Chapter 2. Is the current rate of temperature rise exceptional?

Many observers state that the current rate of temperature rise is higher than any rise experienced on Earth for at least 10,000 years. If true, this observation alone lends support to the hypothesis that mankind’s global activities (specifically our exceptional releases of carbon dioxide) are responsible for the climate change. Here we attempt to employ the Vostok ice-core data to explore the statistics of temperature change. The objective is to determine whether, taking a longer time span, there is anything unusual in the current rate of global warming. We do so by computing the frequency at which temperature rise rates have occurred in the past. Thus, if we take any 100 centuries at random, and we find that 50 showed a temperature rise of more than 1.0 C, our current rise would be unexceptional. On the other hand, if only one century showed a rise of more than 0.5 C, our current situation would be truly exceptional.

Before considering the temperature rise rate; we need to establish whether the current temperature is exceptionally high. Referring to Figure 1 (in Chapter 1), we see that we are approaching the peak of a long period of increasing temperatures. This is the fifth such period shown in the Vostok record. Table 1 gives the measured maximum temperature for each of the four previous cycles together with the maximum observed so far in the current cycle.

Table 1. Temperature maxima for five most recent climate cycles.

There are several reasons why Table 1 underestimates the maximum temperatures quoted:

1. The temperatures are averaged over a finite depth of ice. Any average is less than the relevant maximum.

2. Vostok temperatures are estimated from the concentrations of isotopic species. Over long periods, there is diffusion and mixing. Thus, even at one point in the core, the isotopic species come from a range of times. In this way, derived temperatures are again mean values, which are less than the relevant maxima.

3. The temperatures are interpolated to give values at exactly 1-metre intervals. Again, any interpolation gives a result that is lower than the higher of the two values interpolated. (A more sophisticated interpolation may be employed, for example, a quadratic through a minimum of 5 points. Occasionally, such sophisticated interpolations predict a higher temperature than the maximum of the points interpolated).

4. The table gives one temperature from a time interval. The time intervals between 1-metre depths are given in column 4 of the table. There is a considerable year to year temperature variability. The chance that the one year chosen is the maximum in the interval is small. The table gives the intervals between the temperature measurements quoted. We see that, in the last 10,000 years, there is a 40-year period between measurements. Thus, a-priori, there is a 1 in 40 chance that the temperature quoted is the maximum between successive measurements. As we go further back in time, the probability that the maximum is correctly identified drops to 1 in 450.

Against the above reasons for believing that the maximum temperatures were higher than quoted, we note that they are quoted as “above present”. The “present” temperature was set a few years ago, and temperatures have risen slightly in the meantime. Thus, we should reduce the quoted values by that temperature difference. The table gives only four major temperature cycles. Hence, we have insufficient data to predict confidently the natural maximum for the current climatic cycle. Nevertheless, it is clear that there is nothing particularly exceptional about the temperatures that we are currently experiencing.

What may be exceptional is the rate of rise over the last century (following industrialisation and extensive combustion of fossil fuels). Note that the long-term geological periods of “rapid rise” are very slow compared to the temperature rises experienced in recent years. Thus, the average rise over the most recent 20,000-year period is about 0.075 degrees/century. At that rate, it would take about 40 centuries for the temperature to rise a further 3 degrees C.

Our analysis of the Vostok temperature data is as follows. We pick a period, say 100 years. We then find all the Vostok data points that differ by a time interval of approximately 100 years (say from 80 years to 120 years). We calculate the mean temperature rise (or fall) during that period. Thus, we compute:

r = –100(T₁ – T₂)/(t₁ – t₂) (R1a)

t₁ and t₂ are the times at the beginning and end of the interval (in years before present) and T₁ and T₂ are the measured temperatures at these times. The coefficient 100 gives rise rates per century and the negative sign is used because the Vostok data counts time backwards (years before present).

The rise rates are then assembled as follows:

n(2.0) counts all those centuries for which the rise rate was greater than 2.0 C/century.

n(1.8) counts all those centuries for which the rise rate was greater than 1.8 C/century.

and so on down to:

n(0.1) counts all those centuries for which the rise rate was greater than 0.1 C/century

n(0.0) counts all those centuries for which the temperature rose at all.

