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PHY2502 Course Project:

Experiments concerning the Tropopause

and Thermal Stratification of the

Troposphere with a

Radiative–Convective Model

Andre R. Erler

April 30, 2009

1

Andre R. Erler PHY2502 Course Project April 30, 2009

Contents

1 Introduction 3

1.1 The Tropopause . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Radiative–Convective Models 7

2.1 The SCCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 The Experiments and Model Results 9

3.1 Ozone and Carbon Dioxide . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 Water Vapour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.3 Convective Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Discussion 18

5 Summary and Conclusion 22

A Convective Adjustment Scheme 25

2


1 Introduction

A major focus of atmospheric and climate dynamics is to predict (i.e. explain) the general

circulation of atmospheres as a function of external parameters. General Circulation Models

(GCMs) are now able to simulate a wide range of climates under different forcings, but

they study of GCM results alone gives little theoretical insight.

A problem of particular interest and difficulty is the thermal stratification of the extra–

tropical troposphere. The average stratification (i.e. the temperature profile) of the strato-

sphere is well explained by radiative transfer calculations, as a function of trace gas dis-

tribution (most importantly ozone and CO2). In the lower troposphere radiative equilib-

rium temperature profiles are in general unstable with respect to convective overturning

, so that dynamical/convective adjustment of the thermal stratification occurs. We have

come to believe that the stratification of the extra–tropical troposphere is determined by

baroclinic eddies; this hypothesis is supported by the fact that the two–dimensional (ver-

tical/meridional) distribution of radiative equilibrium temperatures is also baroclinically

unstable. Only we are not able to predict the observed tropospheric temperature profile

from theoretical considerations (based on external parameters). Numerous attempts have

been made (cf. Held , 1982), but Thuburn and Craig (1997) showed that none of the pro-

posed approaches scale correctly when planetary parameters are perturbed.1

1.1 The Tropopause

The tropopause is defined by the WMO as the height where the lapse–rate falls below

the threshold value of ΓTP < 2 K/km and the average over the next 2 km satisfies this

1Under the premise that the thermal stratification obtained from GCM experiments does scale correctly.

The paucity of experimental data has always been one of the major limitations of climate dynamics,

and GCM experiments serve as a substitute.

3


criterion as well. This definition is purely empirical and gives virtually no hint at its

physical significance.

From a theoretical/dynamical point of view the tropopause is generally interpreted as

the layer where dynamical adjustment of the lapse–rate ceases, and radiative equilibrium

dominates (cf. Held , 1982). Implicit in this view is a sense of statistical averaging, in

particular, because the stratosphere is locally not in radiative equilibrium (only on average).

It is customary to divide the problem into a radiative constraint and a dynamic con-

straint. Fig. 1 illustrates the competing mechanism that determine the tropopause height.

The dynamical constraint forces adjustment to a given dynamically stable stratification,

which is in general not in radiative equilibrium anymore; heat is deposited in the tro-

posphere. The tropopause is then said to be the point where the atmosphere attains a

temperature profile which is in radiative equilibrium. Because both, temperature and ra-

diative flux, of the dynamically adjusted region and the radiative equilibrium region have

to match at the boundary (the tropopause), both conditions are not independent and have

to be solved for simultaneously. Two boundary conditions have to be specified (for temper-

ature and radiative flux): the radiative flux either at the top of the atmosphere (common

for global average calculations, cf. Held , 1982) or at the surface (for local calculations, cf.

Thuburn and Craig , 2001), and a temperature.2

Thuburn and Craig (2001) give a very simple interpretation of the tropopause in which

they assume that the stratosphere is essentially optically thin, which means that the ra-

diative flux is relatively homogeneous throughout the layer and the same as the outgoing

longwave radiation. The tropopause can be viewed as the height to which convective adjust-

ment extends; with the constraint that the net upward radiative flux must be continuous,

convective adjustment reaches radiative equilibrium, when it reaches a temperature at

2Which can be coupled to the radiative flux condition via the Stefan–Boltzmann law, if for example the

surface temperature is assumed to be equal to the surface air temperature.

4


which the upward radiative flux approximately equals the outgoing longwave radiation.

From elementary radiative transfer considerations (Goody , 1964) using a two–stream grey

atmosphere model Thuburn and Craig (2001) further show, that approximately half of the

upwelling radiative flux in the optically thick layer (the troposphere) is emitted locally as

blackbody radiation (at the local temperature). Thus convective adjustment extends to

the (blackbody) radiative temperature equivalent to half the outgoing longwave radiation

(which can again be related to the surface emission). In order to deduce the tropopause

height from the tropopause temperature, the surface temperature and the tropospheric

lapse–rate are needed.

