SkS Analogy 10 - Bathtubs and Budgets

Tag Line

Adding 101 gallons to a 100-gallon bathtub causes it to overflow, even if added slowly.

Elevator Statement

No matter how slowly a bathtub is filled, adding more than 100 gallons of water to a 100-gallon bathtub causes it to overflow. This principle is clear, obvious, and one we encounter every time we take a bath.

Similar to the finite volume of a bathtub, Earth’s atmosphere has a finite volume. Because Earth’s atmosphere has a fixed volume, human emissions of greenhouse gases (GHGs) are increasing the atmospheric concentration of CO2, because we keep adding GHGs into this closed system, similar to running water into a bathtub.1 The concentration of CO2 does not depend on how rapidly we add GHGs to the atmosphere, but only on the total amount added.2 Lowering emission rates is important for improving the air quality in a large city, but carbon emission rates are meaningless for determining the global warming we will eventually experience due to a revved-up greenhouse effect: only the total amount emitted matters.3

Climate Science

The atmosphere has a budget of carbon it can accept to keep overall warming below a certain level, and the bathtub in this analogy is intended to be a visual representation of such a budget. Although scientists do not know with certainty how much the climate will warm for a doubling of CO2 (the so-called climate sensitivity), a climate sensitivity of 3°C/doubling of CO2 (see Ocean Time Lag)4 is consistent with the warming observed since the 1970’s, and is mid-range in the IPCC estimates that range from a low of 1.5 to a high of 4.5°C. Using a climate sensitivity of 3°C/doubling of CO2, a rough guide of the carbon budget corresponding to a particular level of warming is as follows5

•    350 ppm CO2 = 1°C warming (a warming we can live with, see
•    400 ppm CO2 = 1.5°C warming (things get worse for an increasing number of people)
•    445 ppm CO2 = 2°C warming (dangerous warming, but the best we can hope for)6 

      2°C warming is a guardrail beyond which changes become catastrophic

•    560 ppm CO2 = 3°C warming (really bad: and positive feedbacks accelerate)
•    700 ppm CO2 = 4°C warming (To be avoided at all costs)7

This means that there is a set amount of carbon we can emit into the atmosphere to keep the global average temperature below specific target temperatures. Emission rates are meaningless in this context. It is only the total emitted carbon that controls the ultimate warming.

However, emission rates are important for two other reasons.

As the world comes out of an ice age the Earth warms about 5°C over 10,000 years, so we know that species currently present on Earth are adapted to that rate of change. The current rate of warming of about  2°C per 100 years is far in excess of natural cycles, and many species will likely disappear, even if we stabilize the warming at 2°C. Anything we can do to slow down the rate of warming will help species adapt and buy us needed time to implement solutions.

However, the core, long-term problem remains the total carbon emitted into the atmosphere, and not the rate of carbon emissions, because assuming that we are able to stabilize the temperature at 2°C warming (which implies that net carbon emissions drop to 0 at that point), 2°C warming will still imply multi-meter sea level rise, which will occur over millennia, and will require costly mitigation strategies for coastal communities.

The current level of CO2 in the atmosphere is already causing problems for many people, and the fraction of the Earth’s population experiencing climate-change-related stress will increase with increasing CO2 concentrations. The "bathtub" has been rapidly filled by developed nations, so it is reasonable that developed nations lead the efforts to stop the carbon buildup in the atmosphere.

The problem with talking about carbon budgets and motivating action needed to keep our total emitted carbon within a 2°C budget is that there is a delay between cause and effect. Specifically…

And there is perhaps a third problem with comprehending the consequences of blowing a budget. People in “developed” countries live in an age of instant credit, refinancing, bankruptcy, welfare, so many variations of handouts and financial forgiveness that it seems that there is always somebody, something, or some kind of safety net that will catch us. The concept of no safety net is foreign to people living in affluent societies. For all intents and purposes there are no natural safety nets that will counteract our GHG emissions.8 At current emission rates that are increasing CO2 at a rate of 2 ppm/year, we will blow past the budget for staying below 2°C warming within 20 years.9 The only possibility for avoiding 2°C warming is for humans to alter their behavior and reduce energy use while we bring low-carbon energy sources online. The Paris Accord is one attempt to motivate governments to move in this direction, but individuals should also move on their own to reduce personal carbon emissions. There simply is no time to waste in waiting for governments to motivate action.

Negative-Emissions Technologies

Every year that we fail to reduce energy use and reduce carbon emissions, we increase the probability that we will blow past 2°C warming. Realizing that our current emissions trajectories will take us past 2°C warming, and yet wanting to maintain a semblance of hope without undue imposition on people in the developed countries, most emission scenarios presented by climate scientists for staying below 2°C global warming rely on negative-emission technologies that allow us to emit now and remove CO2 later. Kind of like the financial equivalent of buy now pay later. The problem, however, is that these negative-emissions technologies have only been demonstrated at the pilot scale, and have not been proven at the scale needed to meaningfully counteract GHG emissions. A good read on the subject is Kevin Anderson, or you can watch him as well. How easy is it to remove cream and sugar from your coffee once stirred in? How easy will it be to suck CO2 out of the atmosphere at a rate that will counteract the current, massive emission rates?

Is a 3°C target more reasonable and achievable?

