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  • View in gallery
    Fig. 1.

    Monthly mean atmospheric concentrations of CFC-12 (CF2Cl2) from the ALE/GAGE network in parts per trillion by volume (pptv). The averages include only data not influenced by pollution events. The station locations are Ireland [Adrigole (52°N, 10°W) and Mace Head (53°N, 10°W)], Oregon [Cape Meares, Oregon (45°N, 124°W)], Barbados [Ragged Point (13°N, 59°W)], Samoa [Point Matatula, American Samoa (14°S, 171°E)], and Tasmania [Cape Grim (41°S, 145°E)]. The current Mace Head station replaced the Adrigole station. Northern Hemisphere stations are plotted with filled markers, Southern Hemisphere stations with hollow markers.

  • View in gallery
    Fig. 2.

    Schematic of the simple two-box model. The tracer mixing ratios in the Northern and Southern Hemispheres are χS and χN, and r represents the mass flux between the hemispheres. Here, SN is the source of tracer in box N.

  • View in gallery
    Fig. 3.

    Solutions of the two-box model without and with a steady source (left- and right-hand panels, respectively).

  • View in gallery
    Fig. 4.

    Steady and time-dependent (seasonal) modes of the Hadley-cell model (top panels). Complete streamfunction at the solstices for ε = 1 (bottom panels). Arrows indicate the flow direction. Solid and hollow arrows in the seasonal mode indicate that the flow reverses direction. Hollow circles indicate hyperbolic fixed points. Solid circles indicate elliptic fixed points. For the individual modes the arrows lie along the coincident stable and unstable manifolds.

  • View in gallery
    Fig. 5.

    Observed and fitted climatological streamfunction of the tropical mean-meridional circulation, here plotted in pressure coordinates.

  • View in gallery
    Fig. 6.

    Initial and final positions of a material line in the Hadley-cell model. The initial location is the short vertical line at 0° latitude from the surface to z = 0.1H. The final position is the long curving line. This shows the folding and stretching that leads to theheteroclinic tangle characteristic of chaotic flows. It also shows that particles initially located close together can be caught in the upper branches of either Hadley cell, leading to divergent trajectories.

  • View in gallery
    Fig. 7.

    Transport of a small, dense blob of particles from its initial location near y = L/2, z = 1.0H. The blob is first sheared and stretched, then repeatedly rapidly folded and stretched to fill the domain.

  • View in gallery
    Fig. 8.

    Poincaré sections using a 1-yr interval for α = 1.0 and three values of β. When the seasonal cycle is relatively weak, particles in the interior of the Hadley cells are trapped and isolated from the surrounding flow. Particles whose trajectories bring them near the ITCZ can be carried upward and across into the other hemisphere.

  • View in gallery
    Fig. 9.

    The shading in each panel indicates the maximum displacement of a particle from its initial location at 1-yr intervals during a 10-yr integration. Light shading indicates that at some time particles are found far from their initial location. Dark shading indicates that particles do not travel far from their initial locations. Compare the lower three panels in the center column with the Poincaré sections from Fig. 8.

  • View in gallery
    Fig. 10.

    Southern Hemisphere mixing ratio as a function of time from the area-filling trajectory calculations (triangles). The solid lines are a nonlinear fit to all 10 yr of values with an exponential function (16). The labels indicate the flow parameters (α and β), the asymptotic value of the fitted curve χf, and the exponential relaxation timescale rf.

  • View in gallery
    Fig. 11.

    Northern and Southern Hemisphere mixing ratios for three cases with a steady source in the Northern Hemisphere. The lag between the two hemispheres is a measure of the interhemispheric exchange rate.

  • View in gallery
    Fig. 12.

    Frequency distribution of residence times for eight particles. The solid line is a fit with an exponential distribution using the average residence time.

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Interhemispheric Exchange by Seasonal Modulation of the Hadley Circulation

Kenneth P. BowmanDepartment of Meteorology, Texas AM University, College Station, Texas

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Peter J. CohenDepartment of Meteorology, Texas AM University, College Station, Texas

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Abstract

Observations of long-lived tracers indicate that the time required to mix tropospheric air between the Northern and Southern Hemisphere extratropics is on the order of 1 yr. Mixing within the extratropical troposphere is known to be quite efficient, with timescales of a few months, in large part due to effective mixing by Rossby waves and extratropical cyclones. The mechanisms for transport through the Tropics, however, are not well understood, and general circulation models have reproduced the observed interhemispheric mixing time with varying degrees of success. A simple two-mode kinematic model of the zonally symmetric Hadley circulation is developed to investigate whether the seasonal oscillation of the Hadley cells could be responsible for a significant part of the interhemispheric mixing. The model explains 78% of the variance in the observed tropical climatological mean-meridional circulation. The flow has two adjustable parameters, α, which determines the strength of the time-mean part of the flow, and β, which determines the strength of seasonal oscillation. Lagrangian trajectories in the model are shown to be chaotic in the formal sense. The mixing properties of the flow are examined for a range of αs and βs around the observed values. The mixing timescale decreases as the amplitude of the seasonal cycle increases, and, in this parameter range, generally decreases as the strength of the time-mean part of the flow increases. For observed values of α and β, the mixing time is on the order of 4 months. Given that time is also needed to mix tracers into the Tropics from the midlatitudes of one hemisphere, and out of the Tropics into the midlatitudes of the other, this timescale is not inconsistent with a 1-yr interhemispheric exchange time. The rapid 4-month mixing time within the Tropics indicates that the seasonal cycle of the Hadley circulation alone may account for a large part of tropical mixing. The residence times of particles in a given hemisphere roughly follow an exponential distribution, and the bulk transport can be well represented by a simple box model, providing further evidence that the mixing in the model is fast and efficient.

