The time-dependent response of the Hadley circulation to a periodic forcing
is explored via a simplified nonlinear axisymmetric model. Thermal forcing
towards a given equilibrium potential temperature drives the model
atmosphere. The vertical stratification of this temperature is forced to
become periodically neutral with a period

The Hadley cell is a specific and observable feature of tropical circulation (Dima and Wallace, 2003).The main driving process of this feature can be seen as a combination of low-latitude convective processes and interactions of tropical circulation and higher-latitude eddies (Charney, 1969). Although tropical circulation follows a seasonal cycle due to the asymmetry of solar forcing during a solar year (Fang and Tung, 1999), it is also subjected to alternating intra-seasonal strong and weak periods (Goswami and Shukla, 1984). The cycles modulate precipitation activity in the equatorial regions.

It is well known that the rainfall distribution over Indian tropical regions varies considerably from day to day with an alternating behavior associated with a few periodic or quasi-periodic cycles. Cycles of 3–7, 10–20 and 30–60 days are present in the atmospheric dynamics of tropics and subtropics. The 3–7 day cycle is related to synoptic-scale convective systems generated over the Bay of Bengal, while the others have been associated with the cycles of monsoon rainfall over the Indian region, as discussed by Kripalani et al. (2004), who showed a prevalence of the 10–20 day cycle during a normal monsoon year and of the 30–60 day cycle during a drought year.

The latter cycle is known as the Madden–Julian oscillation (Madden and Julian, 1972). Observational analyses and modeling studies revealed that there are dominant periods in the tropics and subtropics. Since the publication of that paper, tropical oscillations have been extensively studied with observations (Yasunari, 1979, 1980, 1981; Sikka and Gadgil, 1980; Yoneyama et al., 2013) as well as by means of models (Goswami and Shukla, 1984; Zhu and Hendon, 2015; Wang et al., 2016). The Madden–Julian oscillation has a significant impact on the Indian (Murakami, 1976; Yasunari, 1979) and Australian monsoons (Hendon and Liebmann, 1990). Kessler and McPhaden (1995) for instance suggested that it could play an important role in the onset and development of El Niño events. The relationship between this oscillation and tropical cyclogenesis has also been posited in some works (Maloney and Hartmann, 2000, 2001; Mo, 2000). Moreover, He et al. (2011) showed that Hadley circulation variability is closely related to Madden–Julian oscillation convection. Goswami and Shukla (1984) suggested that this oscillation in the Hadley circulation is due to the interaction between the internal dynamics of tropical circulation with moist convection; in fact, with constant latent heat included in their model, the quasi-periodic oscillation vanished. This led them to consider that latent heat released during the moist processes can play a fundamental role in the dynamics of this cycle.

Tropical atmosphere dynamics can be explored via complex general circulation models as well as rather simpler models like the axisymmetric ones. Axisymmetric models focus on the main processes occurring in the tropospheric region, capturing the central process of the Hadley circulation. In such models, for example, eddies are not allowed, and all the processes involved in the higher latitudes are not considered. In this paper, an axisymmetric model (Cessi, 1998) is used to perform an analysis of Hadley cell behavior when periodic forcing is applied. The model spin up takes less than 100 days. When the imposed forcing is not variable, the developing circulation results in a steady state with the Hadley cell representing a fixed-point solution within the phase space. In this kind of configuration, other atmospheric processes, acting on longer or shorter spatial and temporal scales, like those mentioned earlier, are essentially excluded.

The Hadley circulation is the meridional overturn that develops in response to a temperature that is in radiative–convective equilibrium, and so it is worth analyzing how the model atmosphere behaves when the vertical stratification of equilibrium temperature becomes neutral. Although the equilibrium temperature towards which the model atmosphere evolves is essentially stable, a less stable temperature stratification may occur, example, due to tropospheric heating caused by condensation of water vapor, resulting in a less stable atmosphere. However, it is reasonable to assume that this condition occurs only periodically.

