Journal of Applied Physiology
Vol. 81, No. 4, pp. 1850-1854, October 1996
ENVIRONMENT

Prediction of barometric pressures at high altitudes with the use of model atmospheres

John B. West

Department of Medicine, University of California, San Diego, La Jolla, California 92093-0623

ABSTRACT

INTRODUCTION

METHODS

RESULTS

DISCUSSION

ACKNOWLEDGEMENTS

FOOTNOTES

REFERENCES

ABSTRACT

West, John B. Prediction of barometric pressures at high altitudes with the use of model atmospheres. J. Appl. Physiol. 81(4): 1850-1854, 1996.It would be valuable to have model atmospheres that allow barometric pressures (PB) to be predicted at high altitudes. Attempts to do this in the past using the International Civil Aviation Organization or United States Standard Atmosphere model have brought such models into disrepute because the predicted pressures at high altitudes are usually much too low. However, other model atmospheres have been developed by geophysicists. The critical variable is the change of air temperature with altitude, and, therefore, model atmospheres have been constructed for different latitudes and seasons of the year. These different models give a large range of pressures at a given altitude. For example, the maximum difference of pressure at an altitude of 9 km is from 206 to 248 Torr, i.e., ~20%. However, the mean of the model atmospheres for latitude of 15° (in all seasons) and 30° (in the summer) predicts PB at many locations of interest at high altitude very well, with predictions within 1%. The equation is PB (Torr) = exp (6.63268  0.1112 h  0.00149 h²), where h is the altitude in kilometers. The predictions are good because many high mountain sites are within 30° of the equator and also many studies are made during the summer. Other models should be used for latitudes of 45° and above. Model atmospheres have considerable value in predicting PB at high altitude if proper account is taken of latitude and season of the year.

Standard Atmosphere; high-altitude physiology; maximal oxygen consumption; acclimatization; air temperature; latitude; season of the year

INTRODUCTION

THERE ARE MANY SITUATIONS in high-altitude physiology where it would be useful to be able to predict the barometric pressure from the altitude. Examples include calculating the effect of changing the altitude on the inspired PO₂ and the efficacy of oxygen enrichment of room air at different altitudes.

Unfortunately, in the past, attempts to predict barometric pressure from altitude have fallen into disrepute, mainly because of the inappropriate use of the International Civil Aviation Organization or United States Standard Atmosphere (9, 13). Many early physiologists determined the correct relationship between the barometric pressure and altitudes on mountains from numerous measurements (2, 4, 5, 10, 11, 29). However, with the introduction of the Standard Atmosphere by the aviation industry, a number of physiologists erroneously employed this to predict barometric pressures. The most obvious errors occurred at extreme altitudes such as the summit of Mt. Everest (6, 7, 16, 19), where the predicted pressure was ~17 Torr lower than the true value. This difference almost certainly explains why Mt. Everest can be climbed without supplementary oxygen (23). However, the use of the Standard Atmosphere to predict barometric pressures also resulted in significant errors at lower altitudes (15, 27).

Although the Standard Atmosphere is generally inapplicable in high-altitude physiology, there are a number of other model atmospheres that have been developed by geophysicists. These are generally unknown in high-altitude physiology but could be valuable when used correctly. This article discusses some model atmospheres and shows how the use of the appropriate model for the latitude and season can give accurate predictions for the barometric pressure-altitude relationship, particularly at latitudes within 30° of the equator, where many of the highest mountains are located.

METHODS

The reduction in barometric pressure with increasing altitude is based on simple physical principles. There are two basic equations. The hydrostatic equation is

(1)

where P is pressure,  is density, z is height, and g is the acceleration of gravity at height z.

The second equation is the ideal gas law in the form

(2)

where R is the universal gas constant, T is the absolute temperature, and M is the molecular weight of air, which is constant at all altitudes of interest to us.

From these relationships, it follows that barometric pressure would decrease exponentially as altitude increased if the air temperature remained constant. However, air temperature falls with increasing altitude, and the sea level temperature and rate of fall of temperature with altitude vary according to latitude, season of the year, and atmospheric disturbances such as storms. These temperature differences are the only reasons why the pressure-altitude relationship differs between different regions of the Earth.

