Journal of Geophysical Research: Atmospheres
RESEARCH ARTICLE
10.1002/2013JD020772
Key Points:
• Stratospheric H2 O change with temperature less than expected from
Clausius-Clapeyron
• This residence time effect is result
of spatiotemporal variance of
temperature
• Quantify radiative temperature
response to change in upwelling
Correspondence to:
S. Fueglistaler,
[email protected]
Citation:
Fueglistaler, S., Y. S. Liu, T. J. Flannaghan,
F. Ploeger, and P. H. Haynes (2014),
Departure from Clausius-Clapeyron
scaling of water entering the stratosphere in response to changes
in tropical upwelling, J. Geophys.
Res. Atmos., 119, 1962–1972,
doi:10.1002/2013JD020772.
Received 21 AUG 2013
Accepted 28 JAN 2014
Accepted article online 4 FEB 2014
Published online 25 FEB 2014
Departure from Clausius-Clapeyron scaling of water
entering the stratosphere in response to changes
in tropical upwelling
S. Fueglistaler1 , Y. S. Liu2 , T. J. Flannaghan1 , F. Ploeger3 , and P. H. Haynes4
1 Department of Geosciences/AOS, Princeton University, Princeton, New Jersey, USA, 2 School of Geosciences, University of
Edinburgh, Edinburgh, UK, 3 Institute of Energy and Climate Research, Stratosphere (IEK-7), Forschungszentrum Juelich,
Juelich, Germany, 4 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK
Abstract Water entering the stratosphere ([H2 O]entry ) is strongly constrained by temperatures in the
tropical tropopause layer (TTL). Temperatures at tropical tropopause levels are 15–20 K below radiative
equilibrium. A strengthening of the residual circulation as suggested by general circulation models in
response to increasing greenhouse gases is, based on radiative transfer calculations, estimated to lead to
a temperature decrease of about 2 K per 10% change in upwelling (with some sensitivity to vertical scale
length). For a uniform temperature change in the inner tropics, [H2 O]entry may be expected to change as
predicted by the temperature dependence of the vapor pressure, referred here as “Clausius-Clapeyron (CC)
scaling.” Under CC scaling, this corresponds to ∼1 ppmv change in [H2 O]entry per 10% change in upwelling.
However, the change in upwelling also changes the residence time of air in the TTL. We show with
trajectory calculations that this affects [H2 O]entry , such that [H2 O]entry changes ∼10 % less than expected
from CC scaling. This residence time effect for water vapor is a consequence of the spatiotemporal variance
in the temperature field. We show that for the present-day TTL, a little more than half of the effect is due
to the systematic relation between flow and temperature field. The remainder can be understood from
the perspective of a random walk problem, with slower ascent (longer path) increasing each air parcel’s
probability to encounter anomalously low temperatures. Our results show that atmospheric water vapor
may depart from CC scaling with mean temperatures even when all physical processes of dehydration
remain unchanged.
1. Introduction
Water vapor is the strongest greenhouse gas under current climatic conditions. Its abundance in the
atmosphere is strongly constrained by the near-exponential temperature dependence of the water vapor
pressure, leading to a decrease in saturation mixing ratios by 4 orders of magnitude from the surface to
the very cold tropopause in the tropics. Air enters the stratospheric overworld (terminology following
Hoskins [1991]) predominantly in the tropics, such that the region of the temperature minimum around
the tropical tropopause effectively controls water entering the stratosphere ([H2 O]entry ). While this general mechanism has been known since Brewer’s [1949] seminal analysis of midlatitude stratospheric water,
the details of the processes controlling the last dehydration remain incompletely understood [Jensen and
Pfister, 2004; Luo et al., 2003; Fueglistaler et al., 2009b]. These uncertainties notwithstanding, the observed
relation between [H2 O]entry and tropical mean tropopause temperature variations on seasonal and interannual time scales is testament to the strong constraints of the average temperatures around the tropical
tropopause on [H2 O]entry [Mote et al., 1996; Bonazzola and Haynes, 2004; Randel et al., 2004; Fueglistaler et al.,
2005; Fueglistaler and Haynes, 2005; Schoeberl et al., 2012; Fueglistaler et al., 2013].
The dissipation of upward propagating waves in the stratosphere forces a diabatic residual circulation with
upwelling over low latitudes and downwelling over high latitudes [Holton et al., 1995]. The forced ascent
drives temperatures below radiative equilibrium, establishing in steady state a balance between radiative heating and diabatic ascent. The relation between divergence of the radiative fluxes (producing the
radiative heating/cooling) and the vertical temperature structure is generally complex and only for specific cases does the Newtonian cooling approximation (a linearization of radiative heating in the form
Q(z) = (Te (z) − T(z))∕𝜏(z), where Q is radiative heating, z is altitude, T is temperature, and the two fitting
terms Te and 𝜏 are interpreted as the radiative equilibrium temperature and radiative damping time scale)
FUEGLISTALER ET AL.
©2014. American Geophysical Union. All Rights Reserved.