In Sect. 2 we describe the model we used and how we set up the stratification by modulating the vertical distribution of the temperature so as to periodically reach neutral static stability. In Sect. 3, we describe how modulation of the vertical stratification by a periodic function has an impact on the time evolution of the stream function, which exhibits a periodic response of two main frequencies, suggesting that a periodic response is intrinsic to this model when stratification changes periodically. In this section, we also investigate model sensitivity for some parameters and model configuration (solstitial vs. equinoctial). In Sect. 4 we present our conclusions.

This paper employs the full model described in Cessi (1998), who studied its
analytic and numerical solutions via a power series expansion in the Rossby
number

The horizontal coordinate is defined as

Starting from the dimensional equations of the angular momentum, zonal
vorticity and potential temperature, we obtain a set of dimensionless
equations. The new equations are non-dimensionalized using a scaling that
follows Schneider and Lindzen (1977), but the zonal velocity

The non-dimensional model equations are

The thermal Rossby number

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

A simulation with the forcing given by Eq. (6) leads to a stationary situation where the Hadley cell is a fixed point for the system of Eq. (3). We refer to this simulation as the control experiment. The parameters used in the simulations we describe in this paper are shown in Table 1.

Values of the parameters used in this work

In order to change vertical stratification we can define

This approach was used by Tartaglione (2015), who used constant

Thus, we analyze the response of our model to an equilibrium temperature that
moves periodically from a stable stratification to a neutral one; i.e., we
allow exponent

When the

A cautionary note is necessary. The main role of convection is to bring the atmosphere into a state of neutral
vertical stratification, deleting the effects of the unstable layer created by radiative-only processes. Thus the
dry unstable condition is almost never met in the real atmosphere as there is always an overturn that leads the
atmosphere to be statically stable and the most prevalent instability is the conditional stability due to the
presence of water vapor. However, the neutral condition of

Temporal evolution of the stream function at 3

Using Eq. (7) as forcing, we obtain a vertical stratification of the
equilibrium temperature that becomes periodically neutral. The values of

We now take a closer look at the details in a specific point of the domain,
that corresponding to 3

The time evolution of the stream function, at 3

If we look at the phase space evolution and the plot of
couples(

Time evolution of the solution, for

Thus, if the adiabatic forcing is reached with a relatively fast change of
the stratification, the solution follows adiabatic forcing, increasing the
strength of the circulation (Fig. 3a), which leads to strong subtropical jet
streams and stronger easterly winds in the equatorial region (Fig. 3b). As in
Fang and Tung (1999), who found stronger circulation when they replaced a
fixed Sun (equinoctial Hadley cell) with a moving Sun, here it is the
time-varying stratification stability of

Periodicity with two dominant cycles in the model response is interesting in
light of the oscillations observed in the tropical atmosphere. Madden and
Julian (1971, 1972) were the first to show the existence of an oscillation in
pressure and winds with a predominant peak in the spectrum at a period of
40–50 days. They also showed that the amplitude of this peak was greater in
the tropical station and was weaker in the sub-tropical stations. Yasunari (1979) demonstrated by means of spectral analysis that cloudiness
fluctuations have two dominant periodicities: one of about 15 days and
another of 40 days. Other studies documented 15-day oscillations within the
tropical regions related to monsoons (e.g., Krishnamurti and Bhalme, 1976;
Krishnamurti and Ardanuy, 1980; Krishnamurti and Subrahmanyam, 1982).
Yasunari (1981) showed that even the 40-day oscillation has some relation to
the Asian summer monsoon. Anderson and Rosen (1983) found similar results by
using zonally averaged zonal winds. Thus, the features of these oscillations
suggest that it may be possible to understand them with a zonally averaged
model. Goswami and Shukla (1984) used a symmetric model with hydrology to
study the Hadley circulation and found that it has well-defined strong and
weak episodes. These oscillations of the Hadley circulation occurred in their
model in two dominant ranges of periodicities: one with a period of between
10 and 15 days and another with a period of between 20 and 40 days. Since our
model does not include hydrology, this double period has to be related to the
internal dynamics of the system. In fact, if periodicity is expected with a
time period equal to the forcing time period

