## 1. Introduction

The effect of the sun on seasonal variations in climate has been known for millennia. Seasonal cycles of the atmospheric circulation are obvious, for example, in the occurrence of monsoons, and affect agriculture, economics, and the everyday life of human beings. The latitude where the sun is overhead marches gradually across the equator from the Tropic of Capricorn (23.5°S) to the Tropic of Cancer (23.5°N) before returning to complete the four seasons. Since the seasonal march of the sun is smooth with respect to latitude, it is natural to expect that the transition between seasons in the atmospheric circulation would also be smooth.

However, observations of the annual variation of the intertropical convergence zone (ITCZ) indicate that this is not the case. We find that the maximum of zonal mean precipitation does not move smoothly from one hemisphere to the other but seems to jump over the equator. Using the Tropical Rainfall Measuring Mission (TRMM) daily 1° × 1° gridded data based on satellite measurements (Kummerow et al. 2000), we plot the climatological (1998–2004) zonal mean precipitation as a function of day and latitude along with the latitude where maximum precipitation occurs (Figs. 1a,b). With either indicator of the location of the ITCZ, we see that the ITCZ moves rapidly from one hemisphere to another twice a year and stays between 5° and 10° off the equator between the transitions. The transition from the NH to the SH is especially abrupt. This abrupt transition is also apparent over a longer period (1979–present) in the Global Precipitation Climatology Project (GPCP) pentad (5 day) data (Figs. 1c,d), which are a combination of gauge measurements and satellite estimates of rainfall on a 2.5° latitude–longitude grid (Xie et al. 2003). The advantage of the GPCP dataset is that it has a longer period of record compared to TRMM. The disadvantage is that it is a pentad dataset with coarser spatial resolution and thus less resolution of the seasonal transition. This rapid transition is less apparent in monthly averaged data [compare Dima and Wallace (2003) and Hu et al. (2007)].

The vertical velocity provides similar evidence for the abrupt seasonal transition of the zonal average circulation. European Centre for Medium-Range Weather Forecasts (ECMWF) reanalyses provide daily vertical velocity on a 2.5° × 2.5° grid (Uppala et al. 2005). The vertical velocity at 300 mb was used because it is collocated with deep convection and the maximum vertical velocity with respect to latitude can be used as an indicator of the ITCZ. Data assembled after 1979, when observations became much more abundant following the introduction of satellite retrievals, are used here. A similar annual march of the ITCZ with rapid crossing of the equator is indicated by the maximum vertical velocity (Figs. 1e,f). The same results are obtained from the widely used National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis (Kalnay et al. 1996) and NCEP-2 (Kanamitsu et al. 2002) reanalysis.

The aim of this paper is to explain how axisymmetric atmospheric dynamics can contribute to a rapid seasonal migration of tropical rainfall across the equator. A model with no dynamics but only local radiative and convective processes shows a gradual seasonal transition, with the ITCZ collocated with the warmest sea surface temperature (SST) and never getting far from the equator as a result of the ocean’s thermal inertia. We show that axisymmetric atmospheric dynamics cause a rapid transition from one solstitial circulation to the next. With a two-level, nonlinear model of an axisymmetric atmospheric circulation connected to a slab ocean, we explore the role of angular momentum in the seasonal transition of the ITCZ. Eddy transports of momentum are set to zero, which is made possible by the semi-Lagrangian numerical scheme for advection. Our model is similar to that used in classical studies of the axisymmetric tropical circulation (e.g., Held and Hou 1980; Lindzen and Hou 1988), although we introduce a less idealized representation of diabatic heating using a simple model of latent and radiative heating. Ocean temperature is calculated using a slab ocean and a surface energy balance. We drive a seasonal cycle through the migration of solar heating back and forth across the equator, and our model shows abrupt seasonal transitions as observed.

Recent studies argue that eddy heat and momentum transports to midlatitude are fundamental to the tropical circulation (Schneider 2005; Walker and Schneider 2005). These eddy fluxes are largest in the subtropics, but close to zero near the equator (Peixoto and Oort 1992; Trenberth and Stepaniak 2003). While these fluxes may be important to the width and strength of the Hadley circulation (HC; Walker and Schneider 2005), it remains to be seen whether they are important to seasonal migration of the ITCZ in the deep tropics. We want to see how well a model excluding eddies can simulate the observed seasonal transition of the ITCZ. While eddies may be important to this transition, their potential effect is beyond the scope of this study.

In this paper, we investigate the role of axisymmetric atmospheric dynamics and thermal inertia of the ocean in causing an abrupt seasonal transition of the ITCZ. The formulation of our model is described in section 2. Simulations corresponding to the annual mean solar forcing are given in section 3. In section 4, a time-dependent simulation in response to seasonally varying solar forcing is obtained. The abrupt seasonal transition of the circulation, possible reasons for the abrupt transition, and factors that affect the time scale of the transition are discussed in section 4. Sensitivity of the transition to certain model assumptions is discussed in section 5. Summaries and discussions are given in section 6.

## 2. The model

### a. Model structure

In this model, we start with the set of axially symmetric (i.e., zonal average) primitive equations for a thin fluid on a sphere of radius *a* rotating with angular velocity Ω. Because the Hadley circulation has a simple vertical structure with compensating meridional flow at upper and lower levels, the model consists of an upper and lower layer. The model calculates the upper-layer potential temperature *θ _{T}*, which determines the surface air temperature through a prescribed lapse rate. The assumed fixed lapse rate of 6 K km

^{−1}is based on the correspondence of the observed vertical structure to a moist adiabat (Betts 1982; Xu and Emanuel 1989; Peixoto and Oort 1992), at least within the convecting half of the tropics. The surface layer is attached to a slab ocean of temperature

*T*, with prescribed ocean heat transport that is currently set to zero. The model is forced by solar heating. Diabatic heating consists of latent heat release and radiative cooling by greenhouse gases along with solar absorption. For simplicity, we omit an explicit moisture budget and instead specify relative humidity. A moisture constraint in which the global integral of precipitation equals the global integral of evaporation is added.

_{o}The domain of each layer is from the South Pole to the North Pole. The variable chosen to describe latitude is *y*, defined as sine(latitude). In this way, latitudinal resolution is finer in low latitudes, where the HC exists.