Corresponding author address: Dr. Kenneth P. Bowman, Climate System Research Program, Department of Meteorology, Texas AM University, College Station, TX 77843-3150.

Email: bowman@csrp.tamu.edu

Abstract

Observations of long-lived tracers indicate that the time required to mix tropospheric air between the Northern and Southern Hemisphere extratropics is on the order of 1 yr. Mixing within the extratropical troposphere is known to be quite efficient, with timescales of a few months, in large part due to effective mixing by Rossby waves and extratropical cyclones. The mechanisms for transport through the Tropics, however, are not well understood, and general circulation models have reproduced the observed interhemispheric mixing time with varying degrees of success. A simple two-mode kinematic model of the zonally symmetric Hadley circulation is developed to investigate whether the seasonal oscillation of the Hadley cells could be responsible for a significant part of the interhemispheric mixing. The model explains 78% of the variance in the observed tropical climatological mean-meridional circulation. The flow has two adjustable parameters, α, which determines the strength of the time-mean part of the flow, and β, which determines the strength of seasonal oscillation. Lagrangian trajectories in the model are shown to be chaotic in the formal sense. The mixing properties of the flow are examined for a range of αs and βs around the observed values. The mixing timescale decreases as the amplitude of the seasonal cycle increases, and, in this parameter range, generally decreases as the strength of the time-mean part of the flow increases. For observed values of α and β, the mixing time is on the order of 4 months. Given that time is also needed to mix tracers into the Tropics from the midlatitudes of one hemisphere, and out of the Tropics into the midlatitudes of the other, this timescale is not inconsistent with a 1-yr interhemispheric exchange time. The rapid 4-month mixing time within the Tropics indicates that the seasonal cycle of the Hadley circulation alone may account for a large part of tropical mixing. The residence times of particles in a given hemisphere roughly follow an exponential distribution, and the bulk transport can be well represented by a simple box model, providing further evidence that the mixing in the model is fast and efficient.

Corresponding author address: Dr. Kenneth P. Bowman, Climate System Research Program, Department of Meteorology, Texas AM University, College Station, TX 77843-3150.

Email: bowman@csrp.tamu.edu

1. Introduction

As the potential for human impacts on the chemistry of the troposphere has increased, so has the need for a theory of chemical tracer transport. One reason that atmospheric transport is such a difficult problem is that it occurs over a range of spatial scales from millimeters to thousands of kilometers and over temporal scales from seconds to years. Transport throughout this range of scales is caused by a great variety of disparate phenomena. Constructing a complete theory of transport, even for passive tracers, will require the synthesis of observations and theory from across the range of space and time scales.

Transport in the tropical troposphere poses some particularly difficult challenges. Much of the vertical transport in the Tropics presumably occurs in mesoscale convection, which is impossible to observe in any detailon a global scale. Even the large-scale winds and temperatures are not observed well with conventional meteorological instruments. The lack of a strong dynamical balance relation in the Tropics makes it impossible to infer instantaneous winds from temperatures (heights), although data assimilation techniques may help. General circulation models (GCMs) can be useful, since they do not depend on balance but solve (approximately) the fully nonlinear momentum equations. The models, however, are fundamentally limited by their parameterized representations of convection and subgrid-scale mixing. Different convection schemes produce different tropical circulations and climates. In addition to observations, a theory of atmospheric transport will require models with varying degrees of complexity: mechanistic models to study individual phenomena and comprehensive models to investigate their interactions.

This paper focuses on the very largest scale of horizontal transport in the atmosphere: exchange of air between the Northern and Southern Hemispheres. The key to understanding interhemispheric exchange is an understanding of transport through the Tropics. The tropical circulation can be analyzed conventionally into zonal-mean and eddy parts, which can then be further divided into time-mean and stationary parts. A priori it is difficult to judge the contributions of each of these parts to the total transport in the Tropics. For example, is meridional transport caused mainly by the small-scale, short-timescale convective events, by the zonally asymmetric monsoon circulation, or by the zonally symmetric Hadley circulation? As a first step toward answers to these questions, this paper will attempt to answer the question, could the seasonally varying Hadley circulation be responsible for a significant part of interhemispheric transport?

Some basic information about interhemispheric transport can be obtained from trace gas observations. Because many long-lived anthropogenic trace gases have their sources predominantly in the Northern Hemisphere (NH), they can provide information on the rate of interhemispheric exchange of air and transport through the Tropics. Figure 1 shows monthly mean concentrations of CFC-12 (CF2Cl2) from the Atmospheric Lifetime Experiment (ALE) and the Global Atmospheric Gases Experiment (GAGE) (Prinn et al. 1983; Cunnold et al. 1994; Fraser et al. 1996). Data were obtained via the Internet from the Carbon Dioxide Information Analysis Center at Oak Ridge National Laboratory. The data show that the NH midlatitude stations (Ireland and Oregon) slightly lead the low-latitude NH station (Barbados), followed by the Southern Hemisphere (SH) low-latitude station (Samoa), with the midlatitude SH station (Tasmania) typically having the lowest concentrations at any given time. These data are consistent with a NH extratropical source for CFC-12. The mixing time within northern midlatitudes and into the NH subtropics is a few months. The mixing time through the Tropics into the Southern Hemisphere is longer, however, on the order of 1–1.5 yr. Based on tropical CO2 observations, Heiman and Keeling (1986) estimated an interhemispheric exchange time of 1.4 yr. In a side note, the CFC data also show a leveling off of the upward trends during the last few years of the record, apparently a result of successful restrictions on global chlorofluorocarbons (CFC) production under the Montreal Protocol and its subsequent amendments (Fraser et al. 1996; Montzka et al. 1996). Because the CFC data are point measurements, the information they contain on the mechanisms of interhemispheric transport is limited.

