Pressure Variations on Mount Everest
TAN Yii Hsien, Barnabas
Imperial College London
19th January 2004
1. Abstract
Pressure data from the South Col of Mount Everest (at an altitude of nearly
8000m) recorded at 5-minute intervals for the period between 17th May 2002 and 25th
April 2003 was analysed using harmonic methods. It was found that the monsoon season
had an anomalous plateau in the pressure readings, beginning from about 4 hours before
sunset and lasting until about 2100H, when the pressure started to decrease drastically.
This could be the effect of precipitation since an afternoon peak in rainfall is not unusual
for mountainous regions. Moreover, a nocturnal peak in rainfall, occurring just past
midnight, had been observed to exist most strongly during the summer monsoon in the
Himalayan region. More data is required to firmly establish any link but the preliminary
data so far suggests that precipitation could be a significant factor in affecting pressure
variations at the South Col. It was also found that the main winter season had 2
uncharacteristic features – an early morning peak in the pressure readings and a double
peak in the pressure readings during the course of the day. The former could be the
result of blocking, which occurs frequently in winter when cold, stable air masses are
common and persist for extended periods. It was suggested that the latter could be the
result of the mixing that occurs when the air directly above the earth’s surface is heated
and rises to meet the colder upper air. These explanations remain mere postulations
until a network of barometers is set up in the Himalayan region and readings are taken
over several years to give a more complete understanding of the pressure variations at
such high altitudes. It would then be possible to distinguish between pressure variations
that are regional, and pressure variations that are specific to the immediate topography.
2. Introduction
In May 2002, a climbing expedition successfully placed an instrument containing
a barometer, a temperature sensor and a humidity sensor on the South Col of Mount
Everest. [1] This is located at an altitude of nearly 8000m above sea level. Mount
Everest is the highest mountain in the world and stands on the border of Nepal and Tibet.
The South Col is a rock-strewn wind-swept saddle between Everest and Lhotse. The data
comprising pressure, temperature and relative humidity readings recorded at 5-minute
intervals was retrieved in Spring 2003 and is the first continuous record of atmospheric
pressure at such a high altitude. This is actually part of a larger project to develop a
network of barometers on three mountains in the Himalayan region, namely Mount
Everest, Cho Oyu and Shishapangma. These three mountains form a chain along similar
latitude, spanning from east to west and stretching for about 80 km. Simultaneous
observations on these mountains will allow us to distinguish between what are regional
pressure variations and what are pressure variations specific to the immediate
topography.
1
Climatic changes in this region are poorly understood because the Himalayan
range has the fewest observations, in spite of it being the largest mountain range on earth.
This is likely because the Himalayan range lies within several poor and developing
countries, such as Nepal, that do not have the resources to perform sophisticated
meteorological studies. For instance, before 2001, no radiosondes had been launched in
Nepal for more than 20 years. [2] The Himalayas and the Tibetan plateau play an
important role in regional climate, particularly with respect to monsoon circulation.
Links between the monsoon and other global-scale phenomena extend the implications of
climatic variations in the Himalayas and in the Tibetan Plateau beyond the regional scale.
Climatic changes in the Himalayan region could be a reflection of large-scale climate
changes, or they could even be driving them. [3] It is hoped that the data obtained from
these mountains will help to contribute towards understanding more about the
mechanisms behind the regional climatic changes at these altitudes.
3. Analysis methodology
The pressure, temperature and relative humidity readings were recorded at 5minute intervals. We were interested in the data recorded for specific seasons, each
comprising between 2 to 4 months. For example, for the monsoon season, we were
interested in the months June 2002 to September 2002. For each particular season, the
readings at every 5-minute interval, spanning a 24-hour day, were averaged and binned.
The relative humidity readings were plotted as such. The specific humidity readings
were calculated from the relative humidity readings using the relationship between
relative humidity and specific humidity, given by Equation (1), and the formula for the
saturation vapour pressure, given by Equation (2).
RH 
Q
PS
------ (1)
