Introduction
Water vapour plays several distinct roles in the atmosphere. In the
lower troposphere it is particularly relevant for weather, whereas it
is particularly relevant for climate in the upper troposphere and the
stratosphere . The relevance for climate originates
from the peculiar molecular line spectrum of the H2O
molecule. It has strong spectral lines at wavelengths exceeding
16 µm (rotation band) and at 6.3 µm
(vibration–rotation band). These get optically thick in the upper
troposphere and the stratosphere, that is, a satellite instrument that
observes the Earth in these wavelength bands cannot look deeper into
the atmosphere than into these emitting layers. Radiation from
further below gets absorbed before it can leave the
atmosphere. Although the amount of water vapour in these layers is
only a small fraction of its total amount in the atmosphere, the
contribution of water vapour in the upper troposphere to radiative cooling
of the atmosphere is disproportionately large . In
total, water vapour contributes two-thirds of the natural greenhouse
effect.
There is a long-standing debate on the role of upper tropospheric
water vapour in a changing climate, i.e. under conditions where the
troposphere gets warmer. computed the change of
surface temperature required to restore the net longwave radiation at
the ground following an increase of the atmospheric CO2
content. He assumed a fixed relative humidity which implies an
increase of water vapour amount in all atmospheric levels where the
temperature increases. His results suggested that the increase of
water vapour amount with increasing temperature causes
a self-amplification effect, i.e. he found that water vapour is able
to feed into a positive greenhouse feedback loop. Some years later,
envisaged a world with constant relative humidity in
their radiation-convection model and found that CO2 doubling
led to a surface temperature rise of 2.3 K, whereas a former
version of this model with fixed absolute humidity resulted only in
a surface warming of 1.3 K for the same forcing
. These were the first model manifestations of
a potential water vapour feedback in a global warming scenario.
criticised that the discussion of a potential
climate warming due to CO2 enhancement was focussed solely on
the radiative mode of the cooling of the Earth surface. He argued that,
particularly in the tropics, convective transport of latent heat into
the middle troposphere would short-circuit the radiative resistance
imposed by the bulk of the water vapour column in the lower
troposphere and that convection must be taken into account in an
assessment of the water vapour feedback. The crucial question was
whether convection enhances or diminishes the concentration of upper
tropospheric water vapour. Early attempts to check this consisted of
measuring the vertical distribution of water vapour in regions of more
or less convection or in cold and warm seasons
. It turned out that convective regions are more humid
than non-convective ones with a higher relative humidity over the
entire tropospheric column . Comparing summer vs.
winter values of middle and upper tropospheric water vapour
concentrations using satellite data showed that increased convection
leads to increased water vapour above the 500 hpa level
. Comparison of water vapour profiles above the
tropical western and eastern Pacific regions led to the same
conclusion, namely that increased convection does not lead to a drying
of the upper troposphere . Studies using global
circulation models (GCMs) showed that the specific humidity increases
at all levels throughout the atmosphere in reaction to climate
warming. Absolute (i.e. additive) changes are largest at the ground
and decrease upwards in a more or less exponential manner. However,
relative (i.e. multiplicative) changes are largest in the upper
troposphere, and exceed a factor of 2 . Another
GCM study showed that the feedback on global mean surface temperature
changes, due to extratropical free tropospheric water vapour, exceeds
the corresponding feedback of free tropospheric water vapour in the
tropical zones by 50 % . These findings
motivate further studies of changes of upper tropospheric water vapour
in midlatitude zones.
The old results obtained by and led
to the widely assumed view that the relative humidity, RH, will stay
approximately unchanged in a warmer world . While this
might turn out true in a global-mean sense, it is probably not true
locally. A robust feature of climate models run under the assumption
of surface warming is an increase of RH in the global upper
troposphere above 200 hPa and at ±10∘ around the
equator up to 500 hPa, a decrease in the subtropics and in the
tropics between 500 and 200 hPa and insignificant change of
RH elsewhere see e.g.and references cited
therein. This “elsewhere” includes, in
particular, the free troposphere of the midlatitudes, where we should
thus expect small changes of RH, at most.
However, a recent comparison
of decadal means of upper tropospheric humidity (UTH, a radiance-based
quantity defined later in Eq. 2), for the decades 1980–1989 and
2000–2009, respectively, and performed for the 30 to 60 ∘N
latitude belt, showed a moderate (few percent) but statistically
significant increase of UTHi (UTH with respect to ice) over large
regions in this zone .