For those who see a 2°C target as too ambitious, and prefer a longer-term approach of focusing on 3°C as a more attainable target, it is important to note that we may not have an option to stop the warming at 3°C. By that point positive feedbacks may drive the temperature higher, no matter what we do. There are large uncertainties about how the climate will respond as global temperatures increase, but natural feedbacks will certainly increase with increasing warming, making our emissions reductions less effective at controlling global warming as the planet continues to warm. There is the additional concern that although an increase from 2 to 3°C seems like a small step, in fact it represents significantly increased ocean acidification, increased storm intensity, increased sea-level rise, and most of all, significantly increased uncertainty about our ability to put the brakes on at all.10

This is why there is an urgency to meeting a 2°C target. A carbon budget for staying below 2°C is the last carbon budget for which there is a consensus among scientists that we have a reasonable expectation of controlling our own destiny. Beyond that warming, in addition to escalating negative effects, it is unclear how much control we will have over the climate.

No matter how much we justify our need for fossil fuels, continued consumption will lead to only one result: continued warming to … and then beyond the danger point. If you have not done so already, please read the NAS booklet on Climate Change: Evidence and Causes and then determine how you can begin/continue to cut back your use of fossil fuels. Let your elected officials know how you feel.

If you've read this far, you must be hungry for more detail. Check out Andy Skuce's article on the subject: Are we overestimating our global carbon budget?

Is it too late?

There is a growing realization that holding warming to 2°C warming may not be possible (see Glen Peter's excellent summary on this topic). Does this mean it’s too late and we should just give up? If you are driving a car and you suddenly realize you are going to run into a brick wall, no matter what you do, at what point is it “too late” to put on the brakes? At the point you put on the brakes you will avoid something worse. So even if it is too late to stop the warming at 2°C, remember that these are just numbers, there is uncertainty about the effects associated with these numbers, and that we will always avoid something worse later, by taking action now.




1. The difference between adding CO2 to the atmosphere and adding water to a bathtub is that CO2 added to the atmosphere has a negligible effect displacing air, whereas water added to a bathtub displaces all of the air previously in the bathtub. The point of the analogy, however, is to emphasize that to keep warming below the 2°C target set by the Paris Accord, there is a set amount of carbon we can emit before we exceed 2°C of warming, and the bathtub is intended as a visual representation of that budgeted amount of carbon.

2. Bathtub analogies have been formulated where water runs out of the drain as it enters from the faucet. The idea is that the natural system has sources and sinks, and that the rate at which the bathtub fills up is the difference between the sources and the sinks. In such analogies human emissions cause the inflow to increase, so that the human contribution is the difference between a large inflow and a large outflow. In this analogy the drain of the bathtub is plugged, so that the inflow represents the entire human contribution to the buildup of greenhouse gases. Therefore, in this analogy shutting off the faucet represents reducing net human greenhouse gas emissions to zero.

3. It is well known that land and ocean currently absorb roughly 50% of the CO2 that we emit. Emission scenarios account for this, so that when discussing how much CO2 we can emit to stay below a particular level of warming, the uptake by natural systems is accounted for. One way to assess the net effect of how well we're doing to reduce CO2 emissions is to look at the rate of CO2 increase per year, because the measured increase includes the difference between the CO2 emitted and the CO2 absorbed by natural systems. The problem, however, is that as the earth warms, the amount of CO2 taken up by natural systems may decrease, reducing the amount of CO2 we can emit to stay below target temperature levels. See Andy Skuce's excellent article on this issue.

4. 3°C increase of global average temperature for a doubling of CO2 concentration compared to the pre-industrial level of 280 ppm.

5. Dumping a fixed amount of carbon into the closed system represented by the atmosphere results in a measurable amount of CO2 in the atmosphere, which is quantified as molecules of CO2 (i.e., parts of CO2) per million molecules of air (ppm).

6. (PDF download) Hansen et al. state that 2°C warming could be dangerous.

7. (PDF download) Turn Down The Heat: Why a 4°C Warmer World Must be Avoided.

8. In the physical world there are “safety nets” of sorts, systems that maintain/restore balance, referred to by scientists as thermostats that maintain Earth’s climate within habitable bounds, but these processes act far too slowly to counteract the rate at which we are dumping CO2 into the atmosphere. Kind of like comparing the rate of evaporation of water from a bathtub compared to the rate at which it's being filled. The thermostats that control CO2 levels in the atmosphere require 10's to 100's of millennia to restore/maintain balance.

9. The CO2 concentration we will reach and the temperature we will lock in is a bit more complicated than the simplistic statement of multiplying 2 ppm/year x 20 years to say that we will be at 445 ppm (or thereabouts) in 20 years, because if we were to magically reduce our emissions to 0 after 20 years, the CO2 levels would decrease thereafter to a new equilibrium point, determined by the ability of land and oceans to absorb some of the CO2 we emit. For an interesting read on this subject that delves into more detail about stabilization dynamics, see Andy Skuce’s article Global Warming: Not Reversible, But Stoppable. Because CO2 emissions are controlled by many factors related to population, rise of affluence in the developing countries, deforestation, etc., I have assumed in my simplistic calculations that a reasonable expectation is that for the next 20 years we more or less maintain current emission levels. Whether this is optimistic, pessimistic, or realistic depends on how we respond to the call to action to reduce emissions. Realistic extrapolations of current trends are at best, flat. Getting our CO2 emissions pointing downward is a work in progress.

10. I do not have a good reference stating that warming to 3°C will cause uncontrollable feedbacks, but scientists often verbally state this in talks. A good source that infers the probability of uncontrollable feedbacks at 3°C warming is from Hansen et al., where they state "Cumulative emissions of ~1000 GtC, sometimes associated with 2 degrees C global warming, would spur “slow” feedbacks and eventual warming of 3-4 degrees C with disastrous consequences." The implications are that if we target warming for 2°C, and slow feedbacks take us to 3-4°C, then targeting warming for 3°C will likely activate feedbacks that will take us past 4°C, a warming that climate scientists say must be avoided at all costs (See footnote 7 above).


Posted by Evan on Thursday, 12 October, 2017

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