Schneider (2004) also proposed an alternative definition of the tropopause based on

isentropic mass flux: he defines the tropopause to be the isentropic level, at which 90% of

the isentropic mass flux closes. Note however that this definition can only be evaluated in

a meaningful way in a statistically averaged sense.

This study will use a radiative transfer model to study the thermal stratification and the

tropopause height. However, the convective adjustment scheme will also be a major focus.

In particular the relationship between convective adjustment and heat flux convergence will

be considered: dynamical heat flux convergence effected by baroclinic eddies, will be viewed,

in a statistical average on long time scales, as a form of convective adjustment which leads

to the observed extra–tropical tropospheric stratification. The magnitude of the heat flux

convergence, or equivalently the magnitude of the column–integrated convective heating

rate, must be in accordance with the heat transport in the general circulation.

Schneider (2004) proposed a dynamical constraint for the tropopause height based on the

consideration of entropy transport along the hemispheric isentropic overturning cell. It is

along these lines that I am hoping to establish a connection between heat flux convergence,

convective adjustment, and the tropopause height: the dynamical constraint by Schnei-

5


Figure 1: A schematic illustrating the mechanisms that determine the tropopause height:

solar insolation is received primarily at the surface and radiated to space at the top of the

atmosphere; eddies transport heat toward the poles, and deposit heat in the troposphere

in mid–latitudes. The extra-tropical tropopause is thought to be at the height where

radiative heat transfer begins to dominate over quasi–horizontal eddy fluxes.

der (2004) relates dynamic/convective adjustment to entropy fluxes, and by radiative–

convective equilibrium considerations it should be possible to relate convective adjustment

to heat flux convergence, which places an additional constraint on global heat/entropy

transport. This would be a step towards closing the general circulation problem and pre-

dicting/explaining the thermal stratification of the extra–tropical troposphere. This is of

course a very ambitious goal and this work can only be a small step in that direction. Parts

of this s work will be dedicated to a numerical “proof of concept”: an estimate of heating

rates implied by convective adjustment.

6


2 Radiative–Convective Models

For this study a radiative–convective transfer model was used. Such models couple a radia-

tive transfer model with a convective adjustment component The radiative transfer model

solves the radiative transfer problem, typically discretized in spectral bands based on the

radiatively active species under consideration, and in pressure levels in the vertical; ra-

diative transfer in the horizontal is not considered explicitly, it is merely accounted for in

an increased length of the effective optical path (the slanted path length). The convective

adjustment component typically forces the temperature profile below a prescribed critical

lapse–rate, which is deemed convectively unstable.

The radiative equilibrium temperature profile of the earth’ atmosphere is unstable with

respect to vertical convection in the lower troposphere; this was first found by Manabe

and Moller (1961) using a radiative transfer model. Manabe and Strickler (1964) used

convective adjustment in order to force the temperature lapse–rate below the dry adia-

batic lapse–rate.3 However, the observed tropospheric lapse–rate on earth is below the

dry adiabat, and in fact close to the moist adiabat. An accepted critical lapse–rate for

convective adjustment also used by Manabe and Wetherald (1967) is 6.5 K/km; the major

break–through in radiative–convective modeling of the troposphere was achieved by Man-

abe and Wetherald (1967): instead of prescribing the absolute humidity, they prescribed

a fixed relative humidity, so that the water vapour mixing ratio changes with tempera-

ture. A constant relative humidity of about 80% is also in much better agreement with

climatological observations. The simulations of Manabe and Wetherald (1967) were the

first to reproduce a realistic temperature profile of the atmosphere, including troposphere

and stratosphere. They also found that due to fixed relative humidity (i.e. temperature

3This can indeed be accurate in the absence of strong moist convection or baroclinic eddies, for example

in the lower atmosphere of Saturn’s moon Titan.

7


dependent water vapour mixing ratio) the radiative relaxation time (time to attain equi-

librium) was significantly increased, and the general sensitivity of the temperature profile

to external perturbations (e.g. solar insolation) was roughly doubled.

Although radiative transfer column models were initially developed for use in GCMs,

they were extensively applied to studies of climate sensitivity. Traditionally in such studies

the single–column models were usually meant to be representative for a mean global cli-

mate. Here a fundamentally different approach will be taken: in order to investigate how

radiatively active constituents in concert with convective adjustment affect the height of

the tropopause, a fixed surface temperature is prescribed and the concentration of chemical

constituents and the critical lapse–rate for convective adjustment varied. Furthermore a

setup explicitly representative for the mid–latitudes will be chosen, because the physical

processes determining the tropopause differ in the tropics and the extra–tropics and polar

regions, so that a global average would only complicate the interpretation.