The data for this study had
been obtained from 30 years of intercalibrated satellite data
from the High-resolution Infrared Radiation Sounder (HIRS) instruments
on the NOAA polar orbiting satellite series . This
finding was in accordance with previous studies detecting a moistening
of the upper troposphere, both globally and in the zonal mean
. Those trends were based both on HIRS and Microwave
Sounder Unit (MSU) data. It was later shown that the global mean upper
tropospheric moistening could not be explained by natural sources and
resulted primarily from an anthropogenic warming of the climate
.
The apparent contradiction between an increasing UTH and a virtually
constant RH in the midlatitudes inspired us to investigate how the
upper tropospheric humidity can change while the relative humidity is
constant. It is possible to treat this question with analytical
methods and radiative transfer calculations. This paper is only
intended to demonstrate the principles and to give rough estimates.
Therefore, the main aim of our study is to understand if and how the
UTH can change in cases where the RH will remain constant in a warming
environment. As such, our methods are valid only over regions where RH
remains constant. Our findings show that the UTH can still change
under constant RH because of the weighting function that defines UTH.
The weighting function can change because of changes in two quantities
that define it: the peak emission altitude and the water vapour scale
height. We describe the mechanisms with which the two properties of
the weighting function can modify the UTH under the assumption of
constant RH.
Generic weighting functions K(z,z‾,H) for water
vapour spectral transitions of various line strengths (parameter
z‾) and for various exponential vertical profiles of
water vapour concentration (scale height H). The left panel
shows the dependence of K(z) on z‾ for
a fixed scale height of 2 km. The labels at the curves are
the corresponding values of z‾ in kilometres. Notice that the
half widths of the weighting functions are almost independent of
z‾. The right panel shows the dependence
of K(z) on the water vapour profile, i.e. on H. The numbers at
the curves indicate the chosen value of H in kilometres.
Analytical calculations
The UTH, as obtainable from radiation
measurements from nadir sounders, such as HIRS or
the SEVIRI instrument on Meteosat is a weighted
mean over a vertical profile of relative humidity,
RH(z). There is some freedom in the choice of weighting
function also termed weighting kernel or Jacobian,
see. For the present purpose,
it suffices to use a generic weighting function of the form
cf.:
K(z,z‾,H)=H-1e-(z-z‾)/Hexp-e-(z-z‾)/H,
where z is altitude, z‾ is the altitude where the
weighting function peaks (which is the altitude where the optical
depth down from the top of the atmosphere reaches unity) and H is
the scale height of an exponential water vapour profile. A derivation
of this generic kernel function is given in Appendix A. The shape
of this function is illustrated in Fig. for various choices
of peak altitude and scale height.
The UTH is then given by the following integral:
UTH=∫0∞RH(z)K(z,z‾,H)dz.
If a profile RH(z) is fixed, UTH can still change whenever
the scale height and/or the altitude of peak emission are changing.
Effect on scale height
Fixed relative humidity under warming implies that the actual water
vapour pressure, e, and the saturation vapour pressure, e∗,
change in the same proportion:
dlne=dlne∗=LRwT2dT,
where Rw is the gas constant of water vapour.
There are two types of relative humidity at subzero temperatures, with
respect to supercooled liquid water (RHw) and
with respect to ice (RHi). In the equation above,
we have allowed two possibilities for the latent heat, L,
which can be latent heat of evaporation
(Lw=2.50 MJkg-1) or sublimation
(Li=2.84 MJkg-1). Obviously, vapour pressure
can change in proportion to the saturation vapour pressure only for
one of these versions. If RHi(z) is constant,
then RHw(z) would change on warming, and vice
versa. Thus, only one version of relative humidity can be constant
under warming conditions (which might be a little surprising, because
this has never been explicitly stated to the authors' knowledge; see
the Appendix B). The following derivation is valid for both forms of
the Clausius–Clapeyron equation, thus we will not show an index “i”
or “w”.
The vapour pressure scale height is defined as
H=-dlnedz-1.
This means
-d(H-1)dt=ddtdlnedz=ddzdlnedt=ddzLRwT2⋅dTdt=LRwddz1T2⋅dTdt+1T2⋅d2Tdzdt=LRw-2T3dTdz⋅dTdt+1T2⋅d2Tdzdt=LRwT2d2Tdzdt-2TdTdz⋅dTdt.