An interhemispheric exchange time of ∼1 yr could be considered either fast or slow, depending on one’s point of view. It iscertainly slower than the mixing time within the midlatitudes of one hemisphere, suggesting that some kind of barrier to mixing exists in the Tropics. On the other hand, the data suggest that there is rather efficient and complete exchange of air between the hemispheres on an annual timescale.

General circulation models have been used to simulate the distribution of a variety of trace substances in the atmosphere, including long-lived tracers such as CFCs (Prather et al. 1987; Hartley et al. 1994) and CO2, as well as shorter-lived substances, such as dust. Prather et al. found that it was necessary to introduce additional diffusive mixing in the Tropics in the GISS GCM in order to reproduce the observed interhemispheric gradient of CFCs. By doing this they were in effect assuming that errors in the model transport were due to deficiencies in simulating the small-scale transient circulation features in the Tropics. Jacob et al. (1987) used the same model to simulate the global distribution of 85Kr. They estimated an interhemispheric exchange time of 1.1 yr with the additional mixing included. Tans et al. (1990) combined transport calculations from the GISS model and global CO2 observations to conclude that a heretofore unknown midlatitude sink of atmospheric CO2 must exist. Rind and Lerner (1996) recently reevaluated interhemispheric transport in a new version of the GISS GCM with updated boundary layer, convective, and land-surface parameterizations. They found that the new version of the model did not require the enhanced tropical mixing to produce a good fit to the surface observations of long-lived tracers. The interhemispheric mixing time decreased from about 2.4 yr in the old version to 1.3 yr in the new version (in both cases without additional diffusion in the Tropics). They attribute most of the improvement to increases in the strength of the transport by the mean meridional circulation.

Pierrehumbert and Yang (1993) used a GCM simulation to study large-scale Lagrangian transport by computing trajectories along isentropic surfaces in the troposphere. They found mixing in the midlatitudes to have timescales of about 1 month, with a significant barrier in the the subtropics (e.g., Figs. 2 and 3 in Pierrehumbert and Yang 1993). They suggested that cross-isentropic (diabatic) processes might be more important for mixing in the Tropics.

While a perfectly steady Hadley circulation would not cause any net transport, the Hadley circulation may contribute to the interhemispheric transport through its seasonal oscillation. As the intertropical convergence zone (ITCZ) moves north and south in response to the solar heating, air that was previously in one Hadley cell can be carried upward and poleward in the other Hadley cell. This “entrainment” into the other Hadley cell is most likely the reason for the observed increase with height in the Southern Hemisphere of those trace species that have Northern Hemisphere surface sources. We can anticipate that the rate of interhemispheric transport will depend on the speed of the Hadley circulation and how far the ITCZ oscillates in its annual cycle. Therefore, we will construct a model that will allow us to determine the dependence of transport on the amplitude of the Hadley cell oscillation.

The transport of tracers in two-dimensional Rayleigh–Bénard convection with flow similar to the zonally symmetric Hadley circulation has been studied theoretically and experimentally. Shraiman (1987) and Young et al. (1989) analyzed the transport of a tracer by molecular diffusion in a theoretical model of steadytwo-dimensional flow with spatially periodic convective cells. They found that the convective transport within the cells increased the effective diffusivity of the flow and increased the rate at which tracer was transported between adjacent convective rolls. If the strength of the vertical temperature gradient is increased in such a configuration, the convective cells become unsteady, resulting in a time-periodic modulation of the velocity field (see Clever and Busse 1974; Bolton et al. 1986). Solomon and Gollub (1988) studied tracer transport in this situation experimentally and found that the effective diffusivity was much greater than for the steady case. In fact, molecular diffusion appeared to play no role in the transport. They also computed the effective diffusivity in a simple kinematic numerical model of the flow. Camassa and Wiggins (1991) analyzed the kinematic model in detail using lobe dynamics and showed that Lagrangian trajectories in the flow are chaotic and that mixing rates can be estimated quantitatively for small-amplitude perturbations. Similar problems have been studied by Bertozzi (1988), Knobloch and Weiss (1987), Weiss (1991), Pierrehumbert (1991), del-Castillo-Negrete and Morrison (1993), Yang and Liu (1994), Liu and Yang (1994), Yang (1996a, b), and others.

2. Methods

a. Models

1) Box model

First we develop an idealized box model to use as a standard of comparison, both for observations and for a more complex model to be developed below. As the simplest possible model of interhemispheric mixing, consider two boxes representing the Southern and Northern Hemispheres, respectively (see schematic in Fig. 2). The mass of air in the two boxes is equal, and the air within each box is well mixed so that the mixing ratio is uniform throughout the box. Exchange between the two hemispheres is represented by two pipes that carry air from one box to the other, one pipe in each direction. The flow through the two pipes is equal, so that the mass within each box remains constant. If the mass mixing ratios of a tracer in the two boxes are represented by χS and χN, respectively, the rates of change of χS and χN are given by
i1520-0469-54-16-2045-e1
Here r is the mass flux through each pipe as a fraction of the mass of a box. Recalling the observational data discussed above, we assume that there is a net source of tracer SN only in box N.
Consider a situation in which all the tracer is in box N at t = 0, that is, χS(0) = 0 and χN(0) = χ0N. Assuming that r is constant, these equations can be solved for χS(t) and χN(t), giving
i1520-0469-54-16-2045-e2
If SN = 0 (no source or sink), the solutions simplify to
i1520-0469-54-16-2045-e3
In this case the tracer mixing ratios in the two boxes converge exponentially toward χS = χN =χ0N/2 with an e-folding time τ = 1/(2r) (Fig. 3, left panel).
If SN is a nonzero constant, then for large time the solutions asymptote to linear functions of time with slope SN/2,
i1520-0469-54-16-2045-e4
(Fig. 3, right panel). Thus, for large time the difference between the two boxes χN(t) − χS(t) approaches the constant value SN/(2r), while the lag between the two solutions approaches 1/r. Significantly, the difference between the hemispheres depends on the source strength, but the lag depends only on the interhemispheric mass exchange rate. If we assume that the source strength of CFC-12 during the 1980s was constant and that this model provides a reasonable approximation to the mixing behavior of the atmosphere, it is possible to estimate the mass exchange rate from Fig. 1 to be ∼0.75–1.0 yr−1.