A major limitation of this study is the use of a prescribed critical lapse–rate, for it is

exactly this what we aim to predict eventually. This would require the solution of the

full three–dimensional problem (i.e. a GCM experiment). However this limitation also

dramatically reduces the complexity of the problem, so that it is my hope that some insight

can be gained along this path.4 As has been alluded to in the introduction, convective

adjustment in this study is not understood as dry or moist convective overturning within

a vertical column, but rather seen as a product of the activity of baroclinic eddies, acting

on larger temporal and spatial scales. It is therefore not immediately obvious which value

should be chosen, for both, the moist and the dry adiabat, are in this sense relatively

meaningless (in fact moist convection will not be considered at all). The value used for

most of the experiments presented here is motivated by climatological observations.

4This approach is somewhat complementary to purely dynamical GCM experiments, where a relaxation

temperature profile is prescribed to simulate the effect of radiative forcing.

8


2.1 The SCCM

The radiative transfer model used for this work is the single–column radiative transfer

model which is part of the NCAR Community Climate Model version 3, SCCM 1.2.5 The

SCCM model was used within a Phyton environment, using the wrapper– and interface–

package CliMT (Caballero, 2009). This approach greatly facilitated the configuration of the

model and the post–processing and graphical display using the Python–package matplotlib.

The NCAR SCCM is a one–dimensional radiative transfer model; it supports compu-

tation of radiative transfer in longwave (infrared) and shortwave (visible/UV) bands of

most major radiatively active trace species. Here only ozone and water vapour will be

considered. The model can be compiled with an arbitrary number of vertical levels. For a

commercially available 2366 MHz processor a good trade–off between execution time and

resolution (smoothness of the solution) was found to be at 90 vertical levels. A single run

from an ICAO–like standard atmosphere towards radiative–convective equilibrium takes

about one minute, with 1600 time steps, equivalent to 258 model days. The time–step was

varied so as to take larger steps initially and the take smaller steps near equilibrium.

3 The Experiments and Model Results

Three sets of experiments were conducted. One investigating the effect of ozone and carbon

dioxide, one varying the relative humidity, and finally one to study the effect of different

critical lapse–rates in the convective adjustment scheme.

5I intended to use the SCAM model, the single–column radiative transfer model of the more recent

Community Atmosphere Model, however due to problems in dynamically linking the netCDF–libraries,

I had to resort to the SCCM for now. The SCAM model does build correctly, as does the SCCM model,

the difference is that the former links dynamically. I expect to solve this problem, given more time, but

currently I cannot spend more time on technical issues.

9


All experiments are compared to a reference calculation (which appears in every plot and

is denoted by “ref.”). The reference profile can be considered the best approximation to

the mean mid–latitude climate: the lapse–rate is set to 6.5 K/km, the relative humidity is

80%. Climatological values are used for CO2 and O3; note that the ozone profile used here

is representative for the tropics and not for the mid–latitudes.6 The surface temperature

was fixed at 285 K, and insolation parameters were set for 45◦ N in all experiments. Note

that surface air temperatures do not necessarily match the surface temperature exactly if

strong radiative cooling is present, because the adjustment occurs only at the beginning of

each time–step.

3.1 Ozone and Carbon Dioxide

The purpose of this experiment is to determine the effect of Ozone (stratospheric mostly)

and CO2 on the tropopause. The following cases were computed and compared to the

reference profile: no ozone, no CO2, and double CO2. The results are displayed in Fig. 3.

The concentration of ozone and water vapour used in the experiments are displayed in

Fig. 2. The ozone profile was prescribed on pressure levels: even though all runs except

the no–ozone experiment have the same profile of ozone on pressure levels, the different

temperature profiles produce different ozone profiles in geometric altitude. The standard

mixing ratio used for CO2 was 380 ppm (and 760 ppm in the double–CO2 experiment) at

all levels.

The reference temperature profile (top left panel, Fig. 3) is characterized by a con-

stant (prescribed) lapse–rate in the troposphere and an almost isothermal profile in the

stratosphere, as expected and found in climatologies. The tropopause height is at 12.7 km,

somewhat higher than the climatological mean (approx. 11 km), but in agreement with the

6The ozone profile is the default profile that comes with the SCCM. Since we only require some shortwave

heating in the stratosphere, this profile should suffice.

10


10-8 10-7 10-6 10-5 10-4

Ozone Mass Mixing Ratio [kg/kg]

0

5

10

15

20

25

Alt

itude [

km]

ref.

w/o CO2

w/o O3

2× CO2

10-5 10-4 10-3 10-2 10-1 100 101

Water Vapor Mass Mixing Ratio [kg/kg]

0

5

10

15

20

25

ref.

w/o CO2

w/o O3

2× CO2

Figure 2: Vertical profiles of mass mixing ratio for ozone O3 and water vapour H2O. Note

that ozone was prescribed ob pressure level and the vertical plot axis is geometric altitude

(not log–p). For water vapour a constant relative humidity of 80% was used on all levels.

