This doesn’t really match up with climate history, but climate models have mostly struggled to do much more than reproduce the stereotyped view. *See Natural Variability and Chaos – Four – The Thirty Year Myth for a different perspective on (only) the timescale.*

In this stereotypical view, the only reason why “long term” (=30 year statistics) can change is because of “external forcing”. Otherwise, where does the “extra energy” come from (*we will examine this particular idea in a future article*).

One of our commenters recently highlighted a paper from Drijfhout et al (2013) -Spontaneous abrupt climate change due to an atmospheric blocking–sea-ice–ocean feedback in an unforced climate model simulation.

Here is how the paper introduces the subject:

Abrupt climate change is abundant in geological records, but climate models rarely have been able to simulate such events in response to realistic forcing.

Here we report on a spontaneous abrupt cooling event, lasting for more than a century, with a temperature anomaly similar to that of the Little Ice Age. The event was simulated in the preindustrial control run of a high- resolution climate model, without imposing external perturbations.

This is interesting and instructive on many levels so let’s take a look. In later articles we will look at the evidence in climate history for “abrupt” events, for now note that Dansgaard–Oeschger (DO) events are the description of the originally identified form of abrupt change.

*The distinction between “abrupt” changes and change that is not “abrupt” is an artificial one, it is more a reflection of the historical order in which we discovered “slow” and “abrupt” change. *

Under a **Significance** inset box in the paper:

There is a long-standing debate about whether climate models are able to simulate large, abrupt events that characterized past climates. Here, we document a large, spontaneously occurring cold event in a preindustrial control run of a new climate model.

The event is comparable to the Little Ice Age both in amplitude and duration; it is abrupt in its onset and termination, and it is characterized by a long period in which the atmospheric circulation over the North Atlantic is locked into a state with enhanced blocking.

To simulate this type of abrupt climate change, climate models should possess sufficient resolution to correctly represent atmospheric blocking and a sufficiently sensitive sea-ice model.

Here is their graph of the time-series of temperature (left) , and the geographical anomaly (right) expressed as the change during the 100 year event against the background of years 200-400:

*Figure 1 – Click to expand*

In their summary they state:

The lesson learned from this study is that the climate system is capable of generating large, abrupt climate excursions without externally imposed perturbations. Also, because such episodic events occur spontaneously, they may have limited predictability.

Before we look at the “causes” – the climate mechanisms – of this event, let’s briefly look at the climate model.

Their coupled GCM has an atmospheric resolution of just over 1º x 1º with 62 vertical levels, and the ocean has a resolution of 1º in the extra-tropics, increasing to 0.3º near the equator. The ocean has 42 vertical levels, with the top 200m of the ocean represented by 20 equally spaced 10m levels.

The GHGs and aerosols are set at pre-industrial 1860 values and don’t change over the 1,125 year simulation. There are no “flux adjustments” (no need for artificial momentum and energy additions to keep the model stable as with many older models).

See note 1 for a fuller description and the paper in the references for a full description.

The simulated event itself:

How did this abrupt change take place?

The main mechanism was a change in the Atlantic Meridional Overturning Current (AMOC), also known as the Thermohaline circulation. The AMOC raises a nice example of the sensitivity of climate. The AMOC brings warmer water from the tropics into higher latitudes. A necessary driver of this process is the intensity of deep convection in high latitudes (sinking dense water) which in turn depends on two factors – temperature and salinity. More importantly, more accurately, it depends on the **competing** **differences** in anomalies of temperature and salinity

To shut down deep convection, the density of the surface water must decrease. In the temperature range of 7–12 °C, typical for the Labrador Sea, the SST anomaly in degrees Celsius has to be roughly 5 times the sea surface salinity (SSS) anomaly in practical salinity units for density compensation to occur. The SST anomaly was only about twice that of the SSS anomaly; the density anomaly was therefore mostly determined by the salinity anomaly.

In the figure below we see (left) the AMOC time series at two locations with the reduction during the cold century, and (right) the anomaly by depth and latitude for the “cold century” vs the climatology for years 200-400:

*Figure 2 – Click to expand*

What caused the lower salinities? It was more sea ice, melting in the right location. The excess sea ice was caused by positive feedback between atmospheric and ocean conditions “locking in” a particular pattern. The paper has a detailed explanation with graphics of the pressure anomalies which is hard to reduce to anything more succinct, apart from their abstract:

Initial cooling started with a period of enhanced atmospheric blocking over the eastern subpolar gyre.

In response, a southward progression of the sea-ice margin occurred, and the sea-level pressure anomaly was locked to the sea-ice margin through thermal forcing. The cold-core high steered more cold air to the area, reinforcing the sea-ice concentration anomaly east of Greenland.

The sea-ice surplus was carried southward by ocean currents around the tip of Greenland. South of 70°N, sea ice already started melting and the associated freshwater anomaly was carried to the Labrador Sea, shutting off deep convection. There, surface waters were exposed longer to atmospheric cooling and sea surface temperature dropped, causing an even larger thermally forced high above the Labrador Sea.

It is fascinating to see a climate model reproducing an example of abrupt climate change. There are a few contexts to suggest for this result.

1. From the context of timescale we could ask how often these events take place, or what pre-conditions are necessary. The only way to gather meaningful statistics is for large ensemble runs of considerable length – perhaps thousands of “perturbed physics” runs each of 100,000 years length. This is far out of reach for processing power at the moment. I picked some arbitrary numbers – until the statistics start to converge and match what we see from paleoclimatology studies we don’t know if we have covered the “terrain”.

Or perhaps only five runs of 1,000 years are needed to completely solve the problem (I’m kidding).

2. From the context of resolution – as we achieve higher resolution in models we may find new phenomena emerging in climate models that did not appear before. For example, in ice age studies, coarser climate models could not achieve “perennial snow cover” at high latitudes (as a pre-condition for ice age inception), but higher resolution climate models have achieved this first step. (See Ghosts of Climates Past – Part Seven – GCM I & Part Eight – GCM II).

As a comparison on resolution, the 2,000 year El Nino study we saw in Part Six of this series had an atmospheric resolution of 2.5º x 2.0º with 24 levels.

However, we might also find that as the resolution progressively increases (with the inevitable march of processing power) phenomena that appear at one resolution disappear at yet higher resolutions. This is an opinion, but if you ask people who have experience with computational fluid dynamics I expect they will say this would not be surprising.

3. Other models might reach similar or higher resolution and never get this kind of result and demonstrate the flaw in the EC-Earth model that allowed this “Little Ice Age” result to occur. Or the reverse.

As the authors say:

As a result, only coupled climate models that are capable of realistically simulating atmospheric blocking in relation to sea-ice variations feature the enhanced sensitivity to internal fluctuations that may temporarily drive the climate system to a state that is far beyond its standard range of natural variability.

Spontaneous abrupt climate change due to an atmospheric blocking–sea-ice–ocean feedback in an unforced climate model simulation, Sybren Drijfhout, Emily Gleeson, Henk A. Dijkstra & Valerie Livina, PNAS (2013) – free paper

EC-Earth V2.2: description and validation of a new seamless earth system prediction model, W. Hazeleger et al, *Climate dynamics *(2012) – free paper

**Note 1**: From the Supporting Information from their paper:

Climate Model and Numerical Simulation. The climate model used in this study is version 2.2 of the EC-Earth earth system model [*see references*] whose atmospheric component is based on cycle 31r1 of the European Centre for Medium-range Weather Forecasts (ECMWF) Integrated Forecasting System.

The atmospheric component runs at T159 horizontal spectral resolution (roughly 1.125°) and has 62 vertical levels. In the vertical a terrain-following mixed σ/pressure coordinate is used.

The Nucleus for European Modeling of the Ocean (NEMO), version V2, running in a tripolar configuration with a horizontal resolution of nominally 1° and equatorial refinement to 0.3° (2) is used for the ocean component of EC-Earth.

Vertical mixing is achieved by a turbulent kinetic energy scheme. The vertical z coordinate features a partial step implementation, and a bottom boundary scheme mixes dense water down bottom slopes. Tracer advection is accomplished by a positive definite scheme, which does not produce spurious negative values.

The model does not resolve eddies, but eddy-induced tracer advection is parameterized (3). The ocean is divided into 42 vertical levels, spaced by ∼10 m in the upper 200 m, and thereafter increasing with depth. NEMO incorporates the Louvain-la-Neuve sea-ice model LIM2 (4), which uses the same grid as the ocean model. LIM2 treats sea ice as a 2D viscous-plastic continuum that transmits stresses between the ocean and atmosphere. Thermodynamically it consists of a snow and an ice layer.

Heat storage, heat conduction, snow–ice transformation, nonuniform snow and ice distributions, and albedo are accounted for by subgrid-scale parameterizations.

The ocean, ice, land, and atmosphere are coupled through the Ocean, Atmosphere, Sea Ice, Soil 3 coupler (5). No flux adjustments are applied to the model, resulting in a physical consistency between surface fluxes and meridional transports.

The present preindustrial (PI) run was conducted by Met Éireann and comprised 1,125 y. The ocean was initialized from the World Ocean Atlas 2001 climatology (6). The atmosphere used the 40-year ECMWF Re-Analysis of January 1, 1979, as the initial state with permanent PI (1850) greenhouse gas (280 ppm) and aerosol concentrations.

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We saw in Part Three that this particular paper ascribed a probability:

We find that the latest observed 30-year trend pattern of near-surface temperature change can be distinguished from all estimates of natural climate variability with an estimated risk of less than 2.5% if the optimal fingerprint is applied.

That paper did note that greatest uncertainty was in understanding the magnitude of natural variability. This is an essential element of attribution.

It wasn’t explicitly stated whether the 97.5% confidence was with the premise that natural variability was accurately understood in 1996. I believe that this was the premise. I don’t know what confidence would have been ascribed to the attribution study if uncertainty over natural variability was included.

In this article we will look at the IPCC 5th report, AR5, and see how this field has progressed, specifically in regard to the understanding of natural variability. Chapter 10 covers *Detection and Attribution of Climate Change*.

From p.881 (the page numbers are from the start of the whole report, chapter 10 has just over 60 pages plus references):

Since the AR4, detection and attribution studies have been carried out using new model simulations with more realistic forcings, and new observational data sets with improved representation of uncertainty (Christidis et al., 2010; Jones et al., 2011, 2013; Gillett et al., 2012, 2013; Stott and Jones, 2012; Knutson et al., 2013; Ribes and Terray, 2013).

Let’s have a look at these papers (see note 1 on CMIP3 & CMIP5).

I had trouble understanding AR5 Chapter 10 because there was no explicit discussion of natural variability. The papers referenced (usually) have their own section on natural variability, but chapter 10 doesn’t actually cover it.

I emailed Geert Jan van Oldenborgh to ask for help. He is the author of one paper we will briefly look at here – his paper was very interesting and he had a video segment explaining his paper. He suggested the problem was more about communication because natural variability was covered in chapter 9 on models. He had written a section in chapter 11 that he pointed me towards, so this article became something that tried to grasp the essence of three chapters (9 – 11), over 200 pages of reports and several pallet loads of papers.

So I’m not sure I can do the synthesis justice, but what I will endeavor to do in this article is demonstrate the minimal focus (in IPCC AR5) on how well models represent natural variability.

That subject deserves a lot more attention, so this article will be less about what natural variability is, and more about how little focus it gets in AR5. I only arrived here because I was determined to understand “fingerprints” and especially the rationale behind the certainties ascribed.

Subsequent articles will continue the discussion on natural variability.

The models [CMIP5] are found to provide plausible representations of internal climate variability, although there is room for improvement..

..The modeled internal climate variability from long control runs is used to determine whether observed and simulated trends are consistent or inconsistent. In other words, we assess whether observed and simulated forced trends are more extreme than those that might be expected from random sampling of internal climate variability.

Later

The model control runs exhibit long-term drifts. The magnitudes of these drifts tend to be larger in the CMIP3 control runs than in the CMIP5 control runs, although there are exceptions.

We assume that these drifts are due to the models not being in equilibrium with the control run forcing, and we remove the drifts by a linear trend analysis (depicted by the orange straight lines in Fig. 1). In some CMIP3 cases, the drift initially proceeds at one rate, but then the trend becomes smaller for the remainder of the run. We approximate the drift in these cases by two separate linear trend segments, which are identified in the figure by the short vertical orange line segments. These long-term drift trends are removed to produce the drift corrected series.

[Emphasis added].

Another paper suggests this assumption might not be correct. Here is Jones, Stott and Christidis (2013) – “piControl” are the natural variability model simulations:

Often a model simulation with no changes in external forcing (piControl) will have a drift in the climate diagnostics due to various flux imbalances in the model [Gupta et al., 2012]. Some studies attempt to account for possible model climate drifts, for instance Figure 9.5 in Hegerl et al. [2007] did not include transient simulations of the 20th century if the long-term trend of the piControl was greater in magnitude than 0.2 K/century (Appendix 9.C in Hegerl et al. [2007]).

Another technique is to remove the trend, from the transient simulations, deduced from a parallel section of piControl [e.g., Knutson et al., 2006]. However whether one should always remove the piControl trend, and how to do it in practice, is not a trivial issue [Taylor et al., 2012; Gupta et al., 2012]..

..We choose not to remove the trend from the piControl from parallel simulations of the same model in this study due to the impact it would have on long-term variability, i.e., the possibility that part of the trend in the piControl may be long-term internal variability that may or may not happen in a parallel experiment when additional forcing has been applied.