where RH is the relative humidity, PS is the saturation vapour pressure and Q is
the specific humidity.
PS  0.611  exp(
17.27T
)
T  237.3
------ (2)
where T is the temperature in degrees celcius and the saturation vapour pressure
PS is in units of kPa.
The pressure and temperature anomaly were obtained by subtracting the mean
reading for that particular season. In addition, the pressure anomaly had the semi-diurnal
component subtracted using harmonic analysis.
Harmonic analysis consists of representing the fluctuations or variations in a time
series as having arisen from the adding together of a series of sine and cosine functions.
[4] About 20% of the solar radiation reaching the top of the atmosphere is absorbed by
the atmosphere. This atmospheric solar heating, combined with upward eddy conduction
of heat from the ground, generates internal gravity waves in the atmosphere at periods of
2
the integral fractions of a solar day (primarily at the diurnal and semi-diurnal periods).
These internal gravity waves cause regular oscillations in atmospheric wind, temperature
and pressure fields, which are often referred to as atmospheric tides. [5] “Both the sun
and moon produce atmospheric tides, and there exist both gravitational tides and thermal
tides. The harmonic component of greatest amplitude, the semi-diurnal solar atmospheric
tide, is both gravitational and thermal in origin. The fact that it is greater than the
corresponding lunar atmospheric tide is due to a resonance in the atmosphere with a free
period very close to the tidal period.” [6] The semi-diurnal component is actually the
second harmonic and the mathematics behind its computation is elaborated on further in
Appendix A. In the surface pressure field, the semi-diurnal oscillation is thought to
predominate over much of the globe, and is regularly distributed in its amplitude and
phase (typically peaking 2-3 hours before noon and midnight). [5] It is relatively well
understood and hence for our purposes, we removed it from our data in order to focus on
the pressure pattern that is distinct from the semi-diurnal component and unique to the
Himalayan region. The Interactive Data Language (IDL) was the programming language
that was used to assist in the analysis of the data.
Fig. 1: Pressure anomaly at South Col as a function of local time for October 2002 to November
2002. The actual pressure anomaly, the semi-diurnal component and the resultant pressure
anomaly minus the semi-diurnal component are plotted on the same axes.
The error analysis for the calculation of the semi-diurnal component and for
harmonic analysis in general is beyond the undergraduate level. For our purposes,
current knowledge about the semi-diurnal component was used as a criterion to determine
3
whether our results were physically sensible. There were 3 questions that were asked of
the function that was calculated to be the semi-diurnal component. Firstly, did the
function complete 2 full cycles? Secondly, did the function peak 2-3 hours before noon
and midnight? And thirdly, was the function the most prominent component that
contributed to the shape of the actual pressure data? The linear Pearson correlation
coefficient was considered as a method to answer the third question but it was found to be
inappropriate because the function in question was calculated from the actual pressure
readings. The linear Pearson correlation coefficient can be used only for functions that
are independent of each other. Hence, we were restricted to visual comparisons to
determine whether the function in question was the most prominent component
contributing to the shape of the actual pressure data. As an example to illustrate this
process of error analysis, the pressure readings for the period from October 2002 to
November 2002 were considered. From Figure 1, it is evident that the semi-diurnal
component was the most prominent component contributing to the distribution of the
pressure data that was observed.
In addition, the sunrise and sunset timings for individual days were obtained from
the US Naval Observatory Astronomical Applications Department website. [7] The
average sunrise and sunset timings calculated over all the days of that particular season
were then labelled on Figures 3, 6, 8 and 9 for the pre-monsoon season, the monsoon
season, the post-monsoon season and the main winter season respectively.
4. Theory
a. Hydrostatic equation
The hydrostatic equation describes the variation of pressure, p z , with height z. It
is given below.
z
p z  p0 exp (
0
Mg
dz )
RT z
------ (3)
where p 0 is the pressure at sea level, M is the relative molecular mass of air, g is
the acceleration due to gravity, R is the universal gas constant and Tz is the temperature
at height z.
dT
The lapse rate  is defined as the vertical temperature gradient 
. If  is
dz
dT
assumed to be constant with height, then by use of a substitution of variables, dz  
,