We conducted the same analysis for those centuries in which the temperature falls. The ratio

p(2.0) = n(2.0)/n(0.0)

gives the proportion of centuries in which the rise rate is greater than 2 C/century. Thus, it gives the probability that, choosing a century at random, we will find a rate of temperature rise greater than 2 degrees/century.

There are several criticisms of this methodology. It can underestimate rise rates because:

1. The temperatures actually measured are the means of temperatures over a number of years (because there is diffusion and because the ice sample comes from a finite depth of ice). Hence, extreme variations tend to be smoothed out.

2. The data has been interpolated to equal 1-metre intervals. This interpolation again smoothes out the variability of the data.

It can overestimate rise rates because:

1. Measurement errors can exaggerate climate variability. For example, consider a climate in which the temperature is rising steadily at 0.2 C per century. Consider that there is a measurement error of 0.1 C. Sometimes this will add to the temperature at the end of the century and subtract from the beginning of the century. In this case, the estimated temperature rise may be (0.2 + 0.1 + 0.1) = 0.4 C. Sometimes the errors will be in the opposite direction. The estimated rise rate will then be zero.

2. There are errors because we are taking end temperatures and not fitting the best line through all the intermediate temperatures. Consider that the temperature follows a cyclic pattern about a constant mean value. Over periods shorter than the cycle period, the end-point difference gives a good approximation to the mean temperature rise over the period. However, over periods that include many cycles, the best-fit straight line tends to have a slope less than that obtained by taking the difference between the temperatures at the end points.

Taking just periods of one century is arbitrary. We happen to be concerned about global warming over the last century. However, there is no statistical merit in taking temperature rises over periods of one century. During the last 50 years, temperatures have been rising faster than in the previous 50 years. Perhaps we should be worrying about the 50-year rate of global warming, not the average over the century. Furthermore, from the practical point of view, restricting our analysis to only periods of one century restricts the Vostok data that we can access. As the core gets deeper, the ice is more compressed, so that a 1-metre depth corresponds to a longer time interval. Beyond 240,000 years before present, successive temperatures are quoted at times greater than 100 years apart. Thus, all the older data is inaccessible to analysis. In order to get maximum value from the measurements, it is desirable to employ a methodology that can access all the data points. Finally, in order to obtain correlations in the way described, we need to take a finite margin of time around the selected interval. Thus, when we calculate the average time interval for our target value of 100 years, it will differ from 100 years depending on the time intervals chosen. Repeating the analysis at a range of time intervals enables us to interpolate to any desired interval.

We expect a relationship between the statistics obtained for differing time intervals. Thus, consider a climate that has a constant mean temperature over a very long period, but makes pseudo-random changes from decade to decade. Consider that the maximum deviation from the mean is 0.5 C. The maximum measured temperature rise rate over a 50-year period is then 100(0.5 + 0.5)/50 = 2.0 C/century. Over a 100 year period, it is 100(0.5 + 0.5)/100 = 1.0 C/century. Thus, the statistical distribution of temperature rise rates will be greater for a shorter period than for a longer period. This conclusion holds for all climate change statistics apart from a uniform rate of change that applies throughout the entire period of the measured data. The correlation between temperature change-rate statistics and period over which the rise is measured is related to the underlying climate variability.

As an example of a very simple temperature-variation model, consider a system in which there is a random, or pseudo-random, variation about a steady state. This temperature distribution is the same every year. It follows that the probability of achieving a given temperature change will be independent of the time interval chosen. Mathematically, we can write:

Probability of temperature change ∆T is independent of time interval, t. The temperature rise rate is

r = ∆T/t

Hence, we would expect probabilities to be related to (rt).

In practice, the temperature in one year is correlated with the temperature in the next. Hence, the temperature change over a short period will be less than over a long period. If there were a steady change in temperature, r would be independent of t. We would expect behaviour between these two extremes. Thus, for any given probability level, we would expect that r would vary at a rate that lies between t^–1 and t⁰.

Table 2. Time intervals employed in analysis.

