This post is really a joint post, written by myself and by Roger Pielke Sr. It’s partly a continuation of a post Roger wrote on Judith Curry’s blog, and partly a consequence of my attempt to answer one of Roger’s question. The latter post resulted in a rather tetchy exchange between myself and Roger which we’ve now resolved, illustrating that it is possible to recover from such exchanges (I sometimes wish more would try to do so – sometimes I don’t, though).
This post is really just intended to be a simple illustration of how one might use the forcing/feedback paradigm to both understand and quantify anthropogenic global warming, and – potentially – as a mechanism for assessing global climate models. It is a work in progress, and this is simply a draft. So, bear that in mind. Comments welcome.
The definition of a forcing is essentially the net change in energy balance (change in net TOA flux) due to external (e.g. solar), volcanic emissions and internally human imposed perturbations (e.g. added CO2) . Typically, it has been defined relative to some baseline time period (IPCC, 2013). This change in energy balance will cause warming/cooling and a temperature response, which will then produce a feedback response. If the change in forcing at time t (relative to ) is , and the change in temperature is (also relative to ) then the radiative imbalance, , at time is
where is the imbalance at , and is the feedback response in Wm-2K-1, is the heat capacity. If we assume that the system is in radiative balance at , then we have
It is important to note that the feedbacks are assumed a linear function of . That this might not be correct is discussed in the post. In addition, there are radiative forcings/feedbacks that are not directly connected to as dicussed in NRC (2005), such as from the input of cloud condensation nuclei and ice nuclei and their subsequent effect on clouds.
We can actually put some numbers in. According to the GISS dataset, the change in forcing between 1880 and 2011 is 1.635Wm−2, and the change in temperature is 0.77K. The feedback response (Soden & Held 2006; Wielicki 2013) is about Wm−2K−1. We assume is positive for a net negative feedback (i.e., minus in Equation (2)). Therefore the radiative imbalance in 2011 should be
If we consider the NOAA Ocean Heat Content data, then it suggests that the oceans are currently accruing energy at the rate of about 1022 J/year. If this is ~93% of the total system heat uptake rate. The radiative imbalance is therefore
We can also estimate how the total system energy changes with time by integrating Equation (2) in time. In other words
Figure 1: Estimate of the change in Ocean Heat Content (OHC) by integrating a simple energy balance formalism.
If we compare the results shown in Figure 1 with the NOAA 0-2000m Global Ocean Heat Content shown in Figure 2, then our results suggest that the system should have accrued about 2 × 1023 J between 1970 and 2011, while the NOAA data suggests about 2.5 × 1023 J over the same period. The NOAA data is also for the oceans only, and so only accounts for about 93% of the change in system energy.
Figure 2: Figure showing the change in 0-2000m and 0-700m Global Ocean Heat Content (OHC) – Levitus et al. (2012).
With respect to other estimates, The heat content of the world’s oceans for the 0-2000m layer increased by 2.4 x 1023 J corresponding to a rate of 0.39 Wm-2, according to Levitus et al 2012. The layer from the surface to 2000 m depth warming rate of 0.39 Wm-2 ± 0.031 Wm-2 per unit area of the Earth’s surface accounts for approximately 90% of the warming of the climate system according to Levitus et al. Thus, if we add the 10%, the 1955-2010 the radiative imbalance is 0.43 Wm-2 ± 0.031 Wm-2. About 1/3 of this heating is at levels below 700m, according to Levitus et al 2012. They concluded that a strong positive linear trend in exists in world ocean heat content since 1955. Since about 2003, the heating rate in the upper 700m was less than in the earlier years back to around 1997 (Figure 3).
Figure 3: Figure showing the 0-700m change in Global Ocean Heat Content (NOAA).
The IPCC reports that the global average radiative imbalance is 0.59 Wm-2 for 1971-2010 while for 1993-2010 it is reported as 0.71 Wm-2. Trenberth and Fasullo (2013) state that the imbalance is 0.5–1Wm−2 over the 2000s.
Our basic calculation would seem to be slightly underestimating the change in total energy. This could be due to errors in the forcings, feedbacks, or temperature response. It is possible that we should lag the temperature response slightly to account for the time it takes for the upper ocean to equilibrate with the atmosphere.
There is also the issue of the heating that has been reported below 700m in the oceans. Levitus et al report [and Figure 2 illustrates] that about 1/3 of the heating has gone into that layer. If correct, this heat is not likely to transfer back to the surface so as to affect weather and other aspects of the climate on multi-decadal time scales. Also, if more goes into the deeper parts of the ocean, there is less to heat the surface – slower surface warming. If less goes into the deeper parts of the ocean, then there is more to heat the surface – faster surface warming. This heat is, however, not sampled when a surface temperature trend is used to diagnose global warming.
Levitus S., et al., 2012, World ocean heat content and thermosteric sea level
change (0–2000 m), 1955–2010, Geophysical Research Letters, 36, L10603.
National Research Council, 2005: Radiative forcing of climate change: Expanding the concept and addressing uncertainties. Committee on Radiative Forcing Effects on Climate Change, Climate Research Committee, Board on Atmospheric Sciences and Climate, Division on Earth and Life Studies, The National Academies Press, Washington, D.C., 208 pp.
Soden B.J., Held I.M., 2006, An Assessment of Climate Feedbacks in Coupled Ocean–Atmosphere Models, Journal of Climate, 19, 3354-3360.
Wielicki B.A., et al., 2013, Achieving Climate Change Absolute Accuracy in Orbit, Bulletin of the American Meteorological Society, 94, 1519-1539
For a bit of fun, I coded up the simple two-degree-of-freedom model that Isaac Held discusses here. The basic model is meant to represent a mixed layer with a temperature and heat capacity , and the deeper ocean with temperature and heat capacity . The basic equations are
In this case I integrated the equations for and using the forcing timeseries. I used and , which gives an ECS of and a TCR to ECS ratio of , giving a TCR of . You can get the change in OHC using the change in temperatures and the heat contents of the two layers. The basic result is shown in the Figure below. The upper figure shows the temperature compared to GISSTemp (dashed line), the lower shows the OHC since 1950. I did this really quickly and am not trying to match anything exactly. This is just meant to be illustrative, so bear that in mind.