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This chapter is under revision. Here we indicate an approach to climate modelling that may provide correlations with sufficiently few parameters to be statistically significant.
The analysis in the previous chapters has failed to establish that increasing carbon dioxide levels are causing the current global warming. The analysis has shown that the current global warming is probably exceptional (more than 50% probability). It has also shown that carbon dioxide is a plausible cause of global warming. However, the correlation is no more than would be expected by coincidence. Thus, proof of the carbon hypothesis of global warming depends ultimately on a mechanistic model linking carbon dioxide concentration directly to global temperatures. We have shown that it is unlikely that a global circulation model can provide the necessary link. In this chapter, we explore simple models that might provide the desired link.
At this stage, we should emphasize the limitations of mathematical models. Mathematical models are extensively employed to design process plant, and to design vehicles and aircraft. They are also used to develop new products. In all cases, the physics and chemistry is much better understood than the physics and chemistry of climate processes. Furthermore, the models are frequently applied to designs based on previous versions for which the mathematical models have been extensively tested against practical measurements. Nevertheless, the majority of designs developed on the basis mathematical models are tested before being used. These may be tested on a small scale (pilot plants) or on a larger scale as prototypes. These tests are needed because every mathematical model is limited by the imagination of the modeller to include all relevant factors, and to include proper interaction between the factors. The models are also limited by our understanding of the basic sciences upon which the models are built. (For example, turbulence models are empirical, and even on a laboratory scale concurrent radiant and convective heat transfer in turbulent flow is difficult to model. Furthermore, laboratory measurements of properties of mixtures of more than three components are extremely limited). In the majority of cases, some changes are needed to refine the models and, in some cases, radical changes are required. For this reason, climate models present a major challenge. Modellers do not have a set of similar planets to test their models on, or to evolve their modelling tools. We must take note of the warnings presented by a range of current models. However, they have not been tested against physical prototypes, or against scale prototypes. Thus, unless the models are extremely simple to verify, we must treat them with caution. To this extent, we remain ignorant of their reliability. Alongside this caution, it should be emphasized that climate change has important implications. In the circumstances, ignorance is no excuse for inaction.
Our goal is to derive the simplest possible model that links carbon dioxide concentration to global warming. The model must have no more than two adjustable parameters and should ideally correlate recorded climatic change better than the trivial “models” of Chapter 3. However, more important than correlating current conditions, the models must reproduce the climate change statistics presented in Chapter 1. Any model that cannot reproduce the natural variation of climate is wrong; it cannot be relied upon to indicate the incremental effect of carbon dioxide on climate.
A model with only two adjustable parameters must almost certainly be a simple model. How can we possibly model the complex climatic conditions on Earth with such a simple model? Weather forecasting only predicts days ahead, but requires massive models and impressive computer power. Climate modelling predicts centuries ahead. Nevertheless, there is hope that we can derive meaningful relationships because we are concentrating on the incremental effect of adding carbon dioxide to the atmosphere. If we had a similar planet with different geography, and a different climate history, adding a greenhouse gas to the atmosphere should have a similar effect. Thus, we could move a few continents, alter the day length, move closer or further from the sun, and greenhouse gases would still warm the planet. All we need to establish is whether the addition makes a significant impact on climate. If carbon dioxide has a significant effect on global temperatures, increasing the atmospheric concentration by about 40% (which is what mankind’s activities have achieved) will also have a significant effect. If carbon dioxide is irrelevant in determining global temperatures, increasing its concentration by a moderate amount will have an insignificant impact on climate. Our approach is to present simple general models that should be applicable to many planets. We can then substitute values similar to those on Earth and scope the effect of our ignorance. We first introduce simple steady-state models. We then explore models that might generate the kind of climate variability found in Chapter 2. Any model that cannot reproduce this variability is inadequate. If it cannot show natural variation of climate, it will inevitably find an unnatural cause for the current period of climate change. With so few external factors to explain climate change, such steady models are almost bound to find that carbon dioxide is the cause of the current period of change. Specifically, consider that we take a model and run it for 10,000 years with zero man-made carbon dioxide releases. If the model settles to a steady temperature and atmospheric carbon dioxide concentration, it is wrong.
We summarize here the succession of models studied:
Black-body planet with a uniform surface temperature.