### b. Fundamental equations

*u, υ, w*) be the zonal mean velocity of the fluid in the longitudinal, latitudinal, and vertical direction, respectively. Assuming no diffusion, which is sometimes taken to represent eddy transports in simple models, the zonally averaged zonal wind equation for the free troposphere iswhere

*a*is the radius of the earth and Ω is the earth’s angular rotation rate. In addition to the meridional coordinate

*y, z*represents the natural logarithm of pressure:

*z*= −

*H*

_{0}ln(

*p*/

*p*), where

_{s}*H*

_{0}=

*RT*

_{0}/

*g*and

*T*

_{0}and

*p*are a (fixed) reference temperature and surface pressure, respectively.

_{s}*δ*= 2 km adjacent to the tropopause (at height

*H*= 16 km). Integrating the angular momentum equation (3) over the free troposphere, that is, from the top of the surface boundary layer (also of depth

*δ*) to the tropopause at

*z*=

*H*, and using mass continuity, we obtain the angular momentum equation for the upper branch of the HC:where

*M*is the angular momentum of the free troposphere, assumed to be independent of height, and

*V*is the meridional wind averaged over the depth of the upper layer:

*V*=

*δ*

^{−1}∫

^{H}

_{H−δ}

*υ dz*. We assume that angular momentum is conserved in the upper branch of the HC in the absence of eddies except in ascending regions where surface air with zero zonal wind is added (Shell and Held 2004). The

*κ*term on the right-hand side represents the effect of mass exchange with the lower layer on the momentum of the upper layer:Here

*M*is the angular momentum of the lower (surface) layer, neglecting the contribution from the surface zonal wind:

_{B}*M*= Ω

_{B}*a*

^{2}(1 −

*y*

^{2}). Air rising from the lower layer carries with it the zero relative angular momentum of that layer. Air that descends from the upper layer to the lower layer carries with it the momentum of the upper layer and thus does not affect the momentum of the upper layer. The ascending region is defined asdeduced from the continuity equationsetting the vertical velocity

*w*= 0 at the tropopause.

*υ*is calculated by neglecting nonlinear advection terms:To replace the geopotential Φ with temperature

*T*, we apply ∂/∂

*z*to Eq. (7) and assume hydrostatic balance:Then, integrating over the depth of the troposphere

*H*while assuming |

*u*(0)| ≪ |

*u*(

*H*)| (Held and Hou 1980) and using the assumption of compensating meridional flow between the upper and lower layers gives uswhere

*U*is the zonal wind averaged over the depth of the upper layer linked to

*M*by (2) andis the column-averaged temperature.

*T*is the temperature of the upper branch of the HC at height 16 km and is related to the prognostic potential temperature of the upper branch

_{T}*θ*byThe right-hand side of Eq. (11) represents thermal wind balance.

_{T}*θ*iswhere

*Q*is the diabatic heating rate per unit mass and

*C*is the specific heat of the atmosphere at constant pressure. Diabatic heating includes latent heating from precipitation, shortwave radiative heating, and thermal radiative cooling of the atmosphere, along with sensible heating from the surface. Their parameterization will be described in the following subsection.

_{p}*θ*(

*z*) =

*θ*+ Δ

_{s}*θ*(

*z*/

*H*), where

*θ*is the surface air temperature (SAT) and Δ

_{s}*θ*is a fixed potential temperature difference between the tropopause and the surface. Integrating (12) over the depth of the troposphere, we get an equation for the potential temperature of the troposphere, written in terms of

*θ*, the value at the tropopause:whereis the pressure-weighted column-integrated diabatic heating rate per unit mass.

_{T}In addition to *θ _{T}*, the ocean surface temperature

*T*is another prognostic variable that we calculate through the surface energy balance, as described in section 2d.

_{o}### c. Diabatic heating

#### 1) Radiation

Radiative heating in the diabatic heating term * Q* in (13) includes heating by absorption of solar radiation and cooling by absorption and reradiation at thermal wavelengths.

The annual variation of daily mean insolation at the top of atmosphere (TOA) for each latitude is obtained from Hartmann (1994). The planetary albedo is gained from the climatological annual mean of the 1985–89 Earth Radiation Budget Experiment (ERBE) dataset, which has a horizontal resolution 4° latitude by 5° longitude (Barkstrom et al. 1989). The SH and NH planetary albedos are averaged about the equator to give a symmetric function. Prescribing planetary albedo eliminates cloud feedbacks from our model, which simplifies its interpretation. The net absorbed shortwave radiation (SW) at the TOA is the only external forcing in our model. Using the daily absorbed SW at the TOA obtained from the above formulation, which peaks at the solstices, the model exhibits an abrupt hemispheric transition of the ITCZ, as described in section 4. To eliminate the possibility of an abrupt transition resulting from enhanced solar forcing near the solstices (Fig. 2a), we normalize the daily TOA SW forcing so that its maximum with respect to latitude is identical at each day throughout the year (Fig. 2b). The atmospheric fractional absorption of the SW is assumed to be 0.27 to be consistent with the annual and zonal mean net surface solar radiative flux from the International Satellite Cloud Climatology Project (ISCCP) data products (Zhang et al. 2004). The combination of prescribed planetary albedo and atmospheric absorption implicitly specifies the surface albedo.

_{2}). A two-stream column model is used to calculate the infrared radiative transfer in the atmosphere. We specify relative humidity

*r*with height, referring to Manabe and Wetherald (1967):where

*p*is pressure and

*p*is surface pressure. The surface relative humidity is assumed to be 0.8. Because relative humidity is specified, column water vapor varies with temperature, which introduces a water vapor feedback. We assume pressure and height are related by

_{s}*p*/

*p*=

_{s}*e*/

^{−}z*H*

_{0}, where

*H*

_{0}is the atmospheric scale height, assumed to be 8 km, and the pressure at the tropopause and the surface are 100 and 1000 mb. Then, we can calculate water vapor density as a function of height, given a temperature profile:where

*ρ*is water vapor density,

_{w}*q*and

*q** are specific and saturated specific humidity,

*ρ*is atmospheric density,

*ϵ*and

*R*are thermodynamic constants, and

*e** is saturated vapor pressure, which is a function of

*T*(

*z*), the atmospheric temperature at height

*z*. The mixing ratio of CO

_{2}is assumed to be constant, 0.056% by mass, corresponding to a present-day volume mixing ratio of 370 ppm. Methane (CH

_{4}) and nitrous oxide (N

_{2}O) are two other long-lived greenhouse gases that contribute radiative forcing to the current climate. We include them in our model only implicitly. For their present-day abundance, their total radiative forcing is comparable with 30% CO

_{2}forcing based on the relation between concentration and radiative forcing (Houghton et al. 1996, 2001). For simplicity, we take their radiative effects into account by increasing the CO

_{2}concentration by a factor of 1.3 in our LW radiative transfer calculation.