equation (3) becomes:
T
p z  p0 exp (
Mg z dT
)
R T0 T
------ (4)
where TZ  T0  z , with T0 being the effective surface temperature.
4
Equation (4) can be rearranged to obtain an expression for T0 :
T0 
z
p
R
1  exp (
ln z )
Mg p0
------ (5)
By inspection, it is seen that as the effective surface temperature, T0 , increases,
pz
also increases. This expression is obtained based on the assumption that
p0
the vertical temperature gradient is constant with height. In reality, this is just an
approximation. However, in general, at hydrostatic equilibrium, a pressure increase at a
mountain site can be a measure of the integral warming below the site and is associated
with an increase in the mean layer temperature below the site. This principle has been
used to derive long-term temperature trends from a number of mountain station pressure
observations. The derived trends are consistent with the record of surface temperature
observations. [8]
the ratio
b. Shortwave and longwave radiation
All objects emit radiation, with wavelength depending on the temperature of the
radiating body. The sun, with a temperature of about 6000 degrees celcius emits most of
its radiation in the wavelength range 0.15 – 3 m . Terrestrial objects, including the
earth’s atmosphere, have much lower temperatures than the sun and so radiate energy at
longer wavelengths (3-100 m ). Hence, shortwave radiation is radiation from the sun
while longwave radiation is radiation from objects or gases at terrestrial temperatures.
Both types of radiation can be directed upward from the ground to the atmosphere or
downward from the atmosphere to the ground. [9]
Incoming solar radiation begins at sunrise and ends at sunset, typically peaking at
midday. Incoming and outgoing longwave radiation continues throughout both night and
day, normally peaking in the early afternoon. At night, there is a net loss of longwave
radiation because the amount of outgoing longwave radiation is greater than the amount
of incoming longwave radiation. The amount of radiation reaching the earth’s surface
would have an effect on the mean layer temperature below a mountain site. A net gain of
radiation by the earth’s surface would consequently lead to an increase in the mean layer
temperature below the mountain site. On the other hand, a net loss of radiation from the
earth’s surface would lead to a decrease in the mean layer temperature below the
mountain site.
5
c. Temperature inversion
In a normal situation, the temperature decreases as the altitude increases. The rate
of decrease varies but the average rate of decrease is taken to be 6.5 degrees celcius per
1000m. This is known as the normal lapse rate. [10] In a temperature inversion, this is a
deviation from the normal situation, the temperature increases as the altitude increases.
This occurs when cold air is near the ground and there is a layer of warmer air above it.
Altitude
TA <
TS
T A > TS
T
Fig. 2: Graph of altitude versus temperature to illustrate what happens within the inversion layer
From Figure 2, it can be seen that within the inversion layer, the temperature
increases as altitude increases but above the inversion layer, the temperature decreases
with increasing altitude. Consider a packet of air. If it were in the lower right region,
where its temperature, TA, is higher than the temperature of its surroundings, TS, it will
rise. If it were in the upper left region, where its temperature, TA, is lower than the
temperature of its surroundings, TS, it will sink. Hence, the packet of air is trapped at the
boundary layer, neither rising nor sinking because it has attained a form of stable
equilibrium.
There are several reasons why a temperature inversion might occur. One reason
is the lack of cloud cover at night. On a clear night, the earth’s surface would radiate heat
away rapidly. This would cause the ground and the air directly above it to be cooler than
the air at higher altitudes. The lack of cloud cover is characteristic of the winter season in
the Himalayan region.
5. Discussion of results
The pressure and temperature data from 17th May 2002 to 25th April 2003 was
divided into four seasons. The pressure and temperature anomaly for each of these four
seasons were considered separately.
6
a. Pre-monsoon season (February 2003 – April 2003)
Fig. 3: Pressure and temperature anomaly at South Col as a function of local time for February
2003 to April 2003. Semi-diurnal component of the pressure readings has been removed.
The pattern of pressure anomaly in Figure 3 was what was expected from the
earlier discussion of how a pressure increase at a mountain site could be a measure of the
integral warming below the site. After sunrise, as the sun warmed the atmosphere below
the site, the pressure at the mountain site increased. This happened until about an hour
before sunset when the amount of incoming solar radiation was reduced. After sunset,
there was a net loss of radiation from the earth’s surface since there was no incoming
solar radiation and moreover, there was a net loss of longwave radiation as the amount of
outgoing longwave radiation exceeded the amount of incoming longwave radiation. As a
result, the mean layer temperature below the mountain site decreased and there was a
