Thereby in the absence of sea level pressure differences, pressure altitude scales linearly with sea level absolute temperature.

We now approximate Equation (13b) to obtain an expression with only constants and linear functions of the arguments,

(14a)

In Equation (14a) f(T) is the fractional increment in absolute temperature on proceeding from T(0,A) to T(0,std). Similarly, f(p) is the fractional increment in sea level pressure on proceeding from p(0,A) to p(0,std). The factor (1 - f(p)) approximates 1/(1 + f(p)), a valid assumption for small f(p), and arises from the 1/p(0,std) factor in Equation (13b). As these fractions never refer to any elevation other than sea level, the "0" indices in f(T) and f(p) are suppressed.

As the cross-term in f(T) x f(p) of Equation (14a) is small, we obtain the approximate expression

(14b)

The terms in Equation (14b) are regrouped,

(14c)

In Equation (14c) H(std) is the scale height assuming a fixed temperature T(0,std), and replaces the factor -T(0,std)/L of Equations (14a,b). H(std) equals 8,435 meters (27,673 feet).

The terms in Equation (14c) have the following interpretations.

• f(T) z(A) is the shift in altitude due to a sea level temperature difference between mountain A and the Standard Atmosphere. As seen in Equation (13c), the fractional altitude shift is identical to the fractional shift in sea level absolute temperature.

• H(std) f(p) is an altitude-independent shift in pressure altitude due to a fractional shift in sea level pressure. Low sea level pressure at A results in f(p) > 0 (to attain standard pressure), and thus yields a positive pressure altitude shift. As this term is nonzero even at sea level, it is akin to "raising" or "lowering" the entire air column by the indicated amount.

• - (z(A)/) f(p) is an altitude-dependent shift in pressure altitude due to a fractional shift in sea level pressure from A to the standard value. These two latter shifts are equal apart from a factor of 1/. Hence a 1% sea level pressure shift results in only a 0.19% shift in pressure altitude as contributed by this term.
  We interpret this term as follows. With rising true altitude the temperature falls. Hence a smaller fractional change in true altitude is required to secure a given fractional change in pressure. At sea level only the first term in H(std) f(p) exists. The additional term in - (z(A)/) f(p) is the correction required to diminish the fractional altitude change to the value appropriate for the lower temperature.
  At the true altitude z(A) = H(std) the two terms cancel completely, having reached the height at which the absolute zero of temperature ensures that no altitude change at all is required to secure a given fractional pressure change. The real atmosphere's temperature rises with altitude at the tropopause - well before this mathematical fiction takes hold at roughly 44,000 meters (145,000 feet).

Using the approximate Equation (14c) rather than the exact Equation (13b), a 5% drop in extrapolated sea level temperature at mountain A relative to the Standard Atmosphere results in a 5% upshift in the Standard Atmosphere altitude, relative to A, to secure an identical pressure reading.

Similarly, a 1% drop in mean sea level pressure at mountain A relative to the Standard Atmosphere results in roughly an 84 meter (277 foot) upshift in the Standard Atmosphere altitude, relative to A, to secure an identical pressure reading. The upshift decreases linearly with true altitude at mountain A as seen above.

Variability in the sea level extrapolated temperatures and pressures at A and B are treated analogously. For example, suppose the empirically determined sea level temperature standard deviation at mountain A is ± 4° C (7 ° F). For typical values of T(0,A) this corresponds to roughly a 1.4% uncertainty in pressure altitude (cf Equation (14c)). This uncertainty increases linearly with true altitude, e.g. ± 14 meters at 1,000 meters and ± 28 meters at 2,000 meters.

Let the empirically determined sea level pressure standard deviation at mountain A be ± 8 millibars (0.23 inches Hg) - roughly a one part in 130 fractional pressure shift. Multiplying by the scale height H(std), there results a 65 meter (213 foot) uncertainty in pressure altitude. This value declines linearly with true altitude owing to the term in - (z(A)/) f(p).

Temperature and pressure variations from their mean values may be correlated at a given location. Denote by ²(T) and ²(p) the variance in pressure altitude from these two sources, respectively; and computed as the squares of the temperature and pressure-derived uncertainties in pressure altitude using Equation (14c). Denote by (T,p) the (possibly nonzero) covariance of temperature and pressure.

The total variance in pressure altitude arising from these sources is

(15)

If temperature and pressure variations are perfectly, positively correlated, then (T,p) = (T) + (p). For perfect negative correlation, (T,p) = (T) - (p).

Consider the effect of temperature on pressure altitude. For Mount McKinley meteorological records from the West Buttress Route have been provided by the National Park Service.⁶ Figure 1 contains temperature graphs from Kahiltna Base Camp at 2,200 meters (7,200 feet) for the 2003 through 2006 climbing seasons. Figure 2 contains temperature graphs from Advance Base Camp at 4,350 meters (14,300 feet) for the same periods.

Using the environmental lapse rate of 6.5° C / km (3.57° F / 1,000 feet), one extrapolates from these observed temperatures to the theoretical, sea level temperatures which would exist at Mount McKinley's base during the climbing season - and as required in the detailed calculations.

