The impact of the equilibrium temperature distribution,

The Earth's atmosphere is driven by differential heating of the Earth's
surface. At the Equator, where the heating is larger than that at other
latitudes, air rises and diverges poleward in the upper troposphere,
descending more or less at 30

There are a few studies suggesting possible causes of these phenomena. One of the theories postulates global warming as a possible cause of Hadley cell widening (Lu et al., 2009). However, the atmosphere is a complex system containing many subsystems interacting with one another, and global warming might not be the only cause that is suggested to explain the widening. Ozone depletion (Lu et al., 2009; Polvani et al., 2011), SST warming (Chen et al., 2013; Staten et al., 2011) and aerosol (Allen et al., 2012) have also been invoked to explain the Hadley cell widening.

Climate models vary to some extent in their response and the relationship between global warming and the Hadley cell is not straightforward. For instance, Lu et al. (2007) found a smaller widening than the observed one. Gitelman et al. (1997) showed that the meridional temperature gradient decreases with increasing global mean temperature, and the same result can be found in recent modeling studies (Schaller et al., 2013).

Much of our understanding of the Hadley cell comes from theories using simple models (Schneider, 1977; Schneider and Lindzen, 1977; Held and Hou, 1980, hereafter HH80), and such a simple model will be adopted here in order to understand how temperature distributions can change the Hadley circulation. How much temperature change impacts the real Hadley circulation is not clear yet, perhaps because of discrepancies between observations, reanalysis (Waliser et al., 1999) and climate model outputs, although these differences are becoming less marked because of newer observational data sets or corrections of the older ones (Sherwood et al., 2008; Titchner et al., 2008; Santer et al., 2008). Hence, it is critical to understand the possible mechanisms behind the cell expansion starting from a simple model.

The objective of this study is to analyze the sensitivity of a model of symmetric circulation to the radiative–convective equilibrium temperature distribution. Our point of departure is the symmetric model used by Cessi (1998), which is a bi-dimensional model considering atmosphere as a thin spherical shell. This model will be briefly described in Sect. 2. The model mainly describes a tropical atmosphere; hence, it does not allow for eddies. Although eddies may play a central role in controlling the strength and width of the Hadley cell (e.g., Kim and Lee, 2001; Walker and Schneider, 2006), a symmetric circulation, driven by latitudinal differential heating, can exist even without eddies, and it is a robust feature of the atmospheric system (Dima and Wallace, 2003). The temperature distributions used in this study represent some paradigms of tropical atmospheres. Among the possible causes that can change temperature distributions are El Niño, global warming and changes in solar activity. We will show, in Sect. 3, that the energy input is not as important as the forcing distribution. Our results are consistent with those obtained both by Hou and Lindzen (1992) (hereafter HL92) and recently by Tandon et al. (2013), who performed experiments similar to those described here. The conclusions will be drawn in Sect. 4.

The model used in this study is a bi-dimensional model of the axisymmetric
atmospheric circulation described in Cessi (1998). The horizontal coordinate
is defined as

The model is similar to the Held and Hou model (HH80), but it prescribes a
horizontal viscosity

The non-dimensional model equations are

The thermal Rossby number

The boundary conditions for the set of Eq. (3) are

The model flow started from an isothermal state at rest and is maintained by
a Newton heating function where the heating rate is proportional to the
difference between the model potential temperature and a specified
radiative–convective equilibrium temperature distribution, which follows the
HH80 one:

Vertical

The meridional and vertical averages of

With the prescribed

Whether global warming makes the equilibrium temperature distribution narrower or wider is beyond the aim of the paper. One can expect that global warming will broaden the temperature distribution, but, at the same time, it could have an impact above all on the sea surface temperature (SST), bringing more water in the upper atmosphere, which changes the vertical distribution, especially of the temperature in the intertropical convergence zone (ITCZ). It is supposed that, in first approximation, oceans force the atmosphere, so we have to allow for the possibility that increasing SST can change the forcing distribution. Increasing uniformly SST the Hadley cell might show a poleward expansion, as showed by Chen et al. (2013) by an aqua-planet model, but in that case, the mechanism was supposed to be related mainly to mid-latitude eddies rather than a tropical forcing. Since other causes can change the temperature distribution of a planet, such as changes in the solar activity for instance, we will focus on the temperature distribution regardless of its cause.