In the two-stream radiative transfer model, we divide the troposphere into 12 layers. Layer thickness is generated in such a way that each layer has the same water vapor content. A flux emissivity table for water vapor and CO_{2} with various densities at different temperatures is given by Staley and Jurica (1970). The temperature at the center of each layer is used in the radiative transfer calculation. As discussed before, we assume a fixed temperature lapse rate with height and, given the temperature at the uppermost layer, we have a temperature profile. With absorber density calculated from the assumed relative humidity profile, the LW radiative flux within the troposphere is obtained.

#### 2) Atmospheric latent heating and moisture constraints

*in the energy equation. We assume that the precipitation has a Gaussian distribution with latitude within the circulation, with a maximum at the ITCZ defined by the warmest SST. We assume that there is only one convection belt over the whole tropics. The ITCZ has a half-width of*Q

*L*, which is set equal to 2.5° based on TRMM and GPCP zonally averaged precipitation (Figs. 1a,c):where

*P*

_{0}is specified as described below and

*y*is sin(lat of the ITCZ). Thus, 95% of the tropical precipitation falls in the narrow 10°-wide band centered over the warmest SST. The ITCZ width, related to

_{I}*L*, is an external parameter in this model that we estimate from the observed precipitation in Fig. 1. Because the circulation strength is sensitive to this width (Hou and Lindzen 1992), we consider the sensitivity of our results to this parameter in section 5.

A more realistic parameterization of ITCZ convection might take into account the stability of the entire column, accounting for upper-tropospheric temperature in addition to SST. As a sensitivity study, we center the ITCZ over the latitude where *h** − *h _{B}* is a minimum. Here

*h** is the saturated moist static energy of the midtroposphere and

*h*is the moist static energy at the surface. The difference

_{B}*h** −

*h*is related to convective available potential energy (e.g., Emanuel 1994). Because our model surface temperature has a constant offset to the upper-tropospheric value (due to our fixed lapse rate assumption), we use SST to evaluate

_{B}*h*and its contribution to column stability. In this parameterization, the location of the ITCZ depends not only on the presence of warm air at the surface (related to a warm underlying ocean) but also cool air aloft. It thus introduces an additional degree of freedom in the location of the ITCZ. Despite this, we find that the seasonal migration of the ITCZ in our model is virtually unchanged, compared to the behavior described in section 4, where the ITCZ is collocated with the warmest SST. This is because latitudinal variations in

_{B}*h*dominate those in

_{B}*h**, mainly because water vapor is much greater at the surface than above the boundary layer. While prescribing the ITCZ location in terms of the difference of moist static energy within the column is slightly more realistic than using only SST, we prefer the latter because it makes the model easier to interpret. The collocation of tropical rainfall and SST allows the ITCZ location to be understood in terms of the relative influence of solar heating and atmospheric dynamics upon ocean temperature.

*y*

_{c}_{1}and

*y*

_{c}_{2}are the southern and northern edges [in units of sin(lat)] of the circulation. In reality, precipitation at any instant is related to evaporation over a range of previous times. However, seasonal variations of evaporation within our model circulation are within a few percent and are approximately invariant with time. By relating precipitation to evaporation, a moisture budget is implicitly included in the model. At high latitudes, outside the circulation, local precipitation equals local evaporation.

### d. Surface energy budget

*ρ*, specific heat

_{o}*C*,

_{p}*, and depth*

_{o}*h*and no dynamical transport of the energy. Then the prognostic equation for SST (

*T*) iswhere the right-hand side represents the net surface energy flux, which includes

_{o}*R*, the net radiative flux at the surface (positive downward), the product

_{s}*L*, the evaporative cooling of the surface, and

_{υ}E*S*, the surface sensible heat flux, all evaluated per unit area. Then storage of energy in the ocean is the excess of the net radiative flux into the surface over the sum of the latent heat and sensible heat fluxes from the surface to the atmosphere. The mixed layer depth is taken to be 50 m, close to the observed tropical average of 56 m (Russell et al. 1985), although we will subsequently allow this parameter to vary.

*R*is the sum of the net SW and LW fluxes at the surface. Turbulent fluxes of sensible and latent heat are calculated with the bulk aerodynamic formulas. The sensible heat flux is written aswhere

_{s}*ρ*= 1.29 kg m

_{s}^{−3}is the surface air density,

*C*= 10

_{DH}^{−3}is the assumed aerodynamic transfer coefficient for temperature,

*U*is the surface wind velocity, assumed to be constant at 8 m s

_{r}^{−1}, and

*T*is the surface air temperature.

_{s}*C*= 10

_{DE}^{−3}is the assumed aerodynamic transfer coefficient for humidity,

*q*is specific humidity at the surface, and

_{s}*q**(

*T*) is the saturated specific humidity at the sea surface temperature

_{o}*T*. If

_{o}*r*is the surface relative humidity, evaporation can be further written asThe coefficient

_{s}*β*is an empirical factor that we adjust to match the observed area-average evaporation rate within the tropics (30°S–30°N) and accounts for the presence of land where the surface supply of moisture is limited. To match surface evaporation as estimated by the Climate Prediction Center (CPC) Merged Analysis of Precipitation (CMAP; Xie and Arkin 1996), we set

*β*= 0.5. Multiplying (22) by the latent heat of vaporization

*L*produces the latent heat flux. Thus, given surface relative humidity, surface density, and values of surface temperature and surface air temperature, the sensible and latent heat fluxes are obtained.

_{υ}### e. Boundary conditions

At the North and the South Poles where *y* = ±1, there is no heat flux, so ∂*θ _{T}*/∂

*y*= 0. Because of axisymmetry, there is no poleward or zonal velocity; that is,

*u*=

*υ*= 0.