Now we set
ΔT=(dT/dt)ΔtandΔH=(dH/dt)Δt,
and compute
the corresponding Δ(H-1) as follows:
Δ(H-1)=-LRwT2ddz(ΔT)-2TΓΔT=-LRwT2ΔΓ-2TΓΔT=-LΓRwT2ΔΓΓ-2ΔTT.
Γ=dT/dz is the temperature lapse rate,
ΔT is the temperature change in a certain altitude and
dΔT/dz is the “lapse rate” of this warming
tendency, or in other words, the change of the lapse rate itself.
Let us make a few estimates: first,
L/RwT2≈0.1 K-1. Then, ΔT/T≈0.01 and Γ≈-0.01 Km-1. Then the
second right-hand side (rhs) term times prefactor is of the order
-10-5 m-1. The change of the lapse rate can be
estimated from the result of a climate model simulating a world under
CO2 doubling their Fig. 4a.
ΔΓ is either zero, if the temperature would change equally
at all altitudes, as in middle latitudes, or we can assume that
ΔT changes approximately in proportion to the actual
temperature (i.e. ΔT(z)∝T(z) with a proportionality
factor of about 0.01 to be consistent with the previous assumptions),
and thus ΔΓ≈0.01Γ. However, for the tropics, the
results of suggest that ΔT changes more
in the upper troposphere than close to the ground. We take this into
account for our estimate by allowing the first factor to have a
magnitude of up to half the second one, that is, we locate ΔΓ/Γ
in the interval [-ΔT/T,+ΔT/T], which gives
Δ(H-1)=LΔTΓRwT3⋅(2+x)withx∈[-1,+1].
Summarising, we estimate Δ(H-1)≈-10-5 m-1. Now
dH=-d(H-1)⋅H2.
H itself is of the order 2 km, thus we have ΔH≈40 m.
The scale height of water vapour in the tropopause can thus be expected to
increase by a few tens of metres as a consequence of tropospheric warming,
even if the profile of relative humidity should be unchanged.
To corroborate these estimates we have analysed the changes in the
scale height of water vapour (ΔH) using observed air
temperatures (T) and observed changes (ΔT) in the past
30 years. The data of T and ΔT were provided by the
study of and refer to the period 1980–2011. Trend
estimates were calculated from NCEP reanalysis data after filtering
out natural variations such as the quasi-biennial oscillation (QBO) and the 11-year solar cycle
and excluding periods following major volcanic eruptions. The trends
were derived for various latitude zones and atmospheric layers. In our
analysis, we have used the observed temperature trends as input to
Eq. (8) to estimate the changes in scale height from 1980 to 2011 in
the northern high latitudes (60–90 ∘N), the midlatitudes
(30–60 ∘N) and the tropics (5–30 ∘N), for the
atmospheric layers of 1000–925, 925–500 and
500–300 hPa. The trends were given per decade so we
multiplied the results by 3 to calculate the overall change in the
past 30 years (ΔT). Table summarises the
observed temperature changes taken from the study by
, as well as the calculated changes in scale height
of water vapour at the three mentioned latitudinal belts.
The ΔH calculations were done using Eqs. (8) and (9). In Eq. (8) we
considered a fixed temperature lapse rate (Γ) of
-0.01 Km-1 and different ratios L/RwT3 for each
layer. Lw, the enthalpy of evaporation, has been used for
the layers of 1000–925 and 925–500 hPa, and
Li, the enthalpy of sublimation, has been used for the
layer of 500–300 hPa. The range given for ΔH
corresponds to the assumptions of ΔΓ=0
(corresponding to the middle value given)
or ΔΓ=±0.01Γ.
In Eq. (9) we considered a fixed water vapour scale height
(H) of 2 km to derive the final ΔH in metres.
From Table it appears that the observed changes in ΔH during the past 30 years were generally small. The largest
changes in the scale height of upper tropospheric humidity (layer
500–300 hPa) were calculated for the high latitudes, where
ΔH increased by 30±15 m. The respective changes in
the middle latitudes and the tropics were estimated to be
15.6±7.8 and 9±4.5 m, respectively.