The Standard Atmosphere assumes a sea-level pressure of 760 Torr (1,013 mbar), sea-level temperature of 15°C, and linear decrease in temperature with altitude (lapse rate) of 6.5°C/km up to an altitude of 11 km. These conditions may never actually exist in any part of the world. However, the Standard Atmosphere was introduced as a reference atmosphere for the aviation industry, which needed some sort of average for calibrating altimeters, low-pressure chambers, and assessing the performance of aircraft under well-defined conditions. The Standard Atmosphere was never meant to be used to predict the actual barometric pressure at a particular location and was in fact developed by people who clearly recognized that there would be substantial local variations caused by latitude and seasons of the year.

Because of the dominant effect of air temperature, model atmospheres have been developed for latitudes of 15, 30, 60, and 75°N because the air temperature decreases with increasing latitude. Model atmospheres in the southern hemisphere are essentially the same if the latitude is taken into account. Because the season of the year also affects air temperature, models have been constructed for each latitude for midsummer (July in the northern hemisphere) and midwinter (January) and sometimes for other months as well. Additional model atmospheres have been developed for both unusually cold and warm conditions in the winter at the higher latitudes.

To derive a model atmosphere, air temperatures at the various altitudes for different latitudes and seasons need to be obtained, e.g., from radio-sonde balloons. These are released every 12 h on every day of the year from many meteorological stations around the world. Figure 1 shows an example of a temperature profile obtained in this way.

RESULTS

Examples of model atmospheres. Table 1 gives the barometric pressures of a number of model atmospheres at altitudes of 0-10 km for every 15° of latitude north in midsummer (July) and midwinter (January). The original data are given in millibars (21) and have been converted to Torr by multiplying by 0.750. The values for southern latitudes are essentially the same but of course the seasons are reversed. The range of 0-10 km covers all the altitudes of interest to the high-altitude physiologist.

Table 1. Barometric pressure-altitude relationships in some model atmospheres Altitude, km Standard Atmosphere 15°N 30°N 45°N 60°N 75°N Jul Jan Jul Jan Jul Jan Jul Jan 0 760 760 760 766 760 764 758 760 759 760 1 674 678 678 680 677 673 672 666 671 663 2 596 604 604 603 601 592 595 583 593 579 3 526 536 536 533 533 520 525 510 523 505 4 462 475 475 471 471 456 462 445 457 438 5 405 419 420 414 415 398 406 387 403 379 6 354 369 370 363 365 347 356 335 351 327 7 308 324 325 317 320 301 310 289 306 281 8 267 283 284 276 279 260 269 248 265 241 9 231 247 248 239 243 224 233 212 229 206 10 199 214 215 207 211 193 201 181 197 176 Values are in Torr.

Other model atmospheres are also available, especially for unusually warm and cold winters for 60 and 75°N latitude. However, the differences in the barometric pressures between these model atmospheres and those shown for midwinter (January) in Table 1 are very small.

The model atmospheres used by geophysicists extend up to much greater altitudes, typically 90 km (Fig. 1). This takes them into regions of the atmosphere known as the stratosphere, mesosphere, and thermosphere. However, the troposphere is the only region of importance in high-altitude physiology. The tropopause that separates the troposphere from the stratosphere has an altitude of ~9 km at the poles and 19 km at the equator and is the region of the atmosphere where the temperature falls predictably at ~6.5°C for every 1 km of ascent.

Figure 2 shows the barometric pressure-altitude relationship for the Standard Atmosphere and for the two extreme supplemental model atmospheres with the highest and lowest pressures at a given altitude. The highest barometric pressures at any altitude are near the equator, and the top line shows the means of the models for 15°N for all seasons of the year and for the latitude of 30°N in midsummer (July). The pressures for these two models generally agree within 1 Torr (Table 2). Note that the pressures considerably exceed those of the Standard Atmosphere. At an altitude of 3 km, the difference is 10 Torr, whereas at an altitude of 9 km, the difference increases to 17 Torr. At these very high altitudes, this is a very significant difference from a physiological point of view; for example, the higher pressure results in significantly higher maximal oxygen consumptions (23).