Representation of the evolution of the trajectory of the stream-function at 3

Difference between the stream function (in dimensionless units) of
the experiment with

The time series of the vertically averaged stream function shows that the observed periodic response involves mainly
low-level processes. Figure 4a shows the time series of the stream function averaged over the lower 3200 m, while
Fig. 4b shows it averaged over all the domain height. In both cases strong and weak patterns in the stream function are
present, and the strong episodes of the stream function at lower levels do not dump suddenly, but persist for a while.
However, when averaged over the entire domain, the time series shows strengthening and weakening of the stream function that
seem to be periodic. This periodic behavior tends to become intermittent with increasing

The interaction of slow parameter variation with the fast rate of motions in the phase space causes a phenomenon known as “dynamic bifurcation”
(Guckenheimer and Holmes, 2002).
Figure 5 shows a one-dimensional bifurcation diagram, i.e., the differences between two stream functions at the same point
of the domain (3

Periodicity is still present, in the sense that chaotic behavior of the model appears periodically. This occurs, for
instance, for

Time evolution of the vertically averaged stream function (non-dimensional unit) over lower levels,
up to 3200 m

It seems that a bifurcation delay might be active when

The one-dimensional bifurcation diagram as a function of

We can say that identification of dynamic bifurcations caused by slow variation can, in general, be a problematic task, because of transitions. During these transitions two situations could occur: bifurcations with abrupt change of the attractor size (and in such a case the dynamic bifurcation could be visible), or transitions occurring with changes in the geometry of the chaotic attractor. In the latter case it may be difficult to observe these variations in the short time that the system is in the chaotic state.

One can argue that the amplification seen in the model solution when

As in Fig. 1g, but halving the time step, and increasing the spatial resolution, i.e., halving
the horizontal

The behavior of the numerical solution for large values of

Moreover, a second circulation is produced at the poleward edge of the Hadley circulation in the case where

Mean stream function in dimensionless units (contour) and zonal wind in m

To investigate whether history affects the value of the internal state of our model, we performed a simulation
where we changed the value of

Temporal evolution of the stream function (in dimensionless units) at 3

The time evolution of stream function (in dimensionless unit) in the ascending branch of the
circulation for

The time evolution of stream function (in dimensionless unit) at 3

The one-dimensional bifurcation diagram as a function of

Although a parameter uncertainty analysis similar to that presented by Knopf et al. (2006) is outside of the aims of this paper, a sensitivity analysis for some parameters was performed to establish whether found features persist under different conditions.

In previous experiments, static stability is forced toward a specific minimum limit, such as neutrality, but whenever it is forced toward values that differ slightly from the neutral condition, the solution behavior remains unaltered.

When the change is significant (e.g., 20 % higher or lower than the neutral
condition), we need to distinguish the model response as a function of

The sensitivity to the parameter

The model configuration has symmetry around the Equator. To determine how the
model responds to the loss of this symmetry, we performed solstitial
experiments with the same parameters used for the equinoctial (symmetric)
configuration with maximum heating located in a hemisphere at 6

The periodic chaotic behavior is even related to vertical viscosity. It even appears for time periods less than 63 days,
when the vertical viscosity is close to zero. When the vertical viscosity is 0.5 m