1962
Journal of Geophysical Research: Atmospheres
10.1002/2013JD020772
provide a physically correct description. That is, in general, radiative heating is a function of the temperature and tracer structure of the entire profile. However, in practice the sign of the relation at a specific
altitude—larger radiative heating at lower temperatures—follows the expectations from the Newtonian
cooling approximation in the stratosphere down to the tropical tropopause layer (TTL) [see, e.g., Fueglistaler
et al., 2009b]. Hence, forcing the system with an increase in diabatic upwelling will lower temperatures
around the tropical tropopause, leading to a drier stratosphere. Interactions between the circulation and
radiatively active trace gases (specifically ozone, see Avallone and Prather [1996]) amplify the temperature
response substantially in the tropical lower stratosphere [Fueglistaler et al., 2011].
Coupled chemistry-climate models predict a robust increase in the strength of the stratospheric residual circulation, in particular in the lower stratosphere in response to increasing greenhouse gases [Garcia
and Randel, 2008; Butchart et al., 2010; SPARC CCMVal, 2010; Shepherd and McLandress, 2011]. Based on
the above argument, this increase in upwelling would lead to lower temperatures and consequently a
drier stratosphere.
Here, we analyze the impact of a prescribed change in the diabatic residual circulation on water entering the stratosphere. We evaluate the time and zonal mean temperature response with radiative transfer
calculations, considering forcing scales corresponding to a change in the entire lower stratosphere and
corresponding to a change confined to the depth of the TTL (these scales correspond to what is sometimes denoted as the “deep” and “shallow” branches of the stratospheric residual circulation, see, e.g.,
Gerber [2012]).
For the case where these dynamically forced changes in temperature are uniform in the inner tropics and
all else being equal, [H2 O]entry would change as expected from the temperature dependence of the water
saturation mixing ratio. We will refer to this as Clausius-Clapeyron scaling of [H2 O]entry .
However, the change in upwelling also corresponds to a change in the transport time scale of air entering
the stratosphere. In this paper, we will show that the change in transport time scale also changes [H2 O]entry ,
i.e., a change in transport time scale changes [H2 O]entry even if the temperature field is not modified. This
residence time effect induces a robust departure from Clausius-Clapeyron scaling of [H2 O]entry in response to
dynamical forcing.
2. Clausius-Clapeyron Scaling
A small uniform change in the tropical average temperature (T ) around tropopause levels due to a change in
upwelling balanced by radiative heating (superscript “rad”),
rad
𝛿TTP
≡
𝜕TTP
⋅ 𝛿Q,
𝜕Q
(1)
where Q is the radiative heating and the subscript “TP” refers to “tropopause level,” implies a change in entry
mixing ratios
𝛿[H2 O]rad
≡
entry
sat
𝜕𝜒TP
𝜕Q
⋅ 𝛿Q,
(2)
sat
sat
where 𝜒TP
≡ pvap (TTP )∕pTP and pvap (TTP ) is the vapor pressure and 𝜒TP
the saturation (volume) mixing ratio at the temperature of the tropical tropopause. For this situation, we say [H2 O]entry changes at
the rate expected from Clausius-Clapeyron, implicitly referring to the temperature change at the tropical
tropopause. (Note that in practice the changes are directly calculated from model experiments and Taylor
series expansion is not required.)
The average mixing ratio of air entering the stratosphere is an average over air masses that experience their
last dehydration at different temperature and pressure conditions [see, e.g., Fueglistaler et al., 2005]. In order
to avoid the complications arising from the nonlinear temperature dependence of the water vapor pressure, we will discuss in this paper all changes in [H2 O]entry in terms of the corresponding change in frost
point temperature.
Equation (2) defines the change in [H2 O]entry due to radiatively induced (tropical mean) temperature
changes. The total change in [H2 O]entry , 𝛿 [H2 O]entry may be also affected by other processes controlling
FUEGLISTALER ET AL.
©2014. American Geophysical Union. All Rights Reserved.
1963
Journal of Geophysical Research: Atmospheres
10.1002/2013JD020772
dehydration that respond to changes in diabatic heating. In particular, a change in radiative heating
inevitably affects the time scale of transport across the TTL. That is, the change in [H2 O]entry from this
residence time effect is
𝛿[H2 O]residence
≡
entry
𝜕[H2 O]entry |
| ⋅ 𝛿Q
|T
𝜕Q
(3)
where the partial derivative is evaluated at a fixed temperature structure. With trajectories (see below), it is
straightforward to express [H2 O]entry as equivalent frost point temperature, and noting that changes in saturation mixing ratio arising from temperature changes are larger than those arising from pressure changes
[see, e.g., Fueglistaler and Haynes, 2005], we can relate the two effects on [H2 O]entry in terms of temperature.
residence
That is, 𝛿[H2 O]residence
∝ 𝛿TLDP
and
entry
residence
𝛿TLDP
≡
𝜕TLDP |
| ⋅ 𝛿Q
𝜕Q |T
(4)
where TLDP is the average temperature of the Lagrangian Dry Point (LDP; see section 3.2) of trajectories
representing troposphere-to-stratosphere transport.