Table 2. Barometric pressure-altitude relationships for two model atmospheres, their means, and a quadratic equation of best fit Altitude, km 15°N, All Months 30°N, July Mean Values From Equation* Difference 0 759.9 760.1 760.0 759.2 0.8 1 678.1 678.5 678.3 678.3 0.0 2 603.6 603.8 603.7 604.2  0.5 3 536.1 536.1 536.1 536.6  0.5 4 474.9 474.8 474.9 475.2  0.3 5 419.5 419.5 419.5 419.5 0.0 6 369.3 369.7 369.5 369.3 0.2 7 324.1 324.8 324.4 324.1 0.3 8 283.4 284.4 283.9 283.6 0.3 9 247.0 248.0 247.5 247.4 0.1 10 214.4 215.5 214.9 215.2  0.3 Values are in Torr. * Equation is barometric pressure = exp (6.63268  0.1112 h  0.00149 h2), where h is altitude in kilometers.

The bottom line in Fig. 2 shows the model with the lowest barometric pressures; it can be seen that the differences in pressures between the two supplemental models are enormous. At an altitude of 3 km, the difference is 31 Torr, and this increases to 42 Torr at an altitude of 9 km. These very large variations reinforce the necessity of using the correct model atmosphere to predict pressure.

Figure 2 also shows barometric pressure-altitude relationships at various sites of interest in high-altitude physiology where both the pressure and altitude are accurately known. These data are also shown in Table 3, which gives the location, latitude, month of the year or season, and source. Of course, a very large number of barometric pressures have been measured at high altitude (3), but in many instances, especially during high-altitude expeditions, the altitude is not accurately known. The measurements shown in Table 3 were selected using this criterion and were also selected to cover the large altitude range in Fig. 2.

Table 3. Measured and predicted barometric pressures at some locations where altitude is accurately known Country Location Latitude Month/ Season Altitude, m Measured Pressure,* Torr Predicted Pressure, Torr Difference, %  Ref. Nos. Chile Collahuasi/Rosario 21°S Oct 4,680 438 437 0.2 Personal measurement Chile Aucanquilcha 21°S Nov 5,950 372 372 0 24 France Vallot observatory 46°N Jul 4,350 452 455 0.7 18 Italy Capanna Margherita 46°N Jul/Aug 4,559 440 443 0.7 12 Nepal Mt. Everest 28°N Sept/Oct 5,400 400 399 0.3 27 Nepal Mt. Everest 28°N Oct 8,050 284 282 0.7 27 Nepal Mt. Everest 28°N Oct 8,848 253 253 0 27 Peru Cerro de Pasco 11°S Dec 4,300 458 458 0 1, 20 Peru Morococha 11°S All year 4,540 446 444 0.4 8 Tibet/China Lhasa 29°N Oct/Nov 3,658 493 495 0.4 28 US Crooked Creek, CA 38°N Summer 3,094 530 531 0.2 14 US Barcroft lab, CA 38°N Summer 3,810 483 486 0.6 14 US Pike's Peak, CO 39°N Summer 4,300 462 458 0.9 17 US White Mountain Summit, CA 38°N Summer 4,342 455 456 0.2 22 * Variations of 2-3 Torr occur around these values because of weather. Predicted from model atmosphere for 15°N for all seasons and 30°N in July. Additional data provided by personal communications.

It is remarkable that most of the measurements listed in Table 3 fall close to the top line in Fig. 2. In other words, it turns out that most of the studies at high altitude give barometric pressures at the extreme top of the large range, which is predicted by model atmospheres at any given altitude. There are basically two reasons for this. One is that many of the high mountains of the world, including those with high mining sites, are within 30° latitude of the equator (26). Exceptions are the European Alps, parts of Tibet, and parts of the Colorado Rocky Mountains and Sierra Nevada in the United States. The other reason for the high pressures is that most expeditions and many scientific studies at high altitude are carried out in the summer when the pressures are highest.

Figure 3 shows the barometric pressure-altitude relationships for the four model atmospheres of most interest in high-altitude physiology. The uppermost line labeled 1 is the same as the top line in Fig. 2 and is the mean for the supplemental model at 15°N for all seasons and at 30°N in July (midsummer) (Table 2). The line labeled 3 is for 30°N in January (midwinter), and it can be seen that although the two lines are almost superimposed at an altitude of 3 km, there is an appreciable difference at an altitude of 9 km, where the pressure difference is 9 Torr. This is one reason why climbing Mt. Everest without supplementary oxygen in midwinter is much more difficult than in midsummer, quite apart from the lower temperatures. The resulting reduction of 9 Torr in barometric pressure will appreciably reduce maximal oxygen consumption (23). Only one successful summit ascent without supplementary oxygen has been made in the winter, and this was on the first day of winter, December 22, 1987 (25).