The horizontal lines indicate the tropopause height (colours match Fig. 3).

radiative transfer calculations of Thuburn and Craig (2001). The profile of the no–ozone

experiment looks radically different. The stratosphere is not isothermal but the temper-

ature is slowly decreasing. The convective adjustment extends almost up to 17 km. The

reason for the comparatively strong negative temperature gradient in the stratosphere is

probably the still high water vapour mixing ratio, which leads to excessive cooling (the

earth’ stratosphere is dryer than 80% relative humidity). The water vapour cooling effect

is probably also accounts for the higher tropopause of the reference simulation: it offsets

stratospheric ozone–warming to an unrealistically large extend. The no–CO2 experiment

exhibits a similar structure as the reference but the tropopause is about 2 km lower and

the isothermal stratospheric profile is consequently shifted towards warmer temperatures.

The reason for this behaviour is most likely the cooling effect of CO2 in the stratosphere,

11


160 180 200 220 240 260 280Temperature [K]

0

5

10

15

20

25

Alt

itude [

km]

ref.

w/o CO2

w/o O3

2× CO2

1 0 1 2 3 4 5 6 7Brunt-Vaisala Frequency N2 [10−4 s−2 ]

0

5

10

15

20

25

ref.

w/o CO2

w/o O3

2× CO2

1.0 0.5 0.0 0.5 1.0 1.5 2.0Convective Heating Rate T conv [K/day]

0

5

10

15

20

25

Alt

itude [

km]

ref.

w/o CO2

w/o O3

2× CO2

0.0 0.1 0.2 0.3 0.4 0.5Convective Heating Rate Qconv [10−3 J m−3 s−1 ]

0

5

10

15

20

25

ref.

w/o CO2

w/o O3

2× CO2

Figure 3: The effect of ozone and CO2 concentrations: temperature profile T (top left),

thermal stratification as measure by N2 (top right), temperature tendencies Tconv due to

convective adjustment (bottom left), and convective heating rate Qconv (bottom right).

The horizontal lines indicate the tropopause height corresponding to the temperature

profile of the same line colour. Note that the reference and the double–CO2 profile are

almost identical, so that the tropopause lines overlap.

12


as CO2–cooling also offsets some of the ozone–warming. The double–CO2 profile is almost

identical to the reference profile, suggesting that the CO2–bands are almost saturated.

The N2–profiles (top right panel, Fig. 3) give a better visual impression of the qualitative

structure of the tropopause: it is clearly visible as a sharp step–like increase in N2. The

region of convective adjustment is discernable as a gently curved (not straight) line. There

are two points of interest: first, the tropopause as determined by the WMO–criterion clearly

fail in the experiment where no ozone was present; the tropopause should be located at

the upper boundary of the convective adjustment region, near 15 km (as discussed in the

preceding section). Second, the reference profile exhibits a maximum just above 15 km,

which is reminiscent of the so–called Tropopause Inversion Layer (Birner , 2006), only it

appears approximately 5 km above the actual tropopause which is much too high.7 The

bottom panels in Fig. 3 shows heating due to convective adjustment (which is balanced

by radiative fluxes): the heating rates (right) and the temperature tendencies (left). The

heating in the form of energy flux convergence is concentrated in the bottom layer, below

5 km, while the temperature tendencies are much more uniform. The reason is, that the

heat capacity is proportional to density, and the upper atmosphere can simply not absorb

so much energy, and maintain convective equilibrium at the same time. The altitude where

convective heating q vanishes, obviously coincides with the vertical extend of convective

adjustment, and ideally also with the tropopause height.

3.2 Water Vapour

The water vapour experiment was motivated by the findings of Manabe and Wetherald

(1967), and investigates the influence of varying humidity on the tropopause height. In

this set of experiments ozone and CO2 mixing ratios were set to zero, so as to isolate the

7The N2–maximum is also much weaker than reported by Birner (2006), but this is to be expected from

models of this vertical resolution (Son and Polvani , 2007).

13


effects of water vapour (except for the reference profile).

The results are displayed in Fig. 4: evidently the tropopause height is not very sensitive to

the value of relative humidity. Values ranging from 20% to 100% all yield tropopause values

within 1 km of each other. Qualitatively only the RH = 0 profile and the reference profile

differ. The higher humidity values are all characterized by a high tropopause and a non–

zero stratospheric lapse–rate (like the no–ozone profile, which is very similar). Generally

profiles with a higher relative humidity have a higher tropopause and a colder stratosphere

(Fig. 4, top left). Both implies weaker radiative cooling (cf. Thuburn and Craig , 2001),

which balances the higher optical thickness. The water vapour mass mixing ratios are

displayed on the top right panel. However the convective heating rate Tconv (bottom left

panel) imply higher convective heating in the troposphere. The reason is again that the

convective adjustment has to balance the higher radiative cooling. The resulting fluxes into

the stratosphere partly compensate the cooling effect of water vapour so that the net effect

is very small.