Here are further comments from Knutson et al 2013:

Five of the 24 CMIP3 models, identified by “(-)” in Fig. 1, were not used, or practically not used, beyond Fig. 1 in our analysis. For instance, the IAP_fgoals1.0.g model has a strong discontinuity near year 200 of the control run. We judge this as likely an artifact due to some problem with the model simulation, and we therefore chose to exclude this model from further analysis

*Figure 1*

Perhaps this is correct. Or perhaps the jump in simulated temperature is the climate model capturing natural climate variability.

The authors do comment:

As noted by Wittenberg (2009) and Vecchi and Wittenberg (2010), long-running control runs suggest that internally generated SST variability, at least in the ENSO region, can vary substantially between different 100-yr periods (approximately the length of record used here for observations), which again emphasizes the caution that must be placed on comparisons of modeled vs. observed internal variability based on records of relatively limited duration.

The first paper referenced, Wittenberg 2009, was the paper we looked at in Part Six – El Nino.

So is the “caution” that comes from that study included in the probability of our models ability to simulate natural variability?

In reality, questions about internal variability are not really discussed. Trends are removed, models with discontinuities are artifacts. What is left? This paper essentially takes the modeling output from the CMIP3 and CMIP5 archives (with and without GHG forcing) as a given and applies some tests.

This was a “Part II” paper and they said:

We use the same estimates of internal variability as in Ribes et al. 2013 [the “Part I”].

These are based on intra-ensemble variability from the above CMIP5 experiments as well as pre-industrial simulations from both the CMIP3 and CMIP5 archives, leading to a much larger sample than previously used (see Ribes et al. 2013 for details about ensembles).

We then implicitly assume that the multi-model internal variability estimate is reliable.

[Emphasis added]. The Part I paper said:

An estimate of internal climate variability is required in detection and attribution analysis, for both optimal estimation of the scaling factors and uncertainty analysis.

Estimates of internal variability are usually based on climate simulations, which may be control simulations (i.e. in the present case, simulations with no variations in external forcings), or ensembles of simulations with the same prescribed external forcings.

In the latter case, m – 1 independent realisations of pure internal variability may be obtained by subtracting the ensemble mean from each member (assuming again additivity of the responses) and rescaling the result by a factor √(m/(m-1)) , where m denotes the number of members in the ensemble.

Note that estimation of internal variability usually means estimation of the covariance matrix of a spatio-temporal climate-vector, the dimension of this matrix potentially being high. We choose to use a multi-model estimate of internal climate variability, derived from a large ensemble of climate models and simulations. This multi-model estimate is subject to lower sampling variability and better represents the effects of model uncertainty on the estimate of internal variability than individual model estimates. We then simultaneously consider control simulations from the CMIP3 and CMIP5 archives, and ensembles of historical simulations (including simulations with individual sets of forcings) from the CMIP5 archive.

All control simulations longer than 220 years (i.e. twice the length of our study period) and all ensembles (at least 2 members) are used.

The overall drift of control simulations is removed by subtracting a linear trend over the full period.. We then implicitly assume that this multi- model internal variability estimate is reliable.

[Emphasis added]. So two approaches to evaluate internal variability – one approach uses GCM runs with no GHG forcing; and the other approach uses the variation between different runs of the same model (with GHG forcing) to estimate natural variability. Drift is removed as “an error”.

The IPCC report also reviews the spatial simulations compared with spatial observations, p. 880:

Figure 10.2a shows the pattern of annual mean surface temperature trends observed over the period 1901–2010, based on Hadley Centre/ Climatic Research Unit gridded surface temperature data set 4 (Had- CRUT4). Warming has been observed at almost all locations with sufficient observations available since 1901.

Rates of warming are generally higher over land areas compared to oceans, as is also apparent over the 1951–2010 period (Figure 10.2c), which simulations indicate is due mainly to differences in local feedbacks and a net anomalous heat transport from oceans to land under GHG forcing, rather than differences in thermal inertia (e.g., Boer, 2011). Figure 10.2e demonstrates that a similar pattern of warming is simulated in the CMIP5 simulations with natural and anthropogenic forcing over the 1901–2010 period. Over most regions, observed trends fall between the 5th and 95th percentiles of simulated trends, and van Oldenborgh et al. (2013) find that over the 1950–2011 period

the pattern of observed grid cell trends agrees with CMIP5 simulated trends to within a combination of model spread and internal variability..

Let’s take a look at van Oldenborgh et al (2013).

There’s a nice video of (I assume) the lead author talking about the paper and comparing the probabilistic approach used in weather forecasts with that of climate models (see Ensemble Forecasting). *I recommend the video for a good introduction to the topic of ensemble forecasting.*

With weather forecasting the probability comes from running ensembles of weather models and seeing, for example, how many simulations predict rain vs how many do not. The proportion is the probability of rain. With weather forecasting we can continually review how well the probabilities given by ensembles match the reality. Over time we will build up a set of statistics of “probability of rain” and compare with the frequency of actual rainfall. It’s pretty easy to see if the models are over-confident or under-confident.

Here is what the authors say about the problem and how they approached it:

The ensemble is considered to be an estimate of the probability density function (PDF) of a climate forecast. This is the method used in weather and seasonal forecasting (Palmer et al 2008). Just like in these fields it is vital to verify that the resulting forecasts are reliable in the definition that the forecast probability should be equal to the observed probability (Joliffe and Stephenson 2011).

If outcomes in the tail of the PDF occur more (less) frequently than forecast the system is overconfident (underconfident): the ensemble spread is not large enough (too large).

In contrast to weather and seasonal forecasts, there is no set of hindcasts to ascertain the reliability of past climate trends per region.

We therefore perform the verification study spatially, comparing the forecast and observed trends over the Earth. Climate change is now so strong that the effects can be observed locally in many regions of the world, making a verification study on the trends feasible.Spatial reliability does not imply temporal reliability, but unreliability does imply that at least in some areas the forecasts are unreliable in time as well. In the remainder of this letter we use the word ‘reliability’ to indicate spatial reliability.

[Emphasis added]. The paper first shows the result for one location, the Netherlands, with the spread of model results vs the actual result from 1950-2011:

*Figure 2*

We can see that the models are overall mostly below the observation. But this is one data point. So if we compared all of the datapoints – and this is on a grid of 2.5º – how do the model spreads compare with the results? Are observations above 95% of the model results only 5% of the time? Or more than 5% of the time? And are observations below 5% of the model results only 5% of the time?

We can see that the frequency of observations in the bottom 5% of model results is about 13% and the frequency of observations in the top 5% of model results is about 20%. Therefore the models are “overconfident” in spatial representation of the last 60 year trends:

*Figure 3*

We investigated the reliability of trends in the CMIP5 multi-model ensemble prepared for the IPCC AR5. In agreement with earlier studies using the older CMIP3 ensemble, the temperature trends are found to be locally reliable. However, this is due to the differing global mean climate response rather than a correct representation of the spatial variability of the climate change signal up to now:

when normalized by the global mean temperature the ensemble is overconfident. This agrees with results of Sakaguchi et al (2012) that the spatial variability in the pattern of warming is too small. The precipitation trends are also overconfident. There are large areas where trends in both observational dataset are (almost) outside the CMIP5 ensemble, leading us to conclude that this is unlikely due to faulty observations.

It’s probably important to note that the author comments in the video “on the larger scale the models are not doing so badly”.

It’s an interesting paper. I’m not clear whether the brief note in AR5 reflects the paper’s conclusions.

It was reassuring to finally find a statement that confirmed what seemed obvious from the “omissions”:

Abasic assumption of the optimal detection analysis is that the estimate of internal variability used is comparable with the real world’s internal variability.

Surely I can’t be the only one reading Chapter 10 and trying to understand the assumptions built into the “with 95% confidence” result. If Chapter 10 is only aimed at climate scientists who work in the field of attribution and detection it is probably fine not to actually mention this minor detail in the tight constraints of only 60 pages.

But if Chapter 10 is aimed at a wider audience it seems a little remiss not to bring it up in the chapter itself.

I probably missed the stated caveat in chapter 10’s executive summary or introduction.

The authors continue:

As the observations are influenced by external forcing, and we do not have a non-externally forced alternative reality to use to test this assumption, an alternative common method is to compare the power spectral density (PSD) of the observations with the model simulations that include external forcings.

We have already seen that overall the CMIP5 and CMIP3 model variability compares favorably across different periodicities with HadCRUT4-observed variability (Figure 5). Figure S11 (in the supporting information) includes the PSDs for each of the eight models (BCC-CSM1-1, CNRM-CM5, CSIRO- Mk3-6-0, CanESM2, GISS-E2-H, GISS-E2-R, HadGEM2- ES and NorESM1-M) that can be examined in the detection analysis.

Variability for the historical experiment in most of the models compares favorably with HadCRUT4 over the range of periodicities, except for HadGEM2-ES whose very long period variability is lower due to the lower overall trend than observed and for CanESM2 and bcc-cm1-1 whose decadal and higher period variability are larger than observed.

While not a strict test, Figure S11 suggests that the models have an adequate representation of internal variability—at least on the global mean level. In addition, we use the residual test from the regression to test whether there are any gross failings in the models representation of internal variability.

Figure S11 is in the supplementary section of the paper:

*Figure 4*

From what I can see, this demonstrates that the spectrum of the models’ internal variability (“historicalNat”) is different from the spectrum of the models’ forced response with GHG changes (“historical”).

It feels like my quantum mechanics classes all over again. I’m probably missing something obvious, and hopefully knowledgeable readers can explain.

Chapter 9, reviewing models, stretches to over 80 pages. The section on internal variability is section 9.5.1:

However, the ability to simulate climate variability, both unforced internal variability and forced variability (e.g., diurnal and seasonal cycles) is also important. This has implications for the signal-to-noise estimates inherent in climate change detection and attribution studies where low-frequency climate variability must be estimated, at least in part, from long control integrations of climate models (Section 10.2).

Section 9.5.3:

In addition to the annual, intra-seasonal and diurnal cycles described above, a number of other modes of variability arise on multi-annual to multi-decadal time scales (see also Box 2.5). Most of these modes have a particular regional manifestation whose amplitude can be larger than that of human-induced climate change. The observational record is usually too short to fully evaluate the representation of variability in models and this motivates the use of reanalysis or proxies, even though these have their own limitations.

Figure 9.33a shows simulated internal variability of mean surface temperature from CMIP5 pre-industrial control simulations. Model spread is largest in the tropics and mid to high latitudes (Jones et al., 2012), where variability is also large; however, compared to CMIP3, the spread is smaller in the tropics owing to improved representation of ENSO variability (Jones et al., 2012). The power spectral density of global mean temperature variance in the historical simulations is shown in Figure 9.33b and is generally consistent with the observational estimates. At longer time scale of the spectra estimated from last millennium simulations, performed with a subset of the CMIP5 models, can be assessed by comparison with different NH temperature proxy records (Figure 9.33c; see Chapter 5 for details). The CMIP5 millennium simulations include natural and anthropogenic forcings (solar, volcanic, GHGs, land use) (Schmidt et al., 2012).

Significant differences between unforced and forced simulations are seen for time scale larger than 50 years, indicating the importance of forced variability at these time scales (Fernandez-Donado et al., 2013). It should be noted that a few models exhibit slow background climate drift which increases the spread in variance estimates at multi-century time scales.

Nevertheless, the lines of evidence above

suggest with high confidence that models reproduce global and NH temperature variability on a wide range of time scales.

[Emphasis added]. Here is fig 9.33:

*Figure 5 – Click to Expand*

The bottom graph shows the spectra of the last 1,000 years – black line is observations (reconstructed from proxies), dashed lines are without GHG forcings, and solid lines are with GHG forcings.

In later articles we will review this in more detail.

The IPCC report on attribution is very interesting. Most attribution studies compare observations of the last 100 – 150 years with model simulations using anthropogenic GHG changes and model simulations without (note 3).

The results show a much better match for the case of the anthropogenic forcing.

The primary method is with global mean surface temperature, with more recent studies also comparing the spatial breakdown. We saw one such comparison with van Oldenborgh et al (2013). Jones et al (2013) also reviews spatial matching, finding a better fit (of models & observations) for the last half of the 20th century than the first half. (As with van Oldenborgh’s paper, the % match outside 90% of model results was greater than 10%).

**My question** as I first read Chapter 10 was *how was the high confidence attained* and *what is a fingerprint*?

I was led back, by following the chain of references, to one of the early papers on the topic (1996) that also had similar high confidence. (We saw this in Part Three). It was intriguing that such confidence could be attained with just a few “no forcing” model runs as comparison, all of which needed “flux adjustment”. Current models need much less, or often zero, flux adjustment.

In later papers reviewed in AR5, “no forcing” model simulations that show temperature trends or jumps are often removed or adjusted.

I’m not trying to suggest that “no forcing” GCM simulations of the last 150 years have anything like the temperature changes we have observed. They don’t.

But I was trying to understand what assumptions and premises were involved in attribution. Chapter 10 of AR5 has been valuable in suggesting references to read, but poor at laying out the assumptions and premises of attribution studies.

For clarity, as I stated in Part Three:

..as regular readers know I am fully convinced that the increases in CO2, CH4 and other GHGs over the past 100 years or more can be very well quantified into “radiative forcing” and am 100% in agreement with the IPCCs summary of the work of atmospheric physics over the last 50 years on this topic. That is, the increases in GHGs have led to something like a “radiative forcing” of 2.8 W/m²..

..Therefore, it’s “very likely” that the increases in GHGs over the last 100 years have contributed significantly to the temperature changes that we have seen.

So what’s my point?

Chapter 10 of the IPCC report fails to highlight the important assumptions in the attribution studies. Chapter 9 of the IPCC report has a section on centennial/millennial natural variability with a “high confidence” conclusion that comes with little evidence and appears to be based on a cursory comparison of the spectral results of the last 1,000 years proxy results with the CMIP5 modeling studies.