### f. Numerical methods

The angular momentum equation (4) includes both advection and a source term representing the injection of air with zero relative angular momentum from the surface into the free troposphere within the ascending branch of the circulation. For such a problem, the combined semi-Lagrangian (Staniforth and Côté 1991) and Crank–Nicholson (Press et al. 1992) schemes are appropriate numerical techniques. Semi-Lagrangian schemes are based on the conservation of physical properties (e.g., angular momentum) along a trajectory if there is no source. It is not diffusive unless it uses a low-order interpolant. To save calculation time, we use linear interpolation in the semi-Lagrangian scheme, which might introduce a little diffusion. However, the strong gradients near the midlatitude jets, described in the next section, suggest that the diffusive effect from the low-order interpolant is small in practice.

For the meridional wind, potential temperature, and SST equations [Eqs. (11), (13), and (19)], the Crank–Nicholson scheme is used, which is semi-implicit and unconditionally stable for simple diffusional problems. Because these three equations do not have nonlinear advection terms, the semi-Lagrangian scheme is not necessary, and would otherwise increase the calculation time.

The total number of grid points between the South and North Poles is 401, with resolution *dy* = 0.005, roughly 0.3° in the tropics; the time step is 864 s.

## 3. Circulation with steady annual mean solar forcing

Despite our interest in the seasonal cycle, we begin with the model response to steady annual mean solar forcing that is a maximum on the equator. This shows the key behaviors of the model and how well it represents the observed circulation despite its simplifications. We analyze the model in relation to other axisymmetric circulation models, most of which have more simple treatments of diabatic heating (e.g., Held and Hou 1980).

### a. Angular momentum, wind, and temperature

The angular momentum, zonal wind, meridional wind, potential temperature of the upper layer, and SST of the steady state at all latitudes are shown in Fig. 3. Westerly winds occur in both hemispheres and fall to zero at the equator. Westerly jets are located at *y* = ±0.7 corresponding to 44° latitude, where the edges of the circulation are defined. The edges of the model HC are closer to the poles, and the jets are stronger, than the observed annual mean state (Peixoto and Oort 1992). Inside the circulation region, angular momentum is nearly uniform, even though air with zero relative angular momentum ascends in the free troposphere over a range of latitudes (Schneider 1977; Held and Hou 1980). The zonal wind is consistent with thermal wind balance.

The meridional wind in the upper level indicates that air moves from the equator toward the poles, with the highest speed of 5 m s^{−1} at *y* = ±0.1 (about 6° latitude). Since *υ* depends on the assumed depth of the upper layer *δ*, a more robust comparison to observations is with the mass flux, Ψ = 2*πa**y*^{2}*ρυδ*. According to this definition, the model maximum mass flux in the upper layer of the HC, with a density *ρ* of 0.3 kg m^{−3}, is 12 × 10^{10} kg s^{−1}, which is about 2 times stronger than the climatological annual mean of 7 × 10^{10} kg s^{−1} (Peixoto and Oort 1992). The strong mass flux comes from the strong precipitation of the model, which we discuss in section 5.

Upper-tropospheric potential temperature and SST exhibit small meridional gradients within the circulation region because of the meridional energy transport by the HC. The warmest SST of 304.4 K occurs right on the equator, thus locating the ITCZ here. Temperature is low outside the circulation region where dynamical transports are zero and radiative processes control the temperature.

### b. Energy and moisture budgets

At the TOA, net incoming SW radiation exceeds outgoing longwave radiation (OLR) in the tropics with the opposite in the subtropics, indicating poleward energy transport by the circulation (Fig. 4). The excess of incoming SW over OLR in the tropics equals the excess of OLR over SW in the subtropics, consistent with the need for energy balance within the circulation. Outside the circulation, net incoming SW equals OLR, which indicates radiative–convective equilibrium. Within the atmosphere, diabatic heating is dominated by latent heat released by precipitation, which has a tropical average of 104 W m^{−2}, equivalent to 3.6 mm day^{−1}, comparable to the CMAP value of 3.4 mm day^{−1} (Xie and Arkin 1996), and about 20% more than measured by TRMM. In the model, evaporation occurs over the whole range of the circulation from 44°S to 44°N, compared to 30°S–30°N for the observations. The maximum latent heating reaches 1300 W m^{−2} at the ITCZ, corresponding to a precipitation rate of 45 mm day^{−1}, which is about 4 times that of TRMM and GPCP (Fig. 1). This overestimate occurs even though the tropical average precipitation is comparable with the observed value. The overestimate results from the larger extent of the model circulation (with a greater surface area available for evaporation) as well as the concentration of model rainfall within the ITCZ (which disallows the dispersion of rainfall into the subtropics, resulting from the intrusion of midlatitude eddies). If we broaden the heating by doubling the ITCZ width (setting *L* = 5° instead of 2.5°), the tropical average precipitation remains roughly the same, while the maximum ITCZ rainfall is reduced by half and the circulation weakens by 15%, consistent with Hou and Lindzen (1992). We discuss the remaining twofold discrepancy with respect to observed ITCZ rainfall in section 5. The latent heating, plus heating from SW absorption, and sensible heating from the surface are balanced by the atmospheric LW cooling within the HC. At the surface of the HC, absorbed SW is balanced mostly by evaporation, consistent with observations. Net upward LW and sensible heat release are comparable, together compensating about one-third of the absorbed SW.

Column water vapor content has a maximum of 46 kg m^{−2} at the equator, which is nearly identical to the zonal annual mean of the ISCCP climatology (Schiffer and Rossow 1983). The water vapor content is much less outside the circulation, consistent with the low temperature, indicating the model’s water vapor feedback. The greenhouse effect of CO_{2} (whose concentration is fixed) warms the polar region where little water vapor is present. Although we ignore the possible insulating effect of sea ice at high latitudes, this does not distort our simulation of the tropics, which are unconnected to high latitudes in the absence of eddy transports.