In contrast to the no–ozone experiment in Fig. 3, the tropopause determination accord-

ing to the WMO–definition appears to be successful: the tropopause coincides with the

highest level of convective adjustment and non–vanishing heating rates, and is generally

well defined. It is also evident as a regime transition in the water vapour profiles (top right,

Fig. 4).

The profile resulting from zero–humidity is warmer than the others, opposite to the ex-

pectation that water vapour is a greenhouse gas. The reason for this is that the surface

temperature is fixed and the main heat source is convective adjustment, so that water

vapour has a cooling effect, as mentioned before. Consequently, since all the main radia-

tively active species were set to zero (water vapour, ozone, and CO2), the temperature

profile quickly adjusts to a state of convective equilibrium and heating rates drop to zero.

Also not that the surface temperature of the RH = 0 experiment is exactly the prescribed

14


160 180 200 220 240 260 280Temperature [K]

0

5

10

15

20

25

Alt

itude [

km]

RH = 1.0RH = 0.5ref.RH = 0.0RH = 0.2

10-5 10-4 10-3 10-2 10-1 100 101

Water Vapor Mass Mixing Ratio [kg/kg]

0

5

10

15

20

25

RH = 1.0RH = 0.5ref.RH = 0.0RH = 0.2


0

5

10

15

20

25

Alt

itude [

km]

RH = 1.0RH = 0.5ref.RH = 0.0RH = 0.2

1 0 1 2 3 4 5 6 7Brunt-Vaisala Frequency N2 [10−4 s−2 ]

0

5

10

15

20

25

RH = 1.0RH = 0.5ref.RH = 0.0RH = 0.2

Figure 4: Equilibrium temperature profiles resulting from different prescribed relative

humidity: temperature profile T (top left), water vapour mass mixing ratio q (top right),

convective temperature tendency Tconv (bottom left), and thermal stratification as mea-

sure by N2 (bottom right). As in Fig. 3, horizontal lines indicate the tropopause heights

of the corresponding temperature profile (several lines almost coincide). Note that the

RH = 0 profile is shown to have zero convective heating rate.

15


surface temperature.

3.3 Convective Adjustment

Convective adjustment is an artificial parameter that is introduced to account for a variety

of processes of high complexity. It is essentially an empirical parameter: the value of the

critical lapse–rate is set to the observed climatological mean lapse–rate. In this set of

experiments the effect of varying critical lapse–rate is investigated. It is of course trivial

to predict the tropopause height when lapse–rate and surface temperature are fixed; the

purpose of this experiment is to investigate the radiative equilibrium above the tropopause

and the amount of convective heating implied by a certain critical lapse rate. As in the

water vapour experiment, ozone and CO2 were set to zero; the relative humidity was again

set to 80%.

The Results are displayed in Fig. 5. As expected, the lower the critical lapse–rate is, the

higher extends the convective adjustment.

Again, the top left panel displays the temperature profiles: it is interesting to note, that

the stratospheric temperature profile only weakly depends on the tropospheric lapse rate,

the main difference being a slight offset to higher temperatures for smaller tropospheric

lapse–rates. Consequently the stratospheric lapse–rates all converge to the same value (top

right panel), regardless of tropopause height.8

As in Fig. 3, the bottom panels compare temperature tendencies Tconv (left) and heating

rates Qconv (right). The vertical dependence of temperature tendencies associated with

different critical lapse–rates is qualitatively relatively uniform: strong heating occurs in

the surface layer, then a layer of relatively uniform temperature adjustment follows, the

adjustment quickly falls off to zero at the tropopause. In terms of heating rates, the drop

8Note that the asymptotic stratospheric lapse–rate, as read off from Fig. 5 (top left), is very close to the

lapse–rate threshold of 2 K/km used in the WMO–definition of the tropopause.