In chapter 10, the executive summary states:

..given that observed warming since 1951 is very large compared to climate model estimates of internal variability (Section 10.3.1.1.2), which are assessed to be adequate at global scale (Section 9.5.3.1), we conclude that it is

virtually certain[99-100%] that internal variabilityalonecannot account for the observed global warming since 1951.

[Emphasis added]. I agree, and I don’t think anyone who understands radiative forcing and climate basics would disagree. To claim otherwise would be as ridiculous as, for example, claiming that tiny changes in solar insolation from eccentricity modifications over 100 kyrs cause the end of ice ages, whereas large temperature changes during these ice ages have no effect (see note 2).

The executive summary also says:

It is

extremely likely[95–100%] that human activities caused more than half of the observed increase in GMST from 1951 to 2010.

The idea is plausible, but the confidence level is dependent on a premise that is claimed via one graph (fig 9.33) of the spectrum of the last 1,000 years. High confidence (“that models reproduce global and NH temperature variability on a wide range of time scales”) is just an opinion.

It’s crystal clear, by inspection of CMIP3 and CMIP5 model results, that models with anthropogenic forcing match the last 150 years of temperature changes much better than models held at constant pre-industrial forcing.

I believe natural variability is a difficult subject which needs a lot more than a cursory graph of the spectrum of the last 1,000 years to even achieve **low confidence** in our understanding.

Chapters 9 & 10 of AR5 haven’t investigated “natural variability” at all. For interest, some skeptic opinions are given in note 4.

I propose an alternative summary for Chapter 10 of AR5:

It is extremely likely [95–100%] that human activities caused more than half of the observed increase in GMST from 1951 to 2010, but this assessment is subject to considerable uncertainties.

Multi-model assessment of regional surface temperature trends, TR Knutson, F Zeng & AT Wittenberg, *Journal of Climate* (2013) – free paper

Attribution of observed historical near surface temperature variations to anthropogenic and natural causes using CMIP5 simulations, Gareth S Jones, Peter A Stott & Nikolaos Christidis, *Journal of Geophysical Research Atmospheres* (2013) – paywall paper

Application of regularised optimal fingerprinting to attribution. Part II: application to global near-surface temperature, ￼Aurélien Ribes & Laurent Terray, *Climate Dynamics* (2013) – free paper

Application of regularised optimal fingerprinting to attribution. Part I: method, properties and idealised analysis, ￼Aurélien Ribes, Serge Planton & Laurent Terray, *Climate Dynamics* (2013) – free paper

Reliability of regional climate model trends, GJ van Oldenborgh, FJ Doblas Reyes, SS Drijfhout & E Hawkins, *Environmental Research Letters* (2013) – free paper

**Note 1**: CMIP = Coupled Model Intercomparison Project. CMIP3 was for AR4 and CMIP5 was for AR5.

At a September 2008 meeting involving 20 climate modeling groups from around the world, the WCRP’s Working Group on Coupled Modelling (WGCM), with input from the IGBP AIMES project, agreed to promote a new set of coordinated climate model experiments. These experiments comprise the fifth phase of the Coupled Model Intercomparison Project (CMIP5). CMIP5 will notably provide a multi-model context for

1) assessing the mechanisms responsible for model differences in poorly understood feedbacks associated with the carbon cycle and with clouds

2) examining climate “predictability” and exploring the ability of models to predict climate on decadal time scales, and, more generally

3) determining why similarly forced models produce a range of responses…

From the website link above you can read more. CMIP5 is a substantial undertaking, with massive output of data from the latest climate models. Anyone can access this data, similar to CMIP3. Here is the Getting Started page.

And CMIP3:

In response to a proposed activity of the World Climate Research Programme (WCRP) Working Group on Coupled Modelling (WGCM), PCMDI volunteered to collect model output contributed by leading modeling centers around the world. Climate model output from simulations of the past, present and future climate was collected by PCMDI mostly during the years 2005 and 2006, and this archived data constitutes phase 3 of the Coupled Model Intercomparison Project (CMIP3). In part, the WGCM organized this activity to enable those outside the major modeling centers to perform research of relevance to climate scientists preparing the Fourth Asssessment Report (AR4) of the Intergovernmental Panel on Climate Change (IPCC). The IPCC was established by the World Meteorological Organization and the United Nations Environmental Program to assess scientific information on climate change. The IPCC publishes reports that summarize the state of the science.

This unprecedented collection of recent model output is officially known as the “WCRP CMIP3 multi-model dataset.” It is meant to serve IPCC’s Working Group 1, which focuses on the physical climate system — atmosphere, land surface, ocean and sea ice — and the choice of variables archived at the PCMDI reflects this focus. A more comprehensive set of output for a given model may be available from the modeling center that produced it.

With the consent of participating climate modelling groups, the WGCM has declared the CMIP3 multi-model dataset open and free for non-commercial purposes. After registering and agreeing to the “terms of use,” anyone can now obtain model output via the ESG data portal, ftp, or the OPeNDAP server.

As of July 2009, over 36 terabytes of data were in the archive and over 536 terabytes of data had been downloaded among the more than 2500 registered users

**Note 2**: This idea is explained in Ghosts of Climates Past -Eighteen – “Probably Nonlinearity” of Unknown Origin – *what is believed and what is put forward as evidence for the theory that ice age terminations were caused by orbital changes*, see especially the section under the heading: **Why Theory B is Unsupportable**.

**Note 3**: Some studies use just fixed pre-industrial values, and others compare “natural forcings” with “no forcings”.

“Natural forcings” = radiative changes due to solar insolation variations (which are not known with much confidence) and aerosols from volcanos. “No forcings” is simply fixed pre-industrial values.

**Note 4**: Chapter 11 (of AR5), p.982:

For the remaining projections in this chapter the spread among the CMIP5 models is used as a simple, but crude, measure of uncertainty. The extent of agreement between the CMIP5 projections provides rough guidance about the likelihood of a particular outcome.

But—as partly illustrated by the discussion above—it must be kept firmly in mind that the real world could fall outside of the range spanned by these particular models. See Section 11.3.6 for further discussion.

And p. 1004:

It is possible that the real world might follow a path outside (above or below) the range projected by the CMIP5 models. Such an eventuality could arise if there are processes operating in the real world that are missing from, or inadequately represented in, the models. Two main possibilities must be considered: (1) Future radiative and other forcings may diverge from the RCP4.5 scenario and, more generally, could fall outside the range of all the RCP scenarios; (2) The response of the real climate system to radiative and other forcing may differ from that projected by the CMIP5 models.

A third possibility is that internal fluctuations in the real climate system are inadequately simulated in the models. The fidelity of the CMIP5 models in simulating internal climate variability is discussed in Chapter 9....

The response of the climate system to radiative and other forcing is influenced by a very wide range of processes, not all of which are adequately simulated in the CMIP5 models(Chapter 9). Of particular concern for projections are mechanisms that could lead to major ‘surprises’ such as an abrupt or rapid change that affects global-to-continental scale climate.Several such mechanisms are discussed in this assessment report; these include: rapid changes in the Arctic (Section 11.3.4 and Chapter 12), rapid changes in the ocean’s overturning circulation (Chapter 12), rapid change of ice sheets (Chapter 13) and rapid changes in regional monsoon systems and hydrological climate (Chapter 14). Additional mechanisms may also exist as synthesized in Chapter 12. These mechanisms have the potential to influence climate in the near term as well as in the long term, albeit the likelihood of substantial impacts increases with global warming and is generally lower for the near term.

And p. 1009 (note that we looked at Rowlands et al 2012 in Part Five – Why Should Observations match Models?):

The CMIP3 and CMIP5 projections are ensembles of opportunity, and

it is explicitly recognized that there are sources of uncertainty not simulated by the models. Evidence of this can be seen by comparing the Rowlands et al. (2012) projections for the A1B scenario, which were obtained using a very large ensemble in which the physics parameterizations were perturbed in a single climate model, with the corresponding raw multi-model CMIP3 projections. The former exhibit a substantially larger likely range than the latter. A pragmatic approach to addressing this issue, which was used in the AR4 and is also used in Chapter 12, is to consider the 5 to 95% CMIP3/5 range as a ‘likely’ rather than ‘very likely’ range.

Replacing ‘very likely’ = 90–100% with ‘likely 66–100%’ is a good start. How does this recast chapter 10?

And Chapter 1 of AR5, p. 138:

Model spread is often used as a measure of climate response uncertainty, but such a measure is crude as it takes no account of factors such as model quality (Chapter 9) or model independence (e.g., Masson and Knutti, 2011; Pennell and Reichler, 2011), and not all variables of interest are adequately simulated by global climate models..

..

Climate varies naturally on nearly all time and space scales, and quantifying precisely the nature of this variability is challenging, and is characterized by considerable uncertainty.

[Emphasis added in all bold sections above]

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It’s an easy paper to read (and free) and so I recommend reading the whole paper.

The paper uses NOAA’s Geophysical Fluid Dynamics Laboratory (GFDL) CM2.1 global coupled atmosphere/ocean/land/ice GCM (see note 1 for reference and description):

CM2.1 played a prominent role in the third Coupled Model Intercomparison Project (CMIP3) and the Fourth Assessment of the Intergovernmental Panel on Climate Change (IPCC), and its tropical and ENSO simulations have consistently ranked among the world’s top GCMs [van Oldenborgh et al., 2005; Wittenberg et al., 2006; Guilyardi, 2006; Reichler and Kim, 2008].

The coupled pre-industrial control run is initialized as by Delworth et al. [2006], and then integrated for 2220 yr with fixed 1860 estimates of solar irradiance, land cover, and atmospheric composition; we focus here on just the last 2000 yr. This simulation required one full year to run on 60 processors at GFDL.

First of all we see the challenge for climate models – a reasonable resolution coupled GCM running just one 2000-year simulation consumed one year of multiple processor time.

Wittenberg shows the results in the graph below. At the top is our observational record going back 140 years, then below are the simulation results of the SST variation in the El Nino region broken into 20 century-long segments.

* Figure 1 – Click to Expand*

What we see is that different centuries have very different results:

There are multidecadal epochs with hardly any variability (M5); epochs with intense, warm-skewed ENSO events spaced five or more years apart (M7); epochs with moderate, nearly sinusoidal ENSO events spaced three years apart (M2); and epochs that are highly irregular in amplitude and period (M6). Occasional epochs even mimic detailed temporal sequences of observed ENSO events; e.g., in both R2 and M6, there are decades of weak, biennial oscillations, followed by a large warm event, then several smaller events, another large warm event, and then a long quiet period. Although the model’s NINO3 SST variations are generally stronger than observed, there are long epochs (like M1) where the ENSO amplitude agrees well with observations (R1).

Wittenberg comments on the problem for climate modelers:

An unlucky modeler – who by chance had witnessed only M1-like variability throughout the first century of simulation – might have erroneously inferred that the model’s ENSO amplitude matched observations, when a longer simulation would have revealed a much stronger ENSO.

If the real-world ENSO is similarly modulated, then there is a more disturbing possibility.

Had the research community been unlucky enough to observe an unrepresentative ENSO over the past 150 yr of measurements, then it might collectively have misjudged ENSO’s longer-term natural behavior. In that case, historically-observed statistics could be a poor guide for modelers, and observed trends in ENSO statistics might simply reflect natural variations....A 200 yr epoch of consistently strong variability (M3) can be followed, just one century later, by a 200 yr epoch of weak variability (M4). Documenting such extremes might thus require a 500+ yr record. Yet few modeling centers currently attempt simulations of that length when evaluating CGCMs under development – due to competing demands for high resolution, process completeness, and quick turnaround to permit exploration of model sensitivities.

Model developers thus might not even realize that a simulation manifested long-term ENSO modulation, until long after freezing the model development. Clearly this could hinder progress.

An unlucky modeler – unaware of centennial ENSO modulation and misled by comparisons between short, unrepresentative model runs – might erroneously accept a degraded model or reject an improved model.

[Emphasis added].

Wittenberg shows the same data in the frequency domain and has presented the data in a way that illustrates the different perspective you might have depending upon your period of observation or period of model run. It’s worth taking the time to understand what is in these graphs:

*Figure 2 – Click to Expand*

The first graph, 2a:

..time-mean spectra of the observations for epochs of length 20 yr – roughly the duration of observations from satellites and the Tropical Atmosphere Ocean (TAO) buoy array. The spectral power is fairly evenly divided between the seasonal cycle and the interannual ENSO band, the latter spanning a broad range of time scales between 1.3 to 8 yr.

So the different colored lines indicate the spectral power for each period. The black dashed line is the observed spectral power over the 140 year (observational) period. This dashed line is repeated in figure 2c.

The second graph, 2b shows the modeled results if we break up the 2000 years into 100 x 20-year periods.

The third graph, 2c, shows the modeled results broken up into 100 year periods. The probability number in the bottom right, 90%, is the likelihood of observations falling outside the range of the model results – if “*the simulated subspectra independent and identically distributed.. at bottom right is the probability that an interval so constructed would bracket the next subspectrum to emerge from the model.*”

So what this says, paraphrasing and over-simplifying: “we are 90% sure that the observations can’t be explained by the models”.

*Of course, this independent and identically distributed assumption is not valid, but as we will hopefully get onto many articles further in this series, most of these statistical assumptions – stationary, gaussian, AR1 – are problematic for real world non-linear systems*.

To be clear, the paper’s author is demonstrating a problem in such a statistical approach.

Models are not reality. This is a simulation with the GFDL model. It doesn’t mean ENSO is like this. But it might be.