## 4. Circulation with seasonally varying solar forcing

### a. Radiative–convective equilibrium

The model with seasonally varying solar forcing shows abrupt transitions of the ITCZ latitude, which we discuss below. To distinguish the effect of atmospheric dynamics on the seasonal migration of the ITCZ, a radiative–convective equilibrium solution is calculated by setting advective terms to zero and requiring longwave radiation emitted by each column to balance the local solar heating at the TOA. In this case, precipitation comes from local evaporation; that is, latent heating within the overlying column equals the evaporative cooling at the surface.

Figure 5 shows the annual cycle of SST at all latitudes (based on an average over the last 5 yr of the 20-yr simulation). The maximum SST is 321 K, much higher than calculated with dynamical transports, which is 303 K (discussed below). While the sun moves between 23.5°N and 23.5°S (corresponding to *y* = ±0.40), crossing the equator twice within a year, the ITCZ hardly deviates from the equator, staying within 1°. This implies that the inertia of the atmosphere–ocean system is large and that local radiative–convective processes by themselves cannot make the ocean temperature adjust to the solar forcing quickly enough for the warmest water to follow the sun.

### b. Effect of atmospheric dynamics

*υ*and

*θ*a small amount of diffusion, written aswhere

_{T}*ν*= 10 and

_{υ}*ν*= 0.5 m

_{θ}^{2}s

^{−1}. We also apply a five-point running mean on

*υ*. Without diffusion, the model gives a similar result, except that

*υ*and

*θ*have very small grid-scale wiggles at any single time. The diffusion does not affect our conclusions. The composite of the seasonal cycle from the last 5 yr is presented here.

_{T}#### 1) Angular momentum and wind

The annual cycle of the upper-level angular momentum, meridional wind, potential temperature, and SST at all latitudes is shown in Fig. 6. Angular momentum is approximately conserved within the circulation throughout the year (Fig. 6a), as was shown by Fang and Tung (1999) in their time-dependent nonlinear axially symmetric HC study, where diabatic heating is parameterized by Newtonian relaxation toward a seasonally varying radiative equilibrium temperature. The angular momentum has small variations within the circulation, where air rising on the poleward side of the ITCZ has lower angular momentum than air rising on the equatorward side. The meridional wind consists of two overturning circulations, a winter and summer circulation (Fig. 6b). The winter cell is associated with strong cross-equatorial flow toward the winter hemisphere with its maximum velocity of about 8 m s^{−1} occurring in the summer hemisphere close to the equator. The summer circulation is much weaker, its maximum meridional velocity being only one-third or less of the maximum velocity of the winter circulation and its spatial domain less than half that of the winter circulation. That the winter circulation is stronger than the summer circulation is in agreement with studies by Lindzen and Hou (1988) and Fang and Tung (1999).

#### 2) Abrupt transitions in the model

The meridional circulation does not adjust smoothly to the seasonal migration of the sun across the equator, but exhibits an abrupt switch around day 140 and a half year later at day 320; the ascending branch of the circulation jumps from one hemisphere to the other within a few days. After the switch, the atmosphere experiences a noisy adjustment for a few weeks and eventually becomes stable. The reason for the wavy behavior is not known, but most likely results from both dynamics and numerics. Since the transition between the winter and summer circulations is abrupt, the observed equatorially symmetric “equinoctial” circulation observed for at least a short time twice each year (Lindzen and Hou 1988; Dima and Wallace 2003) is never obtained.

The temperature field, including the potential temperature at the upper branch and the SST, varies little within the circulation, consistent with the small tropical temperature gradient in the present tropical climate (Peixoto and Oort 1992). The abrupt transition in the meridional wind field corresponds to small changes in ocean temperature (Fig. 6c). The warmest water, where we define the ITCZ and center of the latent heating, stays near 10° latitude for half a year and then jumps to a nearly identical latitude in the other hemisphere around days 140 and 320 (Fig. 7), despite the smooth migration of the sun across the equator. Once the convection and the related ascending branch of the circulation migrates, the meridional circulation reverses, as is shown here. There is slight movement of the ITCZ toward the pole after the jump, but the ITCZ stays around 10° latitude most of the time. The ascending region is about 12° latitude wide centered at the ITCZ, consistent with the imposed width of the precipitation [Eq. (16)].

Why does the ITCZ change hemispheres abruptly despite the gradual migration of the solar forcing? The potential temperature exhibits an equatorial minimum throughout the year (Figs. 6d and 8); that is, each hemisphere at the level of the tropical free troposphere is comparably warm, despite the more intense solar forcing in the summer hemisphere. This is a result of approximate conservation of angular momentum within the circulation (Schneider 1977; Held and Hou 1980; Lindzen and Hou 1988). Under the assumption of uniform angular momentum and thermal wind balance, peak heating off the equator leads to a minimum of atmospheric temperature over the equator, with the temperature in the opposite hemisphere symmetric about the equator (Lindzen and Hou 1988). Since the SAT is closely related to the upper-branch temperature by a moist adiabat and SST is related to SAT by surface fluxes, SST exhibits a local maximum in both hemispheres despite the concentration of solar heating on only one side of the equator. Because convection favors the warmest SST, the minimum in equatorial ocean temperature allows the ITCZ to bypass this latitude and move rapidly toward the relative maximum in the new summer hemisphere. But this raises the question: At the equinox, when the sun is on the equator, why is the warmest water not at this latitude? Note that when the ITCZ is centered on the equator, atmospheric temperature exhibits a single maximum at this latitude that would reinforce the warm SST (Fig. 3). We suggest that thermal inertia of the ocean prevents the equator from warming rapidly enough. We explore this issue in the following subsection.

#### 3) The role of ocean thermal inertia

Because of the possible role of ocean thermal inertia in maintaining the equatorial minimum of SST even around the equinoxes, we calculate whether the abrupt transition depends on the mixed layer depth. Annual variation of the latitudinal position of the ITCZ and the maximum latitude of the ITCZ as a function of mixed layer depth are depicted in Figs. 9 and 10. As the mixed layer depth increases, the ITCZ remains closer to the equator and its jump to the summer hemisphere is increasingly delayed. When the mixed layer depth is greater than 110 m, the ITCZ migrates gradually across, which means that large enough ocean inertia smoothes out the abrupt transition. Note that, though the ITCZ stays close to the equator for deeper mixed layers, it deviates from the equator more than in radiative–convective equilibrium, calculated with a relatively shallow 50-m mixed layer (Fig. 5). This indicates the important effect of atmospheric dynamics in transporting heat meridionally.