2) Hadley cell model

We develop a simple kinematic model of the zonally averaged Hadley circulation by expanding the mean-meridional streamfunction ψ in a space–time Fourier series and then retaining only the two dominant modes. With those simplifications the streamfunction can be written
i1520-0469-54-16-2045-e5
where y is latitude, z is altitude, H is the depth of the troposphere, L is the width of one Hadley cell (30° latitude), and τ is 1 yr. Note that the domain is restricted to the Tropics (30°S–30°N). For convenience we write the equation in geometric coordinates, but z can equally well be thought of as a mass (pressure) coordinate. By defining the ratio ε = ψ1/ψ0, we can treat the time-dependent ψ1 mode as a perturbation to the steady ψ0 mode. The ψ0 mode is the time-mean part of the Hadley circulation; it has one closed cell in each hemisphere (Fig. 4, top left). The ψ1 mode is the seasonally varying part of the circulation (Fig. 4, top right). Thus ψ0 controls the speed of the annual-mean part of the circulation, while ε controls the amplitude of the seasonal cycle. The bottom panels in Fig. 4 show the streamfunction at the “solstices,” that is, at the times of maximum amplitude of the perturbation for ε = 1. The model does not account for mixing between the Tropics and midlatitudes but assumes that quasi-isentropic mixing by eddies will efficiently exchange air between the subtropics and midlatitudes. This model, therefore, is only representative of transport by the mean meridional circulation within the Tropics.

The fidelity of this simple model to the real atmosphere is assessed by fitting the model to the observed climatological monthly mean zonal-mean meridional streamfunction shown in Fig. 3 of Oort and Yienger (1996) and comparing the truncated model to the complete streamfunction. We fit only to the streamfunction between 30°S and 30°N. We include an arbitrary phase in the seasonal mode because the streamfunction lags approximately 1 month behind the solar radiation forcing. (Thus, for this comparison with observations we use a model with three degrees of freedom rather than two.) The results offitting the model to the data are shown in Fig. 5. The ψ0 mode explains 16% of the variance of the data, while the ψ1 mode explains 62%. The two modes together, therefore, represent 78% of the variance of the climatological monthly mean Hadley circulation in the Tropics.

The model circulation has many of the features of the observed zonal-mean Hadley circulation, including the seasonal variation of the strength of the two Hadley cells and the seasonal oscillation of the location of the ITCZ, here identified as the interior line where υ = 0. Typical maximum values for υ are between 2.5 and 3 m s−1 near the surface and in the upper troposphere during the solstitial seasons. The relative magnitude of the perturbation ε ≈ 2.8; therefore, the seasonal modulation of the Hadley cells is not a small perturbation. In the numerical experiments the properties of the model are evaluated for a range of values of ψ0 and ψ1 greater and less than the observed values, indicated by ψobs0 and ψobs1. We define scaling parameters α and β such that ψ0 = αψobs0 and ψ1 = βψobs1.

The meridional and vertical velocity components are related to the streamfunction by
i1520-0469-54-16-2045-e6
which lead to equations of motion for a massless particle in Hamiltonian form
i1520-0469-54-16-2045-e7
with Hamiltonian ψ. This flow can be considered to be periodic in both the y and z direction, but because υ ≡ 0 at ± L and w ≡ 0 at 0 and H, even when ε ≠ 0, the flow is confined at all boundaries and air does not enter or leave the domain. The streamfunction of the Hadley cell flow is similar to the Rayleigh–Bénard flow analyzed by Solomon and Gollub (1988) and Camassa and Wiggins (1991) but differs in the time-dependent part. The differences in the perturbation arise from the differing character of the flows. In the Rayleigh-Bénard convection problem time-dependence arises from instability of the flow, and tracer will continue to spread horizontally from cell to cell as time increases. The Hadley cell flow, by contrast, is forced periodically, and tracer is strictly confined to two cells.

In the absence of a seasonal cycle (ε = 0) the flow is steady, particles remain on streamlines, and there is no interhemispheric exchange of air. In fact there is no transport across, streamlines at all. If tracer distributions are initially only a function of ψ, [i.e., χ = χ(ψ)] then there is also no mixing of tracer within each Hadley cell. If isopleths of tracer are not initially coincident with the streamlines, then there will be steady straining (twisting) of the tracer field resulting from shear across the streamlines.

With the addition of a seasonal cycle the behavior of tracers in this simple model changes dramatically. Most importantly, the two hemispheres are no longer isolated from one another. As the ITCZ moves during the year, air that was previously in the summer hemisphere Hadley cell is now in the winter hemisphere cell (Fig. 4, bottom panels). This air is then carried upward and poleward in the winter hemisphere Hadley cell.(The winter hemisphere Hadley cell actually straddles the equator, with the ITCZ located in the summer hemisphere.) This is the fundamental mechanism of interhemispheric exchange in this model.

b. Numerical solution

While it is possible to determine many of the characteristics of the model analytically, as will be shown in section 3a, the theory is largely limited to small-amplitude perturbations (ε ≪ 1) and answers to some questions must be found numerically. We solve (7) for [y(t), z(t)] using a fifth-order adaptive Runge–Kutta method taken from Press et al. (1986). This method estimates truncation errors by computing two estimates of the new particle location [y(t + δt), z(t + δt)], one using a single time step of size δt, and one using two time steps of size δt/2. Calculations are carried out in double precision, and the time step size is adjusted during integration to satisfy a convergence criterion σ. Table 1 shows the differences between numerical solutions as a function of σ for 1-yr forward integrations using α = β = 1. For example, changing σ from 10−5 to 10−6 changes the numerical solution by 1.4 × 10−6. On the basis of these results, and other integrations with varying convergence criteria, σ was chosen to be 10−6 for the experiments presented below.