The changes in
scale height were larger in the lower troposphere than in the upper
troposphere. This can be explained by the fact that the ratio ΔT/T which is proportional to ΔH, was larger in the lower
atmospheric layers than at 300–500 hPa (see Table 1), and
therefore the ΔH was larger as well. From the analysis it
appears that the high latitudes will probably be the most vulnerable
to UTH changes in a warming climate. However, our calculations, which
were based on observed changes of layer-mean air temperatures, give us
a good indication as to the extent of the changes in the water vapour
scale height that can occur in the atmosphere. These are very small
indeed; it would be very difficult to compute them directly from data
sets of humidity profiles with sufficient precision.
Effect on peak emission altitude
In this section, we show calculations of how the peak emission altitude
(where the optical depth reaches unity) changes with changing
temperature but fixed relative humidity. This change is generally
different for each spectral line, thus it is a function of wavenumber
(or wavelength).
For the calculation, we use SBDART Santa Barbara DISORT
Atmospheric Radiative Transfer,. This code is based
on a LOWTRAN 7 transmission model, having a spectral resolution of
20 cm-1, which suffices for the present purpose. We chose
the wavelength range 4.6–10 µm and used the spectral
resolution of LOWTRAN, 20 cm-1. This wavelength range
contains in particular the strong water vapour vibration–rotation band
at about 6.3 µm, which is the basis for determining UTH
(e.g. channel 12 of HIRS). With this setting we performed three model
runs for a cloud-free midlatitude summer atmosphere, one with the
standard profiles of temperature and water vapour concentration (from
the 1972 compilation of standard atmospheres by McClatchey), and two
that have increased temperature by 0.5 and 1 K, respectively,
up to 12 km altitude and correspondingly increased water
vapour concentration, such that the relative humidity is the same as
before. For the transition between water vapour concentration and
relative humidity we use SBDART's function relhum. From the output of
the model runs we then take the optical depth in each wavelength
interval, τλ, and search by linear interpolation that
altitude, z‾(λ), where τλ=1.
Figure shows the results. The left panel shows the peak
emission altitude for the standard midsummer atmosphere, while the
right panel shows how the emission altitude increases when the
temperature throughout the tropopause increases by 0.5 and
1 K.
There are certain narrow bands for which SBDART computes
a surprisingly strong increase of z‾, amounting to several
hundred metres. These are artefacts; a control run with a slightly
shifted wavelength range (4.5–9.9 µm) produces only two
peaks and at different wavelengths.
Over most of the
vibration–rotation band, z‾ increases by about 30 to
70 m.
Layer-mean air temperatures (T), changes in air temperatures
(ΔT over 30 years), ratios (ΔT/T) and changes in scale
height of water vapour (ΔH) at three latitudinal belts:
60–90, 30–60 and 5–30 ∘N. The data of T and ΔT
were provided by the study of and refer to the
period 1980–2011. The ΔT were calculated from NCEP
reanalysis and filtered from natural variations.
60–90 ∘N
Layer
ΔT (K)
T (K)
ΔT/T
ΔH (m)
1000–925 hPa
2.52
265.0
0.00951
58.6±29.3
925–500 hPa
0.87
255.9
0.00340
22.6±11.3
500–300 hPa
0.75
230.8
0.00325
30.0±15.0
30–60 ∘N
Layer
ΔT (K)
T (K)
ΔT/T
ΔH (m)
1000–925 hPa
0.84
282.6
0.00297
16.2±8.1
925–500 hPa
0.78
269.9
0.00289
17.2±8.8
500–300 hPa
0.45
241.4
0.00186
15.6±7.8
5–30 ∘N
Layer
ΔT (K)
T (K)
ΔT/T
ΔH (m)
1000–925 hPa
0.36
297.1
0.00121
6.0±3.0
925–500 hPa
0.51
282.7
0.00180
9.8±4.9
500–300 hPa
0.30
254.0
0.00118
9.0±4.5
An independent analytical estimate of Δz‾ can be
performed using equations in Sect. 5a; in the following we
use their nomenclature. Their Eq. (15) shows that the
optical depth, at the level where the temperature is 240 K,
increases by 5 % when the temperature increases by
1 K. The reason for this is the corresponding increase of the
air mass factor (water vapour above that level) by approximately
10 % (Eqs. 12 and 8), that is, from u0 to 1.1u0.