corresponding pressure decrease, as predicted by equation (5).
The pattern of temperature anomaly in Figure 3 was typical of all 4 seasons, with
the lag time after sunrise ranging between 2-3 hours and the maximum temperature
occurring between 1-2 hours before sunset. From a picture taken of the instrument at its
final location at the South Col [1], it can be seen that the instrument was placed directly
beside a big rock. This rock might have the effect of shielding the instrument from the
direct rays of the sun, depending on the relative positions of the instrument, the rock and
the sun. Given that this pattern of temperature anomaly was characteristic of all 4
seasons, it is highly probable that the aspect of the instrument was a significant factor in
influencing the temperature anomaly. Knowing whether the instrument has a north-south
or east-west aspect would enable the effect of the rock’s shadow falling on the instrument
7
at different times of the day to be taken into account when attempting to account for this
pattern of temperature anomaly. Unfortunately, for this experiment, the aspect of the
instrument was not known. This should be taken into account for future runs of the
experiment.
After sunset, the temperature anomaly curve fell off more steeply than the
pressure anomaly curve, implying that the temperature readings, which are an indication
of the local temperature at the mountain site, decreased at a faster rate than the pressure
readings, which are an indication of the mean layer temperature below the mountain site.
b. Monsoon season (June 2002 – September 2002)
Himalayan
region
Fig. 4: Mean amount of cloud cover for the period from June to August, averaged over 19 years
from 1983 to 2001. This data is obtained from the International Satellite Cloud Climatology
Project [11].
An abundance of cloud cover and the absence of strong winds characterise the
monsoon season. Cloud cover data was obtained from the International Satellite Cloud
Climatology Project website [11]. The International Satellite Cloud Climatology Project
was established in 1982 as part of the World Climate Research Programme to collect and
analyse satellite radiance measurements to infer the global distribution of clouds, their
properties, and their diurnal, seasonal and inter-annual variations. Figures 4 and 5 show
the amount of cloud cover averaged over 19 years (from 1983 to 2001) for the summer
and winter periods respectively. For the period from June to August, there is on average
at least 75% cloud cover over the Himalayan region. In contrast, for the period from
December to February, there is on average only about 25% cloud cover over the
Himalayan region. There is about 3 times more cloud cover during the monsoon season
8
than during the winter season. During the monsoon season, there is a high pressure
region over the Himalayas, which now lies at the core of the upper level anti-cyclone.
This repels the approach of the sub-tropical jet stream, which can create mean wind
speeds up to 200 mph. The absence of strong winds resulted in an accumulation of snow
at the mountain site, which caused the instrument to be buried under snow. This was
attested to by the relative humidity readings for this season, which ranged between 9899%. The accumulation of snow over the instrument gave rise to relatively smoothed
data as wind fluctuations were damped out.
Himalayan
region
Fig. 5: Mean amount of cloud cover for the period from December to February, averaged over 19
years from 1983 to 2001. This data is obtained from the International Satellite Cloud
Climatology Project [11].
9
Fig. 6: Pressure and temperature anomaly at South Col as a function of local time for June 2002
to September 2002. Semi-diurnal component of the pressure readings has been removed.
There were 2 features in Figure 6 that were different from Figure 3. The first
feature was the lag time of about 5 hours between sunrise and the onset of pressure
increase. This might be explained by the abundance of cloud cover during this season
since the cloud cover could shield the earth’s surface from the incoming solar radiation
by reflecting solar radiation back to space. Hence, it would take a longer time for the
incoming solar radiation to compensate for the net loss of the longwave radiation.
The second feature was the plateau in the pressure readings, which began from
about 4 hours before sunset and lasted until about 2100H, when the pressure started to
decrease dramatically. There are 2 possible reasons to explain this. The first is the
abundance of cloud cover, which could trap the outgoing radiation. This would cause the
mean layer temperature below the mountain site to take a longer time to cool. However,
this does not explain the steep pressure drop that occurs at about 2100H, which suggests a
very rapid rate of cooling. This also does not explain why the plateau begins about 4
hours before sunset. The second possible reason is the presence of precipitation.
Precipitation would act as an alternative heat source, as latent heat would be released
upon condensation. This would compensate for the decrease in the mean layer
temperature below the mountain site, thus keeping it constant. When the precipitation
evaporates, latent heat is absorbed and this would cause the mean layer temperature