3. Results

a. Analytical properties

In this section we describe the general characteristics of the flow field and demonstrate that particle trajectories are chaotic when ε is small and nonzero. We consider first the properties of the unperturbed flow (Fig. 4, top left). The unperturbed velocity field is given by
i1520-0469-54-16-2045-e8
This velocity field has six hyperbolic fixed points on the boundaries: four at the corners of the domain, (±L, 0) and (±L, H), and two at the equator, (0, 0) and (0, H) (see Fig. 4, top left). There are also two elliptic fixed points in the interior of the flow, one at the center of each Hadley cell at (±L/2, H/2).

When ε = 0, the flow is steady and particles remain on streamlines. All of the interior points are, therefore, periodic points. Given an initial location (y0, z0) at t = t0, it is straightforward to compute the value of the streamfunction (ψ(y0, z0, t0)) and then find the shape of the trajectory of the particle by solving (5) for y in terms of z or vice versa. It is also possible to solve for the particle position as a function of time. Because the acceleration is sinusoidal in each direction, the equation for each independent variable is that of a nonlinear pendulum. These can be solved in terms of elliptic integrals; but, as we are primarily concerned with the perturbed system, which has no known analytical solution (at least to the authors), these are of little use here.

It is important, however, to know the motion along the vertical trajectory connecting the two hyperbolic points at the equator. This kind of trajectory is often referred to as a separatrix because it separates the unperturbed flow into two disjointregions. The motion along the separatrix can be found by noting that υ(0, z, t) = 0. From (7) and (8) the equation of motion is
i1520-0469-54-16-2045-e9
With the initial condition z(0) = H/2, this can be integrated directly to yield the position as a function of time
i1520-0469-54-16-2045-e10
We will use this result below in proving that the system is chaotic.
We briefly review the properties of the perturbation part of the streamfunction (Fig. 4, top right). The perturbation velocities are given by
i1520-0469-54-16-2045-e11
This flow field has four hyperbolic fixed points in the corners of the domain, which coincide with four hyperbolic fixed points of the basic state, and a single elliptic fixed point in the center of the domain at (0, H/2). The perturbation flow field has no separatrices or heteroclinic trajectories in the interior of the fluid; if this comprised the complete flow, all particles would circulate between the two hemispheres. The perturbation field has υ ≠ 0 at y = 0 so that particles are displaced across the separatrix of the basic-state flow when ε ≠ 0.
It is possible to prove that the perturbed system (5) is chaotic by using the method of Melnikov (Melnikov 1963; Ottino 1989; Wiggins 1992). We apply the Melnikov method to the heteroclinic trajectory that connects the hyperbolic points at (0, 0) and (0, H) derived above. The displacement of a particle from this trajectory in the perturbed system is dependent upon the Melnikov integral
i1520-0469-54-16-2045-e12
Substituting from (10) for z(tt0) and manipulating the integrand gives
i1520-0469-54-16-2045-e13
Because sech is an even function and tanh is an odd function, the part of the integrand in brackets is an odd function about t0. Because the sine is periodic and symmetric, as t0 varies the integral alternates between positive and negative, depending on the phase relative to the sine function. Therefore, M(t0) has a countably infinite number of zeroes, which means that the stable and unstable manifolds of the perturbed hyperbolic points at the equator intersect infinitely many times. Thus the stable and unstable manifolds associated with the hyperbolic fixed points near the equator give rise to chaotic motion for small ε. Thus we can surmise that particles whose trajectories carry them near the equator (the ITCZ) will likely have chaotic trajectories.

Note that this method only establishes that the flow is chaotic for small ε and in a small part of the domain, that is, near the separatrix. The behavior of the trajectories around the elliptic points in the interior of the flow is described by the Poincaré–Birkhoff and Kolmogorov–Arnold–Moser (KAM) theorems (e.g., Ottino 1989). The trajectories with rational periods break up into a collection of hyperbolic and elliptic fixed points, producing multiple “cat’s-eyes” with their own separatrices (Cantor islands). Most of the irrational trajectories, on the other hand, are conserved, leading to barriers to mixing around the elliptic points. Numerical experiments on a wide range of problems have shown that as the amplitude of the perturbation (ε) is increased, the conserved trajectories and the mixing barriers gradually disappear, leading to mixing throughout the fluid. Thisproblem is addressed numerically below.

A physical picture of the mixing near the separatrix can be found by following a small, vertically oriented, material line initially placed near the surface at the equator (Fig. 6). In this example the magnitude of ψ0 has been reduced by setting α = 0.13 in order to make the effects easier to see. Because w increases upward from the lower boundary, the material line is stretched vertically. At the same time, the oscillating υ velocity of the seasonally varying part of the flow carries the material line back and forth across the equator, while the vertical shear of υ causes the material line to develop meanders. As particles approach the top of the domain, w decreases and some parts of the line are carried northward while others are carried southward. Some particles are caught in the Northern Hemisphere Hadley cell and are carried northward, while others go southward in the Southern Hemisphere cell.

b. Mixing properties

The method of Melnikov establishes that particle trajectories that pass near the ITCZ are chaotic in a Hadley circulation with a weak seasonal cycle. In order to investigate the characteristics of the flow at locations away from the separatrix and for large ε (large β), it is necessary to turn to numerical experiments. Figure 7 illustrates the efficiency of mixing by this simple flow for the case α = 1, β = 1. A small, dense patch of particles (64 × 64 grid) is initialized in the lower center of the northern Hadley cell (top panel). The remaining panels in Fig. 7 show the distribution of particles at succeeding 1-yr intervals. In the first year, the shear stretches the patch into a long narrow filament, part of which remains in the Northern Hemisphere and part of which is transported into the Southern Hemisphere. By the end of the second year, the filament has been repeatedly stretched and folded, and now occupies a large part of the domain. By the end of year 4 there is still some evidence of the filamentary structure, but particles that originated in 0.25% of the domain are now distributed throughout the domain.

