On the Effects of Wind on Road Performances
(This article is a combination of articles appearing in AFR for Weeks 937 and 940 and written by Ken Young.)
Some 20 years ago, I worked out a simple model that demonstrated runners could receive a significant tail wind benefit on a course whose start and finish lay more than 30% of the race distance apart. These calculations were the basis of the present USA rule that defines a course as point-to-point if the straight-line distance between the start and finish is more than 30% of the race distance, e.g., the start and finish for a 10 km road course for record purposes cannot be further apart, as the crow flies, than 3 km. The course could be out 6.5 km and back 3.5 km on the same road or it could be a U-shaped course with the ends of the "U" being 3 km apart (across the top of the U) and still be classified as a standard course, i.e., eligible for records on the same basis as a course whose start and finish lie at exactly the same point. Recently, the IAAF has decided to recognize road records and rather than relying on the 20 years of past experience with road-record keeping in the USA, has decided to re-invent the wheel. Altho they wisely adopt the 1 m/km drop limit in defining a standard (record quality) course, they propose allowing up to a 50% separation between the start and finish. This has prompted me to re-examine the technical basis of the effects of wind on race times as a function of start/finish separation and wind speed. This analysis basically considers two different expenditures of energy while running, the energy required to run through the air (air resistance or "external" energy) and all of the other expenditures of energy associated with running (call this the "internal" energy, i.e., muscular resistance, energy required for the oxygen transport system, etc.). The energy lost due to air resistance is proportional to the square of the wind speed. Altho the exact form for the internal energy as a function of running speed is not known, it is likely to be quadratic or exponential based on observations. Simply, I doubt that I could ever have run a 9.78 for 100 meters, even downhill with a 20 m/s tail-wind. My muscles simply cannot move that fast. I.e., the shape of the internal energy curve as a function of running speed must increase rapidly near the upper limit of the runner's natural speed limit and increase only slowly with increasing pace for slower speeds. As it turns out, the range of speeds resulting from this analysis is quite small (less than 1 m/s) and so the exact form of this curve is not critical to the analysis. Rather than attempt to quantify the actual energy expenditures, all that is necessary for this analysis is to have a reasonable estimate of the ratio of internal to external energy (I/E ratio) under conditions of calm (no air motion except for the movement of the runner thru the air). This ratio was obtained by trying different values of the I/E ratio for an out/back course with the same start/finish location with different wind speeds and having a pretty good idea of how much a particular wind speed would slow a runner capable of a pace of 180 seconds/km (5.55556 m/s or 30 minutes for 10 km) with no wind. E.g., a I/E ratio of 12 produces a slow-down of 11 seconds for a 2 m/s wind and a slow-down of 65 seconds for a 5 m/s wind over a distance of 10 km, running at a 30 minute 10K pace. Table 1. Slow-down for 5 m/s wind at 30 minutes for 10K for different values of the I/E ratio I/E = 12 65 seconds I/E = 15 50 seconds I/E = 20 38 seconds Experience suggests that a slow-down of 50 seconds is not unreasonable for a 5 m/s wind speed. A slow-down of 65 seconds seems slightly excessive while 38 seconds is probably an under-estimate. Hence, an I/E ratio of 15 will be used in the remainder of the analysis. The results of the analysis are not very sensitive to the exact value for this ratio. The basic calculation is to determine the (relative) energy expenditure (internal plus external) expressed as a rate, i.e., per second, required to run say 30:00 for 10K. Then by an interative process, determine the runner's speed with a given tail or head wind, that will equal that rate of energy expenditure. I.e., with a tail-wind, a runner can run faster while expending the energy at the same rate and conversely, the runner will run slower into a head-wind using energy at the same rate. Once the runner's speeds with and against the wind are determined, it is straightforward to calculate the slow-down or speed-up as a function of start and finish separation. The following table is for a 30:00 10K, showing the speed-up (negative numbers) in sec/km as a function of wind speed (in m/s) and start/finish separation (from 10% to 70%).
Table 2. The effects of wind (m/s) and S/F separation (%) on times (sec/km) wind 10% 20% 30% 40% 50% 60% 70% 1.0 -0.08 -0.28 -0.49 -0.69 -0.89 -1.09 -1.30 1.5 0.09 -0.22 -0.53 -0.84 -1.15 -1.46 -1.77 2.0 0.35 -0.05 -0.46 -0.87 -1.28 -1.69 -2.10 2.5 0.63 0.12 -0.39 -0.89 -1.40 -1.91 -2.42 3.0 1.16 0.54 -0.08 -0.69 -1.31 -1.93 -2.55 5.0 3.97 2.90 1.83 0.77 -0.30 -1.37 -2.43 What is an "acceptable" speed-up? Consider that a 1 m/km net drop is considered the maximum allowable benefit resulting from a net change in elevation between start and finish. This equates roughly to a 1 sec/km speed-up. Accordingly, the wind benefit should not exceed -1.00 sec/km and ideally should be less since a course with the maximum allowable drop and an equivalent maximum allowable S/F finish separation could produce a benefit as great as -2 sec/km. It is clear from the above table that a 50% start/finish separation can produce benefits up to -1.4 sec/km, i.e., a 50% S/F separation can produce a greater benefit than a 1 m/km net drop. These results are relatively insensitive to the runner's pace. For a pace of 33:20 for 10K, the maximum benefit at 40% S/F separation is -0.91 sec/km and at 50% S/F separation is -1.54 sec/km. Hence, a 50% S/F separation is not recommended for any pace. There are a number of reasons why one should be conservative in dealing with the effects of wind on times, i.e., a 30% S/F separation would be advised rather than a 40% separation. Of all the meteorological parameters that are commonly recorded, wind speed varies the most on the smallest scales both in time and space. Consider that many road races are held in the morning when wind speeds are normally increasing. A course that is out-and-back with the out-leg into the wind will experience a greater benefit than the above table would suggest since the head-wind on the out-leg would be weaker than the tail-wind on the return. For example, consider a course with a 30% S/F separation and the out-leg experiencing a 1 m/s head-wind while the return leg experiences a 2 m/s tail-wind. Given a runner capable of a 33:20 for 10K under calm conditions, the out-time would be 710 seconds while the return time would be 1274.5 seconds (3 km out and 7 km return) for a total time of 33:04.5 or a speed-up of 15.5 seconds. This is -1.55 sec/km. Thus, under these conditions, even a 30% S/F separation is excessive. A 50% S/F separation would yield a speed-up of 22.5 seconds or -2.25 sec/km.