below the mountain site to suddenly decrease, thus accounting for the steep drop in
pressure. The onset of precipitation in the late afternoon might have the effect of cooling
the mean layer temperature below the mountain site, thus bringing about the plateau in
the pressure readings about 4 hours before sunset. The relative humidity readings for this
10
season were used to calculate the specific humidity and there was a marked rise in the
amount of water vapour in the air towards the late afternoon, peaking at about 1700H, as
seen in Figure 7. This might suggest the occurrence of an afternoon peak in rainfall that
is not unusual for mountainous regions, which often show an afternoon peak associated
with diurnally forced upslope flow. [12]
A hydro meteorological network was installed in the Marsyandi River basin in
central Nepal during the spring of 1999. An interesting nocturnal peak in rainfall, just
past midnight, was observed. Other researchers observed this regional phenomenon as
well. What is interesting about this nocturnal rainfall peak is that it exists most strongly
during the summer monsoon, while the afternoon peak is predominant during other times
of the year. This phenomenon is also observed elsewhere in the Himalayas. [2] While
more data is required to firmly establish any correlation between the effects of
precipitation and the plateau in the pressure readings at the South Col, the data so far
suggests that precipitation could be a significant factor in affecting the pressure variations
at the South Col and should be taken into account for future studies of high altitude
pressure variations.
Fig. 7: Relative and specific humidity at South Col as a function of local time for June 2002 to
September 2002.
The temperature anomaly in Figure 4 did not follow the trend of the pressure
anomaly. The plateau only occurred with the pressure readings and not with the
temperature readings. This is because the pressure readings gave an indication of the
mean layer temperature below the mountain site while the temperature readings gave an
11
indication of the local temperature at the mountain site, at an altitude of 8000m. This
suggests that the abundant layer of cloud cover was at an altitude of less than 8000m.
c. Post-monsoon season (October 2002 – November 2002)
Fig. 8: Pressure and temperature anomaly at South Col as a function of local time for October
2002 to November 2002. Semi-diurnal component of the pressure readings has been removed.
During the post-monsoon season, the high pressure region previously centred on
the Himalayas, starts to shift southwards. Strong winds start to pick up due to the
approach of the sub-tropical jet stream, which creates mean wind speeds up to 180 mph.
[13] According to Bernoulli’s Fluid Equation, a moving fluid exchanges its kinetic
energy for pressure. [14] Hence, the pressure of a fluid is inversely proportional to the
velocity of the fluid flow. The strong winds from the sub-tropical jet stream experience
mean fluctuations in wind speeds of up to 150 mph, thus resulting in noisier pressure
data, as can be seen in Figure 8.
There were 2 developing features in the pressure anomaly that would become
more apparent during the main winter season. They were the early morning peak in
pressure and the double pressure peak during the course of the day.
12
d. Main winter season (December 2002 – January 2003)
Fig. 9: Pressure and temperature anomaly at South Col as a function of local time for December
2002 to January 2003. Semi-diurnal component of the pressure readings has been removed.
It can be seen from Figure 9 that the early morning peak in pressure and the
double pressure peak in the course of the day were more prominent during the main
winter season. The main winter season is characterised by a lack of cloud cover and this
provides ideal conditions for temperature inversion to occur. Consider the schematic
diagram of the topology of the Himalayan region in Figure 10. The Tibetan plateau is
bounded on one side by the Himalayan range. After sunset, as the air is cooled, it would
want to “drain off” to lower ground since cooler air is denser. But on the side of the
Tibetan plateau that is bounded by the Himalayan range, the air is unable to “drain off”
because it is physically blocked by the Himalayan range and moreover, due to the
occurrence of temperature inversion, the air is trapped at the boundary layer and hence, is
unable to rise above the Himalayan range. This blocking effect could result in an
accumulation of air mass and be the likely cause of the peak in pressure in the early hours
of the morning.
After sunrise, the pressure rose very quickly to reach a maximum after only about
3-4 hours. This could be due to a lack of cloud cover to shield the earth’s surface from
the incoming solar radiation, thus causing the earth’s surface to be heated up very
quickly. As the earth’s surface was heated, the air directly above it was also heated and
rose to mix with the colder upper air. It was at this point that the pressure readings,
which are a reflection of the mean layer temperature below the mountain site, started to
13
decrease while the temperature readings, which are a reflection of the local temperature at