Figure 1. Kahiltna Base Camp temperature data. (Click for a detailed view)

Meteorological data from nearby population centers corroborate the estimated sea level temperatures. For Mount McKinley, data is available for both Fairbanks and Talkeetna, Alaska - two communities on opposite sides of the Alaska Range. Figure 3 contains temperature graphs from Fairbanks, Alaska (elevation 138 meters = 453 feet) for May and June of 2003 through 2006; and corresponding to the climbing season at Mount McKinley. Figure 4 contains temperature graphs for Talkeetna, Alaska (elevation 109 meters = 358 feet) with the same time periods.

Figure 2. Advance Base Camp temperature data. (Click for a detailed view)

We obtain 10.3° C (50.5° F) and 15.6° C (60° F) for Mount McKinley's theoretical sea level temperature in May and June, respectively; and 27° C (80° F) for the comparison mountains. Note the 5.3° C (9.5° F) change in one month for Mount McKinley. Values are based on the mean daily temperatures at May 15 and June 15, using the linear least-squares best fits for the eight months of available data (May and June for 2003 through 2006). Daily maxima and minima are typically 5.5° C (10° F) higher and lower than these mean values, respectively.

These mean, extrapolated sea level temperatures are substituted into Equation (13c). The fractional drop in absolute temperature at sea level between the comparison mountains and Mount McKinley yields a 5.8% and 3.8% higher pressure altitude at Mount McKinley, for a given true altitude, compared with the more southerly peaks, for May 15 and June 15, respectively.

Figure 3. Fairbanks temperature data. (Click for a detailed view)

At the 6,194 meter (20,320 foot) summit these fractions correspond to 360 meters (1,175 feet) and 240 meters (780 feet) of additional pressure altitude relative to the same true altitude on the comparison mountains, all peaks being considered during their respective optimal climbing seasons.

At the 5,250 meter (17,200 foot) high camp on the West Buttress Route these fractions correspond to 305 meters (995 feet) and 200 meters (660 feet) of additional pressure altitude relative to the same true altitude on the comparison mountains.

These true altitude differences between locations at identical pressure altitudes are approximate in that variability in temperature and sea level barometric pressure have not been accounted for. This variability is considered in the next section.

Figure 4. Talkeetna temperature data. (Click for a detailed view)

Consider the effect of latitude on pressure altitude. The Himalaya and Karakoram Ranges closely straddle 30° N; and Aconcagua lies near 33° S - roughly the latitude of San Diego, California apart from sign. We use 30 degrees north or south as representative of these mountains.

Substitution of = 30° and = 63° into Equation (12) yields a fractional shift of 0.0029 in the resulting values of g. The acceleration due to gravity at Mount McKinley is 0.29% more than at 30° north or south.

From Equation (7b) a given pressure altitude will exist at Mount McKinley at an elevation 0.29% lower compared with the more southerly mountains due to the effect of latitude.⁷ Equivalently, at a given true altitude, the pressure altitude at Mount McKinley will be 0.29% higher compared with the more southerly mountains - again due to latitude alone.

Temperature and pressure variations contribute to the observed pressure altitude at Mount McKinley and other mountains. Whereas temperature data is available for the West Buttress climbing route, pressure must be estimated based upon meteorological records from the nearby population centers of Fairbanks and Talkeetna, Alaska.

Linear least-squares analyses show no trend with day of season for the mean, sea level-corrected pressures at these communities. Mean daily pressure equals that of the Standard Atmosphere to within the current work's desired accuracy - roughly 15 or 30 meters in altitude (50 or 100 feet).

Figure 5. Fairbanks and Talkeetna 2003 pressure data. (Click for a detailed view)

An essential component of this analysis is the assumption that sea level pressure variations at Mount McKinley are similar to those at nearby population centers for which pressure data is available. Without some estimate of both sea level-corrected pressure, and of its variability, no further analysis is possible.

Figure 5 contains graphs of mean daily pressure, corrected to sea level, at both Fairbanks and Talkeetna, for May and June 2003. Figure 6 contains analogous graphs for the calendar 2006 climbing season. In both graphs the daily and weekly high and low pressure variations are nearly identical upon comparing the two communities. As Mount McKinley lies in-between these two communities, the sea level-corrected pressure there must be nearly identical.

For the detailed calculations we use for Mount McKinley the mean of the Fairbanks and Talkeetna standard deviations in pressure as pooled over the May and June months of 2003-2006. The Fairbanks and Talkeetna pressure standard deviations over these eight months are 7.59 and 7.79 millibars (0.2241 and 0.2301 inches Hg) respectively.

Figure 6. Fairbanks and Talkeetna 2006 pressure data. (Click for a detailed view)

As the mean sea level-corrected pressure at both Fairbanks and Talkneetna equals that for the Standard Atmosphere, Equation (13c) is used to compute the mean pressure altitude at various West Buttress camps on Mount McKinley, and for the summit. The more elaborate Equation (13b) would be used had a significant difference existed between the measured mean sea level-corrected pressure and the Standard Atmosphere.

Standard deviations in both mean daily temperature and mean daily pressure are used to compute the overall deviations in pressure altitudes at various camps on Mount McKinley, and for the summit, using Equations (14c,15,16).

