Delta_T_realized_1.6 = 0.25°C.

Delta_T_realized_1.6/Delta_T_observed = 0.25°C / 0.346°C = 0.72

So, on the assumption that CO2 accounted for 60% of the warming over the last 3 decades, not even the max sensitivity given by Schwartz would be able to account for the part of the observed warming that is attributable to CO2.

Again, we note that our "Assumption I" corresponds to the "minimum" case for warming attributable to CO2. The case where the net contribution of "natural" plus "other anthropogenic" sources was negative (instead of zero) would mean the amount of actual warming due to CO2 would be larger than Delta_T_observed and the mismatch worse (than 0.72)

At this point, let me reiterate the "likely" sensitivity values specified by the IPCC in their recent Fourth Assessment Report: "2 to 4.5°C with a best estimate of about 3°C, and ... very unlikely to be less than 1.5°C".


Attribution again

Assumption II: Not all the warming over the period 1975-2006 is attributable to Greenhouse gas concentration increases

Because of uncertainties associated with attribution analysis, it is possible (though less probable) that increases in greenhouse gas concentrations did not account for all the observed warming over the past 3 decades. In fact, it has been suggested (eg, by Hansen et al) that up to 25% of global warming might have been due to black soot (through its impact on the earth's "albedo", or reflectivity) .

Hansen was actually referring to the entire 20th century period, but for the purposes here, we will assume it applies to the last 3 decades as well.

It should also be noted that the IPCC estimated the contribution from soot (black carbon aerosols) due to its effect on snow and ice albedo to be about + 0.1 ± 0.1 W m-2 and that this was actually included in the “other anthropogenic” group shown in Fig 9.9c, to which the IPCC attributed an overall negative forcing.

But let us assume for the moment that soot made the overall forcing contribution from "other anthropogenic" a net positive and that this accounted for fully 25% of the observed warming over the past 3 decades.

Under that assumption (which, again, is not in line with the IPCC analysis -- see figure 9.9c above) we must first subtract 25% "off the top" of the observed warming (0.576°C) to yield the part of the warming caused by greenhouse gases (0.43°C).

Then we take 60% of that result to get the warming due to CO2 alone, Delta_T_observed = 0.26C. Under that assumption, CO2 therefore accounted for about 45% of the total warming ( 0.6 X 0.75)

Assume climate sensitivity of 1.6°C (and that CO2 caused 45% of total warming)

Under the latter set of assumptions, Schwartz' max sensitivity value (1.6°C) could account for the warming attributable to CO2 over that period:

Delta_T_realized_1.6/Delta_T_observed = 0.25°C / 0.26°C = 0.96

So, it seems that Schwartz' maximum sensitivity (1.6°C) can yield the required "realized warming" over the period 1975-present, but only if about 25% of the warming was due to something other than greenhouse gases.


Conclusions?

So, what conclusions (if any) can be drawn from the above rough "exploratory analysis"?

It is certainly not possible to nail down the precise value of climate sensitivity, since definitive answers regarding sensitivity probably depend on further narrowing the uncertainties in "contributions" to forcing.

But there are nonetheless a few things that we can say:

1) Based on the analysis in the last section above, we can probably safely say that unless something other than greenhouse gases caused significantly more than 25% of the warming over the past 32 years (1975-2006 inclusive), which is not likely, according to the IPCC attribution analysis, Schwartz' central sensitivity value of 1.1C, for which Delta_T_realized_1.1 = 0.20°C, would not account for the warming attributable to CO2 over that period, since even under the "25% assumption"

Delta_T_realized_1.1 / Delta_T_observed = 0.20°C / 0.26°C = 0.77 (77% of warming)

In fact, in order for Schwartz' sensitivity value (1.1°C) to account for the warming attributable to CO2 over that period, something other than greenhouse gases would have to have caused fully 40% of the total observed warming over that period -- making the temperature contribution due to CO2 about 36% of the total.

Based on the attribution analysis given by the IPCC, this does not seem likely.

2) If greenhouse gases accounted for 90% or better of the warming over the period in question (as seems most probable, according to IPCC), that means CO2 accounted for 54% or better of the total warming, implying that Delta_T_observed >= 0.31°C.

In turn, that implies that it is unlikely than any of the sensitivity values in Schwartz' range (not even the maximum of 1.6°C) would account for the warming attributable to CO2. In the latter case,

Delta_T_realized_1.6/Delta_T_observed <= 0.25°C / 0.31°C = 0.81

Accounts for at most 81% of the total due to CO2.

Final remarks

Finally, I would just remind the reader that much of the above discussion makes the basic assumption that Schwartz' simple model is valid -- ie, that the climate system has a single characteristic time constant and heat capacity that can be used to determine sensitivity to CO2 doubling through the equation S = τ / C.

But, as I indicated at the very beginning, I will leave the validity of those assumptions for others to determine.