We now turn to an analysis of how the large-scale mixing characteristics of the flow depend on the model parameters α and β. Figure 8 shows Poincaré sections that give a general picture of the global mixing structure of the perturbed flow for three cases with relatively weak perturbations, α = 1, β = ¼, ⅛, and ⅙. (Remember that β = 1 is equivalent to ε = 2.7.) Eight parcels are initialized at the locations in the lower half of the domain at y = L/2 (marked by the crosses). The locations of all eight parcels are plotted at 1-yr intervals for 1000 yr. When the perturbation is very weak (lower panel, β = ⅙), most particles (six of the eight) are trapped in the interior of the NH Hadley cell. The two crescent shapes that appear to be drawn with dashed lines are, in fact, from a nearly periodic orbit with a period close to 2 yr. The particle alternates between the two crescents each year. A longer Poincaré section would trace out the complete shape of the crescents. The two lowest particles, which have unperturbed trajectories that carry them close to the ITCZ, are displaced back and forth across the equator by the seasonally oscillating part of the flow. These particles create the cloud of points surrounding the isolated interiors of the two Hadley cells. When β increases to ⅛, the latitudinal amplitude of the oscillation of the ITCZincreases and the size of the trapping region decreases (middle panel). In this case only four particles are trapped. The development of new cat’s-eyes around the periodic points of a rational orbit can be faintly seen in the dark, fuzzy line of points between the trapped and chaotic regions. When β is increased again to ¼, evidence of trapping nearly disappears, except for two small islands (white areas), one in each hemisphere. Examination of the Poincaré sections for each individual particle in this case reveals that all particles show the same qualitatively random distribution throughout the domain, with the exception of the two islands. The islands are the remaining locations of nearly periodic orbits. In this case the orbits are trapped within a single Hadley cell. In cases with larger perturbations, small invariant regions still appear. In some cases these invariant regions result from orbits that form figure eights, first crossing the equator into the other hemisphere, then looping back to near their starting point 1 yr later. In other cases with large perturbations, there is no evidence of any invariant regions.

It is difficult to relate Poincaré sections to the mixing properties of the flow, so we turn to to several other methods. One approach is to fill the domain densely with particles, integrate forward in time, and analyze the distribution and location of particles as a function of time. For these calculations a grid of 128 × 128 particles is initially located in one hemisphere, and the trajectories of the particles are integrated forward for 10 yr. From these results we calculate how far each particle is displaced from its initial location and how many particles are in each hemisphere as a function of time (interhemispheric mass flux).

If particle i has initial location x0i = (y0i, z0i) (superscripts indicate time) and at 1-yr intervals is located at xni, n = 0, 1, . . . , N, we compute
i1520-0469-54-16-2045-e14
where xmaxi = (ymaxi, zmaxi) is the farthest point in the domain from the initial location of particle i (that is, the farthest corner of the domain). In the numerator, max indicates the maximum of the N annual samples. The final result is a value between 0 and 1. (The corners are actually fixed points, so parcels cannot, in fact, travel all the way to the corner in finite time.) The dis calculated in this way are shown in Fig. 9 for three values of α between ½ and 2 and a six values of βs between ⅙ and 2. The magnitude of di is indicated by shade of gray; small values are dark while large values are white. This figure reveals much more of the detailed structure of the flow than the Poincaré sections. Particles around the edges of the domain have trajectories that carry them near the ITCZ so they are likely to jump into the other hemisphere and have large values of di. When the perturbation is small, trapping inside the Hadley cells is clearly visible. For intermediate values of β, some details of the structure of the quasiperiodic orbits are visible. For large perturbations the fine-grained structure of the flow is apparent, with the surprising persistence of smallperiodic islands. It is important to remember that particles do not simply sit stationary within these islands, but follow sometimes complex paths through the domain to return to their initial locations annually. The case with realistic flow parameters (α = 1, β = 1) has two such islands that are also visible in the Poincaré section (not shown). The significance of these islands for the real atmosphere is probably minimal, since the addition of any noise would tend to destroy them.
The area-filling trajectory calculations also allow us to compute the number of particles in each hemisphere as a function of time. This is expressed as a “mixing ratio,” with the mixing ratio initially 1 in the Northern Hemisphere and 0 in the southern, as in the box model. The SH mixing ratio during the first two years of the integrations is show in Fig. 10 for the same set of experiments in Fig. 9. For comparison with the box model solutions, the results in Fig. 10 are fitted with curves of the form
i1520-0469-54-16-2045-e15
which have the same form as (3), where the subscript f indicates that the coefficients are obtained from a nonlinear fitting procedure using the IDL procedure CURVEFIT (Research Systems Incorporated 1995). If the Hadley-cell model behaved in the same way as the box model, χf would be 0.5 and rf would give the interhemispheric mass exchange rate. The curves in Fig. 10 are fit to the entire 10 yr of simulated flow. Only the first 2 yr are shown, however, since most of the change occurs in that time period. Some experiments with weak seasonal cycles and weak mixing require several years to approach their asymptotic values. The values of χf and rf from the curve fit for each experiment are given in each panel of Fig. 10.

For the experiments with β ≤ 0.25, the Poincaré sections and displacement plots show that there is some trapping with the Hadley cells. As a result, some particles never enter the Southern Hemisphere. This is clearly seen in these mixing ratio graphs also. When β ≤ 0.25 the SH mixing ratios never reach 0.5, but asymptotically approach levels between 0.08 and 0.44.

