Wind–Evaporation Feedback and Abrupt Seasonal Transitions ... · 1999; Marshall et al. 2004), but...

Wind–Evaporation Feedback and Abrupt Seasonal Transitions of Weak,Axisymmetric Hadley Circulations

WILLIAM R. BOOS AND KERRY A. EMANUEL

Program in Atmospheres, Oceans and Climate, Massachusetts Institute of Technology, Cambridge, Massachusetts

(Manuscript received 31 August 2007, in final form 7 December 2007)

ABSTRACT

For an imposed thermal forcing localized off the equator, it is known that conservation of absoluteangular momentum in axisymmetric flow produces a nonlinear response once the forcing exceeds a criticalamplitude. It is shown here that, for a moist atmosphere in convective quasi-equilibrium, the combinationof wind-dependent ocean surface enthalpy fluxes and zonal momentum advection can provide a separatefeedback that causes the meridional flow to evolve nonlinearly as a function of a sea surface temperature(SST) forcing, even if an angular momentum–conserving response is not achieved. This wind–evaporationfeedback is examined in both an axisymmetric primitive equation model and a simple model that retainsonly a barotropic and single baroclinic mode. Only SST forcings that do not produce an angular momen-tum–conserving response are examined here. The wind–evaporation feedback is found to be inhibited inmodels with linear dynamics because the barotropic component of the Hadley circulation, which is coupledto the baroclinic circulation via surface drag, keeps surface winds small compared to upper-level winds. Inmodels with nonlinear dynamics, the convergence of zonal momentum into the ascending branch of thecross-equatorial Hadley cell can create barotropic westerlies that constructively add to the baroclinic windat the surface, thereby eliminating the inhibition of the wind–evaporation feedback. The possible relevanceof these results to the onset of monsoons is discussed.

1. Introduction

Steady-state, axisymmetric solutions for the circula-tion in a differentially heated fluid on a rotating spherehave been advanced over the past few decades as ameans of understanding the Hadley circulation (e.g.,Schneider 1977; Held and Hou 1980). The associatedtheory emphasizes conservation of absolute angularmomentum in the free troposphere, which results in anonlinear dependence of circulation strength and ex-tent on the imposed thermal forcing.

Results of this theory have been used to explain cer-tain aspects of the seasonal cycle of the Hadley circu-lation. Lindzen and Hou (1988) showed that the steady,axisymmetric response to a heating with a peak dis-placed just a few degrees off the equator exhibits astrong asymmetry in the strength and meridional extent

of the summer and winter Hadley cells, consistent withobservations. Plumb and Hou (1992, hereafter PH92)showed that the strength of steady, axisymmetric me-ridional flow increases nonlinearly as the magnitude ofa thermal forcing localized off the equator is enhancedbeyond a threshold value. They suggested that thisthreshold behavior might be relevant to the seasonalonset of monsoons, but noted that the roles of timedependence and zonal asymmetries need to be as-sessed.

The effect of time dependence on axisymmetrictheory has only begun to be explored. Fang and Tung(1999) examined a version of the axisymmetric drymodel used by Lindzen and Hou (1988), but with atime-dependent forcing. They found that there was noabrupt change in the circulation as the maximum of thethermal forcing was moved meridionally in a seasonalcycle, primarily because the circulation did not achieveequilibrium with its forcing during the course of thisseasonal cycle. The dry axisymmetric models used byFang and Tung (1999) and PH92 equilibrate on timescales on the order of 100 days, which is longer than thetime scale associated with the seasonal cycle of earth’s

Corresponding author address: William R. Boos, MassachusettsInstitute of Technology, Rm 54-1721, 77 Massachusetts Ave.,Cambridge, MA 02139.E-mail: [email protected]

2194 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 65

DOI: 10.1175/2007JAS2608.1

© 2008 American Meteorological Society

JAS2608

insolation forcing.1 However, moist axisymmetric mod-els with stratifications and rates of radiative coolingsimilar to those of the earth’s tropics will equilibratemuch faster, an issue we discuss in an appendix andillustrate in the body of this paper. Therefore, it is notreasonable to apply a time-varying forcing having a365-day period to a model with an equilibration timescale on the order of 100 days and expect the results tobe relevant to the seasonal cycle of earth’s Hadley cir-culation.

The fact that moist axisymmetric models equilibrateon a time scale shorter than that of the earth’s seasonalcycle is illustrated by the results of Zheng (1998), whoextended the study of PH92 using a moist axisymmetricmodel forced by an off-equatorial sea surface tempera-ture (SST) anomaly. Zheng (1998) confirmed that thecirculation entered an angular momentum–conserving(AMC) regime once the magnitude of the SST anomalyexceeded a certain threshold, and found that an abrupt,nonlinear onset of the summer circulation occurredwhen the SST forcing was varied in a seasonal cycle. Itis of interest that Zheng (1998) did not use wind-dependent surface enthalpy fluxes in his model runs,because Numaguti (1995) found that wind-dependentocean evaporation was needed to obtain an abruptpoleward shift in the peak precipitation as a prescribedSST maximum was gradually shifted poleward in athree-dimensional general circulation model (GCM).Based on the diagnostics presented by Numaguti(1995), it is difficult to judge whether the meridionalflow in his model increased nonlinearly as a function ofthe forcing. The forcing used by Numaguti (1995) wasalso not localized off the equator, and so is a betteranalog of that used by Lindzen and Hou (1988) thanthat used by PH92.

Although this paper only explores axisymmetric dy-namics, zonally asymmetric eddies may play an impor-tant role in Hadley circulation dynamics. For thermalforcings centered on the equator in a three-dimensionalmodel with zonally symmetric boundary conditions,Walker and Schneider (2006) found that the strength ofthe Hadley circulation was directly related to the eddymomentum flux divergence, and that scalings based onthe assumptions of axisymmetric angular momentumconservation did not hold. However, when the peakthermal forcing was displaced sufficiently far from theequator, their model produced cross-equatorial flowthat nearly conserved absolute angular momentum, atleast in its free-tropospheric ascending branch.

Schneider and Bordoni (2008) found that abrupt tran-sitions between the eddy-controlled equinoctial regimeand the AMC solsticial regime occurred in a model witha seasonally varying forcing, and discussed the similar-ity between the dynamics of this transition and the on-set and end of monsoons. While eddy transports maythus play an important role in seasonal Hadley dynam-ics, we shall demonstrate that a wind–evaporation feed-back may also play an important role, and that thisfeedback is sufficiently complex to merit initial exami-nation in idealized two-dimensional models.

The desire to explain the seasonal evolution of mon-soons motivates this and many previous works onabrupt seasonal transitions. Several regional summermonsoons do begin abruptly, by which we mean theyevolve faster than can be explained by a linear responseto their forcing. The start of precipitation and the re-versal of zonal wind which mark the beginning of theSouth Asian and Australian summer monsoons occuron time scales shorter than those contained in the in-solation forcing (Murakami et al. 1986; Wheeler andMcBride 2005; Webster et al. 1998). Mapes et al. (2005)found that even when the period of the solar and SSTforcings was increased by a factor of 5 in an atmo-spheric GCM, the onset of summer precipitation overIndia still seemed to occur over the same one- to two-week time scale as in a control run.

In this paper, we use results from axisymmetric mod-els to show how wind–evaporation feedback might pro-duce such an abrupt seasonal transition. Although itmay be difficult to say how an idealized model of theglobal Hadley circulation relates to the various regionalmonsoons, the fact that more than half of the globallow-level cross-equatorial mass flux occurs in the So-mali jet (e.g., Findlater 1969) would seem to ensure thatthe South Asian monsoon, at least, projects stronglyonto the zonal mean Hadley circulation during borealsummer. Wind speeds in the off-equatorial, southwest-erly part of the Somali jet do intensify abruptly near thestart of the Indian monsoon (Krishnamurti et al. 1981;Halpern and Woiceshyn 1999). Nevertheless, furtherwork is needed to reconcile this fact with the findingthat a scalar index of the climatological, monthly-meanHadley circulation projects almost entirely onto a sinu-soid with a period of 365 days (Dima and Wallace2003). Prive and Plumb (2007) examined the applica-bility of nonlinear axisymmetric theories for the Hadleycirculation to steady-state monsoons, and found thatzonal asymmetries in the land surface could cause pro-found changes in the circulation that could not be de-scribed by axisymmetric frameworks. A similar study ofthe process of monsoon onset has yet to be performed,and we see the present paper as a first step in a process

1 A sinusoid with a period of 365 days varies with a time scale� � 58 days, if � is interpreted as the e-folding time.

JULY 2008 B O O S A N D E M A N U E L 2195

which might later employ more complex spatial do-mains representative of real-world monsoons.