The paper illustrates a problem I highlighted in Part Five – observations are only one “realization” of possible outcomes. The last century or century and a half of surface observations could be an outlier. The last 30 years of satellite data could equally be an outlier. Even if our observational periods are not an outlier and are right there on the mean or median, matching climate models to observations may still greatly under-represent natural climate variability.

Non-linear systems can demonstrate variability over much longer time-scales than the the typical period between characteristic events. We will return to this in future articles in more detail. Such systems do not have to be “chaotic” (where chaotic means that tiny changes in initial conditions cause rapidly diverging results).

*What period of time is necessary to capture natural climate variability?*

I will give the last word to the paper’s author:

More worryingly, if nature’s ENSO is similarly modulated, there is no guarantee that the 150 yr historical SST record is a fully representative target for model development..

..In any case, it is sobering to think that even absent any anthropogenic changes, the future of ENSO could look very different from what we have seen so far.

Are historical records sufficient to constrain ENSO simulations? Andrew T. Wittenberg, *GRL* (2009) – free paper

GFDL’s CM2 Global Coupled Climate Models. Part I: Formulation and Simulation Characteristics, Delworth et al, *Journal of Climate*, 2006 – free paper

**Note 1**: The paper referenced for the GFDL model is GFDL’s CM2 Global Coupled Climate Models. Part I: Formulation and Simulation Characteristics, Delworth et al, 2006:

The formulation and simulation characteristics of two new global coupled climate models developed at NOAA’s Geophysical Fluid Dynamics Laboratory (GFDL) are described.

The models were designed to simulate atmospheric and oceanic climate and variability from the diurnal time scale through multicentury climate change, given our computational constraints. In particular, an important goal was to use the same model for both experimental seasonal to interannual forecasting and the study of multicentury global climate change, and this goal has been achieved.

Two versions of the coupled model are described, called CM2.0 and CM2.1. The versions differ primarily in the dynamical core used in the atmospheric component, along with the cloud tuning and some details of the land and ocean components. For both coupled models, the resolution of the land and atmospheric components is 2° latitude x 2.5° longitude; the atmospheric model has 24 vertical levels.

The ocean resolution is 1° in latitude and longitude, with meridional resolution equatorward of 30° becoming progressively finer, such that the meridional resolution is 1/3° at the equator. There are 50 vertical levels in the ocean, with 22 evenly spaced levels within the top 220 m. The ocean component has poles over North America and Eurasia to avoid polar filtering. Neither coupled model employs flux adjustments.

The control simulations have stable, realistic climates when integrated over multiple centuries. Both models have simulations of ENSO that are substantially improved relative to previous GFDL coupled models. The CM2.0 model has been further evaluated as an ENSO forecast model and has good skill (CM2.1 has not been evaluated as an ENSO forecast model). Generally reduced temperature and salinity biases exist in CM2.1 relative to CM2.0. These reductions are associated with 1) improved simulations of surface wind stress in CM2.1 and associated changes in oceanic gyre circulations; 2) changes in cloud tuning and the land model, both of which act to increase the net surface shortwave radiation in CM2.1, thereby reducing an overall cold bias present in CM2.0; and 3) a reduction of ocean lateral viscosity in the extra- tropics in CM2.1, which reduces sea ice biases in the North Atlantic.

Both models have been used to conduct a suite of climate change simulations for the 2007 Intergovern- mental Panel on Climate Change (IPCC) assessment report and are able to simulate the main features of the observed warming of the twentieth century. The climate sensitivities of the CM2.0 and CM2.1 models are 2.9 and 3.4 K, respectively. These sensitivities are defined by coupling the atmospheric components of CM2.0 and CM2.1 to a slab ocean model and allowing the model to come into equilibrium with a doubling of atmospheric CO2. The output from a suite of integrations conducted with these models is freely available online (see http://nomads.gfdl.noaa.gov/).

There’s a brief description of the newer model version CM3.0 on the GFDL page.

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I’ve had a question about the current approach to climate models for some time and found it difficult to articulate. In reading Broad range of 2050 warming from an observationally constrained large climate model ensemble, Daniel Rowlands et al, *Nature* (2012) I found an explanation that helps me clarify my question.

This paper by Rowlands et al is similar in approach to that of Stainforth et al 2005 – the idea of much larger **ensembles** of climate models. The Stainforth paper was discussed in the comments of Models, On – and Off – the Catwalk – Part Four – Tuning & the Magic Behind the Scenes.

*For new readers who want to understand a bit more about ensembles of models – take a look at Ensemble Forecasting*.

The basic idea behind ensembles for weather forecasts is that we have uncertainty about:

- the initial conditions – because observations are not perfect
- parameters in our model – because our understanding of the physics of weather is not perfect

So multiple simulations are run and the frequency of occurrence of, say, a severe storm tells us the probability that the severe storm will occur.

Given the short term nature of weather forecasts we can compare the frequency of occurrence of particular events with the % probability that our ensemble produced.

Let’s take an example to make it clear. Suppose the ensemble prediction of a severe storm in a certain area is 5%. The severe storm occurs. What can we make of the accuracy our prediction? Well, we can’t deduce anything from that event.

Why? Because we only had one occurrence.

Out of a 1000 future forecasts, the “5%ers” are going to occur 50 times – if we are right on the money with our probabilistic forecast. We need a lot of forecasts to be compared with a lot of results. Then we might find that 5%ers actually occur 20% of the time. Or only 1% of the time. Armed with this information we can a) try and improve our model because we know the deficiencies, and b) temper our ensemble forecast with our knowledge of how well it has historically predicted the 5%, 10%, 90% chances of occurrence.

This is exactly what currently happens with numerical weather prediction.

And if instead we run one simulation with our “best estimate” of initial conditions and parameters the results are not as good as the results from the ensemble.

The idea behind ensembles of climate forecasts is subtly different. Initial conditions are no help with predicting the long term statistics (aka “climate”). But we still have a lot of uncertainty over model physics and parameterizations. So we run ensembles of simulations with slightly different physics/parameterizations (see note 2).

Assuming our model is a decent representation of climate, there are three important points:

- we need to know the timescale of “predictable statistics”, given constant “external” forcings (e.g. anthropogenic GHG changes)
- we need to cover the real range of possible parameterizations
- the results we get from ensembles can, at best, only ever give us the
**probabilities**of outcomes over a given time period

Item 1 was discussed in the last article and I have not been able to find any discussion of this timescale in climate science papers (that doesn’t mean there aren’t any, hopefully someone can point me to a discussion of this topic).

Item 2 is something that I believe climate scientists are very interested in. The limitation has been, and still is, the computing power required.

Item 3 is what I want to discuss in this article, around the paper by Rowlands et al.

In the latest generation of coupled atmosphere–ocean general circulation models (AOGCMs) contributing to the Coupled Model Intercomparison Project phase 3 (CMIP-3), uncertainties in key properties controlling the twenty-first century response to sustained anthropogenic greenhouse-gas forcing were not fully sampled, partially owing to a correlation between climate sensitivity and aerosol forcing, a tendency to overestimate ocean heat uptake and compensation between short-wave and long-wave feedbacks.

This complicates the interpretation of the ensemble spread as a direct uncertainty estimate, a point reflected in the fact that the ‘likely’ (>66% probability) uncertainty range on the transient response was explicitly subjectively assessed as −40% to +60% of the CMIP-3 ensemble mean for global-mean temperature in 2100, in the Intergovernmental Panel on Climate Change (IPCC) Fourth Assessment Report (AR4). The IPCC expert range was supported by a range of sources, including studies using pattern scaling, ensembles of intermediate-complexity models, and estimates of the strength of carbon-cycle feedbacks.

From this evidence it is clear that the CMIP-3 ensemble, which represents a valuable expression of plausible responses consistent with our current ability to explore model structural uncertainties, fails to reflect the full range of uncertainties indicated by expert opinion and other methods....Perturbed-physics ensembles offer a systematic approach to quantify uncertainty in models of the climate system response to external forcing. Here we investigate uncertainties in the twenty-first century transient response

in a multi-thousand-member ensembleof transient AOGCM simulations from 1920 to 2080 using HadCM3L, a version of the UK Met Office Unified Model, as part of the climateprediction.net British Broadcasting Corporation (BBC) climate change experiment (CCE). Wegenerate ensemble members by perturbing the physics in the atmosphere, ocean and sulphur cycle components, with transient simulations driven by a set of natural forcing scenarios and the SRES A1B emissions scenario, and also control simulations to account for unforced model drifts.

[Emphasis added]. So this project runs a much larger ensemble than the CMIP3 models produced for AR4.

Figure 1 shows the evolution of global-mean surface temperatures in the ensemble (relative to 1961–1990), each coloured by the goodness-of-fit to observations of recent surface temperature changes, as detailed below.

The raw ensemble range (1.1–4.2 K around 2050), primarily driven by uncertainties in climate sensitivity (Supplementary Information),

is potentially misleading because many ensemble members have an unrealistic response to the forcing over the past 50 years.

[Emphasis added]

And later in the paper:

..On the assumption that models that simulate past warming realistically are our best candidates for making estimates of the future..

So here’s my question:

If model simulations give us probabilistic forecasts of future climate, why are climate model simulations “compared” with the average of the last few years current “weather” – and those that don’t match up well are rejected or devalued?

It seems like an obvious thing to do, of course. But current averaged weather might be in the top 10% or the bottom 10% of probabilities. We have no way of knowing.

Let’s say that the current 10-year average of GMST = 13.7ºC (I haven’t looked up the right value).

Suppose for the given “external” conditions (solar output and latitudinal distribution, GHG concentration) the “climate” – i.e., **the real long term statistics of weather** – has an average of 14.5ºC, with a standard deviation for any 10-year period of 0.5ºC. That is, 95% of 10-year periods would lie inside 13.5 – 15.5ºC (2 std deviations).

If we run a lot of simulations (and they truly represent the climate) then of course we expect 5% to be outside 13.5 – 15.5ºC. If we reject that 5% as being “unrealistic of current climate”, we’ve arbitrarily and incorrectly reduced the spread of our ensemble.

If we assume that “current averaged weather” – at 13.7ºC – represents reality then we might bias our results even more, depending on the standard deviation that we calculate or assume. We might accept outliers of 13.0ºC because they are closer to our observable and reject good simulations of 15.0ºC because they are more than two standard deviations from our observable (note 3).

The whole point of running an ensemble of simulations is to find out what the spread is, given our current understanding of climate physics.

Let me give another example. One theory for initiation of El Nino is that its initiation is essentially a random process during certain favorable conditions. Now we might have a model that reproduced El Nino starting in 1998 and 10 models that reproduced El Nino starting in other years. Do we promote the El Nino model that “predicted in retrospect” 1998 and demote/reject the others? No. We might actually be rejecting better models. We would need to look at the statistics of lots of El Ninos to decide.

Here’s a couple of papers that don’t articulate the point of view of this article – however, they do comment on the uncertainties in parameter space from a different and yet related perspective.

First, Kiehl 2007:

Methods of testing these models with observations form an important part of model development and application. Over the past decade one such test is our ability to simulate the global anomaly in surface air temperature for the 20th century.. Climate model simulations of the 20th century can be compared in terms of their ability to reproduce this temperature record. This is now an established necessary test for global climate models.

Of course this is not a sufficient test of these models and other metrics should be used to test models..

..A review of the published literature on climate simulations of the 20th century indicates that a large number of fully coupled three dimensional climate models are able to simulate the global surface air temperature anomaly with a good degree of accuracy [Houghton et al., 2001]. For example all models simulate a global warming of 0.5 to 0.7°C over this time period to within 25% accuracy. This is viewed as a reassuring confirmation that models to first order capture the behavior of the physical climate system..

One curious aspect of this result is that it is also well known [Houghton et al., 2001] that the same models that agree in simulating the anomaly in surface air temperature differ significantly in their predicted climate sensitivity. The cited range in climate sensitivity from a wide collection of models is usually 1.5 to 4.5°C for a doubling of CO2, where most global climate models used for climate change studies vary by at least a factor of two in equilibrium sensitivity.

The question is: if climate models differ by a factor of 2 to 3 in their climate sensitivity, how can they all simulate the global temperature record with a reasonable degree of accuracy.

Second, *Why are climate models reproducing the observed global surface warming so well?* Knutti (2008):

The agreement between the CMIP3 simulated and observed 20th century warming is indeed remarkable. But do the current models simulate the right magnitude of warming for the right reasons? How much does the agreement really tell us?

Kiehl [2007] recently showed a correlation of climate sensitivity and total radiative forcing across an older set of models, suggesting that models with high sensitivity (strong feedbacks) avoid simulating too much warming by using a small net forcing (large negative aerosol forcing), and models with weak feedbacks can still simulate the observed warming with a larger forcing (weak aerosol forcing).

Climate sensitivity, aerosol forcing and ocean diffusivity are all uncertain and relatively poorly constrained from the observed surface warming and ocean heat uptake [e.g., Knutti et al., 2002; Forest et al., 2006]. Models differ because of their underlying assumptions and parameterizations, and it is plausible that choices are made based on the model’s ability to simulate observed trends..

..Models, therefore, simulate similar warming for different reasons, and it is unlikely that this effect would appear randomly. While it is impossible to know what decisions are made in the development process of each model, it seems plausible that choices are made based on agreement with observations as to what parameterizations are used, what forcing datasets are selected, or whether an uncertain forcing (e.g., mineral dust, land use change) or feedback (indirect aerosol effect) is incorporated or not.

