Duncan Steel: Climate Change And The Shape Of The Earth

This is what scientists are supposed to do, find inconsistencies, point them out, and put them right. And so I gave this a little thought.A favourite teaser in TV and radio quiz programs is to ask contestants which nation contains the point furthest from the Earth’s centre. Most people think the questioner is referring to Mount Everest, but those with a little more knowledge imagine that any difficulty in giving a definitive answer is due to the peak of that mountain forming part of the border between Nepal and Tibet, with the additional complication that the latter is claimed by China. All are wrong.In fact the correct answer is Ecuador, and the mountain in question is an inactive volcano called Chimborazo. Not far behind is a peak in Peru named Huascarán, which is actually taller than Chimborazo, compared to mean sea level; but the Ecuadorian volcano wins by dint of being closer to the equator. Mount Everest is nowhere near these two, in terms of distance from the centre of the Earth, being more than two kilometres below.The above may seem confusing, but it’s quite simple, really. Our planet is not spherical in shape, with the difference between the equatorial and polar radii being more than 21 kilometres. Mountains nearer the equator may have lower heights measured above mean sea level but the sea, like the planet as a whole, has an overall shape best described as a geometrical form known as an oblate spheroid.The World Geodetic System defines a standard geoid or Earth shape for mapping and navigation that is referenced as WGS 84. This is what is used, for example, by the Global Positioning System (GPS), by which many of us navigate when driving. This geoid is an ellipsoid of revolution such that the equator is a circle with radius 6378.137 km, but polar flattening means that the distance from the Earth’s centre to either of the poles is only 6356.752 km.Of course these figures represent only a convenient, usable model, and the mountains mentioned earlier (along with Slartibartfast’s other “lovely crinkly edges”) mean that we all recognise it’s only an approximation which may or may not be good enough, depending on what you are wanting to do, precisely. But the Earth is most definitely not a sphere.What has this to do with the climate? you might ask. My answer follows.Matters of SignificanceWhen constructing any physical model, a scientist must ask him or herself: what is significant? Here I am concerned with climate, and it is obvious that mountains are significant from the perspective of the local climate. For example, I currently live in Canberra, and at an altitude of about 600 metres we have a pseudo-mountainous climate (affected, obviously, by other factors, such as our distance from the ocean) which, at this time of year, leads to clear and cold nights (temperatures typically a few degrees below freezing) but fine, warm days.In the past I have lived at altitude 1,700 metres in Boulder, Colorado, on the slopes of the Front Range of the Rocky Mountains, where night time temperatures are far lower during the winter, resulting sometimes in half-a-metre of snow falling overnight; but in the morning the warm, dry Chinook (or foehn) wind, resulting from airflow over the mountain range to the west, can make that snow disappear in a few hours.Mountains can affect your weather significantly even though you live at sea level. Another city in which I have lived is Christchurch, New Zealand, right next to the ocean, where the summer climate is dominated by a similar foehn wind (locally called the Nor’wester) that is the result of the Southern Alps, 100 km away across the Canterbury Plains. This desiccating air flow means that in Christchurch you need to water your garden almost every summer afternoon even though the temperature seldom climbs above 25−28°C.In this article and my preceding piece, though, I am concerned with climate on a larger scale: globally, or at least at certain latitude bands without worrying about local variations due to differences in compositions (e.g. sea versus land, rock versus soil), albedos (e.g. deserts of ice or sand versus forests or fields), circulation patterns and their seasonal variations (Christchurch may be warm and dry in the summer, but in the winter the prevailing damp and chilly breeze comes from the south, off the Antarctic Ocean), and so on.In short, I am concerned here only with the flux of sunlight and how it varies during the year, without paying any attention to how our planet then absorbs and disperses the incoming energy. I will not worry, then, about mountains and the other crinkly bits. The matter of significance to me here is only the gross shape of the Earth: the geoid mentioned earlier.The Flux of SunlightIn my preceding article I showed how the incoming flux of sunlight on different days of the year and at different latitudes has changed between 1750 (i.e. before the rapid increase in atmospheric CO2 levels due to human activities) and the present, resulting in an expectation that the start of spring should be coming earlier (all other things being equal) due exclusively to variations in Earth’s heliocentric orbit. The dominant shifting parameter in that case was the ecliptic longitude at which perihelion occurs; in simpler language, the date when Earth is closest to the Sun as compared to the dates of the equinoxes and solstices.Let me caution again here that any use of ‘dates’ needs to be a matter where the investigator is very careful, and simple calendar dates (on any calendar) should not be trusted without diligent, detailed consideration. For reference, what I have done