Now consider a "U-shaped" 10K course in which the "U" consists of three straight sections such as would be obtained as on a rectangular road grid. Make the "out" and "back" legs equal to 3.5 km and the "base" leg equal to 3.0 km (S/F separation = 30%). Assume that an "ideal" wind blows parallel to the base section of the "U" and in the direction the runners run (an "ideal" wind would produce no net effect on the "out" and "back" legs but would be a net tail wind for the "base" leg). With a 2.5 m/s tail wind, a runner would be able to run 12 cm/sec faster, producing an advantage of -14.1 seconds or -1.41 sec/km. This is notably more than the -1.0 sec/km limit (equivalent to a 1 m/km drop). It is also much greater than the -0.39 sec/km given in the previous ADR for a straight out-and-back course. An "ideal" wind is one in which the direction and speed are invariant. In the real world, the interaction between the wind and the surface produces eddies that result in changes in the direction and speed on a scale of a few seconds. Thus, a cross-wind that produces no net component in the direction of the runner would still have the effect of slowing the runner. For example, if half of the time the cross-wind had a head-wind component of 1 m/s (representing roughly a 20 degree shift in the wind direction with the same wind speed), this would slow the runner by 7 cm/sec. If the other half of the time, the wind had a tail-wind component of 1 m/s, this would propel the runner faster by 5.5 cm/sec. The net effect would be to slow the runner by 1.5 cm/sec on both the "out" and "back" legs, slowing the runner by a total of 4.6 seconds. This would reduce the advantage from -14.1 seconds to -9.5 seconds (-0.95 sec/km), i.e., barely within the acceptable range. With a 5 m/s wind, the advantage would be -1.13 sec/km. Now consider a course with a 50% S/F separation. The "out" and "back" legs are 2.5 km each and the "base" leg is 5.0 km. The advantage on the base leg is -23.4 seconds. The compensating effects on the "out" and "back" legs are +3.4 seconds. The net advantage is -20.0 seconds or -2.0 sec/km, twice the advantage for a 30% S/F separation in this configuration and twice what is deemed allowable. What if the wind speed is 5 m/s rather than 2.5 m/s? The advantage with a 30% S/F separation would be -11.3 seconds (-1.13 sec/km). For a 50% S/F separation, the advantage is -26.2 seconds or -2.62 sec/km. The 30% separation shows a 19% increase in the advantage for doubling the wind speed while the 50% separation shows a 31% increase. One can conceive of even more advantageous course configurations. Consider a "V-shaped" course with legs of equal length and a wind that blows parallel to a (hypothetical) line connecting the ends of the "V", i.e., that line along which the S/F separation would be measured. A 30% S/F separation would produce a "quartering" tail-wind at an angle of 17 degrees whereas a 50% separation would produce a "quartering" tail-wind at an angle of 30 degrees. Again, with an "ideal" wind with a speed of 2.5 m/s, the 30% S/F separation would yield a net tail wind of 0.75 m/s whereas the 50% S/F separation would yield a net tail wind of 1.25 m/s. The difference from the previous example is that the tail wind component is constant over the entire distance. For the 30% S/F separation, the runner would be able to run 4 cm/sec faster, giving a time advantage of -15.9 seconds or -1.59 sec/km. The 50% S/F separation would allow the runner to run 6.7 cm/sec faster, giving an advantage of -26.4 seconds (-2.64 sec/km). Again, fluctuations in the wind direction and speed would act to reduce these advantages but both would remain above the -1 sec/km allowable. Hence, it is possible to conceive of configurations that would render even the 30% S/F separation excessive. The 50% S/F separation is worse, in all cases. If one were to set up such a course, the ideal location would be near a large body of water. In that way, one could take advantage of the sea breeze which is quite reliable in its direction. Other situations could take advantage of katabatic (downslope) and anabatic (upslope) winds driven by elevation-induced temperature gradients. The 30% S/F separation rule is already pushing the limit of what should be allowed. Going to a 50% S/F separation is quite unwise.