..Second, the question is whether we should be worried about the correlation between total forcing and climate sensitivity. Schwartz et al. [2007] recently suggested that ‘‘the narrow range of modelled temperatures [in the CMIP3 models over the 20th century] gives a false sense of the certainty that has been achieved’’. Because of the good agreement between models and observations and compensating effects between climate sensitivity and radiative forcing (as shown here and by Kiehl [2007]) Schwartz et al. [2007] concluded that the CMIP3 models used in the most recent Intergovernmental Panel on Climate Change (IPCC) report [IPCC, 2007] ‘‘may give a false sense of their predictive capabilities’’.

Here I offer a different interpretation of the CMIP3 climate models. They constitute an ‘ensemble of opportunity’, they share biases, and probably do not sample the full range of uncertainty [Tebaldi and Knutti, 2007; Knutti et al., 2008]. The model development process is always open to influence, conscious or unconscious, from the participants’ knowledge of the observed changes. It is therefore neither surprising nor problematic that the simulated and observed trends in global temperature are in good agreement.

The idea that climate models should all reproduce global temperature anomalies over a 10-year or 20-year or 30-year time period, presupposes that we know:

a) climate, as the long term statistics of weather, can be reliably obtained over these time periods. Remember that with a simple chaotic system where we have “deity like powers” we can simulate the results and find the time period over which the statistics are reliable.

or

b) climate, as the 10-year (or 20-year or 30-year) statistics of weather is tightly constrained within a small range, to a high level of confidence, and therefore we can reject climate model simulations that fall outside this range.

Given that this Rowlands et al 2012 is attempting to better sample climate uncertainty by a larger ensemble it’s clear that this answer is not known in advance.

There are a lot of uncertainties in climate simulation. Constraining models to match the past may be under-sampling the actual range of climate variability.

Models are not reality. But if we accept that climate simulation is, at best, a probabilistic endeavor, then we must sample what the models produce, rather than throwing out results that don’t match the last 100 years of recorded temperature history.

Broad range of 2050 warming from an observationally constrained large climate model ensemble, Daniel Rowlands et al, *Nature* (2012) – free paper

Uncertainty in predictions of the climate response to rising levels of greenhouse gases, Stainforth et al, *Nature* (2005) – free paper

Why are climate models reproducing the observed global surface warming so well? Reto Knutti, *GRL* (2008) – free paper

Twentieth century climate model response and climate sensitivity, Jeffrey T Kiehl, *GRL* (2007) – free paper

**Note 1**: We are using the ideas that have been learnt from simple chaotic systems, like the Lorenz 1963 model. There is discussion of this in Part One and Part Two of this series. As some commenters have pointed out that doesn’t mean the climate works in the same way as these simple systems, it is much more complex.

The starting point is that weather is unpredictable. With modern numerical weather prediction (NWP) on current supercomputers we can get good forecasts 1 week ahead. But beyond that we might as well use the average value for that month in that location, measured over the last decade. It’s going to be better than a forecast from NWP.

The idea behind climate prediction is that even though picking the weather 8 weeks from now is a no-hoper, what we have learnt from simple chaotic systems is that the **statistics** of many chaotic systems can be reliably predicted.

**Note 2**: Models are run with different initial conditions as well. My only way of understanding this from a theoretical point of view (i.e., from anything other than a “practical” or “this is how we have always done it” approach) is to see different initial conditions as comparable to one model run over a much longer period.

That is, if climate is not an “initial value problem”, why are initial values changed in each ensemble member to assist climate model output? Running 10 simulations of the same model for 100 years, each with different initial conditions, should be equivalent to running one simulation for 1,000 years.

Well, that is not necessarily true because that 1,000 years might not sample the complete “attractor space”, which is the same point discussed in the last article.

**Note 3**: Models are usually compared to observations via temperature anomalies rather than via actual temperatures, see Models, On – and Off – the Catwalk – Part Four – Tuning & the Magic Behind the Scenes. The example was given for simplicity.

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And instead for this article I decided to focus on what might seem like an obscure point. I hope readers stay with me because it is important.

Here is a graphic from chapter 11 of IPCC AR5:

*Figure 1*

And in the introduction, chapter 1:

Climate in a narrow sense is usually defined as the average weather, or more rigorously, as the statistical description in terms of the mean and variability of relevant quantities over a period of time ranging from months to thousands or millions of years. The relevant quantities are most often surface variables such as temperature, precipitation and wind.

Classically the period for averaging these variables is 30 years, as defined by the World Meteorological Organization.Climate in a wider sense also includes not just the mean conditions, but also the associated statistics (frequency, magnitude, persistence, trends, etc.), often combining parameters to describe phenomena such as droughts. Climate change refers to a change in the state of the climate that can be identified (e.g., by using statistical tests) by changes in the mean and/or the variability of its properties, and that persists for an extended period, typically decades or longer.

[Emphasis added].

The idea is fundamental, the implementation is problematic.

As explained in Natural Variability and Chaos – Two – Lorenz 1963, there are two key points about a chaotic system:

- With even a minute uncertainty in the initial starting condition, the predictability of future states is very limited
- Over a long time period the statistics of the system are well-defined

(Being technical, the statistics are well-defined in a *transitive* system).

So in essence, we can’t predict the exact state of the future – from the current conditions – beyond a certain timescale which might be quite small. In fact, in current weather prediction this time period is about one week.

After a week we might as well say either “the weather on that day will be the same as now” or “the weather on that day will be the climatological average” – and either of these will be better than trying to predict the weather based on the initial state.

No one disagrees on this first point.

In current climate science and meteorology the term used is the **skill** of the forecast. Skill means, not how good is the forecast, but how much **better** is it than a **naive** approach like, “it’s July in New York City so the maximum air temperature today will be 28ºC”.

What happens in practice, as can be seen in the simple Lorenz system shown in Part Two, is a tiny uncertainty about the starting condition gets amplified. Two almost identical starting conditions will diverge rapidly – the “butterfly effect”. Eventually these two conditions are no more alike than one of the conditions and a time chosen at random from the future.

The wide divergence doesn’t mean that the future state can be anything. Here’s an example from the simple Lorenz system for three slightly different initial conditions:

*Figure 2*

We can see that the three conditions that looked identical for the first 20 seconds (see figure 2 in Part Two) have diverged. The values are bounded but at any given time we can’t predict what the value will be.

On the second point – the statistics of the system, there is a tiny hiccup.

But first let’s review what is agreed upon. Climate is the statistics of weather. Weather is unpredictable more than a week ahead. Climate, as the **statistics of weather**, might be predictable. That is, just because weather is unpredictable, it doesn’t mean (or prove) that climate is also unpredictable.

This is what we find with simple chaotic systems.

So in the endeavor of climate modeling the best we can hope for is a probabilistic forecast. We have to run “a lot” of simulations and review the statistics of the parameter we are trying to measure.

To give a concrete example, we might determine from model simulations that the mean sea surface temperature in the western Pacific (between a certain latitude and longitude) in July has a mean of 29ºC with a standard deviation of 0.5ºC, while for a certain part of the north Atlantic it is 6ºC with a standard deviation of 3ºC. In the first case the spread of results tells us – if we are confident in our predictions – that we know the western Pacific SST quite accurately, but the north Atlantic SST has a lot of uncertainty. We can’t do anything about the model spread. In the end, the statistics are knowable (in theory), but the actual value on a given day or month or year are not.

Now onto the hiccup.

With “simple” chaotic systems that we can perfectly model (note 1) we don’t know in advance the timescale of “predictable statistics”. We have to run lots of simulations over long time periods until the statistics converge on the same result. If we have parameter uncertainty (see Ensemble Forecasting) this means we also have to run simulations over the spread of parameters.

Here’s my suggested alternative of the initial value vs boundary value problem:

*Figure 3*

So one body made an ad hoc definition of climate as the 30-year average of weather.

If this definition is correct and accepted then “climate” is not a “boundary value problem” at all. Climate is an initial value problem and therefore a massive problem given our ability to forecast only one week ahead.

Suppose, equally reasonably, that the statistics of weather (=climate), given constant forcing (note 2), are **predictable over a 10,000 year period**.

In that case we can be confident that, with near perfect models, we have the ability to be confident about the averages, standard deviations, skews, etc of the temperature at various locations on the globe over a 10,000 year period.

The fact that chaotic systems exhibit certain behavior doesn’t mean that 30-year statistics of weather can be reliably predicted.

30-year statistics might be just as dependent on the initial state as the weather three weeks from today.

**Note 1**: The climate system is obviously imperfectly modeled by GCMs, and this will always be the case. The advantage of a simple model is we can state that the model is a perfect representation of the system – it is just a definition for convenience. It allows us to evaluate how slight changes in initial conditions or parameters affect our ability to predict the future.

The IPCC report also has continual reminders that the model is not reality, for example, chapter 11, p. 982:

For the remaining projections in this chapter the spread among the CMIP5 models is used as a simple, but crude, measure of uncertainty. The extent of agreement between the CMIP5 projections provides rough guidance about the likelihood of a particular outcome. But — as partly illustrated by the discussion above —

it must be kept firmly in mindthat the real world could fall outside of the range spanned by these particular models.

[Emphasis added].

Chapter 1, p.138:

Model spread is often used as a measure of climate response uncertainty, but such a measure is crude as it takes no account of factors such as model quality (Chapter 9) or model independence (e.g., Masson and Knutti, 2011; Pennell and Reichler, 2011), and not all variables of interest are adequately simulated by global climate models..

..Climate varies naturally on nearly all time and space scales, and quantifying precisely the nature of this variability is challenging, and is characterized by considerable uncertainty.

I haven’t yet been able to determine how these firmly noted and challenging uncertainties have been factored into the quantification of 95-100%, 99-100%, etc, in the various chapters of the IPCC report.

**Note 2**: There are some complications with defining exactly what system is under review. For example, do we take the current solar output, current obliquity,precession and eccentricity as fixed? If so, then any statistics will be calculated for a condition that will anyway be changing. Alternatively, we can take these values as changing inputs in so far as we know the changes – which is true for obliquity, precession and eccentricity but not for solar output.

The details don’t really alter the main point of this article.

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In Part One and Part Two we had a look at chaotic systems and what that might mean for weather and climate. I was planning to develop those ideas a lot more before discussing attribution, but anyway..

AR5, *Chapter 10: Attribution* is 85 pages on the idea that the changes over the last 50 or 100 years in mean surface temperature – and also some other climate variables – can be attributed primarily to anthropogenic greenhouse gases.

The technical side of the discussion fascinated me, but has a large statistical component. I’m a rookie with statistics, and maybe because of this, I’m often suspicious about statistical arguments.

The foundation of a lot of statistics is the idea of independent events. For example, spin a roulette wheel and you get a number between 0 and 36 and a color that is red, black – or if you’ve landed on a zero, neither.

The statistics are simple – each spin of the roulette wheel is an **independent event** – that is, it has no relationship with the last spin of the roulette wheel. So, looking ahead, what is the chance of getting 5 two times in a row? The answer (with a 0 only and no “00” as found in some roulette tables) is 1/37 x 1/37 = 0.073%.

However, after you have spun the roulette wheel and got a 5, what is the chance of a second 5? It’s now just 1/37 = 2.7%. The past has no impact on the future statistics. Most of real life doesn’t correspond particularly well to this idea, apart from playing games of chance like poker and so on.

I was in the gym the other day and although I try and drown it out with music from my iPhone, the Travesty (aka “the News”) was on some of the screens in the gym – with text of the “high points” on the screen aimed at people trying to drown out the annoying travestyreaders. There was a report that a new study had found that autism was caused by “Cause X” – I have blanked it out to avoid any unpleasant feeling for parents of autistic kids – or people planning on having kids who might worry about “Cause X”.

It did get me thinking – if you have let’s say 10,000 potential candidates for causing autism, and you set the bar at 95% probability of rejecting the hypothesis that a given potential cause is a factor, what is the outcome? Well, if there is a random spread of autism among the population with no actual cause (let’s say it is caused by a random genetic mutation with no link to any parental behavior, parental genetics or the environment) then you will expect to find about 500 “statistically significant” factors for autism simply by testing at the 95% level. That’s 500, when none of them are actually the real cause. It’s just chance. Plenty of fodder for pundits though.

That’s one problem with statistics – the answer you get unavoidably depends on your frame of reference.

The questions I have about attribution are unrelated to this specific point about statistics, but there are statistical arguments in the attribution field that seem fatally flawed. Luckily I’m a statistical novice so no doubt readers will set me straight.

On another unrelated point about statistical independence, only slightly more relevant to the question at hand, Pirtle, Meyer & Hamilton (2010) said:

In short, we note that GCMs are commonly treated as independent from one another, when in fact there are many reasons to believe otherwise. The assumption of independence leads to increased confidence in the ‘‘robustness’’ of model results when multiple models agree. But GCM independence has not been evaluated by model builders and others in the climate science community. Until now the climate science literature has given only passing attention to this problem, and the field has not developed systematic approaches for assessing model independence.

.. end of digression

In my efforts to understand Chapter 10 of AR5 I followed up on a lot of references and ended up winding my way back to Hegerl et al 1996.

Gabriele Hegerl is one of the lead authors of Chapter 10 of AR5, was one of the two coordinating lead authors of the Attribution chapter of AR4, and one of four lead authors on the relevant chapter of AR3 – and of course has a lot of papers published on this subject.

As is often the case, I find that to understand a subject you have to start with a focus on the earlier papers because the later work doesn’t make a whole lot of sense without this background.

This paper by Hegerl and her colleagues use the work of one of the co-authors, Klaus Hasselmann – his 1993 paper “Optimal fingerprints for detection of time dependent climate change”.