Calculating the vertical distance from the 240 K level to that
level where the air mass is 10 % lower should be an appropriate
estimate for Δz‾ (even if z‾ is not
generally at the 240 K level). For this purpose, we combine
Eqs. (11) and (12) of , use the hydrostatic equation
to transform the vertical coordinate from pressure to altitude, use
the gas equation to get rid of the density and arrive at:
Δz‾≈Δuu0T0Raeλβg≈68m,
with the gas constant of air,
Ra=287 J(kgK)-1, the gravitational
acceleration, g=9.81 ms-2, Δu/u0≈0.1,
and the constants λ=23.1 and β=0.1 from the latter
can be found in Eq. 22. Thus, this simple analytical
estimate confirms the result from the radiative transfer simulation,
viz. that the peak emission level rises as a consequence of climate
warming by about 70 mK-1 of temperature increase.
Left: peak altitude (in kilometres) for infrared emission to space
(i.e. altitude where the optical depth reaches unity) as
a function of wavelength (in micrometres) in the spectral region of
the strong v2 water vapour vibration–rotation band. The
calculation has been done for a standard midlatitude summer
atmosphere. Right: change of the peak altitude (in metres) after
a climate warming of 0.5 K (thick line) and 1 K
(thin line) throughout the troposphere (up to 12 km) with
constant relative humidity. The calculation has been performed
for the midlatitude summer atmosphere and the same wavelength
region as in the left panel.
Absolute (left) and relative (right) change of altitude
dependent weights in the kernel function (Eq. 1) after an increase
of z‾ from 7.00 to 7.05 km and a small
increase of H from 2.00 to 2.01 km.
Results and discussion
Impacts on the kernel function
The changes in scale height and peak emission altitude lead to
a change of the factor (z-z‾)/H in the kernel function; it
increases in a warming climate. The corresponding absolute and
relative changes of the kernel function are shown in Fig.
for an assumed increase of z‾ from 7.0 to
7.05 km and a small increase of H from 2.00 to
2.01 km.
The curves show that weights below 7 km were reduced and
weights above 7 km increased. The colder layers thus gain in
weight, while the warmer ones lose importance. It is particularly
noteworthy that not only the immediate neighbourhood of the old or new
z‾ is affected by such a change; instead, the kernel
function is modified everywhere. Far below or above z‾
these changes are negligible because the original values were
negligible anyway. However, closer to z‾ – yet not only in the
immediate neighbourhood – these changes are significant; they lead to
modification of the retrieved UTH values even if the relative humidity
profiles do not change at all. How large these changes are, will now
be tested with real radiosonde data.
Application to real profiles
In this section, we examine the effects of the changes in the kernel
function on the UTH field using real humidity profiles from
radiosondes. We wanted to investigate how UTH changes when both
quantities of the weighting function change according to our previous
calculations. We assume again an increase in z‾ from
7.00 to 7.05 km and an increase in H from 2.00 to
2.01 km
(a more flexible approach is described below).
We have analysed all the humidity profiles for the
period of February 2000 to April 2001 as obtained from the Lindenberg
corrected RS80A routine radiosondes. Measurements were performed 4
times per day corresponding roughly to times 00:00, 06:00, 12:00 and
18:00 UTC. Each radiosonde profile provides information on the
pressure, temperature and relative humidity with respect to liquid
water per height. In total, we analysed 1564 available humidity
profiles. Details on the radiosonde data can be found in the study by
.
Left: profiles of relative humidity (with respect to water)
measured with radiosondes launched at Lindenberg, Germany on 15
July 2000, 00:00 UTC (blue) and 18 July 2000, 00:00 UTC
(magenta); these profiles have been multiplied with the difference
of the two kernel functions of Fig. , resulting in the
red profile for 15 July and the green profile for 18 July. Right:
differences of UTH when the two kernel functions of Fig.
are applied to 1564 profiles of RH, measured from February 2000 to
April 2001 with radiosondes launched at Lindenberg, Germany.
The radiosondes provide relative humidity profiles with respect to
water. A conversion to relative humidity profiles with respect to ice
is only possible at subzero temperatures. Since our radiosonde
profiles contain temperature above 0 ∘C, we illustrate the
impact of kernel-function changes on UTH for UTH with respect to
liquid water.
In our analysis, we used two weighting functions; one weighting
function with standard peak emission altitude and standard scale
height (7.00 and 2.00 km, respectively) and a second
weighting function with increased peak emission altitude and increased
scale height (7.05 and 2.01 km, accordingly). Each
weighting function was multiplied with the profiles of RH from the
surface up to the lower stratosphere (≈16 km). The UTH
was calculated from the integral as given in Eq. (2), using
a trapezoidal rule. Since our target was to estimate the impact on
UTH from different weighting functions, we estimated for each profile
two values of UTH; one value using the first weighting function and
another value using the second weighting function. We then estimated
the differences in UTH per profile to see the results.