the mountain site, started to increase. As the heating of the atmosphere continued in the
course of the day, the pressure readings continued to increase as the mean layer
temperature below the mountain site increased. This continued until sunset when there
was no longer any incoming solar radiation and the net loss of longwave radiation caused
the mean layer temperature below the mountain site and consequently, the pressure
readings to decrease.
Himalayan range
Accumulation of air mass
Tibetan plateau
Fig. 10: Schematic diagram of the Himalayan region to explain the blocking effect
6. Conclusion
This is the first time that anyone had encountered a continuous record of
atmospheric pressure at such a high altitude. While the pressure anomaly for the premonsoon season was in line with what the hydrostatic equation had predicted, the
pressure anomaly for both the monsoon season and the main winter season had unusual
features, which differed even from each other. For the monsoon season, there were 2
unusual features. The first was the lag time of about 5 hours between sunrise and the
onset of pressure increase while the second feature was the plateau in the pressure
readings. The first feature could be explained by the abundance of cloud cover while the
most feasible explanation for the second feature involved the presence of precipitation.
While more data is required to firmly establish any link between the effects of
14
precipitation and the plateau in the pressure readings at the South Col, the preliminary
data so far suggests that precipitation could be a significant factor in affecting the
pressure variations at the South Col and should be taken into account for future studies of
high altitude pressure variations.
For the main winter season, the unusual features were the early morning peak in
pressure and the double pressure peak during the course of the day. The most feasible
explanation for the early morning peak involved blocking, which occurs frequently in
winter when cold, stable air masses are common and may persist for extended periods.
The double pressure peak in the course of the day was probably due to the mixing that
occurred when the air directly above the earth’s surface was heated and rose to meet the
colder upper air.
At present, these explanations for the various unusual features of the pressure
anomaly remain mere postulations. It is only when the network of barometers is set up
on the three mountains in the Himalayan region, namely Mount Everest, Cho Oyu and
Shishapangma, and readings are taken over several years, then will it be possible to
distinguish between pressure variations that are regional and pressure variations that are
specific to the immediate topography. A more complete understanding of the pressure
variations at such high altitudes will help to contribute towards understanding more about
the mechanisms behind the regional climatic changes at these altitudes.
7. Bibliography
[1] Imperial College London, Department of Physics,
Space and Atmospheric Research Group website
http://www.sp.ph.ic.ac.uk/~rtoumi/EVE/eve2002.html
[2] Barros, A. P., and T. J. Lang, 2003: Monitoring the Monsoon in the Himalayas,
Observations in Central Nepal, June 2001, Monthly Weather Review, 131 (7), 1408-1427
[3] Shresta A. B. et al., 1999: Maximum temperature trends in the Himalayas and
vicinity, Journal of Climate, 2775-2786
[4] Daniel S. Wilks, “Statistical Methods in the Atmospheric Sciences”, pg 325,
International Geophysics Series, Academic Press (1995)
[5] Dai, A. and Wang, J., 1999: Diurnal and semidiurnal tides in global surface pressure
fields, Journal of the Atmospheric Sciences, 56, 3874-3891
[6] American Meteorological Society website
http://amsglossary.allenpress.com/glossary/search?id=atmospheric-tide1
[7] US Naval Observatory Astronomical Applications Department website
http://aa.usno.navy.mil/cgi-bin/aa_rstablew.pl
[8] Toumi, R., Hartell, N., Bignell, K., 1999: Mountain station pressure as an indicator of
climate change, Geophysics Research Letters, 26, 1751-1754
15
[9] C. David Whiteman, “Mountain Meteorology: Fundamentals and Applications”,
pg 42-44, Oxford University Press (2000)
[10] C. David Whiteman, “Mountain Meteorology: Fundamentals and Applications”,
pg 31, Oxford University Press (2000)
[11] International Satellite Cloud Climatology Project website
http://isccp.giss.nasa.gov/products/browsed2.html
[12] Banta, R. M., 1990: The role of mountain flows in making clouds, Atmospheric
Processes over Complex Terrain, Meteor. Monographs Vol. 23, 45, American
Meteorological Society, 229-283
[13] C. David Whiteman, “Mountain Meteorology: Fundamentals and Applications”,
pg 63, Oxford University Press (2000)
[14] Plus Internet Mathematics Magazine website
http://plus.maths.org/issue1/bern/tindex.html
8. Appendices
Appendix A: Mathematics behind the calculation of the 2nd harmonic, adapted from
“Statistical Methods in the Atmospheric Sciences” (Daniel S. Wilks)
Appendix B: IDL code used to produce the pressure and temperature anomaly plots
Appendix C: IDL code used to produce the relative humidity and specific humidity plots
16
Appendix A: Mathematics behind the calculation of the 2nd harmonic
A given data series consisting of n points can be represented exactly by adding
together a series of n harmonic functions,
2
n2