The main goal of this paper is to examine the effectof wind-induced surface heat exchange (WISHE) onthe seasonal cycle of meridional flow in the context ofprevious studies of nonlinear axisymmetric theory. Inparticular, we use axisymmetric (non-eddy-resolving)models forced by thermal maxima localized off theequator, as in PH92. One of our main results will bethat WISHE can produce an abrupt, nonlinear intensi-fication of meridional flow as a function of such a forc-ing, by a mechanism distinct from that of a transition toan AMC regime. In examining this effect of WISHE,we will limit our focus to weak SST forcings that do notproduce an AMC circulation. The effect of WISHE forstronger forcings that do produce AMC flow is nottrivial and will be explored in a separate paper.

The next section of this paper begins by presentingresults from a dry primitive equation model to empha-size that meridional flow generated by a near-linearviscous response can be just as strong as that occurringin a nonlinear AMC regime; any feedback between thethermal forcing and the circulation could thus producea rapid change in the circulation without involving non-linear dynamics. A moist primitive equation model isthen used to show that a wind–evaporation feedbackcan produce a circulation that depends nonlinearly onthe SST forcing, but that this feedback requires thenonlinear advection of zonal momentum even thoughthe free-tropospheric circulation is not in an AMC re-gime. An idealized two-mode model of troposphericflow is introduced to explain this interaction betweenwind–evaporation feedback and zonal momentum ad-vection. We conclude by discussing the relevance ofthese results to monsoons and some possible effects ofprocesses omitted from these idealized models. An ap-pendix presents a scale estimate for the equilibrationtime of both dry and moist axisymmetric models.

2. Primitive equation model

The time-dependent behavior of Hadley circulationsis examined in both dry and moist versions of the axi-symmetric (latitude–height) GCM used by Pauluis andEmanuel (2004) and Pauluis (2004). The model dynam-ics are based on the Massachusetts Institute of Tech-nology (MIT) GCM (Marshall et al. 1997; Adcroft et al.1999; Marshall et al. 2004), but a series of alternateparameterizations is used for subgrid-scale physics. Themodel domain is a partial sphere extending betweenrigid walls at 64°N and 64°S, with 1° meridional reso-lution on a staggered spherical polar grid. There are 40pressure levels with 25-hPa resolution, from 1000 hPa

to a rigid lid at 0 hPa, and no representation of orog-raphy. Viscosity is represented by the vertical diffusionof momentum with a coefficient of 100 Pa2 s�1. Fric-tional stresses in the planetary boundary layer are rep-resented by vertically homogenizing horizontal veloci-ties in the lowest 200 hPa of the atmosphere over a timescale of 20 min, and a bulk flux formula for momentumis used to represent surface drag. An eighth-order Sha-piro filter is used to reduce small-scale horizontal noisein the temperature, specific humidity, and horizontalwind fields.

Instead of diagnosing the spatial extremum of theoverturning streamfunction as a metric for circulationintensity, we use the meridional flow integrated bothvertically through the boundary layer (800–1000 hPa)and meridionally over all model latitudes, a quantityhereafter called the PBL flow. The PBL flow has theadvantage of being a single scalar that is continuouslyrelevant during the transition from summer to winter,whereas one must alternately choose the maximum andminimum streamfunction to represent changes in theseasonal strength of the Hadley circulation. This be-comes relevant when the thermal forcing is prescribedto vary in a seasonal cycle. Because the PBL flow in-volves a meridional integral, it does not measure theequatorially antisymmetric component of the stream-function. We chose a dynamical boundary layer depthof 200 hPa in order to constrain most of the meridionalmass flux in the lower branch of the Hadley circulationto move through this boundary layer, as Pauluis (2004)showed that shallower PBL depths in an axisymmetricmodel resulted in a larger fraction of the low-level flowcrossing the equator in the free troposphere.

a. Dry model

1) DRY MODEL CONFIGURATION

In a dry version of the axisymmetric MIT GCM, thecirculation is forced by relaxation of temperature overa spatially uniform time scale of 10 days to a prescribeddistribution Teq similar to that used by PH92:

Teq � T0 � Tmax

�

2sin��

p0 � p

p0� cos2��

2� � �0

�� .

�1�

This is used to specify Teq only between �0 � �� and�0 � �� ; outside this range Teq is set to T0. Here p0 ispressure at the lowest model level, �0 � 25°N, and�� 15°. his gives an equilibrium temperature withno meridional gradient outside the range 10°–40°N andan extremum centered at 25°N and 500 hPa.

Most integrations of the dry model use an isothermal


background state with T0 � 200 K. As discussed in theappendix, use of an isothermal background state willproduce relatively weak meridional velocities and thusan equilibration time scale for absolute angular mo-mentum M of about 165 days, which is much longerthan the time scale of earth’s seasonal insolation forc-ing. This long time scale is purposely chosen in order toprovide a clear separation in time between the equili-brated, AMC state and an initial transient state inwhich the atmospheric dynamics are predominantly lin-ear. Furthermore, this long equilibration time scale issimilar to that used by PH92 and Fang and Tung (1999),and so allows results presented herein to be more easilycompared with their findings. A model integration isalso performed using a background state having near-neutral stratification, with zT0� 0.9g/cp, which reducesthe time scale for M advection to about 15 days. Themoist model, discussed below, also equilibrates on atime scale shorter than that of the forcing.

We examine the equilibrated response to steady forc-ings as well as the time-dependent response to a sea-sonally varying forcing. In all cases, the initial modelstate used was that of an atmosphere at rest with me-ridionally uniform temperature T0. To avoid high-amplitude initial transients, the off-equatorial anomalyof equilibrium temperature was increased from zero forthe steady forcings according to

Tmax � �m�1 � e�t��i�, �2�

with �i � 30 days. Three separate integrations withsteady forcings were performed, using �m � 5, 10, and15 K. For the seasonal forcing, Tmax in (1) was variedaccording to

Tmax � �m cos�2�t

365 days�, �3�

where t is time in days and �m � 15 K.

2) DRY MODEL RESULTS

As in PH92, the response to steady forcings tookhundreds of days to achieve a steady state, at whichtime the equilibrated PBL flow depended nonlinearlyon �m (Fig. 1). The streamfunction maximum (notshown) behaved similarly to the PBL flow, although itachieved a steady state more quickly and did not havean initial transient peak for �m� 15 K. The existence ofthe initial peak in PBL flow around day 80 for all valuesof �m more clearly divides the evolution of the flow intotwo regimes: an initial transient phase where the PBLflow scales nearly linearly with �m, and a steady statewhere the PBL flow depends nonlinearly on �m. Thesmall kink at day 400 in the PBL flow for �m � 15 K is

associated with inertial instabilities that produce rollcirculations of high vertical wavenumber (Dunkerton1989 discusses this behavior in more detail). These in-ertial rolls are symptomatic of the fact that this value of�m was near the maximum for which numerically stablesolutions could be achieved.

The model evolution can be understood in terms ofthe evolution of M. During the first 100 days of inte-gration for �m� 15 K, the M distribution is only slightlyperturbed from its initial state of solid-body rotation,although the streamfunction and PBL flow reach am-plitudes nearly as large as in equilibrium (Fig. 2, leftpanel). After equilibration, which takes more than 500days, the M field has undergone considerable homog-enization in the upper troposphere with the zero con-tour of absolute vorticity displaced to about 20°S (Fig.2, right panel). During both the initial transient peakand the final steady state, most of the lower branch ofthe meridional circulation flows through the boundarylayer, peak ascent occurs slightly equatorward of �0,and a weak summer cell exists in the midlatitude sum-mer hemisphere, all of which are consistent with theaxisymmetric results of Lindzen and Hou (1988) andFang and Tung (1999). The streamfunction maximum isfairly low in the free troposphere, suggesting that theseresults are free of any amplification of the circulationdue to the rigid-lid upper-boundary condition (Walkerand Schneider 2005).