Fingerprints, by the way, seems like a marketing term. Fingerprints evokes the idea that you can readily demonstrate that John G. Doe of 137 Smith St, Smithsville was at least present at the crime scene and there is no possibility of confusing his fingerprints with John G. Dode who lives next door even though their mothers could barely tell them apart.

This kind of attribution is more in the realm of “was it the 6ft bald white guy or the 5’5″ black guy”?

Well, let’s set aside questions of marketing and look at the details.

The essence of the method is to compare observations (measurements) with:

- model runs with GHG forcing
- model runs with “other anthropogenic” and natural forcings
- model runs with internal variability only

Then based on the fit you can distinguish one from the other. The statistical basis is covered in detail in Hasselmann 1993 and more briefly in this paper: Hegerl et al 1996 – both papers are linked below in the References.

At this point I make another digression.. as regular readers know I am fully convinced that the increases in CO2, CH4 and other GHGs over the past 100 years or more can be very well quantified into “radiative forcing” and am 100% in agreement with the IPCCs summary of the work of atmospheric physics over the last 50 years on this topic. That is, the increases in GHGs have led to something like a “radiative forcing” of 2.8 W/m² [*corrected, thanks to niclewis*].

And there isn’t any scientific basis for disputing this “pre-feedback” value. It’s simply the result of basic radiative transfer theory, well-established, and well-demonstrated in observations both in the lab and through the atmosphere. People confused about this topic are confused about science basics and comments to the contrary may be allowed or more likely will be capriciously removed due to the fact that there have been more than 50 posts on this topic (post your comments on those instead). See The “Greenhouse” Effect Explained in Simple Terms and On Uses of A 4 x 2: Arrhenius, The Last 15 years of Temperature History and Other Parodies.

Therefore, it’s “very likely” that the increases in GHGs over the last 100 years have contributed significantly to the temperature changes that we have seen.

To say otherwise – and still accept physics basics – means believing that the radiative forcing has been “mostly” cancelled out by feedbacks while internal variability has been amplified by feedbacks to cause a significant temperature change.

Yet this work on attribution seems to be fundamentally flawed.

Here was the conclusion:

We find that the latest observed 30-year trend pattern of near-surface temperature change can be distinguished from all estimates of natural climate variability with an estimated risk of less than 2.5% if the optimal fingerprint is applied.

With the caveats, that to me, eliminated the statistical basis of the previous statement:

The greatest uncertainty of our analysis is the estimate of the natural variability noise level..

..The shortcomings of the present estimates of natural climate variability cannot be readily overcome. However, the next generation of models should provide us with better simulations of natural variability. In the future, more observations and paleoclimatic information should yield more insight into natural variability, especially on longer timescales. This would enhance the credibility of the statistical test.

Earlier in the paper the authors said:

..However, it is

generally believedthat models reproduce the space-time statistics of natural variability on large space and long time scales (months to years) reasonably realistic. The verification of variability of CGMCs [coupled GCMs] on decadal to century timescales is relatively short, while paleoclimatic data are sparce and often of limited quality...We assume that the detection variable is Gaussian with zero mean, that is, that

there is no long-term nonstationarity in the natural variability.

[Emphasis added].

The climate models used would be considered rudimentary by today’s standards. Three different coupled atmosphere-ocean GCMs were used. However, each of them required “flux corrections”.

This method was pretty much the standard until the post 2000 era. The climate models “drifted”, unless, in deity-like form, you topped up (or took out) heat and momentum from various grid boxes.

That is, the models themselves struggled (in 1996) to represent climate unless the climate modeler knew, and corrected for, the long term “drift” in the model.

In the next article we will look at more recent work in attribution and fingerprints and see whether the field has developed.

But in this article we see that the conclusion of an attribution study in 1996 was that there was only a “2.5% chance” that recent temperature changes could be attributed to natural variability. At the same time, the question of how accurate the models were in simulating natural variability was noted but never quantified. And the models were all “flux corrected”. This means that some aspects of the long term statistics of climate were considered to be known – in advance.

So I find it difficult to accept any statistical significance in the study at all.

If the finding instead was introduced with the caveat “*assuming the accuracy of our estimates of long term natural variability of climate is correct..*” then I would probably be quite happy with the finding. And that question is the key.

The question should be:

What is the likelihood that climate models accurately represent the long-term statistics of natural variability?

- Virtually certain
- Very likely
- Likely
- About as likely as not
- Unlikely
- Very unlikely
- Exceptionally unlikely

So far I am yet to run across a study that poses this question.

Bindoff, N.L., et al, 2013: Detection and Attribution of Climate Change: from Global to Regional. In: Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change

Detecting greenhouse gas induced climate change with an optimal fingerprint method, Hegerl, von Storch, Hasselmann, Santer, Cubasch & Jones, *Journal of Climate* (1996)

What does it mean when climate models agree? A case for assessing independence among general circulation models, Zachary Pirtle, Ryan Meyer & Andrew Hamilton, *Environ. Sci. Policy* (2010)

Optimal fingerprints for detection of time dependent climate change, Klaus Hasselmann, *Journal of Climate* (1993)

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One commenter asked:

Why do we expect that vertical transport of water vapor to vary linearly with horizontal wind speed? Is this standard turbulent mixing?

The simple answer is “almost yes”. But as someone famously said, make it simple, but not too simple.

Charting a course between too simple and too hard is a challenge with this subject. By contrast, radiative physics is a cakewalk. I’ll begin with some preamble and eventually get to the destination.

There’s a set of equations describing motion of fluids and what they do is conserve momentum in 3 directions (x,y,z) – these are the Navier-Stokes equations, and they conserve mass. Then there are also equations to conserve humidity and heat. There is an exact solution to the equations but there is a bit of a problem in practice. *The Navier-Stokes equations in a rotating frame can be seen in The Coriolis Effect and Geostrophic Motion under “Some Maths”*.

Simple linear equations with simple boundary conditions can be re-arranged and you get a nice formula for the answer. Then you can plot this against that and everyone can see how the relationships change with different material properties or boundary conditions. In real life equations are not linear and the boundary conditions are not simple. So there is no “analytical solution”, where we want to know say the velocity of the fluid in the east-west direction as a function of time and get a nice equation for the answer. Instead we have to use numerical methods.

Let’s take a simple problem – if you want to know heat flow through an odd-shaped metal plate that is heated in one corner and cooled by steady air flow on the rest of its surface you can use these numerical methods and usually get a very accurate answer.

Turbulence is a lot more difficult due to the range of scales involved. Here’s a nice image of turbulence:

*Figure 1*

There is a cascade of energy from the largest scales down to the point where viscosity “eats up” the kinetic energy. In the atmosphere this is the sub 1mm scale. So if you want to accurately numerically model atmospheric motion across a 100km scale you need a grid size probably 100,000,000 x 100,000,000 x 10,000,000 and solving sub-second for a few days. Well, that’s a lot of calculation. I’m not sure where turbulence modeling via “direct numerical simulation” has got to but I’m pretty sure that is still too hard and in a decade it will still be a long way off. The computing power isn’t there.

Anyway, for atmospheric modeling you don’t really want to know the velocity in the x,y,z direction (usually annotated as u,v,w) at trillions of points every second. Who is going to dig through that data? What you want is a **statistical description** of the key features.

So if we take the Navier-Stokes equation and average, what do we get? We get a problem.

For the mathematically inclined the following is obvious, but of course many readers aren’t, so here’s a simple example:

Let’s take 3 numbers: 1, 10, 100: the average = (1+10+100)/3 = 37.

Now let’s look at the square of those numbers: 1, 100, 10000: the average of the square of those numbers = (1+100+10000)/3 = 3367.

But if we take the average of our original numbers and square it, we get 37² = 1369. It’s strange but the average squared is not the same as the average of the squared numbers. That’s non-linearity for you.

In the Navier Stokes equations we have values like east velocity x upwards velocity, written as uw. The average of uw, written as is not equal to the average of u x the average of w, written as . For the same reason we just looked at.

When we create the Reynolds averaged Navier-Stokes (RANS) equations we get lots of new terms like. That is, we started with the original equations which gave us a complete solution – the same number of equations as unknowns. But when we average we end up with more unknowns than equations.

It’s like saying x + y = 1, what is x and y? No one can say. Perhaps 1 & 0. Perhaps 1000 & -999.

The Reynolds approach is to take a value like u,v,w (velocity in 3 directions) and decompose into a mean and a “rapidly varying” turbulent component.

So , where = mean value; u’ = the varying component. So . Likewise for the other directions.

And

So in the original equation where we have a term like , it turns into , which, when averaged, becomes:

So 2 unknowns instead of 1. The first term is the averaged flow, the second term is the turbulent flow. (Well, it’s an advection term for the change in velocity following the flow)

When we look at the conservation of energy equation we end up with terms for the movement of heat upwards due to average flow (almost zero) and terms for the movement of heat upwards due to turbulent flow (often significant). That is, a term like which is “the mean of potential temperature variations x upwards eddy velocity”.

Or, in plainer English, how heat gets moved up by turbulence.

..End of Digression

“Closure” is a maths term. To “close the equations” when we have more unknowns that equations means we have to invent a new idea. Some geniuses like Reynolds, Prandtl and Kolmogoroff did come up with some smart new ideas.

Often the smart ideas are around “dimensionless terms” or “scaling terms”. The first time you encounter these ideas they seem odd or just plain crazy. But like everything, over time strange ideas start to seem normal.

The Reynolds number is probably the simplest to get used to. The Reynolds number seeks to relate fluid flows to other similar fluid flows. You can have fluid flow through a massive pipe that is identical in the way turbulence forms to that in a tiny pipe – so long as the viscosity and density change accordingly.

The Reynolds number, Re = density x length scale x mean velocity of the fluid / viscosity

And regardless of the actual physical size of the system and the actual velocity, turbulence forms for flow over a flat plate when the Reynolds number is about 500,000. By the way, for the atmosphere and ocean this is true most of the time.

Kolmogoroff came up with an idea in 1941 about the turbulent energy cascade using dimensional analysis and came to the conclusion that the energy of eddies increases with their size to the power 2/3 (in the “inertial subrange”). This is usually written vs frequency where it becomes a -5/3 power. Here’s a relatively recent experimental verification of this power law.

* Figure 2*

In less genius like manner, people measure stuff and use these measured values to “close the equations” for “similar” circumstances. Unfortunately, the measurements are only valid in a small range around the experiments and with turbulence it is hard to predict where the cutoff is.

A nice simple example, to which I hope to return because it is critical in modeling climate, is **vertical eddy diffusivity** in the ocean. By way of introduction to this, let’s look at heat transfer by conduction.

If only all heat transfer was as simple as conduction. That’s why it’s always first on the list in heat transfer courses..

If have a plate of thickness d, and we hold one side at temperature T1 and the other side at temperature T2, the heat conduction per unit area:

where k is a material property called conductivity. We can measure this property and it’s always the same. It might vary with temperature but otherwise if you take a plate of the same material and have widely different temperature differences, widely different thicknesses – the heat conduction always follows the same equation.

Now using these ideas, we can take the actual equation for vertical heat flux via turbulence:

where w = vertical velocity, θ = potential temperature

And relate that to the heat conduction equation and come up with (aka ‘invent’):

Now we have an equation we can actually use because we can measure how potential temperature changes with depth. The equation has a new “constant”, K. But this one is not really a constant, it’s not really a material property – it’s a property of the turbulent fluid in question. Many people have measured the “implied eddy diffusivity” and come up with a range of values which tells us how heat gets transferred down into the depths of the ocean.

Well, maybe it does. Maybe it doesn’t tell us very much that is useful. Let’s come back to that topic and that “constant” another day.

Back to the original question. If you imagine a sheet of paper as big as your desk then that pretty much gives you an idea of the height of the troposphere (lower atmosphere where convection is prominent).

It’s as thin as a sheet of desk size paper in comparison to the dimensions of the earth. So any large scale motion is horizontal, not vertical. **Mean vertical velocities** – which doesn’t include turbulence via strong localized convection – are very low. **Mean horizontal velocities** can be the order of 5 -10 m/s near the surface of the earth. Mean vertical velocities are the order of cm/s.

Let’s look at flow over the surface under “neutral conditions”. This means that there is little buoyancy production due to strong surface heating. In this case the energy for turbulence close to the surface comes from the kinetic energy of the mean wind flow – which is horizontal.

There is a surface drag which gets transmitted up through the boundary layer until there is “free flow” at some height. By using dimensional analysis, we can figure out what this velocity profile looks like in the absence of strong convection. It’s logarithmic:

*Figure 3 – for typical ocean surface*

Lots of measurements confirm this logarithmic profile.

We can then calculate the surface drag – or how momentum is transferred from the atmosphere to the ocean – using the simple formula derived and we come up with a simple expression:

Where U_{r} is the velocity at some reference height (usually 10m), and C_{D} is a constant calculated from the ratio of the reference height to the roughness height and the von Karman constant.

Using similar arguments we can come up with heat transfer from the surface. The principles are very similar. What we are actually modeling in the surface drag case is the turbulent vertical flux of horizontal momentum with a simple formula that just has mean horizontal velocity. We have “closed the equations” by some dimensional analysis.

Adding the Richardson number for non-neutral conditions we end up with a temperature difference along with a reference velocity to model the turbulent vertical flux of sensible heat . Similar arguments give latent heat flux in a simple form.

At the surface the horizontal velocity must be zero. The vertical flux of horizontal momentum creates a drag on the boundary layer wind. The vertical gradient of the mean wind, U, can only depend on height z, density ρ and surface drag.

So the “characteristic wind speed” for dimensional analysis is called the friction velocity, u*, and

This strange number has the units of velocity: m/s – ask if you want this explained.

