Chapter 4. Global Warming Models.

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:

1. Black-body planet with a uniform surface temperature.

2. Grey-body planet with uniform surface temperature.

3. Grey-body planet with non-uniform surface temperature.

4. Non-grey planet with non-uniform surface temperature.

5. As (4) with effect of polar and winter snow (or other reflector).

6. As (4) with effect of cloud.

7. AS (4) with a layer of greenhouse gases.

8. As (7) using approximate relationships for properties of greenhouse gases.

9. Unsteady models.

4.1. Black body planet.

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:

T_B = 279 K (or 6 C)

This figure can be compared to the current arithmetic mean surface temperature of Earth, namely

T_A = 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.

4.2 Grey body planet.

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.

4.3 Grey planet with non-uniform surface temperature.

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:

T_R = 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.

4.4 Non-grey planet with non-uniform surface temperature.

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:

T_N = 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.

4.5 Effect of Snow Cover.

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.

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.

4.6. Effect of Cloud Cover.

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.

4.7 Greenhouse gases (water vapour alone)

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. The two major greenhouse gases in the Earth’s atmosphere are water vapour and carbon dioxide. We consider first the effect of water vapour alone then, in Section 4.8, consider the additional impact of carbon dioxide.

To date, we have considered only radiation effects but both the atmosphere and oceans exhibit significant convective effects. Convection results from the movement of fluids that takes heat with them. It is convective instability in the atmosphere that gives rise to changeable weather. It may be that the slower convective currents in the oceans impact on climate change. The weather record (and the recent climate record) has been obtained by measurement of temperatures at up to a few metres above local ground level. (A few weather stations are on the roofs of buildings). The radiation calculations presented above give mean temperatures on the surface of the Earth. They apply both to the solid surfaces and to the ocean surface; the absorption/emission properties of ocean water are similar to those of dry land. In the absence of global warming gases, radiation passes straight through the atmosphere in both directions and the air is not warmed at all. The air warms by convective currents; air adjacent to the ground is warmed and convective currents (winds, eddies etc) move the warmed air so that it mixes with adjacent air such that the whole of the air near to the surface of the earth is warmed. Air temperature settles to a level that is an average of that of the surface. Heat transfer from air to surface also reduces the variation in surface temperature. Such variations arise from the inclination of surface irregularities (some point towards the sun and some are shaded) and variation in surface emissivity/absorptivity. Winds also disperse temperature variations over longer distances and air temperature changes more slowly between day and night. Wider scale convective currents arise from locally warmer and cooler volumes of air. Warm air rises and, as it does so cools by adiabatic cooling. This cooling can be observed in the laboratory by suddenly releasing the pressure in a pressurized vessel. Despite the adiabatic cooling, such convective currents disperse energy through the depth of the atmosphere. The inverse effect occurs with descending air. Such air is heated by convective compression. It is noted that, in the absence of greenhouse gases, convection has no effect on mean surface temperatures. It merely reduces the variability of temperature as compared to a planet with no atmosphere

Water vapour is a greenhouse gas that is present in significant quantities. Two thirds of the Earth’s surface is covered in water. The equilibrium water vapour pressure depends on the water temperature. The following table gives pressure as a function of temperature over pure water:

Note that atmospheric pressure is approximately 100 kPa (1 bar). Thus, the pressures also give the approximate percentage volumetric concentrations. Vapour pressure of water is slightly lower over salt water, so that concentrations found on Earth tend to be slightly lower. The relative humidity is the water vapour concentration as a percentage of the pressures in the above table. Note also that vapour pressure approximately doubles for a 10 degree rise in temperature.

The arithmetic mean temperature of the surface of the Earth is estimated as a bit below 15 C. (The mean for radiation calculations is a bit above 15 C). Hence, 1.7 kPa is a reasonable estimate of the mean water vapour pressure over the surface of the Earth.

Water vapour also has a finite concentration over ice at temperatures below freezing. Thus, even over the poles, there is some greenhouse effect.

The effect of water vapour (as any other greenhouse gas) is as follows. The sun’s rays pass through the vapour with very little energy absorbed. However, the longer wavelength radiation from the Earth’s surface is absorbed in the gas to some extent. In this way, the air is warmed by radiation from the surface as well as from convection. This warming is spread by convection, including convection back to the Earth’s surface. By reducing the ability of radiation to escape into outer space, the water vapour effectively reduces the emissivity of the surface of the Earth. The net effect is that the surface emissivity is now less than its absorptivity. It follows that (compare with Section 4.4) instead the non-grey surface causing global cooling, it now causes global warming. The water vapour concentrations noted above may be sufficient to account for the observed global warming and the temperatures that we currently observe.

Note that the water-vapour effect results in a positive feedback that tends to give unstable variable temperatures. Thus, as shown in the table, any increase in ocean temperature increases the water vapour pressure that gives increased global warming. The period for feedback is comparatively rapid. It only needs the surface layers of the ocean to reach a higher temperature in order to generate a higher vapour pressure.