Table 1 presents both the pressure altitudes and their standard deviations for these camps and the summit, expressed as pressure altitude minus true altitude differences. The calculations are performed using mean temperatures on five calendar days spanning the May and June climbing season.

Table 1. Mount McKinley pressure altitude / true altitude differences in meters (feet). Variabilities (after the "±" signs) correspond to one standard deviation.^a

Figure 7. Mendoza, Argentina temperature data. (Click for a detailed view)

Analogous prescriptions for computing pressure altitude are applied to other mountains and mountain ranges as described here.

Mendoza, Argentina and Santiago, Chile are two cities that lie on opposite sides of the Andean Cordillera. Hence they comprise an appropriate set for estimating pressure at Aconcagua just as do Fairbanks and Talkeetna, Alaska for Mount McKinley. Figure 7 contains temperature graphs from Mendoza, Argentina (elevation 704 meters = 2,310 feet) for December through February of four austral summers, corresponding to the Aconcagua climbing season. Figure 8 contains temperature graphs for Santiago, Chile (elevation 476 meters = 1,562 feet) for the same periods.

As the Aconcagua climbing season straddles the southern Hemisphere's summer solstice, a maximum in temperature is observed mid-season with a slightly lower mean temperature to either side - an inverted parabola. Hence both Mendoza and Santiago temperature data are better fit with a quadratic curve.

Figure 8. Santiago, Chile temperature data. (Click for a detailed view)

The Mendoza data exhibits an unusually large, negative pressure-temperature cross-correlation (-0.6430). The cross-correlation is computed using observed deviations from the quadratic least-squares best fits to both pressure and temperature, i.e. it is not computed using deviations from the mean pressure and mean temperature as averaged over the entire season. Nevertheless, as seen in Figure 9 a simple graph of pressure versus temperature shows how striking this correlation is, even though the graph's production did not account for the seasonal-dependence of mean pressure and temperature.

Figure 9. Mendoza, Argentina pressure-temperature cross-correlation. (Click for a detailed view)

The Santiago data exhibits unusually small departures in both pressure and temperature (cf Figure 8). The coastal location, with consequent moderating influence of the ocean on climate, explains this result.

Table 2 presents both the pressure altitudes and their standard deviations for various Aconcagua camps and the summit, expressed as pressure altitude minus true altitude differences. The calculations are performed using mean temperatures on five calendar days covering the December through February climbing season, and are based on the Mendoza, Argentina and Santiago, Chile datasets.

Table 2. Aconcagua pressure altitude / true altitude differences in meters (feet). Variabilities (after the "±" signs) correspond to one standard deviation.^a

Kathmandu, Nepal meteorological data is used for estimating pressure altitudes at Mount Everest - the mountain chosen as representative of High Asia. It was desired to balance these records with analogous data from Tibet. Unfortunately daily meteorological observations are not readily available for this region.

Figure 10 contains temperature graphs from Kathmandu, Nepal (elevation 1,337 meters = 4,386 feet) for April and May of four consecutive Mount Everest climbing seasons. Figure 11 contains pressure graphs for Kathmandu over the same periods.

Figure 10. Kathmandu, Nepal temperature data. (Click for a detailed view)

The Kathmandu data exhibits a marked decrease in sea level pressure as the season progresses (cf Figure 11); and corresponds to the approaching monsoon season. The temperature data exhibits an increase in mean value as the summer solstice is approached (cf Figure 10).

The temperature and pressure are negatively correlated with a cross-correlation of -0.3467 as summed over the four seasons examined. This negative correlation does not correspond to the obvious, secular trend of diminishing pressures and rising temperatures as the seasons progress. Rather, individual deviations of pressure and temperature from their respective moving averages are negatively correlated.

Figure 11. Kathmandu, Nepal pressure data. (Click for a detailed view)

Table 3 presents both the pressure altitudes and their standard deviations for various Mount Everest camps and the summit, expressed as pressure altitude minus true altitude differences. The calculations are performed using mean temperatures on five calendar days spanning the April and May climbing season, and are based on the Kathmandu, Nepal data.

In Table 1 baseline pressure altitudes, i.e. without considering their variabilities, are both higher and lower than the true altitudes for Mount McKinley. However in Tables 2 and 3 baseline pressure altitudes are always lower for both Aconcagua and High Asia (the latter represented by Mount Everest). This point is discussed at length in the next section.

There are 20 entries in Table 1 for the Mount McKinley pressure altitude, as the product of four elevations and five days spaced over the climbing season. Similarly, there are 25 entries in Table 2 for Aconcagua; and 25 in Table 3 for High Asia.

Table 3. High Asia pressure altitude / true altitude differences in meters (feet) as represented by Mount Everest. Variabilities (after the "±" signs) correspond to one standard deviation.^a

Direct tabulation of true elevation differences required for observing the same pressure on pairs of mountains would entail a table with as many entries as the pairwise products of the numbers of entries in Tables 1 through 3. For example, High Camp on Mount McKinley on May 15 and Camp Berlin on Aconcagua on January 31 would be one of 20 x 25 = 500 entries in a Mount McKinley - Aconcagua comparison chart. A different table would be required for each pairwise mountain combination, i.e. Mount McKinley - High Asia and Aconcagua - High Asia.