We have measured statistics for the time intervals given in Table 2. The difference between “intended interval” and “mean interval” arises from the data. For example, to get an intended time interval of 80 years, we took all the Vostok temperature measurements that differed from one another by intervals of from 65 years to 95 years. If the measurements had been uniformly distributed over the interval, we would have achieved a mean interval of 80 years. However, the measurements are not quite uniformly distributed in time. Hence, the recorded “mean interval” differs slightly from the intended mean. Estimating mean temperature rises from equation (R1a), we find that the probability of a given temperature change is related to rt^0.8. Thus, it corresponds reasonably well with the very simple model. It follows that, rather than use equation (R1a), we should expect ∆T to vary with t^0.2. With such a small time-interval dependence, we have used the simplified model for correcting equation (R1a). Thus, we put

r = 100(T₁ – T₂)/t_m (R1b)

In equation (R1b), t_m is the relevant mean time interval. Equation (R1b) overcorrects the temperature rise rate. We analysed temperature rise rates using both uncorrected and overcorrected mean rise rates. We found that there was negligible difference between the uncorrected and overcorrected statistics. Thus, using a finite margin about set time intervals introduces no error into the analysis. We omitted the data set for 27.5 years from further analysis because there are too few points to give a statistically significant distribution. We wish to predict (or measure) the probability of achieving a given rate of temperature rise. We are specifically interested in the probability that the current rise (in the last 50 to 100 years) might occur by chance. Any correlating equation that we apply will be only an approximation to the actual distribution function. It is important that it is accurate at the probability levels significant to us, namely between 5% and 50%. If the estimated probability of the current temperature rise occurring naturally is less than 5%, we know that the conditions are unlikely to have arisen by chance. If the probability is greater than 50%, we know that conditions have probably arisen by chance. It is in the 5% to 50% band that the decision becomes difficult, so that it is important that the statistics in this band are accurate.

The correlation between probability and rate of temperature rise is non-linear. However, it has a simple form that can be fitted using linear regression, namely:

f_p{p} = af₁{r, t} + bf₂{r, t}

To obtain the best least-squares fit to the probabilities, we need to minimize:

S = Σ(p_i – p_ci)²

In the above equation p_i is the i’th measured probability and p_ci is the corresponding calculated probability. The non-linear transformation alters the relative weighting of the points. The resulting distortion can be corrected by suitably weighting the transformed variables, thus:

S = Σw_i(f{p_i} – f{p_ci})²

The weights w_i are chosen such that, as closely as possible, the transformed and original summations are the same. For small discrepancies, we have that:

f{p_i} = f{p_ci} + (df/dp)_i(p_i – p_ci)

Comparing the two summations, we obtain:

w_i(df/dp)_i ² = 1

Hence, we need to set

w_i =( df/dp)_i ^-2

Two of the distributions that we have considered are the Negative Exponential and the Weibull. For both, the final transformation is ln{p}. (In one case, ln(p) vs r, in the other ln(p) vs r^w). Thus, for both, we need to put:

w_i = p_i²

To estimate the Weibull exponent, we fit ln{–ln{p}} vs ln{r}. The appropriate weighting is then:

w_i = (p_i ln{p_i})²

We considered that, for the data set with t < 35 years, there were too few data points and it covered too short a period. Analysis of the remaining data showed that all data sets were best correlated by the Weibull distribution. Namely

p = exp{–kr^w} (R.1)

In the above equation,

r is the temperature rise rate (degrees C per century)

w is the Weibull index, which is close to 1.2 for all data sets.

p is the probability that temperature rises at a rate greater than r.

“w” is sufficiently close to 1.0 to suggest that the simple Negative Exponential distribution might apply, namely:

p = exp{–kr} (R2)

Equation (R2) occurs so widely in physical and life sciences and in engineering, that is temping to believe that an approximate theoretical relationship for climate variability could be derived. However, we have not pursued that possibility here.

The value of “k” depends on the time interval. Small values of “k” give a high probability of large temperature gradients. Large values of “k” give a low probability of large temperature gradients. We have already noted that the longer the time interval, the lower the mean temperature gradient that we expect. Thus, we expect larger values of “k” for larger time intervals. For Weibull (w = 1.2), the following simple linear correlation was obtained:

k = 0.6059 + 0.021485t (R3)

In equation (R3), t is the time interval over which the temperature rise rate is observed. The correlation is plotted in Figure 5. Equation (R3) is found to apply well up to time intervals of thousands of years. However, it clearly does not apply well for intervals much less than 50 years. As the interval approaches zero, the value of “k” should also approach zero because the probability of large temperature gradients over periods as short as 1 year becomes very high. The observed value for the 27.5-year time interval falls below the line, but is not sufficiently reliable to generate a curve joining the origin and the first reliable point at 50 years.