Water vapour plays a second important part in determining climate. In warm wet conditions, water evaporates from the surface to give water vapour. The evaporation process absorbs large quantities of heat (as latent heat of evaporation). This evaporation causes local cooling. (For example, tropical rain forests are cooler than they would have been had the area been completely dry). The relatively high proportion of water vapour reduces the density of the air. (The molecular weight is 18 as against 28 for dry air). The resulting low density causes air to rises. As it rises, it cools by adiabatic expansion. The cooling causes some of the water vapour to condense. As it condenses, it rejects large quantities of latent heat. Thus, the heat of evaporation is recovered, but the water vapour has the effect of transporting heat from the surface to higher altitudes. Overall, the air at those altitudes is some 20 C warmer than it would have been had there been no water vapour. The water vapour condenses out into clouds and ultimately precipitation. The resulting relatively warm dry air returns to Earth by convection. As it returns, it warms by adiabatic compression. We observe that warm wet areas with rising air, such as the tropical rain forests, tend to be banded by relatively hot dry areas, such as the Sahara, Kalahari, Australian and Atacama deserts. This water vapour cycle probably has a relatively small effect on net global warming, but it is relevant when we come to consider the impact of carbon dioxide.

In summary, we do not need to hypothesize global warming effects from other gases. Water vapour has by far the largest concentration of all the greenhouse gases and is one of the most effective. That does not preclude the possibility that other gases also contribute.

4.8 Greenhouse Gases (Carbon Dioxide)

The current increased pressure of carbon dioxide averages out at about 0.038 kPa. (It may get higher before we get around to updating this web page!). Thus, at ground level, it is a relatively small fraction of the water vapour concentration. From the table in Section 4.7, we can derive the following percentages:

At a given concentration, carbon dioxide is a less effective greenhouse gas than water vapour. (It is always difficult to compare the effectiveness of different greenhouse gases because their impact depends on their concentration. For a grey non-greenhouse gas, its effectiveness doubles when the concentration is doubled. For water vapour, it is necessary approximately to quadruple the concentration to double the effect. For carbon dioxide, it is necessary to multiply the concentration by 16 to double the effect. Thus, the greenhouse ranking depends on the concentration considered. All greenhouse gases lose their effectiveness as their concentration increases. The comparative figures published in the press are for small variations about the current levels. These published figures also ignore interaction effects between the gases). Thus, just looking at the table, we would expect carbon dioxide to contribute little to global warming at ground level. In practice, the position is much less favourable to carbon dioxide because of interaction between the two gases. The two gases together are much less effective than the gases on their own. In particular, water vapour severely reduces the ability of carbon dioxide to absorb radiation. It follows that, even at quite low temperatures, carbon dioxide contributes negligibly to global warming. If we also bear in mind also that the Earth radiates about 50% more heat from areas at 30C than at 0C, it is clear how unimportant is the contribution of carbon dioxide.

The above comments apply only at ground level and up to altitudes at which the concentration of water vapour exceeds that of carbon dioxide. Above these altitudes, carbon dioxide becomes the dominant greenhouse gas. We now consider the effect of carbon dioxide at these high altitudes.

At high altitudes, a heat balance for carbon dioxide is as follows:

1) It absorbs some heat that has radiated through the lower layers of water vapour

2) It absorbs (or contains) heat brought up by convection, noting particularly that it is approximately 20C warmer than if there had been no water vapour

3) It returns heat to the Earth’s surface through convection that returns bulk air to the surface

4) It loses heat to outer space by radiation

5) It returns heat to the Earth by radiation

Consider each of these effects in turn:

Carbon dioxide absorbs very little heat radiated from the Earth’s surface. The reason is that water vapour has already absorbed heat energy in those parts of the spectrum that are absorbed by water vapour. The remaining energy is unabsorbed by carbon dioxide and passes straight through into outer space.

The carbon dioxide is considerably warmer at this altitude than it would have been in the absence of water vapour.

The energy returned to the Earth by convection would have been significant if carbon dioxide had absorbed radiation. The energy that it absorbed would diffuse through the remaining atmospheric gases which would thereby be heated. Adiabatic compression would heat the gases further. In this respect, carbon dioxide probably has little, or no, more significance than an inert gas.

As noted earlier, all good absorbers are good emitters. Thus, carbon dioxide is an effective source of radiation. It is free to radiate at the wavelengths not accessible to it for absorption from radiation from the Earth’s surface. Thus, it radiates energy into outer space energy that would otherwise be returned to the Earth by convection.

We would expect it to radiate almost as much energy back to the Earth’s surface as it radiates into outer space. (We do not count this back-radiation at lower altitudes because the net absorptivity values already include allowance for the effect).

We have noted that water vapour provides a strong positive feedback in global warming because it only requires the surface of the oceans to be warmed to release water vapour. However, carbon dioxide does not exhibit such a strong positive feedback. Although over 98% of the environmental carbon dioxide is dissolved in the oceans, the concentration is still very low. As the oceans warm, the partial pressure over carbon dioxide over the oceans increases. (The correlation in the Appendix indicates that it increases by about 50% for every 10C temperature rise). However, as carbon dioxide is released from the oceans, the surface layers become depleted and the process of diffusion from deep layers to the surface is slow; it can take hundreds of years. (The converse case also holds, it will take hundreds of years before the oceans dissolve all the carbon dioxide that mankind has released over the last century).

In conclusion, it is not clear whether carbon dioxide contributes to net global warming, to net global cooling, or has negligible net effect. It contributes to global warming if the small amounts of radiant energy that it absorbs at high altitudes exceed the amount that it radiates to outer space. It limits global warming if the contrary balance applies. If neither effect is great, or if they cancel, it has negligible effect on global warming.

Certainly the case against carbon dioxide has not been proven and much more work needs to be done to challenge the conventional view before that case will be proven either way. In the meantime, the precautionary principle should be applied.

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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