The total change in water entering the stratosphere in response to a change in upwelling is, expressed as
rad
residence
+ 𝛿TLDP
, and the ratio
frost point temperature, 𝛿TTP
rCC ≡
residence
𝛿TLDP
(5)
rad
𝛿TTP
describes the departure from Clausius-Clapeyron scaling (denoted with subscript “CC”) of [H2 O]entry in
response to a perturbation in upwelling. For rCC = 0, [H2 O]entry scales exactly as expected from the change
in tropical average temperature. Below, we show that for a perturbation in upwelling, rCC < 0.
3. The Experiments
The TTL temperature and circulation arise from forcings both in the troposphere and the stratosphere
[Fueglistaler et al., 2009b]. Analysis of the scaling of [H2 O]entry with mean temperatures based on perturbation experiments with general circulation models (GCMs) may be difficult to interpret because (i) it is difficult
to perturb the model to produce a clean forcing in terms of a change in the zonal mean residual circulation
only and (ii) because processes other than those of interest here that affect dehydration may also respond
to the perturbation.
Hence, for the objective of this paper—understanding the fundamental mechanism relating the residual
circulation and [H2 O]entry —we use a different strategy. We calculate the radiatively induced temperature
rad
residence
, see equation (1)) and the residence time effect (ΔTLDP
, see equation (4)) separately,
change (ΔTTP
whereby we use radiative transfer calculations for the former and trajectory calculations for the latter.
In the following, we first describe the evaluation of the radiatively induced temperature change in response
to a perturbation in upwelling (section 3.1), and in section 3.2 we describe the trajectory experiments for the
residence time effect.
3.1. Radiatively Induced Temperature Change (𝚫T rad )
rad
The radiatively induced change in tropical mean temperatures, ΔTTP
(see equation (1)) is calculated using
a radiative transfer model. Since the radiative response critically depends on the vertical scale and structure
of the perturbation [Sasamori and London, 1966; Fels, 1982] in upwelling, our approach of calculating the
dynamical forcing on [H2 O]entry via its impact on residence time and via radiation separately has the additional advantage over a GCM experiment in that we can explore the radiative temperature response for a
wide range of scenarios of the perturbation in upwelling.
Calculations were done with a constant ozone profile and with an ozone profile that reflects the changes in
ozone due to the imposed upwelling [Avallone and Prather, 1996] calculated using the model of Ploeger et al.
[2012]. We found (not shown) that at tropopause levels and below (which is the important region for dehydration) the difference in the temperature change between the two calculations is only a few percent. For
FUEGLISTALER ET AL.
©2014. American Geophysical Union. All Rights Reserved.
1964
Journal of Geophysical Research: Atmospheres
10.1002/2013JD020772
Table 1. Overview of Trajectory Experiments
Experiment
Scaled Q
ΔQ
Perturbation
Qe2
Qe1 = (1 + f ) ⋅ Q, f = ±[0.1]
= Q + ΔQ, ΔQ = f ⋅ ⟨Q⟩, f = ±[0.5, 0.1, 0.05]
Comments
Retains 4-D structure, affects dispersion.
Dispersion unchanged.
simplicity, we therefore show (section 4.1) results for calculations without ozone adjustment. Similarly, calculations were done with a fixed specific humidity and fixed relative humidity profile up to the tropopause. The
temperature response with a fixed specific humidity profile is somewhat larger (a few percent) than when
the humidity profile is allowed to adjust to the temperature changes, but results depend on the assumptions made concerning how the humidity responds. We therefore show in the following the results based on
the fixed specific humidity profile and return to this aspect in the discussion in section 6.
The temperature change required to balance the prescribed perturbation in radiative heating is calculated
with an iterative algorithm similar to the so-called fixed dynamical heating calculations. That is, we calculate the radiative heating of a base profile and then iteratively modify the profile’s temperature until the
difference in radiative heating between the base and modified profile equals the prescribed perturbation in
radiative heating. Further details of the algorithm are provided in Appendix A.
residence
3.2. Residence Time-Induced Change in [H2 O]entry (𝚫TLDP
)
We estimate [H2 O]entry based on back trajectory calculations of trajectories started above the tropical cold
point tropopause at 410 K potential temperature between 30◦ S and 30◦ N on a 2◦ × 2◦ longitude/latitude
grid, using European Centre for Medium-Range Weather Forecasts (ECMWF) ERA-Interim [Dee et al., 2011]
data and a modified version of the trajectory code OFFLINE [Methven, 1997; Methven et al., 2003; Liu et
al., 2010]. In these calculations, the water vapor of each trajectory at the end point in the stratosphere is
given by the minimum saturation mixing ratio during ascent from the troposphere into the stratosphere
[Fueglistaler et al., 2005, 2013]. The location of the minimum saturation mixing ratio is the Lagrangian Dry
Point (LDP). The LDP is only evaluated for trajectories that can be traced back to the tropical midtroposphere
(here defined as reaching a potential temperature of 340 K or less) within 1 year. Typically, more than 90%
of all trajectories started between 30◦ S and 30◦ N at 410 K satisfy this condition [Liu et al., 2010] that is used
to define troposphere-to-stratosphere transport (TST). These model calculations realistically capture the
observed means, and annual and interannual variations [Fueglistaler et al., 2005; Fueglistaler and Haynes,
2005; Liu et al., 2010; Fueglistaler et al., 2013] of water entering the stratosphere.