We have used a dry axisymmetric model to simulate the Hadley circulation and to investigate the role of a changing stratification of the thermal forcing, which simulates moist convection that alters the static stability of the model atmosphere. The bi-dimensionality of the model prevents the generation of any eastward traveling wave. Hence, in our discussion, the influence of eddies' momentum fluxes is not taken into account. We have shown that the stream function representing the Hadley circulation in an axisymmetric model can exhibit a periodic behavior when the vertical stratification of the thermal forcing is periodically forced to become neutral. The question that arises is whether this oscillation can be linked with the observed atmospheric fluctuations within the tropical region. Although the Madden–Julian oscillation is a three-dimensional phenomenon with the development of a Kelvin–Rossby wave (Gill, 1980), it is certainly associated with the evolution of convective anomalies (Hendon and Salby, 1994). It has already been suggested that quasi-periodic oscillation seems to be an intrinsic characteristic of the tropical atmosphere, in accordance with the results of Goswami and Shukla (1984). Bi-dimensional models, while they can give us a framework of the basic physics underlying atmospheric processes, are quite limited as the real atmosphere is naturally three-dimensional. The findings of this work suggest that if a cyclic process perturbs tropical stratification, the Hadley circulation strength may be periodic with possible bursts when the perturbation time is larger than 60 days. These results must necessarily be compared with observations of the real atmosphere or to the results of three-dimensional models.

If the forcing period is up to 63 days, the stream-function evolution shows a periodic behavior. For period forcing longer than 63 days, the slow frequency associated with the forcing period modulates a fast response in the system, generating a chaotic motion that persists for a period of time before returning to a non-chaotic solution. It is not clear whether these high-frequency characteristics are actually present in the meridional circulation of our planet. In fact, when we look for periodicities on the order of tens of days in observations, the higher-frequency signals are usually removed. Moreover, even though we can detect such signals, it is not easy to associate them with the oscillations caused by a change of static stability, as opposed to other processes. The chaotic dynamics observed in the model could exist on other planets where the vertical stratification takes longer to become neutral. The change of vertical stability that in our model simulates the cycle of large-scale convection might be equivalent to the recharge and discharge of moisture that supports the Madden–Julian oscillation (Zhu and Hendon, 2015). This change of stability can be imagined to control the aggregation process of convection, which allows a bistable equilibrium between moist and dry situations (Raymond and Zeng, 2000; Zhang et al., 2003; Arnold and Randall, 2015). An important aspect is the rate at which this process occurs. As we have shown in the model we consider, when this rate is over 63 days it results, in the short term, in a chaotic impact on the Hadley circulation strength.

Many parameters control the numerical solution for the model presented in this paper. We have analyzed model sensitivity
to a few of these parameters with imposed stratification periodic forcing. The forcing imposes static stability to move
toward neutrality, and the model's response does not change if static stability remains close to neutral, but when it
departs from neutrality, there is no change in the model solution for low values of

The model in the equinoctial configuration shows intermittent chaotic behavior when the forcing period is large because of the importance of the stratification forcing imposed on the model atmosphere. However, this static stability becomes less important in creating large spikes in the stream function when a solstitial configuration is adopted. With a single strong cell, the circulation is more stable.

The role of friction in the symmetric circulation, driven by a meridional thermal gradient of a fast rotating planet like the Earth, is contradictory. On the one hand it is an essential ingredient to allow a meridional overturn, instead of a strong zonal wind in cyclostrophic balance only. On the other hand, the value must be close enough to zero to allow angular momentum conservation. Although these conditions are met in the model we consider, the presence of a time-varying stratification alters the classic view of a stably stratified vertical temperature gradient. Other than the meridional thermal gradient, this imposed time-varying stratification represents another nonlinear forcing, which amplifies the model response when the vertical viscosity is small, itself representing a source of amplification of the model response in the inviscid case.

A FORTRAN version of the model is available by email request to the corresponding author.

The author declares that they have no conflict of interest.

The author wishes to thank two anonymous reviewers who were willing to review this paper. One of them provided useful suggestions to remarkably improve the paper. Uni Research Climate is acknowledged to cover the publication cost. Edited by: V. Shrira Reviewed by: two anonymous referees