Grey-body planet with uniform surface temperature.
Grey-body planet with non-uniform surface temperature.
Non-grey planet with non-uniform surface temperature.
As (4) with effect of polar and winter snow (or other reflector).
As (4) with effect of cloud.
AS (4) with a layer of greenhouse gases.
As (7) using approximate relationships for properties of greenhouse gases.
Unsteady models.
Here we consider a planet that rotates so fast, or has such extensive atmospheric mixing that its surface is at a constant temperature, independent of position or season. We consider that the planet is a perfect absorber and a perfect radiator. Thus, both absorptivity and emissivity are 1.0. This is the simplest possible planetary model. We substitute the power of the Sun, namely
W = 3.86 × 1026 watts.
We also substitute the mean distance of the Earth from the Sun, namely
r = 1.496 ×1011 metres
We find that the surface temperature of such a planet would be:
TB = 279 K (or 6 C)
This figure can be compared to the current arithmetic mean surface temperature of Earth, namely
TA = 288 K (or 15 C)
On this basis, Earth shows 9 C of global warming. All further elaborations of the model either add to or decrease the global warming that we see in practice. Thus, the observed global warming is the net result of a number of effects, some that cause global warming, and some that cause global cooling.
An opaque grey surface is one that absorbs a constant fraction of radiant energy for every wavelength. Thus, it has a constant absorptivity. Its emissivity is the same as its absorptivity at all wavelengths. Thus, at all wavelengths, it emits a constant fraction of the radiation that would be emitted by a perfect black body. For an opaque grey body, the energy that is not absorbed is reflected.
It is easy to show that the planet reaches the same temperature as a black body planet.
The heat received from the sun by a planet of non-uniform surface temperature is the same as that for the same planet with a uniform temperature. However, the relationship between radiant heat loss and surface temperature is non-linear. Specifically, consider three equal areas of temperatures 260 K, 280 K and 300 K (-13 C, 7 C, and 27 C). If the rate of heat loss from the 280 K surface is 100%, that from the 260 K surface is 74.4% and that from the 300 K surface is 131.8%. Thus, the mean rate of radiant energy loss from the hotter and colder surfaces is 103.1%. This is the rate of energy loss from a surface at 282 K. Thus, we can calculate the correct heat loss by taking a mean temperature of 282 K instead of the arithmetic mean of 280 K. The calculation of an appropriate mean for a real planet has to take into account the extreme temperatures that arise at the poles and the equator, between day and night, and between summer and winter. For Earth, the appropriate mean is approximately:
TR = 290 K (or 17 C)
Thus, comparing with the results in section 4.1, we seek to explain an overall global warming of about (17 – 6) C or 11 C.
In this section, we ignore seasonal and geographic variations in absorptivity and emissivity. We also defer consideration of snow and cloud cover.
Non-grey bodies have emissivities and absorptivities that depend on the wavelength of radiation. Thus, the emissivities and absorptivities depend on the source temperature of the radiation. Thermodynamic analysis proves that, where the source and destination temperatures are the same, emissivity is identical to absorptivity for every surface (and indeed, for absorbing and radiating gases and clouds). The surfaces of the rocky planets (including Earth) have an absorptivity to solar radiation that is less than its emissivity at ambient temperatures. It follows that, compared to a grey planet, heat is gained less efficiently and lost more efficiently. The result is global cooling compared to a corresponding grey planet. The absolute temperature is reduced by the 4th root of the ratio of absorptivity to emissivity. Under the restrictions introduced above, the absorptivity of both Earth and Mars is about 0.85 and the emissivity is about 0.95. The emissivity is insensitive to surface temperature in the ambient temperature range. Thus, it is valid to compute a mean surface temperature as for the black and grey body cases. This mean should be compared to the actual mean as derived in Section 4.3 above. The calculated surface temperature of Earth is then:
TN = 271 K
We now see that we need to account for about 19C of global warming. Put another way, the non-grey surface of Earth introduces about 8 degrees of global cooling.
It is interesting to note the effect of constructing a planet of a greenhouse gas. Carbon dioxide has an absorptivity for solar radiation that is about a 20 times less than its emissivity at ambient temperatures. Constructing a planet with the optical properties of gaseous carbon dioxide would cause a global cooling of about 150 C. Thus, “greenhouse” gases do not automatically introduce global warming, it depends on how they are deployed.