Figure 5. Dependence of temperature-rise k-value on time interval.

Figure 6. Probability of maintaining temperature rise rates over intervals from 27 years to 324 years considering all Vostok temperature data.

Figure 6 shows probabilities calculated from equations (R1) and (R3) compared to the measured probabilities from the Vostok ice-core data. As anticipated from figure 5, the only data set that is badly correlated is the set for time interval 27.5 years. The probabilities for the remaining time intervals are all predicted within a few percent. The agreement gives us confidence that the correlation can be extrapolated and interpolated.

If we know the current rate of global warming, Figure 6 enables us to compute the probability of it arising from natural climate variability. We have estimated the current warming rate using the data given in Figure 7.

Figure 7. Measured annual and linear correlations.

Figure 7 gives annual global mean temperatures for the period 1866 to 1996, as reported by J Hansen and H Wilson (Climatic Change, Vol 25, p285, 1993). The last few points are as extended by those authors at the Goddard Institute for Space Sciences (GISS). The temperature deviations are corrected to degrees Celsius by assuming a mean temperature for the period 1951 to 1980 of 14.00 C. (The data are given in Section 14 of the Handbook of Physics and Chemistry, 79th Edition, 1999, CRC Press, Washington DC, USA). The preferred way of computing mean rates of temperature change is to fit the best straight line through the data. We have fitted such a line for the 100-year period 1897 to 1996 and for the 50-year period 1947 to 1996. Both of these straight lines are shown on the figure. The slopes are:

100-year period: 0.52 C/century.

50-year period: 0.83 C/century.

In order to make the estimates more closely mimic the way that figure 6 was computed, we have also estimated the temperature rise rates by an alternative method. In this method, we have found the differences between the mean temperatures at the beginning and end of each time interval. Thus, we have computed the mean global temperature in the 10 years preceding the ends of the time intervals. We obtain the following rise rates:

100-year period: 0.64 C/century.

50-year period: 0.52 C/century.

The differences between the two estimates of rise rate illustrate the controversy in estimating exactly what temperature rise rate we are currently experiencing. The variation from one year to the next can reach nearly 0.5 C.

We can now compute the probability that these rise rates will occur in the natural course of events pre-industrialization. The values are given in Table 3 as “p% (all data)”.

Table 3. Probability that current temperature rise rates occur by chance.

Table 3 gives the probability that the observed temperature rise (or more) would occur by chance. If we use the best straight-line method of estimating temperature rise rates, we get the following results. There is a 26% probability that the 50-year temperature rise could occur by chance. There is a 28% probability that the 100-year temperature rise could occur by chance. The parameters in equations (R1) and (R3) were derived from end-point temperature differences over the time intervals. Thus, we should obtain better probability estimates using this measure of temperature rise. On this basis, the global temperature rise observed between 1947 and 1996 occurs at a probability of 46%. The corresponding 100-year probability is 20%. These probabilities are sufficient to raise concern that external factors (for example, mankind’s activities) might be the cause of the rise, but they do not give overwhelming proof that something unusual is happening. If we take 100 successive centuries, the correlation suggests that 20 of these will show temperature changes greater than we are currently experiencing.

Some workers have suggested that the past 10,000 years have been exceptionally quiet. Hence, any current sudden change in global temperature marks the end of a quiet era. This end may have been triggered by mankind’s activities. To test this hypothesis, we have reanalysed the data restricting the analysis to the period up to 10,000 years before present. The restricted time interval results in fewer data points; the numbers are shown in Table 4.

The confidence in the statistical analysis is less because we have fewer data points. However, the reduced data set shows a consistently higher climatic volatility. The corresponding Weibull k-values are included in Figure 5 and are consistently less than those for the full data set. For time intervals above (and including) 80 years, the k-values are well correlated by:

k = 0.021811t – 0.94101 (R3a)

Table 4. Time intervals used in analysing data for last 10,000 years.

As for the full data set, we would expect this line to go through the point (0,0) below a time interval of 80 years. Despite the limited number of data points, the probability functions are well correlated by equation (R1), with w = 1.2, and equation (R3a). The results obtained are illustrated in Figure 8.

Figure 8. Probability of maintaining temperature rise rates over intervals from 27 years to 324 years considering Vostok temperature data for last 10,000 years.