in my calculations is this: I have taken a ‘year’ to be defined as a Gaussian year (365.256898 days long), which is tantamount to taking the semi-major axis of Earth’s orbit to be precisely one astronomical unit. When I label a year as ‘1750’, ‘2000’ or ‘1246’ this does not mean a calendar year (by any definition), but rather a Gaussian year at around that epoch (variously termed ‘AD’ or ‘CE’ [Common Era] year counts).What I then do in my computations is to split that model year into one-thousand equal divisions in time, derive the instant at which the Sun ascends through the equatorial plane (defining the vernal equinox and thus the zero of ecliptic longitude), and place that instant at the start of the 80th day in that year. I will leave it to the reader to consider how that compares to the ecclesiastical equinox as defined in the Gregorian calendar (i.e. the whole of March 21st), or the instant of the astronomically-defined vernal equinox (which shifts over a range of 53 hours contained within March 19th–21st between 1903 and 2096 CE using the leap-year cycle copied from the Gregorian calendar).Having done that − and input to my software the longitude of perihelion, and the appropriate orbital eccentricity, and the obliquity of the ecliptic, for the epoch in question – I can then compute the flux of sunlight at the Earth (assuming that the solar irradiance is unchanging), and also its latitudinal dependence, all as a function of the time of year. My time steps are the one-thousandths of a Gaussian year mentioned above, but I label the abscissae in my plots as the Day of Year based on the above: that is, the vernal equinox occurring at the start of day 80.This may sound complicated, but actually it is quite simple compared to much scientific research. Any reasonably-competent final-year undergraduate in the physical or mathematical sciences should be able to duplicate my calculations. Repeatability is a vital facet of science, and I would hope that others will soon repeat and so confirm my results.Having explained all that, I can now turn to the new matter of significance which is the core point of my present article. This is that although one can calculate fluxes (in units of watts per square metre) of sunlight at the Earth, and above any particular latitude, the power (in watts) available obviously depends on the number of square metres involved, and therefore the size and shape of our planet as defined by the geoid adopted. (Also obvious is the fact that the overall energy delivery [i.e. in joules] will depend on the exposure time to any particular sunlight power, and this I will deal with in a future article; please stay tuned to this station.)The Shape of the EarthPeople who oppose any widely-held belief are often pejoratively-termed flat-earthers. Indeed the history of the myth that until the past few centuries many believed the planet to be flat, and that it was possible to sail off the edge, makes interesting reading. In fact it was obvious to the ancients that the Earth is round from a variety of simple observations, such as the horizon being inarguably curved when you look from an elevated position such as the top of a good-sized hill, or the shape of the terrestrial shadow cast on the Moon during a lunar eclipse.I would propose a new term of derision: spherical-earthers, for those who imagine our planet to be shaped like a sphere, and perhaps use that in their evaluations of the total annual energy delivered to our planet by the Sun. Yes, the Earth is round, but it is not spherical, else the summit of Mount Everest would indeed be the point on the surface that is furthest from the planet’s centre.It is easy to presume that the Earth is a sphere, when so many terms used in the natural sciences end with that word: hemisphere, atmosphere, troposphere, stratosphere, mesosphere (both above and below ground), ionosphere, thermosphere, exosphere, asthenosphere, lithosphere, biosphere, cryosphere, hydrosphere, to cite but a handful. Anyone scanning the various reports of the Intergovernmental Panel on Climate Change (IPCC) using the search term ‘sphere’ will find several of those terms turning up again and again.Turning to the IPCC’s Fourth Assessment Report (2007), my search of the section headed Frequently Asked Questions 1.1: What Factors Determine Earth’s Climate? in the part of the report produced by Working Group I (which dealt with The Physical Science Basis) turned up a paragraph beginning with this sentence:“Because the Earth is a sphere, more solar energy arrives for a given surface area in the tropics than at higher latitudes, where sunlight strikes the atmosphere at a lower angle.”As a passing statement suitable for broad consumption that is fair enough, although it is not strictly accurate, as the reader will recognise. On the other hand, I was unable to find any other mention of the true shape of the planet elsewhere in this part of the report (although I may have missed it, in which case mea culpa).The above piqued my interest: this is what scientists are supposed to do, find inconsistencies, point them out, and put them right. And so I gave this a little thought.The eccentricity that defines the elliptical shape of Earth’s heliocentric orbit is currently about 0.0167, and we know that this affects the climate a great deal. By comparison, the eccentricity of an elliptical cross-section of the WGS 84 geoid with the polar axis representing the semi-minor axis and the equatorial radius the semi-major axis is a little more than 0.0818, nearly five times higher. Flags started metaphorically waving in my head: surely one cannot neglect the non-sphericity of our