*T*−

_{a}*T*, where

_{o}*T*and

_{a}*T*are SAT and SST, respectively. Then the equation for SST iswhere

_{o}*k*is an empirical constant, chosen so that

*k*(

*T*−

_{a}*T*) is comparable to the turbulent plus longwave fluxes in the full model, and

_{o}*F*is the solar forcing at the surface.

*θ*

_{0}is a reference temperature,

*y*is sine of the latitude of the ITCZ (Lindzen and Hou 1988), defined to coincide with the warmest water, and

_{I}*θ*(

_{T}*y*) is the potential temperature of the upper branch of the HC at the ITCZ. Since

_{I}*θ*(

_{T}*y*) varies little in a year (Fig. 6c), it is set constant. Given

_{I}*y*,

_{I}*θ*(

_{T}*y*) can be gained. With the assumption of fixed lapse rate in the troposphere,

*T*can then be obtained.

_{a}With *T _{a}* derived from Eq. (24), the ocean temperature

*T*, calculated according to (23), results from a competition between two processes. Solar forcing favors an SST maximum in the summer hemisphere, while the air–sea flux restores SST toward the hemispherically symmetric atmospheric temperature distribution corresponding to uniform angular momentum. The atmospheric temperature has an off-equatorial maximum in both hemispheres except when the ITCZ is directly on the equator. Solving Eq. (23) with the annually varying forcing

_{o}*F*gives the annual cycle of SST and the ITCZ (Fig. 11). For a shallow mixed layer, in this case 1 m, the warmest water and ITCZ follow the sun with several days lag, moving smoothly from one hemisphere to the other. The small inertia allows the equatorial ocean to warm rapidly, which is reinforced by the atmospheric temperature, before the sun passes into the new summer hemisphere. For mixed layers deeper than 10 m (but more shallow than 300 m), the ITCZ has abrupt transitions between the hemispheres. For this case, the sun moves away from the equator before the local temperature minimum can be erased. For deep mixed layers (greater than 300 m), the ITCZ crosses the equator smoothly again and stays close to the equator. This implies that thermal inertia of the ocean affects the abruptness of the seasonal transitions of the ITCZ. Large enough thermal inertia smoothes out abrupt transitions, while small inertia allows atmospheric dynamics to reinforce equinoctial solar heating and warm the equatorial ocean so that ocean temperature follows the external heat source, the sun. Note that both the kinematic and full models exhibit the observed abrupt migration of the ITCZ into the summer hemisphere for realistic values of the mixed layer depth around 50 m (Figs. 9 and 11). However, for reasons that we do not currently understand, we never observe a gradual migration of the ITCZ across the equator with the full model despite a shallow mixed layer. For a 1-m mixed layer depth, the ITCZ in the full model oscillates rapidly between hemispheres at 30° latitude for a few weeks before finally settling in the summer hemisphere.

#### 4) The role of angular momentum

As discussed above, introducing atmospheric dynamics results in the ITCZ moving roughly 10° off the equator. In contrast, the ITCZ stays on the equator if only radiative–convective processes are dominant (Fig. 5). Also, the introduction of atmospheric dynamics makes the ITCZ change hemispheres abruptly. We speculated that the conservation of angular momentum, which is realized by nonlinear advection by the meridional circulation, makes it possible to develop a local maximum of SST simultaneously in each hemisphere so that the ITCZ jumps between those two maxima as the solar forcing moves across the equator. To test this explanation, we show that abrupt migration of the ITCZ is absent in a linearized model that does not conserve angular momentum.

*and*M

*as its angular momentum and meridional velocity, that is,In radiative–convective equilibrium*V

*= 0. To make a simpler expression of the angular momentum equation, we setSubstituting these into Eqs. (4) and (11) and adding a small amount of diffusion in Eq. (11), as we did in section 4b, the*V

*M*and

*υ*equations of the linear model can be written asIn the absence of friction in (27),

*u*′ becomes unrealistically large because the tendency term, representing annual variations of the zonal wind, must balance the meridional advection (which includes the Coriolis term). We add Rayleigh friction, which is generally used in other linear tropical circulation models (Gill 1980; Lindzen and Nigam 1987), and set

*ϵ*= (20 days)

^{−1}as the drag coefficient. For simplicity, we omit from Eq. (27) the effect of mass exchange between the lower and upper layers within the ascending region. This has a similar effect on

*M*′ as the

*ϵ*term, but cannot balance meridional advection in the descending regions. The other equations are the same as in the nonlinear model.

In contrast to its seasonal excursions within the nonlinear model, the ITCZ of the linear model deviates little from the equator throughout the year (Fig. 12). This shows that, without the nonlinear meridional transport of angular momentum, the ITCZ is restricted to the equator and does not change latitude abruptly.

The inability of linear dynamics to move the ITCZ significantly off the equator is the result of its inefficiency at transporting heat from the tropics to the subtropics compared to its nonlinear counterpart, so temperature falls off rapidly away from the equator in the linear simulation. Although the tropical temperature gradient in the linear model is smaller than that in the radiative–convective equilibrium as a result of meridional heat transport, it is much larger than the gradient in the nonlinear simulation. Even for linear simulations with shallow mixed layers (e.g., 5-m and 1-m depth), the ITCZ stays at the equator. We repeat the linear calculation, replacing the model radiative–convective temperature gradient with the flatter profile from Held and Hou (1980). For a mixed layer of 50-m depth, the ITCZ stays at the equator, while for 1 m the ITCZ migrates between roughly 4° latitude while crossing the equator smoothly.

For realistic mixed layer depths, migration of the ITCZ off the equator only in the fully nonlinear case suggests the importance of the small meridional temperature gradient associated with near-uniform angular momentum. This uniformity creates an annual mean state for which modest increases in SST can create a temperature maximum off the equator (Fig. 6c). In contrast, temperature falls off rapidly away from the equator in the radiative–convective equilibrium, which is the unperturbed state for the linear calculation. The linear circulation never overcomes the large meridional temperature gradient and is unable to move the warmest water off the equator, except when it starts with a small temperature gradient and small thermal inertia. Even in this case, it moves the warmest SST and ITCZ only gradually off the equator.