The time evolution thus seems to be a superposition

FIG. 1. Time evolution of the PBL flow (defined as the merid-ional wind integrated vertically and meridionally in the boundarylayer) for the dry GCM with steady forcing. The three curvescorrespond to forcings with different values of �m, which sets themagnitude of the off-equatorial equilibrium temperatureanomaly.


of a relatively rapid and linear thermally dominatedadjustment followed by a slow, nonlinear adjustmentdominated by M advection. Although this in many waysrestates some results of PH92, we include this discus-sion to emphasize a particular point: changes in thethermal forcing can produce a linear response in thestrength of the circulation within a few inertial periods(the initial transient peak occurs after 80 days becauseof the spinup used for the forcing). If the thermal forc-ing is not externally specified but depends on the cir-culation itself, feedbacks could occur because of inter-actions between the circulation and the forcing. Latersections of this paper will examine the particular case ofa wind–evaporation feedback, but coupling betweenthe circulation and the thermal forcing could also occurvia radiation or other processes. Depending on the na-ture of these interactions, such feedbacks could operatein a regime where the atmospheric dynamics are pre-dominantly linear and M advection is not important.

For the seasonally varying forcing, the circulationtook less than 100 days to achieve a regular periodiccycle (not shown). After this initial adjustment time,the streamfunction and PBL flow peaked each yearshortly before Teq reached its maximum. We will referto this time of peak PBL flow as the summer solstice,using the convention of boreal summer for Tmax � 0.The streamfunction and M distribution at this summersolstice, shown in Fig. 3, closely resemble those seenduring the initial transient peak of the runs with steadyforcing, without the folding over of M surfaces seen inthe equilibrated response to the steady forcing. Con-

tours of M tilt very slightly against the flow in the uppertroposphere of the winter hemisphere, suggestive oftheir deformation by flow during the previous solstice.Also, the meridional width of the M peak near theequator is slightly larger for the seasonally varying forc-ing than for the steady forcing, suggesting that somehomogenization is accomplished by the seasonally re-versing meridional winds.

The PBL flow exhibits no abrupt transitions overtime, although a slight nonlinearity is apparent betweensummer and winter seasons when the PBL flow is plot-

FIG. 3. As in Fig. 2, but for the dry GCM with seasonallyvarying forcing, at the time of largest PBL flow.

FIG. 2. Absolute angular momentum (shading) and meridional streamfunction (thin contours) for the dry GCMwith steady forcing and �m � 15 K (left) at the initial transient peak and (right) after the model achieved a steadystate. Thick solid line is the zero absolute vorticity contour. Streamfunction contour interval is 1 1010 kg s�1,starting at 0.5 1010 kg s�1, with negative contours (denoting clockwise rotation) dashed. Angular momentumcontour interval is 0.2 109 m2 s�1.


ted against the quantity Tmax�/2, which is the equilib-rium temperature anomaly at 25°N (Fig. 4). This phase-space trajectory would be a perfect ellipse if the PBLflow varied linearly with Tmax with only a phase lagbetween the two quantities. The equilibrated PBL flowfor the steady forcings, also plotted on this phase dia-gram, increases nonlinearly with Tmax.

The PBL flow in the seasonally forced run leadsTmax, as shown by the clockwise phase-space trajectory.It is straightforward to show that this phase relationshipis expected when the thermal relaxation time is shortcompared to the forcing period. If Tmax� sin(At) and Tis assumed to lag Tmax by �t but to have the samefrequency A, then a trigonometric identity can be usedto write the Newtonian cooling as

Q �Tmax � T

��

2�

cos�At �A�t

2 � sinA�t

2. �4�

This shows that Q will lead Tmax by (�/2 � ��t/2),which is slightly less than one-quarter cycle if �t is muchsmaller than the forcing period. Linear theory for a seabreeze (Rotunno 1983), although formulated on an fplane instead of a sphere, predicts that the circulationwill be in phase with Q so long as f is larger than thefrequency of the thermal forcing, which is true for aforcing period of 365 days everywhere except within atenth of a degree of the equator. Since M advectionseems to be of minor importance for the evolution of

the streamfunction for the seasonally varying forcing,the dynamics might be decently described by this lineartheory, with Q in phase with the PBL flow and bothleading Tmax by one-quarter cycle. However, the PBLflow and Tmax peak at nearly the same time, giving themajor axis of the phase-space ellipse its positive slopeand indicating that part of the circulation must be inphase with Tmax. So it would seem that the phase lagbetween Tmax and the PBL flow explains why the phasetrajectory is an ellipse rather than a straight line, butthat a significant part of the flow is also in phase withTmax. Similar phase relationships were obtained for anincreased Newtonian cooling time scale of 30 days, aswell as for a reduced viscosity of 10 Pa2 s�1. This re-duced viscosity was near the lowest value for whichnumerically stable solutions could be obtained. Integra-tions using smaller values of �0 and �m also producedcirculations that evolved nearly linearly with Tmax.

If the time scale of M advection is smaller than thatof variations in Tmax, then we expect the time evolutionof a seasonally forced model to become nonlinear. Thishypothesis is tested by using the reduced backgroundstratification of zT0 � 0.9g/cp, for which M advectionhas an estimated time scale of around 15 days (see theappendix). When this model is integrated using thesame value of �m � 15 K in (3), the phase diagramshows that the sensitivity of the PBL flow to Tmax doesincrease suddenly shortly before the summer solstice(Fig. 5). This trajectory deviates considerably from theellipse expected for a linear response, and loosely re-sembles the figure “8” expected for two time series

FIG. 4. Phase diagram of the PBL flow and the spatial extremumof the equilibrium temperature anomaly for the dry GCM. Solidline is for the run with seasonally varying forcing, with time pro-gressing in the direction of the arrow. The circles connected by thedashed line denote the equilibrated response to steady forcings.

FIG. 5. As in Fig. 4, but the phase diagram for the dry GCM witha reduced background stratification (T0/z� 0.9 �d) and a strongforcing (�m � 15 K).


where one evolves at twice the frequency of the other.At the time when the sensitivity of the PBL flow toTmax increases abruptly, the absolute vorticity near themodel tropopause approached zero throughout muchof the tropics (not shown), indicating that the abrupttransition is associated with the onset of M-conservingflow at upper levels. While a more comprehensive in-vestigation would estimate model equilibration times asa function of the ratio of zT0 to the Newtonian coolingrate, only these limiting cases are presented here forbrevity.

b. Moist model

1) MOIST MODEL CONFIGURATION

In a moist version of the model, the circulation isforced by parameterized radiative cooling together withsurface fluxes of latent and sensible heat that are ver-tically redistributed by moist convection. The lowerboundary consists entirely of ocean with a specifiedSST given by the one-dimensional analog of (1),

Ts � T0 � Tmax

�

2cos2��

2� � �0

�� , �5�

with �0� 25°� and �� 15°, as in the dry model. Thisis used to specify the SST only between �0 � �� and�0 � �� ; outside this range Ts is set to T0, with T0 �296 K. This provides an SST anomaly centered at 25°Nand confined between 10° and 40°N, with no cross-equatorial SST gradient. The anomaly is prescribed tooscillate with the same seasonal cycle used for the drymodel, given by (3).

Surface evaporation Fq is represented by a bulk for-mula:

Fq � �Ck |V | �q*�Ts� � q�, �6�

where � and q are the density and specific humidity ofair at the lowest model level, Ck is a transfer coefficientset to 0.0012, and q*(s) is the saturation specific hu-midity at surface temperature Ts. Surface fluxes of sen-sible heat and momentum take a similar form, with thesame nondimensional transfer coefficient. The effectivewind speed is

|V | ��u2 � �2 � �g2, �7�

where u and � are the horizontal winds at the lowestmodel level. The parameter �g is included to representthe effects of a range of subgrid-scale variations inwind, and in a more realistic treatment its value wouldvary with the local dynamic and thermodynamic state.Indeed, many surface flux formulas represent the effectof convective gustiness with a state-dependent velocity

that typically has a value near 1 m s�1 for dry convec-tion (Stull 1988) or up to about 5 m s�1 for precipitatingconvection (e.g., Williams 2001). For simplicity, we set�g to the constant value of 4 m s�1. This somewhat largevalue was chosen to suppress the formation of convec-tive anomalies that propagate meridionally on intrasea-sonal time scales, which occur in this GCM for values of�g less than about 3 m s�1. Bellon and Sobel (2008) haveexamined what seem to be similar poleward-propagat-ing anomalies in another axisymmetric model forced bysteady SST, and proposed that they are relevant to theobserved poleward migrations of the convective maxi-mum associated with “active” and “break” episodes ofthe South Asian summer monsoon. The interaction ofthese propagating anomalies with the seasonal cyclemay be relevant to the onset of monsoons, especiallysince the onset of the Indian monsoon is generally co-incident with the first of several poleward migrations ofa zonally elongated convective maximum during borealsummer (e.g., Yasunari 1979; Goswami 2005). Thesepropagating anomalies are purposely suppressed herein order to focus first on the arguably simpler case of aclassical, seasonally reversing Hadley circulation; we in-tend to examine the interaction of such anomalies withthe seasonal cycle in future work.