By presenting the results in terms of pressure altitudes in the Standard Atmosphere, one mountain at a time, this needless proliferation of print is avoided. One must, however, apply the prescription of the Section "Tabulation of Calculated Altitude Differences".

Nevertheless it is instructive to directly compare the true altitudes on pairs of mountains for which the pressure altitudes are identical. This exercise is of inherent interest to high altitude climbers and expeditions. We render it practical within this article by limiting consideration to one true altitude: the summit of Mount McKinley. In so doing the combinatorial explosion described above is obviated.

In Table 4 differences in true altitudes required to match Mount McKinley's summit pressure altitude are presented for Aconcagua as the comparison mountain. To compute these differences the prescription of the Section "Tabulation of Calculated Altitude Differences" is employed.

Table 4. Mount McKinley / Aconcagua true altitude differences in meters (feet) for matching pressures at the summit of Mount McKinley. Rows are calendar days during the Mount McKinley climbing season; and columns similarly for Aconcagua. Variabilities (after the "±" signs) correspond to one standard deviation.^a

In Table 5 differences in true altitudes required to match Mount McKinley's summit pressure altitude are presented for High Asia, using Mount Everest as the comparison mountain.

Table 5. Mount McKinley / High Asia true altitude differences in meters (feet) for matching pressures at the summit of Mount McKinley. Rows are calendar days during the Mount McKinley climbing season; and columns similarly for High Asia. Variabilities (after the "±" signs) correspond to one standard deviation.^a

The entries in Tables 4 and 5 are not simply the difference of the corresponding, relevant entries in Tables 1 through 3. Specifically, Tables 2 and 3 pertain to specific true altitudes on the comparison mountains, corresponding to established camps - and these camps are not located at the exact pressure altitudes for Mount McKinley's summit.

The current work uses a first principles approach to calculate pressure altitude. The resulting variables include both sea-level temperature and pressure; and the acceleration due to gravity.

This approach has the advantage of not introducing error arising from applying a geophysical model developed for a specific latitude/season combination; and applying it to a montane setting at a slightly different latitude and time of year.

For example, models exist for specifying the temperature/pressure profiles of the atmosphere at 30° N during January and July.¹ However neither of these months correspond to the Himalaya climbing season. As Mount Everest is normally scaled in May and October, the use of a July profile to represent conditions in May introduces error - albeit likely a slight one.

The current work prescribes a method for calculating pressure altitude without being straightjacketed into finding that geophysical model which most closely approximates the time and location of interest.

Pressure altitude is computed for various true altitudes on Mount McKinley, with analogous results obtained at Aconcagua and High Asia for comparison. The calculations are performed at two levels of theory - without and with account of variability in temperature and sea level-corrected barometric pressure.

During the May and June climbing season, Mount McKinley's summit pressure altitude declines from a mean value of 157 meters (514 feet) higher than the true altitude, to 70 meters (229 feet) lower than the true altitude (Table 1). This seasonal trend results from rising sea level temperatures as extrapolated from the detailed meteorological data (Figures 1-4). Daily temperature variations modify these values with a standard deviation of roughly 140 meters (450 feet). These pressure altitude / true altitude differences are not particularly large compared with the true altitude of 6,194 meters (20,320 feet).

The summit pressure altitude atop Mount McKinley is on average 115 meters (378 feet) lower June 15 compared to May 15. The average summit pressure altitude is 227 meters (743 feet) lower July 1 compared to May 1. Swings in pressure altitude owing to regional weather patterns (mainly via temperature shifts) will cause variability about these mean values.

These observations should be noted by all those contemplating an ascent of Mount McKinley.

During the December through February climbing season, Aconcagua's summit pressure altitude declines from a mean value of 250 meters (821 feet) lower than the true altitude; to a mean value mid-season of 289 meters (922 feet) lower than the true altitude; and then rises to a mean value of 246 meters (806 feet) lower than the true altitude (Table 2). This trend results from rising and then diminishing sea level temperatures as extrapolated from the detailed meteorological data (Figures 7,8). Daily temperature variations modify these values with a standard deviation of roughly 50 meters (160 feet). These pressure altitude / true altitude differences are not particularly large compared with the true altitude of 6,962 meters (22,841 feet) - but are larger in magnitude than the corresponding differences at Mount McKinley.

During the April and May climbing season, Mount Everest's summit pressure altitude declines from a mean value of 397 meters (1,301 feet) lower than the true altitude; to 560 meters (1,839 feet) lower than the true altitude (Table 3). This seasonal trend results from rising sea level temperatures, and declining sea level-corrected pressures (Figures 10,11). Daily temperature variations modify these values with a standard deviation of roughly 70 meters (225 feet). These pressure altitude / true altitude differences, although not particularly large compared with the true altitude of 8,848 meters (29,028 feet), are larger in magnitude than the corresponding differences at Mount McKinley and Aconcagua.

It is extremely noteworthy that all pressure altitudes at Aconcagua and Mount Everest, during their respective climbing seasons, are lower than the true altitude (Tables 2,3). In contrast, pressure altitudes at Mount McKinley are at first slightly higher than the true altitude; and then, at season's end, slightly lower than the true altitude (Table 1).