 2kt

(A1)
yt  y   C k cos
  k 
 n

k 1 
The cosine wave consisting of the k  1 term in Eq. (A1) is the fundamental, or
first harmonic. The other n  1 terms in the summation of Eq. (A1) are higher
2
2k
harmonics, or cosine waves with frequencies wk 
that are integer multiples of the
n
fundamental frequency w1 . The second harmonic is the cosine function that completes
exactly 2 full cycles over the n points of the data series, with its own amplitude C2 and
phase angle 2 .
Now, Ck 
where Ak 
and Bk 
A
2
k
 Bk2

(A2)
2 n
 2kt 
y t cos


n t 1
 n 
(A3)
2 n
 2kt 
y t sin 


n t 1
 n 
 1
tan

 1
tan

In addition,  k  
tan 1


tan 1

(A4)
Bk
Ak
Ak  0, Bk  0
Bk
 2
Ak
Ak  0, Bk  0
Bk

Ak
Ak  0, Bk  0
Bk

Ak
Ak  0, Bk  0
(A5)
The second harmonic is obtained by setting k  2 and is given by
 2 2t

y k  2  y  C 2 cos 
 2 
 n

(A6)
17
Appendix B: IDL code used to produce the pressure and temperature anomaly plots
Notes:
1. The words in blue are procedures and functions used in IDL.
2. The words in green are comments that were added to describe what different parts
of the code do.
3. The words in red are variables that need to be changed depending on which
season is being considered. In this case, it is the post-monsoon season, ie. the
months of October and November.
PRO himap_harmonic_analysis
; Read in the data.
restore, 'modified_himap1_time_corrected_eot.sav'
data = data (4410:103382) ; These boundaries represent the time on South Col, beginning from 17th May 2002 to 25th
April 2003.
dummy = WHERE (data.month EQ 10 or data.month EQ 11, count) ; This determines the period we will focus on, in
this case, the months of October and November.
IF count NE 0 THEN data_seasonal = data (dummy)
; Set up a postscript plot.
set_plot, 'ps'
!P.MULTI = [0, 1, 1]
device, filename = 'himap_oct_to_nov2002_pressure_&_temp_anamoly.ps', $ ; This is the name of the file to which the
postscript plot is written.
XSIZE = 23.5, YSIZE = 16.0, XOFFSET = 2, YOFFSET = 28, $
BITS = 8, /color, /landscape
bin = 5 ; This sets the bin size, in this case, to 5 minutes.
n = (60/bin) * 24 ; This determines the number of readings over 24 hrs.
pressure_anomaly = FLTARR (n) ; This creates a floating point array with the given dimensions.
temp_anomaly = FLTARR (n)
A_sum = FLTARR (n)
B_sum = FLTARR (n)
hr = 0
m=0
k = 2 ; This determines which harmonic function is represented, in this case, the 2nd harmonic.
; This FOR loop calculates the pressure and temperature anomaly readings for each 5 minute bin.
FOR j=0, n-1 DO BEGIN
dummy = WHERE (data_seasonal.hour EQ hr and data_seasonal.minute GT m and data_seasonal.minute LT m+bin,
count)
IF count NE 0 THEN data_specific = data_seasonal (dummy)
pressure_anomaly (j) = MEAN (data_specific.pressure) - MEAN (data_seasonal.pressure)
temp_anomaly (j) = MEAN (data_specific.temp1) - MEAN (data_seasonal.temp1)
A_sum (j) = pressure_anomaly (j) * cos ( (2*!pi*k*(j+1)) / n )
B_sum (j) = pressure_anomaly (j) * sin ( (2*!pi*k*(j+1)) / n )
m = m + bin
IF m GT 60-bin THEN BEGIN
m = 0 ; This resets m to zero at the end of 1 hour for the next FOR loop.
hr = hr + 1 ; This moves the FOR loop on to the next hour.
ENDIF
ENDFOR
18
y = pressure_anomaly
i = INDGEN (n) + 1
A = (2./n) * TOTAL (A_sum) ; This is the expression to evaluate Eq. (A3) for k=2.
B = (2./n) * TOTAL (B_sum) ; This is the expression to evaluate Eq. (A4) for k=2.
C = SQRT (A^2 + B^2) ; This calculates the amplitude of the 2nd harmonic.
phase = ATAN (B/A) ; This calculates the phase angle of the 2nd harmonic.
dummy = WHERE (A GT 0 and B LT 0, count)
IF count NE 0 THEN phase(dummy) = phase(dummy) + 2.*!pi
dummy = WHERE (A LT 0 and B GT 0, count)
IF count NE 0 THEN phase(dummy) = phase(dummy) + !pi
dummy = WHERE (A LT 0 and B LT 0, count)
IF count NE 0 THEN phase(dummy) = phase(dummy) +!pi
harmonic = mean (y) + C * cos (((2*!pi*k*i) / n) - phase) ; This is the expression for the 2nd harmonic, ie. the semidiurnal component.
less_harmonic = FLTARR (n)
; This FOR loop calculates the pressure anomaly minus the semi-diurnal component.
FOR j=0, n-1 DO BEGIN
less_harmonic (j) = pressure_anomaly (j) - (mean (y) + C * cos (((2*!pi*k*(j+1)) / n) - phase))
ENDFOR
print, 'A =', A
print, 'B =', B
print, 'phase (in radians) =', phase
x = FINDGEN (n)/ (60/bin) ; This sets up the horizontal axis.
plot, x, less_harmonic, $
title = 'Pressure and temperature anomaly at South Col (October - November 2002)', $ ; This is the label for
the title of the plot.
xtitle = 'Local Time (Hr)', $
ytitle = 'Pressure anomaly (mb)', $
yrange = [(min (less_harmonic)), (max (less_harmonic))], xstyle = 1, xrange = [0, 24], $
ystyle = 8 ; This ensures y-axis is drawn on only one side of the plot.
axis, yaxis = 1, yrange = [-2, 3], ystyle = 1, $ ; The AXIS procedure draws a new axis, in this case, on the
other side of the plot.