The impact of changes in residence time is evaluated with diabatic trajectory calculations where we perturb the diabatic heating rates, while retaining for all calculations the four-dimensional temperature
(and horizontal wind) structure of ERA-Interim. The residence time effect is then the difference between
the entry mixing ratio of the perturbed case and the base case, i.e., (in terms of temperature change)
residence
ΔTLDP
= TLDP (Q0 + ΔQ) − TLDP (Q0 ), where Q0 is the radiative heating of the base case, and ΔQ is
the perturbation.
Here we present results from three different perturbation experiments, summarized in Table 1. To keep
the notation simple, TLDP refers to the average temperature at the LDP for all trajectories representing
troposphere-to-stratosphere transport of the periods discussed.
Experiment 1, “Scaled Q.” We change the model diabatic heating rates Q(𝜆, 𝜙, p, t) (where 𝜆, 𝜙, p, and t are
longitude, latitude, pressure, and time, respectively) by a fraction f , i.e.,
Qe1 (𝜆, 𝜙, p, t) = (1 + f ) ⋅ Q(𝜆, 𝜙, p, t)
(6)
where Qe1 is the diabatic heating rate for experiment 1. The advantage of this perturbation is that it retains
the full four-dimensional structure of the diabatic heating field. The disadvantage of this experiment is
that it changes the dispersion of trajectories, which in turn also affects [H2 O]entry in a manner difficult to
quantify correctly (see Liu et al. [2010] for details). All results shown are based on the averages over the
period 2001–2005 of one reverse domain filling calculation per month, which corresponds to about 300,000
trajectories (i.e., 180 × 31 (starting grid) ×12 × 5 (period) ×0.9 (TST-fraction)) for each data point.
Experiment 2, “ΔQ.” We add an offset ΔQ to the model diabatic heating rates. So long as ΔQ is slowly varying in time and space, results will not suffer from biases arising from changes in dispersion. Since we are
interested in changes in upwelling that are primarily an amplification or weakening of today’s zonal mean
FUEGLISTALER ET AL.
©2014. American Geophysical Union. All Rights Reserved.
1965
Journal of Geophysical Research: Atmospheres
p [hPa]
(a)
structure in the TTL, we prescribe ΔQ to be a
modification of Q based on a scaled version of
the zonal mean, time mean diabatic heating.
That is,
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195
Qe2 (𝜆, 𝜙, p, t) = Q(𝜆, 𝜙, p, t) + ΔQ(𝜙, p)
205
215
T [K]
delta-Q [K/day] (dash)
p [hPa]
(b)
-0.2
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-0.1
-3
-2
0.0
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0.1
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0.2
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delta-T [K] (solid)
(c)
p [hPa]
10.1002/2013JD020772
(d)
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30 34 38 42 46 50
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-25 -20 -15 -10 -5
ΔQ(𝜙, p) = f ⋅ ⟨Q(𝜆, 𝜙, p, t)⟩
(7)
(8)
where Qe2 is the diabatic heating rate for experiment 2 and ⟨.⟩ refers to the zonal mean, time
mean quantity. Here we show results only for
ΔQ calculated from the climatological mean,
annual mean time averages. Results using, for
example, seasonally evolving ΔQ are similar,
but more difficult to interpret. Note that the
time and area mean of the perturbation of
experiment 1 is identical with that of experiment 2 for the same factor f , but locally (in
longitude/time) the perturbations differ. As for
experiment 1, all results shown are based on
the averages over the period 2001–2005 of one
reverse domain filling calculation per month.
4. Results
4.1. The Radiatively Induced Change in
Temperatures and Impact on [H2 O]entry
-dT:dQ [day]
T - Te [K]
Figure 1 shows the radiatively induced temperature change in response to a change in
Figure 1. (a) The base temperature profile (black) and the temupwelling, ΔT rad , for three scenarios of the
perature profiles after relaxation against a prescribed change
in radiative heating, for three scenarios modeled after (purupwelling perturbation. (Note: These three
ple) an increase in the deep stratospheric residual circulation,
scenarios should not be confused with the
and an increase only in the TTL (i.e., shallow branch of the
two trajectory experiments; hence, we use
Brewer-Dobson circulation), with the green and brown perturthe terminology “trajectory experiments” and
bations roughly bracketing the range of length scale. (b) The
“upwelling scenarios.”) For all three scenarios,
corresponding temperature difference profiles (solid, lower x
the perturbation is—exactly as for the trajecaxis) and the prescribed change in radiative heating (dashed,
upper x axis). (c) The ratio of temperature change to heattory experiments 1 and 2—a fractional change
ing rate change (corresponding to the damping timescale
of the base radiative heating profile, with a
in Newtonian cooling). (d) The difference between observed
tapering function to control the vertical scale
temperature (T ) and equilibrium temperature (Te ) of the Newof the perturbation. The base radiative heattonian cooling approximation for the prescribed radiative
ing rate profile for the results shown here is
heating perturbations.
the climatological mean, tropical mean profile.