Most mountains are climbed during the warmer months of their respective hemispheres. Hence is it anticipated, and observed, that pressure altitudes will tend to be lower than true altitudes because during these months temperatures are higher than the Standard Atmosphere. The higher temperatures raise the scale height (cf Equations (7c,8)) and hence diminish the perceived pressure altitude.

Mount McKinley, lying so far north, forms an exception in that even during its optimal season temperatures are tolerably similar to the Standard Atmosphere. Thus pressure altitudes are either marginally higher (May) or lower (late June) than the corresponding true altitudes.

Dr. West was aware of the role warmer months play in skewing the availability of pressure data from montane settings.³ He was also aware that the Standard Atmosphere tends to understimate the calculated pressure atop high mountains; in turn because they are generally climbed in their respective warmer months when the temperature exceeds that predicted by the Standard Atmosphere.

In Table 4 direct comparison is made of true elevations at which the same pressure altitudes are observable on Mount McKinley and Aconcagua. The central entry of that table represents the middle of their respective climbing seasons. Using that entry, one must be an average of 301 meters (989 feet) higher on Aconcagua to observe the same pressure as at the summit of Mount McKinley - 6,495 meters (21,309 feet). Earlier days in the Mount McKinley climbing season, such as May 1, result in larger true altitude differences approaching 400 meters (1,300 feet).

In Table 5 direct comparison is made of true elevations at which the same pressure altitudes are observable on Mount McKinley and, representing High Asia, Mount Everest. The central entry of that table represents the middle of their respective climbing seasons. Using that entry, one must be an average of 396 meters (1,301 feet) higher on Mount Everest to observe the same pressure as at the summit of Mount McKinley - 6,590 meters (21,621 feet). Earlier days in the Mount McKinley climbing season, such as May 1; and/or later days in the Mount Everest climbing season, result in larger true altitude differences approaching 600 meters (1,950 feet).

For both Aconcagua and High Asia, variabilities of these true altitude differences are roughly ± 145 meters (475 feet) at one standard deviation (Tables 4,5).

Noting this variability, from the bottom row of Table 4 it is possible in late June for the pressure altitude at Mount McKinley's summit to be observed on Aconcagua's slopes in December and February at the same 6,194 meter elevation (20,320 feet).

Noting this variability, from the top row of Table 5 it is possible in early May for the pressure altitude at Mount McKinley's summit to be observed at 6,900 meters (22,600 feet) on Mount Everest's slopes in May - 700 meters (2,300 feet) higher than Mount McKinley's summit elevation.

These true altitude differences for matching summit pressure are not due to colder than standard temperatures at Mount McKinley. The differences arise from warmer than standard temperatures at the comparison mountains.

Recall that Dr. Moore considered the effect of low temperature upon barometric pressure for Earth's highest mountains in a 1968 paper.² Comparing Mount McKinley summit pressure with the Himalaya, he reported pressure altitudes similar to those calculated here. In the summer Mount McKinley's summit pressure was estimated equivalent to being 400 meters (1,300 feet) higher in the summer Himalaya - 6,600 meters (21,650 feet). This pressure altitude is equivalent to that obtained in Table 5 as the central entry.

In the winter Mount McKinley's summit pressure was estimated equivalent to being a full 950 meters (3,100 feet) higher in the summer Himalaya - 7,150 meters (23,450 feet). This latter value is obtained in the present formulation by using a sea level temperature T(0) = -13° C (8° F) for interior Alaska by winter, and 27° C (80° F) for the comparison Asian mountains.⁸

Although it is possible than in his pioneering paper Dr. Moore used a formulation similar to the current one based on Equations (7c,11b), there is nothing readily evident in the manuscript to suggest this.² The values of pressure altitude as a function of latitude appear to be interpolated from tabulated data.

If Dr. Moore's article was limited by interpolation of known data, then the present work both provides a theoretical basis for his, and corroborates it through independent calculation.

This temperature-dependence of pressure arises anywhere and anytime - not just on Mount McKinley. Ascending a 14,000 foot mountain in Colorado by winter will meet with a lower atmospheric pressure enroute and at the summit compared with a summer hike. The effect's magnitude is given by the fractional decrease in extrapolated sea level temperature relative to summer.

An extreme example of temperature influencing pressure altitude is given by winter on the Antarctic plateau. Vostok Station has recorded the coldest surface temperatures on Earth. At 3,420 meters (11,220 feet), Vostok has ample elevation and sufficiently low temperatures to support a large pressure altitude / true altitude difference (cf Equation (13c)). From roughly April through September the mean daily temperature is typically -65° C (-85° F). Substitution into Equation (13c) yields a pressure altitude of 4,280 meters (14,035 feet) - fully 860 meters (2,815 feet) higher than the true altitude!

Latitude, via a shift in the acceleration due to gravity, causes a change in pressure altitude from a combination of centrifugal force and mass-distribution effects arising from Earth's eccentricity.

Account of latitude raises the pressure altitude at Mount McKinley's summit 17 meters (55 feet) relative to Aconcagua (cf footnote Table 4). Account of latitude raises the pressure altitude at Mount McKinley's summit 19 meters (63 feet) relative to Mount Everest (cf footnote Table 5). These changes in pressure altitude are minor, and have little consequence for those climbing the world's higher mountains.