Snow cover is a particular case of a planet with two distinct types of surface. It is interesting that the emissivity of snow is similar to that of soil (about 0.95). However, it has a very low absorptivity to solar radiation (typically 0.1 to 0.2). Thus, radiation from the surface of the planet can be calculated as in section 4.4. We need to adjust the heat received by the planet to account for the low absorptivity of snow (and ice). Snow is found predominately near the poles and in winter. Thus, looked at from the sun, the snow-covered area is made up of two small rims towards the North and South poles. Furthermore, much of the snow-covered area is on the dark side of the Earth and cannot be seen by the sun at all. (The nights are long in the winter). The following table gives an approximate relationship between mean latitude of snow cover and the resultant global cooling.
The table is calculated based on an absorptivity of 0.85 for parts of the planet not covered in snow.
We can give a rough estimate of the effect on Earth by assuming the mean snow line is at around 60 degrees. We then get an estimate of global cooling of about 3 degrees C.
Global cooling caused by snow cover.
|
Mean snow latitude (degrees) |
Global cooling. (degrees C) |
|
0 |
82 |
|
10 |
55 |
|
20 |
36 |
|
30 |
23 |
|
40 |
14 |
|
50 |
7 |
|
60 |
3 |
|
67 |
1.5 |
|
70 |
1.0 |
|
80 |
0.1 |
|
90 |
0.0 |
Consideration of snow cover introduces a factor not seen in the previous sections. Thus, we see a positive feedback position. If the planet becomes warmer, the snowline retreats, and the planet warms further. Conversely, if the planet cools, the snowline advances, and the planet cools further. We also observe that, if the planet were completely snow covered, it would remain sufficiently cold to remain snow covered. Thus, there is more than one solution to any climate model. At the least, there is a very cold solution and an ambient solution.
In summary, snow cover currently contributes about 3 C to global cooling. Allowing for this effect, we now need to account for around 22 C of global warming.
RICHARD BRANSON’S PRIZE. Richard Branson has announced a prize for anyone who can cure global warming. We will forego the honour of collecting that prize. However, it is clear that if you cover a few million square miles of ocean with a thin layer of material having similar surface properties to snow, substantial cooling will be achieved. We make no comment on the effect on ocean life below this layer, or on the energy cost to produce the layer. However, our method compares favourably to biological methods, and methods that project fine particulates into the upper atmosphere. Specifically, you can reverse our method if you “overcook” the cure and accidentally initiate a disastrous period of global cooling.
Clouds reflect, absorb, emit and transmit radiation. To date, we have omitted consideration of cloud cover, despite the large amount of solar energy reflected from clouds on Earth. The rationale for this omission can be explained by considering the following very simple model. Consider a planet uniformly covered in patchy cloud. Thus, the proportion of the sky covered in cloud is the same everywhere. However, the cloud is patchy, so that there are holes through the cloud everywhere. Further, consider that the cloud is perfectly reflective. For the proportion of the planet that is cloud covered, no radiation reaches the surface. However, no radiation can leave the surface under such clouds (all the radiation is reflected back). In this way, the cloud-covered proportion is perfectly insulated; it maintains a uniform temperature, determined by convection from adjacent cloud-free areas. For the cloud-free areas, the heat balance is identical to that for a non-grey planet. Thus, heat gain and heat loss are reduced by the same proportions. It follows that the surface temperature of such a partially cloud-covered planet would be identical to that of a completely cloud-free planet. It is for this reason that we initially estimated the temperature of planet Earth assuming a surface absorptivity of 0.85, rather than allowing for the measured albedo of the planet.
From the albedo of planet Earth, we can calculate that the planet has the equivalent of roughly 30% full cloud cover. This cover is made up partly of opaque clouds and partly of clouds that transmit part of the light falling on them, and reflect part.
Having this estimate of cloud cover, we are in a position to estimate the effect of cloud cover on global temperatures. We need to consider two effects. The first is the effect of uniform semi-transparent cloud cover. The second is the effect of non-uniform cloud cover.