In the experiments with β ≥ 0.5, there are some small islands that suggest that the asymptotic mixing ratio will not quite reach 0.5 asymptotically. The interhemispheric exchange is vigorous enough, however, that in some cases the SH mixing ratio intermittently exceeds 0.5. For these generally well-mixed cases, the mixing ratios asymptotically approach values between 0.44 and 0.5. The roughly periodic variations in some of the panels are related to the time required for blobs of air to circulate around the Hadley cells. This time is generally less than the perturbation period of 1 yr. The case with realistic flow parameters reaches ∼0.5, with some fluctuations, in less than 1 yr and settles down to values very close to 0.5 in slightly more than 1 yr. The mass exchange rate rf estimated from the curve fit gives a lag between the two hemispheres for a tracer of 0.35 yr, or about 4 months.

In order to simulate a NH source, it is possible to add particles to the NH half of the domain at each time step. The new particles at each time step are initially distributed randomly throughout the northern half of the domain. The number of particles in each hemisphere is plotted as a function of time in Fig. 11. The lag between the two time series is estimated using results from t≥ 2 yr by computing the average difference between the values of hemisphere, fitting a straight line to the points, and then dividing by the average of the slopes. This procedure seems to work better than extrapolating to zero (taking the difference between the y intercepts). The resulting estimates of the lag 1/r agree reasonably well with the estimates from the area-filling trajectory calculations shown in Fig. 10.

Box models assume that the fluid within a box is well mixed; that is, the position of a given particle carried into the box is randomized. Therefore, the probability that any given particle will be transported to the other hemisphere is a constant equal to the flow rate r, and the interval between jumps from one hemisphere to the other (residence times) should follow an exponential distribution. Whether the particles in the Hadley-cell model behave like a hypothetical particle in the box model can be tested by computing the probability distribution of residence times. This distribution is computed from the same integrations that were used to make the Poincaré sections. These are 5000-yr integrations of 8 particles with particle locations saved 10 times per year. The exponential distribution has only a single parameter, λ, which is the number of escapes per unit time, to characterize both the mean and the variance. The average residence time (analogous to the mixing time 1/r) is equal to 1/λ.

The frequency distributions and exponential fits are shown in Fig. 12. The solid lines show fitted exponential distributions that use only the average residence time, 1/λ. The fits are poor for the cases with weak seasonal cycles (not shown), as some particles are trapped and have infinite residence times. In the well-mixed cases, particles tend to be swept back and forth from one hemisphere to the other frequently, and the probability that a particle will remain in one hemisphere for a lengthy period is small. Therefore, occurrences of long residence times are infrequent, and sampling fluctuations lead to uncertainties in the tail of the distributions. The fits in the cases with better mixing are generally good. Once again the mixing timescales agree reasonably well with the values estimated from the bulk-mixing and steady-source approaches (Figs. 10 and 11). In addition, the frequency distributions show that the probability of a particular particle jumping from one hemisphere to the other is essentially random.

4. Discussion and conclusions

A simple two-mode kinematic model of the Hadley circulation is used to investigate interhemispheric transport by the zonally averaged circulation. The two-mode model is compared to a box model. The basic-state mode consists of two steady Hadley cells antisymmetric about the equator. The perturbation mode is a single seasonally oscillating cell that extends across the equator. The two-mode model explains 78% of the variance of the observed climatological Hadley-cell streamfunction. By using the method of Melnikov, it is shown that Lagrangian motion in the two-mode model is chaotic for small perturbations. In reality, the observed perturbation is large. Numerical experiments show that for realistic model parameters the transport is strongly chaotic and that mixing due to the seasonal oscillation of the Hadley cell is rapid. Mixing in the Hadley-cell model is so efficient that for realistic parameters the mixing time is actually less than the observed lag between the hemispheres. The model, however, does not include the time required to mix tracers in or out of the subtropics of each hemisphere. This suggests that much of the observed interhemispheric mixing of atmospheric tracers in the Tropics may be due to the seasonal cycle of the zonally symmetricHadley circulation alone. Apparent deficiencies in GCM simulations of long-lived atmospheric tracers have been attributed to errors in the simulation of small-scale, high-frequency features such as tropical convection. It is possible that the deficiencies may be due instead to errors in the simulation of the large-scale zonally symmetric circulation or its seasonal cycle. One interesting speculation is that planets with small axial tilt, where the seasonal cycle of heating is weak, may have much more isolated hemispheres than Earth.

Despite the complex and chaotic nature of transport in the Hadley-cell model, the bulk properties of transport bear many similarities to simple two-box models in which one box represents each hemisphere. The hemisphere-averaged mixing ratio as a function of time in initial value problems agrees quite well with the exponential behavior produced by box models. In addition, the probability distribution of residence or waiting times for individual particles is roughly exponential and in agreement with the interhemispheric mass transport rates computed from bulk calculations.

This model is of course highly idealized. A number of directions for future research are possible. One is to consider the transport in dynamical (rather than kinematic) models of the Hadley circulation, such as those of Held and Hou (1980) or Fang and Tung (1996). One limitation here is that analytical results will be difficult to obtain. Experiments must be primarily numerical. Another research area that we are pursuing is to investigate the impact of noise on the transport. The purpose is to represent such effects as the effectively random vertical transport of particles in convective systems. A third area is to analyze the contributions to interhemispheric mixing by all the scales of motion, including the zonally averaged, slowly varying Hadley circulation, the zonally asymmetric monsoon circulation, and the high-frequency convective events. For these purposes we are using a global climate model and global reanalyses from which we can obtain three-dimensional velocities.

Acknowledgments

G. Carrie and Y. Hu helped both authors to understand aspects of this problem. K. Bowman was funded in part by NASA Grant NAGW-3442 to Texas A&M University. The authors would also like to thank the reviewers for a number of suggestions that helped improve this paper.