Line 2 in Fig. 3 is for 45°N in July (midsummer). Note that the pressures are slightly higher than those for line 3, indicating that season of the year is more important than latitude over this range of latitudes. This point is emphasized even more by line 4, which is for 45°N in January (midwinter). Note the relatively large differences in barometric pressure as a result of the change from midsummer to midwinter at 45°N. At an altitude of 3 km, the pressure is 13 Torr lower, and the difference increases to 19 Torr at an altitude of 9 km.

Figure 4 shows the variation of barometric pressure with latitude in both summer and winter according to the model atmospheres. It can be seen that at the surface of the Earth, there is very little variation, although there is a slight increase in barometric pressure between latitudes of 30 and 50°N in the winter. With increasing altitudes, the equatorial bulge of pressures becomes more prominent and is most marked at the highest altitude of 10 km.

Predicting barometric pressure pressure from a model atmosphere. In both Figs. 2 and 3, it can be seen that the lines are very nearly straight when the logarithm of pressure is plotted against altitude. Actually the lines are slightly convex upward, i.e., the slope becomes increasingly negative at high altitudes. However, a quadratic equation using the relationship between logarithm of pressure and altitude gives minimal errors over a large range of altitudes, as shown in Table 2. The equation is barometric pressure = exp (6.63268  0.1112 h  0.00149 h²), where h is altitude in kilometers and exp refers to the natural logarithm.

This equation was tested on the 14 locations listed in Table 3. The column labeled "difference" shows the amount by which the predicted pressure differs from the measured pressure. All the differences were <1%. Note that the two values in the European Alps at the highest latitude of 46°N both gave predicted pressures that were too high by 3 Torr when the mean model atmosphere for 15°N for all seasons and 30°N in July was used. However, when the correct model for 45°N in July was employed, the differences for both locations decreased to 1 Torr.

DISCUSSION

The main conclusion of this study is that the model atmospheres prepared by the geophysicists can be of considerable value in high-altitude physiology for predicting barometric pressure from altitude if they are used correctly. The key is to choose the model for the correct latitude and season. When this is done, the models have considerable predictive value as demonstrated by the results shown in Table 3 and the data points near the top line in Fig. 2.

The main reason why model atmospheres fell into disrepute was the inappropriate use the Standard Atmosphere. As pointed out earlier, the Standard Atmosphere is a compromise that does not fit any latitude or season very well and was never intended to be used to predict the barometric pressure at any particular location. However, the fact that the Standard Atmosphere has poor predictive value should not discredit other model atmospheres when they are used correctly.

As Figure 2 shows, it is remarkable that many of the barometric pressures measured at high altitude conform to one pressure-altitude relationship, making this model particularly useful. This is the mean of the models for a latitude of 15°N for all seasons and 30°N for midsummer. As Fig. 3 shows, the pressure-altitude relationship for the considerably higher latitude of 45°N is also close to the summer (line 2). The relationship for 30°N in midwinter is also close, with the pressures a little lower. When we get to midwinter at a latitude of 45°N, the barometric pressures start to deviate considerably more. However, note that up to altitudes of 5,000 m, which covers most of the commercial activities such as mines and many scientific activities such as telescope sites, the maximum difference in barometric pressure for lines 1-3 is 6 Torr. This is the maximum amount of variation of barometric pressure with season and latitude for all seasons up to a latitude of 30°N, and even up to 45°N in the summer (but not the winter).

As noted in Table 3, weather variations will cause small alterations in barometric pressure that are not accurately predictable. Even if the pressure alteration due to weather at a lower altitude is accurately known, the effect on the pressure at high altitude depends on the rate of fall of temperature in the column of air, and this is usually not available. Nevertheless, the relationships between barometric pressure and altitude presented here have considerable value in solving problems in high-altitude physiology.

ACKNOWLEDGEMENTS

This work was supported by National Heart, Lung, and Blood Institute Grant R01-HL-46910.

FOOTNOTES

Address for reprint requests: J. B. West, Univ. of California, San Diego, Dept. of Medicine 0623-A, 9500 Gilman Drive, La Jolla, CA 92093-0623.

Received 5 February 1996; accepted in final form 21 May 1996.

REFERENCES