So dimensional analysis suggests that should be a constant – “scaled wind shear”. The inverse of that constant is known as the Von Karman constant, k = 0.4.

So a simple re-arrangement and integration gives:

where z_{0} is a constant from the integration, which is roughness height – a physical property of the surface where the mean wind reaches zero.

The “real form” of the friction velocity is:

, where these eddy values are at the surface

we can pick a horizontal direction along the line of the mean wind (rotate coordinates) and come up with:

If we consider a simple constant gradient argument:

where the first expression is the “real” equation and the second is the “invented” equation, or “our attempt to close the equation” from dimensional analysis.

Of course, this is showing how momentum is transferred, but the approach is pretty similar, just slightly more involved, for sensible and latent heat.

Turbulence is a hard problem. The atmosphere and ocean are turbulent so calculating anything is difficult. Until a new paradigm in computing comes along, the real equations can’t be numerically solved from the small scales needed where viscous dissipation damps out the kinetic energy of the turbulence up to the large scale of the whole earth, or even of a synoptic scale event. However, numerical analysis has been used a lot to test out ideas that are hard to test in laboratory experiments. And can give a lot of insight into parts of the problems.

In the meantime, experiments, dimensional analysis and intuition have provided a lot of very useful tools for modeling real climate problems.

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Are we sure that over Connecticut the parameter C

_{DE}= 0.004, or should it be 0.0035? In fact, parameters like this are usually calculated from the average of a number of experiments. They conceal as much as they reveal. The correct value probably depends on other parameters. In so far as it represents a real physical property it will vary depending on the time of day, seasons and other factors. It might even be, “on average”, wrong. Because “on average” over the set of experiments was an imperfect sample. And “on average” over all climate conditions is a different sample.

Interestingly, a new paper has just shown up in JGR (“accepted for publication” and on their website in the pre-publishing format): *Seasonal changes in physical processes controlling evaporation over an inland water*, Qianyu Zhang & Heping Liu.

They carried out detailed measurements over a large reservoir (134 km² and 4-8m deep) in Mississippi for the winter and summer months of 2008. What were they trying to do?

Understanding physical processes that control turbulent fluxes of energy, heat, water vapor, and trace gases over inland water surfaces is critical in quantifying their influences on local, regional, and global climate. Since direct measurements of turbulent fluxes of sensible heat (H) and latent heat (LE) over inland waters with eddy covariance systems are still rare, process-based understanding of water-atmosphere interactions remains very limited..

..Many numerical weather prediction and climate models use the bulk transfer relations to estimate H and LE over water surfaces. Given substantial biases in modeling results against observations, process-based analysis and model validations are essential in improving parameterizations of water-atmosphere exchange processes..

Before we get into their paper, here is a relevant quote on parameterization from a different discipline. This is from *Turbulent dispersion in the ocean*, Garrett (2006):

Including the effects of processes that are unresolved in models is one of the central problems in oceanography.

In particular, for temperature, salinity, or some other scalar, one seeks to parameterize the eddy flux in terms of quantities that are resolved by the models. This has been much discussed, with determinations of the correct parameterization relying on a combination of deductions from the large-scale models, observations of the eddy fluxes or associated quantities, and the development of an understanding of the processes responsible for the fluxes.

The key remark to make is that it is only through process studies that we can reach an understanding leading to formulae that are valid in changing conditions,

rather than just having numerical values which may only be valid in present conditions.

[Emphasis added]

Latent heat transfer is the primary mechanism globally for transferring the solar radiation that is absorbed at the surface up into the atmosphere. Sensible heat is a lot smaller by comparison with latent heat. Both are “convection” in a broad term – the movement of heat by the bulk movement of air. But one is carrying the “extra heat” of evaporated water. When the evaporated water condenses (usually higher up in the atmosphere) it releases this stored heat.

Let’s take a look at the standard **parameterization** in use (adopting their notation) for latent heat:

LE = ρ_{a}LC_{E}U(q_{w} −q_{a})

LE = latent heat transfer, ρ_{a} = air density, L = latent heat of vaporization (2.5×10^{6} J kg^{–1}), C_{E} = bulk transfer coefficient for moisture, U = wind speed, q_{w} & q_{a} are the respective specific humidity in the water-atmosphere interface and the over-water atmosphere

The values ρ_{a} and L are a fundamental values. The formula says that the key parameters are:

- wind speed (horizontal)
- the
**difference**between the humidity at the water surface (this is the saturated value which varies strongly with temperature) and the humidity in the air above

We would expect the differential of humidity to be important – if the air above is saturated then latent heat transfer will be zero, because there is no way to move any **more** water vapor into the air above. At the other extreme, if the air above is completely dry then we have maximized the potential for moving water vapor into the atmosphere.

The product of wind speed and humidity difference indicate how much mixing is going on due to air flow. There is a lot of theory and experiment behind the ideas, going back into the 1950s or further, but in the end it is an over-simplification.

That’s what all parameterizations are – over-simplifications.

The **real formula** is much simpler:

LE = ρ_{a}L<w’q’>, where the brackets denote averages,w’q’ = the turbulent moisture flux

w is the upwards velocity, q is moisture; and the ‘ denoting eddies

*Note to commenters, if you write < or > in the comment it gets dropped because WordPress treats it like a html tag. You need to write < or >*

The key part of this equation just says “how much moisture is being carried upwards by turbulent flow”. That’s the real value so why don’t we measure that instead?

Here’s a graph of horizontal wind over a short time period from Stull (1988):

*Figure 1*

And any given location the wind varies across every timescale. Pick another location and the results are different. This is the problem of turbulence.

And to get accurate measurements for the paper we are looking at now, they had quite a setup:

Figure 2

Here’s the description of the instrumentation:

An eddy covariance system at a height of 4 m above the water surface consisted of a three-dimensional sonic anemometer (model CSAT3, Campbell Scientific, Inc.) and an open path CO2/H2O infrared gas analyzer (IRGA; Model LI-7500, LI-COR, Inc.).

A datalogger (model CR5000, Campbell Scientific, Inc.) recorded three-dimensional wind velocity components and sonic virtual temperature from the sonic anemometer and densities of carbon dioxide and water vapor from the IRGA at a frequency of 10 Hz.

Other microclimate variables were also measured, including Rn at 1.2 m (model Q-7.1, Radiation and Energy Balance Systems, Campbell Scientific, Inc.), air temperature (Ta) and relative humidity (RH) (model HMP45C, Vaisala, Inc.) at approximately 1.9, 3.0, 4.0, and 5.5 m, wind speeds (U) and wind direction (WD) (model 03001, RM Young, Inc.) at 5.5 m.

An infrared temperature sensor (model IRR-P, Apogee, Inc.) was deployed to measure water skin temperature (Tw).

Vapor pressure (ew) in the water-air interface was equivalent to saturation vapor pressure at Tw [Buck, 1981].

The same datalogger recorded signals from all the above microclimate sensors at 30-min intervals. Six deep cycling marine batteries charged by two solar panels (model SP65, 65 Watt Solar Panel, Campbell Scientific, Inc.) powered all instruments. A monthly visit to the tower was scheduled to provide maintenance and download the 10-Hz time-series data.

I don’t know the price tag but I don’t think the equipment is cheap. So this kind of setup can be used for research, but we can’t put one each every 1km across a country or an ocean and collect continuous data.

That’s why we need parameterizations if we want to get some climatological data. Of course, these need verifying, and that’s what this paper (and many others) are about.

When we look back at the parameterized equation for latent heat it’s clear that latent heat should vary linearly with the product of wind speed and humidity differential. The top graph is sensible heat which we won’t focus on, the bottom graph is latent heat. Δe is humidity, expressed as partial pressure rather than g/kg. We see that the correlation between LE and wind speed x humidity differential is very different in summer and winter:

*Figure 2*

The scatterplots showing the same information:

*Figure 3*

The authors looked at the diurnal cycle – averaging the result for the time of day over the period of the results, separated into winter and summer.

Our results also suggest that the influences of U on LE may not be captured simply by the product of U and Δe [humidity differential] on short timescales, especially in summer. This situation became more serious when the ASL (atmospheric surface layer, see note 1) became more unstable, as reflected by our summer cases (i.e., more unstable) versus the winter cases.

They selected one period to review in detail. First the winter results:

*Figure 4*

On March 18, Δe was small (i.e., 0 ~ 0.2 kPa) and it experienced little diurnal variations, leading to limited water vapor supply (Fig. 5a).

The ASL (see note 1) during this period was slightly stable (Fig. 5b), which suppressed turbulent exchange of LE. As a result, LE approached zero and even became negative, though strong wind speeds of approximately around 10 ms

^{–1}were present, indicating a strong mechanical turbulent mixing in the ASL.On March 19, with an increased Δe up to approximately 1.0 kPa, LE closely followed Δe and increased from zero to more than 200 Wm

^{–2}. Meanwhile, the ASL experienced a transition from stable to unstable conditions (Fig. 5b), coinciding with an increase in LE.On March 20, however, the continuous increase of Δe did not lead to an increase in LE. Instead, LE decreased gradually from 200 Wm

^{–2}to about zero, which was closely associated with the steady decrease in U from 10 ms^{–1}to nearly zero and with the decreased instability.These results suggest that LE was strongly limited by Δe, instead of U when Δe was low; and LE was jointly regulated by variations in Δe and U once a moderate Δe level was reached and maintained, indicating a nonlinear response of LE to U and Δe induced by ASL stability. The ASL stability largely contributed to variations in LE in winter.

Then the summer results:

*Figure 5*

In summer (i.e., July 23 – 25 in Fig. 6), Δe was large with a magnitude of 1.5 ~ 3.0 kPa, providing adequate water vapor supply for evaporation, and had strong diurnal variations (Fig. 6a).

U exhibited diurnal variations from about 0 to 8 ms

^{–1}. LE was regulated by both Δe and U, as reflected by the fact that LE variations on the July 24 afternoon did not follow solely either the variations of U or the variations of Δe. When the diurnal variations of Δe and U were small in July 25, LE was also regulated by both U and Δe or largely by U when the change in U was apparent.Note that during this period, the ASL was strongly unstable in the morning and weakly unstable in the afternoon and evening (Fig. 6b), negatively corresponding to diurnal variations in LE. This result indicates that the ASL stability had minor impacts on diurnal variations in LE during this period.

Another way to see the data is by plotting the results to see how valid the parameterized equation appears. Here we should have a straight line between LE/U and Δe as the caption explains:

*Figure 6*

One method to determine the bulk transfer coefficients is to use the mass transfer relations (Eqs. 1, 2) by quantifying the slopes of the linear regression of LE against UΔe. Our results suggest that using this approach to determine the bulk transfer coefficient may cause large bias, given the fact that one UΔe value may correspond to largely different LE values.

They conclude:

Our results suggest that these highly nonlinear responses of LE to environmental variables may not be represented in the bulk transfer relations in an appropriate manner, which requires further studies and discussion.

Parameterizations are inevitable. Understanding their limitations is very difficult. A series of studies might indicate that there is a “linear” relationship with some scatter, but that might just be disguising or ignoring a variable that never appears in the parameterization.

As Garrett commented “..having numerical values which may only be valid in present conditions”. That is, if the mean state of another climate variable shifts the parameterization will be invalid, or less accurate.

Alternatively, given the non-linear nature of climate process, changes don’t “average out”. So the mean state of another climate variable may not shift, the mean state might be constant, but its variation with time or another variable may introduce a change in the real process that results in an overall shift in climate.

There are other problems with calculating latent heat transfer – even accepting the parameterization as the best version of “the truth” – there are large observational gaps in the parameters we need to measure (wind speed and humidity above the ocean) even at the resolution of current climate models. This is one reason why there is a need for reanalysis products.

I found it interesting to see how complicated latent heat variations were over a water surface.

*Seasonal changes in physical processes controlling evaporation over an inland water*, Qianyu Zhang & Heping Liu, JGR (2014)

*Turbulent dispersion in the ocean*, Chris Garrett, *Progress in Oceanography* (2006)

Note 1: The ASL (atmospheric surface layer) stability is described by the Obukhov stability parameter:

ζ = z/L_{0}

where z is the height above ground level and L_{0} is the Obukhov parameter.

L_{0} = −θ_{v}u*^{3}/[kg(w’θ_{v}‘)s ]

where θ_{v} is virtual potential temperature (K), u* is frictional velocity by the eddy covariance system (ms^{–1}), k is Von Karman constant (0.4), g is acceleration due to gravity (9.8 ms^{–2}), w is vertical velocity (m s^{–1}), and (w’θ_{v}‘)s is the flux of virtual potential temperature by the eddy covariance system

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If there were no radiatively-active gases (aka “GHG”s) in the atmosphere then the atmosphere couldn’t cool to space at all.

Technically, the **emissivity** of the atmosphere would be zero. Emission is determined by the local temperature of the atmosphere and its emissivity. Wavelength by wavelength emissivity is equal to **absorptivity**, another technical term, which says what proportion of radiation is absorbed by the atmosphere. If the atmosphere can’t emit, it can’t absorb (note 2).

So as you increase the GHGs in the atmosphere you increase its ability to cool to space. A lot of people realize this at some point during their climate science journey and finally realize how they have been duped by climate science all along! It’s irrefutable – more GHGs more cooling to space, more GHGs mean less global warming!

Ok, it’s true. Now the game’s up, I’ll pack up Science of Doom into a crate and start writing about something else. Maybe cognitive dissonance..

Bye everyone!

Halfway through boxing everything up I realized there was a little complication to the simplicity of that paragraph. The atmosphere with more GHGs has a higher emissivity, but **also** a higher absorptivity.