16


160 180 200 220 240 260 280Temperature [K]

0

5

10

15

20

25

Alt

itude [

km]

ref.LR = 5.0LR = 8.0LR = 6.5LR = 9.5

4 2 0 2 4 6 8 10Lapse--rate [K/km]

0

5

10

15

20

25

ref.LR = 5.0LR = 8.0LR = 6.5LR = 9.5


0

5

10

15

20

25

Alt

itude [

km]

ref.LR = 5.0LR = 8.0LR = 6.5LR = 9.5

0.0 0.1 0.2 0.3 0.4 0.5Convective Heating Rate Qconv [10−3 J m−3 s−1 ]

0

5

10

15

20

25

ref.LR = 5.0LR = 8.0LR = 6.5LR = 9.5

Figure 5: The effect of convective adjustment on tropopause height: temperature profile T

(top left), lapse–rate Γ (top right), convective temperature tendencies Tconv (bottom left),

and thermal stratification as measure by N2 (bottom right). As before, horizontal lines

indicate the tropopause height of the corresponding temperature profile (the reference

tropopause and the LR = 8.0–tropopause almost coincide).

17


at the tropopause blends into a more uniform decline throughout the troposphere, so that

only two regimes can be distinguished clearly.

The effective total convective heating is similar (but not equal) for all critical lapse–

rates, so that the radiative flux at the tropopause is roughly the same, resulting in almost

identical stratospheric profiles. The colder tropopause temperature of lower critical lapse–

rate experiments is compensated for by higher fluxes from the interior atmosphere (due to

higher temperature and humidity).

4 Discussion

Tab. 1 summarizes the tropopause height and the total column heating rate of all experi-

ments, sorted according to sections (in the preceding chapter).

The reference profile is among those with the lowest column heating rate. The reason

is that stratospheric ozone has a heating effect, so that it helps to maintain a stable

stratification in the troposphere and smaller convective heating is necessary (for the same

critical lapse–rate). The “water vapour” and “lapse–rate” sets were computed without

ozone, so that a higher rate of convective adjustment is necessary to maintain the same

lapse–rate.

The heating rate shows the strongest dependence on relative humidity, because water

vapour is a strong radiator and cools the troposphere, hence a higher rate of adjustment

is necessary when the water vapour concentration is high. As expected, the heating rate

vanishes when water vapour is absent (the atmosphere is transparent to IR radiation).

The tropopause height itself shows only a very weak sensitivity to relative humidity, this

however is a consequence of the fixed lower boundary condition and lapse–rate and the

fact that convective heating balances radiative cooling. The tropopause simply extends to

the level where the temperature is such as to achieve radiative flux balance (cf. Thuburn

18


Trace gas

Experiment TP–height Heating rate

reference 12.7 km 1.07 J m−2 s−1

w/o O3 N/A 1.14 J m−2 s−1

w/o CO2 10.9 km 1.29 J m−2 s−1

2× CO2 12.7 km 1.05 J m−2 s−1

Water Vapour


RH = 1.0 17.0 km 1.52 J m−2 s−1

RH = 0.2 16.3 km 0.86 J m−2 s−1

RH = 0.5 17.0 km 1.12 J m−2 s−1

RH = 0.0 N/A -0.02 J m−2 s−1

Lapse–rate


LR = 5.0 22.9 km 1.44 J m−2 s−1

LR = 6.5 17.0 km 1.37 J m−2 s−1

LR = 8.0 13.4 km 1.32 J m−2 s−1

LR = 9.5 10.8 km 1.29 J m−2 s−1

Table 1: Tropopause height and energy flux convergence (column–integrated heating rate)

of all experiments, sorted by sets (corresponding to sections in the text). “N/A” indicates

that the tropopause as defined by the WMO is considered to have no meaning.

19


and Craig , 2001).

The critical lapse–rate also affects the column heating rates, although not as strongly

as the tropopause height (as expected). A lower critical lapse–rate leads to a significantly

higher tropopause and higher heating rates (note that the tropopause temperature changes

only slightly). The initial hypothesis that a higher heat flux convergence leads to a higher

tropopause height can tentatively be confirmed, however, the results are not very signifi-

cant. The strong response of the tropopause height with only a small increase in heating

rate is likely due to the small density at high altitudes, so that a small change in column

energy affects high altitudes strongly.

Thuburn and Craig (2001) conducted similar experiments where the tropopause height

was investigated as a function of lapse–rate and surface temperature; surface temperature

variations were not considered here but the qualitative dependence on lapse–rate agrees

with Thuburn and Craig (2001); however, the absolute height is generally about 3 km above

that of Thuburn and Craig (2001); from their description is is not clear whether shortwave

heating due to stratospheric ozone was considered, they only discuss the longwave effect

of lower stratospheric ozone. The inclusion of shortwave heating in the stratosphere would

explain the discrepancy. Another possibility is the unrealistically moist stratosphere leading

to too low stratospheric temperatures and higher convective adjustment.9

With regard to the thermal stratification and tropopause height the following can be

said.

Radiative effects, in particular due to the presence of water vapour and ozone, are cer-

tainly of great importance, if not even or primary importance. The presence of stratospheric

ozone was found to severely affect the tropopause height and the impact of heat flux con-

vergence in the troposphere; however, it is possible that both effects partly cancel each

9I did not investigate vertically inhomogeneous variation of trace gases and water vapour, because I have

not yet implemented the necessary modifications to the model.