We estimate that the effect of uniform semi-transparent clouds on global temperature is small. The effect is to introduce a small amount of global warming. The maximum extent for Earth is about 2 C. (Thus, the net global warming that still needs to be accounted for is about 20 C). This calculation ignores the effect in snow-covered areas. Clouds reflect back energy reflected from snow cover. Thus, the effective absorptivity of the snow is increased. With sufficient cloud-cover, the global cooling effect of snow can be eliminated. It is probably cautious to eliminate 1 C of the 3 C snow-cooling effect.
The effect of non-uniform cloud cover is significant. Over the oceans and maritime areas, cloud cover is thicker and more extensive at night and in winter. A significant proportion of the population lives on large continents remote from the sea. Thus, the differential cloud cover effect that applies over the majority of the planet is not seen by the majority of the population. The effect of this differential cloud cover is to let in solar radiation during the day, and during the summer, but to trap it during the night and during the winter. Thus, differential cloud cover contributes significantly to global warming. It may even account for the majority of the global warming required to reach observed temperatures.
Greenhouse gases preferentially absorb radiation at particular wavelengths. They generally have negligible absorption in the visible spectrum, so are virtually transparent to solar radiation. They absorb in the infrared spectrum, which includes the wavelengths radiated from the surface of the Earth. These gases have negligible reflectivity. For simplicity, we will assume that the greenhouse gases absorb no solar radiation. We can then picture the effect of a layer of greenhouse gases as shown in Figure 12.
Figure 12. Heat Balance at Greenhouse Gas Layer.
In Figure 12, the symbols have the following meanings:
S is solar radiation. It passes straight through the greenhouse layer to reach the Earth.
R is reflected solar radiation from the surface of the Earth. It passes straight through the greenhouse layer and radiates into space.
E is radiation from the surface of the Earth. It is partially absorbed by the greenhouse layer and warms the gases in that layer. The unabsorbed fraction passes through the layer and radiates into space.
C is heat carried from the surface of the Earth up to the greenhouse layer by convection. The dotted line shows heat returned to the surface of the Earth by convection.
G is radiation from the greenhouse layer. It radiates approximately equally upwards into space and downwards back to the Earth’s surface. A small proportion is reflected from the Earth’s surface where part is reabsorbed into the greenhouse layer. This quantity is so small compared to the other uncertainties that we can ignore it.
In the absence of the greenhouse layer, the temperature of the surface of the Earth can be calculated from a heat balance between the solar energy absorbed (S – R) and the energy (E) radiated from the surface of the Earth. The higher the surface temperature is, the more the radiation from the Earth’s surface. Thus, the surface temperature continues to rise until the heat loss matches the heat received.
In the absence of the greenhouse layer, convection has no net effect. Where the planet has no water (or other evaporating liquid), the air convected upwards cools by adiabatic expansion. The returning air is reheated by adiabatic compression and returns to Earth at approximately the same temperature that it left. For a planet with a large surface of water, the effect of convection is subtler. Where the air rises from the surface, water evaporates and the latent heat absorbed cools the surface somewhat. The moist, warm air rises. As it rises, the air cools by adiabatic expansion and the water vapour in the air condenses. Eventually, the condensed water returns to Earth as rain, hail or snow. The condensation releases latent heat so that the air temperature may be 20C higher than it would have been if cooled by adiabatic expansion alone. The dry air returns to Earth. As it descends, it is heated by adiabatic compression. Thus, the returning air is drier, but much warmer than the ascending air. The enhanced sensible heat of the descending air balances the latent heat in the ascending air. The effect can be seen on Earth in the tropics and immediately adjacent desert areas. Thus, over the tropics, warm moist air ascends with consequent high rainfall. This air returns as descending hot dry air over the adjacent deserts (Sahara, Australian etc).
In the absence of convection, the effect of the greenhouse layer is to absorb part of the energy, E. The greenhouse layer radiates upwards, into space, and downwards, back to the surface of the planet. In the absence of convection, the planet warms up until it is hot enough for the energy, E, to match the sum (S – R + G). The addition of the re-radiated energy G can result in significant warming; it can be more than sufficient to account for the global warming seen on Earth.
With both convection and radiation, the energy G radiated into space can also carry away part of the energy C carried up by convection. Thus, the convection currents returning to Earth carry back less heat, which results in a global cooling. The net effect of the greenhouse layer is a balance between the heating caused by re-radiation of absorbed radiation and the cooling caused by the radiation into space of convected heat that would otherwise be returned to Earth.