Moist convection in this GCM is represented by thescheme of Emanuel and Zivkovic-Rothman (1999). Ra-diation is calculated by the longwave scheme of Mor-crette (1991) and the shortwave scheme of Fouquartand Bonnel (1980). Insolation is independent of lati-tude and moves through a diurnal cycle, although thediurnal cycle has an effect only through atmosphericabsorption of shortwave radiation because of the use ofan entirely oceanic lower boundary with prescribedSST. Integrations are performed with no representationof cloud radiative effects (i.e., clear sky), and the spe-cific humidity used for the radiative calculations is fixedat a reference profile that does not vary with latitude ortime. Integrations with radiatively interactive specifichumidity produced similar results, but with additionalvariability on time scales of 2–3 days that seemed ex-traneous to the physics of seasonal transitions. Whilethe radiative effects of clouds are expected to alter thesolutions, they are neglected here in order to betterisolate the physics involving wind-dependent surfacefluxes.

2) RUNS WITHOUT WISHE

We first discuss model integrations with |V | in thesurface flux formulas for latent and sensible heat fixedat 5 m s�1, thereby eliminating any effects of WISHE.The results are consistent with the findings of Zheng(1998), in that an AMC circulation occurs only when


the amplitude of the off-equatorial SST anomaly ex-ceeds a certain threshold. For example, using �m� 10 Kproduces a phase-space trajectory resembling that ob-tained for the seasonally forced dry model with near-neutral stratification, in that the sensitivity of PBL flowto the amplitude of the SST anomaly increases abruptlyas the anomaly nears its summer maximum (Fig. 6).Shortly after this increase in sensitivity (at a time indi-cated by the dot on the phase trajectory), M contoursare strongly deformed from a resting state, with thezero contour of absolute vorticity � shifted to nearly20°S near the tropopause (Fig. 7). The lower branch ofthe circulation crosses the equator well above theboundary layer, roughly following M contours in thefree troposphere. This cross-equatorial jumping behav-ior presumably has such high amplitude because of in-teractions with moist convection, as discussed by Pau-luis (2004).

When a much weaker forcing is used, with �m � 1 Kand no WISHE, no abrupt intensification is seen (Fig.8, gray line in left panel). This phase trajectory closelyresembles the ellipse obtained for the seasonally forceddry model with an isothermal background state. Thecirculation near the time of peak SST is weak and con-fined entirely to the summer hemisphere, with M con-tours deviating only very slightly from the vertical inthe upper troposphere (Fig. 9, right panel).

3) RUNS WITH WISHE

Now we examine a series of model integrations withsurface enthalpy fluxes that depend on wind speed ac-

cording to (6) and (7). As stated in the introduction,runs with WISHE will only be performed for forcingsthat are not strong enough to produce an AMC re-sponse. The effect of WISHE in an AMC regime isqualitatively different and will be explored in a separatepaper.

For a run with WISHE and �m � 1 K, the circulationexhibits a rapid onset and withdrawal of the summercirculation, as indicated by the sudden increase in theslope of the phase trajectory near the time of maximumSST (Fig. 8, black line in left panel). Halfway throughthe abrupt intensification of the summer circulation inthis WISHE run, M contours are slightly deformedfrom a resting state (Fig. 9, left panel), although theyare far from the near-horizontal state in the upper tro-posphere characteristic of an AMC regime (cf. Fig. 7).The circulation at this time exists mostly in the summerhemisphere, consistent with a response that has not en-tered an AMC regime.

Given that the deformation of M contours is weak inthe WISHE run, one might hypothesize that the abruptintensification occurs because of a linear feedback be-tween the strength of the Hadley circulation and thesurface enthalpy flux. However, the abrupt onset of astrong solsticial circulation occurred only when bothWISHE and nonlinear momentum advection were rep-resented in the model. This was found by conducting athird model run with WISHE but without the advectionof either zonal or meridional relative momentum. Thephase trajectory of this run was nearly elliptical with aweak solsticial circulation and no abrupt transitions

FIG. 6. As in Fig. 4, but the phase diagram for the moist GCMwith strong forcing (�m � 10 K) and wind-independent surfaceenthalpy fluxes. The dot denotes the model state for which M and� are shown in Fig. 7.

FIG. 7. As in Fig. 2 but for the moist GCM with strong forcing(�m � 10 K) and wind-independent surface enthalpy fluxes, at thetime represented by the dot in Fig. 6. Streamfunction contourinterval is 4 1010 kg s�1, which is 4 times that used in Figs. 2and 3.


(Fig. 8, right panel). Eliminating both WISHE and mo-mentum advection in a fourth model run also produceda nearly elliptical phase trajectory. The use of WISHEwithout momentum advection did make the summercirculation almost twice as strong as the winter circula-tion, but this occurred without any rapid change in theslope of the phase trajectory and was a weak effectcompared to that produced by the combination ofWISHE and momentum advection.

Omitting the advection of relative momentum fromthe model equations without also omitting the advec-tion of humidity and potential temperature results innonconservation of total energy in the model. How-ever, omitting all of these advection terms in an addi-tional run produced a phase trajectory that differed

only slightly from that of the run in which only relativemomentum advection was omitted (not shown). Thisestablishes that the combination of nonlinear advectionand WISHE are both required for the abrupt intensifi-cation, and suggests that it is the nonlinear advection ofmomentum, in particular, that plays a central role in thefeedback. An understanding of this interaction betweenWISHE and momentum advection will prove to be fa-cilitated by use of an even simpler model, which is in-troduced in the next section.

3. Two-mode model

To better understand the nature of the abrupt inten-sification in the GCM run that employed both WISHE

FIG. 9. As in Fig. 7, but for the moist GCM with weak forcing (�m � 1.0 K), at times denoted by the dot in Fig.8. Runs (left) with and (right) without WISHE are shown (both included nonlinear momentum advection).

FIG. 8. As in Fig. 4, but the phase diagram for the moist GCM with weak forcing (�m � 1.0 K). Runs (left) withand (right) without nonlinear momentum advection are shown. The black and gray lines are for runs with andwithout WISHE, respectively. Dot in (left) denotes the time at which M and � are shown in Fig. 9.


and nonlinear momentum advection, we implement adifferent axisymmetric model in which the assumptionof convective quasi equilibrium is used to simplify thethermodynamics and in which only a barotropic modeand one baroclinic mode of the horizontal wind arerepresented.

a. Model formulation

1) GENERAL PHYSICS

This model is highly similar to the Quasi-EquilibriumTropical Circulation Model (QTCM) of Neelin andZeng (2000), but with the thermodynamics phrased interms of moist entropy instead of temperature and hu-midity. This is done for consistency with previous the-oretical studies of purely baroclinic WISHE modes byEmanuel (1987, 1993); an overview of the thermody-namics is given here, and the reader is directed to thosepapers for a more detailed derivation.

Moist convection is assumed to keep variations in s*,the saturation moist entropy of the free troposphere, con-stant with height. This allows variations in the geopo-tential � to be partitioned into a barotropic component��0 that is invariant with height, and a baroclinic com-ponent that can be written in terms of variations in s*:

p� � ��T

p�s*s* �

p�0, �8�

where the derivative of T is taken at constant s*. In thisparticular model we assume no variations in surface pres-sure, so that the above can be integrated to obtain ��:

� � �b � �Tb � T�s*, �9�

where the b subscript denotes a property at the top ofthe subcloud layer and T serves as a vertical coordinate.Using (9) together with the property that the verticalintegral of purely baroclinic � perturbations must van-ish gives

�� T � T �s*, �10�

where T is a mass-weighted vertical mean tropospherictemperature.

Conservation equations, phrased on an equatorial �plane, for axisymmetric horizontal wind can then bewritten in terms of the fluctuating component of s* (the� symbol is henceforth omitted):

u

t� �

u

y� w

u

z� �y� � Fu

�

t� �

�

y� w

�

z� �yu � �T � T �

s*y�

y�0 � F� ,

�11�

with the F terms representing both surface drag anddiffusion.