## 5. Sensitivity to model assumptions

### a. Prescribed width of the ITCZ

Compared to observations, model precipitation at the center of the ITCZ is about 2–3 times too strong. There are several reasons for the strong precipitation. First, in the real world, there is net moisture transport out of the tropics by atmospheric eddies. Over the whole tropics, evaporation estimated by NCEP exceeds CMAP and TRMM precipitation by about 13%–25%. Within our model’s tropical circulation, moisture balance requires that the total evaporation equal the total precipitation. So, although the model’s evaporation rate is about the same as the observation [partly because we tune tropical mean evaporation through the parameter *β* in Eq. (21)], the precipitation should be more than observed. Second, the area coverage of the model’s tropical circulation is 40% wider than observed. Thus, the model’s total tropical evaporation (and precipitation) is larger than observed, even though its areal average is in good agreement. Third, the model concentrates precipitation within the ITCZ and there is no precipitation outside this zone within the circulation, whereas, in the real world, ascent associated with midlatitude cyclones reaches into the subtropics, which precipitates out moisture before it can be supplied to the ITCZ (Trenberth and Stepaniak 2003). We want to investigate the HC without eddies, so all evaporated moisture is assumed to precipitate within the ITCZ. This also neglects the small contribution to tropical precipitation by drizzle within subtropical low clouds.

Hou and Lindzen (1992) show that concentrated diabatic heating strengthens the meridional circulation. Conversely, the circulation is weaker for dispersed heating, which might reduce the ability of the circulation to maintain an angular momentum and temperature distribution that are symmetric about the equator despite asymmetric radiative forcing. To see if the abrupt seasonal transition is sensitive to this, we double the width of the ITCZ, setting *L* = 5°. This results in an ITCZ width of roughly 20° latitude and is probably the largest value of *L* consistent with the daily distribution of zonally averaged rainfall according to TRMM and GPCP (Fig. 1). The ITCZ maximum precipitation drops by about half and the maximum meridional wind speed decreases, as expected, from 8 to 6 m s^{−1}. The annual cycle of the ITCZ is shown in Fig. 13. The transition is slightly less abrupt (compared to Fig. 9) for the same mixed layer depth. The ITCZ jumps across the equator into the summer hemisphere and then slowly moves toward the pole. The nearly instantaneous transition of the ITCZ disappears for mixed layers deeper than roughly 70 m, but the ITCZ still spends less time at the equator compared to the sun. The frequency distribution of the latitude of the sun and the ITCZ better demonstrates this (Fig. 14). The solar latitude has a sinusoidal dependence on time, thus having a nearly even distribution of frequency with latitude (Fig. 14a), spending equal amounts of time around the equator and other latitudes, except the peak latitude. Although driven by the sun, the ITCZ has two sharp peaks at its maximum latitude for mixed layer depths less than roughly 80 m (Figs. 14b,c), indicating that it spends more time at these two latitudes than at any other latitude, and a minimum amount of time at the equator, indicating a rapid transition across this latitude. For a 200-m mixed layer, the ITCZ crosses the equator gradually, spending more time around this latitude (Fig. 14d). In summary, doubling the width of the precipitation zone smoothes the seasonal transition of the ITCZ for deep mixed layers, but, for realistic tropical mixed layer depths ranging from 20 to 90 m (Russell et al. 1985), the abrupt transition across the equator persists. We also note that the histogram of ITCZ latitude corresponding to a 50-m mixed layer better resembles the observed histogram (Hu et al. 2007) than the frequency distribution of the solar latitude (Fig. 14a).

We also do an extreme experiment by setting precipitation equal to the local evaporation everywhere so that heating is not concentrated within an ITCZ. For a mixed layer depth of 50 m, the warmest water varies between about 10°N and 10°S, crossing the equator rapidly within a few days, although not instantaneously. The frequency distribution of the latitude of the warmest water is similar to Fig. 14c, indicating a rapid transition across the equator.

The observed atmospheric circulation transports water vapor from the subtropics to the deep tropics so that precipitation is not ubiquitous throughout the circulation. Though the model does not simulate this dynamical transport, the assumed concentration of precipitation within the ITCZ mimics its effect.

### b. Wind–evaporation feedbacks

Observed surface winds are stronger in the subtropics than in the tropics, with wind speeds of 6–8 m s^{−1} in the subtropics and 4–6 m s^{−1} in the tropics (Peixoto and Oort 1992). In the model we assume a constant surface wind velocity of 8 m s^{−1} over the globe in the calculation of the latent and sensible heat fluxes at the surface [Eqs. (20) and (21)]. This spatially uniform wind is a concern, because larger surface winds in the subtropics would make SSTs cooler than in the present model. The wind–evaporation feedback could weaken or destroy the equatorial minimum in SST by cooling the subtropics and thus inhibit jumping of the ITCZ in the model.

There is no rigorous way to include wind–evaporation feedbacks in a zonally averaged model because wind speed is a quadratic quantity and the speed of the zonal average wind (which is calculated) is not necessarily the zonal average of wind speed (which is central to the zonally averaged evaporation). In addition, the bulk aerodynamic coefficient *C _{D}* in Eqs. (20) and (21) is smaller over cold nonconvecting regions, where the boundary layer is relatively stable. This would partly compensate the effect of the stronger observed surface wind on evaporation over the subtropical ocean, although it is difficult to calculate in a simple way. Nonetheless, we attempt to incorporate a wind–evaporative feedback by augmenting the 8 m s

^{−1}base wind (defined as the local turbulent wind speed in the absence of any large-scale circulation) by the surface wind calculated in the model. The zonal component is computed, following Held and Hou (1980), such that the zonal wind stress at the surface balances the export of angular momentum aloft. With this representation of the wind–evaporative feedback, we still see an abrupt seasonal migration of the ITCZ into the summer hemisphere as long as the base wind is high enough (>4 m s

^{−1}). For a smaller surface base wind, the ITCZ stays around the equator, the effect similar to decreasing the drag coefficient or deepening the mixed layer.

## 6. Conclusions

We have shown that the major features of the HC can be simulated with our axisymmetric circulation model. Compared to previous axisymmetric models (Held and Hou 1980; Lindzen and Hou 1988; Fang and Tung 1999), our model has an ocean mixed layer and a more physical representation of diabatic heating where only solar heating at TOA is externally prescribed.