The resulting probabilities of observing the current rates of global warming are given as the last line of Table 3. Note that, since the correlations below 80-year intervals are uncertain, the 50-year figures given in Table 3 have been computed from the measured k-value for 46.9 years rather than from the correlation. This value (k = 1.081) should be conservative because it is well above the correlating line. We see that if we take account of the climate volatility over the last 10,000 years, the currently observed rise rates would be expected to occur with probabilities of around 50%.

Having found that the 10,000-year statistics differ from the 420,000-year statistics, we explore how climate volatility varies over time. We have divided the 420,000 year data set up and repeated the analysis for the following periods of time:

Periods used in exploring climate variability. (Thousands of years before present).

For Vostok values from the distant past, there is a large time interval between successive 1-metre depths. Hence, we modify the earlier analysis to include longer time intervals. For intervals over 300 years, the probability of sustained large temperature rise rates becomes low. Hence, we modify the analysis to include much smaller rise rates (from 0.02 C/century). The following time intervals were used for each of the periods given above:

Target time intervals for Weibull analysis of climate variability (years).

For the periods given, many of the intervals shown above have very few data points. Thus, the cumulative probability statistics may be derived from many less than 100 points. It is then inevitable that we get more scattered correlations and less reliable estimates of parameters. Nevertheless, the complete set of derived Weibull k-values is plotted in Figure 9. The figure also shows equation (R3a), which was derived earlier for the 10,000 year BP data set. That correlation was fitted to periods ranging from 80 to 270 years. The periods used for Figure 9 are quite different. Nevertheless, the same correlation fits the new data well for periods from 60 to 1,000 years. This result confirms that the correlation is not an artefact of the way that the analysis was conducted.

Figure 9. Temperature-rise k-values as a function of time interval derived from ice-core data for eight geological periods ranging to 400,000 years before present.

As anticipated, the parameters derived from limited data sets are more scattered. However, they have been ranked as follows. The k-values have been adjusted to the target intervals by making the adjustment

k’ = k + 0.02(t_T – t_A)

The correction is small in all cases. The coefficient 0.02 is taken from the coefficients of equations (R3) and (R3a) because all data sets show a similar dependence on time interval. For each data set, the values for each period were ranked for each time interval. Additionally, the k-values for each interval were divided by the corresponding mean, and the mean of all the resulting normalized k-values was taken. The resulting overall mean values are given in Table 5. The table also gives the mean number of years per metre of ice for each period.

Table 5. Relationship between k-values and time period.

The ranking in Table 5 also corresponds to the majority ranking for each of the individual intervals. Thus, there is consistency in the data. We would expect that, at greater depths the snow is more compressed, so that the number of years per metre would increase. However, the number of years per metre decreases for the period 20,000 YBP to 150,000 YBP. Closer inspection of the tabulated Vostok data shows that the decrease actually extends to about 130,000 YBP, when there are about 50 year/metre. Further back in time, the number of years per metre increases rapidly. A plausible explanation is that temperatures were falling steadily from 130,000 YBP to 20,000 YBP and that the falling temperatures produced a drying atmosphere with reduced snowfall. A reduced annual snowfall would give a greater number of years per metre of snow depth. At first sight, there is no obvious correlation between k-value and climate cycle. We plotted k-value with number of years per metre and obtained a reasonable correlation. The rationale for this correlation was that, if more years are compressed into a given depth, there is greater mixing between years and hence dilution of temperature gradients. However, the best correlation was obtained between k-value and time. There is a theoretical basis to the correlation. Over time, mixing and diffusion processes tend to reduce all compositions to a common mean composition. Temperature is deduced from isotopic ratios. Thus, any initial deduced temperature difference will decline with the standard diffusion/mixing law as follows:

ΔT = ΔT₀exp{–cy} (R4)

In equation (R4), ΔT₀ is the initial temperature difference, and ΔT is the derived temperature difference after “y” years. “c” is a constant.