The right panel of Fig. shows the differences in UTH
resulting from the change in the two quantities of the weighting
function, i.e. of z‾ by 0.05 km and of H by
0.01 km. The results indicate small differences in UTH
between 0.3 and -1.2 %. Mean differences were of the order
-0.2 % with standard deviation of differences of about
0.2 %. Evidently, it can be inferred that a change in the
properties of the weighting function, i.e.
Δz‾=0.05 km and ΔH=0.01 km,
can result in small changes in UTH of not more than roughly
±1 %.
A similar exercise has been conducted with two more
radiosonde stations, one tropical station in Abidjan (Côte d'Ivoire,
5.25 ∘N) and one polar station on Bear Island (Norway,
74.5 ∘N). We retrieved profiles from January and July
2015 for both stations from the radiosonde archive at the University
of Wyoming (http://weather.uwyo.edu/upperair/sounding.html). It is
not appropriate to dwell further on details of these data because,
unlike the Lindenberg data, the upper tropospheric relative humidity
values of the Abidjan and Bear Island data are not corrected. We
just took them at face value and found results that do not
qualitatively differ from those presented above, that is, most
changes are negative and their magnitudes do not exceed 1 %. The
results of this exercise are presented in Fig. .
Before closing this paragraph, it is worth noting the discussion on the left panel
of Fig. , which shows two individual profiles of RH over
Lindenberg (blue and magenta) along with the changes in the integrands
of Eq. (2) (red and green). The blue line shows the profile of RH for
15 July 2000, taken at 00:00 UTC while the line with purple colour
shows the respective profile for 18 July 2000. We note here that these
specific profiles resulted in UTH differences with opposite sign,
e.g. negative difference in UTH from the profile of 15 July and
positive difference from the profile of 18 July. Comparing the two
individual profiles in general, we can clearly see that the first
profile had higher RH than the second profile up to 7.5 km
height, lower RH between 7.5 and 8 km height, higher RH up
to 10 km and much lower RH above 10 km
height. Furthermore, the very dry layer between 2 and 3 km
altitude on 18 July is noteworthy.
UTH differences for changes in the peak emission altitude and
water vapour scale height for tropical station Abidjan and polar
station Bjørnøya (Bear Island). Peak emission altitude is assumed
to rise by 50 m, scale height by 10 m.
As stated above, the change in the kernel functions gave the lower
layers less weight and the upper layers more weight. For the present
examples, this means that the very dry layer in the lower troposphere on 18 July
was reduced in weight, and instead the moist layer between 10 and
12 km gained weight. The result of this is an increase of UTH
for 18 July. For 15 July, in turn, the main effect is the gain in
weight of the strong humidity decrease at 10 km altitude which
resulted in a decrease of UTH.
We see that the change in UTH depends not only on “climatological”
changes of scale height and peak emission altitude. For an individual
profile of relative humidity it depends strongly on the shape of that
profile. We have seen mostly negative changes at Lindenberg, but at
other locations the conditions may be different such that positive
changes would prevail.
We tested profiles from two more stations, one
in the tropics (Abidjan, Côte d'Ivoire) and one in the Arctic (Bear
Island, Norway) and found similar, mostly small changes in the
negative direction.
Whether positive or negative changes prevail depends
also strongly on the choice of an appropriate z‾, that is,
it depends on the filter function of the instrument detecting the
upper tropospheric water vapour. A high peak emission altitude (approximately
9 km or so) would already mean that much dry stratospheric air
is seen and after an increase of z‾ this would be the case even more,
such that UTH would decrease mostly. A more neutral
partition of signs of UTH changes is only possible if z‾
is located in the middle troposphere. Anyhow, the UTH changes under
the condition of constant relative humidity are small, which implies
that their detection with statistical significance needs very long
homogeneous time series.
Further discussion
In the previous section we treated the weighting function and the RH
profile as quantities that can be changed independently. We took a
standard weighting function and applied it to all RH profiles and
then we did the same for a modified weighting function. However, in
reality the weighting function for the actual radiance measurement is
a function of the RH and temperature profile. For each radiosonde
measurement, there is exactly one corresponding weighting function
that determines the radiance that reaches the satellite.