planet?Defining ParametersIn my previous article I wrote that the total energy intercepted by the Earth over the course of a year is 5.4971 × 1024 joules, from my computations based on a Solar Constant of 1367 watts per square metre. I also stated that I got the same value for 1750 and 2000, which is true, if considering figures rounded to the fourth decimal place as above.In fact there are slightly different values (5.497134 and 5.497121) derived for those two epochs, as is to be expected, with two distinct contributions producing the tiny difference. One is the change in the obliquity or tilt of our spin axis (23.4723 degrees in 1750, 23.4398 in 2000). The other is the change in orbital eccentricity (e=0.016804 in 1750, 0.016704 in 2000): the annual solar energy received varies as 1/√(1 – e2) and was therefore slightly greater in 1750.For the calculations presented in this article I will take the earlier era to be used in the computations to be 1246, which is the last epoch when the longitude of perihelion was aligned with any of the equinoxes or solstices: it is therefore a more interesting era to use in terms of investigating how the Earth’s orbit and shape affects the climate, rather than the artificial 1750 epoch. (An apology to geologists: I recognise that my loose, alternating usage of the terms era and epoch must be driving you mad, but it is astronomy that is the subject here.)Around 1246 perihelion passage occurred at the same longitude as the winter/December solstice, resulting in the Northern Hemisphere having autumns and winters the same length as each other; and similarly for spring and summer, the latter pair being longer than the former pair; and in the Southern Hemisphere the effect was the reverse. In 1246 the obliquity was 23.5376 degrees, and the eccentricity 0.017003. The longitude of perihelion was 270 degrees, the same as the longitude of the winter solstice (as aforesaid) and 90 degrees ahead of the vernal equinox. Just for completeness, the value I calculated for the total annual insolation in 1246 was 5.497161 × 1024 joules, very slightly greater than the values for 2000 and 1750, as to be expected on the basis of the obliquity and eccentricity both being slightly higher 700+ years ago. The Effect of non-SphericityMy computational model includes the effect of the Earth being non-spherical. It is straightforward to visualise what this means. If you imagine yourself to be located on the Sun and looking towards the Earth at the time of either of the equinoxes, you would see an elliptically-shaped planet with equatorial and polar radii of a=6378.137 and b=6356.752 kilometres respectively. At that time (at either equinox) the Earth presents its minimum cross-section to the Sun, and its value is easily derived; it’s simply πab.Now think about the solstices. At those instants one or other of the Earth’s poles in tipped towards the Sun by the maximum angle possible, that angle being the obliquity. To consider this in another way, the Sun has reached its maximum declination (either north or south). Viewed from the Sun the semi-major axis (a) of the elliptical shape of the Earth is still that equatorial radius, but the semi-minor axis (b) is greater than the polar radius: it is now the distance from the centre of the Earth to the Arctic (or Antarctic) Circle. That distance is about 6360 km, but obviously it changes slightly with the obliquity (i.e. over centuries). For the interested and pedantic reader, the value was 6360.148 km in 1246, 6360.130 km in 1750, and 6360.121 km in 2000, a decrease of 27 metres between 1246 and 2000.In view of the above, we anticipate that the cross-sectional area of the Earth presented to the Sun will vary in a predictable way during each year. In the graph below I show my results for the epochs 1246 and 2000, plotted as the fractional extra area at any time of year compared to the minima occurring at the equinoxes.As might be expected from the input data as stated above, the peaks of the derived curves occur at a little above 0.00053. To verify this, get out your pocket calculator and evaluate [(6360.1xy − 6356.752)/6356.752], which is the fractional change in the cross-sectional area because this scales directly as the fractional change in the semi-minor axis, b, which maximises as half the length of the line joining the Arctic and Antarctic circles while also passing through the centre of the Earth.There are other detailed points to note, though. First, the peaks for 1246 (red dashed line) are very slightly higher than for 2000 (blue continuous line). This is due to the obliquity having decreased by almost 0.1 degrees (just over 0.4 percent) between 1246 and the present.Second, because the computations for the two epochs were both registered in the same way with the vernal equinox occurring at the start of day 80 in each year, the curves are in phase at that juncture, and the two plotted lines of finite thickness are essentially on top of each another between about day 0 and day 160. After that the blue line is slightly in advance (to the left of) the red line until the autumnal equinox (on day 267), following which the blue line lags (is to the right of) the red line. This is as is to be expected from the changing angular speed of the Earth in its orbit, with perihelion in 1246 occurring on day 356 and in 2000 on day 4.If you really can’t resist trying to convert those to calendar dates, then a word to the wise: be aware of the distinction between ordinal and cardinal numbers. That is, January 1st is Day-of-Year 0 (zero).Climatic SignificanceAre the curves shown in the above plot, defined solely by the shape of the Earth, of any significance