Initially, the probability of finding a temperature difference ΔT₀ is “p”. This probability is given by the Weibull equation:

p = exp{–k(ΔT₀/t)^w} (R5)

After “y” years, the deduced temperature difference is ΔT. Thus, from equation (R4), the probability of finding that deduced temperature difference is:

p = exp{–k(ΔTexp{cy}/t)^w} (R6)

Equation (R6) can be expressed as:

p = exp{–k’(ΔT/t)^w} (R7)

In equation (R7),

k’ = k.exp{cyw} (R8)

Equation (R7) says that the probability of finding any given deduced temperature gradient also follows a Weibull distribution. Equation (R8) relates the value of “k” for the deduced temperature gradient to a value of “k” that applied when the layer of snow was first laid down. On this basis, the logarithm of deduced values of “k” should correlate linearly with the age of the ice. We should also find that the ratio of values of “k” is independent the time interval. This deduction justifies taking a mean ratio for “normalized k” in Table 5. In practice, it seems that the lines tend to have a common difference rather than a common ratio. However, this simple theory enables us to make a reasonable estimate of the climatic variability throughout the past 400,000 years. The correct value of “k” describing the climatic variability can be deduced by extrapolating back to y = 0.0. The correlation for equation (R8) is shown in Figure 10. If we exclude the outlier for the period 10,000 years to 20,000 years (mean y = 15,000), the correlation is reasonably consistent with the theory. The rationale for omitting the outlier is that it is based on very few points and is hence unreliable. (There is also a large underlying rate of temperature rise, which has the effect of reducing calculated k-values). This theoretical treatment suggests that the apparent change in climate variability is an artefact of the way that temperature estimates are derived from isotopic proportions.

This explanation seems to be more plausible than the alternative explanation that the observed effect is real. If the effect were real, it would imply that climate has steadily become less stable over the last 400,000 years. There seems to be no physical explanation for such increasing climatic instability. Indeed, such a steady deterioration would imply that we should expect increasingly extreme climate variability even without any disturbance from mankind.

Figure 10. Apparent Weibull k-value as function of ice age.

If we accept that the apparently increasing climatic variability is an artefact of the method used to estimate temperature, the results for the last 10,000 years are representative of the whole of the last 400,000 years. Thus, we can conclude that there is about a 50% probability that natural processes alone would generate the current rate of temperature rise.

In the Annexe to this chapter, we explore the extent to which this explanation is plausible.

This analysis also reflects on the results reported in Table 1. If current temperature rise statistics apply equally hundreds of thousands of years ago, temperature extremes must be greater than indicated by the averaged Vostok measurements. Thus, the true maxima for the earlier temperature cycles would be several degrees higher than those estimated.

It may be suggested that we have only included data for periods in which temperature rises. Thus, if we included periods in which temperature falls, we would halve the probability of the observed temperature rises. However, that suggestion would introduce false statistics. We would be equally concerned (or more so, since the effects would be more drastic) if temperatures were falling at the rate that they are currently rising. Thus, the remarkable observation is the rate at which climate is changing. It is not simply the direction of change. Discounting the other equally remarkable observation is making a conclusion after the event. It is like claiming that the probability of a coin landing heads after it has already landed heads is 25%. The probability of two heads in sequence is 25%, but the probability after we already know the first outcome is 50%. Thus, we have already observed that the temperature is rising. What might be remarkable is the rate of rise.

In conclusion, we have an indicator that there may be something unusual in the current rate of global warming. However, the statistics give only weak support to the hypothesis that there is something climatically unusual.

Annexe to Chapter 2. Effect of diffusion on apparent Vostok temperatures and carbon dioxide concentrations.

We have hypothesized that, over centuries, chemical species diffuse in ice cores. Thus, the concentration at a given depth corresponds to a mean concentration over a period of years. It is not a concentration at one point in time. It follows that measured carbon dioxide concentrations do not correspond to concentrations at a specific time, but concentrations averaged over a number of years. Estimates of time and temperature are both derived from isotopic proportions. These proportions are also averaged over a number of years. Hence, the derived temperatures are values averaged over a number of years. The further we go back in time, the greater the number of years over which the average is taken. In this way, extreme variations in temperature and in carbon dioxide concentration are eliminated because only mean values are recorded in the ice cores. (Time is also derived from isotopic proportions, so that the measured times are also averaged values).