Tendencies of s* and subcloud-layer entropy sb are

s*t� �

s*y� �N2�w � Mc� � R � �

2s*

y2

Hb�sb

t� �

sb

y � � E �MIN�0, �wb �Mc��sb � sm�

� Hb�2sb

y2 , �12�

where Mc is the net upward mass flux in convectiveclouds, Hb is the depth of the subcloud layer, wb is thevertical velocity at the top of the subcloud layer aver-aged over clear and cloudy areas, � is a bulk precipita-tion efficiency, E is the surface entropy flux, and R isthe rate of radiative cooling, which is fixed at 1 K day�1.The second term on the right-hand side of the sb equa-tion represents the downward advection of low-entropymidtropospheric air into the boundary layer, with thedifference between sb and the midtropospheric entropysm hereafter assumed to be a constant, denoted �. TheMIN function is used to eliminate this term when thenet mass flux is upward. Horizontal diffusion is repre-sented with a constant diffusivity �. A dry static stabil-ity is defined

N2 � cp�T�d

�m

ln�

z, �13�

with �d and �m the dry and moist adiabatic temperaturelapse rates, respectively. The surface entropy flux isgiven by a bulk transfer formula:

E � Ck |V |�s*o � sb�, �14�

where s*o is the saturation moist entropy at the tempera-ture of the sea surface. Here |V | is given by (7) as forthe primitive equation model, with the same surfacegustiness of �g � 4 m s�1. For lower values of �g, themodel exhibits what seem to be meridionally propagat-ing WISHE modes that are more intense than in themoist GCM. As discussed in the previous section, thesetransients may be relevant to actual monsoon circula-tions, and we hope to examine their interaction with theseasonal cycle in future work.

The subcloud-layer equilibrium is used to specify anequilibrium cloud-base mass flux:

Meq � w �1� �E � Hb��

2sb

y2 � �sb

y ��. �15�


The actual convective mass flux is relaxed toward thisequilibrium value over a time scale of 3 h, and is con-strained to be nonnegative. We also set Mc � 0 if sb s*, so that there is no convection where the atmosphereis stable.

2) MODAL DECOMPOSITION

The horizontal wind is projected onto a barotropicmode v0, and a baroclinic mode v1 that uses tempera-ture as a vertical coordinate:

v � v0 � T �v1, �16�

where T ! is the temperature anomaly normalized by�T � Ts � T :

T � �T � T

�T. �17�

The axisymmetric continuity equation

w

z�

�

y� 0, �18�

combined with the assumptions of no topography andconstant surface pressure, gives a nondivergent baro-tropic wind. Under the constraint that � vanishes atthe meridional boundaries, the barotropic wind mustbe purely zonal so that � � !�1. The horizontal windis thus completely specified by u0, u1, and �1, whichdepend on y and time only. No dynamical boundarylayer is used in the momentum equations, and u1 and �1

can be taken to represent baroclinic winds at the sur-face.

With these assumptions, the vertical velocity musthave a vertical structure independent of latitude andtime:

w�y, z, t� � w�y, t��z�, �19�

with the vertical structure obtained from the tempera-ture profile,

��z� � �0

z

T � dz. �20�

In the axisymmetric two-mode framework, continuity isthen

w � ��1

y. �21�

Prognostic equations for u1 and �1 are obtained bysubstituting (16) into (11), multiplying by T !, and inte-grating from the surface to the tropopause:

u1

t� �1

u0

y�"T �3#

"T �2#�1

u1

y�"�T �pT �#

"T �2#

�1

yu1

� �y�1 �1

"T �2#�CD

H|V |�u0 � u1��

2u1

y2

�1

t�"T �3#

"T �2#�1

�1

y�"�T �pT �#

"T �2#

�1

y�1

� �Ts*y� �yu1 �

1

"T2#�CD

H|V |�1��

2�1

y2 ,

�22�

where angle brackets represent mass-weighted verticalintegrals, which are represented in pressure coordinatesfor simplicity. Quantities on the left-hand side are thetime tendency and advective terms. Quantities on theright-hand side are the Coriolis, drag, and horizontaldiffusion terms, with the �1 equation also containing apressure gradient forcing phrased in terms of s*. Hori-zontal diffusion is used to represent some effects ofeddies and to numerically stabilize the model. Verticaldiffusion is neglected. The drag terms have been writ-ten using the same bulk flux formula for momentum asin the primitive equation model. The depth of the tro-posphere H is assumed constant.

A prognostic equation for the barotropic wind is cal-culated by integrating the zonal momentum equation in(11) in the vertical and taking the curl to give a baro-tropic vorticity equation:

�0

t� "T �2#

2

y2 ��1u1� �

y �CD

H|V |�u0 � u1��

� �2�0

y2 . �23�

The left-hand side contains the time tendency term andthe advective terms written in flux form. The right-handside contains the drag and horizontal diffusion terms,which use the same coefficients CD and � as for thebaroclinic modes. The barotropic wind u0 is obtained byinverting $0.

The conservation equation for s*, given in (12), mustalso be vertically integrated to eliminate the height de-pendence of w and Mc :

s*t� �N2"�#�w � Mc � � R � �

2s*

y2 , �24�

where R is assumed to be a prescribed, vertically uni-form constant. This is the fully nonlinear conservation


equation for s*, where the horizontal advection termhas been eliminated by the assumption that s* is con-stant with height. The conservation equation for sb doesnot need to be modified, although care must be taken touse the values of w and Mc at the top of the subcloudlayer and the mean value of � within that layer:

�Hb

sb

t� �b

sb

y � � E �MIN�0, �w �Mc��b�

� Hb�2sb

y2 . �25�

Here %b is the value of % at the top of the subcloudlayer, and �b is the mass-weighted meridional wind ver-tically averaged within the subcloud layer. The equilib-rium mass flux then becomes

Meq � w �1

��b�E � �Hb�

2sb

y2 � �1

sb

y�b��,

�26�

with Mc relaxed toward Meq over a 6-h time scale.The model consists of (13), (14), (17), and (21)–(26),

which include the time tendency equations for the prog-nostic variables s*, sb, u1, �1, and $0. The parametersdependent on temperature were derived assuming asurface temperature of 296 K at 1000 hPa, a dry adiabatup to 900 hPa, and a moist pseudoadiabat from 900 to150 hPa. The values used for model parameters arelisted in Table 1. This model employs many simplifica-tions even when compared to other reduced models of

the tropical atmosphere, and because of this cannotrepresent a number of processes. In particular, it omitswave radiation into the stratosphere (Yano and Eman-uel 1991), dynamical boundary layer effects (Sobel andNeelin 2006), moisture–radiation feedbacks, and someeddy transports. We do find, however, that it can rep-resent the fundamental physics of the WISHE-inducedabrupt seasonal transition seen in the moist GCM.

b. Linear properties

The above system includes nonlinear advectionterms, which, because of the GCM results, we expect tobe required for a WISHE feedback. It is of interest thatEmanuel (1993) found purely baroclinic zonally sym-metric circulations to be unstable to WISHE in a lin-earized version of the above system that employed noSST gradient and a basic state with easterly surfacewinds. While a basic state having a poleward SST gra-dient and westerly surface winds would be more rel-evant to the onset of the solsticial Hadley circulation,we discuss in the next section how the presence of acoupled barotropic circulation can stabilize WISHEmodes in a linear system. Indeed, the Hadley circula-tion cannot be represented in terms of a purely baro-clinic mode; a barotropic mode must be added to obtainzonal winds that are simultaneously strong at upperlevels and weak at the surface, as discussed in detail byBurns et al. (2006). For this reason, a linear stabilityanalysis of a purely baroclinic system is not presentedhere.

However, some insight can still be gained by exam-

TABLE 1. Parameters used in the two-mode model.