Let’s draw a little diagram. Here are two “layers” (see note 3) of the atmosphere in two different cases. On the left 400 ppmv CO2, on the right 500ppmv CO2 (and relative humidity of water vapor was set at 50%, surface temperature at 288K):

*Figure 1*

It’s clear that the two layers are both emitting more radiation with more CO2.More cooling to space.

For interest, the “total emissivity” of the top layer is 0.190 in the first case and 0.197 in the second case. The layer below has 0.389 and 0.395.

Let’s take a look at all of the numbers and see what is going on. This diagram is a little busier:

*Figure 2*

The key point is that the OLR (outgoing longwave radiation) is **lower** in the case with more CO2. Yet each layer is emitting **more** radiation. How can this be?

Take a look at the radiation entering the top layer on the left = 265.1, and add to that the emitted radiation = 23.0 – the total is 288.1. Now subtract the radiation leaving through the top boundary = 257.0 and we get the radiation absorbed in the layer. This is 31.1 W/m².

Compare that with the same calculation with more CO2 – the absorption is 32.2 W/m².

This is the case all the way up through the atmosphere – each layer **emits more** because its emissivity has increased, but it also **absorbs more** because its absorptivity has increased by the same amount.

So more cooling to space, but unfortunately more absorption of the radiation below – two competing terms.

Emission of radiation is a result of local temperature and emissivity.

Absorption of radiation is the result of the incident radiation and absorptivity. Incident upwards radiation started lower in the atmosphere where it is hotter. So absorption changes always outweigh emission changes (note 4).

If it’s still not making sense then think about what happens as you reduce the GHGs in the atmosphere. The atmosphere emits less but absorbs even less of the radiation from below. So the outgoing longwave radiation increases. More surface radiation is making it to the top of atmosphere without being absorbed. So there is less cooling to space from the atmosphere, but more cooling to space from the surface **and** the atmosphere.

If you add lagging to a pipe, the temperature of the pipe increases (assuming of course it is “internally” heated with hot water). And yet, the pipe cools to the surrounding room via the lagging! Does that mean more lagging, more cooling? No, it’s just the transfer mechanism for getting the heat out.

That was just an analogy. Analogies don’t prove anything. If well chosen, they can be useful in illustrating problems. End of analogy disclaimer.

If you want to understand more about how radiation travels through the atmosphere and how GHG changes affect this journey, take a look at the series Visualizing Atmospheric Radiation.

Note 1: For more on the details see

- Part Three – Average Height of Emission – the complex subject of where the TOA radiation originated from, what is the “Average Height of Emission” and other questions
- Kiehl & Trenberth and the Atmospheric Window

Note 2: A very basic point – absolutely essential for understanding anything at all about climate science – is that the absorptivity of the atmosphere can be (and is) totally different from its emissivity when you are considering different wavelengths. The atmosphere is quite transparent to solar radiation, but quite opaque to terrestrial radiation – because they are at different wavelengths. 99% of solar radiation is at wavelengths less than 4 μm, and 99% of terrestrial radiation is at wavelengths greater than 4 μm. That’s because the sun’s surface is around 6000K while the earth’s surface is around 290K. So the atmosphere has low absorptivity of solar radiation (<4 μm) but high emissivity of terrestrial radiation.

Note 3: Any numerical calculation has to create some kind of grid. This is a very course grid, with 10 layers of roughly equal pressure in the atmosphere from the surface to 200mbar. The grid assumes there is just one temperature for each layer. Of course the temperature is decreasing as you go up. We could divide the atmosphere into 30 layers instead. We would get more accurate results. We would find the same effect.

Note 4: The equations for radiative transfer are found in Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations. The equations prove this effect.

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As DeWitt Payne noted, a post with a similar problem posted on Wattsupwiththat managed to gather some (unintentionally) hilarious comments.

Here’s the problem again:

**Case 1**

Spherical body, A, of radius ra, with an emissivity, εa =1. The sphere is in the vacuum of space.

It is internally heated by a mystery power source (let’s say nuclear, but it doesn’t matter), with power input = P.

The sphere radiates into deep space, let’s say the temperature of deep space = 0K to make the maths simpler.

1. What is the equation for the equilibrium surface temperature of the sphere, Ta?

**Case 2**

The condition of case A, but now body A is surrounded by a slightly larger spherical shell, B, which of course is itself now surrounded by deep space at 0K.

B has a radius rb, with an emissivity, εb =1. This shell is highly conductive and very thin.

2a. What is the equation for the new equilibrium surface temperature, Ta’?

2b. What is the equation for the equilibrium temperature, Tb, of shell B?

**Notes**:

The reason for the “slightly larger shell” is to avoid “complex” view factor issues. Of course, I’m happy to relax the requirement for “slightly larger” and let Bryan provide the more general answer.

The reason for the “highly conductive” and “thin” outer shell, B, is to avoid any temperature difference between the inside and the outside surfaces of the shell. That is, we can assume the outside surface is at the same temperature as the inside surface – both at temperature, Tb.

This kind of problem is a staple of introductory heat transfer. This is a “find the equilibrium” problem.

How do we solve these kinds of problems? It’s pretty easy once you understand the tools.

The **first tool** is the first law of thermodynamics. Steady state means temperatures have stabilized and so energy in = energy out. We draw a “boundary” around each body and apply the “boundary condition” of the first law.

The **second tool** is the set of equations that govern the movement of energy. These are the equations for conduction, convection and radiation. In this case we just have radiation to consider.

For people who see the solution, shake their heads and say, this can’t be, stay on to the end and I will try and shed some light on possible conceptual problems. Of course, if it’s wrong, you should easily be able to provide the correct equations – or even if you can’t write equations you should be able to explain the flaw in the formulation of the equation.

In the original article I put some numbers down – “*For anyone who wants to visualize some numbers: ra=1m, P=1000W, rb=1.01m*“. I will use these to calculate an answer from the equations. I realize many readers aren’t comfortable with equations and so the answers will help illuminate the meaning of the equations.

I go through the equations in **tedious detail**, again for people who would like to follow the maths but don’t find maths easy.

Energy in, Ein = Energy out, Eout : in Watts (Joules per second).

Ein = P

Eout = emission of thermal radiation per unit area x area

The first part is given by the Stefan-Boltzmann equation (σT_{a}^{4}, where σ = 5.67×10^{-8}), and the second part by the equation for the surface area of a sphere (4πr_{a}²)

Eout = 4πr_{a}² x σT_{a}^{4} …..[eqn 1]

Therefore, P = 4πr_{a}²σT_{a}^{4} ….[eqn 2]

We have to rearrange the equation to see how Ta changes with the other factors:

T_{a} = [P / (4πr_{a}²σ)]^{1/4} ….[eqn 3]

If you aren’t comfortable with maths this might seem a little daunting. Let’s put the numbers in:

T_{a} = 194K (-80ºC)

Now we haven’t said anything about how long it takes to reach this temperature. We don’t have enough information for that. That’s the nice thing about steady state calculations, they are easier than dynamic calculations. We will look at that at the end.

Probably everyone is happy with this equation. Energy is conserved. No surprises and nothing controversial.

Now we will apply the exact same approach to the second case.

First we consider “body A”. Given that it is enclosed by another “body” – the shell B – we have to consider any energy being transferred by radiation from B to A. If it turns out to be zero, of course it won’t affect the temperature of body A.

Ein(a) = P + E_{b-a} ….[eqn 4], where E_{b-a} is a value we don’t yet know. It is the radiation from B absorbed by A.

Eout(a) = 4πr_{a}² x σT_{a}^{4} ….[eqn 5]- this is the same as in case 1. Emission of radiation from a body only depends on its temperature (and emissivity and area but these aren’t changing between the two cases)

– we will look at shell B and come back to the last term in eqn 4.

Now the shell outer surface:

Radiates out to space

We set space at absolute zero so no radiation is received by the outer surface

Shell inner surface:

Radiates in to A (in fact almost all of the radiation emitted from the inner surface is absorbed by A and for now we will treat it as all) – this was the term E_{b-a}

Absorbs all of the radiation emitted by A, this is Eout(a)

And we made the shell thin and highly conductive so there is no temperature difference between the two surfaces. Let’s collect the heat transfer terms for shell B under steady state:

Ein(b) = Eout(a) + 0 …..[eqn 6] – energy in is all from the sphere A, and nothing from outside

= 4πr_{a}² x σT_{a}^{4} ….[eqn 6a] – we just took the value from eqn 5

Eout(b) = 4πr_{b}² x σT_{b}^{4} + 4πr_{b}² x σT_{b}^{4} …..[eqn 7] – energy out is the emitted radiation from the inner surface + emitted radiation from the outer surface

= 2 x 4πr_{b}² x σT_{b}^{4} ….[eqn 7a]

And we know that for shell B, Ein = Eout so we equate 6a and 7a:

4πr_{a}² x σT_{a}^{4} = 2 x 4πr_{b}² x σT_{b}^{4} ….[eqn 8]

and now we can cancel a lot of the common terms:

r_{a}² x T_{a}^{4} = 2 x r_{b}² x T_{b}^{4} ….[eqn 8a]

and re-arrange to get Ta in terms of Tb:

T_{a}^{4} = 2r_{b}²/r_{a}² x T_{b}^{4} ….[eqn 8b]

T_{a} = [2r_{b}²/r_{a}²]^{1/4} x T_{b} ….[eqn 8b]

or we can write it the other way round:

T_{b} = [r_{a}²/2r_{b}²]^{1/4} x T_{a} ….[eqn 8c]

**Using the numbers given, T _{a} = 1.2 T_{b}. So the sphere is 20% warmer than the shell** (actually 2 to the power 1/4).

We need to use Ein=Eout for the sphere A to be able to get the full solution. We wrote down: Ein(a) = P + E_{b-a} ….[eqn 4]. Now we know “E_{b-a}” – this is one of the terms in eqn 7.

So:

Ein(a) = P + 4πr_{b}² x σT_{b}^{4} ….[eqn 9]

and Ein(a) = Eout(a), so:

P + 4πr_{b}² x σT_{b}^{4} = 4πr_{a}² x σT_{a}^{4} ….[eqn 9]

we can substitute the equation for T_{b}:

P + 4πr_{a}² /2 x σT_{a}^{4} = 4πr_{a}² x σT_{a}^{4} ….[eqn 9a]

the 2nd term on the left and the right hand side can be combined:

P = 2πr_{a}² x σT_{a}^{4} ….[eqn 9a]

**And so, voila:**

T’_{a} = [P / (2πr_{a}²σ)]^{1/4} ….[eqn 10] – I added a dash to Ta so we can compare it with the original value before the shell arrived.

**T’ _{a} = 2^{1/4 }T_{a} ….[eqn 11] – that is, the temperature of the sphere A is about 20% warmer in case 2 compared with case 1**.

Using the numbers, T’_{a} = 230 K (-43ºC). And T_{b} = 193 K (-81ºC)

In case 2, the inner sphere, A, has its temperature increase by 36K even though the same energy production takes place inside. Obviously, this can’t be right because we have created energy??.. let’s come back to that shortly.

Notice something very important - Tb in case 2 is almost identical to Ta in case 1. The difference is actually only due to the slight difference in surface area. Why?

The **system** has an energy production, P, in both cases.

- In case 1, the sphere A is the boundary transferring energy to space and so its equilibrium temperature must be determined by P
- In case 2, the shell B is the boundary transferring energy to space and so its equilibrium temperature must be determined by P

Now let’s confirm the mystery unphysical totally fake invented energy.

Let’s compare the flux emitted from A in case 1 and case 2. I’ll call it R.

- R(case 1) = 80 W/m²
- R(case 2) = 159 W/m²

This is obviously rubbish. The same energy source inside the sphere and we doubled the sphere’s energy production!!! Get this idiot to take down this post, he has no idea what he is writing..

Yet if we check the energy balance we find that 80 W/m² is being “created” by our power source, and the “extra mystery” energy of 79 W/m² is coming from our outer shell. In any given second no energy is created.

When we snapped the outer shell over the sphere we made it harder for heat to get out of the system. Energy in = energy out, in steady state. When we are not in steady state: energy in – energy out = energy retained. Energy retained is internal energy which is manifested as temperature.

We made it **hard for heat to get out**, which accumulated energy, which increased temperature.. until finally the inner sphere A was hot enough for all of the internally generated energy, P, to get out of the system.

Let’s add some information about the system: the heat capacity of the sphere = 1000 J/K; the heat capacity of the shell = 100 J/K. It doesn’t much matter what they are, it’s just to calculate the transients. We snap the shell – originally at 0K – around the sphere at time t=100 seconds and see what happens.

The top graph shows temperature, the bottom graph shows change in energy of the two objects and how much energy is leaving the system:

At 100 seconds we see that instead of our steady state 1000W leaving the system, instead 0W leaves the system. This is the important part of the mystery energy puzzle.

We put a 0K shell around the sphere. This absorbs all the energy from the sphere. At time t=100s the shell is still at 0K so it emits 0W/m². It heats up pretty quickly, but remember that emission of radiation is not linear with temperature so you don’t see a linear relationship between the temperature of shell B and the energy leaving to space. *For example at 100K, the outward emission is 6 W/m², at 150K it is 29 W/m² and at its final temperature of 193K, it is 79 W/m² (=1000 W in total).*

As the shell heats up it emits more and more radiation inwards, heating up the sphere A.

The mystery energy has been revealed. The addition of a radiation barrier stopped energy leaving, which stored heat. The way equilibrium is finally restored is due to the temperature increase of the sphere.

Of course, for some strange reason an army of people thinks this is totally false. Well, produce your equations.. (this never happens)

All we have done here is used conservation of energy and the Stefan Boltzmann law of emission of thermal radiation.

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