for the climate? For year 2000 the variation in cross-section over the year amounts to about one part in 1,887 (i.e. 1/0.00053007), while the slightly-higher amplitude in 1246 results in a variation by about one part in 1,872 (i.e. 1/0.00053425). We might take one part in 1,880 as a suitable average measure.As discussed in my previous article, the IPCC’s Fourth Assessment Report states that “The only increase in natural forcing of any significance between 1750 and 2005 occurred in solar irradiance,” with a best-guess of 0.12 watts per square metre. Compared to the implicitly-inconstant Solar Constant of 1367 watts per square metre, that is one part in 11,400.On that basis it is clear that the shape of the Earth, and the slowly-altering orientation of its spin axis, is significant in climate change considerations. Let me not fool you, though, with sleight-of-hand: the one-part-in-1,880 I derived above is a repeated (but slowly-modulated) up-and-down variation across a year in the total cross-section of the Earth as presented to the solar flux, whereas any systematic change in solar radiation output (i.e. the 0.12 watts per square metre mentioned above) will act consistently across the whole year.The change in the amplitude of the annual variation curves (i.e. the difference seen in the peaks in the above plot) is only 0.78 percent. That’s small; but it may be significant. I cannot answer here the question of whether it is significant or not in terms of climate change, because I am considering only the astronomical-control of the incoming solar flux, and not the response of the terrestrial climate system to such a change. All I can say at this stage is that this is another parameter that is certainly changing over some centuries, and so it should be considered and accommodated in any detailed climate change modelling.Earth’s Solid AngleI will close with a further cautionary tale. One should not imagine that the intra-annual variation in terrestrial cross-section as seen from the Sun (i.e. our planet’s physical cross-section, measured in square kilometres) is the dominant parameter in terms of the total solar flux intercepted by the Earth. The variation in our heliocentric distance between perihelion and aphelion has a far larger effect, the solar flux at Earth changing by approximately seven percent during each orbit. Consider the following plot.Here I have computed the solid angle occupied by the Earth as seen from the Sun. All of the effects discussed previously (i.e. the WGS 84 geoid as a model of the Earth’s shape, and the eccentricity, obliquity and longitude of perihelion as appropriate for both 1246 and 2000) have been included.The two curves have a substantial phase difference due to the longitude of perihelion having shifted, from 270 degrees in 1246 to 282.895 degrees in 2000. The amplitudes of the two curves are also different, the major influence now being the reduction in eccentricity (from 0.017003 in 1246 to 0.016704 in 2000).For those who have not met the term previously, a steradian is the standard unit of solid angle, defined in a similar way to the radian as a unit of angle in planar geometry. One radian is the angle which is subtended by an arc of a circle of length equal to the radius of the circle; thus there are 2π radians in 360 degrees, and a radian is equivalent to just less than 57.3 degrees. The parallel definition for a steradian is that it is the solid angle subtended at the centre by an area on the surface of a sphere equal to the square of the radius (r) of that sphere; thus the solid angle of a complete sphere is 4π steradians (because its surface area is 4πr2).The plot above encompasses two factors affecting the solid angle as a function of the epoch and also the day-of-year: (A) The inverse-square (1/r2) drop-off in solar flux with Earth’s heliocentric distance, which has a range varying as [(1+e)/(1–e)]2 and so currently amounts to 6.91 percent although in 1246 the figure was 7.04 percent; and also (B) The change in the physical cross-section that the Earth presents to the Sun at different times of year, which we have seen is about one part in 1,880 or 0.053 percent and was slightly higher in 1246 than in 2000/the present.The point to take away from this is that if one is going to accommodate in any comparative climate modelling the change in Earth’s orbital eccentricity over the past millennium, then it is also necessary to include the effect imposed by the Earth’s physical shape, because the latter causes intra-annual variations (o.o53 percent) that are substantial compared to the changes in the former (a change from 7.04 percent in 1246 to 6.91 percent now, the difference between those being 0.13 percent).Finally, note that the data plotted in the above graph may be used in a direct calculation of the total power of the sunlight striking the Earth by making use of the solar luminosity, a parameter well-known to astrophysicists and generally taken to be 3.839 × 1026 watts. Selecting a sample instantaneous value from the plot of 5.7 nano-steradians, and assuming that the solar luminosity is evenly emitted over all 4π steradians, the solar power delivery to Earth is a little above 1.741 × 1017 watts at that instant.How that power is split up between different latitudes obviously must depend to some extent upon the primary factor I have been discussing in this article: the shape of the Earth. This latitudinal variation, and how it varies in time as Earth’s orbital parameters and obliquity gradually shift, will be dealt with in a future article. Dr Duncan Steel lives in Canberra, Australia. He occasionally checks his email on TMA1@grapevine.com.au