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Fig. 1.
Fig. 1.

Monthly mean atmospheric concentrations of CFC-12 (CF2Cl2) from the ALE/GAGE network in parts per trillion by volume (pptv). The averages include only data not influenced by pollution events. The station locations are Ireland [Adrigole (52°N, 10°W) and Mace Head (53°N, 10°W)], Oregon [Cape Meares, Oregon (45°N, 124°W)], Barbados [Ragged Point (13°N, 59°W)], Samoa [Point Matatula, American Samoa (14°S, 171°E)], and Tasmania [Cape Grim (41°S, 145°E)]. The current Mace Head station replaced the Adrigole station. Northern Hemisphere stations are plotted with filled markers, Southern Hemisphere stations with hollow markers.

Citation: Journal of the Atmospheric Sciences 54, 16; 10.1175/1520-0469(1997)054<2045:IEBSMO>2.0.CO;2

Fig. 2.
Fig. 2.

Schematic of the simple two-box model. The tracer mixing ratios in the Northern and Southern Hemispheres are χS and χN, and r represents the mass flux between the hemispheres. Here, SN is the source of tracer in box N.

Citation: Journal of the Atmospheric Sciences 54, 16; 10.1175/1520-0469(1997)054<2045:IEBSMO>2.0.CO;2

Fig. 3.
Fig. 3.

Solutions of the two-box model without and with a steady source (left- and right-hand panels, respectively).

Citation: Journal of the Atmospheric Sciences 54, 16; 10.1175/1520-0469(1997)054<2045:IEBSMO>2.0.CO;2

Fig. 4.
Fig. 4.

Steady and time-dependent (seasonal) modes of the Hadley-cell model (top panels). Complete streamfunction at the solstices for ε = 1 (bottom panels). Arrows indicate the flow direction. Solid and hollow arrows in the seasonal mode indicate that the flow reverses direction. Hollow circles indicate hyperbolic fixed points. Solid circles indicate elliptic fixed points. For the individual modes the arrows lie along the coincident stable and unstable manifolds.

Citation: Journal of the Atmospheric Sciences 54, 16; 10.1175/1520-0469(1997)054<2045:IEBSMO>2.0.CO;2

Fig. 5.
Fig. 5.

Observed and fitted climatological streamfunction of the tropical mean-meridional circulation, here plotted in pressure coordinates.

Citation: Journal of the Atmospheric Sciences 54, 16; 10.1175/1520-0469(1997)054<2045:IEBSMO>2.0.CO;2

Fig. 6.
Fig. 6.

Initial and final positions of a material line in the Hadley-cell model. The initial location is the short vertical line at 0° latitude from the surface to z = 0.1H. The final position is the long curving line. This shows the folding and stretching that leads to theheteroclinic tangle characteristic of chaotic flows. It also shows that particles initially located close together can be caught in the upper branches of either Hadley cell, leading to divergent trajectories.

Citation: Journal of the Atmospheric Sciences 54, 16; 10.1175/1520-0469(1997)054<2045:IEBSMO>2.0.CO;2

Fig. 7.
Fig. 7.

Transport of a small, dense blob of particles from its initial location near y = L/2, z = 1.0H. The blob is first sheared and stretched, then repeatedly rapidly folded and stretched to fill the domain.

Citation: Journal of the Atmospheric Sciences 54, 16; 10.1175/1520-0469(1997)054<2045:IEBSMO>2.0.CO;2

Fig. 8.
Fig. 8.

Poincaré sections using a 1-yr interval for α = 1.0 and three values of β. When the seasonal cycle is relatively weak, particles in the interior of the Hadley cells are trapped and isolated from the surrounding flow. Particles whose trajectories bring them near the ITCZ can be carried upward and across into the other hemisphere.

Citation: Journal of the Atmospheric Sciences 54, 16; 10.1175/1520-0469(1997)054<2045:IEBSMO>2.0.CO;2

Fig. 9.
Fig. 9.

The shading in each panel indicates the maximum displacement of a particle from its initial location at 1-yr intervals during a 10-yr integration. Light shading indicates that at some time particles are found far from their initial location. Dark shading indicates that particles do not travel far from their initial locations. Compare the lower three panels in the center column with the Poincaré sections from Fig. 8.

Citation: Journal of the Atmospheric Sciences 54, 16; 10.1175/1520-0469(1997)054<2045:IEBSMO>2.0.CO;2

Fig. 10.
Fig. 10.

Southern Hemisphere mixing ratio as a function of time from the area-filling trajectory calculations (triangles). The solid lines are a nonlinear fit to all 10 yr of values with an exponential function (16). The labels indicate the flow parameters (α and β), the asymptotic value of the fitted curve χf, and the exponential relaxation timescale rf.

Citation: Journal of the Atmospheric Sciences 54, 16; 10.1175/1520-0469(1997)054<2045:IEBSMO>2.0.CO;2

Fig. 11.
Fig. 11.

Northern and Southern Hemisphere mixing ratios for three cases with a steady source in the Northern Hemisphere. The lag between the two hemispheres is a measure of the interhemispheric exchange rate.

Citation: Journal of the Atmospheric Sciences 54, 16; 10.1175/1520-0469(1997)054<2045:IEBSMO>2.0.CO;2

Fig. 12.
Fig. 12.

Frequency distribution of residence times for eight particles. The solid line is a fit with an exponential distribution using the average residence time.

Citation: Journal of the Atmospheric Sciences 54, 16; 10.1175/1520-0469(1997)054<2045:IEBSMO>2.0.CO;2

Table 1. 

Difference between two numerical solutions with different convergence criteria σ. The difference is defined as the Euclidean distance between the endpoint of a 1-yr trajectory and the endpoint of a trajectory computed using the same initial condition and a smaller value σ.

Table 1. 
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