Parameter name Symbol Value

Surface minus mean atmospheric temperature �T 35 KMean boundary layer depth Hb 910 mMean tropopause height H 14 kmDry static stability N2 1.0 J kg�1 m�1

Normalized second moment of temperature "T !2# 0.57Normalized third moment of temperature "T !3# �0.36Correlation function for vertical advection "%T !pT !# �0.18Vertical mean structure function "%# 1.8 104

Structure function at top of subcloud layer %b 420 mMean subcloud-layer meridional wind �b 0.89 �1

Horizontal diffusivity � 1.0 105 m2 s�1

Surface transfer coefficient for momentum CD 0.0012Enthalpy exchange coefficient Ck 0.0012Surface gustiness �g 4.0 m s�1

Radiative cooling rate R 1 K day�1

Bulk precipitation efficiency � 0.85Entropy drop at top of subcloud layer �eb � �m 15 KGradient of Coriolis parameter � 2.28 10�11 m�1 s�1

Specific heat of air at constant pressure cp 1010 J kg�1 K�1

Convective response time �c 3 h


ining such a highly simplified, purely baroclinic systemwith dynamics linearized about a resting state. To do so,we neglect horizontal diffusion and represent surfacedrag as Rayleigh damping over a time scale 1/r. Forsimplicity, we assume radiative cooling operates on thesame time scale as this mechanical damping, so that Rin (12) can be written as rs*. We set the precipitationefficiency � equal to unity, which corresponds to anatmosphere in which evaporatively driven downdraftsdo not occur and free-tropospheric temperatures arecontrolled by boundary layer processes (Emanuel1993). The system consisting of (11) and (12) can thenbe rewritten, assuming modal perturbations in u, � ands* proportional to e�i&t:

�r � i��u � f�, �27�

�r � i�� T � T �s*y� fu, and �28�

�r � i��s* �N2�T � T �

�E. �29�

These can be combined into a single expression:

� � � 1

�r � i��2 � f 2�N2�T � T �

�

E

y. �30�

If the time scale of the damping is short compared tothat of the forcing (i.e., r k &), then the real part of (30)is well approximated by

� � � 1

r2 � f 2� N2�T � T �

�

E

y, �31�

which is the same as the solution for the steady re-sponse to a time-invariant forcing. For & correspondingto earth’s seasonal cycle of insolation, this limit willhold if viscous losses are dominated by the strong ver-tical mixing of momentum in the planetary boundarylayer, so that 1/r is on the order of a few days. Thisresult then shows that the meridional gradient of thesurface entropy flux is the relevant forcing for meridi-onal flow, and that the flow will be nearly in phase withthis forcing. If the circulation is instead controlled bynearly inviscid free-tropospheric dynamics, then such alinear treatment will likely not provide an appropriatedescription.

c. Numerical results

The fully nonlinear two-mode model is forced by thesame time-varying SST used for the GCM, consisting of(3) and (5) with �m � 1 K, and runs were performedwith and without WISHE, and with and without non-linear momentum advection. Although no explicit dy-namical boundary layer exists in the two-mode model,the PBL flow was calculated by integrating meridionalflow through the lowest 200 hPa of the atmosphere, toease comparison with the GCM results. These phase-space trajectories are in many ways similar to thosefrom the moist GCM, with an abrupt, high-amplitudeintensification of the summer circulation occurring onlywhen both WISHE and nonlinear momentum advec-tion are used (Fig. 10). The two-mode model also ex-hibits an abrupt intensification of the winter circulationnot seen in the GCM, although even in the two-modemodel the winter circulation has about half the peak

FIG. 10. As in Fig. 8 but for the two-mode model. (right) The dashed line is for a run with no barotropic windand no nonlinear momentum advection, but with WISHE.


amplitude of the summer circulation. This abrupt inten-sification of the winter circulation can be eliminated(while preserving the abrupt summer intensification) bymodifying several parameters of the two-mode model,in particular by using larger values of the surface gusti-ness, static stability, and horizontal diffusivity. The two-mode model in general seems more sensitive to theeffects of WISHE, as even without momentum advec-tion WISHE produces a fivefold increase in the peakPBL flow, although this increase occurs more slowlyand is much weaker than that achieved in the modelthat includes both WISHE and momentum advection.Also, the phase-space trajectory for the run with bothWISHE and momentum advection exhibits high-frequency oscillations about the mean trajectory duringthe rapid intensifications. While these oscillations arenot associated with propagating instabilities, they mightresult from the same physics that would cause propa-gating WISHE modes at lower values of the surfacegustiness parameter.

In the no-WISHE integrations of the two-modemodel, the boundary layer flow peaked at least one-quarter cycle before the SST. This would seem to occurbecause of the relatively short phase lag between SSTand subcloud-layer entropy. The surface entropy fluxrelaxes the entropy of the boundary layer toward thatof the sea surface over the time scale �E:

E

Hb�

1�E�s*o � sb�, �32�

with �E � Hb /(Ck |V | ). For typical values of these pa-rameters, �E is on the order of 10 days, so that the phaselag between sb and s*o is expected to be small compared

to the 365-day period of s*o . Thus, the relation of E tothe prescribed s*o is analogous to the relation of New-tonian cooling to the prescribed equilibrium tempera-ture in the dry model. By the same trigonometric rela-tionships used for the dry model to derive (4), E shouldthen lead s*o by slightly less than one-quarter cycle, andthe meridional circulation should be in phase with E. Insuch a simple linear theory, the major axis of the phase-space ellipse should be horizontal, as it nearly is for thetwo-mode model without WISHE. Although this con-trasts with the GCM results where the major axis has apositive slope, such differences between the two-modemodel and the GCM seem to result from phase lags ina predominantly linear response. Both models exhibitan abrupt intensification of solsticial flow due to theinteraction of WISHE and nonlinear momentum ad-vection.

4. Physics of the WISHE feedback

This section explains how the combination ofWISHE and nonlinear momentum advection producesan abrupt intensification of the summer circulation inboth the GCM and the two-mode model.

In the GCM run using both WISHE and nonlinearmomentum advection, the surface enthalpy flux peaksabout 8° south of the SST maximum, and this enthalpyflux peak is meridionally sharper than the SST peak(Fig. 11, left panel). Given that the meridional gradientof the surface enthalpy flux is the relevant forcing in alinear regime, it seems natural to ask how linear theatmospheric response is when viewed as a function of

FIG. 11. Surface enthalpy flux for the moist GCM with weak forcing (�m � 1.0 K). Runs (left) with and (right)without WISHE are shown. Time is the number of days after SST has zero gradient, with maximum SST occurringat day 91 and 25°N. Contour interval is 10 W m�2.


this quantity.2 We compute the mean value of yE be-tween 10°N and the latitude at which E peaks, and plotthis quantity as a function of the SST at 25°N for theGCM runs with momentum advection and �m � 1 K(Fig. 12). The resulting phase trajectories closely re-semble the corresponding trajectories for meridionalPBL flow in the moist GCM (cf. left panel of Fig. 8).Thus, the meridional circulation does not seem to be farfrom a linear response to yE, even though nonlineardynamics are needed to achieve the positive feedback.This is an important point, because WISHE could beviewed simply as a way to reduce the critical SST gra-dient needed to obtain a nonlinear intensification fromM advection, so that the same physics explored byPH92 and Zheng (1998) would apply at a lower SSTthreshold. The near-linear relationship between yEand PBL flow is one piece of evidence that WISHEactually provides a separate mechanism for a nonlinearresponse of the PBL flow to the SST. Another piece ofevidence is the small deformation of M contours in theruns with WISHE; the upper-tropospheric absolutevorticity is not uniformly zero over any nonvanishingregion in either the GCM or the two-mode model forthe forcing with �m � 1 K (not shown).

If the net meridional mass flux exhibits a near-linearresponse to the surface enthalpy flux, why does WISHEnot produce an abrupt onset when momentum advec-tion is omitted from the model? The reason for this wasbriefly mentioned above in discussion of the linearproperties of the two-mode model. Emanuel (1993)showed that zonally symmetric, purely baroclinic circu-lations can be linearly unstable to WISHE. The Hadleycirculation, however, projects strongly onto a barotro-pic mode, with surface winds that are generally weakcompared to upper-tropospheric winds. Since WISHEinstabilities arise from enthalpy flux anomalies drivenby surface winds, a reduction in the magnitude of sur-face winds by the partial cancellation of the baroclinicand barotropic components of zonal wind at the surfacewould be expected to reduce or even eliminate anyinstability. This hypothesis was tested by eliminatingthe barotropic mode from a version of the two-modemodel linearized about a resting state. The resultingcirculation exhibited an abrupt increase in its sensitivityto SST as soon as the SST gradient changed sign tobecome positive in the tropics (Fig. 10, right panel).While such a change in sensitivity should not necessar-ily be interpreted as an instability, WISHE clearly has astronger effect on circulations with linear dynamics in

the absence of a coupled barotropic mode. WhenWISHE was turned off with the barotropic mode elimi-nated, the phase trajectory was nearly elliptical with amuch weaker meridional flow (not shown).