Note that, near the surface of the Earth, where convection dominates, greenhouse gases have no effect. At these levels, the atmospheric temperature is similar to the temperature of the surface of the Earth.
It is difficult to produce simple models to represent the balance between convection and radiation. Indeed, even detailed models are difficult to develop and verify. Computing combined convection and radiation heat transfer presents modelling challenges even in controlled laboratory experiments. For this reason, it is only possible to scope the range of possible effects.
Both carbon dioxide and water vapour are greenhouse gases. Thus, both may contribute to global warming on Earth. Calculations are complicated by the interactions between the two gases. The principal absorption lines for carbon dioxide almost coincide with major absorption lines for water vapour. Then, at low carbon dioxide concentrations, the carbon dioxide absorption is swamped by the water vapour absorption. Thus, adding small amounts of carbon dioxide to water vapour has negligible effect on the absorptivity of the mixture. At ground level on Earth, the concentration of carbon dioxide is less than 1/30 of the average concentration of water vapour. As we move up through the atmosphere, water vapour condenses out and its concentration gets less. Thus, the proportion of carbon dioxide increases. It follows that the contribution of carbon dioxide to global warming depends on the altitude of the greenhouse gas layers. If they are very high, where water vapour concentration is low, carbon dioxide may be the major contributor to greenhouse global warming.
Calculations have been undertaken for greenhouse layer effects with no convective heat transfer. They show that to achieve global warming of between 10 C and 20 C, it requires a greenhouse layer absorptivity of between 0.3 and 0.5. If this warming results entirely from carbon dioxide, we can estimate the effect of incremental increases in carbon dioxide concentration.
To good approximation, the emissivity of carbon dioxide is given by:
ε = 1 – exp{–cx0.25}
“c” is a constant that depends on gas temperature and total pressure.
“x” is carbon dioxide concentration.
During the past 100 years, atmospheric carbon dioxide concentrations have probably increased by 40%. Substituting into the equation, we find that the resulting emissivity (and absorptivity) increase by about 0.03. The corresponding surface temperature increases by just over 1 C. Thus, under this scenario, the observed increase in atmospheric carbon dioxide concentration accounts for the observed increase in temperature.
If we have significant heat convection, a higher emissivity is required to dissipate the heat. The resulting change in emissivity resulting from the incremental increase in carbon dioxide concentrations is then less. However, the incremental effect on temperature is similar.
If a substantial fraction of the greenhouse gases is water vapour, increased carbon dioxide concentration has a proportionately smaller effect. The effect of carbon dioxide on global warming can be negligible.
Taken together with the more sophisticated global circulation models, it is possible that the increase in carbon dioxide concentration does account for the observed global warming. However, this conclusion cannot be proven beyond reasonable doubt.
The analysis of chapter 2 shows that the world has never had a stable climate. The statistics show that the planet is always subject either to global warming or global cooling. It follows that any realistic climate model should reproduce this behaviour. Any model that does not reflect this behaviour is useless in determining the effect of carbon dioxide on climate change. It is useless because it cannot possibly give a natural explanation for climate change, when we see that natural processes have driven climate change for hundreds of millennia. Without the possibility of a natural explanation, an “unnatural” explanation, such as carbon dioxide is inevitable. There would be no need to write the model, the conclusion would be pre-ordained in the model specification.
There is a particular problem with models approved by the Intergovernmental Panel on Climate Change. The IPCC only approves models that accurately reproduce climate change over the last few decades. This approval seems to be quite a natural requirement if you look on climate modelling as an extension of weather forecasting. However, we see from the Appendix that a defined carbon dioxide concentration does not correspond to a specific temperature. For a given concentration, temperatures can differ by over 5C. Furthermore, a given carbon dioxide concentration does not determine a specific rate of temperature rise. At a given temperature, there is a tendency for low carbon dioxide concentrations to be associated with global warming and high concentrations with global cooling. However, particularly for “short” periods of a few centuries, the correlation is weak. Thus, a “correct” climate model would give an indeterminate temperature that cycles erratically. In order to make a model match both the current temperature and the current rate of temperature rise, two parameters must be fitted from measured climate data. We see from Chapter 3 that a model in which two parameters are fitted from measured climate data is not statistically significant unless is gives a mean error significantly less than 0.1C over 40 years or more. A more promising approach would be to develop models that show the natural climate variability found in Chapter 2. We can then model the addition of carbon dioxide to the environment and determine its incremental effect. It took weather modellers thousands of days (around 10,000) before they had sufficient experience to predict 5 to 7 days ahead. If we adopt a similar approach to climate modelling, we can anticipate that it will take thousands of years before we have the experience and data to predict climate confidently.