Therefore, in our simple calculations of UTH from Eq. (2) we have
neglected the fact that the kernel function itself depends on the
profile of relative humidity and assumed that the weighting function
and the RH profile are separable parameters. This trick was motivated
by our assumption of constant relative humidity. However, “constant”
is meant here in a climatological sense only; individual radiosonde
launches still give individual RH(z) profiles and thus the weighting
function changes from profile to profile. The question is then whether
this dependence could lead to modifications of our conclusions. In
general, this is not the case, as we argue in the following. To
address the issue we have calculated the UTH differences by using
different weighting functions for each RH profile of the Lindenberg
data.
Although typically there are strong variations in each profile
of relative humidity, a corresponding profile of absolute humidity
would still be more or less an exponential one superposed with minor
wiggles. This would cause merely a small correction to the shape of
the kernel function (details in Appendix A).
Left panel: peak emission altitude (kilometres) at Lindenberg for
1560 radiosonde profiles from February 2000 to April 2001. Right
panel: UTH differences for changes in the peak emission altitude and
water vapour scale height. Peak emission altitude is assumed to rise
by 50 m, scale height by 10 m.
The altitude of the kernel's peak may shift considerably, for instance
from dry to moist days, and the effect of such shifts have been
tested. We use again the equations of together
with profiles from McClatchey's (1972) US standard atmosphere to
estimate that an optical thickness of unity is approximately reached
at that level where the temperature is 242 K in the tropics
and 236 K in the midlatitudes. For this calculation, we assume
climatological mean relative humidities of 25 % in the tropical upper
troposphere and 45 % in the extratropical one their Fig. 4,
top right. With this information, we use the profiles from
Lindenberg again, but this time with individual selection of
z‾, such that T(z‾)=236 K, and then
with the assumption that z‾ would rise 50 m due to
climate change. Although the peak altitude, z‾, shows
large seasonal and daily variations (Fig. , left panel), as
expected, the UTH differences do not qualitatively differ from
the case where we simply assume a constant peak altitude at
7.00 km (Fig. , right panel). Of course, the
individual UTH values depend strongly on the individual
z‾, but from the present analysis it appears that their
differences do not. Still we find predominantly negative changes with
a magnitude of less than 1 %.
Summary and outlook
In this paper, we treated the question how the upper tropospheric
humidity can change in regions where relative humidity will only
marginally change as a consequence of tropospheric warming. This is
possible since UTH is a weighted mean over the profile of relative
humidity. This mean can change once the weights change even when the
RH(z) profile stays constant.
Two quantities in the weighting function can change: the scale height
of the water vapour concentration profile and the peak emission
altitude (which varies with wavelength). We showed that the change of
the water vapour concentration scale height is rather small, of the
order 10 m, with latitudinal and vertical variations. In the
midlatitude upper troposphere it might have been increased by
10 m between 1980 and 2010, which is a relative change of less
than 1 %.
The peak emission altitudes in the 6.3 µm band of water vapour
generally increase by around 30 to 70 m for a temperature
increase throughout the troposphere of 0.5 to 1 K.
An analytical calculation using
empirical formulae provided by led to a similar
increase of about 70 m for the whole band.
We applied the computed changes of the kernel function to 14 months of
real radiosonde profiles of relative humidity and found that mostly
the resulting UTH is smaller than before after increases of scale
height and peak emission altitude. The absolute changes of UTH due to
changes in the kernel functions are however very small, typically
smaller than 1 %. Such changes would hardly be detectable even in
long humidity time series. The detection of larger changes, in turn,
implies that the condition assumed in this paper, that is, constant
relative humidity, is violated, that is, absolute UTH changes of more
than 1 % or so point to systematic (climatological) changes in
relative humidity.
Determining decadal changes (2000–2009 vs. 1980–1989) of UTHi for
the northern midlatitudes, 30–60 ∘N, from intercalibrated
HIRS data resulted in statistically highly
significant increases of more than 2 % in a large fraction of this
latitude belt. As we see from the present analyses, such an increase
would be unexpected under the assumption that the relative humidity
would have stayed nearly constant during these 30 years, both
for the size of the effect (exceeding 1 %) and its main direction
(positive). Based on the observed increase of UTHi we may
conclude that the relative humidity itself must have increased as well
between 1980 and 2009 in large parts of the northern midlatitudes.