In the two-mode model without WISHE but withmomentum advection, the barotropic zonal wind does,in general, nearly cancel the low-level baroclinic windeverywhere (Fig. 13, right column). In the upper tropo-sphere, the two modes must then add to produce strongeasterlies south of the SST maximum during “summer”conditions. In the two-mode model with both WISHEand momentum advection, the barotropic wind opposesthe surface baroclinic wind nearly everywhere exceptjust south of the SST extremum during solstice condi-tions, during the time when the rapid intensification ofPBL flow occurs (Fig. 13, left column). During summer,the two modes add to produce a narrow region ofstrong surface westerlies, which are in turn associatedwith strong surface enthalpy fluxes. The conservationequation for barotropic zonal wind (23) shows that in aregion of baroclinic surface westerlies, the only processthat can make the barotropic wind more westerly isconvergence of the meridional flux of zonal momen-tum. A similar though somewhat weaker effect is seenduring winter, when barotropic and baroclinic easter-lies add to produce a narrow band of strong surfaceeasterlies near 12°N.

Similar dynamics seem to occur in the moist GCM,although the low-level constructive superposition of

2 At the SSTs used here, the enthalpy flux is very close to aconstant linear multiple of the entropy flux, so we use the twosomewhat interchangeably in this section.

FIG. 12. Phase diagram, for the moist GCM with weak forcing(�m � 1.0 K), of the meridional mean meridional gradient ofsurface enthalpy fluxes, with the mean taken between the enthal-py flux peak and 10°N, plotted against the SST at 25°N. The blackand gray lines are for runs with and without WISHE, respectively,both with nonlinear momentum advection.


barotropic and baroclinic modes occurs only duringsummer. A barotropic wind was defined as the mass-weighted vertical mean zonal wind between the surfaceand the tropopause, with the tropopause defined as thelevel at which the static stability increased sharply tostratospheric values. Instead of a particular baroclinicmode, we examine the residual obtained by subtractingthe barotropic wind from the total zonal wind. In therun conducted without WISHE but with momentumadvection, the barotropic zonal wind and this baroclinicresidual nearly cancel at low levels (Fig. 14, right col-umn). In the run with both WISHE and momentumadvection, the meridional structure of these modes issimilar to that seen in the two-mode model: the baro-

tropic wind adds to the low-level baroclinic residualonly near the ascent branch of the circulation just southof the SST maximum (Fig. 14, left column).

In both the two-mode model and the GCM, then, apositive feedback seems to occur when the convergenceof the meridional flux of zonal momentum becomessufficiently strong to produce a barotropic wind thatadds to the surface baroclinic wind instead of opposingit, thereby eliminating the damping effect of the baro-tropic mode on the wind–evaporation feedback. Themean meridional circulation will, in a vertical mean,converge zonal momentum into the ascending branchas long as M contours tilt toward the equator withheight. Because such a tilt is brought about by advec-tion in a model that conserves M, convergence of zonalmomentum into the ascending branch will be accom-plished by any mean meridional flow as long as thisflow is directed poleward at low levels, a fairly generalcondition for any thermally direct, solsticial flow. To

FIG. 13. Decomposition of the low-level zonal wind in the two-mode model. Runs (left column) with and (right column) withoutWISHE are shown. Both of these runs included nonlinear mo-mentum advection. (top row) The total zonal wind at 850 hPa,(middle row) the barotropic component, and (bottom row) thebaroclinic component at 850 hPa are shown. Black lines denotewesterlies and gray lines easterlies, with a contour interval of 5m s�1 starting at 2.5 m s�1. Time is the number of days after theSST has zero gradient, with maximum SST occurring at day 91 and25°N.

FIG. 14. As in Fig. 13, but for the moist GCM with weak forcing(�m� 1.0 K). The barotropic component is a vertical troposphericmean, and the baroclinic residual was obtained by subtracting thisbarotropic component from the total zonal wind.


create barotropic westerlies, this momentum conver-gence must overcome surface drag and any other hori-zontal transports:

��u�

y � � �CD

H|V |u � F. �33�

Here H is the depth of the troposphere and F repre-sents horizontal transports including diffusion and mo-mentum transports by zonally asymmetric eddies(which do not occur in the models examined in thispaper).

In the simplified case where F is zero and surfacedrag is a linear Rayleigh damping, a scale estimate ofthe terms in (33) provides

V

�y� r, �34�

where V is the velocity scale of the meridional wind, ris the damping coefficient, and �y is the meridionalscale over which the circulation changes. If V respondslinearly to the forcing, as in (31), then this amounts toa condition on the curvature of the surface entropy flux.

While this scaling is likely too crude to be quantita-tively compared with the numerical model results, itdoes show that a critical surface entropy flux gradientshould exist below which no wind–evaporation feed-back will occur. We test for the existence of such athreshold by changing the peak amplitude of the SSTforcing in the two-mode model (integrated with bothWISHE and nonlinear momentum advection). Thecase examined thus far with �m � 1 K seems to nearlycoincide with this threshold. A run using �m � 0.5 Khad no abrupt intensification, and a run using �m � 1.5K had an abrupt intensification reaching a much higheramplitude (Fig. 15). The run with �m � 1.5 K exhibitedhigh-amplitude transient instabilities, of somewhat dif-ferent form than those found by Bellon and Sobel(2008), and their effect on the phase trajectory was re-duced by applying a 10-day moving average. Althoughthis smoothing also reduced the apparent abruptness ofthe onset of summer flow in the resulting phase trajec-tory, it is interesting that the smoothed trajectory has asimilar shape to that seen for weaker forcings which donot produce transient instabilities. That is, it might bepossible to use the mechanisms described in this paperto describe a smoothed version of the seasonal dynam-ics even when such instabilities occur, a hypothesis thatmay merit further examination in future work. Notethat the forcing with �m � 1.5 K did not produce an

AMC response, as indicated by nonzero values of up-per-tropospheric absolute vorticity in this modelthroughout its seasonal cycle (not shown).

5. Concluding remarks

A seasonal cycle that is long compared to the timescale of meridional M advection should be a necessary,albeit not sufficient, condition for the scalings andthresholds of nonlinear axisymmetric theories (e.g.,Held and Hou 1980; PH92) to apply to an arbitraryplanetary atmosphere. We have found that the dry axi-symmetric models used by some previous studies (e.g.,Fang and Tung 1999) equilibrate too slowly to be usedfor study of the time-dependent response to earth’s sea-sonal insolation forcing, but that moist axisymmetricmodels equilibrate much faster because of their char-acteristic stratification and radiative cooling rate. Thismakes examination of the seasonal cycle in a moist at-mosphere, which when fully represented will includeeffects of WISHE, a logical extension of previous workon nonlinear axisymmetric flow.

This paper focused on fairly weak SST forcings thatdid not produce an AMC response. In the axisymmetricmodels used here, WISHE produced meridional flowthat increased nonlinearly as a function of the season-ally evolving SST gradient, even though the flow nevertransitioned into an AMC regime. For Hadley circula-tions with a strong barotropic component, this WISHE

FIG. 15. As in Fig. 8 but for the two-mode model with SSTforcings of different strengths. All runs included both WISHE andnonlinear momentum advection. Dashed line is for �m � 1.5 ',solid black line for 1.0 K, and solid gray line for 0.5 K. The solidblack line is the same trajectory appearing in (left) of Fig. 10.


feedback was found to occur only when the verticallyintegrated convergence of zonal momentum into theascent branch of the circulation was larger than themomentum sink due to surface drag and any other hori-zontal transports. The effect of WISHE for strongerSST forcings that do produce an AMC response will beexamined in a separate paper. Examination of SSTforcings with nonzero gradients on the equator is alsoleft for this separate work, as such forcings are expectedto produce AMC flow regardless of their amplitude (asdiscussed in PH92).