We consider three cases:

1. Temperature measurements modified by solid/solid diffusion of isotopes through the ice.

2. Carbon dioxide measurements modified by gas diffusion through porous snow/ice.

3. Temperature measurements modified by gas diffusion through porous snow/ice.

1) Solid/solid diffusion. We refer to equation (R8) and Figure 10. A best straight line has a slope of about 0.6y^–1. From equation (R8),

c = 2×10^–6/w = 1.67×10^–6 y^–1

To a good approximation, the coefficient “c” is given by:

c = D/l² (R9)

In equation (R9), “D” is the diffusivity of the isotopic ice through pure water ice, and “l” is a characteristic vertical distance through the ice core. Thus, “l” is a distance over which averaging applies. We expect this distance to be a small fraction of the ice-core depth because it is only local fluctuations in temperature that are smoothed, not the major changes that occur over climatic cycles. Diffusivities are quoted in seconds, rather than years. Thus, in appropriate units:

c = 5.28×10^–14 s^–1

We have not found diffusivity data for isotopes of ice. However, solid/solid diffusivities tend to be of the order 10^–15 cm²s^–1. On this basis:

l² = 10^–15/5.28×10^–14 cm²

Hence,

l = 0.14cm

The result implies that this mechanism will only smooth temperatures over a few millimetres. The corresponding time interval is about 0.1 year. Hence, the maximum affect would be to smooth temperatures by a decimal of a degree. It follows that this mechanism cannot account for the results observed in figure 10.

2) Gas diffusion modifies carbon dioxide concentrations. It is observed that the apparent age of gas bubbles in the Vostok ice can be several thousand years younger than the ice surrounding the bubble. When the bubble is first formed, it is clear that the ice and bubble have the same age. Initially, the snow is highly porous and gas can diffuse freely through the snow. Thus, counter-diffusion is established with older gas previously trapped in the snow diffusing to the surface and gas at the surface diffusing back in to replace it. Consider gas within ice that is 2,000 years old. Some of the 2,000 year old gas will have diffused towards the surface and been replaced by fresh gas (of “zero” age) nearer the surface. Thus, the isotopic composition of the mixture corresponding to an age of 1,000 years, may be a mixture of gas 2,000 years old and of gas from the atmosphere of age zero. If the gas initially had an exceptionally high carbon dioxide concentration, that gas will now be diluted with gas of lower carbon dioxide concentration. In this way, peaks of carbon dioxide concentration become smoothed. As time passes, diffusion continues to mix gases from adjacent levels (ages) and concentration differences decline according to an equation similar to equation (R4). The apparent age of the gas depends on two conflicting mechanisms. As the snow/ice gets deeper, it becomes compressed. The compression reduces the size of the passages for diffusion. The reduced passage sizes tend to force gas upwards. This movement forces deeper and older gas into the bubble spaces. This mechanism tends to give an apparent bubble age that is older than the ice in which it settles. On the other hand, the larger passages above a given level make it easier to exchange gas with shallower levels than with deeper levels. From the measured apparent gas bubble ages, it is clear that this latter mechanism dominates. From the apparent difference between the age of the bubbles and the ice within which they sit, it is clear that diffusion occurs over tens of metres and gives carbon dioxide concentrations averaged over thousands of years. This averaging can reduce peak carbon dioxide concentrations by tens of parts per million. Minimum concentrations are increased by similar amounts. If we extend the peaks and troughs of the carbon dioxide record by such amounts the “L” value (equation V2) derived by Westerlund fits the data better than the “best fit” value. This observation tends to support Westerlund’s analysis.

3) Gas diffusion modifies derived temperature. The differences between cabon dioxide age and ice age at a given depth confirm that there is considerable gas diffusion within the snow/ice layers. At Antarctic ice temperatures, the concentration of water vapour in the gas phase is of order 0.1%. The gas is at the same temperature as the ice. Thus, the water vapour will sublime from the gas to the solid and back to the gas phase. In this way, the surfaces of passageways within the ice will be coated with ice that differs in age from the bulk of the ice by thousands of years. These surface layers will counter-diffuse with ice in the bulk of the snow/gas. The specific surface of snow is extremely high for fresh snow, but decreases with age and compaction. In this way, the isotopic composition of snow/ice will change as the snow ages and gas diffuses in and out of the passageways in the snow. The effect is obviously at least an order of magnitude less than the effect on gas composition. However, it seems large enough to make the mechanism a plausible way of accounting for the results obtained in Figure 10. The snow becomes older and more compact as we go deeper into the ice core. Thus, we would expect the gas diffusion effect to become less as the snow gets older. The consequence for Figure 10 is that we would expect the line to become flatter as we go back in time. The number and precision of the points is insufficient to test this hypothesis.

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