In the climbing literature on Mount McKinley (and apart from Dr. Moore's work), the claim is made that breathing is more arduous for a given altitude because centrifugal force due to Earth's rotation is less than for more temperately located mountains. The atmosphere is "whipped outwards" at the equator, raising the scale height with a resulting higher pressure for a given true altitude.

A centrifugal effect indeed manifests itself by a change in the effective value of g, with g diminished slightly at the equator where the speed of rotation is greatest. From Equation (12) g rises only 0.53% between equator and either pole; with 0.34% of this increase due to diminished centrifugal force with latitude. From Equations (7b,8) all pressure altitudes shift this small fraction as a result of Earth's rotation - roughly one part in 300.

For the summits of two 6,000 meter (19,700 foot) mountains, one at the equator and one at either pole, this fraction corresponds to a 20 meter (66 foot) pressure altitude difference. The pressure altitude correction is normally even smaller still since most mountains are not located exactly on the equator or in the polar regions.

At extreme altitudes even a slight shift in pressure altitude elicits a large shift in physiological performance.⁹ Hence a prudent choice of atmospheric model is critical. The key variables are inspired oxygen partial pressure PI,O₂, and the rate of maximal oxygen consumption per kilogram O_{2 max}.

The equations relating these variables are derived in Appendix C. Equation (C.3d) is the key result, expressing the total derivative of O_{2 max} with respect to the atmospheric scale height H using the simplified expressions of Equations (7c,8). The more accurate Equation (11b) is not used because measured O_{2 max} values are of insufficient detail to warrant such an accurate description.

The derivative of Equation (C.3d) is the sum of two terms. One is the explicit derivative of O_{2 max} with respect to PI,O₂, and is given by the first term on the right hand side thereof. The second term, a sum, is the implicit derivative of O_{2 max} with respect to PI,O₂ changes via the accompanying nonlinear shifts in physiological variables. PI,O₂ is effectively a surrogate for pressure altitude, there being a simple exponential relation between the two (cf Appendix C footnote [1]).

Equation (C.3d) is completely general, making no assumptions about the complicated internal physiological mechanisms at-play upon sojourning to high altitudes.

In a landmark research effort, several fit volunteers were subjected to a simulated 40 day "ascent" of Mount Everest in a decompression chamber.¹⁰ Maximum exercise performance was measured at various pressure altitudes.

Equation (C.3d) is applied to that data, the required total derivatives of O_{2 max} with respect to PI,O₂ being simply the differences of O_{2 max} at adjacent data points, divided by the corresponding differences in PI,O₂.

In that study, the lowest PI,O₂ values were 49 and 43 Torr; and correspond to altitudes in the ICAO model atmosphere of 7,622 meters (25,000 feet) and 8,382 meters (27,500 feet) respectively. The 43 Torr pressure was intentionally selected to correspond to the 253 Torr barometric pressure observed at the summit of Mount Everest - a value considerably higher than predicted using the ICAO model.

For these two data points one obtains from Equation (C.3d) a fractional shift in O_{2 max} of 4.98 x 10^-4 for every meter shift in the atmospheric scale height H.^11,12

Using the simple prescription of Equations (7c,8), the observed summit pressure of 253 Torr corresponds to an atmospheric scale height of 8,036 meters (26,365 feet).¹¹ In contrast, the scale height of the ICAO standard atmosphere is 7,685 meters (25,213 feet). Multiplying by 4.98 x 10^-4 / meter, the 351 meter difference yields a substantial 17.5% decrement in O_{2 max} were Mount Everest to be scaled under ICAO standard conditions.

As it is obtained by a numerical first-difference, this 17.5% decline corresponds to climbing at a simulated altitude midway between the 49 and 43 Torr data points - 8,391 meters (27,530 feet) under the test conditions applied. As the slope of the O_{2 max} vs PI,O₂ curve steepens with altitude, a slightly larger fractional decline is expected at the summit.

From Equation (8) the absolute temperature T and scale height H are related by the constant factor R / mg = 29.272 meters / °K. Multiplying the right hand side of Equation (C.3d) by this factor, one obtains the fractional shift in O_{2 max} for every degree centigrade shift in the ambient temperature.

In the current example this yields a 1.46% / °C fractional shift in O_{2 max}. This value is substantial, pointing out yet another obstacle to high altitude winter ascents; while underscoring the importance of using the correct atmospheric model for high altitude physiology studies.

Dr. West reached similar conclusions for the decrement in O_{2 max} under ICAO standard conditions, using data from the 1981 American Medical Research Expedition to Everest II (AMREE),^13,14 and from a 1997 NOVA expedition.¹⁵ The following excerpt from [15] is highly relevant, reproduced here with author permission.

"The extreme steepness of the line relating O2 max to inspired PO2 has some interesting physiological implications. For example, if the barometric pressure on the summit were 236 Torr, as predicted from the International Civil Aviation Organization Standard Atmosphere (2), the inspired PO2 on the summit would be only 39.5 Torr. Therefore, the reduction in inspired PO2 from the value of 43 Torr, which would exist for a summit pressure of 253 Torr, would be 3.5 Torr. The reduction of oxygen consumption would therefore be about 222 ml/min to a value of about 848 ml/min, or a reduction of about 21%. It seems very unlikely that the mountain could be climbed under these conditions."