ytitle = 'Temperature anomaly (Deg C)', /save ; SAVE keyword saves the new scale for use by subsequent
overplots.
oplot, x, temp_anomaly, linestyle = 3 ; The OPLOT procedure overlaps a plot of the data on an existing axis.
xyouts, 1, CEIL (max (temp_anomaly)) - 0.2, '_______ : Pressure anomaly (minus semi-diurnal component)'
xyouts, 1, CEIL (max (temp_anomaly)) - 0.4, '_ . _ . _ : Temperature anomaly'
; This marks and labels the sunrise timing on the plot.
sr = 6.15 ; This is calculated as a fraction of 60, for eg. 0.15 = 9 / 60.
oplot, [sr, sr], [FLOOR (min (temp_anomaly)) - 1, CEIL (max (temp_anomaly)) + 1], linestyle = 1
xyouts, sr - 1.4, FLOOR (min (temp_anomaly)) + 0.2, 'Sunrise 06:09'
; This marks and labels the sunset timing on the plot.
ss = 17.28 ; This is calculated as a fraction of 60, for eg. 0.28 = 17 / 60
oplot, [ss, ss], [FLOOR (min (temp_anomaly)) - 1, CEIL (max (temp_anomaly)) + 1], linestyle = 1
xyouts, ss -1.4, FLOOR (min (temp_anomaly)) + 0.2, 'Sunset 17:17'
device, /close
END
19
Appendix C: IDL code used to produce the relative humidity and specific humidity
plots
Notes:
1. The words in blue are procedures and functions used in IDL.
2. The words in green are comments that were added to describe what different parts
of the code do.
3. The words in red are variables that need to be changed depending on which
season is being considered. In this case, it is the main winter season, ie. the
months of December and January.
PRO himap_humidity
; Read in the data.
restore, 'modified_himap1_time_corrected_eot.sav'
data = data (4410:103382) ; These boundaries represent the time on South Col, beginning from 17th May 2002 to 25th
April 2003.
dummy = WHERE (data.month EQ 12 or data.month EQ 1 , count) ; This determines the period we will focus on, in this
case, the months of December and January.
IF count NE 0 THEN data_seasonal = data (dummy)
; Set up a postscript plot.
set_plot, 'ps'
!P.MULTI = [0, 1, 1]
device, filename = 'himap_dec2002_to_jan2003_relative_&_specific_humidity.ps', $ ; This is the name of the file to
which the postscript plot is written.
XSIZE = 23.5, YSIZE = 16.0, XOFFSET = 2, YOFFSET = 28, $
BITS = 8, /color, /landscape
bin = 5 ; This sets the bin size, in this case, to 5 minutes.
n = (60/bin) * 24 ; This determines the number of readings over 24 hrs.
relative_humidity = FLTARR (n) ; This creates a floating point array with the given dimensions.
saturation_vapour_pressure = FLTARR (n)
specific_humidity = FLTARR (n)
hr = 0
m=0
; This FOR loop calculates the relative humidity and specific humidity readings for each 5 minute bin.
FOR j=0, n-1 DO BEGIN
dummy = WHERE (data_seasonal.hour EQ hr and data_seasonal.minute GT m and data_seasonal.minute LT m+bin,
count)
IF count NE 0 THEN data_specific = data_seasonal (dummy)
relative_humidity (j) = MEAN (data_specific.rh)
saturation_vapour_pressure (j) = 0.611 * exp ((17.27 * MEAN (data_specific.temp1)) / (MEAN (data_specific.temp1) +
237.3))
specific_humidity (j) = relative_humidity (j) * saturation_vapour_pressure (j)
m = m + bin
IF m GT 60-bin THEN BEGIN
m = 0 ; This resets m to zero at the end of 1 hour for the next FOR loop.
hr = hr + 1 ; This moves the FOR loop on to the next hour.
ENDIF
ENDFOR
20
x = FINDGEN (n)/ (60/bin) ; This sets up the horizontal axis.
plot, x, relative_humidity, $
title = 'Relative and specific humidity at South Col (December 2002 - January 2003)', $ ; This is the label for
the title of the plot.
xtitle = 'Local Time (Hr)', $
ytitle = 'Relative humidity (%)' , $
yrange = [ (min (relative_humidity)), (max (relative_humidity))], xstyle = 1, xrange = [0, 24], $
ystyle = 8 ; This ensures y-axis is drawn on only one side of the plot.
axis, yaxis = 1, yrange = [FLOOR (min (specific_humidity)), CEIL (max (specific_humidity))], ystyle = 1, $ ;
The AXIS procedure draws a new axis, in this case, on the other side of the plot.
ytitle = 'Specific humidity', /save ; SAVE keyword saves the new scale for use by subsequent overplots.
oplot, x, specific_humidity, linestyle = 3 ; The OPLOT procedure overlaps a plot of the data on an existing axis.
xyouts, 1, CEIL (max (specific_humidity)) - 0.2, '_______ : Relative humidity'
xyouts, 1, CEIL (max (specific_humidity)) - 0.3, '_ . _ . _ : Specific humidity'
; This marks and labels the sunrise timing on the plot.
sr = 6.75 ; This is calculated as a fraction of 60, for eg. 0.75 = 45 / 60.
oplot, [sr, sr], [FLOOR (min (specific_humidity)), CEIL (max (specific_humidity))], linestyle = 1
xyouts, sr - 1.4, FLOOR (min (specific_humidity)) + 0.2, 'Sunrise 06:45'
; This marks and labels the sunset timing on the plot.
ss = 17.23 ; This is calculated as a fraction of 60, for eg. 0.23 = 14 / 60
oplot, [ss, ss], [FLOOR (min (specific_humidity)), CEIL (max (specific_humidity))], linestyle = 1
xyouts, ss - 1.4, FLOOR (min (specific_humidity)) + 0.2, 'Sunset 17:14'
device, /close
END
21