Numerous physical processes omitted from the mod-els used here may affect the physics of the WISHEfeedback. Using an SST anomaly as a proxy for thethermal forcing of a land surface allows WISHE to op-erate over the entire domain, whereas the relativelysmall thermal inertia of a land surface should eliminateWISHE for periods longer than 5–10 days (e.g., Neelinet al. 1987; Maloney and Sobel 2004). How the WISHEfeedback might alter monsoon onset when it operatesonly over the ocean equatorward of a coast could beeasily explored in a variation on the axisymmetric mod-els used here. Of course, the effect of a land surfacemay extend beyond that of its surface enthalpy flux, aselevated topography in the South Asian monsoon isknown to strongly organize the flow (e.g., Hoskins andRodwell 1995) and to increase free-tropospheric tem-peratures (e.g., Molnar and Emanuel 1999).

The radiative effects of variable clouds and watervapor were also omitted from our models. Ackermanand Cox (1987) estimated horizontal variations in at-mospheric radiative flux divergence of about 100 Wm�2 between clear and cloudy oceanic regions in the1979 Asian summer monsoon, which is similar to themagnitude of wind-induced surface enthalpy flux varia-tions in our models. Bony and Emanuel (2005) foundthat moisture–radiation feedbacks can have a scale-selecting effect on WISHE modes and also excite ad-ditional small-scale instabilities. Both of these resultssuggest that the role of moisture–radiation feedbacks inmonsoon onset merit further study.

SST in our models was treated as a prescribed forc-ing, and it is known that surface winds alter SST bydriving ocean evaporation, dynamical transports, andthe entrainment of thermocline water into the oceanmixed layer. While a common practice in idealizedmodels is to treat the ocean as a dynamically passivebulk mixed layer, and use of such a mixed layer wouldbe a useful extension of this work, it is not obvious thatthese processes are more important than dynamicalocean transports in monsoon regions. Indeed, Websterand Fasullo (2003) proposed that the seasonally revers-ing, cross-equatorial heat flux in the Indian Ocean

regulates the strength of the South Asian monsoon.While they focused on interannual variability, dynami-cal ocean transports might play a role in modulatingany abrupt seasonal transitions.

We noted in the introduction that eddy momentumtransports may play a role in monsoon onset. Eddytransports might even play a direct role in the WISHEfeedback because this feedback requires the conver-gence of zonal momentum to overcome the dampingeffects of the barotropic wind. However, in the drymodel used by Schneider and Bordoni (2008), the as-cending branch of the cross-equatorial cell was associ-ated almost entirely with momentum transports by themean meridional circulation, with eddies extractingzonal momentum primarily from the subsiding branchof the Hadley circulation and depositing it in higherlatitudes of the winter hemisphere. There would thusbe no a priori reason to believe that eddy momentumtransports would oppose the momentum convergencerequired for the WISHE feedback.

Finally, the physics of WISHE may change pro-foundly with both the structure of the forcing and thebreaking of axisymmetry. Prive and Plumb (2007) illus-trated the large effect zonal asymmetries in a forcingcan have on the atmospheric response. Yet even with azonally symmetric forcing, Emanuel (1993) showed thatgrowth rates for WISHE modes peak at a nonzerozonal wavenumber, which suggests that WISHE feed-backs in three dimensions might possess a different sen-sitivity to SST gradients than in axisymmetric models.Thus, to apply the ideas in this paper to actual monsooncirculations, it will eventually be necessary to under-stand the behavior of WISHE modes in a three-dimensional domain with ocean, land, and radiative in-teractions.

Acknowledgments. This research constituted part ofthe first author’s doctoral thesis, and was supported bythe National Science Foundation under Grant ATM-0432090. We thank Arnaud Czaja and Nikki Privé forassistance with the GCM, which was originally imple-mented by Olivier Pauluis and Sandrine Bony. Discus-sions with Alan Plumb, Tapio Schneider, and MasahiroSugiyama were valuable in the preparation of thiswork. Comments by Adam Sobel and an anonymousreviewer greatly improved the presentation and contentof this paper.

APPENDIX

Equilibration Time Scale of Axisymmetric Models

The nonlinear dependence of circulation strength onthe forcing in axisymmetric theory results from advec-


tion of absolute angular momentum in the free tropo-sphere, so the relevance of this nonlinearity in a time-dependent scenario would seem to depend on the mag-nitude of the momentum advection time scale relativeto the forcing time scale. PH92 found that it took aslong as 400 days for their dry axisymmetric model toreach a steady state, and suggested that this resultedfrom the roughly 100-day time scales for viscous dissi-pation and meridional overturning in their model. Pre-sumably, the equilibration rate would be set by theoverturning time scale in the inviscid limit, so here weuse a simple scaling argument to explicitly estimate thetime scale for angular momentum advection in axisym-metric models.

The circulation in the interior of an inviscid atmo-sphere will conserve absolute angular momentum M:

M � �ea2 cos2� � ua cos�, �A1�

where %e is the planetary rotation rate, a is the plan-etary radius, u is zonal velocity, and � is latitude. In anatmosphere near a resting state, advection will be al-most entirely meridional until M surfaces deviate con-siderably from the vertical. A scale estimate for themeridional wind � can be obtained from the thermody-namic equation under the assumption that the domi-nant balance is between the diabatic heating Q andvertical advection of potential temperature �:

�S (Q

cp, �A2�

where & is vertical velocity in pressure coordinates, S�p�, and cp is the specific heat at constant pressure. Themeridional circulation will adjust quite rapidly to satisfy(A2): Eliassen (1951) showed that the thermally forcedmeridional circulation in an axisymmetric vortex equili-brates on a time scale near that of the local inertialperiod. The bulk of the free-tropospheric meridionalflow will occur over some vertical distance �p and somehorizontal distance �y � y2 � y1 (we use Cartesiancoordinates for simplicity). By continuity in a zonallysymmetric fluid,

� � ��y1

y2 �

pdy (

Q�y

Scp�p. �A3�

The time scale �) for meridional advection is then

�M (Scp�p

Q. �A4�

Although �p and Q are part of the solution, approxi-mate values can be estimated based on previous modelresults. For example, a value of half the troposphericdepth seems reasonable for �p, unless flow becomesconcentrated in a boundary layer. Similarly, Q is con-strained by the equilibrium temperature and relaxationtime imposed by the Newtonian cooling scheme in a drymodel, or by radiation in a moist model.

Estimates of �M for several models are presented inTable A1. The fact that the integrations conducted byPlumb and Hou (1992) took hundreds of days to equili-brate is consistent with the parameters of their model:�p � 400 hPa and values of S and Q appropriate forrelaxation over a 10-day time scale to a temperatureprofile that was isothermal in the vertical and had maxi-mum horizontal anomalies of about 10 K. The drymodel of Fang and Tung (1999) used a stratificationcharacteristic of a moist adiabat, but prescribed tem-perature relaxation over a 20-day time scale to producetypical Newtonian cooling rates of 0.3 K day�1 (in-ferred from their figures). It should be noted that �) isthe time scale for M advection, and that the M field willlikely take several times this long to fully equilibrate.Thus, although �M for the Fang and Tung (1999) modelis only slightly larger than the time scale of their appliedthermal forcing (365 days/2� � 58 days), the M field intheir model cannot be expected to achieve equilibriumwhen subjected to such a forcing.

In contrast, a value of �) � 25 days is obtained if oneuses the same value of �p but with a stratification typi-cal of a moist adiabat and a radiative cooling rate of 1K day�1, typical values for the tropical troposphere.Such an average radiative cooling rate is the appropri-ate value for Q if the intensity of meridional flow isrequired to be consistent with a balance between radia-tive cooling and adiabatic heating in the subsidingbranch of the circulation (e.g., Nilsson and Emanuel1999; Emanuel et al. 1994). The value of �) may bereduced below the value given in Table A1 becausefree-tropospheric Hadley flow typically occurs within

TABLE A1. Estimated equilibration time scales.

Model Temperature profile Stratification (S) Heating rate (Q) �M

Plumb and Hou (1992) Isothermal �2 10�3 K Pa�1 0.5 K day�1 165 daysFang and Tung (1999) � � 6 K km�1 �5 10�4 K Pa�1 0.3 K day�1 75 daysZheng (1998)/Moist GCM Moist adiabat �5 10�4 K Pa�1 1 K day�1 25 daysNear-neutral stratification � � 0.9 g/cp �2 10�4 K Pa�1 0.5 K day�1 15 days


about 200 hPa of the tropopause (Peixoto and Oort1992). In any case, the time scale of earth’s insolationforcing is at least 2–3 times greater than �) for param-eters relevant to a moist tropical atmosphere, whichexplains why Zheng (1998) achieved large M deforma-tion on seasonal time scales in his moist model.

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