20


other: the point here is whether heat flux convergence is viewed as cause or effect;10 most

likely it is something in–between. Having said this, under the presence of ozone a given

lapse–rate can be maintained with a lower column heating rate, and as was found in the

lapse–rate experiment, higher heat flux convergence will increase the tropopause height. If

the heat flux convergence is constant, the effects may cancel to first order. However, GCM

experiments by Thuburn and Craig (2001) have shown that the ozone layer indeed reduces

the climatological tropopause height.

The most significant result of this experiment is probably the strong dependence of heat

flux convergence on the amount of tropospheric water vapour. The implication is that the

radiative properties of the atmosphere are of extraordinary importance for the meridional

transport of heat, due to radiative cooling in the upper troposphere, along the poleward

branch of the isentropic overturning cell (Kallenberg et al., 2005).11

The hypothesis that the tropospheric lapse–rate is maintained by baroclinic eddies in the

process of equator–to–pole heat transport appears compatible with the present results: the

isentropic meridional overturning cell exhibits a downward slope towards the pole, which

implies diabatic cooling, i.e. a heat flux convergence everywhere in the extratropics. The

column heating rate prevents radiative equilibrium in the upper troposphere and translate

into radiative cooling, which implies a smaller lapse–rate (as compared to the radiative

equilibrium). The fact that most of the energy flux convergence is concentrated in the lower

troposphere appears contradictory, however, the conclusion is still valid, because the near–

surface layers are dominated by the euqatorward branch, along which no heating would

occur; in this experiment no vertical structure of convective adjustment was assumed.

If heat flux convergence was concentrated in the upper troposphere, this would lead to

10Since we are dealing with a system in equilibrium, it is not possible to infer cause and effect relationships

directly; they have to be derived from other considerations.11This is approximately reflected in the equator–to–pole temperature difference in idealized GCMs (Schnei-

der , 2004).

21


dry–adiabatic adjustment in the boundary layer and possible strong inversions, which is

compatible with observations.12

5 Summary and Conclusion

In this paper results from radiative transfer calculations with varying convective adjustment

and trace gas concentrations were presented. A single–column radiative transfer model with

externally applied hard adjustment torwards a prescribed critical lapse–rate was used. The

subject of investigation was the tropopause height and the implied convective heating rates

of the equilibrium temperature profile. Three major experiments were conducted: a trace

gas experiment, where ozone and CO2 were removed, a water vapour experiment, where

the prescribed relative humidity was varied, and a lapse–rate experiment, where the effect

of the critical lapse–rate for convective adjustment was investigated. The first result was

that stratospheric ozone affects the tropopause significantly, however it was also found that

it only affects the height of the tropopause, but not the qualitative nature or sharpness.

(Only the tropopause cold point disappeared when stratospheric ozone was removed.)

The major result concerns the heating rates associated with the prescribed parameters

relative humidity and critical lapse–rate: convective adjustment implies heat flux conver-

gence in the troposphere, with a strong peak in the lowermost surface layer. (The strong

peak at the surface can be explained by the hydrostatic density profile.) Due to longwave

cooling, high water vapour content was found to have a strongly amplifying effect on the

heating rates implied by convective adjustment. A caveat to the results presented here is

that the surface temperature was fixed, which together with the prescribed lapse–rate de-

12The equatorward branch of the isentropic circulation, as envisioned by Schneider (2005) would in fact

be concentrated very close to the surface, because a significant part of the isentropic mass flux would

close below the mean surface potential temperature.

22


termines the tropospheric temperature profile completely, so that for example greenhouse

gases cannot effect warming in this experimental setup.

The results on implied column heating rates were put into the context of the hemispheric

isentropic overturning cell. The integrated heating rates of a single atmospheric column can

be interpreted as convergence of the meridional equator–to–pole heat fluxes: the isentropic

overturning circulation is subject to diabatic cooling along the poleward branch, in the

middle to upper troposphere. This diabatic cooling was argued to be equivalent to a lapse–

rate adjustment away from radiative equilibrium and towards a more stable stratification.

It was shown, that convective adjustment implies net heating of the extra–tropical tro-

posphere (which is to be expected). In principle it should be possible to derive an analytic

relationship between stratification/convective adjustment and heat flux convergence from

relatively simple radiative transfer models (a two–stream grey atmosphere model may even

suffice). But in order to be of any use, such a relationship would have to be combined with

a theory of meridional heat transport. The isentropic mass flux approach proposed by

Schneider (2005) appears promising, and may be fruitfully combined with a heat flux

convergence constraint envisioned above.