The slope of the O_{2 max} vs PI,O₂ curve increases with altitude. A graph of the fractional change in O_{2 max} vs PI,O₂ is expected to have an even larger magnitude slope for extreme altitudes, having divided by the additional factor O_{2 max} which diminishes with altitude.

Thereby it is fitting that at Mount Everest's summit the estimated fractional decline in O_{2 max} were ICAO standard conditions to prevail, as some 21%, is greater still than the estimated fractional decline calculated in the current work for an altitude intermediate between the final two data points of the Mount Everest simulation study - 8,391 meters (27,530 feet).¹⁰

An independent estimate of the fractional decline in O_{2 max} under ICAO standard conditions, relative to a typical mid-May summit day at Mount Everest, is provided by applying the relevant entry of Table 3 (summit, May 15): a 515 meter (1,690 foot) average decline in pressure altitude at the summit relative to its true altitude. The corresponding increase in atmospheric scale height is 475 meters (1,559 feet). Multiplication by 4.98 x 10^-4 from Equation (C.3d) yields a 24% decline in O_{2 max} were ICAO standard conditions to prevail.

The air pressure experienced at a given true altitude is influenced to a modest degree by the extrapolated sea level temperature. Deviations in pressure altitude due to unusually low or high temperatures are sufficiently large as to be of interest to mountaineers.

Most high altitude ascents occur during the warmer months, and, as such, pressure altitude is lower than true altitude. Mount McKinley is an exception - there, temperatures during the climbing season are only marginally colder (May) or warmer (late June) than the standard atmosphere. Hence pressure altitude is tolerably similar to true altitude. Thereby a given pressure on Mount McKinley is observed at higher altitude elsewhere, all mountains being compared during their respective climbing seasons.

High latitude exerts its influence on air pressure indirectly through colder temperatures. Earth's rotation exerts a minor effect on air pressure via altering the effective value of the acceleration due to gravity. The resulting shift in air pressure is minor.

The warmer, high pressure conditions of ideal summit days provides High Asia mountaineering expeditions a considerably greater O_{2 max} than is predicted using the ICAO standard atmosphere. Indeed, climbing Mount Everest without supplemental oxygen is rendered feasible in large part to this observation. The present work provides a mathematically rigorous prescription for evaluating this effect.

¹ Valley, S. L.. Handbook of Geophysics and Space Environments McGraw-Hill, New York pp 2-11 to 2-19 (1965).

² Moore, Terris. The World's Great Mountains: Not the Height You Think. American Alpine Journal pp 109-116 (1968).

³ West, John B.. Prediction of barometric pressures at high altitudes with the use of model atmospheres. Journal of Applied Physiology 81: 4 pp 1850-1854 (October 1996).

⁴The scale height as realized in the atmosphere differs from that calculated using Equation (8) because of diminishing temperatures with true altitude. Use of sea level temperature for the standard atmosphere as T in Equation (8) yields a scale height of 8,435 meters (27,673 feet). The observed scale height is typically 8,000 meters (26,000 feet), and may be predicted using the more accurate model described in the Section "Modified Hydrostatic Equation".

⁵The actual expression which replaces gz is g(0)R_e /(R_e + z). Division of numerator and denominator by R_e and expansion of the term 1/(1 + z/R_e) yields g(0)z(1 - bz) upon neglect of higher order terms.

⁶Meteorological data from near Denali Pass, at 5,600 meters (18,400 feet), is available from an automated weather station operated by a Japanese Alpine Club / University of Alaska consortium. It remains to be seen what ridiculously low temperatures it will record this coming 2006-2007 winter.

⁷The 0.29% fractional shift is obtained by expanding exp(1+x) for x = 0.0029, and retaining terms up to first-order in x.

⁸ -13.5° C lies in-between the mean January temperatures for Talkeetna and Fairbanks, Alaska - two communities on either side of Denali National Park and Mount McKinley. Extrapolation to sea level and use of Equation (11b) results in the cited equivalent altitude in the summer Himalaya.

⁹ West, John B.. Human Limits for Hypoxia: The Physiological Challenge of Climbing Mt. Everest. Ann. N.Y. Acad. Sci. 899: pp 15-27 (2000).

¹⁰ Sutton, et al.... Operation Everest II: oxygen transport during exercise at extreme simulated altitude. Journal of Applied Physiology 64: 4 pp 1309- (1988).

¹¹The value H used in Equation (C.3d) is that corresponding to the warmer than standard atmosphere of a Mount Everest May summit bid. It is calculated by solving Equations (7c,8) for H, using the observed p = 253 Torr at the summit altitude of z = 8,848 meters. We obtain H = 8,036 meters (26,365 feet).

¹²The value of z used in Equation (C.3d) is obtained by solving (p^* + 47)/760 = exp (-z/H) for z, where p^* = PI,O₂ / r (O₂) and r (0₂) = 0.22 is the oxygen mole fraction. p^* is taken as the average of p^* for the two final data points (220.5 Torr). We obtain z = 8,391 meters (27,530 feet).

¹³A review of the 1981 American Medical Research Expedition to Everest (AMREE) is available on-line as this PDF document.