In this model, the atmosphere is dry, as in many other studies
(e.g., Schneider, 1977, HH80, Caballero et al., 2008); changing the

The Brunt–Väisälä frequency, when the atmosphere reaches the
equilibrium, will be

Starting from Eq. (7), a set of experiments were performed changing

Maximum non-dimensional stream function

This section is divided into three subsections, the first showing the results
of the model applying the equinoctial condition, when the Sun is assumed to
be over the Equator. The solution is steady, as already shown for instance in
Cessi (1998). The second subsection will show the results of the model having
a

The axially symmetric circulation is forced by axially symmetric heating as
in HH80 and many others and as prescribed by Eq. (7). The model started from
an isothermal state and was run for 300 days, even though it reached its
equilibrium approximately after 100 days, in order to be sure that the model
does not have instabilities in the long run. The stream function values
obtained when

The absolute value of the maximum stream function intensity under the
equilibrium conditions for the 36 experiments is shown in Fig. 2. When

The dependence on

Figure 2b shows the maximum zonal wind speed as a function of

Latitude (degrees)

Some observational studies define the border of the Hadley cell where the
stream function goes to 0 at 500 hPa (e.g., Frierson et al., 2007). Since,
in our model, the zero stream function is at the poles, it is problematic to
define an edge of the Hadley cell based on the zero stream function.
Moreover, circulation intensity changes greatly in our experiments, so it is
puzzling to define an edge of the Hadley cell based on an absolute value of
the circulation itself. Hence, we will look at the location of the maximum
stream function, and we will analyze its poleward shift as a function of the
two parameters

The latitude of the maximum stream function value shows a general dependence
on

The height of the maximum stream function value is confined for almost all
the simulations to under 2200 m and the general rule is that, when

In general, the location of the maximum zonal wind speed does not show any
evident relationship with the parameters

Latitudes (in degrees) of the maximum wind speed for the equinoctial
and time-dependent solutions when

The difference between

We can understand these findings in the light of Cessi (1998), who analyzed
the model described by the set of Eq. (3) by using an asymptotic expansion of
the variables

Vertically averaged

When

With

In order to explain equable climates like those supposed to have occurred in
the Cretaceous and Eocene, Farrell (1990) formulated an axisymmetric model
starting from that of Held and Hou and used a forcing with

Figure 5 shows the stream function and the zonal wind speed for the
experiments

Since heating depends on solar irradiation, it is of interest to analyze the
solutions obtained by the annually periodic thermal forcing and to compare it
with the steady solutions described previously in this paper. Starting from
Eq. (7), we can formulate an equilibrium temperature distribution having the
maximum heating off the Equator at latitude

Non-dimensional stream function (contours) and zonal wind speed
(m s

Maximum of the annually averaged non-dimensional stream
function

The annual averages of the time-dependent and equinoctial circulations show that maximum stream functions and zonal wind speeds behave quite similarly (Fig. 6); nevertheless, the instantaneous Hadley circulation almost never resembles the modeled circulation (Fang and Tung, 1999) or the real one (Dima and Wallace, 2003).

The maximum stream function is obtained here when

The meridional position and the height of the maximum stream function show
that there is no clear dependency on

Latitude (degrees)

More than the steady solution, it is evident that the height of the maximum
stream function is lower when

The position of the jet stream is almost similar to the one observed in the
steady solution. It is confined between 28 and 30

Annually averaged non-dimensional stream function (contours) and
zonal wind speed (m s

Figure 8 shows the annually averaged circulation for the same cases as shown in Fig. 5, which is obtained by annually averaged heating. It is impressive how the steady and time-dependent solutions resemble each other. As in Fang and Tung (1999), the annual mean meridional circulation has the same extent, but different from them, the strength of the annual mean circulation of the time-dependent solution is almost the same as the steady solution.

Maximum of the non-dimensional stream function

Boreal winter circulation, non-dimensional stream
function

When the heating center is off the Equator, the intensity of the winter cell
is stronger, whereas the cell of the summer hemisphere is weak and sometimes
almost absent. Figure 9 shows the maxima of the stream function and zonal
wind speed at the winter solstitial as a function of

We can inspect a couple of simulations when the stream function reaches its
maximum in the boreal hemisphere. Figure 10 shows the stream function and the
zonal wind speed when

When

Non-dimensional stream function (contours) and zonal wind speed
(m s

With the parameters used for equinoctial and time-dependent simulations, we
performed an experiment like that of Lindzen and Hou (1988), with

Finally, we notice that comparing the time-dependent solution with

The forcing of an Earth-like planet can change for several reasons. For instance, a change in forcing distribution can be caused by different factors such as global warming or long-term variation of solar activity.

Under the assumption of an equal Equator–pole difference at the surface, we
used an axisymmetric model to study the sensitivity of the tropical
atmosphere to different

The results provide evidence that concentrated equilibrium temperature distributions enhance the meridional circulation and jet wind speed intensities, confirming findings of Lindzen and Hou (1988), even though these authors imposed the same energy input. However, in the present study, the concentrated distribution at the Equator has lower energy input.

The width of the Hadley cell is proportional to

Vertical stratification is important in determining the position and
intensity of the Hadley cell and jet when

The jet stream position does not show any dependence with

The author thanks two anonymous reviewers for their insightful comments on the paper, which helped to improve the manuscript, and the editorial staff of Nonlinear Processes in Geophysics for their valuable work. Helpful discussions with Antonio Speranza, Valerio Lucarini and Renato Vitolo are gratefully acknowledged. The School of Science and Technology, the School of Advanced Studies of the University of Camerino and Fabio Marchesoni are kindly acknowledged for partially funding this publication. Edited by: V. Perez-Munuzuri Reviewed by: two anonymous referees