Compared to a radiative–convective model, the addition of atmospheric dynamics allows the ITCZ to migrate significantly off the equator. This is because the small meridional temperature gradient associated with nearly uniform angular momentum allows even small temperature perturbations to become the warmest location in the tropics, favoring convection and the arrival of the ITCZ. Twice each year, the model exhibits an abrupt seasonal transition in the diabatic heating and tropical wind field, and there is an abrupt migration of the ITCZ into the new summer hemisphere. This migration resembles the abrupt transition of the observed zonal average precipitation in the tropics. Both TRMM and GPCP climatological zonal mean precipitation show that the ITCZ experiences two jumps from one hemisphere to the other each year. By calculating diabatic heating as a function of seasonally varying incident SW at the TOA, we find that diabatic heating itself exhibits abrupt transitions. The occurrence of this abrupt transition of the model does not depend on temporary maxima in TOA SW forcing around the solstices, which contributes to off-equatorial heating in the real world (Fig. 2a). Although our model strictly applies to the zonally averaged circulation, the abrupt cross-equatorial migration of tropical rainfall may contribute to the observed abrupt onset of longitudinally confined circulations like the Asian monsoon (Krishnamurthy and Shukla 2007).

There are two factors contributing to the abrupt seasonal transition of the model ITCZ: nonlinear dynamics and ocean thermal inertia. Because of nonlinear advection, angular momentum of the free troposphere is approximately conserved within the circulation. For an ITCZ displaced from the equator, a tropical circulation with uniform angular momentum and thermal wind balance has two maxima in free-tropospheric temperature that are symmetric about the equator despite incident SW that is much larger in one hemisphere; that is, the equator is a region of minimum atmospheric temperature in our model. The boundary layer air temperature is closely related to the free-tropospheric temperature by a moist adiabat and interacts with SST through air–sea surface flux exchanges. The model shows a similar equatorial minimum in SST. If convection favors the warmest region, then the ITCZ is likely to avoid the equator and move abruptly from one hemisphere to the other in response to small hemispheric asymmetries in temperature driven by seasonal variations in incident SW. Without nonlinear dynamics, the abrupt transition disappears.

The upper ocean, which provides thermal inertia, is the other factor having an important effect on the seasonal transition. Large enough thermal inertia smoothes out the transition, keeping the ITCZ migration gradual and close to the equator. For the present- day range of the tropical ocean mixed layer depth (20–90 m) and realistic ITCZ width, the model exhibits an abrupt transition.

Among the model assumptions, the fixed lapse rate is hardest to justify over the subtropical oceans. As indicated by the trade inversion observed at these latitudes, variations in the upper-tropospheric temperature are not as tightly coupled to temperature variations at the surface compared to convecting regions. Whereas surface air temperature in our model responds instantaneously to temperature changes aloft, the observed adjustment time is closer to 10 days (Betts and Ridgway 1989). This is still rapid compared to the seasonal time scale of solar forcing and may only act to smooth slightly our model’s abrupt migration of the ITCZ. The prescription of constant relative humidity at the surface also contributes to the abrupt transition. Warming of the surface air as a result of a rising temperature aloft would reduce the surface relative humidity. Restoration of the surface humidity by evaporation would delay the warming of the mixed layer and arrival of the ITCZ on this same boundary layer adjustment time scale. A preliminary calculation with an explicit budget for the boundary layer moisture and relative humidity shows exactly this modest smoothing of the hemispheric transition. We are also in the process of incorporating a prognostic budget of boundary layer temperature to eliminate the need for an imposed lapse rate.

Our collocation of the ITCZ over the warmest ocean is obviously a simplistic description of convection. As noted in section 2c(2), we performed a sensitivity test in which the ITCZ was centered over the latitude of minimum column stability. This allows the relatively cold atmospheric temperature over the equator to decrease atmospheric stability and favor rainfall at this latitude, but the ITCZ persisted off the equator owing to the dominant effect of SST on column stability. Ultimately, it would be better to relate precipitation to local column stability according to a convective parameterization as in Betts and Miller (1986). This has the additional advantage of allowing the ITCZ width to be calculated rather than prescribed. While explicit boundary layer budgets may result in a more realistic model, they complicate the model’s interpretation. Parameterization of the ITCZ location in terms of SST is conceptually convenient because interpreting the migration of the ITCZ is reduced to understanding the interplay of atmospheric dynamics and solar heating and how they combine to determine the region of warmest SST.

The results presented here suggest that the abrupt transition of the ITCZ can be modeled with a simple axisymmetric atmospheric circulation coupled to a slab ocean model. However, we have not included the potentially important effects of atmospheric eddies and ocean dynamics. Atmospheric eddies break the conservation of angular momentum within the circulation. They exchange heat and moisture between the tropics and midlatitude by mixing potential vorticity between the adjacent regions. They also bring cold midlatitude air to the subtropics in the winter hemisphere, preventing the winter hemisphere from having a temperature comparable with that of the summer hemisphere. This may inhibit the formation of an equatorial temperature minimum. Nonetheless, Schneider and Bordoni (2008) show that upper-level easterlies shield the deep tropics from eddy forcing as the ascending region migrates into the summer hemisphere; they attribute the abrupt appearance of strong cross-equatorial flow originating in the new summer hemisphere to the dynamics of the axisymmetric circulation. Ocean dynamics may also contribute to the abrupt seasonal transition of the ITCZ. Equatorial upwelling resulting from easterly trade winds keeps equatorial SST at a minimum compared to poleward latitudes, forming a “cold tongue,” which favors a jump of the ITCZ across the equator. How the effects of the axisymmetric circulation, ocean dynamics, and eddies combine to result in the observed rapid migration of the ITCZ to the summer hemisphere remains to be studied.

We thank W. Boos, I. Dima, K. A. Emanuel, A. Del Genio, I. Held, Y. Hu, A. Lacis, R. A. Plumb, R. Pierrehumbert, A. Sobel, M. Spiegelman, and K. K. Tung for many useful discussions on this topic. We are also grateful to J. Chiang and an anonymous reviewer, who gave insightful comments on this article. This project was supported by the Climate Dynamics Program of the National Science Foundation through ATM-06-20066.

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