¹⁴ West, John B., et al.... Maximal exercise at extreme altitudes on Mount Everest. Journal of Applied Physiology 55: 3 pp 688-698 (1983).

True altitude corresponds to the normal sense of what we think of as elevation above sea level. It differs slightly from geopotential height which is graduated in such a manner that changes in gravitational potential scale linearly in that height measure.

Pressure altitude is that altitude in the standard atmosphere which corresponds to the observed air pressure. When the barometer is reading low, the pressure altitude is higher than the true altitude - and conversely when reading higher than normal.

Pressure altitude is useful in aviation. When flying designated elevations to avoid collision, pilots adjust their cockpit altimeters to the same local airport pressure reading. This enables flight at specific pressure altitudes. In the continental United States all flights above 18,000 feet set altimeters to 29.92 inches of mercury, so allowing separation of levels as well. This value is not arbitrarily chosen - it is the sea level pressure of the standard atmosphere.

Density altitude is that altitude in the standard atmosphere which corresponds to the observed air density. As cold air is heavier than warmer air, the density altitude on a cold day is lower than the true altitude.

Density altitude is critical in flight operations. Wing lift is a function of density altitude, as is engine output power. The length of a runway for takeoff must be long enough to accomodate hot days with their higher density altitudes. Time-to-climb records are performed on very cold winter mornings for increasing performance.

Strong equatorial solar heating forces air to rise, turn poleward both north and south, and descend around the thirtieth parallels - the Hadley cell. The trade winds return this air equatorward prior to rising again in a complete cycle. Owing to the Coriolis force (see below), trade winds travel west as well as poleward.¹

Similar albeit less powerful air currents occur over the polar regions, constituting the Polar cells. Air at the sixtieth parallel is heated to some extent, rises, travels to the poles in the upper troposphere, descends, and then returns to the sixtieth parallel as the west-trending polar easterlies.¹ Owing to the Coriolis force these winds have a west component.

The topmost extents of these air currents define the tropopause - in effect, the upper boundary of the troposphere and forming a cap on that atmospheric layer.

These convection cells create semipermanent regions of low and high air pressure at the latitudes of rising or converging air, and hence are of interest here.

A secondary effect dependent on the Hadley and Polar cells is the Ferrel cell of temperate latitudes. Much as a ball bearing is set in motion when placed in-between two larger, rotating bearings, the Ferrel cell exists owing to the Hadley and Polar cells. The Ferrel cell is responsible for the east-trending prevailing westerlies of these latitudes.¹

Jet streams are narrow bands of eastward flowing air located between these convection cells - the subtropical jet (near the thirtieth parallels) and the polar jet (near the sixtieth parallels). They result from the large temperature and pressure differences between air in adjacent convection cells; occur near the tropopause; and are, apart from hurricanes and tornadoes, the strongest winds on Earth.

These global currents serve to redistribute heat and, through horizontal pressure and temperature gradients, form the weather patterns commonly seen at mid-latitudes. In contrast, the tropics and polar regions experience less daily variability in weather.

The Coriolis effect is an apparent deflection of a moving object in a rotating reference frame, e.g. Earth's surface. The deflection is fictitious: to an observer in a non-rotating (i.e., inertial reference frame), there is no deflection. However to an observer fixed to the rotating reference frame, such as Earth's surface, moving objects appear to be deflected from straight-line paths.

Consider a parcel of air traveling due north from the equator. To a distant observer fixed in space the parcel also contains a large eastern component equal in magnitude to the speed of rotation at the equator - roughly 1,670 kilometers per hour (1,030 miles per hour). As the parcel crosses parallels of latitude the underlying surface is traveling slower than at the equator. Hence the air parcel will, in the absence of any external force, assume an easterly component to it's motion relative to the underlying surface.

However to an Earth-based, co-rotating observer a ficitious force appears to be pushing the air parcel east. This is the Coriolis effect.

The Coriolis effect caused by Earth's rotation is responsible for the direction of rotation of weather patterns. The effect deflects objects moving along Earth's surface to the right in the North Hemisphere and to the left in the South Hemisphere. Thus winds around the center of a cyclone rotate counterclockwise in the North Hemisphere and clockwise in the South Hemisphere.

Global atmospheric convection cells are altered by the Coriolis effect. Equator-bound trade winds (see last section) assume a westerly component, as do the subpolar easterlies descending from the poles. Similary, mid-latitude prevailing westerlies assume an easterly component.

Consider the combined effects of pressure gradients and the Coriolis effect associated with a low pressure region. Due to the lower pressure, air will flow into such a region, and, in the absence of Earth's rotation, would assume straight paths. However the Coriolis force deflects these paths perpendicular to the direction of motion. There results an equilibrium of pressure and Coriolis-generated motions wherein a circular pattern is established - a cyclonic flow.

Cyclonic flow is one instance of the more general geostrophic balance wherein air flows along isobars. On a non-rotating planet the air would flow along the straightest possible line, quickly zeroing pressure gradients.

In the Northern Hemisphere air motion around a low-pressure center is counterclockwise. High pressure creates an anticyclone with clockwise motion. In the Southern Hemisphere these senses of rotation are reversed.

The Coriolis force is zero at the equator, and cyclones cannot form there. The Coriolis force is greatest in mid-latitudes.