We need a model that never settles, but moves like a frictionless pendulum. (The final movement will be somewhat more erratic than a pendulum because of other disturbances that tend to be amplified by the system). To be consistent with the results of the Appendix, the model must explain why, at a given temperature, global warming periods have lower carbon dioxide concentrations than global cooling periods.
Two alternative models are put forward here. In both, we hypothesize that measured carbon dioxide concentrations are an indicator of ocean temperature. This hypothesis is consistent with the conclusions of Chapter 1. In both models, we deduce that global warming is driven by a situation in which the ocean is cooler than the land. Conversely, global cooling occurs when the ocean is warmer than the land. In the first model, we assume that the prime driver for global warming is differential cloud cover. In the second model, we assume that the prime driver for global warming is greenhouse gases.
Model 1. Clouds form more strongly at night, and in the winter, when the cooler conditions cause water vapour in the atmosphere to condense. Where the bulk of the ocean is cooler than the surface layers, surface temperatures are likely to fall more sharply at night (and in the winter). Thus, differential cloud cover is amplified. In this way, a relatively cool ocean gives lower atmospheric carbon dioxide levels and greater day/night differences in cloud cover. (Also, greater summer/winter differences arise). Thus, the differential warming effect drives temperatures upwards. Temperatures continue to move upwards until an upper limit is approached. (The upper limit arises because radiant energy loss to space increases more rapidly with temperature than convective heat transfer that mainly determines cloud cover). As warming slows down, ocean temperature catches up with surface temperature and the differential effect reduces. Ultimately, the effect ceases. Without the differential effect supporting a higher temperature, the temperature now begins to fall. The inverse situation arises as we approach a minimum temperature. In this way, the temperature cycles indefinitely. Because of complex, and chaotic, weather patterns, the chaos is promoted into the climate cycles. In this way, the potentially regular climate cycles also become chaotic and irregular.
Model 2. In this model, global warming is primarily caused by greenhouse gases, but the effect of greenhouse gases may be reduced by convection currents. When the bulk of the ocean is warmer than the corresponding atmospheric temperatures, warm, less dense air forms adjacent to the oceans. These conditions drive strong upward convection currents. Convection currents are driven by the difference between surface temperatures and the adjacent bulk atmospheric temperatures. When the ocean is warmer than the atmosphere, there are strong convection currents carrying moist air and energy upwards. When the oceans are cooler than the adjacent atmosphere, these convection currents are suppressed. We have already seen how convection reduces the global warming effect of greenhouse gases. Thus, when the ocean is warmer than the atmosphere, global warming is reduced. Surface/atmospheric temperature continues to fall until it approaches a natural minimum. At this point, ocean temperatures continue to fall (because they are still warmer than the bulk of the atmosphere). As the difference between atmosphere and ocean temperatures reduces, the convection currents become weaker. These weaker currents are less effective at suppressing greenhouse gas global warming. Thus, surface and atmospheric temperatures now begin to rise again. The convection currents are further suppressed and a period of global warming sets in. According to this model, global temperatures also cycle. It is also the case that carbon dioxide concentrations always lag the temperature cycle. It may seem anomalous that, in this model, greenhouse gases are the primary cause of global warming, but carbon dioxide concentrations lag temperatures. However, we have already seen, in Section 4.8, that small changes in carbon dioxide concentrations have a negligible effect on the emissivity and absorptivity of the greenhouse gases. The cooling effect of convection currents dominates over the marginal increase in gas emissivity and absorptivity resulting from higher water vapour and carbon dioxide concentrations.
If this model contributes significantly to natural climate variation, man-made carbon dioxide releases are particularly risky. Increasing carbon dioxide concentrations would increase the greenhouse gas effectiveness without the usual increase in convection currents associated with higher carbon dioxide concentrations.
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