The three scenarios differ in their vertical scale L: Scenario A corresponds to a deep vertical structure from
the TTL up to around 10 hPa; this scenario is representative of changes in upwelling throughout the lower
stratosphere (the deep branch of the Brewer-Dobson circulation). Scenarios B and C roughly bracket the
length scale of a perturbation in upwelling due to waves dissipating in the TTL (i.e., tropical waves, [see, e.g.,
Norton, 2006; Ortland and Alexander, 2013] and synoptic waves [Plumb, 2002]).
0
The results shown in Figure 1 were obtained with a perturbation in Q of +10% of the ECMWF ERA-Interim
radiative heating rates, and while not strictly in the limit where Newtonian cooling is valid [see, e.g.,
Hitchcock et al., 2010], the ratio −dT rad ∕dQ (corresponding to the relaxation time scale 𝜏rad in Newtonian
cooling) evaluated for this specific case provides a reasonable estimate for the relation between upwelling
perturbation and temperature. As expected from the vertical length scale, the scenario A for the deep
Brewer-Dobson circulation gives the largest temperature response of about 3 K at the levels of interest (Figure 1b) for a 10% change in upwelling (corresponding to a radiative time scale 𝜏rad ∼40 days; see
Figure 1c). For the TTL length-scale scenario B, the response is about 2.5 K (𝜏rad ∼35 days), and for scenario C
FUEGLISTALER ET AL.
©2014. American Geophysical Union. All Rights Reserved.
1966
Journal of Geophysical Research: Atmospheres
1.5
delta-TLDP [K]
the response is about 1.5 K (𝜏rad ∼ 30 days). For
completeness, Figure 1d shows the departure
of observed temperatures from the Newtonian
cooling approximation equilibrium temperature evaluated for the three perturbation
scenarios. The figure shows that this temperature difference—an estimate for how far temperatures are below radiative equilibrium—is
in the range between −20 K and −15 K at
the levels of interest [see also Fueglistaler
et al., 2009a, 2009b, Jucker et al., 2013, and
references therein].
Experiments:
(1) scaled-Q dTLDP /dQ ~ 3.4[K/K/day]
(2) delta-Q dTLDP /dQ ~ 3.6[K/K/day]
1.0
0.5
0.0
-0.5
-1.0
-1.5
-0.3
-0.2
-0.1
0.0
0.1
0.2
10.1002/2013JD020772
For the case evaluated here—a change in
mean upwelling without modification of the
zonal structure—the temperature changes
shown in Figure 1b are homogeneous in the
inner tropics, and [H2 O]entry will change as discussed in the section on Clausius-Clapeyron
scaling (section 2). The sensitivity of the water
saturation mixing ratio to temperature at typical temperature and pressure conditions of
the LDP is about 0.5 ppmv per 1 K, such that
the temperature changes shown in Figure 1b
correspond to changes in [H2 O]entry of order
1 ppmv.
0.3
delta-Q @96hPa [K/day]
Figure 2. The residence time effect for [H2 O]entry (expressed
as Lagrangian Dry Point temperature change, ΔTLDP ) calculated from the two trajectory experiments (see Table 1), scaled
Q (red) and ΔQ (blue), as function of the change ΔQ of the
tropical mean diabatic heating rate at 96 hPa. For the scaled Q
experiment, results are shown for perturbations of ±10% (red
triangles). For the ΔQ experiment, results are shown for perturbations of ±5%, ±10%, and ± 50% (blue diamonds). The
slopes are determined with a linear regression to quantify the
sensitivity to small perturbations.
4.2. The Residence Time Effect on [H2 O]entry
Figure 2 shows the results from the two trajectory experiments as function of the forced change in diabatic
heating rates at 96 hPa (i.e., close to the tropical tropopause) in the tropics.
The figure shows that both experiments show a clear response in [H2 O]entry (TLDP ) to the change in
upwelling, with an increase in [H2 O]entry in response to an increase in upwelling (i.e., decrease in residence
time). Quantitatively, the experiments show a change in TLDP to the imposed change in Q of ΔTLDP ∕ΔQ ∼ 3.4
K/K/day for experiment 1, and ∼3.6 K/K/day for experiment 2.
The results from experiment 2 with a range of values for the factor f show that for small perturbations (order
±10%), the response in TLDP is approximately linear with ΔQ, but for larger perturbations the nonlinearity
has to be considered (Liu et al. [2010, Figure 13] show that for experiments of type 1, the functional form of
the relation between ΔTLDP and the scaling factor f is logarithmic for the range f = [0.5 − 10]).