23


References

American Meteorological Society (2000), AMS glossary of meteorology,

http://amsglossary.allenpress.com/glossary/.

Birner, T. (2006), Fine-scale structure of the extratropical tropopause region, J. Geophys.

Res., 111.

Caballero, R. (2009), CliMT 0.6.5, http://maths.ucd.ie/~rca/climt/, an object–

oriented Climate Modeling and diagnostics Toolkit.

Caballero, R., R. T. Pierrehumbert, and J. L. Mitchell (2008), Axisymmetric, nearly invis-

cid circulations in non–condensing radiative–convective atmospheres, Q. J. R. Meteorol.

Soc., 134, 1269–1285.

Goody, R. M. (1964), Atmospheric Radiation: Theoretical Basis, Claredon Press.

Hack, J. J., J. A. Pedretti, and J. C. Petch (1999), Sccm user’s guide, http://echorock.

cgd.ucar.edu/cms/sccm/userguide.html.

Held, I. M. (1982), On the Height and the Static Stability of the Troposphere, J. Atmos.

Sci., 39, 412–417.

Kallenberg, P., P. Berrisford, B. Hoskins, A. Simmons, S. Uppala, S. Lamy-Thepaut, and

R. Hine (2005), ERA–40 Atlas, ECMWF.

Manabe, S., and F. Moller (1961), On the Radiative Equilibrium and Heat Balance of the

Atmosphere, Mon. Weather Rev., 89 (12), 503–531.

Manabe, S., and F. Strickler (1964), Thermal Equilibrium of the Atmosphere with a Con-

vective Adjustment, J. Atmos. Sci., 21, 361–385.

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Manabe, S., and R. T. Wetherald (1967), Thermal Equilibrium of the Atmosphere with a

Given Distribution of Relative Humidity, J. Atmos. Sci., 24 (3), 241–259.

Ramanathan, V., and J. A. Coakley (1978), Climate Modeling Through Radiive–

Convective Models, Rev. Geophys. Space Phys., 16 (4), 465–489.

Schneider, T. (2004), The Tropopause and the Thermal Stratification in the Extratropics

of a Dry Atmophere, J. Atmos. Sci., 61, 1319–1339.

Schneider, T. (2005), Zonal Momentum Balance, Potential Vorticity Dynamics, and Mass

Fluxes on Near–Surface Isentropes, J. Atmos. Sci., 62, 1884–1900.

Son, S.-W., and L. M. Polvani (2007), The dynamical formation of an extra-tropical

tropopause inversion layer in a simple atmospheric general circulation model, Geophys.

Res. Lett.

Thuburn, J., and G. C. Craig (1997), GCM Tests of Theories for the Height of the

Tropopause, J. Atmos. Sci., 54, 869–882.

Thuburn, J., and G. C. Craig (2001), Stratospheric Influence on Tropopause Height: Ra-

diative Constraint, J. Atmos. Sci., 57, 17–28.

A Convective Adjustment Scheme

The SCCM is meant to be run in conjunction with the convection parametrization and

turbulence scheme of the CCM model suit. The single column radiative transfer model

itself does not have a convective adjustment component.

To realize convective adjustment to an arbitrary prescribed lapse–rate, both the tur-

bulent diffusion scheme and the moist convection scheme were unsuitable and were not

25


used. Instead a simple hard adjustment module was applied to the temperature profile

after each time–step. The original code for the hard adjustment module is credited to

Kerry Emanuel and was provided in the CliMT package. However, the hard adjustment

module only adjusts to the dry adiabat, so that the code had to be modified to allow for

arbitrary critical lapse–rates. Both, the SCCM and the hard adjustment module are based

on pressure coordinates; the adjustment criterion used was actually a negative potential

temperature gradient instead of the lapse–rate.

In order to specify arbitrary critical lapse–rates in pressure coordinates I introduced an

effective adiabatic exponent χ to replace κ in

θ = T

(p0

p

)κ

=⇒ φ = T

(p0

p

)χ

, (1)

so that the gradient of the effective potential temperature φ would vanish at the desired

critical lapse–rate. The effective adiabatic exponent χ is thus defined as

χ = Γc ×Rair

g, (2)

where Γc is the desired critical lapse–rate, Rd is the gas constant for air, and g is the

gravitational acceleration.

There is a caveat: because the convective adjustment scheme only sees the effective

potential temperature φ, heating rates computed from the convective adjustment scheme

can not be used. Instead, since the final model state was in radiative–convective equilibrium,

convective heating rates were inferred from the radiative heating rates.

An extension of the hard adjustment scheme to variable lapse–rate as implemented by

myself was apparently planned but was not included in the final release of the CliMT–

package (R. Caballero, personal communication).

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