4.3. Results for Clausius-Clapeyron Scaling
Table 2 summarizes the results in terms of departure from Clausius-Clapeyron scaling rCC (equation (5)), i.e.,
it relates the change in TLDP to the radiative change in temperature in response to the same heating rate
perturbation. The table shows that the departures from Clausius-Clapeyron scaling are of order −10%.
Whether a departure from Clausius-Clapeyron scaling of order 10% is detectable in the observational record
is an open question, but it is important to realize that this effect is of similar magnitude to that of other
Table 2. Departure From Clausius-Clapeyron Scaling of [H2 O]entry With Zonal Mean Temperatures
Around the Tropopause (rCC , See Equation (5)) for Experiments 1 and 2 for the Three Scenarios of Vertical
Scale of Upwelling Perturbationa
Upwelling Scenario (Forcing Length Scale)
Deep B-D
TTL scale
Shallow
a Note
that rCC ≡
in Figure 1.
FUEGLISTALER ET AL.
residence
ΔTLDP
rad
ΔTTP
=
dTLDP ∕dQ
rad ∕dQ ,
dTTP
rCC , Experiment 1
rCC , Experiment 2
3.4/(−40) = −8.5%
3.4/(−35) = −9.7%
3.4/(−30) = −11.3%
3.6/(−40) = −9%
3.6/(−35) = −10%
3.6/(−30) = −12%
rad ∕dQ is shown
where dTLDP ∕dQ is shown in Figure 2 and dTTP
©2014. American Geophysical Union. All Rights Reserved.
1967
Journal of Geophysical Research: Atmospheres
processes (for example, cloud microphysical properties [Jensen and Pfister,
2004; Notholt et al., 2005] or changes
in the zonal asymmetry of circulation and temperature [Fueglistaler
and Haynes, 2005]) that could induce
departures from Clausius-Clapeyron
scaling of [H2 O]entry with mean tropical
tropopause temperatures.
190
T LDP [K]
10.1002/2013JD020772
188
186
184
5. Discussion
5.1. The Process Leading to Departure
From Clausius-Clapeyron Scaling
The previous section has shown that
Figure 3. The climatological mean (2001–2005) annual cycle in TLDP
[H2 O]entry robustly responds to changes
of the base calculation (black), and the perturbations following experiment 2 (ΔQ) with f = +50% (red) and f = −50% (blue). The thin colored in residence time. Closer inspection of
lines show the results of the perturbations shifted in phase, and with
the data shows that the effect varies
an offset in the mean (see text).
somewhat from month to month. When
plotted as function of month, the differences between the calculations with different diabatic heating rates tend to be largest during boreal
summer/fall. The question therefore arises whether the situation during this season is particularly susceptible to changes in diabatic heating. In the following, we analyze the results of the ΔQ experiments (see
Table 1, experiment 2) in more detail to establish the process that leads to the sensitivity of [H2 O]entry on
residence time scale.
J
F
M
A
M
J
J
A
S
O
N
D
month
Figure 3 shows the mean annual cycle (i.e., averaged over the period 2001–2005) in TLDP for the base calculation, and the ΔQ = (±50%) ⋅ ⟨Q⟩ calculations. The figure shows that shifting the results from the perturbed
calculations by their average difference in TLDP , and a phase shift of −1.5∕ + 0.7 months recovers the unperturbed calculations well. The phase shifts were empirically determined but are in qualitative agreement with
expectation from the fractional change in heating rates (the phase information is dominated by the mode
of the distribution of time since LDP, for which the mean time since LDP, or the average diabatic heating, are
only approximations). Note that the amplitude of the seasonal cycle of the long-residence time perturbation
(i.e., the calculation with ΔQ = −50% ⋅ ⟨Q⟩) is less than in the base case. Conversely, the amplitude is a
(b)
(a)
strat
pot. temperature
THa
THb
trop
x0
x1
horizontal dimension
t0
t1
time
Figure 4. Schematic for thought experiment (see text). (a) A stationary temperature field (colored contours, blue corresponding to “colder”) with two pathways corresponding to a systematic relation between temperature field and flow
(black lines, with dotted black line representing faster ascent), and with two pathways (brown and green) corresponding to a random walk (see text). The black pathways are assumed to be symmetric about the center of the horizontal
dimension. Also shown are two isentropes (TH), with temperatures at first intersection of these isentropes, T𝜃 , given by
the temperatures at the position marked with the open diamond for THa , and the positions of the LDPs marked with
open triangles. (b) The random walk trajectories as function of time, with the positions of first intersection of THa and
the positions of the LDPs indicated as in Figure 4a.
FUEGLISTALER ET AL.
©2014. American Geophysical Union. All Rights Reserved.
1968
Journal of Geophysical Research: Atmospheres
(a)
little larger than in the base case in the ΔQ =
+50% ⋅ ⟨Q⟩ calculation. This effect is expected,
as explained in Fueglistaler et al. [2013]: with
longer residence times, the spectra of time
since LDP widen, such that the mean annual
cycle in [H2 O]entry becomes an increasingly
low-pass filtered version of the corresponding
annual cycle in saturation mixing ratios around
the tropical tropopause.
410
pot. temperature [K]
400
390
380
370
360
[] @LDP, * @theta
350
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
delta T [K]
(b)
1.5
delta T @LDP [K]
1.0
10.1002/2013JD020772
slope ~ 1.6 [K/K]
0.5
0.0
-0.5
-1.0
-1.5
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
max(delta T) @theta [K]
Figure 5. (a) The temperature difference in experiment 2
between the base calculation and modified diabatic heating
for ΔQ = +50% ⋅ ⟨Q⟩ (red) and ΔQ = −50% ⋅ ⟨Q⟩ (blue)
for the temperature of back trajectories first intersecting with
an isentrope (T𝜃 ) as function (y axis) of the isentrope (𝜃 ). Also
shown is the corresponding temperature difference for the
Lagrangian Dry Point (LDP, open square) at the average pot.
temperature level of the LDP. (Dashed lines for better visual
comparison with the T𝜃 profile.) (b) Scatterplot for all perturbations following experiment 2 (red: ΔQ = +[5%, 10%, 50%] ⋅ ⟨Q⟩,
blue: ΔQ = −[5%, 10%, 50%] ⋅ ⟨Q⟩). The x value is the maximum temperature difference of T𝜃 for each perturbation (i.e.,
the maximum of the profile shown in Figure 5a of each perturbation), and the y value is the difference in the Lagrangian
Dry Point temperature ΔTLDP associated with the perturbation.
Each data point is the average over all trajectories in the period
2001–2005, with one reverse domain filling calculation per
month. Also shown is the total least squares fit, with a slope of
∼1.6 K/K.
The fact that a well-understood phase shift and
amplitude modification explains most of the
differences between the mean annual cycle in
[H2 O]entry of the base case and the perturbed
cases implies that the residence time effect
shows little dependence on season. Identification of the cause of the residence time effect,
however, is challenging because of the complexity of transport in the TTL [Fueglistaler et al.,
2005; Tzella and Legras, 2011; Bergman et al.,
2012] and its sensitivity to a perturbation in any
component of the flow field.
Consider the following thought experiments,
illustrated in Figure 4. We assume a stationary
temperature field (all subsequent arguments
are also valid for nonstationary fields but are
more difficult to describe) with a spatial structure (the example has a minimum at the center
of the domain). The two black lines (dotted and
dashed) have a deterministic relation between
horizontal position and vertical position, and as
such with the temperature field. The only difference is their vertical velocity (larger for the
dotted path), with the flow field corresponding to the smaller vertical velocity focusing a
larger fraction of the flow into the coldest part
of the domain. The full circulation pattern (not
shown) is assumed symmetric around the center of the spatial domain (i.e., x axis), such that
for this example the flow is divergent out of the
coldest region.
We refer to this situation as having a systematic
relation between flow field and temperature
field. It is evident from the figure that for this specific setup the flow field with a faster vertical velocity has
a higher average TLDP because a smaller fraction of the total flow is exposed to the minima temperatures.
This situation is reminiscent of the zonal mean structure of the circulation and temperature field in the TTL,
and hence, we may expect that for troposphere-to-stratosphere transport the sign of the relation is as in
this example.
Now consider the situation for a flow field without systematic relation between flow field and temperature, exemplified here with a random walk in the horizontal dimension, and the vertical position modeled
with an AR-1 process. The two “random” pathways (green and brown) differ in their vertical velocity, with
the brown path representing faster ascent (illustrated in Figure 4b). For this scenario, the flow field with
slower ascent will produce a lower average TLDP because the average sampling length of each trajectory
FUEGLISTALER ET AL.
©2014. American Geophysical Union. All Rights Reserved.
1969
Journal of Geophysical Research: Atmospheres
10.1002/2013JD020772
increases, which increases the range of temperatures sampled and hence lowers on average the minimum
temperature encountered.
While the first effect—a systematic relation between temperature and circulation—can increase or decrease
the TST-ensemble average TLDP , the second effect will always decrease the TST-ensemble average TLDP , but
from analysis of TLDP alone the two effects cannot be distinguished. The two effects can be distinguished,
however, from evaluation of temperatures at first crossing a specific level, here, for example, an isentrope. If
we denote this temperature T𝜃 , we can see that for the situation of a systematic relation between flow and
temperature field, the behavior of T𝜃 closely matches that for TLDP . However, this is not true for the random
walk case: because there is no preferred direction, the probability density distribution of the horizontal position at first encounter of the specified level is uniform. That is, for an individual back trajectory started at
a given end point, a change in ascent velocity may result in a positive or negative change in T𝜃 (as can be
seen for the green and brown pathways), but averaged over the TST-ensemble these changes must evaluate
to zero.
We conclude that the residence time effect may arise from two processes: (i) a systematic relation between
flow and temperature structure and (ii) an increase in sampling length of random temperature variations
which always decreases the average TLDP . Analyzing the behavior of the average temperature upon first
crossing a fixed level (here an isentrope) allows to distinguish the two effects.
Figure 5a shows the results of this calculation for trajectories of the ΔQ experiment (i.e., experiment 2). The
figure shows the difference in temperature averaged over all TST trajectories between the base calculation,
and the calculation with perturbed diabatic heating, for temperatures at first intersection of isentropes (T𝜃 ,
evaluated for potential temperatures of 360 K, 365 K, …, 405 K; y axis of the figure). Also shown is the corresponding change in TLDP (open square; line given for better comparison with the T𝜃 difference profile) at the
approximate potential temperature of the LDP. The figure shows that changing the diabatic heating of the
trajectories changes the average temperature of trajectories upon first intersecting with a given isentrope
and that this change is about half of the total residence time effect (Figure 5, open blue/red squares). This
shows that the systematic relation between flow and temperature field is an important part of the effect in
the TTL.
Figure 5b shows the change in TLDP as function of the maximum temperature change at first intersection
with isentropes, i.e., max(ΔT𝜃 ) determined from the data shown in Figure 5a, for all perturbation factors
f of experiment 2. The slope of the linear regression is about 1.6 K/K. This implies that (as already seen in
Figure 5a) the systematic relation between flow and temperature field is the slightly more important contributor to the observed dependence of [H2 O]entry on upwelling, with an additional 60% increase in effect
due to the effect of changing the length of the sampling interval.
6. Summary and Conclusions
We have analyzed the impact of a change in upwelling across the tropical tropopause on water entering
the stratosphere. The radiatively induced temperature change implies a drying for an increase in upwelling.
We find that a 10% change in upwelling (corresponding to a heating rate change of ∼0.05 K/d) leads to
a temperature change of about 2 K (with some dependency on the vertical length scale of the perturbation; and a somewhat smaller response if the relative humidity rather than specific humidity profile is
held fixed). This temperature change is equivalent to a change in [H2 O]entry of about 1 ppmv (based on the
saturation mixing ratio evaluated at the relevant temperature and pressure conditions) [see Fueglistaler and
Haynes, 2005].
We have quantified a second effect, the residence time effect. Based on trajectory calculations, we find that
this effect reduces the impact of the radiatively induced tropical mean temperature change on [H2 O]entry by
about 10% (and slightly more if the radiative transfer calculation assumes fixed relative humidity). In other
words, in response to changes in upwelling [H2 O]entry scales with tropical average temperatures about 10%
less than expected from the temperature dependence of the water vapor pressure. The residence time effect
is a consequence of the variance (in space and time) of the temperature field, whereby two aspects play a
role: the systematic relation between flow and temperature field, and the impact of sampling interval length
on the average minimum temperature of air traversing the TTL into the stratosphere. We find that for the
FUEGLISTALER ET AL.
©2014. American Geophysical Union. All Rights Reserved.
1970
Journal of Geophysical Research: Atmospheres
10.1002/2013JD020772
TTL of the present climate, the two effects are of similar importance. Note that the residence time effect also
implies that model biases in transport time scales induce a bias in the models’ [H2 O]entry .
The residence time effect demonstrates that atmospheric water vapor does not have to scale with mean
temperatures as expected from Clausius-Clapeyron, and for the case of [H2 O]entry a robust departure of order
−10% from Clausius-Clapeyron scaling can be expected.
Appendix A: Radiative Temperature Adjustment
The radiative temperature adjustment in response to a perturbation in the diabatic residual circulation is
determined by radiative transfer calculations similar in spirit to fixed dynamical heating calculations. We
define a perturbation heating rate ΔQ and then find the temperature change ΔT required to perturb the
heating rate by ΔQ. ΔT is found by integrating to steady state the equation
dT
= Q(T) − Q(T0 ) − ΔQ,
dt
(A1)
where Q(T) is the radiative heating rate associated with temperature profile T , and T0 is the background
temperature profile about which we make the ΔT perturbation, so T = T0 + ΔT . We integrate from the initial
condition T = T0 using the Euler method with a time step of 0.1 days.
We define T0 to be the 10◦ S–10◦ N average of zonal mean temperature taken from ERA-Interim data averaged from 2001 to 2010. The radiative heating rate Q(T) is computed using the Rapid Radiative Transfer
Model [Mlawer et al., 1997]. ERA-Interim data are used to specify water vapor concentration, averaged over
the same region and period as temperature. The ozone climatology provided by Fortuin and Langematz
[1994] is used to specify the ozone profile and is averaged over 10◦ S–10◦ N and averaged over the annual
cycle. All quantities are interpolated onto a high-resolution grid with 120 levels between the surface and
0.2 hPa using cubic interpolation.
Acknowledgments
This research was supported in part
by DOE grant SC0006841. We thank
ECMWF for providing access to
ERA-Interim data. We thank Tao Wang
and three anonymous reviewers for
constructive and helpful comments
and suggestions.
FUEGLISTALER ET AL.
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1972